Routledge Handbook of Sports Technology and Engineering [1 ed.] 0415580455, 9780415580458

Table of contents :
Contents
Figures
Tables
Contributors
Preface
Acknowledgements
Part I: Sustainable sports engineering
1 Sustainable design of sports products • Aleksandar Subic, Adrian Mouritz and OlgaTroynikov
2 Sustainable manufacturing of sports products • Aleksandar Subic, Bahman Shabani, Mehdi Hedayati and Enda Crossin
Part II: Instrumentation technology
3 Instrumentation of sports equipment • Franz Konstantin Fuss
4 Smart devices and technologies for sports applications • Franz Konstantin Fuss
5 Instrumentation of athletes • Daniel Arthur James
6 Technologies in exertion games • Florian Mueller
Part III: Summer mobility sports
7 Design of racing bicycles • LachlanThompson
8 Mountain bike technology • Robin C. Redfield
9 Rowing equipment technology • Alastair Campbell Ritchie and John Dominy
10 Sport wheelchair technologies • Franz Konstantin Fuss and Aleksandar Subic
Part IV: Winter mobility sports
11 Cross-country ski technology • Leonid Kuzmin and Franz Konstantin Fuss
12 Snowboard technology • Patrick Clifton, Aleksandar Subic and Franz Konstantin Fuss
13 Ice hockey skate design and performance • Rene A.Turcotte and David J. Pearsall
Part V: Apparel and protection equipment
14 Design and mechanics of running shoes • Franz Konstantin Fuss
15 Sports apparel • Aleksandar Subic, Firoz Alam, OlgaTroynikov and Len Brownlie
16 Sports helmets • Aleksandar Subic, Firoz Alam and MonirTa
17 Design and mechanics of mountaineering equipment • Franz Konstantin Fuss and Günther Niegl
Part VI: Sports implements
18 Golf club construction, design and performance • Martin Strangwood and Carl Slater
19 Tennis racquet technology • Franz Konstantin Fuss, Aleksandar Subic and Rod Cross
20 Mechanical behaviour of baseball and softball bats • Lloyd Smith
21 Ice hockey stick mechanics and designs • David J. Pearsall and Rene A.Turcotte
Part VII: Sports balls
22 The science and engineering of golf balls • Martin Strangwood
23 Solid mechanics and aerodynamics of cricket balls • Franz Konstantin Fuss, Aleksandar Subic and Rabindra Mehta
24 Mechanical and aerodynamic behaviour of baseballs and softballs • Lloyd Smith and Jeff Kensrud
25 Hockey balls • Dan Ranga and Martin Strangwood
26 Oval-shaped sports balls: aerodynamics, friction and bounce • Firoz Alam, Franz Konstantin Fuss and Aleksandar Subic
27 Aerodynamics and court interaction of tennis balls • Firoz Alam, Franz Konstantin Fuss, Rabindra Mehta and Aleksandar Subic
28 Aerodynamics and construction of modern soccer balls • Firoz Alam,Takeshi Asai, Rabindra Mehta and Aleksandar Subic
Part VIII: Sports surfaces and facilities
29 Artificial turf • Peter Sandkuehler, Allan McLennaghan and Thomas Allgeuer
30 Natural turf sports surfaces • IainT. James
31 Design of sports facilities • Franz Konstantin Fuss
Index

Citation preview

ROUTLEDGE HANDBOOK OF SPORTS TECHNOLOGY AND ENGINEERING From carbon-fibre racing bikes to ‘sharkskin’ swimsuits, the application of cutting-edge design, technology and engineering has proved to be a vital ingredient in enhanced sports performance. This is the first book to offer a comprehensive survey of contemporary sports technology and engineering, providing a complete overview of academic, professional and industrial knowledge and technique. The book is divided into eight sections covering the following topics: • • • • • • • •

sustainable sports engineering instrumentation technology summer mobility sports winter mobility sports apparel and protection equipment sports implements (racquets, clubs, bats, sticks) sports balls sports surfaces and facilities.

Written by an international team of leading experts from industry, academia and commercial research institutes, the emphasis throughout the book is on innovation, the relationship between business and science, and the improvement of sports performance.This is an essential reference for anybody working in sports technology, sports product design, sports engineering, biomechanics, ergonomics, sports business or applied sport science. Franz Konstantin Fuss is Professor of Sports Engineering and Technology at the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Australia. He is joint Editor-in-Chief of Sports Technology and Chair, Co-Chair and Organiser of Asia-Pacific Congress on Sports Technology (APCST) conferences. He was also a member of the Executive Committee of the International Sports Engineering Association (ISEA) and a Co-Director of the Australian Sports Technology Network (ASTN). Aleksandar Subic is Professor and Head of the School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Australia. He is joint Editor-in-Chief of Sports Technology and the Journal of Sustainable Design, and Chair, Co-Chair and Organiser of APCST conferences. He is also the past president of the ISEA. Martin Strangwood is Senior Lecturer at the School of Metallurgy and Materials, Birmingham University, UK. He is Editor-in-Chief of Sports Engineering. Rabindra Mehta is a research scientist in the Fluid Mechanics Laboratory at NASA Ames Research Center, USA, and a sports aerodynamics consultant. He is also joint Editor-in-Chief of Sports Technology.

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ROUTLEDGE HANDBOOK OF SPORTS TECHNOLOGY AND ENGINEERING

Edited by Franz Konstantin Fuss, Aleksandar Subic, Martin Strangwood and Rabindra Mehta

First published 2014 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Simultaneously published in the USA and Canada by Routledge 711 Third Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2014 Franz Konstantin Fuss, Aleksandar Subic, Martin Strangwood and Rabindra Mehta The right of the editors to be identified as the authors of the editorial material, and of the authors for their individual chapters, has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Routledge handbook of sports technology and engineering / edited by Franz Fuss ... [et al.]. p. cm. 1. Sports—Technological innovations. 2. Sporting goods. 3. Performance technology. 4. Sports sciences. 5. Biomechanics. I. Fuss, Franz. GV745.R69 2013 688.76—dc23 2012047339 ISBN: 978-0-415-58045-8 (hbk) ISBN: 978-0-203-85103-6 (ebk) Typeset in Bembo by FiSH Books Ltd, Enfield

CONTENTS

Figures Tables Contributors Preface Acknowledgements

ix xviii xx xxiii xxviii

PART I

Sustainable sports engineering

1

1

Sustainable design of sports products Aleksandar Subic, Adrian Mouritz and Olga Troynikov

3

2

Sustainable manufacturing of sports products Aleksandar Subic, Bahman Shabani, Mehdi Hedayati and Enda Crossin

27

PART II

Instrumentation technology

41

3

Instrumentation of sports equipment Franz Konstantin Fuss

43

4

Smart devices and technologies for sports applications Franz Konstantin Fuss

59

5

Instrumentation of athletes Daniel Arthur James

82

6

Technologies in exertion games Florian Mueller

94

v

Contents PART III

Summer mobility sports

109

7

Design of racing bicycles Lachlan Thompson

111

8

Mountain bike technology Robin C. Redfield

130

9

Rowing equipment technology Alastair Campbell Ritchie and John Dominy

141

10 Sport wheelchair technologies Franz Konstantin Fuss and Aleksandar Subic

156

PART IV

Winter mobility sports

169

11 Cross-country ski technology Leonid Kuzmin and Franz Konstantin Fuss

171

12 Snowboard technology Patrick Clifton, Aleksandar Subic and Franz Konstantin Fuss

189

13 Ice hockey skate design and performance Rene A.Turcotte and David J. Pearsall

202

PART V

Apparel and protection equipment

213

14 Design and mechanics of running shoes Franz Konstantin Fuss

215

15 Sports apparel Aleksandar Subic, Firoz Alam, Olga Troynikov and Len Brownlie

233

16 Sports helmets Aleksandar Subic, Firoz Alam and Monir Takla

252

17 Design and mechanics of mountaineering equipment Franz Konstantin Fuss and Günther Niegl

277

vi

Contents PART VI

Sports implements

293

18 Golf club construction, design and performance Martin Strangwood and Carl Slater

295

19 Tennis racquet technology Franz Konstantin Fuss, Aleksandar Subic and Rod Cross

306

20 Mechanical behaviour of baseball and softball bats Lloyd Smith

325

21 Ice hockey stick mechanics and designs David J. Pearsall and Rene A.Turcotte

339

PART VII

Sports balls

347

22 The science and engineering of golf balls Martin Strangwood

349

23 Solid mechanics and aerodynamics of cricket balls Franz Konstantin Fuss, Aleksandar Subic and Rabindra Mehta

361

24 Mechanical and aerodynamic behaviour of baseballs and softballs Lloyd Smith and Jeff Kensrud

386

25 Hockey balls Dan Ranga and Martin Strangwood

399

26 Oval-shaped sports balls: aerodynamics, friction and bounce Firoz Alam, Franz Konstantin Fuss and Aleksandar Subic

410

27 Aerodynamics and court interaction of tennis balls Firoz Alam, Franz Konstantin Fuss, Rabindra Mehta and Aleksandar Subic

423

28 Aerodynamics and construction of modern soccer balls Firoz Alam,Takeshi Asai, Rabindra Mehta and Aleksandar Subic

439

PART VIII

Sports surfaces and facilities

453

29 Artificial turf Peter Sandkuehler, Allan McLennaghan and Thomas Allgeuer

455

vii

Contents

30 Natural turf sports surfaces Iain T. James

467

31 Design of sports facilities Franz Konstantin Fuss

480

Index

495

viii

FIGURES

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 2.1 2.2 2.3 2.4 2.5 2.6 3.1 3.2 3.3 3.4

3.5 4.1 4.2

Environmental impact categories Adidas environmental sustainability framework Inventory tree for composite tennis racquets Comparison of environmental impacts of the four composite tennis racquets of Table 1.1 Environmental impact breakdown across racquets’ components Environmental impact breakdown for racquet frames Value of the advanced materials used in sports equipment for 2004, 2005 and 2010 in US$ millions Recycling system for polyester products Closed-loop supply chain: main processes and constraints Armhole detail of the seam free integral garment Sustainable manufacturing framework Continuous improvement model for sustainability Overview of the hot-spot analysis (HSA) process Typical process flow diagram for sports footwear manufacturing Typical factory layout for sports footwear manufacturing (tier-2 supplier) Electrical energy flow in manufacturing of sports footwear (tier-2 supplier) Application of instrumented equipment in training and competition Three angles of instrumented equipment Scanning electron microscope image of the MEMS structure found on an Analogue Devices ADXL202 dual axis iMEMS® accelerometer die Principle of the longitudinal piezoelectric effect (a) unloaded crystal; (b) loaded crystal and method of increasing the charge yield; (c) by connecting several crystal plates mechanically in series and electrically in parallel Free-body diagrams of skier (a) and ski; (b) with boot and insole; (c) with instrumented binding vLinkTM shuttles (a, arrow) mounted on skis and (b) receiver Smart tenpin bowling ball: (a) design; (b) instrumentation and assembled ball; (c) force vector diagrams of a hook shot (right and top views)

ix

5 6 8 10 10 11 13 20 21 23 30 31 32 35 36 37 46 47 50

53 56 61 63

Figures

4.3 4.4 4.5

4.6 4.7 4.8 4.9

4.10

4.11 4.12 5.1

5.2 5.3

6.1 6.2 6.3 6.4 7.1 7.2 7.3 7.4 7.5 7.6

Instrumented cricket ball: (a) first prototype; (b) assembled ball; (c) performance parameters (first flight phase of the ball from release r to impact i) 65 Instrumented American football and electronic components 66 Instrumented Australian Rules football: (left) angular velocities of a torpedo punt against time (ωx, ωy, ωz: angular velocity about the three coordinate axes of the ball; ωr: resultant angular velocity; ωp: angular velocity of the spin axis about the precession axis); (right) three-dimensional visualisation of the spin axis (torpedo punt data shown in the left image) and its precession (axis cone) relative to the ball 67 Representation of force vector diagrams on the curved surface of an instrumented climbing hold, Singapore National Championships 2002 71 Vector diagrams of a more (left) and less (right) experienced climber, Climbing World Cup, Singapore 2002, ladies quarter finals 72 Fully instrumented climbing wall; motion sequence and force vector diagrams of two different climbers superimposed on the holds 74 Speed profiles of two 100-metre wheelchair racers (Paralympians, classified as T53 and T52); in the second race, the T52 athlete’s acceleration is discontinuous (arrow), resulting in a loss of 2 seconds over 100 metres 76 Push frequencies of three wheelchair racers over 100 metres; (left) consistent and decreasing frequency; (centre) consistent and constant frequency; (right) inconsistent frequency 77 Acceleration during a wheelchair rugby match 78 Classification of activities during a wheelchair rugby match 79 Inertial sensor platform used in swimming (left); together with sample sensor outputs for four strokes of freestyle swimming from a wall push off in the forward, sideways and up axis, respectively (right) 89 Data fusion during skiing: centre of mass trajectory and force vectors obtained from down hill skier during turns 90 Physiological load and arousal are measured during ski jump; jump phases are determined by the use of inertial sensors; arousal (obtained from heart rate) is clearly different during practice and competition 91 Jogging over a Distance between London, UK, and Melbourne, Australia 102 A jogger running at night owing to the time differences between London, UK, and Melbourne, Australia 103 Jogging over a Distance’s mini-computer, mobile phone, headset and heart rate belt as worn by the jogger 104 Jogging alone, yet together through technology 105 A typical design drawing required for UCI Technical Committee evaluation 115 Designing for road and time trial events in Europe requires determination of the frame loads from racing often on a cobble road surface 116 Olympian Garry Niewand mapping the load spectra with a strain-gauge-fitted RMIT Superbike 120 The 2004 concept bicycle fitted with off-the-shelf bicycle components and wheels in a trial fit-up exploring conversion of the type to a road racer 122 Superbike showing torsion effect (as described by Wilson 1995) of a time trial standing start 123 Lightweight Styrofoam legs enable designers to model in a wind tunnel the influence of the riders moving legs on the complete system aerodynamics, note the tufts on the frame indicating flow direction 124 x

Figures

7.7 7.8 7.9

Computation fluid dynamic model of frame cross-section aerodynamics The tooling concept for the Superbike Mark 2 The RMIT Superbike and Superbike 2 were designed to minimise tooling change; CAD overlay of the head tube inserts, facilitating a single production tool 7.10 Production Superbikes awaiting paint finishing at Techniques Sport factory 7.11 The commercial production RMIT Superbike 8.1 (a) Single pivot and (b) four-bar linkage suspensions 8.2 (a) Two-leg fork; (b) Rear shock schematic 8.3 (left) Brake lever with master cylinder; (right) rotor and caliper 8.4 Drive-train components 9.1 Progression of design of rowing shells; top row: (left) Wooden eight, built by Sims of Nottingham in 1980, (right) Composite four with wooden shoulders, built by Sims of Nottingham circa 1990; bottom row: (left) Janousek all-composite pair/double scull, built by Janousek of Byfleet circa 1995, (right) composite, shoulderless four with wing riggers 9.2 (left) The effect of movement of the pivot on the efficiency of a lever; (right) the effect of oar stiffness on propulsive efficiency 9.3 Hull velocity (solid line) and handle force (dashed line) for a single stroke by a heavyweight single sculler (M1x) 9.4 Path of the blade for a single stroke; the direction of motion is to the right of the page; black lines indicate the drive phase of the stroke, when the blade face is in the water; time interval between frames is 0.04 seconds 9.5 Pocock monocoque shell with wing riggers 9.6 Blade geometries: (left) modern Macon oar (1980s); (right) Smoothie (Big Blade) type oar (1991–present) 9.7 Forces acting on the blade during the drive phase of the stroke 9.8 Bending moment (solid line, units in Nm) and shear force (dashed line, units in N) along the oar for a male heavyweight sculler; forces correspond to maximum force applied during the drive in Figure 8.3 10.1 Track racing wheelchairs with standard spoke (top), aero-spoke (centre) and disc wheels (bottom) 10.2 Ball sports wheelchairs: tennis (left), rugby (centre), basketball (right) 11.1 Skiing styles: (left) classic; (right) skate (freestyle) 11.2 Triaxial carbon construction: 1 cap; 2 linear fibreglass; 3 unidirectional carbon fibre; 4 core; 5 triaxial woven fibreglass; 6 fibreglass veil; 7 base 11.3 Modern cross-country bindings. a) New Nordic Norm, Rottefella AS; b) Salomon Nordic System, Salomon SAS; c) the principle construction 11.4 Generalised Stribeck curve applied to and modified for the ski glide 11.5 Ski running surface and snow crystals 11.6 Kinetic coefficient of friction as a function of snow hardness 11.7 Unwaxed stoneground ski running surface at magnification of 150 11.8 Dynamic wetting (sliding) of water droplet on a solid surface 11.9 Stena Airmax ship – cavity in ship’s hull filled with pumped air (‘air cushion’) 11.10 Shear and tensile hydrophobicity 11.11 Running surface without kick wax 12.1 General snowboard cross-sectional structure

xi

125 125

127 128 128 132 133 137 139

142 143 146

147 149 151 151

152 159 160 171 173 174 175 177 179 181 182 182 183 184 190

Figures

12.2

Snowboard design parameters: (a) chord length, (b) self-weighted geometry, (c) top view (symmetric snowboard), (d) self-weighted camber, (e) thickness geometry, (f) edge sharpness angle, (g) top view (asymmetric snowboard) 192 12.3 Distribution of bending and torsional stiffness 197 12.4 Frequency response functions of freely suspended boards A–E (as described in the text) 199 13.1 Typical modern skate design 203 13.2 Skate fit: (top) pressure in different areas of skate boot measured in PSI while standing in skates that are smaller, regular or larger than normal fit; (bottom) perception of pressure (PP) and perception of comfort (PC) for individuals in skates that are smaller, regular or larger than normal skate fit 205 13.3 Pressure exerted during left tight turns on the outer lateral calcaneus in recreational and elite ice hockey players 206 13.4 Passive and active torque generation versus range of motion in standard and 209 modified skates measured in situ using the BiodexTM apparatus 13.5 Range of motion, work and power in standard and modified ice hockey skates during the performance of forward skating and crossovers on-ice; skates trended to higher values on all variables and tasks in modified skates 210 14.1 Directional energy return; (a) different surface models; (b) finite element analysis; (c) and (d) absolute energy return 225 14.2 Directional energy transfer in Adidas Bounce shoes; energy profiles from foot-flat to toe-off; (a) un-rotated, (b) counter-clockwise rotation, (c): clockwise rotation 227 15.1 Sports apparel manufacturing process 234 15.2 Woven (left) and knitted (right) fabrics 235 15.3 Measurement of evaporative resistance: (left) with thermal manikin and (right) at different garment zones 236 15.4 Test cylinder set up in wind tunnel 239 15.5 Experimental set-up in wind tunnel 240 15.6 Microscopic structure of Speedo LZR swimsuit; (left) swimsuit fabric structure under electron microscope [100⫻ magnification]; (right) polyurethane panel, swimsuit fabric and glue under electron microscope [100⫻ magnification] 241 15.7 Experimental set-up for TYR swimsuit materials 241 15.8 Microscopic structure of TYR swimsuit; (left) polyurethane panel [all black painted area]; (right) fabric on the red strip superimposed on polyurethane panel 242 15.9 Drag coefficient variation with Reynolds numbers for Speedo LZR and TYR Sayonara swimsuits 242 15.10 Typical cycling and ski-jump suits 244 15.11 Surface structures for cycling suit, ski-jump suit and test cylinder; (left) fabric used for cycling garments [skinsuits]; (centre) fabric used for ski-jump suits; (right) external surface of bare cylinder 245 15.12 Drag coefficient variation with Reynolds numbers for cycling and ski-jump suit materials 245 16.1 Road racing (left) and time trial (right) helmets 254 16.2 Recreational bicycle helmets 254 16.3 Mountain biking and BMX helmets 255 16.4 A plan view of the six helmets studied 257 16.5 Experimental set up in the RMIT research wind tunnel 258

xii

Figures

16.6 16.7 16.8 16.9 16.10 16.11 16.12 16.13 16.14 16.15 16.16 16.17 17.1 17.2

17.3 17.4 17.5 18.1 18.2

18.3

18.4 19.1

19.2 19.3 19.4

Temperature variation with wind speeds at 45-degree pitch and 0-degree yaw angle Drag coefficient (CD) variation with wind speeds at 45-degree pitch and 0-degree yaw angle Dynamic pressure distribution around helmet with and without vents using CFD modelling Typical cricket (left) and baseball (right) helmets An exploded view of a cricket helmet assembly Cross-section of a cricket ball (left) and the corresponding finite element model (right) Finite element model of the helmet hard shell Finite element model of a selected headform Geometry of the polycarbonate face guard Exploded view of the finite element (impacted) model Deformation of the face guard von Mises stress distributions in both the face guard and attachment Karabiners (B, D, X, H, K, Q = classification according to Table 17.1) Belay devices; (1) figure of eight; (2) brake bar karabiner; (3) rappel rack; (4) belay ring; (5) sticht plate; (6) magic plate; (7) tube brake without magic plate function; (8) tube brake with magic plate function; (9) hybrid brake; (10) linear semi-automatic brake; (11) angular semi-automatic brake Belay devices; (a) karabiner; (b) figure 8; (c) tube brake; (d) magic plate; (e) angular semi-automatic brake; (f) hybrid brake Cams: (a) spring-loaded camming device placed in a crack; (b) Abalakov cam; (c) uni-axial cam; (d) dual-axial cam (arrows indicate the axles) Cam mechanics; (a) logarithmic spiral and pitch angle θ; (b) free-body diagrams; (c) uni-axial spring-loaded camming device at different crack widths Plot of hardness against Young’s modulus for metallic alloys used in golf driver heads Schematic diagram of sectioned driver heads showing how a face could be forged in a ‘cupped’ form so that the weld between it and the crown can be reduced in size and moved back away from the face to increase its likelihood of being ‘good’ (a) Section through a complete CFRP head showing thickness at start of crown; (b) adhesive bonding overlap on composite crown of mixed composite/titaniumbased alloy head; (c) corresponding overlap on titanium-based alloy face of same head (a) Sand-blasted wedge showing damage to the groove profiles (b); (c) milled AerMet 100 face insert and clean groove edges (d) Power of the Head Youtek IF Speed 18 ⫻ 20 racquet: (a) power (per cent) at impact locations (51 per cent corresponds to a CoR of 0.51); (b) power zones; (c & d) Head Youtek IF Speed 18 ⫻ 20 compared with Dunlop Maxply Fort (wooden racquet) Impact between ball and racquet Hand force FH against time of an accelerated racquet for impacts at various locations (percentage of racquet length measured from the butt) Tension loss against stiffness

xiii

259 260 261 263 264 267 268 269 270 271 272 274 279

283 283 287 288 298

298

299 302

308 311 314 316

Figures

19.5

19.6 19.7 19.8 19.9 20.1 20.2 20.3 20.4

21.1 21.2

22.1

22.2

22.3

23.1

23.2 23.3

Video images taken at 600 frames per second showing a ball travelling right to left across the strings; image (1) was taken just before the ball strikes the strings and image (4) shows the ball just leaving the strings; in between, the ball pushes the main strings to the left and then the strings snap back while the ball is still on the strings; the snap back imparts additional spin to the ball 317 Typical mode shapes and frequencies of a tennis racquet 319 Typical impulse response on the handle of a tennis racquet 320 Effectiveness of the dynamic absorber 321 Frequency response function (FRF) showing the effect of a tuned absorber on mode 1 322 The effect of I on the hit ball and swing speed (left) and on the collision efficiency (right) 328 The bat–ball coefficient of restitution e as a function of the added weight location 328 Results for normalising e 334 Comparison of a bat profile (cross-section) in the original undeformed shape (dotted line), from impact with a stiff ball (thick line, 11,180 lb/in or 1,963 kN/m), and from impact with a softer ball (thin line, 4,528 lb/in or 795 kN/m) 335 Regions and dimensions of the ice hockey stick: (a) and blade (b); specific blade pattern properties of curvature (c) and twist (d) 340 Representative trial of a slap shot: shaft linear and angular displacement recorded at 1000 frames per second; maximum bend angle (θ ) measured between upper and lower shaft segments 344 The variation in average drive distance for the around 200 professionals on the US PGA tour (two holes for each tournament each year), 1980–2006; the increase marked A is associated with player conditioning; B is from the introduction of titanium-based alloy hollow oversized drivers; C is from the replacement of the Titleist wound ball by the ProV1 and ProV1X 351 (a) Variation in loading/unloading behaviour for samples from the edge and centre of a PBD core; (b) comparison of loading and unloading behaviour for golf ball core samples at 0.00154 s–1; (c) comparison of loading and unloading behaviour for golf ball core samples at 0.0154 s–1; (d) loading and unloading curves for full cores; (e) loading and unloading curves for full balls based on the cores in (d) 353 (a) Sectioned ball showing cover thickness variations; (b) reheated golf ball showing cover flow lines; (c) seams in compression moulded core and mantle; and (d) seam on cover and thin polyurethane cover 355 Construction of cricket balls: (a–c) schematic cross-sections of cricket balls (a) non-layered; (b) three cork layers; (c) five cork layers; (d) Regent Match (white) – non-layered; (e) cork layer element of Sanspareils-Greenlands Tournament rubber core ball; (f–g) cross-sections of layered balls (SanspareilsGreenlands Tournament subtypes): (f) rubber core version; (g) cork core version 362 Velocity independent elasticity parameter R against deflection (a) and viscosity 366 parameter η against compressive force (b) in different types of cricket balls Box and whisker plots of the coefficient of restitution (CoR) of different cricket ball models 367

xiv

Figures

23.4 23.5 23.6 23.7 23.8 23.9 23.10 23.11 23.12 23.13 23.14 23.15 23.16 23.17

24.1 24.2 24.3 24.4 25.1

25.2

25.3

25.4 26.1

Hardness of cricket balls; acceleration against time (British Standard hardness test) and hardness of different cricket ball models 367 Coefficient of restitution (CoR) values determined with the bounce (CoR1) 368 and hardness tests (CoR2); CoR1 plotted against CoR2 Force against deflection and dynamic stiffness of different cricket ball models 369 Contact interaction properties 370 Experimental setup for cricket ball drop test 371 High-speed impact test installation 372 Experimental impact force results for a three-layer cricket ball with impact speed from 5.5–25 m/s 373 Fast-solving mathematical models of cricket balls; (left) single-element model (right) three-element model 373 Single-element model simulation results for impact speed of 27 m/s 374 Three-element model simulation results for impact speed of 27 m/s 375 Cross-section of a cricket ball and the corresponding finite element model 375 Comparison of impact forces for the five-layer cricket ball under an impact speed of 20.8 m/s 376 378 Coefficient of drag CD against the Reynolds number Re Explanation of flow regimens generating different types of swing; note that, in the CD–Re diagram, the CD is not of importance, as it simply reflects the positions of the separation points, the flow velocity on either side of the ball, the local pressure, the pressure difference and thus the direction of the side force (the smaller the CD, the further to the rear the separations points are located, the higher the flow velocity and the lower the local pressure); the further the CD curves are apart, the higher is the pressure difference and thus the side force; note that curve B is not necessarily the same for conventional and contrast swing 382 Representative force-displacement curve from a softball impacting a rigid cylinder at 36 m/s (80 mph) 390 A comparison of the cylindrical coefficient of restitution ec of baseballs and softballs as a function of temperature (a) and relative humidity (b) 392 A comparison of the dynamic stiffness of baseball and softballs as a function of temperature (a) and relative humidity (b) 392 Baseballs, golf balls and smooth spheres: (a) aerodynamic drag CD and (b) lift CL; Re = Reynolds number, S = spin factor 394 Examples of hockey balls: (a) hollow rotationally moulded polyurethane (PU) ball; (b) multi-layer PVC-covered ball; (c) multi-layer cork-cored ball (thin PU cover); (d) cork-resin, cork agglomerate, PU; and (e) cork-resin core with thick PU cover 402 Representative scanning electron micrographs of (a) cork aggregate (as seen in the outer core layer of Figure 25.1d); and (b) cork-resin composite (used in the core of Figure 25.1e and the inner of Figure 25.1d) 404 Experimental and finite element modelled stress–strain curves for cork–resin composite: (a) ε· = 0.018s–1, maximum ε = 0.125; (b) ε· = 0.0059s–1, maximum ε = 0.075; Exp = experiment 406 Variation in coefficient of restitution (CoR) values with impact speed for three hockey ball types 407 Three oval-shaped balls used for experimental study: (a) rugby ball; (b) Australian Rules football; (c) American football 412 xv

Figures

26.2 26.3

Experimental set-up in the test section of RMIT industrial wind tunnel Drag coefficient (CD) as a function of yaw angles and speeds; (a) rugby ball, (b) Australian football, (c) American NFL football, (d) for three oval-shaped balls at a speed of 100 km/h 26.4 Computational fluid dynamics of a simplified rugby ball; velocity contour at 100 km/h and 0 degrees (a) and 90 degrees; (b) yaw angles 26.5 Contour plot and isotribes (lines of identical friction) of CoF as a function of speed and load 26.6 Coefficient of friction (CoF µk) against roughness (Ra) of different surfaces (A–E) and test directions 26.7 Impact of an oval ball 27.1 Orientation of seam towards wind direction,Wilson US Open 3 ball: (a) position 1 (0 degrees); (b) position 2 (90 degrees); (c) position 3 (180 degrees); (d) position 4 (270 degrees) 27.2 Flow visualisation of 28-cm diameter tennis ball model with no spin (Reynolds number 167,000) at NASA Ames Research Center; flow is from left to right 27.3 Flow visualisation on ball with underspin (clockwise rotation at 4 rps, Reynolds number 167,000) at NASA Ames Research Center; flow is from left to right 27.4 Drag coefficient plotted against Reynolds number for new tennis balls; 1: different tennis balls, 2: smooth sphere (subcritical regimen), 3: bald tennis ball (all flow regimens) 27.5 Drag coefficient versus Reynolds number for used Wilson US Open balls 27.6 Dynamics of a bouncing ball 27.7 Impact analysis with SwingerPro 2.0 (Websoft Technologies, Scoresby, Australia) 27.8 Coefficient of friction (CoF) and spin rate; (a) CoF against time (duration of impact); (b) CoF against time and spin rate 27.9 Horizontal coefficient of restitution (CoRx); left: CoRx against spin rate; right: coefficient of friction (CoF) against CoRx 28.1 Transformation of soccer ball design from 1900 to 2010: (a) soccer ball 1900, eight panels, leather; (b) soccer ball 1920, 12 panels, leather; (c) soccer ball 1958, 18 panels, leather; (d) soccer ball 1960s, 32 panels, leather; (e) Adidas Telstar ball, 1970, 32 panels, leather; (f) Adidas Fevernova ball, 2002, 32 panels, synthetic; (g) Adidas Teamgeist ball, 2006, 14 panels, synthetic; (h) Adidas Jabulani ball, 2010, 8 panels, synthetic 28.2 Teamgeist III soccer ball schematics and a production replica ball 28.3 Jabulani soccer ball 28.4 Drag coefficient plotted against Reynolds number for non-spinning soccer balls, golf ball and smooth sphere 28.5 Drag coefficient as a function of spin parameters for different Reynolds numbers (Re) 28.6 Adidas soccer balls: (a) 32-panel Fevernova ball; (b) 8-panel Jabulani ball; (c) 14-panel Teamgeist II ball; (d) 14-panel Teamgeist III ball 28.7 Wind tunnel experimental set-up at RMIT University 28.8 Flow structure around soccer balls: (a) 32-panel Nike ball; (b) 14-panel Teamgeist III ball 28.9 Drag coefficients of Jabulani,Teamgeist II,Teamgeist III and Fevernova balls and a smooth sphere 28.10 An erratic flight trajectory of a soccer ball xvi

413

414 415 416 416 420

426 429 429

430 431 433 436 436 437

440 441 442 443 444 444 446 446 447 448

Figures

28.11 Surface variation of seams, stitches and grooves; (left) Jabulani ball; (centre) Teamgeist III ball; (right) Fevernova ball 449 29.1 Example of a state-of-the-art construction of artificial turf systems 456 29.2 A monofilament extrusion line; left side: extruder, spinneret, water bath; right side: multi-storey ovens for fibre stretching and relaxation 457 29.3 Artificial turf carpet with fibres, primary backing and secondary backing (polyurethane coating) 460 29.4 Artificial turf surface with fibres, granular infill of coated sand and rubber 461 29.5 Simplified exemplary forces (arrows) illustrating player–surface interactions with multidirectional forces and stresses acting during play on turf surfaces 464 30.1 A simplified diagram of the essential requirements of the turfgrass plant, which include solar light and heat, above ground exchange of carbon dioxide (CO2) and oxygen (O2), sufficient water (H2O) from precipitation and/or irrigation to infiltrate into the soil rootzone and be drawn into the plant through its root system, together with essential plant nutrients; there must also be sufficient air-filled porosity to allow respiration within the rootzone 468 30.2 The classification of soils commonly used in sports surfaces by the proportion 470 of sand (2 mm – 63 µm), silt (63 µm – 2 µm) and clay (< 2 µm) particle sizes 30.3 Shear stress at failure with increasing saturation ratio in triaxial testing of two soils used in sports surfaces at 100 kPa confining stress 472 31.1 Whitewater parks: (a) Penrith, Australia (Sydney 2000 Olympic Games); (b) Athens, Greece (2004 Olympic Games); (c) Charlotte NC, USA (US National Whitewater Centre); (d) Shunyi, China (Beijing 2008 Olympic Games); boat lifts are marked with small arrows and water pumps with large arrowheads) 482 31.2 The Water Cube: outdoor (a) and indoor (b) views 483 31.3 Insulated greenhouse effect of the Water Cube: (1) fan-assisted preheated fresh air returned to pool; (2) ETFE cladding can be switched ‘on’ or ‘off ’ to shade interior; (3) controlled natural daylight and radiant heat to passively heat and light the pool; (4) ETFE pillows act as greenhouse and avoid condensation; (5) fresh external air preheated in vented cavity 484 31.4 Table-top design compared with optimised design 485 31.5 Gradients of various landing points within a given xy plane for a kicker angle 488 of θ0 = 25º and a fall height of h = 1 m 31.6 Components of ball management systems: (a) ball picker; (b) Range Servant ball management system; 1: drive unit; 2: ball picker; 3: ball ditch; 4: elevator; 5: ball washing unit; 6: blower; 7: hose; 8: feed hopper; 9: blower manifold; 10: tee-up; 11: dispenser unit; 12: tee mat 490 31.7 Golf balls inseparably melted together 492

xvii

TABLES

1.1 1.2 1.3 2.1 2.2 5.1 7.1 7.2 7.3 7.4 7.5 9.1 9.2 11.1 12.1 12.2 15.1 16.1 17.1 18.1 20.1 22.1 22.2 23.1 25.1 26.1

Average resource consumption per pair of sports shoes Materials inventory for four tennis racquets Comparison of the energy efficiency for carbon-epoxy composite and aluminium Environmental and social impacts across the life cycle of sport performance apparel Specific energy consumption of injection-moulding machines used for benchmarking (case study) Sensor types and technologies Aerodynamic drag and rolling friction design data used for the RMIT Superbike Definition of the terms and symbols used in the performance simulation model Validation of the simulation performance model against back-to-back test rides Frame size chart (Thompson and McLean 1995) for road and road-time-trial élite athlete frames Progressive performance improvement of RMIT-AIS Superbike Makes of boat used by medallists at the World Championships, November 2010 FISA minimum weights for competition boats Hardness at room temperature of ski base materials (top four) and of some glide waxes (bottom 4) intended for the cold and dry snow conditions (Kuzmin 2010) Objective snowboard design parameters Key objective design and subjective performance data Speed sports requiring engineered apparel (all body fitted) Vents and their locations for helmets selected for the study Classification of karabiners; type, usage, design and strength data according to EN 12275 and UIAA 121 Summary of selected metallic alloy mechanical properties Properties of common woods used to make bats Summary of criteria for conforming golf balls Summary of properties used in golf balls Stiffness clusters Summary of International Hockey Federation testing criteria for hockey balls suitable for international games Dimensions of the three balls tested in the RMIT industrial wind tunnel xviii

6 9 16 29 37 84 117 118 119 127 127 144 148 179 192 197 237 258 280 301 325 352 354 363 400 413

Tables

27.1 International Tennis Federation rating of coefficient of restitution (CoR), coefficient of friction (CoF) and surface pace rating (SPR) 27.2 International Tennis Federation rating of court pace (CPR) 27.3 Variation of surface pace rating (SPR) and court pace rating (CPR) with coefficients of restitution (CoR) and friction (CoF) 28.1 Physical parameters of the four FIFA-approved soccer balls used in the study by the Sports Aerodynamics Research Group at RMIT University 29.1 Parameters relevant for production of high-quality turf yarn 29.2 Factors influencing player–surface interaction and biomechanics

xix

434 435 435 445 458 464

CONTRIBUTORS

Firoz Alam RMIT (Royal Melbourne Institute of Technology) University, Melbourne, Australia Thomas Allgeuer Dow Europe GmbH, Horgen, Switzerland Takeshi Asai University of Tsukuba,Tsukuba-city, Ibaraki, Japan Len Brownlie Aerosports Research,Vancouver, British Columbia, Canada Patrick Clifton Australian Football League, Melbourne, Australia Rod Cross Sydney University, Sydney, Australia Enda Crossin RMIT (Royal Melbourne Institute of Technology) University, Melbourne, Australia John Dominy University of Nottingham, Nottingham, UK, and Carbon Concepts Ltd, Clay Cross, Derbyshire, UK Franz Konstantin Fuss RMIT (Royal Melbourne Institute of Technology) University, Melbourne, Australia Mehdi Hedayati RMIT (Royal Melbourne Institute of Technology) University, Melbourne, Australia

xx

Contributors

Daniel Arthur James Griffith University, Brisbane, and Queensland Academy of Sport, Brisbane, Australia Iain T. James TGMS Ltd, Cranfield, Bedfordshire, UK Jeff Kensrud Washington State University, Pullman,WA, USA Leonid Kuzmin Kuzmin Ski Technology AB, Karlstad, Sweden Allan McLennaghan Dow Europe GmbH, Horgen, Switzerland Rabindra Mehta Sports Aerodynamics Consultant, Mountain View, California, USA Adrian Mouritz RMIT (Royal Melbourne Institute of Technology) University, Melbourne, Australia Florian Mueller RMIT (Royal Melbourne Institute of Technology) University, Melbourne, Australia Günther Niegl Climb On Marswiese Climbing Gym,Vienna, Austria David J. Pearsall McGill University, Montreal, Quebec, Canada Dan Ranga University of Birmingham, Birmingham, UK Robin C. Redfield United States Air Force Academy, Colorado Springs, Colorado, USA Alastair Campbell Ritchie University of Nottingham, Nottingham, UK Peter Sandkuehler Dow Chemical Iberica,Tarragona, Spain Bahman Shabani RMIT (Royal Melbourne Institute of Technology) University, Melbourne, Australia Carl Slater University of Birmingham, Birmingham, UK

xxi

Contributors

Lloyd Smith Washington State University, Pullman,WA, USA Martin Strangwood University of Birmingham, Birmingham, UK Aleksandar Subic RMIT (Royal Melbourne Institute of Technology) University, Melbourne, Australia Monir Takla RMIT (Royal Melbourne Institute of Technology) University, Melbourne, Australia Lachlan Thompson RMIT (Royal Melbourne Institute of Technology) University, Melbourne, Australia Olga Troynikov RMIT (Royal Melbourne Institute of Technology) University, Melbourne, Australia Rene A. Turcotte McGill University, Montreal, Quebec, Canada

xxii

PREFACE

Sport is a major global business with supply chains spreading across all geographical borders. Sport is also a fundamental human activity that impacts people on all continents, of all races, cultures and abilities. Like arts or music, it represents the ultimate equaliser between people and nations. The sports and leisure industry globally is as big as the aerospace industry. The size of the entire sports industry involving both sports products and sports services is estimated at over US$800 billion annually. The world sporting goods market is approximately US$120 billion retail, with footwear accounting for US$30 billion, apparel US$50 billion and equipment US$40 billion. The market growth of the global sports industry is 50 per cent faster than the growth corresponding to the general consumer spending. More specifically, the growth of the sports equipment sector is 2.5 times faster, health and fitness equipment twice as fast and apparel (clothing and footwear) 1.5 times faster than the growth of the general consumer spending.The consumer spending on sports equipment accounts for approximately one-third of the overall sports spending. The global sporting events market is typically dominated by soccer, tennis, golf, American football, baseball and Formula 1. The global sports industry is growing faster than the overall gross domestic product, while the market growth for sports equipment is more than twice that of the global economy. The long-term growth prospects remain strong. Investment in research and innovation, design and manufacturing of sports products is significant.This is driven by increasing demands for performance enhancement, injury prevention, personalised design and mass customisation and, perhaps most importantly, by the need to integrate sustainability across the entire supply chain. The sports industry has been quick to adopt new technologies stemming from research in other industry sectors, such as from aerospace, defence and biomedical sectors. These include, for example, high-performance lightweight materials such as aluminium alloys, titanium, carbon-fibre composites and Kevlar®; laser technology, vision and advanced imaging technology; implants and prosthesis; nutritional supplements and electrolyte containing beverages; surgical interventions, artificial atmospheres and other biomedical enhancements. The manufacturing intelligence and practices developed by the automotive industry for high-volume production involving vast supply chains that are distributed across different geographical regions are also increasingly being adopted by the sporting goods industry. Furthermore, like in the automotive industry, the sporting goods xxiii

Preface

industry will in the future most likely adopt higher levels of automation in the manufacturing of sporting goods. Research and innovation associated with sports technology and engineering is multidisciplinary and interdisciplinary. It utilises and integrates through design and analysis many disciplines, including materials science and engineering, aerodynamics and hydrodynamics, impact mechanics, tribology, biomechanics, thermodynamics, electronics, computer science and software engineering, chemical engineering and others. For example, research and development of sports balls is typically aimed at maximising the speed and distance of flight, maximising the energy storage and return (while minimising energy loss), maximising friction for improved grip and generation of spin (while minimising friction for better sliding and rolling effects). This is achieved primarily through advances stemming from the application of aerodynamics, impact mechanics and tribology principles. The wide range of sports and sports technologies covered in this book are used to describe and explain how the different disciplines are manifested in sports and utilised in the design and development of sports equipment. Research, design and development of sports equipment must consider the interaction between the athlete, the equipment and the sport environment. For example, the muscle energy that athletes produce enables them to do work in sports. The energy released to the environment accounts for the performance of the athlete. The mechanical muscle energy is thereby converted for example into: (i) kinetic energy: linear velocity (running, cycling, wheelchair racing, ball kicking, etc.) and angular velocity (figure skating, platform diving, etc.); and (ii) potential energy (high jump, pole vault, weight lifting, climbing, etc.). In addition to the energy released to the environment, energy can be absorbed and is non-recoverably lost (nonconservative energy). Furthermore, energy can be stored in the equipment and is returned to the athlete (conservative energy), thereby increasing the athlete’s performance. There are a number of ways to optimise the energy transfer between the athlete, equipment and environment, such as by: (i) maximising the (conservative) energy which is returned; (ii) minimising the (non-conservative) energy which is lost; (iii) optimising the musculoskeletal system; and (iv) maximising the non-conservative energy.The book describes in detail a number of engineering solutions to achieving these objectives. At its most basic, anything that is able to deform elastically can store and return energy. If the structure or material is non-viscous, then the energy return is ideally 100 per cent. The general requirements for optimal energy return, however, are more complex.To make effective use of the returned energy, the forces from the surface must be exerted at the right location, in the proper direction, at the appropriate time and with the right frequency. This explains why elastic deformation alone does not necessarily guarantee full energy return. If, for instance, the fundamental frequency of a running prosthesis does not match the runner’s stride frequency and the runner enters the flight phase at that point when the prosthesis is maximally deflected, then no energy is returned at all.Typical examples of performance enhancement in sports based on energy return of the sport equipment include: • area elastic and point elastic sports surfaces, which deform and return energy to the athlete; • vaulting poles which improved energy return over the last century and enabled higher jumps; before 1910, poles were made of solid wood, of bamboo before WWII and finally of metal (aluminium, steel) and fibre composites (fibreglass in the 1960s and of carbon fibre today); • running shoes with directional energy return and directional energy transfer; • trampolines and spring boards; • compliant bicycle cranks; xxiv

Preface

• ice hockey sticks, which are pre-bent by the athlete when hitting the ice surface and, when snapping back, add a velocity component to the blade and ultimately to the puck. Other engineering solutions are used in sports to minimise non-conservative energy, for example, by minimising the resistive forces, such as aerodynamic and hydrodynamic drag, and friction forces. Drag increases with the projected area, with the drag coefficient, with the square of the velocity and with the fluid density.The density cannot be changed (except when moving to higher altitudes) and the speed is actually the parameter which is intended to be increased by reducing the drag.The target parameters are therefore the projected area and the coefficient of drag.The projected area can be reduced in various ways, by for example crouching/tucking as far/deep as possible (e.g. downhill skiing ‘eggshell position) or by compressing the body sideways with special suits, that move the body mass into the projected area (e.g. speed skating, swimming). Adverse friction forces affect the ease of sliding and rolling in sports. As explained in this book, sliding friction is important in winter sports, on ice and snow, and is improved by structuring the surface (e.g. grinding), by its material (formerly wax, nowadays hydrophobic running surfaces such as ultra-high molecular weight polyethylene and Teflon) and by the contact area (sharp edges). Furthermore, edge friction was a common source of energy and therefore speed loss in skiing, before the invention of the carving ski and snowboard. Rolling friction occurs in wheels, casters and bearings. In roller bearings, the hardness of the contact surfaces (e.g. diamond-coated) and the lubricant significantly reduces friction. In wheels, the contact area and viscosity of the tyre-surface pair determines the rolling friction force. The less the tyre and/or surface deforms (that is, the harder, more inflated, less loaded and less viscous it is) the smaller is the friction force. Lift typically has a positive effect in sports, provided that means of increasing lift does not worsen drag or weight.Aerodynamic lift reduces the body weight, which, in turn, decreases the friction force. Hydrodynamic lift moves the body out of the water, thereby reducing hydrodynamic drag (but also increasing the comparatively insignificant aerodynamic drag). The book describes this phenomenon in the case of the Speedo swim suits that were found to add to the buoyancy of the body, thus producing a lift force. As a consequence, FINA (the International Swimming Association) regulated the buoyancy of swim suits in 2009. Energy that is non-recoverably absorbed by and/or released from the sports equipment comprises internal friction, vibrations, heat, sound, and angular speed. Internal material friction absorbs and dissipates energy and leads to the hysteresis effect of a material. Internal friction depends on the degree of viscosity of a material and its loss tangent (ratio of energy lost to energy stored). Elastic energy is thereby converted to thermal energy (heat).The internal friction also accounts for the coefficient of restitution of a ball; that is, the ratio of rebound speed to incident speed. Cyclic movements (bicycles, wheelchairs) and impacts (implements, balls) can lead to vibrations if frequency modes are excited. Mechanical muscle energy is thereby converted to vibration energy, which is in most cases non-recoverable and, if damped, converted to heat. Impacts are characterised by sudden deformation, internal friction and vibrations that result in thermal and sound energy, the latter of which produce the characteristic impact sound (e.g. between an implement and a ball). Both energy effects (thermal and sound) are used to detect ball–bat impacts in cricket. An infrared camera reveals the bright spot of the impact location (‘hotspot’) and the ‘snickometer’ records the impact sound. Maximising non-conservative energy is the standard strategy for the design of protective sports equipment (e.g. for absorbing kinetic and impact energy when falling or being punched).These types of equipment comprise: safety fences and netting for skiing; crash mats xxv

Preface

for martial arts and bouldering; personal protective equipment (vests, knee and elbow pads, teeth protectors, mouth guards, helmets, boxing gloves, bungee ropes); cushioning mechanisms in sports shoes; and belay devices and rope brakes in climbing and mountaineering.The energy absorbing mechanisms make use of internal friction (e.g. foams) as well as external friction (rope brakes). Protective equipment indirectly (psychologically) improves performance as a higher degree of safety improves decision making and risk taking. Minimising non-conservative energy in human-powered sports does not mean that the athlete produces more energy. It rather means that the athlete loses less energy and is therefore faster. For example, the monocoque frame design of the RMIT Superbike (shown on the cover of this Book) was banned after several years of successful racing and the rules reverted back to the traditional triangle/diamond design.Whether a revolutionary improvement, when inaccessible to everyone and kept a national secret, has to be considered unfair is debatable; still, the principle applies that the improved equipment design does not allow the athlete to produce more energy. Furthermore, in gravity-powered sports (such as alpine skiing, or more specifically downhill racing), the performance-related energy is not directly produced by the athlete. In addition to the athletes’ skills of decelerating as little as possible, reducing energy loss is the first priority, in terms of low gliding friction, optimal racing position and using low-drag ski suits. Considering the engineering principle that the square of the terminal speed is proportional to the ratio of drag area to body mass, lighter athletes have a decisive disadvantage, which is far more critical than the drag reduction.This is why wearing lead belts or using heavy equipment is considered unfair and is banned by the rules. Maximising conservative energy does not allow the athletes to produce more energy but rather to recycle a part of their energy that is returned by the sport equipment.This improves their performance and is considered unfair, especially when infringing the rules of sport.This is why the maximal sole height of athletic shoes is regulated to set a limit to energy return. A classical case of unfair advantage was sought after when Oscar Pistorius wanted to participate in the 2008 Beijing Olympics. The research to shed light on this issue was commissioned by the International Athletics Federation (IAAF) and conduced by Brüggemann at the German Sports University in Cologne in 2007. The results revealed that Oscar’s running prostheses (Össur Cheetah blades) returned 97.5 per cent of the energy stored, in contrast to the 67 per cent returned by a human foot. Consequently, the IAAF banned Pistorius from participating in able-bodied competitions. This decision was overruled by the Court of Arbitration in Sports, as Oscar has a decisive disadvantage at the start and during the acceleration phase of a sprint competition. Nevertheless, maximising the conservative energy is advantageous if transparent and accessible to everyone: it enables grouping of élite athletes into standard élites and superélites. The design of new sports equipment inherently involves research and innovation. Innovative sports products must be tested scientifically, whereby appropriate test methods have to be embedded in the development process. In addition, extensive testing with élite athletes is indispensable, as the experience and expertise of athletes in terms of recognising that subtle differences of equipment behaviour provides invaluable feedback for optimisation of the product. Omitting an important aspect of equipment technology may result in a lopsided, if not flawed, product. For example, the design of the Soccer World Cup balls was changed from the traditional 32-panel buckyball design to 14 panels in the Teamgeist ball (2006) and eight panels in the Jabulani ball (2010). Fewer seams means a more uneven seam distribution over the ball’s surface, resulting in side forces, generally known from baseball and cricket ball aerodynamics. As a result, the new soccer balls showed a pronounced knuckle-ball effect, which goalkeepers complained strongly about. Surprisingly, it appears that no wind-tunnel tests were conducted xxvi

Preface

in the development process and no experimental aerodynamics results were published by the developers and/or manufacturers before the release of the new balls. This book provides a detailed analysis of this case study, as well as of other relevant experimental studies used in the development of contemporary sports products. The Routledge Handbook of Sports Technology and Engineering is the first comprehensive reference book in this field that covers both sports technology and the engineering research and development associated with the design and manufacture of sports equipment. To provide an in-depth review of the different types of sports equipment used in contemporary sports, the book describes in detail the underpinning theoretical principles associated with the particular design features of the sports equipment and how they affect their performance. It is worth noting that this is the first book of this type that addresses the topics of sustainable design and manufacturing of sports products, as well as instrumented and smart equipment. The many different aspects of sports technology that have been summarised above are covered in 31 chapters by 33 authors, all renowned international experts in their respective fields. Around one-third of the authors are experts from the sports industry while the remaining three-quarters are leading researchers from universities worldwide. The main sections of the book include sustainable sports engineering (design and manufacturing), instrumentation technology (equipment, athletes, exertion games), technology for summer mobility sports (bikes, rowing boats and sports wheelchairs), technology for winter mobility sports (cross country skis, snowboards, ice hockey skates), apparel and protective equipment (shoes, garments, helmets, mountaineering), sports implements (racquets, clubs, bats, sticks; for golf, ice hockey, baseball, tennis), sport balls (cricket, tennis, oval, golf, hockey, baseball, soccer), and sports surfaces and facilities (artificial and natural turf, sports facilities). This book is intended to be used as a textbook by senior undergraduate students and postgraduate students undertaking studies and research in areas of sports engineering, sports technology and sports science, at any level beyond an introductory course. In general, the book provides engineers and scientists with an entry point that gets them into the profession of a sports engineer/technologist, be this the sports industry or sports organisations. Furthermore, the book serves as a reference text for anyone who is involved in sport, such as managers, entrepreneurs, coaches, manufacturers and sports enthusiasts. Franz Konstantin Fuss, Aleksandar Subic, Martin Strangwood and Rabindra Mehta October 2012

xxvii

ACKNOWLEDGEMENTS

The Editors would like to thank Nicole Onslow for editorial assistance.

PART I

Sustainable sports engineering

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1 SUSTAINABLE DESIGN OF SPORTS PRODUCTS Aleksandar Subic, Adrian Mouritz and Olga Troynikov

Introduction The sporting goods industry is composed of sports apparel, sports footwear and sports equipment. This dynamic global industry sector has grown significantly over the years while continuously trying to meet the growing demand for new and improved sports products.According to a market research report (Global Industry Analysts 2008), in 2006, the consumption of sporting goods in the global market was more than US$250 billion, with a percentage breakdown of value sales by product category of 45.45 per cent sports apparel, 33.93 per cent sports equipment and 20.62 per cent sports footwear.While the global sporting goods market is large and growing, the majority of the sales still result from the most economically developed countries, with emerging markets in Eastern Europe, Asia and South America slowly catching up. New sports products innovations are rapidly developed and brought to market by the manufacturers to accommodate the diverse needs and changing personal preferences of users. To grasp the growing commercial opportunities, the industry has diversified by developing and producing lifestyle products in addition to performance products. This has resulted, over the years, in increased consumption of sports products, shorter life cycles and increased disposal rates and waste. Many innovations in sports products are associated with the application of new materials and processes, and with the rapid diffusion of advanced technologies developed by other industry sectors. However, new materials and processes used in sports products carry with them potential environmental risks. Looking at sports products innovations in the past, such risks have occurred; for example, with ski boots, sports apparel and packaging using polyvinylchloride (PVC)-based materials and athletic footwear using petroleum-based solvents and other potentially damaging compounds like sulphur hexafluoride in air bladders for cushioning and impact shock absorption (Subic 2007). Also, composites, such as carbon fibre-reinforced polymers, typically used in tennis racquets, hockey sticks, skis and other sports equipment, involve particular technological challenges at the end of their lives, as they cannot be readily recycled at an acceptable cost or value. Similarly, it is not possible to cheaply recycle equipment made using fibreglass composites. Advances in sports products have unintentionally placed additional burdens on the environment and on societies that are forced to deal with such products at the end of their lives. 3

Aleksandar Subic, Adrian Mouritz and Olga Troynikov

It is estimated that around 80 per cent of the environmental burden of a product is determined during the design stage, when most of the decisions are made about the selection of materials and processes for the new product. Hence, in modern design, environmental issues are given high priority. This has resulted in the development and application of new sustainable design methods and practices that are applicable to sports products and that can be effectively integrated in the design process (Subic and Paterson 2006, 2008; Subic 2007) Additionally, governments and relevant agencies (particularly in Europe) have introduced a wide range of environmental legislations to help reduce the environmental burden associated with manufactured products, indicating clearly that it is no longer acceptable for products to be incinerated or dumped in landfill after their useful life irrespective of the consequences. Sustainable design implies a lower social cost of pollution control and a higher level of environmental protection through more efficient use of resources, reduced emissions and waste. The move towards sustainable design practices raises some critical questions that must be addressed by the sports industry. Is the development of new sports products informed by state-of-the-art research and practice in sustainable design? What are the implications of this for the sporting goods industry? Are advances in recycling and reuse technologies for sports equipment and apparel keeping abreast of the growing requirements for sustainable design? Is ‘green’ design actually sustainable? This chapter addresses these questions by providing a detailed review of the current research in sustainable design and the environmental impacts of materials used in sports products. Special attention is given to composite materials in sports equipment and synthetic fibres used in sports apparel, both of which pose major sustainability issues for the industry going forward. The chapter encompasses three main sections that are perceived to be of highest interest in this context: sustainable design framework, composites in sports equipment and advanced textiles in sports apparel. The issues covered in this chapter highlight the key technological challenges and opportunities facing sports products and the sporting goods industry in general in its quest to embrace the sustainable design paradigm.

Sustainable design framework There is an ever-increasing need to develop, produce and use products that are robust, reliable and of high quality, supportable, cost effective and environmentally sustainable from a total lifecycle perspective, and that are able to respond to the needs of the user/customer, industry and society in a more sustainable manner. Different definitions of sustainability have been used in literature, with up to eight or more dimensions of sustainability reported to date, including: physical, environmental, economic, social, equity, cultural, psychological, ethical. Nevertheless, it is widely accepted today that sustainability encompasses three main domains: social, economic and environmental. Figure 1.1 for example shows the main environmental impact categories identified by Adidas across the entire value chain that supports the overall environmental sustainability strategy. Achieving a sustainable framework across the entire sports products business requires a strategic approach that captures all dimensions of sustainability (economic, social and environmental) and relates these to specific targets and timelines for achieving them. For example, Nike has established such approach within a corporate responsibility framework, whereby their success is being measured by the extent to which businesses meet their milestones for corporate responsibility in addition to achieving business growth. This is an example of a more enlightened approach whereby excellence in sustainability is seen as a prerequisite for business excellence and business leadership. 4

Sustainable design of sports products

Environmental Impact Categories

SUSTAINABLE RESOURCE USE

1

Energy efficiency & climate change

2

Water conservation

3

Materials

EMISSIONS

L2J

a

m

HAZARDS & RISKS

Water discharge

7

Chemicals

5

Waste

8

Sail & groundwater contamination

6

Air emissions

Figure 1.1 Environmental impact categories

Similarly, Adidas also reports on its commitment to corporate responsibility. Key focus areas of the Adidas strategy include: • • • •

environmental sustainability (product, production, own sites); supply chain (direct, indirect, business entities); stakeholder engagement (internal, external); and employees (succession planning, recruitment, employer of choice).

The environmental strategy is central to managing and reducing the environmental footprint across all business operations.The core areas of the Adidas environmental strategy are products, production and own sites, whereby the sustainability framework (Adidas 2008) is concerned with the entire product life cycle (Figure 1.2). Within such a framework, it is essential to establish specific measurable sustainability indicators (including environmental indicators), to conduct carbon and energy footprint analyses and a life cycle assessment, so as to make informed business decisions in relation to all relevant operations. Waste is of particular concern, including waste arising from the company’s own controlled operations and from the supply chains. For example, Nike has reported that it has measured the amount of waste generated across the entire supply chain in 2005/06, which has enabled the company to identify the total cost of waste for footwear alone to be US$844 million. Reducing the waste and hence the cost and environmental impact associated with this waste is clearly of strategic importance. Thus, the Nike corporate responsibility targets incorporate particular targets in terms of waste reduction (for example, footwear waste reduction of 17 per cent from 2007 baseline equating to 155 grams/pair in 2011). Adidas has reported an average resource consumption per pair of sports shoes (Table 1.1; Adidas 2008) against three main indicators: energy use, waste water and average volatile organic compounds. By not only accurately measuring resource consumption but also identifying potential savings and reduction targets, the company is able to progress with realising these savings in own operations as well as assisting relevant vendors in the supply chain. For example, the data shown in Table 1.1 indicate an increase in energy use and average volatile organic compounds in 2007 compared with 2006, when less environmentally efficient Reebok footwear factories joined the Adidas Group system. Clearly, it is critical that all business entities within such a system achieve comparable efficiencies against the common targets. 5

Aleksandar Subic, Adrian Mouritz and Olga Troynikov

/

t /

ft.

m fa

i

%

#

%\ %

3 te)

Figure 16.8 Dynamic pressure distribution around helmet with and without vents using CFD modelling; (top left) rear view with no vents, dynamic pressure (Pa); (top right) side view with no vents, dynamic pressure (Pa); (bottom left) rear view with vents, dynamic pressure (Pa); (bottom right) top view with vents, dynamic pressure [Pa]

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Aleksandar Subic, Firoz Alam and Monir Takla

Most commercially manufactured bicycle helmets are not optimally designed to address all three important criteria of impact safety, thermal comfort and aerodynamic efficiency. While there has been significant focus by the manufacturers on safety performance of recreational and road racing helmets to date, there has been limited effort in achieving better combined performance in terms of thermal comfort and aerodynamic efficiency.The case study described in this section indicates that there are ample opportunities to improve aerodynamic efficiency without compromising impact safety and thermal comfort of bicycle helmets. Generally, with the increase in the number and size of vents, the aerodynamic efficiency reduces. However, with a more appropriate design and placement of vents on the helmet, aerodynamic quality and thermal efficiency can significantly be enhanced.The case study also showed that the time trial helmets perform better aerodynamically than other helmets but their thermal comfort was poor. The bare head does not necessarily reduce aerodynamic drag compared with the head with helmet. In addition, aerodynamic efficiency and thermal comfort can severely be affected by the crosswind and head angle with the oncoming winds.

Modelling and simulation of impact performance of sports helmets Current head protection equipment typically comprises helmets with a high-impact grade polymeric shells, high density EPS liners and face guards.The main objective of the equipment used for head protection is to reduce the acceleration of the head during impact and to spread the applied impact load over a greater surface area to reduce the pressure and likelihood of skull fracture. Modern helmets are designed to achieve precisely this by combining a stiff shell that provides an impact face which spreads the impact force across a larger surface area and soft internal padding which cushions the blow and absorbs the energy of impact. Most helmets today use a fibreglass or ABS shell and uses low-density polyethylene for the padding. In addition, heat-formed liners are included with the padding, which can reshape to give a personal fit under body heat generated during play (Subic and Cooke 2003). Some sports helmets, where ball impact to the face is likely, are fitted with a face guard or ‘grill’, typically made of powder-coated steel (in cricket) or polycarbonate (in ice hockey). Protective helmets should be comfortable without compromising safety, whereby comfort depends on factors such as fit and weight. While new helmets have become lighter in recent years to improve comfort, they still need to provide effective impact absorption and satisfy stringent safety standards requirements. New materials, such as polycarbonate or light metals, have been trailed to replace the steel grill structure used as a face guard to further reduce the weight of the helmet. Some polycarbonate materials have excellent transparency and can withstand high-impact forces. Figure 16.9 shows examples of traditional cricket and baseball helmets. Virtual design and analysis techniques based on numerical methods (such as finite element modelling) have been progressively used in the development and evaluation of new-generation protective equipment in sports.This approach helped to increase the understanding of impact dynamics, which has the potential to significantly improve sports equipment designs. The virtual-design approach reduces development time (typically based on extensive physical prototyping and testing), while achieving higher levels of compliance with governing standards. Structural analysis of the strength and safety of sports helmets is mostly based on finite element modelling, which can be implemented in the development and testing of helmets and face guards for cricket, ice hockey, baseball, skiing, and so on. A finite element model developed for this purpose would typically incorporate a headform, helmet, face guard (where applicable) and the impacting object. The following section describes appropriate modelling and simulation 262

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Figure 16.9 Typical cricket (left) and baseball (right) helmets

strategies and key design attributes for the development of new head protection equipment using the finite element method.

Finite element modelling of sports helmets In a wide range of sports, players are prone to head injuries primarily from high-speed head impact to the ground or to other objects (in, for example, cycling, skiing, ski jump) or from direct impact of a ball to the head of the player (as in cricket, baseball, ice hockey). The most severe injuries typically occur to the skull and face.The main objective of equipment used for head protection is to reduce the acceleration of the head during impact and to spread the applied impact load over a greater surface area to reduce the pressure and likelihood of skull fracture. As discussed, modern helmets are designed to achieve precisely this by combining a stiff shell, which provides an impact face that spreads the impact force across a larger surface area, and a softer internal padding, which cushions the blow and absorbs the energy of impact. A typical helmet would comprise an outer polypropylene shell, inner EPC liner and foam padding. An exploded view of a typical cricket helmet assembly is shown in Figure 16.10. In some sports, protective helmets are also fitted with eye protection goggles and/or a face guard typically made of powder coated steel. Protective helmets and their attachments should be comfortable without compromising safety, whereby comfort depends on factors such as fit and weight. In recent years, research has focused on developing new face guards from alternative materials, such as polycarbonate materials, to replace the heavier steel grill and reduce the overall weight of the helmet and so improve the thermal comfort of the player (Subic et al. 2005).The finite element modelling and analysis approach used in this research has been implemented effectively in the development of helmets and face guards for cricket, ice hockey, baseball and other sports (Subic et al. 2002; Mitrovic and Subic 2000; Knowles et al. 1998; Ko et al. 2000) as well as on impact studies involving face guards (Subic et al. 2005). In a numerical simulation of an object striking the head of a player, it is highly desirable to simulate realistically all components involved in the impact. Simulating real-life impact is a very complex task. It involves modelling complex geometries, non-linear material properties of all components of the hybrid assembly and simulation of high-velocity impact, which involves a 263

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Polypropylene shell

Foam padding

Figure 16.10 An exploded view of a cricket helmet assembly

significant amount of uncertainty owing to the dynamic nature of the impact. It is impossible to predict the exact velocity of the object or the nature of its motion, whether it is pure translation or includes rotation. Also, it is impossible to predict exactly the position of the player or the location of impact with the moving object. In addition, unless the impacting object is a ball, the shape of the object is unpredictable. An important factor affecting the outcome of such simulation is the player’s dynamic response to the impact; that is, their awareness and flexibility. Most of these difficulties apply also to the experimental evaluation of new designs of helmets and face guards.Therefore, to consistently evaluate new designs, standard tests are specified by which such uncertainties can be eliminated or at least controlled in a more predictable manner. Safety standards generally specify striker drop tests for different types of helmets, which imply specific impact velocities as well as specific impact sites to a specified headform with and without protection. Although these tests are less complex than real-life impact, the simulation of the striker drop test still represents a combination of challenges that must be addressed. All the components involved, in addition to the behaviour of the surrounding structure, should be considered in the analysis. All helmet components, including face guard assembly, and their respective material properties should be properly modelled. The interactions between all components have to be realistically modelled.When considering impact with a ball, the behaviour of the ball and its interaction with the helmet and face guard must also be adequately simulated.The impact drop test, in which contact takes place for a brief period of time, should be simulated using explicit dynamic finite element analysis. Modelling and analysis should accurately represent the components and impact events specified in the respective safety standards, so that they can be used as a virtual compliance test for the design of new helmets and face guards. Such models are invaluable when developing new helmet designs. 264

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Case study: structural analysis of cricket helmets In professional cricket, bowlers can reach speeds of up to 160 km/hour. At such speeds, batsmen are prone to injuries from direct impact with the ball, which at test level weighs around 156 grams.The most severe injuries typically occur to the head and face so, in such conditions, head protection is vital.At present, head protection in cricket consists typically of a helmet with a high-impact grade polypropylene shell, a high density EPS liner and a face guard. A finite element model required for structural analysis of the impact between ball and head in cricket must include a headform, helmet, face guard and the impacting ball as key elements of the hybrid model. To achieve realistic results from the virtual impact tests, a fast-solving numerical finite element model of the cricket ball is required (Cheng et al. 2008). Modern cricket helmets are designed to achieve head protection by combining a stiff shell, which provides an impact face that spreads the impact force across a larger surface area, and a softer internal padding that cushions the blow and absorbs the energy of impact. Most helmets today use a fibreglass or ABS shell with low-density polyethylene padding. In addition, heatformed liners are included with the padding, which have the ability to reshape to personal fit under body heat generated during play (Subic and Cooke 2003). Cricket helmets are also fitted with a face guard or ‘grill’ typically made of powder-coated steel. Cricket helmets used, for example in Australia, must comply with Australian and New Zealand standards (Standards Australia/Standards New Zealand 1997a, 2009). Validation of helmet designs is typically conducted according to a separate standard, which describes the specific requirements for the striker drop test (Standards Australia/Standards New Zealand 1997b). The major concern for any face guard or helmet design is passing the standard impact tests.

Finite element simulation of the cricket helmet drop test Whilst the Australian and New Zealand standard (Standards Australia/Standards New Zealand 1997b) specifies the dimensional requirements and material of the headform, it does not provide specific information relating to the stiffness of the supporting structure used in the drop tests.The bare headform, therefore, can be modelled as a stationary rigid body with the geometry as specified in the standard. To ensure compliance of the new face guard design with the standard, ball decelerations resulting from impact between the ball and the bare headform must be compared with those resulting from impact between the ball and face guard.The equation governing the condition of compliance is given by: Mean deceleration Maximum deceleration – (bare headform) Percent difference = –––––––––––––––––––––––––––––––––––––– ⫻ 100 Mean deceleration (bare headorm)

(2)

The percentage difference calculated in equation (2) should be 25 per cent or higher for a new design to pass the compliance test. Equation (2) implies that the accuracy of simulating the cricket ball impact with the bare headform is a major factor in the validation of the numerical results obtained using the finite element method. In the event of impact between two elastic bodies or a rigid body and an elastic body, the principle of energy conservation prevails. Hence, kinetic energy is partially or fully transformed 265

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into elastic energy during the short period of contact between the impacting bodies and then recovered back as kinetic energy during separation. This allows for finite decelerations and, accordingly, finite forces to take place during the contact period. On the other hand, in a hypothetical impact between two rigid bodies, no part of the energy can be transformed into elastic energy.Therefore, the kinetic energy would have to remain constant at all times. Consequently, the reverse in direction of the impacting velocity would have to occur instantaneously, resulting in infinite decelerations/accelerations and associated forces. If both the headform and cricket ball are modelled as rigid bodies, a numerical singularity would result in the simulation of the impact event as an infinite force would be applied for zero time. In other words, ignoring the elasticity of the cricket ball would result in infinite decelerations during the impact between the ball and the rigid headform, thus resulting in artificial compliance with the standard, which could be misleading. When simulating the drop test according to the standard (Standards Australia/Standards New Zealand 1997b), the bare headform is considered as a stationary rigid body.Thus, the elasticity of the impacting ball plays a major role in the simulation and has to be taken into consideration. In simulating the impact with a deformable ball, part of the kinetic energy is transformed into elastic energy and later recovered during the impact event, thus producing realistic decelerations/accelerations and, accordingly, realistic impact forces.

Cricket ball model A cricket ball represents a complicated multilayered structure. Different designs of cricket balls have different internal structures. Therefore, developing an accurate micro model of a cricket ball is a long and tedious process involving a range of experimental, theoretical and numerical tasks.The impact of a cricket ball with a rigid surface was investigated experimentally by Carré et al. (2004), where the ball was dropped to impact a rigid surface on the seam and perpendicular to the seam. The experimental results were used to create a simple model based on a mass-spring-damper system. A new approach developed by Subic et al. (2005) involved a macro model of the ball, which emulates the effects of the interaction of the ball as a whole with other objects.The behaviour of the cricket ball was used to create the finite element model for the ball.The Kookaburra Tuf Pitch cricket ball was used in this analysis. Several experimental quasi-static axial compression tests were conducted under loading velocities varying from 1–500 mm/minute, using an INSTRON Universal testing machine. The experimental results were extrapolated to predict the performance of the ball at the striking speeds planned for the simulation of the faceguard drop test (6.26 metres/second). An exponential relationship between the loading velocity and the ball stiffness was found to prevail, which was used in the extrapolation. The numerical model for the cricket ball was developed as a soft surface encasing a rigid body. Ball stiffness properties obtained from experimental tests were used to calculate the properties of the soft surface of the ball, assuming homogeneous pressure distribution over the contact area and neglecting the friction effect. To calculate pressure values from the experimentally measured force values, the area of contact between the ball surface and the anvils of the testing machine was estimated as a function of displacement. Finally, the non-linear ball stiffness was calculated as a relationship between the contact pressure and the radial deformation of the soft surface of the ball. This modelling approach reduced the complexity of the numerical analysis.As the rigid body of the ball does not contain any deformable elements, the total number of deformable elements in the model is reduced significantly, thus reducing the computing time and reducing the need for computational resources.This adopted definition of 266

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a soft ball surface allowed for the simulation of the non-linear stiffness of the ball. It also allowed for contact pressure distribution analysis based on the geometry and the material properties of the impacted surface of the bare headform or the helmet–face guard assembly. The experimental tests required to create the ball model can be limited to a simple axial compression test. To validate the adopted cricket ball modelling approach and refine the ball model, several compression tests between the developed ball model and a plane rigid surface were simulated, which reproduced the experimentally obtained force displacement diagram with reasonable accuracy. At a subsequent stage, several numerical models for the cricket ball, including a fast-solving model, were developed and incorporated in further simulations (Cheng et al. 2008).The impact behaviour of two-, three- and five-layer cricket balls was measured experimentally using a dynamic signal analyser and high-speed video analysis software. The ball properties obtained experimentally were then used to develop two mathematical models: a single-element model and a three-element model.These models were developed so that they captured the key characteristics of ball impact behaviour while allowing for fast-solving dynamic simulation. The stiffness and damping properties of both models were determined using a novel, fast-solving genetic algorithm. These models predicted the force–time diagram during impact with very little computing cost. However, developing a mathematical model with a reasonable level of accuracy is still a challenge.The simulation of the ball model impact with a flat surface achieved reasonable agreement with the experimental results for both the single-element and the threeelement models.A genetic algorithm approach proved to be more efficient and convenient than directly solving the differential equations. A universal finite element ball model was also developed within the ABAQUS CAE environment. This model can be seen as a combination of a finite element model template and a material parameter selection tool based on an artificial neural network model.Time-dependent material properties of the different sections of the ball have been measured experimentally and included in the finite element ball model. Figure 16.11 shows a cross-section of a five-layer cricket ball and the corresponding finite element model developed in this research. The presented approach allows for rapid model development while producing accurate results at different impact speeds. Sets of real test data obtained from a five-layer cricket ball at impact velocity of 25.01 metres/second were used to examine the described artificial neural network model.A comparison of results shows good agreement between the simulation results and the experimental results. An important feature of the developed universal finite element

Cork

wine

^Leather layer

Inner core

Leather cover

Midsole layer

Cork - rubber

Figure 16.11 Cross-section of a cricket ball (left) and the corresponding finite element model (right)

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model was its flexibility. It was shown that the developed finite element–artificial neural network model could be used to predict the impact behaviour of different types of cricket balls under various dynamic conditions.This flexibility represents an advantage that can be utilised by sports equipment developers to rapidly develop different cricket ball models needed for inclusion in larger simulations involving the impact of a cricket ball with helmet or face guard. The developed finite element–artificial neural network model and the corresponding training process represents an invaluable tool for facilitating design, analysis and structural optimisation of protective equipment in cricket and other impact sports.

Cricket helmet model As an example of a typical cricket helmet, the geometric model of the hard shell of the Albion Test Series cricket helmet was created using a contact digitizer. The three-dimensional points where read into the three-dimensional CAD modeller SolidWorks (Dassault Systèmes,VélizyVillacoublay, France), which produced a surface model. Subsequently, a finite element model was developed using the finite element model pre-processor HyperMesh (Altair HyperWorks, Melbourne, Australia) and analysed in Abaqus/CAE (Dassault Systèmes, Vélizy-Villacoublay, France) environment. Appropriate polypropylene material properties were introduced to the finite element model. Standard four-node shell elements were employed in the FE model, as they represented the most appropriate finite elements for this type of contact analysis. The developed model of the helmet hard shell is shown in Figure 16.12. The material properties of the different components of the helmet, including the non-linear properties of the polypropylene helmet shell, must be included correctly in the hybrid assembly model to obtain accurate results from the impact test simulations. The interior of the helmet, consisting of an EPS liner and foam padding, was also modelled and included in the impact simulation. Appropriate material properties have been used for the individual components. The interactions between the different components of the helmet model were modelled using appropriate contact definitions that allowed contact between two deformable bodies, as well as suitable multi-point constraints.

Figure 16.12 Finite element model of the helmet hard shell

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Headform model A headform was modelled using the dimensions provided in the Standards Australia/Standards New Zealand (1997b) standard. Subsequently, a finite element model of the headform was developed using the finite element model pre-processor, HyperMesh (Altair HyperWorks, Melbourne, Australia). A combination of rigid elements of types R3D3 (triangular) and R3D4 (quadrilateral) were used in the finite element model.The model of a selected headform (size M) is shown in Figure 16.13.

Figure 16.13 Finite element model of a selected headform; RP = reference point

Face guard model The presented face guard finite element model has been developed by Subic et al. (2005) for a novel polycarbonate face guard as an alternative to the standard steel face guard.The structure of the proposed design is essentially a transparent shell with stiffening ribs. The layout of the ribs is similar to that of the Albion C1 protective steel grill. Two additional steel plates were introduced at the sides of the face guard to improve the side stiffness of the face guard and to allow for ease of connection to the helmet. The face guard is connected to the helmet using the same metallic plates as with the conventional steel grill.The transparent plate is 3 mm thick, with outwards facing ribs of five 5-mm2 square cross-section.The geometry of the face guard is shown in Figure 16.14. The material properties of the polycarbonate face guard were introduced in the model, taking into account the material properties in the elastic-plastic range to allow for the occurrence of small local plastic deformations on severe impacts. 269

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Faceguard

Figure 16.14 Geometry of the polycarbonate face guard

Impact analysis using finite element modelling The finite element model consists of the helmet (including hard shell, liner and padding), face guard (including main body, stiffening plates and connecting brackets), headform and the striking ball. An exploded view of the (impacted) model is shown in Figure 16.15. The interactions between the different components of the helmet model are modelled using appropriate contact definitions that allow contact between two deformable bodies, as well as suitable multi-point constraints. Spatial relations and interactions between the different components of the assembly and the impacting ball must be considered carefully, allowing for contact between combinations of rigid bodies and deformable bodies.The rigid body representing the headform is fixed in all degrees of freedom; that is, rigidly connected to the ground. General contact interactions are defined between the headform and helmet components, which are attached to the headform using suitable multi-point constraints.The face guard is connected to the outer shell of the helmet using rigid couplings. The ball is allowed to move freely in the impact direction, with all rotational degrees of freedom locked in place at the immediate preimpact moment. For the protective assembly to be viable, it must withstand the impact test using a standard striker.The intention of the striker test is to check and measure the energy attenuation of the protective assembly.The test simulates a cricket ball of 0.156 kg mass striking the face guard at 20 metres/second (72 km/hour).The associated kinetic energy is equivalent to 30 Joules.The striker test simulates this impact by using a mass of 1.5 kg (combined mass of the ball and supporting striker assembly) dropped from a height of 2 metres, achieving an impact velocity of 6.26 metres/second.The protective assembly must be able to absorb this energy without failing any of the tests. Several drop tests involving impacts to five standard sites on the face guard are required according to the Standards Australia/Standards New Zealand (1997b) standard. These tests encompass striking the areas of the cheek, temple, eye, chin and attachment area, whereby the face guard must not fail and should not come into contact with the headform during impact. In addition, ball decelerations resulting from impact between the ball and the bare headform 270

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Figure 16.15 Exploded view of the finite element (impacted) model

should be compared with those resulting from impact between the ball and face guard. Clause 6.5.3 of the standard states that ‘with an initial striker deceleration within the range of 400g to 500g, the difference between the mean deceleration of the striker impacting the bare headform and the maximum deceleration recorded at any of the test sites on the face guard shall not be less than 25%’ (Standards Australia/Standards New Zealand 1997b). This criterion is used to assess the impact condition of the design with respect to the deceleration levels of the striker (ball) and the impact location on the face guard. In the finite element model, these conditions are simulated by assigning a point mass of 1.5 kg to the rigid body reference node of the rigid ball along with an initial velocity of 6.26 metres/second, which is the velocity that the striker would have attained prior to impact.

Numerical results Several drop tests involving impacts to all five sites of the face guard as specified by the standard (Standards Australia/Standards New Zealand 1997b) have been simulated. The deformation of the face guard for different tests are shown in Figure 16.16. In this figure, the largest global displacement of the face guard caused by an impact is referred to as the ‘maximum’ displacement, while the local displacement at the point of impact is referred to as the ‘impact’ displacement. Results show that the cheek area, although passing the impact test, is the most critical part of the face guard design. Although large displacements resulting from impact to this area are depicted by the analysis, they result in no contact between the face guard and the headform, owing to the large initial clearance at the areas of concern. Impacts to all other sites in this simulation have been found to be safe, with a good margin. The same tests were also simulated with the ball striking the bare headform and the resulting decelerations were recorded for comparison according to equation (1). 271

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1

1

U. Magnitude

U. Magnitude

+2.1309+01 +1.9949+01 +1.8589+01 +1.7229+01 +1.5879+01 +1.4519+01 +1.3159+01 +1.1799+01 _ +1.0439+01 — - +9.0759+00 — - +7.7169+00 — +6.3589+00 — - +5.0009+00 —I—+0.0009+00

+1.6238+01 +1,529e+01 +1.4359+01 +1.3429+01 +1.2489+01 +1.1559+01 +1.0619+01 +9.6779+00 _ +6.7429+00 — - +7.8069+00 — - +6.8719+00 — - +5.9359+00 — - +6.0009+00 —I—+0.0009+00

Attachment

Temple

1

U. Magnitude

-

Displacement (mm)

+3.5479+01 +3.3359+01 +3.1239+01 +2.9109+01 +2.6989+01 +2.4869+01 +2.2749+01 +2.0619+01 +1.8499+01 +1.6379+01 +1.4259+01 +1.2129+01 +1.0009+01 +0.0009+00

Test Site

Impact

Maximum

Attachment

6.47

21.98

Cheek

31.38

35.47

Chin

20.24

25.52

Eye

18.78

18.78

Temple

14.01

16.47

Cheek

1

1

U, Magnitude

I, Magnitude +2.5649+01 +2.3929+01 +2.2209+01 +2.0489+01 +1.8769+01 +1.7049+01 +1.5329+01 +1.3609+01 +1.1889+01 +1.0169+01 +6.4409+00 +6.720e+00 +5.0009+00 0009+00

Chin

Figure 16.16 Deformation of the face guard

272

+2.9919+01 +2.7839+01 +2.5769+01 +2.3689+01 +2.1619+01 +1.9539+01 +1.7459+01 +1.5389+01 +1.3309+01 +1.1239+01 — - +9.1529+00 — - +7.0769+00 —L +5.0009+00 ——+0.0009+00

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The von Mises stress distributions in both the face guard and attachment for different tests are shown in Figure 16.17.The highest values of the von Mises stress in the face guard resulting from different drop tests vary from 43 MPa from impact at the attachment to 66 MPa from impact at the chin location.These stresses are within the elastic limit of the material. Although the data used represent the material properties at quasi-static loading, they are still valid at the considered impact rates (10–100) S–1. In fact, these data are slightly conservative owing to the fact that the yield stress of the material increases at higher strain rates. This increase, however, can be considered as negligible at the considered impact rates.

Conclusion The chapter has provided a detailed study of the key performance characteristics of sports helmets. In particular, the experimental and numerical methods used to determine the aerodynamic performance, thermal comfort performance and safety performance of sports helmets were discussed. This study complements the sports helmet research reported in Subic (2007), which was primarily concerned with the biomechanics of head injuries, head protection requirements, design and construction of helmets for safety. Here, we have introduced contemporary issues in design of sports helmets that are concerned with the reduction of aerodynamic drag, improved heat transfer and cooling and reduction of helmet weight while not compromising safety performance. The chapter introduced novel concepts and methods currently explored in engineering that aim at achieving a better balance between these competing requirements to produce a more advanced customised sports helmet. For example, the case study dealing with the aerodynamics and venting in bicycle helmets showed that conventional vents may reduce aerodynamic performance of helmets, owing to increased drag, while not contributing to higher levels of thermal comfort.The presented aerodynamic studies conducted in a research wind tunnel and using CFD technique produced results that provide scope for more effective design of vents in sports helmets. The chapter also presented a computer-aided engineering design approach based on the finite element method, which allows for rapid virtual development and virtual testing of sports helmets in accordance with the governing design standards.This approach can also be used to assess the compliance of each design concept during the design stage against the relevant national or international standard. Whilst the Standards Australia/Standards New Zealand (1997b) standard specifies the dimensional requirements and material of the headform, it does not provide specific information relating to the stiffness of the supporting structure used in the drop tests.Therefore, the bare headform must be modelled at present as a stationary body, thus assuming infinite stiffness for the supporting structure. While this assumption represents a worst-case scenario, it imposes uncertainty on any experimental or numerical testing results. According to equation (1), the ball decelerations resulting from impact between the ball and the bare headform need to be compared with those resulting from impact between the ball and face guard. The measured (or calculated) values of these decelerations depend heavily on the stiffness or flexibility of the headform together with the supporting structure of the test rig. In other words, the same helmet/face guard assembly may pass the compliance test on a certain test rig and fail the same test on a stiffer rig. Therefore, it is the authors’ opinion that the test standards need to be revised to include a more accurate specification of the test rig, particularly its three-dimensional stiffness in response to impacts from different directions. Future research should consider the stiffness and constraints of the experimental testing equipment and their influence on the validation of sports helmets structural performance.

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Figure 16.17 von Mises stress distributions in both the face guard and attachment

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References Alam, F.,Watkins, S. and Zimmer, G. (2003) ‘Mean and time-varying flow measurements on the surface of a family of idealized road vehicles’, Experimental Thermal and Fluid Science, 27(5): 639–54. Alam, F., Subic, S., Akbarzadeh, A. and Watkins, S. (2007) ‘Effects of venting geometry on thermal comfort and aerodynamic efficiency of bicycle helmets’, in F. K. Fuss, A. Subic and S. Ujihashi, The Impact of Technology on Sport II,. London:Taylor and Francis, vol. 1, pp. 773–80. Alam, F., Subic,A. and Akbarzadeh,A. (2008) ‘Aerodynamics of bicycle helmets’, in M. Estivalet and P. Brisson (eds) The Engineering of Sport 7. Paris: Springer, vol. 1, pp. 337–44. Alam, F., Chowdhury, H., Elmir, Z., Sayogo, A., Love, J. and Subic, A. (2010),‘An experimental study of thermal comfort and aerodynamic efficiency of recreational and racing helmets’, Procedia Engineering, 2(2): 2413–18. Bicycle Helmet Safety Institute (2012) Helmet Related Statistics From Many Sources. Arlington, VA: BHSI. Available online at http://www.bhsi.org/stats.htm (accessed 2 January 2013). Bicycle Institute of America (2009) Bicycling Reference Book: Transportation Issue. Washington DC: Bicycle Institute of America. Bicycle Network Victoria (2012) Melbourne: Bicycle Network Victoria. Available online at http://www.bicyclenetwork.com.au/ (accessed May 2012). Blair, K. B. and Sidelko, S. (2008) ‘Aerodynamic performance of cycling time trial helmets’ in M. Estivalet and M. Brisson (eds) The Engineering of Sport 7 (P76). Paris: Springer, vol. 1. pp. 371–7. Brühwiler, P. A., Buyan, M., Huber, R., Bogerd, C. P., Sznitman, J., Graf, S. F. and Rösgen, T. (2006) ‘Heat transfer variations of bicycle helmets’, Journal of Sports Sciences, 24(9): 999–1011. Carre, M. J., James, D. M. and Haake, S. J. (2004) ‘Impact of a non-homogeneous sphere on a rigid surface’, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 218(3): 273–81. Cheng, N., Subic, A. and Takla, M. (2008) ‘Development of a fast-solving numerical model for the structural analysis of cricket balls’, Sports Technology, 1(2–3):132–44. Consumer Product Safety Commission (1999) ‘Bike helmets’, Consumer Product Safety Review, 4(1): 1–4. Available online at http://www.cpsc.gov/cpscpub/pubs/cpsr.html (accessed 3 January 2013). Insurance Institute for Highway Safety (2012) Fatality Facts 2010 Bicycles. Available online at http://www.iihs.org/research/fatality.aspx?topicName=Bicycles&year=2010 (accessed 2 January 2013). Knowles, S., Fletcher, G., Brooks, R. and Mather J. S. B. (1998) ‘The development of a superior performance cricket helmet’, in Proceedings of the 2nd International Conference on the Engineering of Sport. Oxford: Blackwell Science, pp. 405–13. Ko, C.W., Ujihashi, S., Inou, N.,Takakuda, K., Ono, K., Mitsuishi, H. and Nash, D. (2000) ‘Dynamic responses of helmets for sports in falling impact onto playing surfaces’ in E. Eckehard and S. Haake (eds), Engineering of Sport: Research, Development and Innovation. Oxford: Blackwell Science, pp. 399–406. Kyle, C. and Bourke, E. (1984) ‘Improving the racing bicycle’, Mechanical Engineering, 106(9): 34–45. Mitrovic C. and Subic A. (2000) ‘Simulation of energy absorption effects during collision between helmet and hard obstacles’ in Proceedings of the 3rd International Conference on the Engineering of Sport. Sydney, 10–12 June. Oxford: Blackwell Science, pp. 389–97. NHTSA (2012) Traffic Safety Facts 2010 Data. Bicyclists and Other Cyclists. Washington DC: US Department of Transportation, National Highway Traffic Safety Administration. Available online at http://www–nrd.nhtsa.dot.gov/Pubs/811624.pdf (accessed 2 January 2013). Standards Australia/Standards New Zealand (1997a) Australian/New Zealand Standard Protective Headgear for Cricket Part 3: Faceguards. AS/NZS 4499.3:1997. Prepared by Joint Technical Committee CS/95, Helmets for Ball Games. Homebush: Standards Australia and Wellington: Standards New Zealand. Standards Australia/Standards New Zealand (1997b) Methods of Testing Protective Helmets – Determination of Impact Energy Attenuation – Striker Drop Test. AS/NZS 2512.3.2:1997. Prepared by Joint Technical Committee CS/95, Helmets for Ball Games. Homebush: Standards Australia and Wellington: Standards New Zealand. Standards Australia/Standards New Zealand (2009) Methods of Testing Protective Helmets – Definitions and Headforms. AS/NZS 2512.1:2009. Prepared by Joint Technical Committee CS/95, Helmets for Ball Games. Homebush: Standards Australia and Wellington: Standards New Zealand. Subic, A. (2007) Materials in Sports Equipment Volume 2. Cambridge:Woodhead Publishing and CRC Press. Subic A. and Cooke A. (2003) ‘Materials in cricket’ in M. Jenkins (ed.) Materials in Sports Equipment. Cambridge:Woodhead Publishing and CRC Press, pp. 342–72.

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Aleksandar Subic, Firoz Alam and Monir Takla Subic,A., Mitrovic, C. and Takla, M. (2002) ‘Comparative structural evaluation of protective helmets using the finite element method’, in H. P. Lee and K. Kumar (eds) Recent Advances in Computational Science and Engineering: Proceedings of the International Conference on Scientific & Engineering Computation (IC–SEC), Singapore, 3–5 December. London: Imperial College Press, pp. 506–9. Subic, A.,Takla, M. and Kovas, J. (2005) ‘Modelling and analysis of alternative face guard designs for cricket using finite element modelling’, Sports Engineering, 8(4):209–23.

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17 DESIGN AND MECHANICS OF MOUNTAINEERING EQUIPMENT Franz Konstantin Fuss and Günther Niegl

Introduction Climbing is one of the fastest growing extreme sports and this is reflected in the exponentially evolving equipment design. Mountaineering equipment is focused mainly on safety, prevention of injuries and protection from falls. Fall protection equipment comprises ropes, belay devices (rope brakes), energy absorbers and harnesses, as well as safety mats for bouldering. Shoes, crampons, ice axes and chalk improve friction and grip. Helmets protect the head primarily from falling rocks and secondarily from impacts in the event of a fall. Traditional alpine mountaineering has evolved into different climbing disciplines, such as sport climbing on natural rock faces and artificial climbing centres, characterised by routes equipped with permanent anchors; lead climbing, speed climbing and bouldering, the classical disciplines of climbing competitions; top roping; high-altitude mountaineering; ice climbing; dry tooling (combination of ice and rock climbing); ‘via ferrata’ climbing (with pre-installed safety equipment such as ladders and iron cables); and free climbing and soloing disciplines, where the strength of a rope is merely replaced by mental toughness. The growth rate and popularity of sport climbing was triggered by the introduction of artificial outdoor climbing centres, climbing gyms and even indoor ice-climbing facilities, which make the sport more accessible. Modern design of climbing equipment is driven by innovation and constrained by safety standards. The general design principle is finding the optimum balance of minimising the weight and improving the strength of equipment. This chapter covers selected aspects of ropes and karabiners and discusses the mechanics and tribology of belay devices, friction anchors and chalk.

Ropes Ropes are manufactured from ultraviolet-resistant polyamide and are available in dynamic and semi-static versions. Dynamic ropes are less stiff and stretch considerably under load, thereby converting kinetic to elastic energy and reducing the fall force to a tolerable value, which is maximally 12 kilonewtons (kN) in single ropes. Dynamic ropes are divided into single, half and twin ropes. Single ropes are used as a single strand with a diameter of 9–11 mm and a mass of 277

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55–85 g/m, primarily for sport climbing. For traditional alpine climbing, half ropes (8–9 mm, 45–55 g/m) and twin ropes (diameter and mass of a single strand: 7–8 mm, 35–45 g/m) are applied. A single strand of a half rope is still strong enough to sustain falls without failure. Both twin and half ropes are used in pairs but only twin ropes are clipped in pairs. Half ropes are clipped independently, alternately in anchors on the right and left side of the route, which offers a straighter rope path and thus less friction at the anchors. Half ropes also allow simultaneous belaying of two second climbers. Twin and half ropes provide full rope-length abseils when tied together; in single ropes, the abseil distance is half the rope length. In standardised testing, single ropes must be able to withstand at least five falls with a mass of 80 kg at a fall factor (see below) of 1.75, single-strand half ropes at least five falls with 55 kg and double-strand twin ropes at least 12 falls with 80 kg. Twin ropes have thus a higher safety factor than single and half ropes. Semi-static ropes are stiffer and thus more durable, resistant to cutting and abrasion, but would produce higher shock loads in a climber fall and are thus not suitable for climbing. Semistatic ropes are used specifically for situations where rope stretch is unwanted, such as hauling gear, canyoning, rappelling and caving. The length of the rope between the last anchor (a karabiner, for instance) and the climber is related to the shock force experienced when a fall is stopped by the rope. The brutality of arrest of a fall is expressed by the fall factor f, defined as h – f=L

(1)

where h is the fall height and L is the unloaded length of rope out. Equation (1) assumes that there is no rope slip through a belay device. L is the length of the rope connecting the belayer (more precisely, the rope brake or belay device) and the climber; h is twice the distance between the last anchor and the climber, plus the elastic stretch, plus the difference between L and the distance from belayer to climber (this difference is greater than zero if the rope is slack), plus the displacement of the belayer. From equation (1), 0 ≤ f ≤ 2. In the best case, the distance between last anchor and climber is zero. In the worst case scenario, the distance between belayer and last anchor is zero and thus the climber falls twice the rope length. The higher the h, the larger is the force peak Fmax of the fall arrest. The longer the rope, the smaller Fmax, as longer ropes generate a larger absolute elastic stretch and thus more kinetic energy is absorbed as elastic strain energy.The larger the f, the larger the Fmax: EA kh Fmax ––– –– –– W =1+ 1+2 W f=1+ 1+2 W

(2)

(Wexler 1950), where W is the weight of the climber (mass m times gravitational acceleration g), EA is the rope modulus, the product of tensile modulus E and cross-sectional area A, and k is the stiffness of the rope. Equation (2) assumes that the rope is purely elastic, E is independent of strain and strain rate and friction at karabiners is zero. In reality, ropes are viscoelastic and the higher the h, the faster the rope stretch and the larger is E.The ratio of Fmax to W, the impact load factor, defines the magnitude of the peak deceleration in g. In addition to the peak deceleration, the deceleration rate or load rate is closely connected to the severity of injuries. The · load rate F (Custer n.d.) is estimated by f · – =k h F ⬀ EA L

(3) 278

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The load rate increases with stiffness and fall height. Apart from the fact that moisture increases the weight of ropes, wet ropes have a higher impact load but a smaller number of cycles to failure. The standard EN 892 (CEN 2004) sets the minimum number of drops without break to five (at a fall height of 4.8 metres and a mass of 80 kg) and the dynamic impact load of the first drop must be smaller than 12 kN.According to Spierings et al. (2007), the number of drops sustained falls on average below five at a rope humidity higher than 25 per cent. Specially dried ropes run this risk as well.The best performance is expected at a humidity of four per cent. At 40 per cent humidity, the impact force increases by 15–20 per cent on average (Spierings et al. 2007). In addition to the higher impact force, the jerk (force rate over mass) is also higher in wet ropes and climbers would reach the critical jolt of 120 g/s at only the third drop (Nikonov et al. 2009). This jerk value is considered the critical safety limit in automotive accidents. Furthermore, wet ropes exhibit a smaller viscoplastic component of rope deformation compared with dry ropes; they dissipate less energy than dry ropes and are thus less safe and are, on average, less stiff and stretch more (Nikonov et al. 2009). The higher impact force of wet ropes seems counterintuitive but is simply because the stiffness of wet ropes in the terminal loading segment is larger than that of dry ropes.The impact force and the jerk of both dry and wet ropes increase with the number of drop cycles (Nikonov et al. 2009).

Connectors (karabiners) Connectors, quick release links or karabiners (Figure 17.1; Table 17.1) are loop-shaped metal devices with a sprung or screwed gate, used for attaching to ropes, slings, screws, pitons, and so on. Handling, strength, safety and mass are critical parameters. Karabiners are usually manufactured from 7075-T6 aluminium alloy, which is both lightweight and strong, as well as from steel for extra strength in specific applications. The gate of a karabiner is the weak point, as it can open accidentally during a fall or be activated by rope vibrations.

Design of gates and their locks Gates of karabiners are oscillating and are mounted by a rivet on to the karabiner’s C-shaped body, with a keylock for engaging the gate in the locked position. A locking sleeve is optional. The gate is sprung to keep it in the locked position. Gates can be straight or bent (type D; Figure 17.1).Thinner wire gates do not easily open when hitting the rock face, owing to their small mass.

Figure 17.1 Karabiners (B, D, X, H, K, Q = classification according to Table 17.1)

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Franz Konstantin Fuss and Günther Niegl Table 17.1 Classification of karabiners; type, usage, design and strength data according to EN 12275 and UIAA 121 Karabiner type

Use

Design

B

Basic

Normal use

Offset-D shape (transition from D-shape to pear-shape)

20

7

7

D

Directional

Connector for quickdraws

D-shaped with bent gate; the design prevents transverse loading

20

–1

7

X

Oval shape

Connector for aid climbing

Oval shaped; less strong than B karabiners

18

7

5

H

HMS Connector for Pear-shaped with (Halbmastsicherung, belaying (rope broad gate-opening Munter hitch) brake) for HMS end; twist lock gate not recommended

20

7

6

K

Klettersteig

25

7

8

25

10

–2

Q Quick link

Connector for via ferrata (Klettersteig) suitable for steel cables

Minimum Minimum Minimum strength in strength in strength in main direction transverse main direction (kN) with direction (kN) with closed gate (kN) open gate

Offset-D shape with optional eye; automatic locking required

Connector for D-shaped, oval, extra safety rectangular or quick link triangular; screw lock (maillon rapide)

Notes: 1 Transverse load impossible; 2 Never loaded with open gate

Non-locking karabiners have a sprung gate without locking mechanism. Locking karabiners are classified according to their locking principle (manual or auto) and safety issues (unsecured or secured; Albert 2008). Unsecured karabiners (screw lock, twist lock), in contrast to secured karabiners (safebiners and Belay Master), can be opened by the rope sliding over the barrel or sleeve of the lock.

Manual locking Single-lock ‘screw locks’ require the user to screw the sleeve on to the gate to lock it.The sleeve or barrel winds up and down a thread on the gate. Problems of these locks include incomplete locking and the gate opening when the sleeve is unscrewed by tangential rope forces. The double-lock ‘Belay Master’ (DMM International, Llanberis, UK) incorporates an additional plastic clip, which can only be locked into place when the sleeve is tightened. 280

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Auto-locking In auto-locking, the gate locks automatically. In two-way action ‘twist locks’ (quick lock), the barrel twists through 90 degrees which opens the gate. On release, the barrel automatically springs back 90 degrees to the closed position. The risk of opening is high, as a mere quarter revolution opens the lock, potentially leading to lethal accidents. In three-way action locks (‘safebiners’), the barrel opens via a three-way action: on release, the barrel automatically springs back and rotates through 90 degrees to the closed position. In the ‘push and twist’ lock (lock safe or Triact lock by Petzl, Crolle, France), the barrel has to be raised and turned by 90 degrees such that the gate opens. In the ‘ball lock’ (by Petzl), after pushing the spherical button, the barrel is turned by 90 degrees and the gate opens. Ball locks and the Belay Master are considered very safe.The push and twist is considered safe, the screw lock still marginally safe and the twist lock unsafe (Albert 2008). The Austrian Alpine Association recommends refraining from using twist-lock karabiners.

Failure of karabiners Karabiners are required to have a minimum failure load of 18–25 kN in the longitudinal direction with a closed gate, 7 kN with an open gate and 6–8 kN when transversely loaded (tensile test with metal dowels; British Standards Institution 1998; Table 17.1). The disadvantage of this standard is that it neither reproduces the actual strain rate when falling nor reflects the actual loading (metal dowel versus rope) or cyclic stress.According to Hairer and Hellberg (2009), karabiners tested with a slack line fail at smaller loads which are close to, if not below, the prescribed minimum loads. In aluminium alloys, the tensile strength increases with the strain rate (work hardening; Altenpohl 1965). The fatigue stress, however, generally decreases with the number of load cycles. Blair et al. (2005) tested aluminium type-B karabiners under cyclic load. The relationship between the number N of cycles to failure and peak load L in kN follows 4

()

C – N= L

(4)

where C is 85.4 and 39.3, for closed and open conditions, respectively (Blair et al. 2005). At a repeated load of 10 kN, the karabiner type investigated would fail after approximately 5,320 and 240 cycles, with closed and open gates, respectively. Hairer and Hellberg (2009) tested closed aluminium type-H karabiners with cyclic loads.Their data suggest that C is about 130. Aluminium karabiners used to wear out from abrasion with ropes polluted by sand and small rock particles. In closed condition, failure load does not decrease if up to 40 per cent of an elbow is worn off, whereas it decreases proportionally with the degree of wear if the gate is open (Schambron and Uggowitzer 2009). Karabiners tend to break at the transition from curved elbow to the straight spine (but not at the elbow) with a disengaged gate (Blair et al. 2005), even when the elbow is indented by sharp-edged pitons (Schubert 2002).

Belay devices and rope brakes Rope brakes are an integral part of standard belaying equipment in rock climbing.A rope brake must both allow rope feed to the ascending climber and block rope slip in the event of a fall. As the direction of the fed rope and the slipping rope is the same in lead climbing, the design 281

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of a brake must enable quick and continuous feeding of the rope as well as fast switching between the feeding and locking mode. In general, we can distinguish between two different types of brakes (Würtl 2009 Fuss and Niegl 2010a). In dynamic brakes, the kinetic energy of the falling climber is absorbed by the friction between the brake and the rope slipping through the brake. Friction takes energy out of the falling climber – in addition to the braking hand – by converting kinetic energy to thermal energy. Dynamic brakes are characterised by the rope being considerably deflected by the brake, thereby reducing the tension on the belayer’s side (belt friction). Friction results in different rope tensions on either side of the brake: the tension on that side of the brake towards which the rope slips – the fall side – is larger than on the side of the braking hand of the belayer, thereby multiplying the belayer’s hand force and allowing the belayer the application of relatively small and controllable forces compared with the force produced by the fall arrest. The friction force produced by the brake slows down the slippage of the rope, enabling a gentle deceleration of the falling climber, thereby avoiding the dangerously brutal effects of two extreme conditions with high decelerations: • an unbraked rope slip with subsequent impact on the ground; or • a short rope directly connecting the rock face and the climber, which again results in highimpact forces. In static brakes, the kinetic energy of the falling climber is absorbed by the moving mass of the belayer and by the elasticity of the rope as the slip of the rope through the brake is reduced to a minimum.As fall heights in second climbers are extremely small, belaying of second climbers is consequently also static. In contrast to lead climber belaying, the brake is connected to, for example, a rock anchor and not to the belayer. Static brakes are characterised by the rope being both deflected and compressed.

Design of belay devices and rope brakes (Fuss and Niegl 2010a) Type-H karabiners (Figures 17.2 and 17.3) are used in dynamic belaying for applying the HMS (‘Halbmastsicherung’, ‘Halbmastwurfsicherung’ or Munter hitch, Italian hitch, nœud de demi-cabestan, nodo mezzo barcaiolo). The rope is deflected three-dimensionally by the karabiner and by itself (rope-on-rope contact).The deflection angle of the rope is, depending on the method – the rope held by the belayer’s hand leads either towards, or away from, the belayer’s body – between 490 degrees and 615 degrees, respectively, with approximately 200 degrees rope-on-rope contact. Figure-of-eight (F8) brakes, used for dynamic belaying, deflect the rope three-dimensionally with no significant rope-on-rope contact (Figures 17.2 and 17.3). Ropes deflected in three dimensions tend to twist and coil. F8 brakes are primarily abseil devices and their use as belay devices was overtaken by tube-type brakes. F8 brakes, according to their name, have two openings, a main hole and an eyelet.The main hole is usually circular but can also be polygonal. In F8 brakes with a self-locking design, the main hole is designed with a V-slot, tapering towards the eye, which allows for pinching the rope with subsequent self-locking. Special designs, such as those with lateral hooks, are preferred for canyoning and rescue purposes.The rope deflection angle in F8 brakes is 515 degrees (Fuss and Niegl 2010b) to 630 degrees (Attaway 1999). Brake bar karabiners (Figure 17.2) were mainly used for abseiling, with the karabiner’s gate inserted through the eyelet of a peg, piton or a specially manufactured brake bar (buckle).The peg or bar generated the necessary belt friction by two-dimensionally deflecting the rope, required for controlling the rate of descent. 282

Design and mechanics of mountaineering equipment

10

c i ^

11

Figure 17.2 Belay devices; (1) figure of eight; (2) brake bar karabiner; (3) rappel rack; (4) belay ring; (5) sticht plate; (6) magic plate; (7) tube brake without magic plate function; (8) tube brake with magic plate function; (9) hybrid brake; (10) linear semi-automatic brake; (11) angular semi-automatic brake

Figure 17.3 Belay devices; (a) karabiner; (b) figure 8; (c) tube brake; (d) magic plate; (e) angular semiautomatic brake; (f) hybrid brake Sources:(a, b and c from Fuss and Niegl 2010b; © 2010 Elsevier, reproduced with kind permission; d, e and f from Fuss and Niegl 2010a; © 2010 Routledge, reproduced with kind permission

283

Franz Konstantin Fuss and Günther Niegl

Rappel racks (Figure 17.2) evolved from brake bar karabiners by adding more bars and thus increasing belt friction. In a six-bar rack, the rope deflection angle is 560–800 degrees (Attaway 1999). Belay ring brakes (Figure 17.2) evolved from simple chain rings, with karabiners serving as a brake bar, deflecting the rope. Belay rings are the precursors of plate brakes, with two basic design variants, the Sticht plate and the Magic plate. The Sticht plate (Figure 17.2), used for dynamic belaying, had one or two slots and the rope was threaded in one slot, guided through a karabiner and finally out through the same slot.This principle is common to all plate and tube-type brakes.The rope is thus deflected by the brake (twice) and the karabiner, however only in a single plane (two dimensionally). In the Magic plate (Figure 17.2), with a design similar to the Sticht plate, however, one of the two slots merges with a circular hole located in the centreline.This hole was connected to an anchor via a karabiner. The Magic plate was exclusively designed for static belaying of second climbers. In contrast to the principle of the Sticht plate, the rope shanks, entering and leaving the Magic plate, point in the same direction.This difference allowed for a return stop of the rope and thus worked like the locking mechanism of a luggage strap or aircraft safety belts. In the Magic plate, the rope is thereby compressed between the frame of the brake and the karabiner. Plate brakes evolved into tube-type brakes (Figures 17.2 and 17.3), which are available either without or with Magic plate function.The latter allows for both dynamic and static belaying, depending on the use of the brake. Brakes with Magic plate function have an additional eyelet which serves for static belaying of second climbers. Tube-type brakes cannot be used for dynamic and static belaying at the same time or for switching between these two functions. In dynamic belaying, the tube brake is connected to the belayer’s harness, in static belaying to the rock face. Tube brakes show considerable differences in the design of edges which deflect the rope on the brake hand side: (a) smooth and rounded edges; (b) rounded edge with brake ridges (pinion design); (c) rounded edge with tapered walls (V-groove); (d) curved profile with vertical brake ridges incorporated in tapered walls.These design variations serve for increasing the friction. In tube brakes, the angle of rope deflection is about 280–300 degrees (Fuss and Niegl 2010b). Hybrid brakes (Figures 17.2 and 17.3) combine tube-type brakes with and without Magic plate function in a single device, which allows for manual switching between dynamic and static belaying, namely between manual tube type braking and semiautomatic braking with rope blockage. Manual switching depends on the angle at which the brake is held. The karabiner, which is freely hanging in plate and tube brakes, is constrained by, and moves inside, a non-linear (mostly curved) slot which allows switching between the two functions. Beyond a specific angle, the rope tension pulls the karabiner towards the frame of the brake and thus blocks rope slippage. Linear semi-automatic brakes (Figure 17.2) evolved from hybrid brakes and brake bar karabiners and kept the bar and plate principle. However, the karabiner of hybrid brakes is replaced by a linearly moving (prismatic joint or pin in a slot mechanism) mobile link (brake bar) which automatically compresses the rope once it is tightened by the falling climber. In this event, the brake hand is not required to produce any force (unlike manual braking) as rope compression generates the necessary friction force. Semi-automatic brakes are exclusively used for static belaying. In angular semi-automatic brakes (Figures 17.2 and 17.3), the mobile link is a rotating cam. The tensioned rope applies a torque to the mobile link, which moves the link towards the frame of the brake and compresses and blocks the rope.The mobile link can be supported by 284

Design and mechanics of mountaineering equipment

a spring (sprung cam), which keeps the cam in the blocking position. In the GriGri brake, the angle of rope deflection is about 195 degrees.

Mechanics of dynamic brakes (Fuss and Niegl 2010a,b) The force of the rope on the falling climber’s side is denoted T1, the force in the other end of the rope controlled by the brake hand is denoted T2. It is evident and inherent that T1 is greater than T2. In dynamic belaying, T2 is greater than 0; that is, the belayer has to control the deceleration of the climber manually by using the brake as a force multiplier for his or her own hand force. In static belaying, T2 = 0 by definition. For dynamic belaying, the rope is deflected about the rope brake; consequently, the belt friction equation (1) applies T1 – = β = eµθ T2

(5)

T1 – = eηvθ + µθ T2

(6)

where β is the brake factor (or force multiplication factor fmf ), µ is the static friction coefficient, and θ is the angle of contact or deflection. Including the velocity-dependent viscous friction η in equation (5) yields

according to (Makin and Acarnley 2004), where v is the velocity of the rope sliding through the brake. This principle applies even to ropes deflected by karabiners (anchors). Rearranging equation (6) allows calculating the slip velocity log(T1/T2) – µθ v = –––––––––––– ηθ

(7)

where log denotes the natural logarithm. If v is greater than 0, T2 becomes HFmax, the maximal hand force produced by the belayer. The brake force, FB, transferred from the brake to the karabiner and from there via a sling to the belayer’s harness, follows from FB = T1 – T2

(8)

Note that the direction of FB and T2 is opposite to v and thus both the belayer’s hand and the brake take energy out of the falling climber. The brake factor β increases with decreasing hand forces. Smaller hand forces result in more rope slip and thus in a smooth fall arrest. Increased rope slip is caused by a large slip velocity which in turn increases β according to equation (6). Belayers with smaller hand forces should therefore avoid rope brakes with small viscous friction coefficients, such as F8 brakes, and use tube brakes or the HMS (Fuss and Niegl 2010a,b). Analysing the data of Thomann and Semmel (2007) and extrapolating the brake factor to zero rope slip reveals the static friction coefficient µ, which is approximately 0.2 in the HMS and the F8, and approximately 0.3 in the ATC (tube brake). Fuss and Niegl (2010b), using their own brake model and analysing the experimental data of Thomann and Semmel (2007), estimated the viscous friction of the HMS and the ATC to be approximately twice as much as the one of the F8. This is possibly attributable to the rope on rope contact of the HMS and the lateral ridges of the ATC’s rope outlet on the belayer’s hand side. The latter outlet is tapered 285

Franz Konstantin Fuss and Günther Niegl

such that the rope is compressed laterally.The lateral contact of the rope with the ridges adds another friction force to the brake’s belt friction. Dynamic belaying devices are characterised by four braking conditions (Fuss and Niegl 2010b): • • • •

no rope slip at high µ (static condition), rope slip velocity = 0 rope slip and stop; rope slip velocity returns to zero with discontinuous deceleration critical braking: rope slip velocity and deceleration return asymptotically to zero continuous rope slip (no stop), terminal rope slip velocity is greater than 0.

The smaller the fall velocity v0 at the beginning of rope tightening, the smaller is the window of the friction coefficients between no slip and critical braking.The friction coefficients at critical braking are independent of v0. Equally, the range of friction coefficients between no slip and critical braking is independent of the hand force. However, the smaller the hand force, the larger are the friction coefficients at no slip and critical braking.

Mechanics of semi-automatic brakes Semi-automatic brakes can be operated without hand force T2. According to equation (5), T2 approaching zero is only possible if µ or θ becomes infinite.This principle explains the concept of semi-automatic brakes: the belt friction force FB on the concave side of the deflected rope is augmented by linear friction FL, if the rope is compressed: T1 = FB + FL

(9)

For example, the GriGri’s key element is a sprung eccentric cam with a lever for manual operation (Petzl and Hede 1996).The spring keeps the brake in the open position. FB provides the torque and rotates the cam towards the brake housing such that the rope becomes the more compressed, the larger is T1. The torque produced by FB is in equilibrium with the torque applied by the compressive force FC.The design optimisation results in a short moment arm of FC and a long one of FB, such that small FB produces large FC. The larger FC, the larger is FL, which is destined to replace T2 according to equations (5) and (9).This principle explains the phenomenon that the (minimal) rope slip is independent of hand forces; for example, the GriGri slips 8–9 cm at hand forces between 100 and 400 N (Thomann and Semmel 2007). Semi-automatic brakes, however, are prone to handling errors. Manually moving the lever into the unlocked position is strictly confined to feeding the rope to the climber. If the climber falls, human reflexes tend to pull the lever instead of releasing it (Britschgi 2004), leading to uncontrolled falls and severe injuries (Haslwanter 2001). Implementing panic functions and overriding the human action solves this problem. Incorrect handling of the Cinch caused several serious injuries, especially when the rope is not sufficiently deflected by the brake (Semmel and Hellberg 2010). In the GriGri, the moment arm of the rope leading to the falling climber is far longer than that on the belayer’s side.The eccentric cam produces a large torque required for compressing the rope. In the Cinch, however, the moment arms vary depending on the deflection of the rope, and the rope may well be in line with the pivot of the cam, generating no torque at all.This explains why the Cinch is prone to handling errors.

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Rock protection equipment Rock protection devices tie the climber’s equipment (ropes, karabiners, harnesses) to the rock face and reduce fall height.These devices include nails, pins, pegs, screws, pitons, rock anchors, chocks and frictional anchors.The design criteria for this type of equipment are strength and stability on the one hand and lightweight construction on the other, as climbers need to take all rock anchors with them for ascents which might last for several days. Camming devices in general comprise chocks and frictional anchors, which keep damage to rock faces to a minimum. They are placed in cracks or rock gaps and jam once loaded. Chocks are available in three main design versions: trapezoid, wedge-shaped, and hexagonal (‘Hexentrics’, derived from standard hexagonal nuts). Chocks require tapered cracks and are produced in different dimensions for covering a wide variety of crack dimensions. In contrast to chocks, frictional anchors (Figure 17.4), better known as cams, spring-loaded camming devices, friends, sliders, the CamalotTM (Black Diamond Equipment, Reinarch, Switzerland), and so on, adapt to a larger range of crack widths. They are characterised by a unique design based on an equiangular logarithmic spiral profile (Figure 17.5) which maintains a constant pressure angle. The logarithmic cam profile is well known from cam and follower mechanisms and allows designing the smallest possible cam for a given pressure angle, which remains constant at its maximum when the follower is moving (De Ronde Furman 1916).The development of constant angle cams for climbing purposes is attributed to Vitali Abalakov (Bertulis 1976a, 1976b; Dill 1978), which were known as cam nuts or Abalakov cams (Figure 17.4b), essentially nothing but advanced wedge-shaped chocks with a logarithmic profile.

Figure 17.4 Cams: (a) spring-loaded camming device placed in a crack; (b) Abalakov cam; (c) uni-axial cam; (d) dual-axial cam (arrows indicate the axles)

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Figure 17.5 Cam mechanics; (a) logarithmic spiral and pitch angle θ ; (b) free-body diagrams, FR = resultant force, Fx, Fy = components of the resultant force in horizontal and vertical direction, FN, FT = normal and tangential forces; (c) uni-axial spring-loaded camming device at different crack widths, IC = instant centre, COP = centre of pressure, FE = external force

The next design step was a spring-loaded camming device with unilateral or opposing lobes mounted on a single axle (Figure 17.4c), developed by Greg Lowe (1975) and Ray Jardine (1980).Tony Christianson (1987) finally designed a cam with two parallel axles (Figure 17.4d) which considerably increases the expansion range by the distance between the two axles. The polar equation of the logarithmic spiral (Figure 17.5a) is r = r0 ebϕ

(10)

where r and ϕ are polar coordinates, r0 is the radius at ϕ = 0 as well as the angular position parameter (which rotates the spiral about its origin) and b is the tangent of the pitch angle θ (Figure 17.5a) b = tanθ

(11)

The pitch angle or pressure angle θ is defined as the angle between the radius vector (the direction of which is opposite to the force resultant FR originating at the centre of pressure) and the common normal at the centre of pressure (cam-rock contact point; Figure 17.5c). FT tanθ = b = –– = COF FN

(12)

where FT and FN are the tangential and normal forces (Figure 17.5b) and COF is the coefficient of friction. 288

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Aluminium on granite starts to slip if the resultant contact force approaches 18 degrees (Foster 2002), indicating a static friction coefficient µ of 0.325. Ray Jardine originally used 15 degrees (b = 0.268) on his ‘friends’ prototypes (Foster 2002), which was still too large for some rock types. After the testing phase, Jardine and Wild Country decided on 13.75 degrees (b = 0.245), which worked well on most rock types, including grit stone, and ‘which has become internationally acknowledged as the definitive camming angle’ (Foster 2002). In general, the design parameter b of the logarithmic cam profile must not exceed the static friction coefficient µ. The condition of b is less than µ prevents slippage. Reducing b, however, increases FR, leading to structural failure of the camming device. If a cam with opposing lobes is loaded by the external force FE, the components of the bearing forces at each axle are Fx and Fy (Figure 17.5) FE Fy = –– = – FT = FR sinθ 2 Fx =

(13)

F ––y = – FN = FR cosθ tanθ

(14)

The resulting force FR at the centre of pressure equals 1 FE FR = Fx2 + Fy2 = FT2 + FN2 = –– 1 + ––2 b 2

(15)

According to standard EN 12276 (British Standards Institution 1999), the minimum strength requirement for all sizes and types of friction anchors is FE = 5 kN. From equation (15), we calculate FR of 10.52 kN (approximately 2.1 FE) on each side of the cam at θ = 13.75 degrees. This implies that each of the opposing lobes should not deform at compressive loads FR greater than 11 kN or in the case of twin lobes on either side, at FR greater than 5.5 kN.The cam lobes do not necessarily fail first but deformation is one of the failure modes (Schubert 2005). Standard EN 12276 tests only the strength of the friction anchor in a parallel crack but neither the friction itself nor the strength in flaring cracks. Manufacturers are required to stamp the minimum guaranteed FE in kN (≥ 5 kN) on the friction anchor. In flaring cracks, FR depends on the inclination angle ψ of either crack wall (positive ψ if flaring upward); ψ might even be different on the right and left side of the cam. If ψ is known, FR can be recalculated and checked whether it still meets the minimum requirement of 11 kN on either side, especially when ψ is negative, which in turn increases FR. FE Fy = –– = FR sin(θ + ψ) 2

(16)

Fy Fx = ––––––––– = FR cos(θ + ψ) tan(θ + ψ)

(17)

Note that, in flaring cracks, equation (12) remains unchanged. For example, if the minimum guaranteed FE is 6 kN according to the manufacturer, the minimum guaranteed compressive FR at θ = 13.75 degrees corresponds to 12.62 kN. What is the minimum crack wall angle ψ which produces an FR of 12.62 kN if FE is 4 kN (last anchor load in the event of a fall)?

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1 ψ = –θ + tan–1 ––––––––––– 2FR 2 ––– –1 FE

(18)

( )

Equation (18) delivers ψ = -4.63 degrees (flaring downward). Even small angles can cause cam failure, resulting in catastrophic effects. For expert witness reports, in the event of cam failure leading to injuries or even death, it is paramount to determine the direction of FE and the angle between FE and the crack walls. Cam lobes are usually computer numerical control-machined from 7075 aluminium alloy (Foster 2002), which is both lightweight (2,810 kg/m3; www.matweb.com) and strong (7075T6 UTS and yield strength: 572 MPa and 503 MPa, respectively; www.matweb.com).To reduce the mass, cam lobes are manufactured with holes and partial-depth cut-outs. Stresses were investigated by Custer (n.d.), using Hertzian contact stress analysis, and by Bradshaw (2003) applying finite element modelling. Bradshaw (2003) suggested minimising the amount of material at low stress areas in order to optimise cam design.

Chalk ‘Chalk’ is the minimum standard equipment a climber needs. In free soloing, all equipment the climber relies on is chalk and special climbing shoes. Chalk increases the friction between rocks and artificial handholds and absorbs sweat. Chalk is usually made from magnesium carbonate (MgCO3); for sandstone, climbers prefer rosin (colophony, pine tree sap), which does not pollute the rock. Magnesium-based chalk is available in powder form (‘chalk bag’) or in liquid form.The latter is a suspension of powder chalk in alcohol; when left to dry, the alcohol evaporates and the remaining chalk coats the hand and fingers like a glove. Better climbers produce a higher coefficient of friction COF (up to 1, measured during a Climbing World Cup on a hold of a downward slope of at least 22 degrees; Fuss and Niegl 2008a), because they are confident to approach the limit from their experience. Using chalk, climbers can hold slopers of an angle of up to 47 degrees (Fuss and Niegl 2008b), which requires a minimal COF of 1.072 to prevent slippage. In fact, the maximal COF measured in this study was 1.175. Li et al. (2001) claimed that chalk reduced (sic!) the COF and measured static friction coefficients of 2.5 and 3.0 for conditions with and without chalk, respectively (fingers on granite, sandstone and slate). Such high COF figures are unrealistic and would correspond to freely hanging on a slope of 68.2 and 71.6 degrees, respectively, without slippage. Fuss et al. (2004) analysed the effect of different type of chalk on a resin-based standard artificial handhold for sport climbing.The static friction coefficient µ for different conditions was (mean ± standard deviation): • • • • • •

dry hand on clean hold: 0.722 ± 0.087 wet hand on clean hold: 0.675 ± 0.164 dry hand + powder chalk on clean hold: 0.958 ± 0.145 dry hand + liquid chalk on clean hold: 0.763 ± 0.155 dry hand on hold polluted with chalk: 0.925 ± 0.072 dry hand + powder chalk on hold polluted with chalk: 0.650 ± 0.089. 290

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The difference between a dry and a wet hand on the artificial hold was not significant, which proves that a wet hand (wetted with pure water) does not really impair the grip on artificial holds.The difference between dry and powder-chalked hand on the artificial hold was highly significant, which proves that powder chalk provides a significantly better µ and grip on the hold. The difference between dry and liquid-chalked hand on the artificial hold and was marginally significant; the advantage of liquid chalk over a dry hand is questionable.The difference between a powder-chalked and a liquid-chalked hand was highly significant; powder chalk is consequently superior to liquid chalk, because the coating produced by liquid chalk cracks and flakes off when bending the fingers and disintegrates to small chalk grains that roll between the surfaces and reduce friction.The difference between a dry and a powder-chalked hand on a surface polluted with chalk was highly significant.This result indicates that climbers with higher start numbers have a clear disadvantage if the holds are not completely cleaned several times in the course of a competition. The conditions of a powder-chalked hand on a clean hold and a dry hand on a polluted hold were statistically identical, so it does not matter to which surface the chalk adheres.

Summary Mountaineering equipment is ruled by two principles, performance and safety, and is driven by innovation. Performance improving factors involve enhancing grip (for example, chalk and climbing shoes) and reduction of the weight of equipment. Safety improving factors are directed towards superior strength and extreme durability, as well as user-friendly equipment, less prone to human errors.There is usually a trade-off between less weight and superior strength of climbing equipment, such that equipment optimisation is a decisive design component.

References Albert, P. (2008) ‘Verschluss-Sache’, CLIMB! 4/5: 42–7. Altenpohl, D. (1965) Aluminium und Aluminiumlegierungen. Berlin: Springer. Attaway, S.W. (1999) ‘The mechanics of friction in rope rescue’, paper presented at the International Technical Rescue Symposium (ITRS 99), Fort Collins, CO, November 5–7. Bertulis, A. (1976a) ‘C.C.C.P. spells “FRIENDSHIP”’, Off Belay, 25: 19-25. Bertulis, A. (1976b) ‘A Soviet first ascent in the North Cascades’, American Alpine Journal, 20: 340–4. Blair K. B., Custer D. R., Graham J. M. and Okal M. H. (2005) ‘Analysis of fatigue failure in D-shaped karabiners’, Sports Engineering, 8 (2):107–13. Bradshaw, R. C. (2003) Material Stresses Induced During a Lead Climber Fall in Spring Loaded Camming Devices. Technical Report, University of Massachusetts. Available online at http://www.ecs.umass.edu/mie/labs/ mda/fea/fealib/bradshaw/bradshawReport.pdf (accessed 3 January 2013). British Standards Institution (1998) Mountaineering Equipment. Connectors. Safety Requirements and Test Methods. BS EN 12275:1998. London: BSI. British Standards Institution (1999) Mountaineering Equipment. Friction Anchors. Safety Requirements and Test Methods. BS EN 12276:1999. London: BSI. British Standards Institution (2012) Mountaineering Equipment. Dynamic Mountaineering Ropes. Safety Requirements and Test Methods. BS EN 892:2012. London: BSI. Britschgi, W. (2004) ‘Sicher Partner sichern 1: Elementare Sicherungsfehler und die 3-Bein-Logik’, Bergundsteigen, February: 64–9. Christianson, T. (1987) Mechanically Expanding Climbing Aid. United States Patent Number US4643377. Alexandria,VA: United States Patent and Trademark Office. Custer, D. (n.d.) An Elastic Model of the Holding Power of Spring Loaded Camming Devices Used As Rock Climbing Anchors. Available online at http://web.mit.edu/custer/www/rocking/cams/cams.body.html (accessed 3 January 2013). De Ronde Furman, F. (1916) Cams, Elementary and Advanced. (Reprinted 2008) Charleston, SC: BiblioLife.

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Franz Konstantin Fuss and Günther Niegl Dill, B. (1978) ‘Abalakov cams’, Off Belay, 37: 9–11. CEN (European Committee for Standardization). Mountaineering equipment—dynamic mountaineering ropes—safety requirements and test methods (EN 892:2004; UIAA 101). European Standard; 2004. Foster, S. (ed.) (2002) The Wild Country Cam Book. Tideswell: Wild Country. Available online at http://www.wildcountry.co.uk/videos-and-downloads/catalogues/2010-11-catalogues/ (accessed 3 January 2013). Fuss, F. K. and Niegl, G. (2008a) ‘Instrumented climbing holds and performance analysis in sport climbing’, Sports Technology, 1(6): 301–13. Fuss, F. K. and Niegl, G. (2008b) ‘Quantification of the grip difficulty of a climbing hold’, in M. Estivalet and B. Brisson (eds) The Engineering of Sport 7. Paris: Springer, pp. 19–26. Fuss, F. K. and Niegl, G. (2010a) ‘Design and mechanics of belay devices and rope brakes’, Sports Technology, 3(2): 65–84. Fuss, F. K. and Niegl, G. (2010b) ‘Understanding the mechanics of dynamic rope brakes’, Procedia Engineering, 2(2): 3323–8. Fuss, F. K., Niegl, G. and Tan, M. A. (2004) ‘Friction between hand and different surfaces under different conditions and its implication for sport climbing’, in M. Hubbard, R. D. Mehta and J. M. Pallis (eds) The Engineering of Sport 5. Sheffield: International Sports Engineering Association, vol. 2, pp. 269–75. Hairer, H. and Hellberg, H. (2009) ‘Karabiner im Slacklineeinsatz’, Bergundsteigen, 2/09: 32–5. Haslwanter, T. (2001) ‘Denn erstens kommt es anders…Hallen-Kletterunfall – eine Analyse’, Bergundsteigen, 3/01: 19–20. Jardine, R. D. (1980) Climbing Aids. United States Patent Number US4184657. Alexandria,VA: United States Patent and Trademark Office. Li F.-X., Margetts S. and Fowler, I. (2001) ‘Use of “chalk” in rock climbing: sine qua non or myth?’ Journal of Sports Sciences, 19: 427–32. Lowe, G. E. (1975) Anchor Device for Mountain Climbers. United States Patent Number US3877679. Alexandria,VA: United States Patent and Trademark Office. Makin E. and Acarnley, P. (2004) ‘Speed estimation and control in a belt-driven system for a web production process. Part 1: System modelling and validation’, Proceedings of the Institution of Mechanical Engineers: Part E, Journal of Process Mechanical Engineering, 218(1): 15–24. Nikonov, A., Zupancic, B., Florjancic, U. and Emri, I. (2009) ‘Influence of moisture on functional properties of climbing ropes’, in F.Alam, L.V. Smith,A. Subic, F. K. Fuss and S. Ujihashi (eds) The Impact of Technology on Sport III. Melbourne: RMIT Press, pp. 551–5. Petzl, P. and Hede, J. M. (1996) Disengageable Descender With Self-Locking of the Rope. United States Patent Number US5577576. Alexandria,VA: United States Patent and Trademark Office. Schambron,T. and Uggowitzer, P. J. (2009) ‘Effects of wear on static and dynamic failure loads of aluminiumbased alloy climbing karabiners’, Sports Engineering, 11(2): 85–91. Schubert, P. (2002) ‘Hänger gegen Karabiner, können Kerben Karabiner killen?’, Bergundsteigen, 1/02: 19–20. Schubert, P. (2005) ‘Klemmmaschinen, Normprüfung von Klemmgeräten’, Bergundsteigen, 2/05: 76–81. Spierings, A. B., Henkel, O. and Schmid, M. (2007) ‘Water absorption and the effects of moisture on the dynamic properties of synthetic mountaineering ropes’, International Journal of Impact Engineering, 34: 205–15. Semmel C. and Hellberg, F. (2010) ‘Handarbeit oder Automatisierung?’ DAV Panorama, 2010(3): 66–69. Thomann A. and Semmel C. (2007) ‘Die Bremskraftverstärker. Bergundsteigen, 2/07: 60–5. UIAA (2000) EN and UIAA standards for mountaineering and climbing equipment’, UIAA Journal, 3: 23. Wexler, A. (1950) ‘The theory of belaying’, American Alpine Journal, 7: 379–405. Würtl,W. (2007) ‘Sichern 09’, Bergundsteigen, 2/09: 76–81.

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PART VI

Sports implements

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18 GOLF CLUB CONSTRUCTION, DESIGN AND PERFORMANCE Martin Strangwood and Carl Slater

Introduction Golf clubs consist of a head, a shaft and a grip.The grip is designed to fit the golfer’s hands to provide the balance between control of the shaft during the swing, feedback during the swing and after impact with the ball and damping of excessive impact forces.These viscoelastic leather or synthetic leather items do not contribute in a major way to the performance of the swing (provided that the golfer feels ‘satisfied’).The differing performance of the 14 permitted clubs in a golf bag is dominated by changes in the shaft and head dimensions and characteristics. These can be divided into three main groups: • drivers –designed to give maximum distance from the tee; • irons and wedges – designed to provide more control in approach or recovery shots, for example, from hazards such as bunkers; • putters – designed to give controlled rolling of the ball over the green. Within these groups, there have been myriad designs and variations with associated performance claims. These, however, have been dominated by the manufacturers with, until relatively recently, little independent scientific study reported. The earliest principal study was Cochran and Stobbs’ Search for the Perfect Swing (1968) and, while much of the basic science holds, the move to ‘solid’ construction balls, oversized drivers, cavity back clubs and horseshoe putters means that the details need to be updated.Within the scientific literature, most work has been reported into the design and performance of drivers, with fewer studies being reported on aspects such as peripheral weighting and the mechanics of putting. The action of the shaft in relation to the golfer’s swing has, however, been an active area in biomechanics.

Energy and momentum balance As for the golf ball, golf clubs are governed by the laws of physics (Penner 2003) and so energy and momentum for all the modes in which a ball moves must come from the golfer, through the energy and momentum of the club head at impact.The variation in club design (shaft type and length along with club head design) is primarily to accommodate the different golfer swings, stances and level (usually accuracy in striking the ball with the sweet spot of the club). 295

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Golf shafts The role of the golf shaft is to deliver the club face to the ball to achieve the desired launch conditions (velocity and spin rates) for that shot. The dynamic nature of the impact (approximately 0.5 ms duration) means that the shaft has no influence or effect during that event and the shaft should be tailored to the golfer and their (down) swing.The greater mass of the head than that of the shaft (200–300 g compared with 50 g) means that the centre of mass of the club is situated in the head at a position not in line with the shaft.As the club is accelerated from rest at the top of the backswing/start of the downswing, the main forces acting are gravity and inertia. The inertial forces will cause back-bending of the shaft in the swing plane and twisting around the shaft, whilst gravity initially bends the shaft back but, once the shaft passes through the vertical, gravity causes forward bending.Thus the shaft at impact can be forwardly or backwardly bent (club head leading or lagging the hands) and open or closed; that is, not hitting the ball with the face square-on to the intended direction of the shot (Butler and Winfield 1994). The dynamic shape of the shaft at impact thus depends on the swing of the golfer: whether the acceleration of the hands is uniform or concentrated early or late in the swing, whilst any wrist cock will lead to further loading of the shaft just before impact with the ball. Dynamic measurements of the shaft during the swing and club face loft at impact are difficult to determine (requiring strain gauges or high-speed cameras, which can be intrusive and may affect the golfer during their swing) and so the numbers of reported measurements are limited (Lee et al. 2002; Mather and Jowett 2000). Published studies have tended to be single-subject trials or trying to average over a range of golfers.These approaches both give some correlations but the relationships established for single subjects have not been strong enough for generalisation so that the a priori design of a shaft for a given swing is not yet possible. Shaft properties are often represented by the ‘flex’ (static bending stiffness under a tip load when held at the butt) and for ‘torque’ (the angular displacement for a butt-gripped shaft loaded at the tip).Whilst the angular deflection is quoted, the ‘flex’ is often described qualitatively; for example, ladies’, regular, stiff and extra stiff.There is no agreed standard for this and shafts of the same rating show a range of stiffness values, with these ranges often overlapping with those from different flex ratings from the same or other manufacturers.The bending stiffness is more quantitatively described by the frequency of the fundamental bending vibration (for butt gripping) quoted in cycles per minute. Analytical solutions exist for the bending and torsional stiffness of a parallel-sided tube of constant wall thickness but most shafts are tapered (carbon-fibre composite shafts usually have a simple taper, whilst metallic shafts can be stepped) and can have varying wall thicknesses (especially for composite shafts).This leads to a variation of stiffness along the shaft, as revealed by Brouillette’s (2002) analysis and the concept of the ‘bend’ or ‘kick’ point. The effect of the ‘bend/kick’ point on performance is not well established although, as for the golfball moment of inertia, the variation in this parameter in commercial shafts is quite limited (Huntley 2007). Greater effects in shaft characteristics are seen for inter- and intra-batch stiffness and for the presence of seams that arise from the manufacturing process (Huntley, 2007). There are four main methods of manufacturing shafts: • Seamless tubes (metallic) – a pierced billet is drawn through a die with a floating (free) mandrel to govern the inner diameter. Lateral movement of the mandrel can give a stiffness seam that tends to run along the shaft giving a noticeable effect. • Welded tubes (metallic) – metallic sheets are cut and curved to form a tube with a (usually) tungsten inert gas (TIG) weld forming a physical axial seam. The sheet provides constant 296

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wall thickness, whilst the weld bead is removed by tapering or stepping and polishing so that there is little stiffness variation unless the weld is contaminated by oxygen for titaniumbased alloy shafts. • Filament wound (composites) – these give the most consistent shafts with respect to seams, but cannot apply fibres parallel to the shaft leading to lower stiffnesses unless thicker walls are used (greater mass) (Strangwood 2007). • Sheet wound (composites) – these use unidirectional plies in a range of orientations (Strangwood 2007) wrapped around a mandrel to balance bending and torsional stiffness. The ends of the plies are resin-rich leading to lower stiffness; if these run along the shaft axis then a stiffness seam occurs but spiralling sheet edges do not give a noticeable seam. The technology and design of shafts is relatively mature, so that most research is directed to the relationship between the golfer’s swing and shaft properties.

Driver heads In the late 1990s, there was much research regarding the ‘trampoline’ effect associated with oversized, hollow driver heads, particularly those fabricated from titanium-based alloys. The author summarised the mechanistic understanding in 2003 (Strangwood 2003), which was that increased linear elastic deformation of the whole head, particularly the large and thin crown, reduces the deformation in the ball and hence the associated viscoelastic energy losses giving higher speed off the face.At the time, the United States Golf Association and the R&A (Royal and Ancient Golf Club) coefficient of restitution (CoR) restrictions for drivers were being introduced (initially 0.86 and then 0.83) and a relationship between the crown dimensions and material properties (equation 1) was derived from frequency testing and sectioning. 1 CoR × Young’s modulus 1n ––––––––––––––––– = Constant – –– (Crown thickness) 3 Crown area × hardness

(1)

These studies highlighted the importance of the material strength/modulus ratio to driver performance (Figure 18.1), which was reflected in the development of high-performance drivers, which moved from investment cast α + β titanium-based alloys to forged β-titanium-based alloys, which have higher strength and lower modulus. Forging is a process that results in higher and more consistent strength, allowing thinner sections to be fabricated. Subsequent research (Adelman et al. 2006) has indicated the importance of the joint between face and crown in deformation transfer and so in determining performance. Most joints are TIG or electron beam welds in titanium-based alloy heads and these could be defined as ‘good’ or ‘bad’. ‘Good’ welds had dimensions close to those of the adjacent metal components as well as similar modulus values; for these conditions, the difference in stiffness between the weld and the other components is small, allowing deformation to be readily transferred across the weld; that is, the crown can play its full role in enhancing the elastic deformation of the head and so energy loss reduction. ‘Bad’ welds have either large dimensional differences from the surrounding material or different modulus values – usually by oxygen contamination during TIG welding of titanium-based alloys causing modulus to rise. The difference in constraint causes shear waves to be reflected at the weld so the deformation is concentrated more in the face rather than spreading to the crown. ‘Cupping’ the face and moving the weld between it and the crown back a little way along the crown (Figure 18.2) is a successful route to forming ‘good’ welds and maintaining high CoR values. 297

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Steels - high strength, but high modulus pTi-based alloys Better alloys

Amorphous Zr a+ p Ti-based alloys

Al-based alloys - low modulus, but low strength Figure 18.1 Plot of hardness against Young’s modulus for metallic alloys used in golf driver heads

Plane face

Cupped face w i t h

Figure 18.2 Schematic diagram of sectioned driver heads showing how a face could be forged in a ‘cupped’ form so that the weld between it and the crown can be reduced in size and moved back away from the face to increase its likelihood of being ‘good’

As noted above, the CoR limitations, followed by a limitation on head volume (to 460 ± 10 cm3; R&A 2010) has reduced development aimed at just increasing ball speed off the sweet spot so that the use of more efficient materials, such as amorphous zirconium, has become more of an academic exercise, whilst greater use has been seen of carbon fibre-reinforced polymers (CFRPs) in driver head design. Unlike the CFRPs used in shafts, those in driver heads are 298

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usually twill to give more uniform, in-plane biaxial properties.The large loads (up to 6 kN for a 47 m/s swing speed) and the possibility of defects, such as pores and inclusions, between the many plies means that strength is not high so that the components tend to be thick.Within the strain rate range experienced by driver components, the CFRPs should be acting in a linear elastic manner but the head stiffness is increased owing to the greater thickness (up to 5 mm; Figure 18.3a) needed for a CFRP crown (wear resistance of CFRPs is too poor for the durability needed from face and sole). In addition, the CFRP crown needs to be adhesively bonded (using the composite matrix epoxy resin) to the metallic (usually titanium-based alloy) components, which requires a certain overlap (approximately 10–15 mm; Figures 18.3b and 18.3c). This results in greater constraint and reduces the extent to which the crown can contribute to elastic deformation of the head on impact and so leads to a lower CoR value (as for ‘bad’ welds). The principal advantage gained from the use of CFRPs is a similar modulus to titaniumbased alloys but at around one-quarter of the density. Even with the greater thickness of the CFRP, there is a marked reduction in mass of the crown. This mass can then be redistributed around the head to allow repositioning of the centre of mass in the head to modify the ball launch trajectory. The individual components, such as the face, can also be modified to have differential thickness across them; for example, the centre of the face can be thicker than the edges. In such a way, the stiffness of the face centre can be increased to provide a legal sweet spot impact CoR but the stiffness can be graded to the outside of the face so that the penalty for an off-centre hit can be reduced with more energy being available for corrective sidespin and translational speed off the face. The use of CFRPs noted above results in more ‘forgiving’ drivers that do not benefit accurate golfers that strike the ball at the sweet spot but reduce the penalty for less accurate golfers. This was also the result of the high moment of inertia clubs developed in the late 2000s.These drivers had squarer club heads that placed more mass around the periphery of the head (this needed thinner high-strength metallic alloys for the face and crown but with ‘poorer’ welds to prevent exceeding the CoR criterion for a sweet spot strike; or the use of lower density materials to allow larger crown, sole and skirt areas without taking the club head mass outside the 185–205 g range).The higher head moment of inertia would result in less twisting during the downswing and so should result in reduced dispersion of impact points on the club face, owing to the face being squarer to the ball at impact. Again, an accurate ball striker would not benefit but a less accurate one would suffer a reduced penalty in terms of distance and dispersion in the shots.

Figure 18.3 (a) Section through a complete CFRP head showing thickness at start of crown; (b) adhesive bonding overlap on composite crown of mixed composite/titanium-based alloy head; (c) corresponding overlap on titanium-based alloy face of same head

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Approach and recovery heads (irons and wedges) Grooves and backspin generation The more highly lofted clubs (15–60 degrees) are designed to generate progressively greater launch angles combined with greater backspin rates to generate control in the flight and roll distance of the ball. Following the investigation into high CoR drivers, attention has been focused on the effectiveness of grooves on more lofted clubs. The club regulations in force before 2010 placed an upper limit on the radius of curvature of the groove edges, together with restrictions on groove dimensions, orientation and spacing. This prevented extensive ‘wavy’ grooves, but did allow ‘U’ grooves with very sharp radii coupled with a greater groove crosssectional area compared with more traditional ‘V’ grooves. One particular aspect of the ‘U’ grooves appeared to be their greater efficiency in dealing with wet grass, so that the difference between backspin developed in dry conditions and from wet rough were small; ‘V’ grooves showed a major reduction in backspin rate under the same conditions. In dry conditions, the ball cover will tend to fill both groove types for lower lofts (see Chapter 22) and, as the loft increases the launch angle and backspin rate also increase.The latter increases as the torque from the impact force increases. As loft increases, the normal compressive force decreases. Initially, this will still fill enough of the groove that the tangential force generated between the ball and the groove side will ensure rolling and not sliding of the ball up the face. Eventually, however, the loft will reach a high enough value that the normal impact force does not cause enough interaction between the cover and the groove side and the reduced tangential force results in an element of sliding rather than complete rolling up the face.The backspin rate then starts to decrease as the loft angle increases.This phenomenon occurs in dry and wet conditions. In wet conditions, the continuous nature of the grooves allows the ball to displace water, resulting in similar behaviour to dry conditions. In the wet, rough grass is often present between the club face and the ball.The blades of grass will be trapped between the ball and the club face on the ‘land’ between the grooves as the ball compresses on to the face of the club. For grooves with a sharp radius of curvature, the blades of grass will be sheared and cut so that the short sections of grass can be compressed into the base of the groove, leaving the rest of the groove for the ball cover to fill and generate sufficient tangential force to prevent sliding, so maintaining backspin rates.The greater edge radius of the ‘V’ grooves means that the grass is not sheared as easily and, if still intact, prevents the cover interacting as effectively with the groove. This, coupled with the smaller, volume of the ‘V’ groove compared with the ‘U’ groove causes the former to generate less backspin from the wet rough than in dry conditions. From 1 January 2010, the requirements for conforming grooves were expanded to include a minimum edge radius and a limitation on the effective groove volume.

Wear resistance The role of grooves in backspin generation means that durable clubs will require wear resistance against abrasion from sand and grit particles. Abrasive wear occurs when the surface asperities (two-body wear) or abrasive particles (three-body wear) are at least 20 per cent harder than the surface being worn; for a sand wedge, where the major abrasive is silica, this is only around 650 Hv (Vickers hardness), which can be achieved for a case-hardened steel. Coastal and river sands that have been rounded by wave/water action give less wear, owing to their tendency to roll between the surfaces rather than ploughing into the softer. For general soils, alumina is often present and this abrasive has a hardness of 2,000 Hv; only ceramics, such as an 300

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alumina, rutile or boron nitride, would give a hardness of 1,600 Hv but they have poor impact toughness making them unsuitable for the body of the club head. Ceramics would therefore be used as wear-resistant coatings, together with decorative coatings such as ‘hard’ chromium. Even if a coating is used, it is important to optimise the strength and hardness of the underlying metallic component (this will need to be a balance of strength and toughness). Hence, the metallic alloys that would be used for durable clubs that retain their grooves would be the higher strength alloys, such as β-titanium, AISI 4340 or AerMet 100 (Table 18.1). To achieve the complex head shape coupled with high strength would require forging (often in multiple stages, as there is a limit to the amount of deformation that can be achieved in each forging stage).The grooves (but not any surface finish on the land between them) can be incorporated into the forging die design to form them at this stage but more accurate groove profiles can be obtained by machining them into the face. The machinability of most alloys decreases as the hardness increases so that wear resistant alloys are generally difficult to machine. Most of the high-strength grades in Table 18.1 require heat treatment to achieve their full strength so that they can be forged and solution-treated to a soft condition, allowing easier and faster groove machining before heat treatment to full strength.This would be carried out before any plating or coating and so it is necessary to ensure that the latter does not remove any of the beneficial effects of heat treatment. For irons and wedges, steels are preferred on cost and density grounds, as it is easier to redistribute mass for higher-density steel than for titanium-based alloys. A number of cavity-backed clubs use a carbon fibre composite (twill) insert adhesively bonded or co-cured to the rear of the club. This provides similar stiffness characteristics but at much reduced mass (density of carbon fibre composite generally 0.2–0.25 that of steel) and so allows greater mass redistribution. A high moment of inertia results in reduced rotation of the club head during the downswing and so a squarer impact of the club face on the ball. As for drivers, the poor wear resistance of the epoxy resin matrix in most composites precludes its use on the face itself.

Table 18.1 Summary of selected metallic alloy mechanical properties Alloy

Density, ρ (g/cm3)

Young’s modulus, E (GPa)

Yield stress, σy (MPa)

Tensile strength, UTS (MPa)

Ductility (%)

C-Mn (mild) steel

7.85

210

210–350

400–500

15–35

High-strength steel, e.g. 4340

7.85

207

860–1620

1280–1760

12

AerMet 100

7.89

192

1000–1780

1250–2080

14

316 stainless steel

7.85

195

205–310

515–620

30–40

Cu-Be

8.25

128

200–1200

450–1300

4–60

Al-Cu

2.77

73

75–345

185–485

18–20

Mg-Ti

1.78

45

200–220

260–290

15

Ti-3 Al-2.5 V

4.50

105–110

750

790

16

Ti-6 Al-4 V

4.43

110–125

830–1100

900–1170

10–14

Ti-15 V-3 Al-3 Sn- 3 Cr (β-titanium)

4.71

85–120

800–1270

810–1380

7–16

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Some steel irons and wedges are designed to rust, which will lead to some micron-scale roughening of the land between grooves; this, together with sand blasting, has been claimed to increase the effective coefficient of friction and backspin rates. Sand blasting in the legal range (Ra up to 5.25 µm) has been shown to have no noticeable effect on backspin rates (Monk et al. 2005), whilst the variable nature of rust formation and retention (striking the ball can dislodge poorly adhering oxide) make this unsuitable for consistent play.The degree of roughening induced by sand blasting can also be variable as this is often a manually controlled process (Monk et al. 2005) and so the degree of impact exposure can vary spatially. Sand blasting requires the face material to deform plastically and so is not appropriate to very hard, highstrength face materials, where the shot will just impact elastically and so will bounce off. As noted above, there is no major effect of the conforming surface roughness range on backspin rates and the process of sand blasting can round off the groove edges (as can corrosion) leading to a reduction in their effectiveness, Figure 18.4(a) and (b). A sand blasted finish (as it is applied to softer materials) will be worn away more readily than a milled finish, Figure 18.4(c) and (d) on a material subsequently hardened.

(a)

(b)

(c)

(d)

Figure 18.4 (a) Sand-blasted wedge showing damage to the groove profiles (b); (c) milled AerMet 100 face insert and clean groove edges (d)

Gear effect Off-centre hits lead to the ball flying to the left or right of the intended flight direction, which can result in missing the fairway or green with the penalty of a recovery shot usually using a 302

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highly lofted club that gives a short distance to the shot.As the impact force constitutes a torque about the centre of mass of the club head, it will rotate (clockwise for a toe hit for a righthanded golfer). Conservation of angular momentum would cause an anticlockwise rotation of the ball in this situation. This rotation is sidespin and generates a restoring lateral force acting to bring the ball back towards the desired flight direction (right-to-left for the situation described above). Ideally, the restoring force should vary with the distance of the impact point from the sweet spot, the shot distance and the impact speed so that the ball is brought back to the original intended landing point. However, conservation of energy means that if more rotational kinetic energy is needed, translational kinetic energy is reduced, leading to a reduction in shot distance, a penalty that increases for greater impact distances from the sweet spot.The degree of restorative sidespin is greater for drivers and cavity-backed clubs than for blades, which has been related to the distance of the centre of mass of the club head behind the face. The ball is then treated as a rigid gear that meshes with the face (treated as second rigid gear) so that the speeds of the face and ball at contact are equal.Whilst this qualitatively explains the spin direction of the ball, it does not account for the trend in sidespin rate values as the centre of mass is moved further back from the face (the moment of inertia should increase, leading to lower rotation speeds of the head for the same impact force and so a lower speed for face and ball at the point of contact). In addition, applying the gear effect vertically (to backspin) gives inconsistent and incorrect spin behaviour (Chou et al. 1994; Mahaffey and Melvin 1994).The treatment of the ball and club face as rigid bodies is too simplistic and the sidespin generated arises from face flexing, differential ball compression and asymmetric unloading (the side of the ball further from the sweet spot compresses less and unloads earlier than that closer). These features will be influenced by face thickness and (for drivers) bulge and roll of the face. A full explanation of these phenomena would require finite element modelling incorporating ball hyper- and viscoelastic effects.

Putters Putting differs from the other golf shots in that the ball generally should not leave the ground, merely roll over it. In its simplest form, the putter can be assumed to have zero loft and so, on striking the ball horizontally, causes it to slide initially over the ground.The base of the ball will be in contact with the green, causing a frictional force to act that slows down that part of the ball. The speed differential between the ball’s centre of mass and its contact with the green generates a torque that results in the motion being a combination of sliding and rolling. Eventually, as the ball’s centre of mass slows, the ball will enter a pure rolling phase. Reduced dispersion in putting has been related to the earlier transition to the pure rolling phase (Lindsay 2003; Hurrion and Hurrion 2002) and claims have been made for the effects of loft, vertical position of centre of mass and face groove patterns (putter grooves do not have to be straight and horizontal as for other club types). Often, putters have a positive loft (two to four degrees), which will tend to produce backspin on the ball as well as launching the ball, so that the time to pure rolling is extended by any time in flight as well as the time needed for friction between the ball and the green to remove the backspin produced at impact.Topspin can be generated during impact through the use of a putter with negative loft, which would lead to a shorter transition period or even rolling from impact reducing dispersion. The limit to using negative loft is the risk that the ball will be projected into the ground and bounce up, causing an increase in dispersion, particularly for higher impact speeds when the impact forces are greater. Studies have confirmed the importance of vertical position of head centre of mass and the lack of surface groove effects (Lindsay 303

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2003). A comprehensive study of ball and loft effects (George 2011) has confirmed the benefits of negative (–2 degrees) loft on dispersion through reduced transition distances using imaging at 600 frames per second, as well as the steady increase in sliding/sliding plus rolling for increasing positive loft up to four degrees.That study also identified the effect of ball type that was dominated by the cover material and hardness through variation in coefficient of friction with the putting surface. Commercially, recent trends in putters have been towards ‘horseshoe’ putters that have much of the mass set back from the face, which increases the moment of inertia of the head and results in the face striking the ball more squarely. The centre of mass of the head is kept low for ease of swinging, even though a higher head centre of mass would induce a more rapid transition to pure rolling and so less dispersion (Lindsay 2003).The ‘horseshoe’ mass requires a buttress to attach it to the face.The buttress can be used for marks to assist the golfer (for example, two-ball and three-ball putters) and will also reduce the differential straining and unloading of the ball so that sidespin on off-centre impacts and any corrective action associated with it are reduced.

Acknowledgements The authors wish to thank Professor P. Bowen for the provision of laboratory facilities and EPSRC and The R&A for continued support of their research in this field.

References Adelman, S. C., Strangwood, M. and Otto, S. R. (2006) ‘Modelling vibration frequency and stiffness variations in welded Ti-based alloy golf driver heads’ in E. F. Moritz and S. J. Haake (eds) The Engineering of Sport 6. London: Springer-Verlag, pp. 323–8. Brouillette, M. (2002) ‘On measuring the flexural rigidity distribution of golf shafts’ in E.Thain (ed.) Science and Golf IV: Proceedings of the World Scientific Congress of Golf. London: Routledge, pp. 387–401. Butler, J. H. and Winfield, D.C. (1994) ‘The dynamic performance of the golf shaft during the downswing’ in A. J. Cochran and M. R. Farrally (eds) Science and Golf II. London: E & FN Spon, pp. 259–64. Chou, A., Gilbert, P. and Olsavsky, T. (1994) ‘Clubhead designs: how they affect ball flight’ in A. J. Cochran (ed.) Golf The Scientific Way. Hemel Hempstead: Aston, pp. 15–25. Cochran, A. J. and Stobbs, J. (1968) Search for the Perfect Swing. Philadelphia, PA: Lippincott. George, A. (2011) The Influence of Swing Speed, Dynamic Loft and Ball Construction on Rolling Characteristics of a Golf Ball during Putting. Applied Golf Management Studies Dissertation, University of Birmingham. Huntley, M. P. (2007) Comparison of Static and Dynamic Carbon Fibre Composite Golf Club Shaft Properties and their Dependence on Structure. PhD Thesis, University of Birmingham. Hurrion, P. and Hurrion, R. D. (2002) ‘An investigation into the effect of the roll of a golf ball using the Cgroove putter’ in E. Thain (ed.) Science and Golf IV: Proceedings of the World Scientific Congress of Golf. London: Routledge, pp. 531–8. Lee, N., Erickson, M. and Cherveny, P. (2002) ‘Measurement of the behavior of a golf club during the golf swing’ in E. Thain (ed.) Science and Golf IV: Proceedings of the World Scientific Congress of Golf. London: Routledge, pp. 374–84. Lindsay, N. M. (2003) ‘Topspin in putters – a study of vertical gear-effect and its dependence on shaft coupling’, Sports Engineering, 6(2): 81–93. Mahaffey, S. and Melvin,T. (1994) ‘Metal woods or wooden woods?’ in A. J. Cochran (ed.) Golf The Scientific Way. Hemel Hempstead: Aston, pp. 27–30. Mather, J. S. B. and Jowett, S. (2000) ‘Three dimensional shape of the golf club during the swing’ in A. J. Subic and S. J. Haake (eds) The Engineering of Sport – Research Development and Innovation. Oxford: Blackwell Science, pp. 77–85. Monk, S.A., Davis, C. L., Otto, S. R. and Strangwood, M. (2005) ‘Material and surface effects on the spin and launch angle generated from a wedge/ball interaction in golf ’, Sports Engineering, 8(1): 3–11. Penner, A. R. (2003) ‘The physics of golf ’, Reports on Progress in Physics, 66(2): 131–71.

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Golf club construction, design and performance R&A (2010) A Guide to the Rules on Clubs and Balls. St Andrews:The Royal and Ancient Golf Club. Strangwood, M. (2003) ‘Materials in golf ’ in M. J. Jenkins (ed.) Materials in Sports Equipment. Cambridge: Woodhead Publishing, pp. 129–59. Strangwood, M. (2007) ‘Computational modelling of materials for sports equipment’ in A. J. Subic (ed.) Materials in Sports Equipment. Cambridge:Woodhead Publishing, vol. 2, pp. 3–34.

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19 TENNIS RACQUET TECHNOLOGY Franz Konstantin Fuss, Aleksandar Subic and Rod Cross

Introduction Implements such as racquets, clubs and bats serve several purposes: • they increase the arm length and allow reaching balls outside the arm span; • they improve the velocity when swung, as the linear velocity v equals the product of angular velocity ω and the radius r (r is longer when using a racquet); and • they increase the mass m and the moment of inertia I of the arm.

Evolution of tennis racquets Before the 1980s, tennis racquets were made of wood, with relatively small elliptical hoops (compared with modern racquets). Wooden racquets are quite flexible and have small sweet spots. Subsequently, racquets underwent considerable changes in material: • metals such as steel (Wilson T2000) and aluminium; • fibre composites, such as fibreglass and carbon fibres, which stiffened the racquets; • first-generation graphite racquets, such as Head Prestige and Wilson Prostaff 6.0, which were stiffer but heavier, with not much vibration damping; • second-generation graphite racquets such as the Dunlop 200G, consisting of a hollow graphite structure injected with foam, making them lighter and enabling more vibration damping; • modern titanium and hypercarbon racquets, such as Head TiS6 and Wilson Hypercarbon, which are ultralight, powerful and vibration damping.

Landmarks on a tennis racquet The performance of tennis racquets (which in general applies to any racquet, bat or club) is described among other parameters (such as the moment of inertia and the mass) by several landmarks. Among these are:

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• a geometrical point, the centroid of the string face; • two mechanically important points: the centre of mass of the racquet and the conjugate point at the handle; and • four functional points: the power spot (or best bounce spot), the centre of percussion, the vibration node (node point), and the dead spot. At least one of these four functional points is referred to as ‘sweet spot’, which is commonly the node point (vibration node; sweet spot, sensu strictori) or the centre of percussion; however, the term ‘sweet spot’ is applicable to all functional points, as each of them offers advantages. In general, ‘the sweet spot’ is that region of the string face which offers the best combination of feel and power. Unfortunately, the four functional points do not coincide, which is anyway mechanically impossible, and are not even extremely close. In modern racquet design, the centre of mass and the power spot are close to the throat of the racquet, separated from the vibration node and the centre of percussion, at the centre of the string face, and the dead spot at the tip. The power spot (or best bounce spot) is the most powerful spot on the string face, with the greatest coefficient of restitution (CoR). This means that a maximum amount of that kinetic energy, transferred by a ball to the racquet, is returned to the ball by the racquet.The racquet can be stationary (V equals 0) or moving (V is greater than 0) in this case; in which case, the power spot shifts toward the tip as V increases.The linear speed of a swung racquet is maximal at the tip and therefore the speed of the outgoing ball increases if the ball is struck towards the tip.This stands in contrast to the case where V = 0 and the speed of the outgoing ball is largest close to the throat. The dead spot, in contrast, is the least powerful spot on the string face with zero CoR.This means that a ball dropped on the dead spot of a stationary racquet does not bounce back at all. The reason for this phenomenon is that the kinetic energy of the ball is entirely transferred to the racquet such as the metal spheres in a Newton’s cradle.The misleading term ‘dead spot’ is not as disadvantageous as it sounds.When hitting a stationary ball at the dead spot of a swung racquet, then a large fraction of the kinetic energy of the racquet is transferred to the ball (the racquet does not come to a dead stop). The node point is the vibration node of the first mode in a freely suspended racquet and the node of the second mode in a clamped-free racquet.The hand is not rigid enough to clamp the handle. Therefore, a handheld racquet vibrates essentially as a free racquet but the hand damps the vibrations. It is impossible to excite the diving-board mode (20 Hz, fundamental clamped-free mode) when the racquet is handheld. If the node point of a swung racquet is hit by the ball, then vibrations (fundamental free-free mode, second clamped-free mode) is not excited and the player does not feel any uncomfortable vibration on impact with the ball. The centre of percussion is usually defined by the force equilibrium of the impact force at the string face and the racquet’s inertial force.Therefore, if a ball hits the centre of percussion, the player does not feel any shock force at the hand, as the two forces cancel each other out and the reaction force at the hand becomes zero. The shock force is another uncomfortable component experienced by the player. The power spot and the dead spot (serve only!) are related to the performance of the player. The vibration node and the centre of percussion contribute to the feel of the racquet. The shock forces at the handle are too high and short in duration, such that the athlete cannot compensate them.The muscles cannot react that fast and are thus stretched, together with their tendons. The contact between ball and string face lasts for about 5 ms (± 2 ms), whereas the damping of vibrations takes 30–50 ms. The unpleasant feel of repeated shock forces and 307

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vibrations affect the soft tissues and result in overuse syndromes such as the tennis elbow.The functional landmarks are discussed in detail in the following sections.

Power spot The Power Spot is defined as that point where the CoR of the ball is at its maximum. The CoR is the ratio of rebound to incident speed, which equals the ratio of the square roots of rebound to drop height.The maximal CoR is unity but can be less than one if energy stored within an object is released; for example, angular kinetic energy transformed to linear (in oval balls) or vibrational energy transformed to kinetic energy. Strings vibrate at 500–600 Hz and are softest when hit in the centre of the string face and return most energy.Thus, the power spot of the strings is in the centre of the string face.The stiffer the racquet frame, the higher is the CoR.The shorter a beam, the stiffer it is.Thus, the frame is stiffest at its handle, where the frame power spot is located.The compromise of these two power spots in a racquet (combined string face and frame) is half way in between; that is, at the throat (Figure 19.1a and b).There are three strategies of increasing the CoR: • moving the power spot closer to the handle by enlarging the head of the racquet (and keeping the overall length and stiffness constant); this increases the CoR of a ball on impact, if the racquet is translated; • making the frame stiffer through a larger cross-section and a stiffer material; • reducing the tension of strings, which results in less deflection of the ball; this effect, however, is very small. Comparing vintage wooden racquets with modern ones, the difference in head size becomes evident as the throat of the modern racquet extends further down the handle (Figures 19.1c and d).

Figure 19.1 Power of the Head Youtek IF Speed 18 ⫻ 20 racquet: (a) power (per cent) at impact locations (51 per cent corresponds to a CoR of 0.51); (b) power zones; (c & d) Head Youtek IF Speed 18 ⫻ 20 compared with Dunlop Maxply Fort (wooden racquet) Source: (a & b from Lindsey 2012; © 2012 by Crawford Lindsey and Tennis Warehouse University, reproduced with kind permission)

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Dead spot The dead spot, as mentioned above, is that area of the string face close to the tip, where the ball, hitting a stationary racquet, does not bounce back well. The kinetic energy is almost entirely transferred to the racquet and not returned to the ball.This is because the effective mass of the racquet, Me, at the dead spot is close to the mass m of the ball. If we clamp a racquet horizontally to a table, such that 100 mm of the handle are in contact with the table, and try to determine the dead spot, we would not find one, as the (tennis) ball (m = 57.5 g) always bounces back. Yet the closer to the tip the ball hits the string face, the smaller is the bounce height and thus the CoR. The reason for this is that the racquet is too heavy for the tennis ball to completely absorb the ball’s energy, and vice versa.The experiment, however, works perfectly with a cricket ball (m = 160 g). The effective mass Me of a racquet at any impact point is calculated from r Me = M –––– b+r

(1)

where M is the mass of the racquet, b is the distance between the centre of mass and any impact point, and r is the distance between the centre of mass and the centre of pressure between hand and handle (usually 100 mm off the butt, a standardised point about which the swing weight of the racquet is calculated).The effective mass means that the racquet behaves like a point mass M at the end of a massless rod of length r + b (Cross and Nathan 2009). The position of the dead spot results from

( )

M bDS = r –– –1 m

(2)

where bDS is the distance between the centre of mass and the dead spot. Calculating bDS from equation (2) when using the parameters M and r of different racquets delivers that the dead spot would be beyond the tip of the racquet at a distance of 0.5–1.0 m. For cricket balls, the dead spot is within the string face, about 0.1–0.15 m off the tip. Even if the dead spot does not physically exist in tennis racquets, the closer to the tip that a ball hits a stationary racquet, the larger is the energy transferred to the racquet. Similarly, if a moving racquet strikes a stationary ball close to the tip, then most of the kinetic energy of the racquet is transferred to the ball.This explains why hitting the ball close to the tip when serving results in high ball speeds: • the ball is stationary with respect to the direction of the moving racquet, resulting in maximal energy transfer; • the linear velocity is maximal at the tip, as it is furthest away of the shoulder joint.

Vibration node A racquet vibrates in three different modes of deflection: the bending mode, the torsional mode and the hoop mode.The bending mode is the most important and the node point refers to this mode. If the ball is hit off centre, then the torsional mode is also excited. The hoop mode is always present, owing to the pull at the strings on impact. Although a racquet is not a uniform beam, it can be considered to behave like one, since the centre of mass is roughly in the middle of the racquet.A freely suspended racquet has two node points in the fundamental frequency mode, one at the string face and the other at the transition 309

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from handle to throat, each located roughly one-fifth of the racquet length from tip and butt. In the clamped-free condition, the racquet deflects like a springboard in the fundamental mode (without a node point) and has a single node point at the string face in the second mode, the position of which is identical to the node of the fundamental mode of a free–free racquet.The stiffer a racquet, the higher is the frequency of the fundamental mode of a free–free racquet, which is the same as the frequency of the second mode of a clamped–free racquet. This frequency ranges from 100 Hz in wooden racquets to 200 Hz in very stiff racquets.These vibrations cause an undesirable feeling, loss of control and fatigue.The next mode (second free–free mode) has a frequency 2.75 times the fundamental frequency (free–free condition). It is, however, not excited significantly, as the ball–string contact time is only 5 ms; thus, the amplitude of this mode is very small and moreover this mode dampens out quickly.The fundamental mode of a clamped-free racquet vibrates at 25–40 Hz, depending on its stiffness. The node point refers to the centre line of the racquet. At the hoop, the node points are located closer to the tip compared with the point at the centre, such that the resulting ‘node line’ is curved.This curvature occurs because the hoop deflects not only about an axis perpendicular to the centre line of the racquet (bending mode of a beam) but also about the centre line (as if the right and left halves of the hoop were wings and flap up- and downward). If the string face is hit at the node point or node line, the fundamental vibration mode (free–free condition) is not excited.As this is not always the case, since the performance-related points are located close to the throat and tip of the racquet, vibration damping becomes all the more important. Damping in racquets can be achieved in different ways: • viscous material with a high damping ratio incorporated into the frame; • damped vibration absorbers included in the racquet; for example, small voids filled with energy-absorbing material with a centre mass such as lead beads; • special cushioning materials covering the handle; • small additional masses added to the frame; • less string tension; • piezo shunts. Piezo shunts are layers of piezoelectric material, such as ceramic fibres (HEAD intellifiberTM; Lammer 2005, 2007), which generate electrical power when strained. Layers on opposite sides of the racquet are connected with wires (‘shunt’) and stiffen each other when deflected with the electrical power produced. With electrical bandpass and bandstop filters, a specific frequency band can be filtered out and a specific frequency mode dampened (Lammer, 2005, 2007). String dampers are small weights inserted between cross and mains strings close to the throat. Their effect is treated controversially in the literature. Stroede et al. (1999) found no evidence to support the contention that string vibration dampers reduce hand and arm impact discomfort. Equally, Li et al. (2004) claim that string dampers do not reduce the amount of racquet frame vibration received in the forearm and that they rather serve a psychological purpose. Mohr et al. (2008) found that different damper configurations do have an effect on the amplitude and frequency of the vibration modes.

Centre of percussion If the ball hits the racquet at the centre of percussion, the impact force is in equilibrium with the inertial force at the centre of mass.The two forces cancel each other out and the player feels 310

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no shock force at his or her hand.The two forces, a force couple separated by the distance bCP, produce a torque which is in equilibrium with the racquet’s moment of inertia times its angular acceleration.The linear acceleration a of the centre of mass (which causes the inertial force) and the angular acceleration α are produced by the impact (Figure 19.2).

Figure 19.2 Impact between ball and racquet; ω = angular velocity; v = linear velocity; r = distance between instant centre and centre of mass; b = distance between centre of mass and impact point; FA = impact force by the ball; FI = inertial force; FR = reaction force at the hand (here FR = 0); α = angular acceleration; a = linear acceleration; M = moment produced by the force couple FI and FR; T = product of moment of inertia and α

The distance bCP between the centre of percussion and the centre of mass is calculated from the conservation of linear and angular momentum, ∆p ∆L v = –––– and ω = –––– M ICOM

(3)

where v and ω are the linear velocity of the centre of mass and the angular velocity of the racquet, respectively; p and L are the linear and angular momentum; and M and ICOM are the racquet’s mass and moment of inertia (about the centre of mass).The racquet is rotated about an instant centre located at the handle, which is set to a point 100 mm off the butt for standardisation purposes (the point about which the swing weight is calculated).This rotation about the instant centre is a combined motion consisting of a rotation about the centre of mass and a translation of the centre of mass.The angular velocity ω is the same about any point. As ∆L = ∆p bCP

(4)

We obtain from equations (3) and (4): ∆L ∆p b ω = –––– = ––––CP ICOM ICOM

(5)

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The distance r between the instant centre and the centre of mass is obtained from v ICOM ∆p / M r = –– = –––––––––– = –––– ω ∆p bCP / ICOM bM

(6)

The position of the centre of percussion, defined by the distance bCP, results from ICOM k2 = –– bCP = –––– rM r

(7)

where k is the radius of gyration about the centre of mass, k = r bCP

(8)

The force and moment equilibrium can now be calculated, considering equation (8). FR, the resultant force at the handle is zero. Thus, FA = FI, which are the force applied by the ball to the string face and the inertial force.The moment T1 produced by the force couple is: T1 = FA bCP = FI bCP. T1 is in equilibrium with T2 = ICOM α. As ICOM = M k2 = M r bCP, FI = M a, a = rα, and r bCP = k2, T1 = FA bCP = FI bCP = Ma bCP = M bCP rα T2 = ICOM α = M k2 α = M bCP r α

(9a) (9b)

T1 = T2, q.e.d. As ICOM and r are constants, and when considering FA constant for comparative reasons: the longer b, the larger a, α and FI.The larger the forward pointing FI with respect to the backward pointing FA (considered constant), the larger is the resultant force FR, directed backward and applied by the hand to the handle, if the racquet is the free-body diagram, or directed forward and applied by the handle to the hand, if the hand is a part of the free-body diagram. If the ball is hit between the throat and the centre of pressure (b < bCP; power spot), the hand is subjected to a backward shock force directed towards the palm. If the ball is hit between the tip and the centre of pressure (b > bCP; ‘dead spot’), the player’s hand experiences a forward shock force, directed towards the fingers. The latter case is worse, as the area of the fingers is smaller, resulting in a higher pressure, and the flexor muscles are stretched on impact. To calculate the bCP, we need to know the ICOM. The ICOM is conveniently determined by suspending the racquet and let it swing freely.The distance between the suspension point P and the centre of mass is bP. The gravitational force of the swinging racquet is FG = M g. The moment of FG, TFG = M g sinθ bP. For small angles, sinθ = θ.The torque at P, TP = IP α. From the parallel axis theorem, IP = ICOM + M bP2. From the moment equilibrium we obtain: –M g θ bP = (ICOM + M bP2) α = (ICOM + M bP2) d2 θ/dt2

(10)

The solution of this ordinary differential equation is M g bP sin (ωt) = sin (ωt) ω2 (ICOM + M bP2)

312

(11)

Tennis racquet technology

Solving for IP by simplifying yields MgbP IP = ICOM + M b2P = –––– ω2

(12)

ICOM thus is

(

)

g ICOM = M bP = ––2 – bP ω

(13)

After simplifying further g ω2 = –––– r + bP

(14)

In wooden racquets, the centre of percussion is located about 30 per cent of the racquet length off the tip; in modern racquets, about 20 per cent. As the string face is enlarged towards the handle in modern racquets, the centre of percussion is located roughly at the transition from the middle to the upper third of the string face and, in wooden racquets, roughly at the transition from the lower to the middle third. However, the problem with the centres of percussion in the above calculations is that they apply only to a stationary racquets or racquets moving at a constant speed, as they ignore the moment imparted to the racquet once it is accelerated (Fuss 2011). From equations (9a) and (9b) we obtain ICOM α – FI bCP = 0

(15)

for the static case. Considering that the racquet is accelerated by the torque T, T + ICOM α – FI bCP = 0

(16)

Solving for bCP

T + ICOM α T + ICOM α bCP = –––––––– = –––––––– FI FA

(17)

Equation (17) indicates that the larger T, the larger bCP and, thus, the centre of percussion moves towards the tip of the implement. Even if the torque is known, FA, FI and α depend on various input parameters and are moreover time dependent (Fuss 2011). Figure 19.3 shows the hand force with time for a racquet accelerated with a free moment T of 7.6 Nm and a centre of percussion located at 72.16 per cent (relative to the racquet length, measured from the butt) in static conditions. In a dynamic condition (that is, with the torque T accelerating the racquet) the hand force is only instantaneously zero and the minimum (positive or negative) hand force results from an impact location at 85 per cent (Fuss 2011).

Swing weight The swing weight, IA, is the racquet’s moment of inertia about the conjugate point located 100 mm off the butt. The swing weight is the most important performance parameter of a tennis racquet as it correlates highly with the apparent CoR (CoRA; Cross and Nathan 2009). The racquet mass M, however, has a comparatively small influence on the performance (apart from the fact that the moment of inertia is proportional to M). 313

Franz Konstantin Fuss, Aleksandar Subic and Rod Cross

75

100% 50 9 5 %

- 90%

25

^

160 140 57–62

0.90 1.21–1.25 0.94–0.96

48–55 34–40 55–73

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The science and engineering of golf balls

Covers are generally injection moulded around the solid core(s) (with or without mantle) using a high strength tool steel mould and high clamping forces; high injection pressures are needed to ensure complete filling of the dimple pattern in the die. Thinner covers (approximately 0.8–0.95 mm thick) are often seen for the polyurethane on three- and four-piece balls, whilst two-piece balls have cover thicknesses of approximately 1.4–1.9 mm (both polyurethane and ionomer).The thinner covers will solidify more rapidly than thicker ones and these show little variation in thickness around the core, whereas thicker covers can be liquid for long enough for the core to float or sink under buoyancy/gravity effects (driven by density differences) leading to unequal cover thicknesses around the core. Unequal cover thickness does not have a strong effect on impact but the non-coincidence of ball centre of mass and geometric centre leads to potential draw/bias when rolling in putting.To reduce this asymmetry, multiple injection points for the liquid polymer are used, which can be revealed (along with polymer flow lines) by re-melting the cover (Figure 22.3).The need to separate the metal die halves after cover solidification often leads to a circumferential seam that has no dimples or shallow ones that allow ball removal through elastic deformation of the land between the dimples (Kai 2008). Mantles, if present, are generally hard ionomers and are plain shells so that compression moulding of two hemispherical shells and then joining these (by localised melting and resolidification of the hemispherical shell edges) around the core to give consistent thickness (0.8–1.2 mm) but leaving a seam (Figure 22.3).The presence of seams on the core, mantle and cover has not been shown to have a significant effect on ball performance.

(a)

(b)

(c)

(d)

Figure 22.3 (a) Sectioned ball showing cover thickness variations; (b) reheated golf ball showing cover flow lines; (c) seams in compression moulded core and mantle; and (d) seam on cover and thin polyurethane cover

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Golf ball aerodynamics The importance in ball interaction with the club face is to generate backspin to increase drive distances and give rapid stopping on landing for approach shots.A non-spinning ball will experience a drag force but, owing to the symmetric airflow around the ball, no lift force. In contrast, a ball with backspin (top surface of ball travelling in the same direction as the airflow) will shift the high pressure region from directly in front of the line of flight to in front, but slightly below; the corresponding low pressure region moves up, whilst remaining on the rear half of the ball.This movement of pressure regions leads to the generation of a lift force in addition to the drag force.This allows the ball to be launched at a lower launch angle, which is more suited to a golfer’s swing, giving the ball a higher horizontal speed after impact.The lift force then gives a trajectory that rises, but, as the lift coefficient CL is a function of speed and spin rate, there is a steep descent after the peak height in the trajectory as spin rate and horizontal speed have decreased. The action of dimples on a golfball is to form microvortices that bring the minimum in drag coefficient CD down from the Reynolds number for a smooth ball (Re ~400,000) to that generated by the average golfer (Re ~200,000). By doing this, the lift coefficient values range around 0.5 but the CD (also a function of speed and spin rate) is reduced to approximately 0.2.A smooth ball will travel around one third the distance of a dimpled ball for the same launch conditions.

Ball characterisation Testing by the R&A and USGA for the conforming nature of golf balls is on the basis of passing or failing the criteria in Table 22.1. Those balls that conform are listed (USGA 2012) but no quantitative results are given.A number of quantitative measures are available to characterise balls and their component materials but these do not represent the conditions under which the balls operate in use.

Shore D The polymer properties in Table 22.2 are represented by their Shore D values.This test involves indenting the polymer (component or ball) with a needle and determining the resistance against a calibrated spring. Shore D testing may penetrate up to 5 mm into the polymer (including the surrounding strain field) with an affected area around the needle of 1–2 mm. This technique is well suited to measuring local properties, such as the variation in hardness with radius in a golfball core (Strangwood et al. 2006), which can be related to modulus (Qi et al. 2003). However, the small strains involved and low strain rate (durometers can be applied by hand) mean that these tests do not even approximate to the conditions of a ball during putting.

Compression testing Ball manufacturers often quote compression values for the ball, which, unlike the Shore D hardness, subject the ball to amounts of deformation closer to those that would be experienced in golf shots. Compression values are usually in the range 100–180, with the higher numbers being associated with harder ‘distance’ balls. There are two principal ways of measuring the compression of a ball (Dalton 2002).

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Compression to a fixed amount This is the basis of the PGA of America compression and the Atti compression tester, which uses a piston to compress a ball against a spring of known resistance and so is essentially a scaled-up durometer.The piston travels a set distance, which is shared between the ball and the spring with the spring deforming more for a harder ball and the deformation of the spring is used to define the compression of the ball. These are hand-operated machines and so apply a variable deformation rate of 1.5–15 mm/s, compared with drive deformation rates of around 20,000 mm/s.

Application of a fixed load These are also slow deformation rate tests with a load being applied, by a lever arm (Riehle tester) or directly on to the ball.Various loads are applied, typically 90 (Riehle), 100 or 130 kilogram force (kgf), and the amount the ball deforms is measured from which the compression number is calculated. Some measures (Riehle and 100 kgf compression) use the direct deformation, whilst the difference between the deformation under 130 kgf and 10 kgf is used for the 130–10 kgf compression values. The lower the amount of deformation the higher the compression number as the ball is harder. As for the Atti test these are low strain rate tests and at loads well below those of typical drives (600–1,000 kgf).

Load/deflection testers Compression/tension testing machines can be used to generate load/deflection curves (Figure 22.2) for load as the deflection is increased at a constant rate or for deflection as the load increases, also at a constant rate.The deformation rate is limited by the mechanism in the tester (hydraulic pump or servo-electric motor), but the maximum is around 25 mm/s. The advantage of these tests is that they can replicate the loads and deflections experienced during actual golf shots for both loading and unloading, thus giving representative hysteresis values for the component materials and for full balls.This was the basis of rationalising the coefficient of restitution (CoR) behaviour of two and three piece balls by Strangwood et al. (2006). For testing of some component materials, such as cylinders extracted from golf ball cores or strips cut from mantles, few problems arise in extracting representative samples. Problems exist when trying to determine the properties of cover materials, owing to their thin nature, low modulus and/or curved shape. Mase and Kersten (2004) reshaped cover materials into tensile test pieces but this may have altered the long chain interactions and with it the properties of the polyurethane/ionomer. In materials testing, high strain rates are usually achieved at the expense of large strains; for polymer testing, strain rates similar to those experienced during impact can be obtained from dynamic mechanical thermal analysis. This technique uses strip specimens that can be cut from the core, although mantle samples would need flattening, whilst the covers are generally too compliant to give reliable results.The strain involved in these tests are of the order of one to two per cent and so do not replicate the amounts of deformation experienced during impact. This is a limitation of other high strain rate tests, such as the split Hopkinson bar, which also operates at strain rates much greater than experienced during a golf drive. In addition, the Hopkinson bar measures the propagation of a stress wave pulse from the impact and so unloading data (necessary for hysteresis determination) are not readily available unless coupled with high-speed video imaging.The split Hopkinson bar has been used to show that solid multi-piece balls can show Hertzian behaviour on impact at speeds between 1.4 m/s and 357

Martin Strangwood

67 m/s (Jones 2002). The unloading behaviour of viscoelastic materials, such as PBD, can be characterised by their stress relaxation; the decrease in stress required to maintain a given strain with time at that strain. The inertia of most testing machines means that load overshoot is common leading to inaccuracies in the starting load. Most stress relaxation values are fitted to the shallow decrease in stress at longer times well beyond the 0.5 ms impacts experienced by golf balls. Stress relaxation tests are therefore not suitable for sports balls (Ranga and Strangwood 2010).

Gas cannon tests High speeds (up to 75 m/s–1) can be achieved by firing golfballs under a sudden release of gas pressure in a cannon. If these balls are fired at an immovable target, such as a 200 kg steel plate, and the incoming and return speeds measured then the ratio of these can be used to calculate the CoR, which can be related to the hysteresis and energy losses during loading and unloading. The use of accelerometers on the target plate allows impact forces to be measured, although, as for the split Hopkinson bar, high-speed video imaging (Johnson 2005) is needed to determine stress and strain response, whilst finite element modelling is often needed to deconvolute the signal for multi-piece balls.

Aerodynamic characterisation The importance of lift and drag during flight of a golfball and the dependence of lift and drag coefficients on spin rate mean that these need to be characterised for trajectory modelling. Environmental factors, such as temperature and wind speed, will affect the trajectory for outdoor testing and so uniform temperature and still air conditions are used to determine the flight characteristics in indoor test ranges (Zagarola et al. 1994). Indoor test ranges use arrays of light screens, cameras or radars to measure the velocity and spin rate during the trajectory to give information about lift and drag coefficients but also aspects such as spin decay that could not be readily determined from wind tunnel tests (although the lift and drag coefficients for specific spin rates and angles of attack could be).

Modelling golf ball behaviour Aerodynamics Aerodynamic and computational fluid dynamic (CFD) models are being used to model the trajectory of golfballs, based on wind tunnel or indoor test range data (Smits and Smith 1994; Kai 2008). The models for the whole ball are generally fitted to the wind tunnel/indoor test range data as the scale differences preclude (currently) modelling all the dimples on a ball as well as the ball flight itself.The specialised testing needed to supply data for these models means that most are based within ball manufacturing companies, who often use them to provide trajectory predictions based on these models for launch monitors (as used in driving ranges) based on launch velocities and spin rates. The monitoring of golfballs in flight has become easier more recently through the introduction of radar systems, such as TrackMan, that use Doppler-based techniques to measure the ball path as well as spin rates during the flight. These systems allow more ready access to ball data and so easier trajectory fitting and modelling.

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Mechanical behaviour Much of the modelling of golf balls is associated with their impact behaviour in terms of CoR (speed off the face), launch angle and spin rate. CoR is dominated by the viscoelasticity of the core and so a number of researchers have used spring and dashpot-based models (Cochran 2002; Lieberman and Johnson 1994) with some success. These models, although simple in application, do rely on accurate material data and so the aspects of characterisation noted above with respect to strain and strain rate limit their accuracy; some of the materials data are obtained at strain rates 106 times slower than experienced during a drive. Also Voigt- and Maxwell-type elements are poor at representing non-linear behaviour, which is particularly important in the unloading phase of elastomeric rubbers such as PBD. Finite element models (Tanaka et al. 2006; Tavares et al. 1998b; Cornish et al. 2008b) have generally used hyperelastic loading data (Mooney-Rivlin, Ogden or, more appropriately, Arruda-Boyce). The use of hyperelastic models is suitable, owing to the domination of strain effects over strain rate effects for the PBD grades used in golfballs (Strangwood et al. 2006). While most models work well for the loading phase, the unloading phase is less well modelled as it requires the use of viscoelastic parameters. Prony series are often used to represent the viscoelastic behaviour through the viscoelastic modulus ratio of shear relaxation modulus (g–p) and the relaxation time (τG). Most models use single values for these parameters in their finite element models; for example, Tanaka et al. (2006) use 0.4 for g–p but both are likely to show varying strain and strain rate dependence. Accurate prediction of ball behaviour from component materials rather than fitting to ball characteristics will require strain and strain rate dependent parameters to be used in the Prony series. The CoR represents normal impact behaviour with friction between the face and ball cover needed for oblique impacts. Given the domination of cover deformation into grooves (Cornish et al. 2008a), this can be treated as a constant up to the loft angle corresponding to maximum backspin rate. Models based on this approach do give reasonable predictions (Cornish et al. 2008b) but can be expanded to predict cover interaction at one level and ball compression and rolling at another. The precise role of the mantle in reducing energy losses for greater core deformation is one area that needs addressing along with effective clubface/ball frictional forces.

Acknowledgements The author wishes to thank Professor P. Bowen for the provision of laboratory facilities and Engineering and Physical Sciences Research Council and the R&A for their continued support of his research in this field.

References Cochran,A. J. (2002) ‘Development and use of one-dimensional models of a golf ball’ Journal of Sports Science, 20: 635–41. Cornish, J. E. M., Otto, S. R. and Strangwood, M. (2008a) ‘The influence of groove profile; ball type and surface condition on golf ball backspin magnitude’ in F. K. Fuss, A. Subic and S. Ujihashi (eds) The Impact of Technology on Sport II. London:Taylor & Francis, pp. 229–34. Cornish, J. E. M., Otto, S. R. and Strangwood, M. (2008b) ‘Modelling the oblique impact of golf balls’ in M. Estivalet and P. Brisson (eds) The Engineering of Sport 7. Paris: Springer, vol. 1, pp. 669–75. Dalton, J. L. (1998) ‘The curious persistence of the wound ball’ in M. J. Farrally and A. J. Cochran (eds) Science and Golf III. Champaign, IL: Human Kinetics, pp. 415–22. Dalton, J. L. (2002) ‘Compression by any other name’ in E. Thain (ed.) Science and Golf IV. London: Routledge, pp. 319–27.

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Martin Strangwood Johnson,A. D. G. (2005) The Effect of Golf Ball Construction on Normal Impact Behaviour. PhD Thesis, University of Birmingham. Johnson, A. D. G., Otto, S. R. and Strangwood, M. (2004) ‘Radial property variations in “solid” golf balls and their effects on impact performance’ in M. Hubbard, R. D. Mehta and J. M. Pallis (eds) The Engineering of Sport 5,. Sheffield: International Sports Engineering Association, vol. 2, pp. 10–16. Jones, I. R. (2002) ‘Is the impact of a golf ball Hertzian?’ in E. Thain (ed.) Science and Golf IV. London: Routledge, pp. 501–14. Kai, M. (2008) ‘Science and engineering technology behind Bridgestone Tour golf balls’ Sports Technology, 1: 57–64. Lieberman, B. B. and Johnson, S. H. (1994) ‘An analytical model for ball–barrier impact. Part 1: models for normal impact’ in A. J. Cochran and M. R. Farrally (eds) Science and Golf II. London: E. & F. N. Spon, pp. 309–14. Mase, T. and Kersten, A. M. (2004) ‘Experimental evaluation of a 3-D hyperelastic, rate–dependent golf ball constitutive model’ in M. Hubbard, R. D. Mehta and J. M. Pallis (eds) The Engineering of Sport 5. Sheffield: International Sports Engineering Association, vol. 2, pp. 238–44. Nesbitt, R. D., Sullivan, M. J. and Melvin,T. (1998) ‘History and construction of non-wound golf balls’ in M. J. Farrally and A. J. Cochran (eds) Science and Golf III. Champaign, IL: Human Kinetics, pp. 407–14. Penner, A. R. (2003) ‘The Physics of golf ’, Reports on Progress in Physics, 66: 131–71. Qi, H. J., Joyce, K. and Boyce, M. C. (2003) ‘Durometer hardness and the stress–strain behavior of elastomeric materials’, Rubber Chemistry and Technology, 76: 419–35. Ranga, D. and Strangwood, M. (2010) ‘Finite element modelling of the quasi-static and dynamic behaviour of solid sports balls based on component material properties’, Procedia Engineering, 2(2): 3287–92. Smits, A. J. and Smith, D. R. (1994) ‘A new aerodynamic model of a golf ball in flight balls’ in A. J. Cochran and M. R. Farrally (eds) Science and Golf II. London: E. & F. N. Spon, pp. 340–7. Strangwood, M., Johnson, A. D. G. and Otto, S. R. (2006) ‘Energy losses in viscoelastic golf balls’ Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 220: 23–30. Tanaka, K., Sato, F., Oodaira, H., Teranishi, Y. and Ujihashi, S. (2006) ‘Construction of the finite-element model of golf balls and simulations of their collisions’, Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 220: 13–22. Tavares, G., Shannon, K. and Melvin,T. (1998a) ‘Golf ball spin decay model based on radar measurements’ in M. J. Farrally and A. J. Cochran (eds) Science and Golf III. Champaign, IL: Human Kinetics, pp. 464–72. Tavares, G., Sullivan, M. and Nesbitt, D. (1998b) ‘Golf ball spin decay model based on radar measurements’ in M. J. Farrally and A. J. Cochran, Science and Golf III. Champaign, IL: Human Kinetics, pp. 473–80. USGA (2012) Conforming Golf Ball List. United States Golf Association. Available online at http://www.usga.org/ConformingGolfBall/conforming_golf_ball.asp (accessed 4 January 2013). Zaragola, M.V., Lieberman, B. and Smits, A. J. (1994) ‘An indoor testing range to measure the aerodynamic performance of golf balls’ in A. J. Cochran and M. R. Farrally (eds) Science and Golf II. London: E. & F. N. Spon, pp. 348–54.

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23 SOLID MECHANICS AND AERODYNAMICS OF CRICKET BALLS Franz Konstantin Fuss, Aleksandar Subic and Rabindra Mehta

Introduction Cricket balls are unique pieces of sports equipment, manufactured exclusively from biological materials (with the exception of plastic cores of cheap, practice balls).They consist of a leather cover, the core and the seams. The uniqueness of the ball comes from its deterioration with time throughout an innings, a property inherent to the game, which is caused by the construction of the ball and decisively affects its aerodynamic behaviour.The curved flight path of the ball (the swing) is an intrinsic property, depending on the relative location of the seam and on the surface smoothness or roughness and is used by bowlers as a strategic element to deceive the batsman. In addition to aerodynamics, the ball impacts on the turf and the bat. The ball’s mechanical properties influence its hardness, bounciness and speed. The faster a ball flies, the harder it is on impact.The less energy is lost on impact, the faster a ball bounces off the bat and the more likely a batsman hits a six. The performance criteria of a cricket ball can therefore be summarised under aerodynamics, (the flight path) and solid mechanics (hardness, viscosity, energy return and rebound speed). These performance criteria will be discussed subsequently in detail.

Construction and manufacture of cricket balls The cover of a cricket ball is traditionally made from cow hide. Rump leather is thicker and thus used for high-quality balls, whereas belly leather is reserved for practice balls.The leather cover consists of either two or four pieces (two- and four-piece ball, respectively) of leather. For two-piece balls, circular leather patches are punched out of cow hide and extruded into a hemispherical shape. In four-piece balls, a hemisphere consists of two diagonal patches sewn together with internal stiches in such a way that the thread pierces only the inner surface.The thread bulges the outer surface by producing a rippled pattern, which is of aerodynamic importance. This apparently seamless connection is termed the quarter or secondary seam. The primary seam joins the two hemispheres together and consists of six rows of stitches, three on either side of the seam. In traditional manufacturing, rows one and six (or one and five), two and five (or two and six) and three and four are double hand-stitched in first class balls (grade 1 county balls). In automatic stitching, only rows three and four are double stitched.The leather 361

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is either dyed red (red balls) or bleached white (white balls, with additional surface finish), the latter is exclusively used for day matches extending into night time. For full night matches, yellow, orange or pink balls are intended to be used.The construction of the core is not standardised, as several variations exist, such as single core balls, and three- and five-layered balls. Usually, the layers of a core follow a concentrically spherical pattern, although rolled-core constructions exist as well, such as ‘Swiss-roll’ construction (non-homogenous sphere such as Readers rolled-core ball). The nucleus of cricket balls comes in different sizes, depending on the number of cork layers surrounding it.The nucleus is usually moulded from a cork–rubber mixture.Top-quality balls consist of five cork layers surrounding the nucleus, high-quality balls of three layers and all other balls comprise of a large nucleus only, without any cork layers. A cork layer is constructed from two cork patches, ideally hour-glass shaped and joined together like the leather patches of a baseball. Each layer is surrounded by tension-wound wet yarn, which shrinks when left to dry and thus compresses the layer.The layer thickness differs from model to model; for example, Kookaburra Turf and Sanspareils-Greensland Test are both fivelayered balls with a nucleus diameter of 35 mm and 40 mm, respectively; Kookaburra Regulation is a three-layered ball with a nucleus of 40 mm in diameter (Table 23.1). The construction of different cricket balls is shown in Figure 23.1.

Figure 23.1 Construction of cricket balls: (a–c) schematic cross-sections of cricket balls (a) non-layered; (b) three cork layers; (c) five cork layers; (d) Regent Match (white) – non-layered; (e) cork layer element of Sanspareils-Greenlands Tournament rubber core ball; (f–g) cross-sections of layered balls (Sanspareils-Greenlands Tournament subtypes): (f) rubber core version; (g) cork core version

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Solid mechanics and aerodynamics of cricket balls Table 23.1 Stiffness clusters according to Figure 23.2a; ‘hard’ and ‘soft’ refer to hard and soft subtypes of specific cricket ball models; ‘2p’ and ‘4p’ refer to the number of pieces of the leather cover; the diameter of the nucleus is indicated in mm (non-layered balls are referred to as ‘60–65mm’) Hard cluster

Medium-hard cluster

Medium cluster

Medium-soft cluster

Soft cluster

Regent white Match hard (2p, 60–65 mm), Wisden Royal Corinthian (2p, 60-65 mm), SanspareilsGreenlands Tournament hard (cork nucleus, 4p, 40–50 mm), Gunn & Moore Clubman (4p, 40 mm), Regent red Match hard (4p, 50 mm)

Regent red Match soft (4p, 50 mm), Kookaburra Turf (4p, 35mm), SanspareilsGreenlands Test (4p, 40mm)

Kookaburra Regulation (4p, 40mm), Gray-Nicolls Super Cavalier hard (4p, 42–46 mm), SanspareilsGreenlands Tournament soft (rubber nucleus, 4p, 40 mm)

Gray-Nicolls Super Cavalier soft (4p, 54–60 mm)

Regent white Match soft (2p, 60–65 mm), Kookaburra Club Match (4p, 60–65 mm), Kookaburra Senator (4p, 60–65 mm), Kookaburra Tuf Pitch (2p, 60–65 mm), Kookaburra Special Test (2p, 60–65 mm), Kookaburra Red King (2p, 60–65 mm)

Many Indian and Pakistani balls are inconsistently manufactured. For example, a package of cricket balls can contain balls of different construction and hardness which are externally indistinguishable (Fuss 2008).These harder and softer subtypes exist in the following models (Fuss 2008): • Regent white Match: different rubber-cork mixtures and lacquer surface finish (harder type: mass greater than 157.5 g, softer type: mass less than 154 g); • Regent red Match: different nucleus stiffness; • Gray-Nicolls Super Cavalier: two different nucleus sizes (harder ball with smaller nucleus and more cork layers); and • Sanspareils-Greenlands Tournament: different nucleus materials (harder cork nucleus, softer rubber nucleus) and different layer thickness (harder construction with thinner cork layers; the mass of harder and softer types is less than and greater than 158.5 g, respectively).

Laws of cricket The size and mass of cricket balls is regulated by the Laws of Cricket (Marylebone Cricket Club 2010). The circumference of men’s game balls is 224–229 mm, that of women’s game balls is 214–224 mm and for junior games 210–220 mm.The mass of men’s game balls is 156–163 g, that of women’s game balls is 140–150 g and for junior games 133–143 g.

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Performance criteria of cricket balls The ultimate performance criterion of a sport ball is speed. Speed depends on the coefficient of restitution (CoR) and the aerodynamic drag, both of which are the main sources of energy loss. In cricket, the speed off the bat is certainly an important performance criterion, as the higher the rebound speed, the more likely the batsman will hit a six. In bowling, however, high speed is not necessarily required to deceive the batsman; bowlers rather make use of their skills and the condition of the ball in order to produce a perfect swing. The performance criteria are: • Shape retention and restitution: when colliding with the bat, the ball is considerably deformed. Permanent deformation is unwanted and the shape of the ball should be restored as quickly as possible. The cork layers, separated by wound yarn, enable movements of the layers against each other and thus facilitate fast restoration of the shape, whereas single, moulded core balls show the tendency of permanent deformation. • Abrasion resistance: even if deterioration of the ball is inherent to the game, the ball must last at least for one innings.The leather cover can show considerable wear but must remain intact and the seams should not split open. • Energy loss and return: energy is lost from inner fiction, owing to viscosity, and from buckling of the leather cover and the cork cell edges. The less energy is lost, the higher is the CoR, (the ratio of rebound to incident velocity) and the faster the ball bounces off the bat. • Stiffness of the ball: hardness is not directly a performance parameter but harder balls produce a higher impact force, stay on the bat for a shorter time during impact, deform less and have a smaller contact area. Harder balls are thus more difficult to handle. • Surface roughness: the shape and height of the seams as well as a rough and worn-off surface layer are important for aerodynamics and the swing behaviour of the ball.

Solid mechanics of cricket balls Viscoelasticity Cricket balls are non-linear viscoelastic objects with a power law stress relaxation (Fuss 2008); that is, they exhibit a power decay of force F with time t: F –– = R t–η (1) xO where x0 is the constant deflection of the ball, R is the velocity-independent stiffness parameter, and η is the viscosity parameter. Considering that stress relaxation is modelled by applying a constant deflection x0 through a Heaviside function, the constitutive equation of a power law material or structure results from applying the Laplace transform to equation (1) and replacing the constant deflection x0 by the transformed deflection multiplied by the complex parameter s. F = sηx^ R Γ (1 – η) ^

(2)

where the caret (^) denotes the transformed variable, and Γ denotes the gamma function (Fuss 2008, 2012). Equation (2) reveals the intrinsic properties of the power law model (Fuss 2008, 2012): 364

Solid mechanics and aerodynamics of cricket balls

• the force F is the ηth derivative of the deflection x, times the constant RΓ(1–η); and • 0 ≤ η < 1, as the gamma function in equation (2) approaches infinity when η approaches 1. The velocity dependency of the stiffness of a power law material is calculated from equation (2) in the following way.The transformed deflection x^ is replaced by a ramp function x·0t, where x·0 is the constant deflection rate.Taking the inverse Laplace transform of the modified constitutive equation, and replacing the time t by t = x/x·0 yields F as a function of x at a specific deflection rate x·0. Taking the deflection derivative of F yields the stiffness k and, after exchanging the constant and variable x, expresses the stiffness k at variable deflection rate x· at a specific strain xi: kx· = Rx· η xi–η

(3)

Equation (3) can be rewritten as log kx· = log R + η [log(x· ) – log(xi )]

(4)

Equation (4) exhibits a linear structure, with log R as the intercept and η as the gradient. Instead of testing the stress relaxation of each ball at only one x0, compression testing of each ball at a specific x· allows calculating R and η at any recorded x; that is, as a function of x.The more consistently a ball is manufactured, the smaller is the number of balls required for this procedure. Moreover, the elasticity (stiffness) parameter R is independent of the deflection rate. Figure 23.2 shows R and η of different new cricket ball models, compressed perpendicularly to the seam (Fuss 2008, 2011a). The stiffness of cricket balls can be ranked in clusters of hard, medium-hard, medium, medium-soft and soft balls (Fuss 2011a; Figure 23.2a;Table 23.1).The three initially stiffest balls of the hard cluster (Regent Match white hard, Wisden Royal Corinthian, Regent Match red hard) show a distinct decrease in stiffness between 4 mm and 9 mm of deflection.The Regent Match white (hard subtype) reaches even the lowest values of all balls listed in Table 23.1 at a deflection of 12.5 mm with a subsequent increase in stiffness. A decrease in stiffness is an indicator of material softening, which, in the case of cricket balls, is caused by buckling. The structures that buckle are the cork (closed cell foam, buckled cell edges at the collapse plateau of the foam) and the spherical leather shell.The contribution of each these two factors to the decreasing stiffness is difficult to differentiate but, in the Regent Match white (hard subtype), it seems to be the leather shell, reinforced externally with a surface finish of hard lacquer and internally with a plastic insert. In the Wisden Royal Corinthian, it seems to be the cork, as the non-layered core of this ball consists of cork only, instead of the typical moulded cork–rubber core.The two aforementioned buckling mechanisms are the main sources of energy absorption in cricket balls. The energy loss due to viscous inner friction is comparatively small, as the viscosity η of cricket balls ranges only from 0.035 to 0.105 (Fuss 2011a; Figure 23.2c).

Impact mechanics Slow-speed impact The British specification for cricket balls (BSI 1994) prescribes four impact tests: CoR test; hardness test; impact resistance test and wear resistance test. The impact resistance test applies the hardness test six times to ensure that the seams do not fail. The wear resistance test is a 365

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1 Kookaburra Turf 2 SG Test 3 Kookaburra Regulation 4 Kookaburra Club Match 5 Kookaburra Senator 6 Kookaburra Tuf Pitch 7 Kookaburra Special Test 8 Kookaburra Red King 9 G&M Clubman 10 Wisden Royal Corinthian

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Figure 23.2 Velocity independent elasticity parameter R against deflection (a) and viscosity parameter η against compressive force (b) in different types of cricket balls

durability test that mimics the deterioration of the ball.The outcome of both impact and wear resistance test is judged solely by inspection.The first two tests deliver measurable variables: the CoR test (Figure 23.3; Fuss 2011a) determines the height of bounce after dropping the ball 366

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in

f i.-'i SG Test Turf Regulation Club Match Senator Tuf Pitch Red King Practice Practice

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Figure 23.3 Box and whisker plots of the coefficient of restitution (CoR) of different cricket ball models

from a height of two metres.The CoR is the square root of the ratio of bounce height to drop height. Dropping the ball from a height of two metres results in an incident speed of 6.26 m/s or 22.55 km/h, which is far below the average slow bowling speed (80–90 km/h).This procedure tests only the superficial layers of a cricket ball at a minimal energy loss and is thus not representative. The bounce height must be within 28–38 per cent of the drop height, which equals a CoR of 0.529–0.616.The hardness test (Figure 23.4; Fuss 2011a) measures the deceleration of a five-kg mass dropped from a height of 1.1 metres on a cricket ball. Only the peak deceleration, expressed in multiples of g, is decisive and represents the hardness. The peak

time (ms)

Figure 23.4 Hardness of cricket balls; acceleration against time (British Standard hardness test) and hardness of different cricket ball models

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deceleration must be within the range 150–210 g in conventionally constructed balls and 100–140 g in bonded balls. Dropping a five-kg mass from a height of 1.1 metres results in an incident speed of 4.65 m/s and a kinetic energy of 54 J.This kinetic energy corresponds to the one of a 0.16-kg ball flying at 26 m/s or 93.6 km/h (slow bowling speed). Even if the kinetic energies are the same, the speeds are different and so are the impact forces and the dynamic stiffness.When recording the deceleration of the five-kg striker mass as a time series (for example, at 50 kHz) then further parameters can be calculated in addition to determining the peak deceleration: peak dynamic stiffness, CoR, maximal deflection, peak force, energy loss and return and dynamic stiffness. Comparing the two CoR values (Figure 23.5; Fuss 2011a), determined with the bounce (CoR1) and hardness tests (CoR2), the CoR1 of the Sanspareils-Greenlands Test ball (CoR1 = 0.571–0.593, 0.583 ± 0.007) does not reflect the CoR at higher-impact energies, compared with Kookaburra balls (CoR1 = 0.602–0.624, 0.613 ± 0.005; Fuss 2011a), as the bounce test results in an impact energy of 3.14 J, compared with the 54 J of the hardness test. The hardness test returns a smaller CoR (CoR2 = 0.436–0.562, 0.505 ± 0.030) as the ball deflects more (on both sides). Kookaburra Turf exhibits the highest CoR2 (0.540 ± 0.011) and Kookaburra Tuf Pitch the lowest (0.464 ± 0.018). The relative energy loss, which equals 1– CoR2, ranges between 68.5 per cent and 81 per cent in the hardness test.The maximal impact force results from the hardness multiplied by 9.81 m/s2 and 5 kg. The maximal impact force of the impact test ranges from 4.7 to 8.8 kN in Kookaburra balls (Fuss 2011a).The dynamic stiffness (Figure 23.6; Fuss 2011a) reflects the principle of the hardness test: layered balls are stiffer than single-core bonded balls.

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Figure 23.5 Coefficient of restitution (CoR) values determined with the bounce (CoR1) and hardness tests (CoR2); CoR1 plotted against CoR2

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SG Test Turf Regulation Club Match Senator Tuf Pitch Red King Practice Practice

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Figure 23.6 Force against deflection and dynamic stiffness of different cricket ball models

High-speed impact Singh (2008) investigated the impact behaviour of cricket balls at high speeds, shot with air cannon. The two balls investigated and compared were models with harder cork (such as the Gunn & Moore Clubman or Sanspareils-Greensland [SG] Tournament/harder type) and softer rubber nuclei (for example, SG Test or SG tournament/softer type). The CoR at 27 m/s (97 km/h) of the rubber and cork core ball was about 0.5 and 0.48, respectively. The CoR drops with speed, owing to more energy absorption at higher deflections, and reaches about 0.45 at 36 m/s (130 km/h) in both balls.The impact force at 27 m/s and 36 m/s was measured at about 20 kN and 27 kN, respectively (Singh 2008).

Computer-aided engineering models of cricket balls The ability to model and simulate accurately the dynamics of high-speed impacts that occur between the cricket ball and the batsman is critical to the design and optimisation of protective equipment in cricket. Furthermore, such models support the development of high-performance cricket balls with improved structural properties, such as durability.Virtual design and analysis techniques using numerical methods have been increasingly used to enhance the understanding of impact dynamics in sports in recent years. This type of analysis is mostly based on the finite element method. A finite element model developed for this purpose typically incorporates a dummy and/or a headform, together with appropriate models of protective equipment and of the impacting ball. In most cases, the analysis focuses on the behaviour and the performance of the developed protective gear when impacted by the cricket ball (a numerical model that needs to be included in the impact analysis).Therefore, development of a universal finite element ball model that can be imported and incorporated in impact analysis is of great importance from the sports engineering and technology perspective. This enabling technology allows cost-effective simulations of different impact tests required in the development of protective equipment, such as cricket helmets, face guards. 369

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Simplified cricket ball models Cricket balls represent a complicated multi-layered structure. Different designs of cricket balls have different internal structural configurations.Therefore, developing an accurate micromodel of a cricket ball is a long and tedious process involving a range of experimental, theoretical and numerical tasks.A universal three-dimensional model, when included in numerical simulations involving different impact scenarios, should simulate the energy exchange mechanisms between the ball and the interacting object, calculate energy lost from damping or friction and should determine stress distributions and time-dependent deformations. Carré et al. (2004) investigated experimentally the impact of a cricket ball with a rigid surface, where the ball was dropped to impact the surface on the seam and perpendicular to the seam. From the experimental results obtained, they created a simple model based on a massspring-damper system. Subic et al. (2005) adopted a simplified approach to develop an approximate finite element macromodel of the ball, which emulated the impact properties of the cricket ball as a whole. The properties of the Kookaburra Tuf Pitch cricket ball were first obtained by conducting quasi-static axial compression tests on the entire ball under loading velocities varying from 1 mm/minute to 500 mm/minute using an INSTRON Universal testing machine.The experimental results were then extrapolated to predict the behaviour of the cricket ball at actual impact speeds used in standard tests (6.26 m/s) involving protective equipment. The measured change of the ball stiffness with the loading velocity was modelled by an exponential relationship, which was used in the extrapolation. A finite element model of the cricket ball was developed, based on the concept of a rigid ball encased by a soft surface.The ball properties calculated from experimental tests were used to calculate the properties of the soft surface of the ball, considering the average value of the pressure over the contact area and neglecting the friction effect.The area of contact between the ball surface and the anvils of the testing machine, which was estimated as a function of the radial displacement, was used to calculate the pressure values from the experimentally measured force values. The non-linear stiffness of the ball was calculated as a relationship between the contact pressure and the radial displacement of the outer surface of the ball (Figure 23.7).

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Figure 23.7 Contact interaction properties

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This modelling approach, while approximate by definition, reduced the complexity of the numerical analysis as the experimental tests needed to create the ball model could be limited to a simple axial compression test. Also, the total number of deformable elements in the model is reduced significantly as the rigid body of the ball does not contain any deformable elements. This type of model allows simulation of non-linear impact with reasonable accuracy, where the distribution of contact pressure calculation is based on the geometry and the stiffness of the impacted surface. To validate this approach and to refine the ball model, several compression tests between the developed ball model and a plane rigid surface were simulated, which reproduced the experimentally obtained reference response with reasonable accuracy. Several numerical models of the cricket ball, including a detailed finite element model and a fast-solving model were later developed and incorporated in further simulations (Cheng et al. 2008, 2011).These more advanced models can be effectively used to simulate dynamic impact properties and to determine compliance with relevant standards that govern face guard and/or cricket helmet design, as described later in this section.

Laboratory tests required for computer-aided engineering modelling The impact behaviour of two-, three- and five-layer cricket balls is measured experimentally. To obtain a suitable ball speed for testing, the ball can be dropped from a predetermined height or it can be fired through a cannon gun or a ball pitching machine. A typical drop-test setup shown in Figure 23.8 consists of a load cell mounted on a heavy brass rod and a dynamic signal analyser, coupled with high-speed video analysis software. Figure 23.9 shows an experimental set-up for high-speed impact tests involving cricket ball collision with a rigid surface at high speeds.The cricket ball pitching machine is used in these tests to accelerate the ball at required levels at impact speeds of up to 30 m/s. It is important in such tests to secure alignment of the shooting direction. All measuring instrumentation remains the same as in the drop test setup shown in Figure 23.8.

Dynamic signal analyser

PCB power unit

Figure 23.8 Experimental setup for cricket ball drop test Source: from Cheng et al. 2008; © 2008 Routledge, reproduced with kind permission)

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Figure 23.9 High-speed impact test installation Source: from Cheng et al. 2008; © 2008 Routledge, reproduced with kind permission)

Figure 23.10 shows the impact force results obtained for a typical Kookaburra three-layer cricket ball at impact speeds 5.5–25 m/s.The impact force wave, impact speed and the rebound speed of the ball were also recorded in a similar manner.To obtain the immediate speed readings before and after impact with a better level of accuracy, the recorded images were analysed using the image analysis software, Image-Pro Plus (Media Cybernetics, USA).

Numerical models of cricket balls Fast-solving mathematical models The ball properties obtained experimentally as described in the previous section are used to develop two numerical models: a single-element model, which consists of a single non-linear spring-damper unit; and a three-element model, which is a more complicated spring-damper system, in which three Maxwell units are connected in parallel (Figure 23.11). These models are developed in such a way as to capture the key characteristics of ball-impact behaviour, while 372

Solid mechanics and aerodynamics of cricket balls

25 m/s 20.8 m/s 13.8 m/s 10 m/s 9.09 m/s 7.14 m/s 6.25 m/s 5.5 m/s

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Figure 23.11 Fast-solving mathematical models of cricket balls; (left) single-element model (right) three-element model Source: from Cheng et al. 2008; © 2008 Routledge, reproduced with kind permission

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allowing for fast-solving dynamic simulation. The stiffness and damping properties of both models are determined using a novel, fast-solving genetic algorithm (Cheng et al. 2008). Each of the model parameters influences the final simulation result.The genetic algorithm is used to search the optimal point where the simulation result is in best fit with the experiment. The model parameters are defined through a genetic algorithm operator.The objective here is to represent a solution to the problem as a chromosome.This requires the estimation of six fundamental parameters: chromosome representation, creation of the initial population, fitness evaluation function, selection function, genetic operators and termination criteria. Detailed description of these parameters is provided by Cheng et al. (2008). While stiffness and damping parameters are assumed to be functions of the impact speed, masses are allowed to change freely while remaining as a total mass of 160 g. The fast-solving mathematical models of cricket balls presented here can be used to predict the impact force with very little computing cost.The simulation of the ball model impact with a flat surface using these models has achieved reasonable agreement with the experimental results for both the single-element and the three-element models. Figures 23.12 and 23.13 (Cheng et al. 2008) present simulation results for the two developed models. Comparison of the RMSE values indicates that, under the same conditions, the threeelement model has a better accuracy than the single-element model. The three-element model provides a fast-solving method for calculation of required model parameters. On average, it needs less than one second for solving of an individual cricket ball model using a standard desktop personal computer. It emulates full-impact speed ranging from 4 m/s to 28 m/s and can be easily extended to multi-element models based on its programming structure. In conclusion, the genetic algorithm approach presented here proved to be more efficient and convenient than the conventional approach that involves direct solving of the differential equations.

Figure 23.12 Single-element model simulation results for impact speed of 27 m/s

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Figure 23.13 Three-element model simulation results for impact speed of 27 m/s

Detailed finite element model of cricket ball A universal finite element model of a cricket ball has been developed within the Abaqus/CAE (Dassault Systèmes,Vélizy-Villacoublay, France) environment (Cheng et al. 2011). This model can be seen as a combination of a finite element model template and a material parameter selection tool based on an artificial neural network (ANN) model. The structure of the cricket ball, which typically comprises of multiple layers, and the developed detailed finite element model are shown in Figure 23.14 (Cheng et al. 2011). A central cork–rubber core, cork-and-twine packing and a stitched-leather cover usually constitute the three major layers of a cricket ball.The materials used in manufacturing cricket balls are highly

127mm

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Figure 23.14 Cross-section of a cricket ball and the corresponding finite element model

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rate dependent and, accordingly, impact-speed dependent.The time-dependent material properties of the different sections of the ball measured experimentally are included in the finite element cricket ball model.The model includes the detailed ball geometry, the verified material parameters and the surface interactions between the different components of the ball that occur during the simulation. This approach allows for rapid model development while producing accurate results at different impact speeds. Sets of real test data obtained for a five-layer cricket ball at impact velocity of 20.8 m/s were used to examine the ANN model. Comparison of results (Figure 23.15) shows good agreement between the simulation results and the experimental results. An important feature of the presented universal finite element model is its flexibility. The developed finite element–ANN model of the cricket ball can be used to predict the impact behaviour of different types of cricket balls under various dynamic conditions.This flexibility represents an advantage that can be used by sports equipment designers to rapidly develop different cricket ball models needed for inclusion in larger simulations involving the impact of a cricket ball with other relevant objects.The developed finite element–ANN model and the corresponding training process represents an invaluable tool for facilitating design, analysis and structural optimisation of sports equipment used in cricket. The computer-aided engineering cricket ball model presented in this section has been validated by simulating a cricket ball impacting a cantilever beam (Pang et al. 2011).The developed model was validated through experimental impact tests, where the ball was made to impact a cantilever beam at inbound velocities of approximately 17, 23, 38 and 44 m/s.The simulation results agree reasonably well with the experimental results. The friction effects between the cricket ball and the cantilever beam have been considered, whereby a friction coefficient of 0.2 was adopted for the model. — ~

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Figure 23.15 Comparison of impact forces for the five-layer cricket ball under an impact speed of 20.8 m/s

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Aerodynamics of cricket balls The flight path of a cricket ball in fast bowling is characterised by a unique feature that is inherent to the game: the swing.The ball curves horizontally late in its flight and serves to deceive the batsman ‘into playing one line, only to have the ball swing off that line, leaving him groping’ (Woolmer and Noakes 2008).The same strategy, with the ball curving in different planes (horizontally, vertically), is employed in other ball sports as well, such as baseball, soccer (free kick) and tennis (for example, the drop shot).The term ‘swing’ in cricket is a very generic one and actually refers to four different types of swing, which will be explained subsequently.

Aerodynamics of smooth and rough spheres Figure 23.16 shows the basic principles of fluid flow past a sphere.The coefficient of drag CD is plotted against the Reynolds number Re Ud Re = –– v

(5)

where U is the velocity (m/s), d is the diameter (metres) of the sphere, and v is the kinematic viscosity (m2/s), the ratio of dynamic fluid viscosity µ to fluid density ρ. The CD – Re graph can be divided into four distinct regimes (Achenbach 1972).At smaller velocities, the fluid flow at the boundary layer is laminar, with regular, steady and smooth flow, nearly parallel to the surface.At higher velocities, the boundary layer becomes turbulent with the mean fluid motion still roughly parallel to the surface but with rapid, random and chaotic fluctuations in velocity magnitude and direction (Mehta 2008). The transition from laminar boundary layer to turbulent occurs at smaller velocities, the bigger the sphere, the smaller the kinematic fluid viscosity (for example, at higher altitudes) and the rougher is the surface. The pressure difference between the front and back of the sphere results in a drag force that takes energy out of the flying sphere and slows it down. This pressure difference is directly related to the diameter of the wake at the back of the sphere. The larger the diameter, the smaller is the pressure at the back and thus the larger is the pressure difference, and the drag force.The diameter of the back wake is directly related to the positions of the separation points; that is, where the flow leaves the surface. The further to the rear this fluid separation occurs, the smaller is the rear wake and the drag force. The large pressure at the rear connected with late separation points seems counterintuitive, as increasing pressure causes the boundary layer to separate earlier. However, a turbulent boundary layer has higher momentum than a laminar layer; hence, it is continually replenished by turbulent mixing and is therefore better able to withstand the adverse pressure gradient building up towards the back of the sphere (Mehta 2008).These separation points are different in each flow regime and are reflected in the variation of CD with Re, as shown in Figure 23.16. The four distinct flow regimes are divided into: • subcritical regime: the CD of a smooth sphere is almost constant at 0.5 at Re of 4 ⫻ 104 – 2 ⫻ 105 (Achenbach 1972), the flow at the boundary layer is laminar and the separation point is at an angle from the front stagnation point of about 80 degrees (Figure 23.16). • critical regime: flow transition from laminar to turbulent occurs in the critical region and the CD drops to less than 0.1 at Recrit of 4 ⫻ 105 – 5 ⫻ 105 with the separation point moving backwards from 80 degrees to 95 degrees where a separation bubble is established at this 377

Franz Konstantin Fuss, Aleksandar Subic and Rabindra Mehta

Figure 23.16 Coefficient of drag CD against the Reynolds number Re; flow regimens, separation points (S) and wake diameters (↔); solid curve = smooth surface; dashed curve = rough surface

location, whereby the laminar boundary layer separates. The transition occurs in the freeshear layer and the boundary layer reattaches to the sphere surface in a turbulent state and pushes the separation point back to 120–140 degrees at Recrit, the end of the critical regime, marked by the minimum CDmin. • supercritical regime: in the supercritical region, the CD increases from CDmin to about 0.2 and the separation point moves back upstream to 100–120 degrees. • transcritical regime: in the transcritical region, for Re > 5 ⫻ 106, the CD remains relatively constant at about 0.2 (for subsonic speeds). Surface roughness affects both Recrit and CDmin: the rougher, the smaller Recrit and the larger CDmin. The reason for this phenomenon is that roughness ‘trips’ the boundary lager, which becomes turbulent at smaller Re than in circular objects with smooth surfaces. The term ‘rougher’ depends on two factors: 378

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a) the relative mean surface roughness height ε, defined by Ra ε = –– d

(6)

where d is the diameter of the sphere. Ra, the arithmetic average of the absolute height of the surface asperities, is defined as

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where zi is the height of the ith data point of the profile, n is the number of points measured, L is the length of the surface profile evaluated, and f(x) is the height of the surface profile as a function of x, the direction parallel to the profile. The larger ε, the rougher the surface. b) the skew of the surface roughness.The skewness parameter Rsk defined as n Rsk = –––––––––––––3 (n – 1)(n – 2)Rq

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Zero Rsk reflects an equal distribution of peaks and valleys; positive Rsk results from a profile with dominating valleys (that is, narrow peaks and wide valleys) and negative Rsk is characterised by dominating peaks (wide peaks and narrow valleys). The skew is also related to the density of valleys and peaks: the less frequently the peaks are in positive Rsk or the valleys are in negative Rsk, the larger is the absolute Rsk (independent of being positive or negative).Thus, if Rsk is +∞ or –∞, the surface has the least possible roughness density and is consequently perfectly smooth by definition. Haake et al. (2007) postulated that Recrit decreases and CDmin increases, with increasing Rsk, by linearly fitting data taken from the literature. Fuss (2011b) confirmed this hypothesis with wind tunnel experiments but rejected a linear relationship. The supercritical/postcritical CD increases from a smooth profile (Rsk = ∞) with increasing Rsk from – ∞ to + ∞ until it reaches the subcritical CD of a smooth surface, which, close to Rsk = + ∞ reverts back to laminar flow. This relationship explains why very few ‘trip wires’ on the surface (high positive skew, less dense roughness elements) are ‘rougher’; that is, they decrease Recrit and increase CDmin more, than more frequent and denser elements (with small positive skew) of the same relative roughness height ε.This fact is important for cricket balls, as the roughness elements of new or less-deteriorated cricket balls at the beginning of an innings are less dense and are concentrated at the narrow seam zone, consisting of six rows of stiches and the prominent seam in their middle (small positive Rsk, large Ra). With the ball deteriorating, abrasion adds to the previously smooth hemispheres with a roughness characterised by smaller Ra and larger positive Rsk. The flow regimes of a sphere with a rough surface are basically the same as in a sphere with a smooth surface. 379

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• Subcritical regime: the CD of a rough sphere is 0.5 or slightly higher at small Re. • Critical regime: the critical regime starts and ends at smaller Re the larger ε and Rsk are. CDmin at Recrit correlates positively with ε and Rsk. • supercritical and transcritical regimes (postcritical regime): the CD of these regimes is larger compared with that of a smooth surface. At more pronounced roughness, the transition between the rising CD of supercritical regime and constant CD of the transcritical regime disappears, such that the two regimes are conveniently combined as a ‘postcritical’ regime.

Basic cricket ball aerodynamics In a new cricket ball, with the seam held at an angle from the front stagnation point of 20 degrees, the critical regime is located at Re from 105 to 2 ⫻ 105 and CDmin is 0.1 (Mehta and Pallis 2001; Figure 23.16). By instrumenting a cricket ball with multiple pressure taps and the seam held at 20 degrees from the front stagnation point, Mehta (2005) measured the pressure distribution along the circumference of the ball and the locations of the separation points. At small Re, the separation point was located at 60 degrees on the seam side; at Re = 1.25 ⫻ 105 (25 m/s, 90 km/h) at 120 degrees; and at Re = 1.45 ⫻ 105 – 1.83 ⫻ 105 (29.0–36.5 m/s, 104.4–131.4 km/h) at 130–140 degrees, corresponding to the end of the critical regime with Recrit at about 12 ⫻ 105. Cricket balls curve in two directions: • In the sagittal plane, the ball curves upwards, owing to the Magnus effect. A backspin is imparted to the ball by the bowler, to stabilise the location of the seam during flight (gyro effect).This avoids wobbling of the seam and guarantees consistent aerodynamics.The spin rates of national spin bowlers were recorded at 25–35 revolutions per second in finger spinners and 34–42 revolutions per second in wrist spinners (Spratford and Davidson 2010). As the top surface of a backspun ball advances with respect to the free-stream velocity of the air flowing past the ball, the relative free-stream velocity of the top surface is larger than the one of the bottom surface, which in turn affects the pressure difference between both surfaces.The fluid pressure is smaller on the top surface and the sideward force vector points upwards (Magnus effect). • In the transverse plane, parallel to the ground, the ball curves sideways (swings).The swing is caused by pressure differences between the right and left sides of the ball, which are simply due to different flow regimes on either side of the ball. The rule of the thumb is, the longer the boundary layer stays attached to the surface, the faster the free-stream velocity and the smaller the pressure.Thus, the ball swings towards that half of the ball which has the smaller CD.The CD is usually measured for a whole sphere and not for a hemisphere but each hemisphere can be mirrored to obtain a ball of symmetrical surface properties. Consequently, different CD will occur at the same Re and, in fact, in a ball with asymmetrical surface properties (roughness) such as a cricket ball, each half of the ball may be at a different flow regime at any point of time.This principle is the key to swing mechanics. If the separation points on the left and right sides of the ball are at 120 and 90 degrees from the front stagnation point, respectively, then the wake is deflected rightwards and the ball curves leftwards. In this scenario, the free-stream velocity of the ball is faster, and thus the pressure lower, on the left side (separation point at 120 degrees), producing a pressure differential and consequently a sideward force with a leftward directed vector.

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Aerodynamics of the four different swings of a cricket ball Based on the direction of the swing and the orientation of the seam, four different swings can be distinguished. If the ball is not deteriorated too much and the leather hemispheres are still relatively smooth, the ball is bowled with the seam at an angle to the direction of the pitch.The ‘conventional swing’ occurs at slower velocities and the ball swings in the direction the seam is pointing (the rough side).The ‘reverse swing’ occurs at higher velocities, and the ball swings away from the seam (towards the smooth non-seam side). At higher degrees of surface deterioration, at later stages of an innings and with one side of the ball polished and the other side kept rough (or even illegally made rougher), the seam is pointing towards the batsman.The ‘standard contrast swing’ occurs at slower velocities and the ball swings towards the rough side.The ‘reverse contrast swing’ occurs at higher velocities and the ball swings towards the smooth side.

Conventional swing In the conventional swing, with the seam held at an angle to the initial line of flight, the ball swings towards the seam at relatively slow velocities.This is due to the fact that the seam trips the boundary layer and thus the flow regimes on either side of the ball are different: • subcritical on the smooth non-seam side and critical on the seam side; or • subcritical on the smooth non-seam side and supercritical on the seam side; or • critical on the smooth non-seam side and supercritical on the seam side. This fact results in different locations of the separation points (further backward boundary layer separation on the seam side), different free-stream fluid velocities (faster on the seam side) and thus a pressure differential producing a sideward force towards the seam side. This effect becomes evident, when comparing the CD–Re curves of smooth and rough spheres, shown in Figure 23.17.The CD on either side of the ball is not the decisive factor, as the CD is usually for a whole sphere and not separately for each hemisphere, but the CD rather represents the flow regimes and thus the location of the separation points and the magnitude of the pressure on each side of the ball, as well as the pressure gradient reflected in distance between the two curves. From the distance between the curves, it becomes evident that there is no swing if the flow past both hemispheres is subcritical.With increasing velocity, the seam side enters the critical regime first and produces a pressure gradient.The two CD–Re curves intersect if the flow on the non-seam side is close to the end of the critical regime (Recrit) and the flow on the seam side close to the end of the supercritical regime. It is evident that the pressure gradient at the intersection point is zero and thus no swing occurs. The velocity range for the conventional swing is defined from 13 m/s (46.8 km/h) to 35.5 m/s (127.8 km/h) in a new ball with quarter seams (Mehta 2005). In an older and thus rougher ball with quarter seams, the velocity range ends already at 32 m/s (115.2 km/h) whereas in a new two-piece ball, the velocity range would end beyond the maximal bowling speed ever measured (160 km/h). The maximum swing can be expected at a certain ball speed, where the two CD–Re curves are furthest apart. In fact, maximal sideward force was measured (Imbrosciano 1981) and computed (Bentley et al. 1982) at about 30 m/s (108 km/h), at a seam angle of 20 degrees and a spin rate of 14 revolutions per second. Mehta (2005) measured the maximal sideward force of a new ball without quarter seams at 34 m/s (122.4 km/h), with quarter seams (rougher) at 29 m/s (104.4 km/h) and of an old, worn-off ball with quarter seams (roughest) at 24 m/s (86.4 km/h). 381

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Figure 23.17 Explanation of flow regimens generating different types of swing; note that, in the CD–Re diagram, the CD is not of importance, as it simply reflects the positions of the separation points, the flow velocity on either side of the ball, the local pressure, the pressure difference and thus the direction of the side force (the smaller the CD, the further to the rear the separations points are located, the higher the flow velocity and the lower the local pressure); the further the CD curves are apart, the higher is the pressure difference and thus the side force; note that curve B is not necessarily the same for conventional and contrast swing

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Reverse swing The reverse swing emerged in the mid-1980s as a new bowling concept, where the ball swung in the opposite direction (outswinger); that is, towards the non-seam side. This new form of swing bowling was first demonstrated with astonishing success by the Pakistani bowlers, in particular Imran Khan and Sarfraz Nawaz in the early years, followed by Wasim Akram and Waqar Younis (Mehta 2008). Considered a mystery first, the reverse swing can be easily explained from Figure 23.17. The conventional swing occurs before the intersection point of CD–Re curves and the reverse swing afterwards, at higher velocities. The sideward force of different balls measured by Mehta (2005) predicts that a new ball with quarter seams starts to reverse-swing at 35.5 m/s (127.8 km/h) and an old, worn-off ball with quarter seams at 32 m/s (115.2 km/h). New two-piece balls, which are generally not used in competitive cricket matches, are expected to reverse swing at about 50 m/s, which is beyond maximal bowling speed. The rougher the ball (preferably on the seam side), the smaller is the bowling speed required to produce the reverse swing.This results in the fact, that bowlers, able to produce a conventional swing (outswingers) at the beginning of an innings, can bowl a reverse swing (inswingers) with a more deteriorated ball at later stages of an innings at the same conditions; that is, without any change in grip or bowling action (Mehta 2008). Again, the flow regimes are different on either side of the ball: supercritical on the smooth non-seam side, and transcritical on the seam side.

Standard contrast swing The contrast swing is generated by the difference in roughness between the two hemispheres. The roughness of the seam does not count, as the seam is held at the front stagnation point. At later stages of an innings, the leather surface of the ball shows considerable abrasion including the primary seam, which can be attenuated by the bowler on one side and enhanced on the other. Bowlers polish one side of the ball on their trousers, sometimes with the help of polishing agents allowed by the Laws of Cricket (Marylebone Cricket Club 2011); that is, natural substances such as sweat or saliva. Illegal use of use of petroleum jelly, BrylcreemTM or sunscreen lotion was often at the centre of a ball tampering controversy, especially in the 1970s (Mehta 2008). Enhancing the roughness on the other side is illegal, for example by rubbing the ball on the ground or tampering the quarter seam with fingernails or bottle tops. The contrast in surface roughness causes the boundary layer flow to be laminar on the smoother side and turbulent on the rougher side. Thus, the ball swings towards the rough side. Similar to the conventional swing, the flow regimes on either side of the ball are different: • subcritical on the smooth non-seam side and critical on the seam side; or • subcritical on the smooth non-seam side and supercritical on the seam side; or • critical on the smooth non-seam side and supercritical on the seam side. Conventional swing and contrast swing are bowled at roughly the same ball speed, with maximal sideward force at about 110 km/h.

Reverse contrast swing The reverse contrast swing is bowled at the same conditions as the standard contrast swing, with the exception that the ball speed is higher; that is, speeds after the intersection point of curves 383

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A and B in Figure 23.17. Owing to the difference in relative roughness and skewedness between conventional/reverse swing and standard/reverse contrast swing, the intersection points for the transition from conventional to reverse swing and from standard contrast to reverse contrast swing are not identical. The intersection point marking the transition from standard contrast to reverse contrast swing occurs earlier than the one from conventional to reverse swing. Thus, the reverse contrast swing is bowled at slower speeds than the reverse swing.The flow regimes are different on either side of the ball: supercritical on the smooth side, and transcritical on the rough side.Thus, the ball swings towards the smooth side.

Effects of surface finishing and weather conditions Allegedly, white balls swing more than red ones. The clue to this common belief lies in the difference of coating. For ‘cherries’, the leather is dyed red, greased and polished with a shellac topcoat. The final polish is worn off quickly during an innings and the grease in the leather produces the shine when the ball is polished by the bowler (Mehta 2008). For white balls, the leather is bleached and sprayed with a polyurethane white paint-like fluid and then heat treated so that it bonds to the leather like a hard skin. For the final treatment, a coat of clear polyurethane-based topcoat is added to further protect the white surface (Mehta 2008). In white balls, the surface stays smooth for a longer time; therefore, white balls swing more at higher bowling speeds compared with red balls, in which the protective surface layer is readily worn off. The reverse swing will occur at higher bowling speeds with a new white ball and later in the innings at more reasonable bowling speeds (Mehta 2008). Cricket balls are said to swing more on humid days. Humid air is less dense than dry air but this difference is too insignificant to have an effect on the swing. Mehta (2005) measured the side force on a cricket ball in wet, humid and dry conditions and found no significant difference. The only indirect influence of a rainy day is that the ball will slip more on the soft ground and wet grass which delays surface wear. Consequently, the laminar boundary layer is maintained for a longer time on the non-seam side.

Summary Over the last 25 years, cricket ball aerodynamics have been investigated extensively and the mechanisms of different types of swing are now fully understood. Solid mechanics of cricket balls is lagging behind and optimal tests for standardised assessment of cricket ball performance are still wanting. Low energy tests such as slow speed or little deflection do not represent the actual behaviour of the ball when struck by the bat.

References Achenbach, E. (1972) ‘Experiments on the flow past spheres at very high Reynolds number’, Journal of Fluid Mechanics, 54: 565–75. Bentley, K.,Varty, P., Proudlove, M. and Mehta, R. D. (1982) An Experimental Study of Cricket Ball Swing.Aero Technical Note 82-106. London: Imperial College. BSI (1994) Specification for Cricket Balls BS 5993:1994. London: British Standards Institution. Carré, M. J., James, D. M. and Haake, S. J. (2004) ‘Impact of a non–homogeneous sphere on a rigid surface’, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 218(3): 273–81. Cheng, N., Subic, A. and Takla, M. (2008) ‘Development of a fast-solving numerical model for the structural analysis of cricket balls’, Sports Technology, 1(2–3): 132–44.

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Solid mechanics and aerodynamics of cricket balls Cheng, N.,Takla, M. and Subic,A. (2011) ‘Development of an FE model of a cricket ball’, Procedia Engineering, 13: 238–45. Fuss, F. K. (2008) ‘Cricket balls: construction, non-linear visco-elastic properties, quality control and implications for the game’, Sports Technology, 1(1): 41–55. Fuss, F. K. (2011a) ‘Mechanics of cricket balls’, in Proceedings of the 4th World Congress of Science and Medicine in Cricket, Chandigarh, 30 March – 1 April. Chandigarh: Postgraduate Institute of Medical Education and Research. Fuss, F. K. (2011b) ‘The effect of surface skewness on the super/postcritical coefficient of drag of roughened cylinders’, Procedia Engineering, 13: 284–9. Fuss, F. K. (2012) Nonlinear visco-elastic materials: stress relaxation and strain rate dependency. In: Dai L, Jazar RN (eds), Nonlinear Approaches in Engineering Applications. New York, Springer, pp. 135–170. Haake, S. J., Goodwill, S.R. and Carré, M. J. (2007) ‘A new measure of roughness for defining the aerodynamic performance of sports balls’, Journal of Mechanical Engineering Science, Part C, 221: 789–806. Imbrosciano, A. (1981) The Swing of a Cricket Ball. Project Report. Newcastle, Australia: Newcastle College of Advanced Education. Marylebone Cricket Club (2010) Laws of Cricket, 4th ed. Available online at http://www.lords.org/ laws–and–spirit/laws–of–cricket/laws/ (accessed 7 January 2013). Mehta, R. D. (2005) ‘An overview of cricket ball swing’, Sports Engineering, 8(4): 181–92. Mehta, R. D. (2008) ‘Sports ball aerodynamics’, in H. Nørstrud (ed.) Sports Aerodynamics. NewYork: Springer, pp. 229–331. Mehta, R. D. and Pallis, J. M. (2001) ‘Sports ball aerodynamics: effects of velocity, spin and surface roughness’, in F. H. Froes and S. J. Haake (eds) Materials and Science in Sports. Warrendale: Minerals, Metals and Materials Society, pp. 185–97. Pang,T.Y., Subic,A. and Takla, M. (2011) ‘Finite element analysis of impact between cricket ball and cantilever beam’, Procedia Engineering, 13: 258–64. Singh, H. (2008) Experimental and Computer Modelling to Characterize the Performance of Cricket Balls. Master’s of Science Thesis,Washington State University. Spratford, W. and Davison, J. (2010) ‘Measurement of ball flight characteristics in finger-spin bowling’, in Proceedings of the Conference of Science, Medicine and Coaching in Cricket, Gold Coast, 1–3 June. Melbourne: Cricket Australia, pp. 140–3. Subic, A.,Takla, M. and Kovacs, J. (2005) ‘Modelling and analysis of alternative face guard designs for cricket using finite element modelling’, Sports Engineering, 8: 209–22. Woolmer, B. and Noakes,T. (2008) Bob Woolmer’s Art and Science of Cricket. London: New Holland Publishers.

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24 MECHANICAL AND AERODYNAMIC BEHAVIOUR OF BASEBALLS AND SOFTBALLS Lloyd Smith and Jeff Kensrud

Introduction The behaviour of the ball is an important part of understanding, playing and watching the games of baseball and softball.The first baseballs were small medicine balls constructed from a rubber pill, wrapped in string and covered by horsehide.The ball has evolved a number of times over the years in size and material, including kapok, rubber and cork. In the 19th century, the game had few restrictions and the baseball itself varied among ball clubs. By the end of the 19th century, the ball was confined to between 5 and 5.5 ounces (142 and 156 g) and 9.25 inches (235 mm) in circumference. This baseball, with its rubber pill, absorbed much of the energy, resulting in dull, short hits. In 1920, a cork pill was implemented to increase the liveliness of the ball, which tripled the number of 0.300 hitters (Johnson 2010). In 1931, a thin layer of rubber was wrapped around the cork pill, to slightly reduce the liveliness of the ball. Except for a change from a horsehide to a cowhide cover in 1974, baseball construction has remained largely unchanged. The softball has evolved in similar fashion to the baseball. Early softballs were made from boxing gloves, which gave way to wound string and cork designs.Today, softballs are nominally 7 ounces (200 g), 12 inches (305 mm) in circumference and made from a polyurethane core with a thin leather or synthetic cover.The formulation of the polyurethane is relatively advanced where ball stiffness and elasticity are controlled independently. In the case of baseballs, the design of the pill and tension of the wool windings effect elasticity and stiffness, respectively. Ball elasticity and stiffness interact with the performance of the bat in a complex way.Thus, their accurate control is essential toward reliable performance regulation. This chapter summarises standard test methods that are used to characterise baseballs and softballs and reviews their behaviour to drag, temperature, humidity and impact.

Ball coefficient of restitution Ball elasticity is measured from its coefficient of restitution (CoR or e), which is a measure of the energy that is lost during impact. Balls with higher e will be more lively in play.While e is close to 0.5 for baseballs and softballs, ideally it can range from one (no energy lost) to zero (all energy lost). 386

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The ball CoR is measured from impacts with a rigid surface. It is defined as the ratio of the rebound speed vr to the incoming speed vi as vr e = –– vi

(1)

A standard test method has been developed to measure the e of softballs and baseballs (ASTM 2009a). In this method, a ball is pitched at 60 mph toward a steel plate mounted to a massive, rigid wall. A softball e of 0.44 is perhaps the most common but can be as high as 0.52 and as low as 0.40. Until recently, slow-pitch softball tended to use low-e balls, while fast-pitch softball tended to use high-e balls. As will be discussed in the next section, there is now a general trend in softball toward high-e balls.The e of baseballs is generally higher than softballs and is around 0.55. To describe the energy that is dissipated, consider a ball impacting a massive and rigid wall as described above.The ball e is related to the fraction of dissipated energy, de, according to de = 1 – e2

(2)

This means that a ball with e = 0.50 loses 75 per cent of its initial energy upon impact with a rigid wall. Tests show that the e is a relatively uniform ball property when measured in controlled laboratory conditions. Field conditions can have a significant effect on e, however. In general, conditions that increase ball deformation will reduce e.The ball e, for instance, tends to decrease linearly with increasing speed. Balls impacting a cylindrical surface (such as a bat) will also have lower e than balls impacting a flat wall (approximately 0.02 lower), since the ball will deform more impacting an uneven surface.The ball e should, therefore, be viewed more as a measure to compare the relative liveliness of balls and not as an absolute measure of energy dissipated in a bat–ball collision.

Ball compression Ball compression is a quasi-static measure of the ball stiffness. Ball compression is a significant factor in the performance of hollow bats but is relatively unimportant in the performance of solid wood bats. Ball compression is measured from the force to compress a ball 0.25 inches (6.3 mm) in 15 seconds between flat platens. A standardised method has been developed to measure this property (ASTM 2009b). It should not be confused with the compression of golf balls, however, which concerns the displacement to compress a ball to a specified force. Major League baseballs have a compression of just over 200 lbs (0.9 kN), while college baseballs are stiffer, having a compression approaching 300 lbs (1.3 kN). Softball compression increased steadily with the advent of hollow bats, exceeding 700 lbs (3 kN) in the 1990s. To help to control the hit-ball speed, the compression has been lowered and now ranges between 300 and 500 lbs (1.3 and 2 kN). The interest in ball compression is related to the so-called ‘trampoline effect’ observed in thin-walled hollow bats (Nathan et al. 2004). The barrel of a hollow bat will deform during impact with a ball on the order of 0.12 inches (3 mm).The barrel deformation is nearly elastic, returning most of its energy to the rebound speed of the ball. Recall that ball e is measured from an impact with a rigid surface and that energy loss increases with ball deformation. Now consider two balls of the same e but made in a way that they have different stiffness (or 387

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compression). Next, allow these two balls to impact the same hollow bat. Upon impact, the stiffer ball will produce more barrel deformation and consequently less ball deformation than the softer ball. Essentially, the stiffer ball exchanges some of its ball deformation (which dissipates energy) with barrel deformation (which does not dissipate energy). It is for this reason that the hit-ball speed generally increases with increased ball stiffness and decreased barrel stiffness. Note that the trampoline effect is negligible for bats with very stiff barrels.These include hollow bats with thick walls or solid wood bats. Because of its simple nature, ball compression is used to compare the stiffness of many types of balls. Unfortunately, the ball compression test is sensitive to the ball diameter, which leads to some surprising results. As noted above, the compression of baseballs is less than the compression of softballs. Some conclude that since the compression of a softball is larger than that of a baseball, softballs are not ‘soft’. Even some experts with considerable experience do not realise that it is inappropriate to compare the compression of balls of different size. To illustrate the effect of diameter on ball compression, compare an ideal 9-inch (230-mm) circumference baseball to an ideal 12-inch (305-mm) circumference softball made from the same material. Now consider their contact area with the loading platen after 0.25 inches (6.3 mm) of displacement. The contact areas would be approximately 1.07 in2 and 1.45 in2 (690 mm2 and 934 mm2) for the baseball and softball, respectively (the larger ball produces a 35 per cent larger contact area). The spherical shape of the ball means that most of the deformation will be concentrated near the platen (the increasing ball diameter away from the platen results in less deformation toward the ball centre).This means that the pressure over the contact areas for the two balls would be similar.The ball compression is the product of the pressure and the contact area. For this idealised case, one would expect the compression of the softball to be 35 per cent higher than the baseball, even though they are made of the same material. For the softball to be stiffer than the baseball on this quasi-static scale, therefore, its compression would need to be more than 35 per cent higher than the baseball. While compression of some softballs is more than 35 per cent higher than baseballs, the next section describes an improved ball stiffness test that shows that softballs truly are softer than baseballs. The polyurethane core of softballs allows great latitude in controlling the ball’s liveliness and stiffness. Some have advocated the use of low compression softballs to reduce the trampoline effect with bats and the severity of injury. In fact, low compression balls (200 lbs or 0.9 kN) are commercially available but are not widely used.While low compression balls lower the hit-ball speed, they are also more difficult to field. Apparently, the low compression results in large ball deformation that does not recover during the ball flight.The deformed balls are difficult to field in the air and off the ground.While they may reduce the injury severity, they can also increase the injury frequency. Recently, low compression balls have again been under consideration. These balls have a slightly higher compression than the early low compression balls (300 lbs or 1.3 kN) but are still softer than standard softballs (400 lbs or 1.8 kN). They are able to reduce the trampoline effect with the bat and sufficiently retain their spherical shape so they can be fielded comparable to traditional softballs.These modern low compression balls have another novel feature in that they are produced with a high e (0.52). Thus, low performing bats (that are not sensitive to ball compression) are more lively with these balls than the traditional ball. On the other hand, high performing bats (that are less sensitive to ball e) are less lively with this ball than the traditional ball. Thus, this relatively new high-e, low-compression ball reduces the difference between high- and low-performing bats.

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Dynamic stiffness Dynamic stiffness is a measure of ball stiffness, similar to compression. It is measured against a cylindrical surface at a high rate of speed, to simulate a bat impact. Test results show that it correlates better with bat performance than compression. In the compression test discussed above, one may observe that the 0.25 inches (6.3 mm) of displacement is applied diametrically and produces only 0.12 inches (3.2 mm) of radial displacement. The ball can displace radially nearly one inch (25 mm) in a bat–ball collision, however. The contact duration in the compression test (15 seconds) is also much longer than occurs in a bat–ball collision (0.001 seconds). These differences in contact duration and displacement between the compression test and a bat–ball impact result in a difference in deformation rate of five orders of magnitude. This difference in the rate and magnitude of ball deformation limits the ability of ball compression to describe ball response in a bat–ball impact. A test has been developed to better approximate the deformation rate and magnitude of a ball in a bat impact (Duris and Smith 2004).The test configuration is similar to that described above to measure ball e.An essential difference is that the impact surface is cylindrical (intended to approximate the bat) and load cells (between the cylinder and rigid wall) are used to measure the impact force. Ball e can be obtained from this test but it is lower than the flatplate ball e. It is designated as ec to distinguish it from the flat plate measure of ball e. The value of ec is lower than e, owing to the higher impact speed (as explained below) and the cylindrical impact surface. Upon impact of the ball with the rigid cylinder, the load cells record the impact force as a function of time. Knowing the incident ball speed vi and weight m, the stiffness of the ball k may be found from the peak impact force F as 1 F 2 – – . k=m vi

( )

(3)

The expression is obtained by equating the known kinetic energy of the incoming ball (½mvi2) with its stored potential energy at maximum displacement (½kx2).The unknown ball displacement x is obtained from the peak impact force assuming the ball behaves as a linear spring (F = kx).This leaves only one unknown, k, representing the ball stiffness. A potential shortcoming of this approach is the assumption that the ball deforms as a linear spring. The accuracy of this assumption may be tested by examining a plot of the ball forcedisplacement response during impact. Unfortunately, the nature of the ball impact is unsuitable for a direct measure of ball displacement. An indirect measure of the ball centre of mass displacement can be obtained by integrating its acceleration twice.The acceleration, in turn, is found by dividing the impact force by the ball mass.A representative example of the ball forcedisplacement curve is presented in Figure 24.1, which shows a relatively linear response during the loading phase. (Hysteresis and non-linearity during the unloading phase determine the energy loss and ec but do not affect the linear spring assumption associated with the loading phase.) The dynamic stiffness test was developed to describe both the deformation rate and magnitude of a bat–ball impact. A relatively wide range of impact speeds would approximate the deformation rate reasonably well.To capture the deformation magnitude, however, the impact speed must be chosen more carefully. One might ask: at what speed should a rigid cylinder be impacted to simulate an impact with a recoiling bat? One plausible answer would be the speed where the impulse is the same for the two cases. Impulse is the area under the force–time curve and is the momentum change at impact producing the rebound ball speed. Another plausible 389

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Displacement (mm) 0

0.0

10

0.2

0.4

20

0.6

0.8

Displacement (in) Figure 24.1 Representative force-displacement curve from a softball impacting a rigid cylinder at 36 m/s (80 mph); FEA = finite element analysis, Exp = experiment Source: from Smith and Duris 2009; © 2009 Taylor and Francis, reproduced with kind permission)

answer would be the speed where the centre of mass energy is the same for the two cases. A relation between the recoiling vr and fixed vf cases can be found as vr n – vf = (1 + c)

(4)

where c = mQ2/I and the exponent n is unity for constant impulse and ½ for constant centre of mass energy.The term c is referred to as the effective mass, where Q is the distance from the bat pivot point to the impact location and I is the mass moment of inertia of the bat about the pivot point. Experiments have shown that the constant centre of mass case (n = ½) produces the same ec and F between fixed and recoiling cylinders (Smith et al. 2009). In general, vf is about 85 per cent of vr for most cases involving softball and baseball bat impacts. To compare the compression and dynamic stiffness test, a static stiffness ks can be obtained by dividing the ball compression by its centre of mass displacement, 0.12 inches (3.2 mm). As expected, kd > ks, where for softballs kd/ks about 2.3 and for baseballs kd/ks about 7.5.The difference in this comparison between baseball and softball stiffness measures is largely due to the sensitivity of ball compression to ball diameter.The lower deformation rate and magnitude also limit the ability of compression to describe the effect of ball stiffness on bat performance.Tests that were conducted to compare the effect of ball stiffness on bat performance confirmed the insensitivity of solid barrel wood bats to ball stiffness (Smith et al. 2009). For hollow bats, dynamic stiffness showed a strong correlation, where the bat performance increased with ball stiffness.The correlation between bat performance and ball compression was less apparent.This illustrates that care should be exercised when comparing the compression of different balls. 390

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Hygrothermal response Most manufacturers of hollow bats have a temperature limit, below which the bat warranty is not valid. While many believe the temperature limit concerns the bat design, it is actually related to the ball. At low temperature, balls become stiffer. This increases the impact force, which, in turn, can cause premature failure of the bat.The opposite is also true.At high temperature the ball becomes softer, which lessens the trampoline effect and the hit ball speed. Humidity can also affect the response of the ball; in fact, baseballs are now stored in humidors in Major League parks with dry climates, to keep the offensive statistics on par with other parks. In Major League Baseball, the concern over ball humidity is not related to its stiffness (wooden bats do not have a trampoline effect) but the CoR.At low humidity, baseballs become more elastic, while at high humidity the opposite occurs. The combined effect of temperature and humidity is referred to as ‘hygrothermal response’. While it can have a large effect on the performance of the ball, it is hard to measure and rarely reported. The works of Meyer and Bohn (2008), Kagan and Atkinson (2004) and Drane and Sherwood (2004), for instance, represent nearly all that is known concerning the effect of temperature and humidity on baseballs.The following presents some recent results of temperature and humidity on baseballs and softballs that support and expand on this previous work. The humidity was controlled by suspending the balls over saturated salt solutions in sealed containers.The balls were placed in separate conditioning chambers at a relative humidity ranging from 11 to 97 per cent for six weeks.While balls reach their steady state temperature in a matter of minutes, humidity requires many weeks to reach steady state. In this work, the time to saturation varied from two weeks to over a month for the extreme humidity environments. The weight change in the balls was proportional to the humidity level. Comparing the 11 and 97 per cent relative humidity environments, the softballs gained ½ ounce (14 g), while the baseballs gained one ounce (28 g). The softballs were made of a polyurethane core, while the baseballs were made from wound wool yarn. These differences in construction contribute to their respective moisture tolerances. The ec is presented as a function of the relative humidity and temperature in Figure 24.2. The ec is approximately 20 per cent higher for baseballs than softballs, which is comparable to their difference found from the flatplate e.The ec increased with increasing temperature, while the ec decreased with increasing relative humidity. As was observed with weight gain, the ec of the wool wound baseballs was more sensitive to temperature and relative humidity than the polyurethane softballs.The elasticity of both wool and polyurethane apparently increased with temperature. Moisture tends to act as a plasticiser in many materials, so it is not surprising to see the ec decrease with increasing relative humidity. Ball stiffness is shown as a function of relative humidity and temperature in Figure 24.3, where the baseballs were approximately twice as stiff than the softballs; confirming that softballs are indeed soft.The stiffness of the baseballs was nearly immune to temperature, while the softball stiffness was reduced by 50 per cent going from 40 degrees F to 100 degrees F (4–38 degrees C).The stiffness of the baseballs and softballs showed a similar decrease with increasing relative humidity. Comparing the relative significance of the hygrothermal effects on ec and stiffness, baseballs are most sensitive to changes in ec from humidity, while softballs are most sensitive to changes in stiffness from temperature. The ec and k of a ball can be used to estimate the resulting bat–ball performance (Nathan and Smith 2009).Thus, for a baseball whose ec changes from 0.56 to 0.46 going from 11 to 97 per cent relative humidity, the distance of a fly ball will decrease by 11 per cent. Similarly, for a softball whose stiffness changes from 8,700 to 4,100 lb/in (1,527 to 720 kN/m) going from 391

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Figure 24.2 A comparison of the cylindrical coefficient of restitution ec of baseballs and softballs as a function of temperature (a) and relative humidity (b)

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Figure 24.3 A comparison of the dynamic stiffness of baseball and softballs as a function of temperature (a) and relative humidity (b)

40 to 100 degrees F (4–38 degrees C), the distance of a fly ball decreases by six per cent.While these changes in humidity and temperature are at the extreme of what would be encountered in play, they illustrate that even relatively small changes in temperature and humidity can have a measureable effect on the game.

Ball aerodynamics The study of aerodynamics of sports balls dates back to before the game of baseball was created. In the 17th century, after noticing a tennis ball curve, Isaac Newton explored the path of a ball’s flight through the air and correctly described the cause. This interest continued with others 392

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through the years. As technology improved, experiments were conducted with greater accuracy. Today, the aerodynamics of spheres continues to be a topic of interest. Some might be surprised to learn, however, that much is still not known concerning the aerodynamics of baseballs and softballs. The geometry of the stitch pattern used in softballs and baseballs plays an important role and significantly complicates their aerodynamic response. The next section reviews the aerodynamic properties of baseballs and softballs. Integral to this discussion is an understanding of drag, which acts parallel to the ball path, slowing the ball down (Adair 2002) and lift, which acts normal to the ball path, changing the ball’s direction and gravity. These concepts are reviewed and discussed in terms of the behaviour of baseballs and softballs. Drag and lift are commonly discussed in terms of a non-dimensional speed, known as the Reynolds number, Re, which is defined by VD Re = –– υ

(5)

where D is diameter of the ball, V is velocity of the ball, the and υ is kinematic viscosity of air. Comparing aerodynamic behaviour with Re allows objects of different size to be compared on the same scale. This is convenient for baseballs and softballs, which share a similar surface and stitch pattern but whose diameter can differ by almost a factor of two. Part of the difficulty in understanding aerodynamic effects is measuring drag or lift without interrupting the air flow. Wind tunnels represent a convenient tool but require some type of attachment to the ball to suspend it in the air flow. Stationary balls can be supported on their downstream surface to minimise the effect of the attachments. Lift requires ball rotation, however, for which air flow interrupting side attachments are needed. Despite these experimental challenges, wind tunnel measurements provide remarkably good measures of lift and drag on a variety of objects, including sports balls. In some cases, wind tunnels can be used to study lift while avoiding the effect of ball constraint. If the wind tunnel is oriented vertically, for instance, a spinning ball can be dropped in the air flow. By comparing the release and landing position, the effect of lift can be quantified. In situ methods of studying lift and drag are becoming common with modern video and radar techniques.These devices can track the motion of a ball in three dimensions, from which lift, drag, and even spin rate can be determined. The technologies are sufficiently mature that they are commonly used in broadcasts of golf, cricket, tennis and baseball. Drag is influenced by the size of the wake created by the ball. Consider a ball without rotation traveling through air at a low speed. Air approaching the centre of the ball will stop, forming an upstream stagnation point. At a very low Reynolds number (Re less than four) the air flow around the ball is completely laminar where the flow divides and reconnects on the back side of the ball. The drag is highest in this Reynolds number region. As the Reynolds number is increased, drag decreases before plateauing over the range 103 < Re < 3 ⫻ 105. At these speeds (which are representative of play), the laminar boundary layer separates from the ball (at approximately 80 degrees from the upstream stagnation point) creating eddies and a relatively large wake. As the Reynolds number increases further to 3 ⫻ 105 < Re < 3 ⫻ 106 a so-called ‘drag crisis’ occurs. In the drag crisis region, the boundary layer after the separation point, quickly becomes turbulent, reconnects to the ball surface and separates on the back side of the ball at approximately 120 degrees from the stagnation point (Panton 2005).This reduces the size of the ball’s wake. As a result, drag decreases dramatically, sometimes by as much as 70 per cent. At higher speeds (Re > 3 ⫻ 106), the flow becomes turbulent just after the upstream stagnation point and stays turbulent until finally separating just before 120 degrees. The wake in this region is smaller than in the low-speed case (103 < Re < 3 ⫻ 105) but larger than in 393

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the drag crisis region. Thus, at high speeds (Re > 3 ⫻ 106) ball drag increases relative to the drag crisis region but remains less than the low-speed region (103 < Re < 3 ⫻ 105). Laminar flow in the boundary layer is sensitive to surface roughness.A small scratch or seam on a ball can cause laminar flow to become turbulent. In the 103 < Re < 3 ⫻ 105 region, the added surface roughness perturbs the boundary layer adding turbulence before the typical 80degree separation point.The added turbulence increases the momentum of the fluid and allows the fluid to stay connected up to 120 degrees from the upstream stagnation point. Thus, the drag crisis from balls with a rough surface occurs at a lower Reynolds number (Panton 2005). In addition to roughness, drag is also affected by velocity and ball rotation. As was done for speed, drag can be expressed in non-dimensional form as a coefficient of drag CD defined as (Crowe et al. 2005). 2FD CD = ρ–––– AV 2

(6)

where FD is force due to drag, ρ is the density of air and A is the ball cross sectional area.The drag coefficient is plotted as a function of the Reynolds number for a number of sports balls in Figure 24.4a. The data were obtained by projecting balls without rotation through still air. The drag force was obtained from the ball deceleration by measuring the change in ball speed over a fixed distance (Kensrud 2010).The range in Re for each ball type is representative of play conditions. In Figure 24.4a, the golf ball and smooth sphere show a strong drag crisis behaviour. The tuned and dimpled surface of the golf ball induces a drag crisis at a low Re, which gives the ball low drag for much of its flight.The drag crisis for the smooth sphere is equally strong and, as expected, occurs at a higher Re. Smooth-seam baseballs (Major League Baseball, MLB) and softballs (slow-pitch) differ in diameter but have a relatively flat surface. Raised-seam baseballs (National Collegiate Athletic Association, NCAA) and softballs (fast-pitch) have increased drag and a delayed or muted drag crisis.The drag of smooth- and raised-seam baseballs is compared in Figure 24.4a, where the raised seam increases drag at high Re.

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size distribution, arrangement and mineralogy of these particle size fractions have a significant effect on the physical and mechanical behaviour of the soil, which ultimately determines pitch performance.There is also an organic fraction of the soil formed from living soil microbiology and dead and decaying plant and animal matter. The organic matter content of soil can be determined by mass balance following combustion at 450 degrees Celsius.The organic matter fraction provides nutrients to the plant and retains water and nutrients in the soil. Packing of mineral and organic particles in a soil results in the formation of a pore network within the granular bulk soil. It follows that this pore network has a size distribution and connectivity that is a function of the particle size distribution and the packing density of the particles. Following rainfall, water will infiltrate into the pore network of a soil under gravity and a tension created by capillary action as water is attracted to the dry surface of the pore. Infiltrating water displaces gas within the soil.When all the pores are filled with water, the soil is termed saturated and is anoxic; that is, there is no oxygen for plant root or soil microbe respiration and the turf plant becomes stressed. Typically, the water content of a soil is in flux in response to climatic variables including precipitation and temperature from evapotranspiration (the combined effects of evaporation and plant transpiration). Soil water movement is determined by hydraulic potential within the soil, a balance of gravitational potential and a pore potential that is either negative in unsaturated pores or positive when the soil is saturated and pressure heads build up.Typically, the larger the pore space, the more easily water moves downwards, owing to gravitational potential and the greater the rate of water movement.Water flow rates within the soil are proportional to the hydraulic conductivity of a soil, a rate constant that is a function of pore size (and therefore particle size and packing), pore connectivity and the degree of saturation.Thus, soils with a large proportion of sand particles (Figure 30.2) are more freely draining and have greater infiltration rates than soils with a large proportion of clay particles. This is advantageous, in that it is easier to maintain aerobic conditions within the soil, but has two drawbacks, in that it is difficult to retain sufficient water for plant turgor and health and soluble nutrients are easily removed through drainage water from the soil (a process known as leaching). To reduce leaching, it is necessary to either reduce drainage from the soil or decrease the solubility of nutrients into the soil solution. Cation exchange capacity is a property of a soil that describes its capacity to exchange plant-essential nutrient cations with the soil solution. The cations are held by electrostatic attraction to the surface of certain clay minerals that, because of their mineralogical structure and weathering, have a negative surface charge, and also in complexes with organic matter.The cation exchange capacity of a sand soil is extremely low and as a consequence sand soils have low nutrient status.Amendment of sand soils with organic matter is a common strategy to improve the fertility of the soils but inevitably higher quantities of nutrient fertiliser are required compared with clay soils, which exhibit greater cation exchange capacity. Coated fertiliser products reduce the rate at which the fertiliser dissolves into solution, reducing the risk of nutrient supply exceeding plant demand. So, in terms of water and nutrient retention, it might seem that clay would be the ideal soil selection for a natural turf sports surface. The natural turf surface has to provide traction and impact energy absorption for the player when performing movements. To do this, it needs to have sufficient shear strength for boot or shoe traction. Figure 30.3 shows how the shear strength of a two soils, a sand soil (98:1:1 per cent sand:silt:clay) and a clay loam soil (29:45:26 per cent sand:silt:clay) changes as the saturation ratio (the water content of the soil divided by its saturation water content) increases.The shear strength of the sand soil remains high over a range of saturation ratio from 0.3 to 0.8.The shear strength of the clay is lower and decreases rapidly with increasing water content.This is the essential difference between sand and clay soils 471

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– their shear strength and the sensitivity of that shear strength to water content. Whilst a clay loam provides a better environment for a grass plant in terms of water and nutrient retention, unless it is very dry, as soon as it is loaded it will fail and it will fail plastically, resulting in turf detachment (divoting), surface deformation and localised compaction. A sand soil will prove more resilient in most precipitation conditions although irrigation and fertiliser inputs will be required to prevent drought and nutrient stress in the plant. Furthermore, the saturation ratio and shear strength of a sand soil are likely to be more spatially consistent because of their resistance to compaction and relatively high infiltration rate and hydraulic conductivity. Rapid transport of water from the surface to drainage at depth means that the water content at the surface is lower after a precipitation event, maintaining shear strength in the contact zone between players and the surface. Where water is allowed to pond, significant plastic deformation of the soil takes place during use, further reducing the infiltration rate, thus a negative cycle of pitch deterioration begins. As a result, for the majority of sports, sand is the preferred rootzone soil material (Figure 30.2). This is not the case in sports such as cricket and tennis, where ball bounce is more important than player traction (Figure 30.2). In these sports, densely packed, dry clay soils are used because they have a high stiffness in this state. Consequently these surfaces can only be used in the summer when precipitation is low and evapotranspiration high, such that water content is low. At very low water contents, the shear strength and stiffness of a clay soil increases significantly. This means that the hardness of the surface increases and plastic deformation on ball impact is reduced significantly, resulting in a greater coefficient of restitution between ball and 472

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surface. Hardness increases to such an extent that cricket players wear spiked (2–4 mm diameter), rather than studded footwear and tennis players use dimple-soled rubber shoes to generate traction between the shoe and the grass-soil surface. Thus, it can be seen that material selection is important in the construction of natural turf surfaces.To describe the performance of a natural turf surface, it is essential to characterise the materials used in its construction. As a minimum, this should comprise determination of the particle size distribution, bulk density (packing), water content and organic matter content. These properties of a surface can vary in three dimensions. In constructed surfaces, spatial variation can be expected to be minimal compared with the majority of natural turf systems, which have been levelled and fashioned from agricultural grassland systems, and where spatial variation in soil type and properties can be significant, even at a 7000-m2 pitch scale.Variation of these properties through the profile should also be considered – a 20-mm layer of sand over a 280-mm clay profile will behave like a clay, not like a 300-mm deep sand profile. It should also be emphasised that to classify a ‘natural turf surface’ without reference to the sport being played is foolhardy. One of the advantages of natural turf is that the components can be varied to suit a particular sport, whether it is the grass type across the different playing surfaces of a golf course or the contrast between a clay surfaces for the cricket pitch, where the ball bounces and a sand surface for the outfield where ball bounce is less critical than player traction and wear resistance.

Design and construction Whilst the majority of focus is on the quality and performance of élite-level natural turf facilities, such as football stadia, baseball parks, cricket grounds, championship golf courses or Wimbledon’s tennis courts, the majority of natural turf surfaces are not constructed from materials offsite but are simply formed from the native soil and seeded or turfed with turfgrass cultivars of suitable grass species. This has the advantage of reducing material costs and transport, a typical sand football pitch construction of 70 m ⫻ 100 m ⫻ 0.4 m depth would require 4,480 tonnes of soil material (at a dry density of 1,600 kg m–3) but has the disadvantage that surface behaviour will be determined by the material characteristics and this is commonly to the detriment of surface quality if the native soil has a high clay content. Constructed profiles typically comprise a sand rootzone overlying a gravel drainage layer which in turn overlies a drainage pipe network.The principle of the design is that in a partially saturated state, water is retained within the sand, owing to pore capillarity. This creates a suspended water table in the sand, over the gravel, allowing the plant to access water. During a rainfall event, a positive water pressure builds over the interface between the sand and gravel and forces water into the freely draining gravel and ultimately into the drainage system, where water is transported to outfall.The relative sizes of the sand and gravel are critical. If the sand is too coarse, it will not retain a sufficient depth of water for the plant, if it is too fine then it will exhibit excessive water retention and could migrate into the gravel layer, providing hydraulic continuity that will remove the suspended water table. Medium sands of similar diameter are used – by keeping the diameter of particles uniform it reduces the risk of compaction. Owing to its high hydraulic conductivity, it is not possible to maintain grass health using this construction approach without an irrigation system.Water is required for plant health but also to maintain surface strength. Look at Figure 30.3 again and note how the shear strength of the sand soil increases marginally from 0.3 to 0.5 per cent saturation.This is caused by the apparent cohesion from pore water under tension, increasing the strength of the soil.This effect 473

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diminishes with further addition of water as the pore water tension decreases and eventually become positive pressures. The quantity of irrigation required for maintaining a constructed sand pitch depends on climate but is considerable.Where the availability of water or the cost of irrigation infrastructure is unaffordable then a sand construction surface is inappropriate. A more affordable solution is to use a native soil and, if this has a low hydraulic conductivity, provide a surface drainage system. A surface drainage system provides a high hydraulic conductivity conduit between the soil surface and a piped drainage system at depth.A traditional agricultural subsurface drainage system commonly does not cure problems on a sports surface because it is only designed to lower a near-surface water table. In a sports pitch, it is essential to transport water from the surface to the drainage outfall as quickly as possible during and following a precipitation event. For low hydraulic conductivity soils, the only way to achieve this is by bypassing the native soil using a system of sand slits of varying dimensions that intersect the sub-surface drains (Ward 1983; James et al. 2007). Sand slits are typically sand-filled 400-mm deep, 40-mm wide slits cut in the soil across the length or width of the pitch at an inter-split spacing of one to two metres, although shallower, narrower slits are sometimes used at 100–200-mm spacing. The slit provides a high hydraulic conductivity route for water to move from the surface, where player–surface interaction takes place, to deeper subsurface drains. For the surface to function, the slits must be connected to the surface, so it is essential to topdress the whole pitch with sand to a depth of 12–20 mm and to routinely provide further sand additions to maintain the hydraulic continuity between surface and sand slit. Without this routine maintenance the slits become capped by the shearing of the native soil during play and the sand slit drains become disconnected from the surface and effectively defunct until the continuity is restored. An alternative approach in suitable clay soils is to form a 50-mm diameter channel at 400-mm depth using a mole plough. This technique is only suitable in stable clay soils but provides a rapid, cheap form of drainage (for further details of this method, see James et al. 2007). Pitch construction method is a trade-off between the costs of installation and operation and required level and consistency of performance. An example of the relative costs can be seen by considering the English Premier League, one of the wealthiest club leagues in football. The majority of stadium pitches in the league are constructed sand profiles but the proportion of pitches in their training facilities constructed in this manner is much lower and restricted to the largest clubs.The training pitches, pitches in lower leagues and the majority of recreational football, are drained native soil or a sand-ameliorated drained native soil. This is because the construction and operating costs of sand profile pitches are so high. The image of the muddy weekend player still exists in sport; it is just not seen as often on television because of the improved quality of stadium pitches – but this does come at a cost.

Hybrid surfaces The problem with sand surfaces is that they have low cohesion; their strength comes from inter-particle friction caused by interlocking, rather than inter-particle cohesion that is more common in clay soils.This means that, whilst confined and compacted, sands have a high shear strength; when loose and less confined, their strength reduces, leaving them susceptible to damage from divoting. The grass plant has a critical role in binding the sand soil together to maintain it strength.A number of technologies have been developed to increase the stability or strength of sand soils and can be divided into two broad categories: soil stabilisation and synthetic turf fibre ‘hybridisation’. Soil stabilisation is commonly achieved by the inclusion of narrow fibres (35-mm length, 0.1-mm diameter) of polypropylene or a mix of polypropylene 474

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and rubber which aims to bind the soil together, mimicking the effect of fine turf roots. This can increase the strength of a soil (Baker 1997; McNitt and Landschoot 2005) but also its hardness and stiffness. Synthetic hybrid methods use synthetic turf fibres that are either punched into the natural turf pitch individually or are tufted into a biodegradable backing, which is then filled with soil and turfed or seeded with grass.The synthetic fibres then sit between the surface and mowing height to reduce the wear on the natural turf fibres and provide some vertical reinforcement of the soil profile.The other advantage of this method is that as the natural turf wears, some green colour remains in the surface.

Defining and characterising performance In synthetic turf surfaces, the development is led by sport-specific performance standards that aim to match the performance of natural turf, yet there are limited comparable performance standards for natural turf. Perhaps because it is seen as the benchmark surface and that it has been used for the majority of modern sports since their inception, it is actually rare to find a prescribed performance standard for a natural turf surface. Generally, decisions on playability are left to the match officials, who use criteria such as ‘can the players move without excessive slipping’ and ‘does the ball roll a reasonable distance’ to determine that a pitch is suitable for play. This is not the case in horse racing in the UK, where limits for surface ‘going’ are provided by the governing body/regulator the British Horseracing Authority, which is aiming to reduce the risk of horse injury and race meeting cancellation. Racecourses are obliged to use the GoingStick® (Dufour and Mumford 2008), which measures penetration resistance and shearing resistance of a 100 mm ⫻ 20 mm ⫻ 6 mm blade inserted into the turf. This objective measurement is used to ‘declare’ the performance of the racecourse, information that is used by horse trainers, jockeys and spectators to manage horse performance. This is a rare example of objective measurement tools being used in the decision of when a natural turf surface is suitable for sport. Other examples include the use of the 2.25-kg Clegg Impact Surface Tester device in Australian Rules Football, where impact decelerations of 120 gravities or over are considered too high (Chivers and Aldous 2004) leading to ground closures, although a study by Twomey (2010) showed that, although there was a relationship between hardness and injury in Australian Rules Football, it was not necessarily with extremely hard or soft conditions. Performance quality standards for natural turf do exist (Institute of Groundsmanship 2001; Football Association 2004; Bartlett et al. 2009). Often, they comprise a large suite of different tests that consider grass coverage, grass colour, incidence of weeds, surface levels, traction, hardness, infiltration rate and thatch content (thatch is accumulated organic matter near the surface of the soil profile that adversely affect ball and player surface interactions). It is not common to use such large arrays of testing in routine natural turf management, as they are too time consuming and many of them are subjective.They are used in benchmarking and for ensuring standardisation of pitch quality across tournaments, as conducted by the Sports Turf Research Institute for the FIFA world cup in South Africa 2010 and by the New Zealand Sports Turf Institute in preparation for the 2011 Rugby World Cup.

Current test methods For sports using a relatively large ball that is passed or carried, testing can be reduced to three simple questions: is the surface too hard or too soft? is there sufficient but not excessive traction? and is the ball-surface interaction suitable? Of course, this varies from sport to sport. Football has particular concern regarding traction and ball–surface interaction; golf and cricket 475

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are predominantly concerned with ball–surface interaction and hardness. Additional testing might be required to explain variation in these parameters or the cause of unacceptable limits but the development of testing devices should aim to enable routine, objective measurement of these parameters. It needs to recognise that appropriate technology is not necessarily the most complicated but that it needs to be objective, reliable, allow rapid measurement and to be affordable. Two commonly used devices are the Clegg Impact Surface Tester (SD Instrumentation Ltd, Bath, UK) and the studded-plate traction device.The Clegg Impact Surface Tester comprises a uniaxial accelerometer within a cylindrical aluminium missile that is dropped from a fixed height and the peak deceleration on impact determined. The mass and drop height of the missile vary but the most common configuration is a 2.25-kg hammer dropped from 450 mm (alternatives include 0.5 kg from 550 mm and 4.5 kg from 450 mm). Recommended suitable hardnesses vary from sport to sport, as discussed above. The studded plate testing device has been in use in sports surface testing since the 1980s (Canaway and Bell 1984).The device is a modification of the bevameter used in terramechanics to determine the trafficability of soils and comprises a steel disc with a uniform stud pattern that is loaded with 46 ± 2 kg dropped from 60 ± 5 mm and then rotated at a constant speed manually and the peak torque recorded (BSI 2007). Adaptations of this device log torque with angle of rotation but the majority of ‘standards’ provide ranges for peak torque. Whilst this device is relatively simple to manufacture, it has the drawback of having a total mass of approximately 55 kg, limiting portability in manual operation. Alternative approaches to measuring traction include using a simple shear vane or devices that measure a shearing resistance such as the GoingStick, although these devices cannot determine the effect of variation in stud type and stud pattern. Quantifying the suitability of ball–surface interaction is more complex and varies from sport to sport. Ball roll behaviour can be determined by rolling a ball down a standard ramp and measuring the distance the ball travels before coming to rest. Additionally, the deviation of the ball from a straight trajectory may also be recorded and more sophisticated devices can measure ball deceleration over a one to two metres’ distance from the base of the ramp. Standards set minimum and maximum limits for ball roll distance, based on the assumption that given a standard potential energy on release, the deceleration of the ball is a function of ball–pitch friction, which is constant across the measured area.The measurement of green speed in golf using the Stimpmeter (USGA 2009), is slightly different in that the ramp is raised manually by the operator.Whilst this reduces the cost of equipment and increases the portability of the device, the control of release height is reduced and momentum can be imparted by the operator, affecting the ball roll distance. Ball rebound behaviour is more complex. Original methods looked at vertical ball rebound using a device that drops the ball normally on to the surface from a standard height (for example, three metres for cricket and football) and measures the rebound height of the ball which could then be compared with standard minimum and maximum rebound values or a range of coefficient of restitution. It is rare that the ball has a perfectly normal impact with the surface, so angled ball rebound behaviour is more representative but more difficult to achieve, commonly using air-cannon devices.

Development of improved devices To ensure that testing and testing devices are relevant, it is important to question the purpose of testing natural turf surfaces. If the purpose of testing is to improve the understanding of injury mechanisms then improved, more complex testing devices that more closely represent human 476

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loading are required. This is because the rate of loading on the human body affects adaptation by the body and the mechanical behaviour of natural turf surfaces is loading-rate dependent (Guisasola et al. 2010a). An example of where more complex devices are required would be in understanding traction.The traditional studded plate device only measures peak torque; that is, the shear stress at failure and the strain rate is constant and relatively slow (12 rev/min). Human loading of natural turf is more dynamic. In a study of turning on natural turf at a controlled speed of just 3.83 m/s, peak horizontal loading rate was 2.98 kN s–1 (Guisasola 2008). The soil must yield for traction to take place but the stress–strain behaviour for yield is critical, as this will determine the rate of loading of the soft-tissues in the lower limb and how they accommodate this load. More complex devices would need to record force and deformation throughout the simulated turning movement, which would have to be more complex to reflect the magnitude of loading and rotation of stresses that are observed in actual sports movements. Such complexity of movement and measurement adds cost, however, so there is a trade-off between improved replication of movement and data capture and the volume, extent and resolution of data sets. Epidemiological studies of injury are limited by the lack of explanatory surface characterisation data. With the condition of surfaces changing over short distances within the pitch in response to variation in weather and management, it is essential to capture this information. Surface surveys with high spatial and temporal resolution are time consuming, so devices need to be rapid, portable and relatively low cost to encourage widespread adoption. This does not mean that data should be entirely irrelevant, more that it could be a simplification of the number of parameters measured, focusing on key information, or that it could be that peak values as opposed to whole force-displacement curves would be appropriate. An example would be in horseracing, where there is regular, systematic and obligatory testing of surfaces with devices such as the Going Stick, providing valuable companion data sets for the epidemiology of racehorse injury (Williams et al. 2001). These data do not explain injury mechanisms; they can only point to associations among injury rates, types of injury and surface parameters but they do inform more detailed examination of horse–surface interaction and more complex simulation of loading in understanding injury mechanisms.There is a need for improved data sets in other sports such as football and rugby, both in terms of explaining injury mechanisms and in quantifying spatial and temporal variation in pitch performance that relate to injury incidence, especially as surface construction and management methods continue to evolve (Stiles et al. 2009). The current and future testing devices discussed here only quantify variation in performance parameters such as hardness and strength; they do not explain this variation. To manage spatial and temporal variation in natural turf sports surface requires detailed data on profile particle size distribution, density, water content and organic matter content, requiring laboratory testing of samples of the soil offline. It is commonly variation in these parameters that is responsible for variation in pitch performance.

Next-generation natural turf It is common to discuss the development of synthetic turf in terms of ‘generations’ – the first being short-pile nylon carpets, the second sand-filled 20–25-mm polypropylene tufted fibres and the third as sand and rubber filled longer-pile (40–65 mm) tufted polypropylene/polyethylene fibre surfaces more suited to football and rugby.The transition from secondto third-generation synthetic turf is evolutionary (lengthening of fibres, inclusion of rubber infill) with the driver being improved replication of natural turf. Within a similar time frame, natural turf has been undergoing its own evolution in terms of construction (increase in sand 477

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profile pitches and hybrid surfaces) and management (increased irrigation and nutrient application, improved grass breeding and the use of supplementary growth lighting). The key difference is that this evolution has been restricted to élite-level stadium surfaces. Some advances in grass breeding and husbandry and also drainage construction have improved the quality of recreational surfaces but these remain cost-limited and therefore the benefits in performance from sand construction are not achievable. Further development in natural turf technology is required in two key areas.The first is to increase wear resistance and carrying capacity of recreational surfaces. Solutions will need to be affordable and sustainable over the lifetime of a surface, which is commonly measured in decades at the recreational level. Surface performance is commonly limited by excess water in winter periods and insufficient water during summer periods. It follows that effective affordable drainage is essential but material selection and grass breeding must limit the effect of water shortage in dry periods, especially considering the projected effects of climate change. Developments in drainage technology must also consider the ongoing sustainability of funding, particularly where the effectiveness of drainage is dependent upon that funding, such as with sand-slitting, for example. The second is improved environmental sustainability of surface management. For natural turf surfaces to remain a competitive solution for sport in the future, when resource availability is expected to be reduced and population increased, it is essential to reduce the consumption of resources such as water, natural gas (used in the production of fertilisers), pesticides and minerals (James 2011). Élite-level surfaces could be engineered to improve performance and sustainability by providing real-time information on surface status. Within-surface sensor arrays could provide real-time information on turf stress and strength at a spatial resolution that reflects spatial variation in pitches. This would require the development of affordable, robust sensor technology for installation within a surface but would provide invaluable information for the improved management of surfaces, leading to reduced consumption and providing real-time and historical performance data. Current sensor technology is too expensive, too large and insufficiently robust for installation outside controlled research environments. Improved data recording could inform other aspects of surface management including the use of machinery. Bartlett and James (2011) showed how 21–54 per cent of the carbon footprint of golf course maintenance is from the use of mowing machinery. Machinery operation on golf courses is typically surfacepresentation optimised, which limits resource efficiency. Improved vehicle sensor technology is needed to inform optimisation of machinery operation in more resource-restricted environments. If participation in sport increases in line with increase in population, then sports surfaces will need to provide greater carrying capacity for sport from a decreasing footprint, as competition for space increases, particularly within urban environments. In parks and urban green spaces, natural turf will continue to have a place in the urban environment, as long as it can evolve to increase carrying capacity with reduced consumption. In schools and specialist sports facilities, synthetic turf will become more prevalent, because it has a greater carrying capacity. For sports such as golf and cricket, where large areas of turf are required and variation in surface conditions are desirable, natural turf will remain the surface of choice at both recreational and élite level. The balance between natural and synthetic turf in the stadium environment for sports such as football is changing and the future will depend on the development of synthetic turf and whether it becomes more acceptable to élite participants but also on the development of natural turf construction and management and whether its current playability and aesthetic advantages can be maintained sustainably. 478

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References Baker, S. (1997) ‘The reinforcement of turfgrass areas using plastic and other synthetic materials: a review’, International Turfgrass Society Research Journal, 8: 3–13. Bartlett, M. D. and James, I.T. (2011) ‘A model of greenhouse gas emissions from the management of turf on two golf courses’, Science of the Total Environment, 409: 1357–67. Bartlett, M. D., James, I.T., Ford, M. and Jennings-Temple, M. (2009) ‘Testing natural turf sports surfaces: the value of performance quality standards’, Proceedings of the Institution of Mechanical Engineers Part P: Journal of Sports Engineering and Technology, 223: 21–9. BSI (2007) Surfaces for Sport Areas. Part 1: Determination of Rotational Resistance. BS EN 15301-1: 2007. London: British Standards Institution. Canaway, P. M. and Bell, M. J. (1986) ‘Technical note: an apparatus measuring traction and friction on natural and artificial playing surfaces’, Journal of the Sports Turf Research Institute, 62: 211–14. Chivers, I. and Aldous, D. (2004) ‘Performance monitoring of grassed playing surfaces for Australian Rules football’, Journal of Turfgrass and Sports Surface Science, 70: 73–80. Dufour, M. J. D. and Mumford, C (2008) ‘GoingStick® technology and electromagnetic induction scanning for naturally turfed sports surfaces’, Sports Technology, 1:125–31. Football Association (2004) Natural Grass Construction Upgrade Performance Quality Standard. London: FA. Guisasola, I. (2008) Human–Natural Sports Surface Interaction. Unpublished thesis, Cranfield University, UK. Available online at http://hdl.handle.net/1826/3487 (accessed 12 January 2013). Guisasola, I., James, I. Stiles,V. and Dixon, S. (2010a) ‘Dynamic behaviour of soils used for natural turf sports surfaces’, Sports Engineering, 12:111–22. Guisasola, I., James, I., Llewellyn, C., Stiles,V. and Dixon, S. (2010b) ‘Quasi-static mechanical behaviour of soils used for natural turf sports surfaces and stud force prediction’, Sports Engineering, 12:99–109. Institute of Groundsmanship (2001) Guidelines for Performance Quality Standards, Part One: Sports Surfaces Natural and Non Turf. Milton Keynes: Institute of Groundsmanship. ISO (2009) Soil Quality: Determination of Particle Size Distribution in Mineral Soil Material; Method by Sieving and Sedimentation. ISO 11277:2009. Geneva: International Organization for Standardization. James, I. T. (2011) ‘Advancing natural turf to meet tomorrow’s challenges’, Proceedings of the Institution of Mechanical Engineers Part P: Journal of Sports Engineering and Technology, 225(3): 115–29. James, I.T., Blackburn, D.W. K. and Godwin, R. J. (2007) ‘Mole drainage as an alternative to sand slitting in natural turf sports surfaces on clays’, Soil Use and Management, 23: 28–35. McNitt, A. S. and Landschoot, P. J. (2005) ‘The effects of soil reinforcing materials on the traction and divot resistance of a sand root zone’, International Turfgrass Society Research Journal, 10: 1115–22. Stiles,V. H., James, I.T., Dixon, S. J. and Guisasola, I. N. (2009) ‘Natural turf surfaces: the case for continued research’, Sports Medicine, 39: 65–84. Twomey, D. (2010) Is There a Link Between Injury and Ground Conditions? A Case Study in Australian Football. Paper presented at the 2nd Science Technology and Research in Sports Surfaces (STARSS II) Conference. Loughborough, UK, 21–22 April. USGA (2009) Stimpmeter Instruction Booklet. Far Hills, NJ: United States Golf Association. Available online at http://www.usga.org/course_care/articles/management/greens/Stimpmeter-Instruction-Booklet/ (accessed 12 January 2013). Ward, C. J. (1983) ‘Sports turf drainage: a review’, Journal of the Sports Turf Research Institute, 59: 9–28. Williams, R. B., Harkin, L. S., Hammond, C. J. and Wood, J. L. N. (2001) ‘Predominance of Tendon Injuries on British Racecourses (Flat Racing and National Hunt Racing)’, in R. B.Williams, E. Houghton and J. F.Wade (eds) Proceedings of the 13th International Conference of Racing Analysts and Veterinarians. Cambridge, Newmarket: R & W Publications, pp. 274–7.

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31 DESIGN OF SPORTS FACILITIES Franz Konstantin Fuss

Introduction Stadia have always been dominated by striking architecture, attractive to both athletes and spectators. Stadium design spans almost three millennia, from the classical facilities of Olympia, the birthplace of the Olympic Games, culminating in mega structures such as the Circus Maximus and the Coliseum in ancient Rome, to modern design wonders such as the Water Cube and the Bird’s Nest of the Beijing 2008 Olympic Games. Yet, even if architecture seems to be predominantly representing the symbolic and iconic character of a stadium, it is inherently connected to four key issues, embedded in the architectural planning phase: improving accessibility of remote outdoor sports, sustainable design, safety and the management and operation of sports facilities. Many outdoor sports disciplines are not easily accessible to the masses if the distance between urban areas and the remote natural outdoor places is unreasonable and time consuming. The classical example is swimming, accessible for people living close to slow-flowing rivers, lakes and coastal areas. The problem was solved as early as the times of ancient Greece and Rome, with public swimming centres (thermal baths) and private indoor and outdoor swimming pools.This principle remains unchanged in modern times, with the trend from outdoors to indoors, thus being independent of weather and temperature, and from remote wilderness to the city centre. Yet, the technical solutions offer more possibilities for replicating natural conditions and improving the accessibility of a wide range of sports disciplines.The associated facilities are numerous; to name just a few: indoor skiing facilities (‘snow cities’); indoor ice skating halls; indoor and outdoor sport (rock) climbing gyms; indoor ice-climbing gyms; outdoor driving ranges; outdoor sliding centres for bobsleigh, luge and skeleton; or outdoor white water parks. A major advantage of artificial sport facilities is the flexible and modular design, allowing a relatively quick and easy change of climbing routes and white water courses. Sustainable design arose as an issue only in the past few years, when organisers of large sporting events became more aware of costs, energy, waste and recycling.The Sydney 2000 Olympic Games, also termed the ‘Green Games’, set the gold standard for future games with sophisticated stadium design solutions. The Stadium Australia of 2000, renamed Telstra Stadium, is distinctly marked by a translucent and hyperbolic roof aerodynamically shaped and sloping down to the pitch, which saves 480

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costs of artificial lighting and improves the quality of grass, reduces aerodynamic drag, minimises heat loss from wind, provides protection to twice the number of spectators compared to cantilever roofs, collects rainwater for pitch irrigation and optimises acoustics (John et al. 2007). Further issues of environmentally friendly design include the following factors (John et al. 2007): solar panels and other means of trapping solar energy; providing energy and hot water; natural renewable materials used for construction; spaces naturally ventilated by air chimneys and wind; playing fields lower than the ground level; saving building costs and movement of vehicles required for construction. Qatar, the winner of the bid for 2022 Soccer World Cup, set a new standard for sustainable development of sporting events, by proposing novel ‘recyclable/reusable’ stadia that will be dismantled after the tournament and offered to poorer countries in the region. Safety issues of stadia and sports facilities encompass evacuation in case of emergency situations, antiterrorism measures, containment of fire, structural failures in cases of overload, riots and earthquakes and prevention of injuries of athletes. The management of sports facilities is crucial for smooth operation and automation contributes to reducing operating costs. Operation can be seriously compromised by the incompatibility of two or more components and can result in severe financial losses. This chapter covers selected case reports of all four issues mentioned above.

Artificial sports facilities – whitewater parks Whitewater parks constitute the artificial counterpart of natural mountain torrents, sought after for canoeing, rafting and kayaking and thus are regarded important tourist attractions. Artificial whitewater facilities offer a range of advantages compared with remote mountain torrents. Among them are: • • • • •

accessibility and reduction of training costs; modular design and water speed for replicating courses of varying degrees of difficulty; safe design and provision for shut down in emergency situations; medical services and transport readily at hand; and a short distance between upstream start and downstream finish, conveniently covered by boat lifts.

Disadvantages are the high construction and operating costs and the enormous energy consumption of water pumps essential for closed-loop designs. Nevertheless, in spite of the high costs involved, whitewater parks emerged as very profitable sports facilities if they were open to the public. For example, the Sydney 2000 Olympic Games Penrith whitewater facility reached its break-even point after five years and remains fully booked for weeks ahead.The major revenue comes from rafting, whereas kayaking and canoeing contribute only little to the profit. Whitewater parks are designed either as closed or open loops. Recirculating closed loops, often associated with lakes, rivers or even the sea, for the ease of water replacement, require pumps for operation.The Athens 2004 Olympic Games whitewater park is the only salt water facility so far and designed as a figure of eight. Open loops rely on natural supply of potential energy by water naturally dropping at least a couple of metres. A typical example is the whitewater park at Foz do Iguaçu, Brazil, which takes advantage of the height of the Iguazu falls, providing unlimited water supply. 481

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Design recommendations encompass (Snel 1999): • the course length: minimally 300 metres, required for international competitions, with a slope of approximately two per cent; • the optimum width profile: 10–12 metres, minimally 8 metres; • the water depth: minimally 0.4 metres and 0.75–0.9 metres on average for canoeing (for safe Eskimo rolls) and 1.5 metres for freestyle kayak. Additionally, the water speed can reach 7.5 m/s (Shunyi Olympic Park, Beijing 2008 Olympic Games) and the flow rate ranges between 15 and 22 m3/s. Water diversions and obstacles can be stationary or mobile, with design elements such as boulders, bollards and dams. Plastic bollards, set in holes in the concrete floor of the water channel, can be easily rearranged and prove to be the most suitable modular design element. The water circulation of the Penrith whitewater course is provided by six 300-kW water pumps with a combined flow rate of 15 m3/s. Boat lifts are usually designed as conveyor belts, producing the necessary friction for pulling the boats upwards. Figure 31.1 shows the design of four whitewater parks: the Olympic courses of Penrith (Sydney 2000) and Shunyi (Beijing 2008) are horseshoe-shaped; the Olympic course of Athens (2004) is coiled and the US National Whitewater Centre has two courses of different length.

Figure 31.1 Whitewater parks: (a) Penrith, Australia (Sydney 2000 Olympic Games) © 2011 Google Imagery/Digitalglobe, Cnes/Spot Image, GeoEye, Sinclair Knight Merz; (b) Athens, Greece (2004 Olympic Games) © 2011 Google Imagery/Digitalglobe, GeoEye; (c) Charlotte NC, USA (US National Whitewater Centre) © 2011 Google Imagery/Digitalglobe, Orbis Inc., GeoEye, US Geological Survey; (d) Shunyi, China (Beijing 2008 Olympic Games); boat lifts are marked with small arrows and water pumps with large arrowheads (© 2011 Google Imagery/Digitalglobe, GeoEye)

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Sustainable design – the water cube The Water Cube (Figure 31.2), the swimming stadium of the Beijing 2008 Olympic Games, was designed by ARUP in a holistic way, by addressing architectural aesthetics, sustainable design and safety issues. The façade and walls are built from foam bubble-like elements, by addressing the fundamental problem of the most efficient way to divide space into cells of equal size with the least surface area between them.This question was already asked, and subsequently answered by Lord Kelvin (Thomson 1887) with 14-sided tetrakaidecahedrons, the surface of which consists of eight hexagons and six squares. Weaire and Phelan (1994) provided a better solution, more efficient by two per cent than Kelvin’s pattern, by creating a regular pattern of four units of 14-sided cells with two opposing hexagonal faces and two belts of six pentagons surrounding these hexagons and one unit of (12-sided pentagonal) dodecahedra, where all units have the same volume.This ‘Weaire-Phelan foam’ was used for constructing the outer shell of the Water Cube. Apparent irregularity was added to the design by tilting the regular WeairePhelan foam three-dimensionally and cutting out a block of the same external size as the Water Cube, 177 metres ⫻ 177 metres ⫻ 31 metres (Carfrae 2007). Nevertheless, regularity is still preserved as the façade pattern repeats itself. The cell edges of the bubbles were constructed from a network of steel tubular members and the cell faces from ethyltetrafluoroethylene (ETFE). This design resulted in 22,000 steel members (90 km in total) and 12,000 nodes; 4,000 bubbles and 100,000 m2 of ETFE bubble cladding (ARUP 2005).The structural stability and earthquake-proof design was simulated and tested with finite-element analysis. ETFE is a recyclable material with a mass of one one-hundredth of an equivalent-sized glass panel. This plastic material is strong, lets in more ultraviolet light than glass and thoroughly cleans itself with every rain shower. It is also a better insulator than glass and is much more resistant to the weathering effects of sunlight (ARUP 2005). The disadvantageous effect of

Figure 31.2 The Water Cube: outdoor (a) and indoor (b) views Source: © 2007 ARUP+PTW+CCDI, reproduced with kind permission

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ETFE lies in its combustibility and thus did not follow the prescriptive rules of the Chinese Building Code. However, ETFE shrinks away from a fire, so is thus effectively ‘self-venting’ and lets smoke out of the building (ARUP 2005). How smoke and heat would spread through the building and how the performance of different smoke exhaust rates would keep smoke away from exiting people, was modelled with ‘fire dynamics simulator’, a computational fluid dynamics-based model of fire-driven fluid flow. Eventually, the Water Cube was the first Olympic site in Beijing to receive approval for fire engineering, owing to thorough performance-based fire engineering tailored to the building (ARUP 2005). The Water cube is designed as a giant greenhouse (Figure 31.3) suited for the extreme climate of Beijing, ranging between –13 degrees and 36 degrees C.The walls and the roof of the Water Cube are constructed from an inner and outer layer of ETFE pillows with a vented cavity in between (Figure 31.3). Fresh air enters the cavity at the base of the walls; it is heated by the sun and rises to the roof, where fans supply the heated fresh air to the interior of the building. The power of the sun passively heats the building and pool water and reduces the energy consumption of the leisure pool hall by 30 per cent. The thermal mass heat storage ensures that heating by the sun during the day is offset by overnight cooling: 20 per cent of the solar energy falling on the building is trapped within the building and is used to heat the pools and the interior area.This is equivalent to covering the entire roof in photovoltaic panels.The leisure pool hall is well lit by the translucent ETFE cladding and saves up to 55 per cent of lighting energy (ARUP 2005).Additionally, rain water is collected and contributes to minimising water consumption.

Figure 31.3 Insulated greenhouse effect of the Water Cube: (1) fan-assisted preheated fresh air returned to pool; (2) ETFE cladding can be switched ‘on’ or ‘off ’ to shade interior; (3) controlled natural daylight and radiant heat to passively heat and light the pool; (4) ETFE pillows act as greenhouse and avoid condensation; (5) fresh external air preheated in vented cavity Source: Carfrae 2007; © 2007 Tristram Carfrae/ARUP, reproduced with kind permission of Tristram Carfrae,ARUP and Ingenia)

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‘. . . the Water Cube . . . solves all the technical problems in one fell swoop. And it achieves the wonderful, if somewhat coincidental, reality of becoming a building filled with water, made from a box of bubbles’ (Carfrae 2007).

Athlete safety – design optimisation of safe jump landing surfaces

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The landing surfaces of acrobatic jumps, often spiced up with extravagant aerial manoeuvres (performed with skis, snowboards, inline skaters, BMX bikes, motorcycles, and so on), traditionally follow a simple design (Figure 31.4), starting with a roughly horizontal tabletop or deck after the kicker, continued by a down slope with nearly constant gradient and a flat or uphill run-out. In motorcycle jumps, the tabletop may be replaced by objects (cars, buses, rivers, canyons) to be jumped over and corresponds to the distance bridged by the jump.Woe betide anyone who under- or overshoots the down slope. Severe injuries are quite common and even deaths occur occasionally, owing to the inappropriate design of the landing area. Even hitting the down slope does not necessarily guarantee a safe landing, which depends on the take-off velocity v0 and the angle ϕ of the down slope.Yet, which angle is safe enough for which v0? This question was addressed by various authors in different ways. Reinke (2006/2007) matched the flight path angle θ with the (constant) slope angle ϕ (Figure 31.4) when landing shortly after the knuckle and calculated and the (maximal) table length L at θ = ϕ for only one specific initial velocity v0, ramp angle θ0 and ramp height H (Figure 31.4). From projectile equations we obtain

Figure 31.4 Table-top design compared with optimised design; L = table length, θ0 = ramp angle, H = ramp height, ϕ = angle of slope A; slope B = optimised slope (landing profile 1 in Figure 31.5); dashed line = flight path

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v cosθ ( –––––––– ) v + 2gH 2gH v cosθ L = ––––––– (sinθ + sin θ + ––––––– ) v g

ϕ = θ = cos–1

0

0

2 0

(1)

2

0

0

2

0

0

2 0

(2)

The disadvantage of Reinke’s (2006/2007) method is that it ignores aerodynamics. Including drag does not guarantee landing shortly after the knuckle (which is a dangerous prerequisite as such) but results in hitting the table at high impact force. Shortening the table length L provides definite landing after the knuckle but, at this point, θ is steeper than ϕ.The next optimisation step is to quantify the danger of impacts and to include aerodynamics. Böhm and Senner (2008) analysed the impact energy based on Reinke’s guidelines, by considering aerodynamic drag and lift at different v0 and various constant slope angles ϕ. The impact energy increases with v0 on the slope (as θ becomes steeper). High energies resulted from impacts on slopes too flat, on the table top and on the run-out. McNeil and McNeil (2009) performed a comparable study by including the drag and liftto-drag ratio. They quantified the magnitude of the impact with the height equivalent of landing impact, h, which corresponds to the ‘equivalent landing height’ used for characterising ski jumping impacts (Müller 1997). Comparable to the results of Böhm and Senner (2008), h increases with v0 on a slope of constant angle ϕ. McNeil and McNeil (2009) suggested replacing the horizontal tabletop by a parabolic profile, thereby reducing the high h on the table top. Yet, even a parabolic landing surface did not guarantee a constant h. Hubbard (2009) solved the constant h problem, which is semi-analytically only possible when neglecting aerodynamics. The fall height h is a function of the velocity component v⊥ perpendicular to the slope v⊥2 h = –– 2g

(3)

Landing with a specific v⊥ corresponds to falling from a height h in a 1-g environment. Skiers hitting the ground with high perpendicular velocities usually suffer from severe injuries, whereas high tangential velocities rarely lead to injuries. v⊥ depends on the difference between slope surface angle ϕ and the angle θ of the flight path at the point of impact v⊥ = v sin(ϕ – θ)

(4)

where v⊥ = 2gh

(5)

dy ϕ = tan–1 –– dx

(6)

gx v θ = tan–1 ––y = tan–1 tanθ0 – ––––––– v02 cos2 θ0 vx

(

) 486

(7)

Design of sports facilities

and gx v = vx2 +vy2 = v02 cos2 θ0 + v0 sin θ0 – ––––––– v02 cos2 θ0

(

2

)

(8)

where v0 results from the inverted flight path parabola v0 =

x 2g –––––––––––––––– 2(x tanθ0 – y) cos2 θ0

(9)

(Hubbard 2009). The take-off velocity v0 is not necessarily parallel to the take-off ramp, nor is the take-off angle θ0 necessarily equal to the kicker angle. In many sports disciplines, the athletes jump off the kicker with an additional velocity component perpendicular to the take-off ramp (‘pops’), which has to be added to the tangential velocity vector and which adds to the kicker angle (Hubbard 2009).The wind speed may alter the take-off velocity too. By solving for ϕ from equation (4) and substituting for v⊥, ϕ, θ0, v and v0 from equations (5) to (9), we obtain dy 2y – x tanθ –– = tan tan–1 –––––––––– + sin–1 dx x

[

(

)

4h (x tanθ – y) cos2 θ –––––––––––––––––––––– 2 x – 4y (x tanθ – y) cos2 θ

]

(10)

where the gradient dy/dx of the slope surface is expressed as a function of the independent variables x and y and the constants h and θ0. Equation (10) defines the soft landing requirement for any given (landing) point (x,y) as a first-order non-linear differential equation, where any point has its unique gradient dy/dx and slope angle ϕ and is reached by a specific initial velocity v0. Both quantities differ from point to point. However, each point has exactly the same perpendicular landing velocity v⊥ and thus exactly the same fall height h. Equation (10) does not contain g, which is not surprising when considering that h is a function of g in equation (3). As any point within a given xy plane is a suitable landing point, the design of an appropriate landing slope depends on selecting one specific point and numerically integrating equation (10) from that very point forward and backward, such that the condition of a continuous gradient dy/dx is fulfilled. By selecting different starting points, we obtain a family of landing profiles, all of which fulfil the criterion of fall height h. However, the fall height h in equation (10) depends on the designer’s choice. Hubbard (2009) suggests a reasonably safe height of about one metre maximally for uncontrolled falls in acrobatic ski jumps. Hubbard’s method is investigated further subsequently, by addressing three issues: • the optimisation of the landing surface; • the accuracy of numerical integration; • design recommendations when considering aerodynamic drag. Out of the above-mentioned family of curves, each of which represents a suitable landing profile, the most optimal profile is determined from the following boundary conditions (Figure 31.5): • maximal v0: any landing point located above the flight path at maximal v0 is unfeasible; • maximal slope angle, depending on the designer’s choice and safety issues.

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Figure 31.5 Gradients of various landing points within a given xy plane for a kicker angle of θ0 = 25º and a fall height of h = 1 m; any flight path of the athlete’s centre of mass starts at 0,0; open circles: no solution for dy/dx; grey circles: feasible landing points where v0 < v0max (maximally possible v0, depending on the in-run design); black circles: unfeasible landing points as v0 cannot be > v0max (as enabled by the in-run design); grey and black circles are separated by the flight path p of the maximally possible v0 (v0max, 65 km/h for this specific jump); grey gradients: acceptable dy/dx; black gradients: unfeasible gradients, too steep slope (–45º for this specific design); grey and black gradients are separated by line s; dashed curves: possible landing profiles (1, 2, 3) for selected points (identified by asterisks; landing profile 1 = slope B in Figure 31.4)

Thus, the area of selectable points can be confined by the maximally possible v0 and by the steepest desirable gradient dy/dx (Figure 31.5).The most optimal landing profile is the longest one, passing through the intersection point of the flight path at maximal v0 and the line or curve separating points of acceptable and unacceptable gradients. Figure 31.5 shows the gradient of different points on the xy plane, some possible slope solutions and the resulting optimised design of a safe landing surface. The accuracy of numerical integration depends on ∆x, when numerically calculating the slope angle ϕ and dy/dx from the flight path angle θ at a specific h and at a specific starting point x,y on the flight path of the maximally possible v0. Intersecting the slope with the flight path at the next smaller v0 generates a new point (x – ∆x, y + ∆y) for which the slope is recalculated. This procedure is repeated until x approaches zero.This method is very forgiving as y values too low or too high (with respect to the positive y axis) are self-correcting as they generate steeper or flatter gradients, respectively. In fact, applying an increment of ∆x = 1 metre from x = 50 m to 7.5 m results in a maximal y error of 16 cm, whereby the ‘inaccurate’ landing profile lies above the ‘accurate’ one, which in turn reduces the fall height. The difference in steepness is less than one degree. 488

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Equations (2) to (10) are only applicable when neglecting aerodynamic forces.Aerodynamic drag influences and changes the flight path and causes a steeper landing or impact angle. This results in a larger fall height, even when the impact velocity is smaller than without drag.The design considerations are: • the fall height is directly proportional to the drag (almost linear proportionality for same landing points). • same landing surface but different flight paths (different v0): larger v0 and landing at lower sections of the slope increases the fall height. • same flight path but different landing surfaces: larger fall height at lower landing surfaces (with smaller y values). If v0 = 18 m/s, θ0 = 25 degrees, the design fall height h = 1 metre and 0.5ρcDA = 1, landing on slope 1 and 2 of Figure 31.5 increases the fall height by 45 per cent and 73 per cent, respectively. It is thus advisable to select that slope with the highest possible y values as well as to reduce the design fall height to account for aerodynamic drag. Summing up, kicker and landing surface design have improved considerably from simple conceptual design to sound scientific and engineering methods and are designed to minimise serious injuries.

Operation of sports facilities – ball management systems A medium-sized driving range with two storeys has an average turnover of 300,000 golf balls every three months.The large number of range balls comes from the fact that the balls remain on the landing area for a longer time, as they are not collected immediately. The replacement every three months is required as the cheap low-quality range balls degrade over time, affected by weather conditions and club head impacts. The annual waste of polymeric products of a medium-sized driving range amounts to about 54 tons and 49 m3. Manual collecting, washing and supplying of range balls to customers is too time and manpower consuming.Automated systems simplify the operation of a driving range.These ball management systems (Figure 31.6) return the balls from the lawn of the landing area right to the tee mat of the customer.This requires a couple of steps, embedded in the design of a ball management system (such as Range Servant, Halmstad, Sweden): • a motorised drive unit, operated by an employee of the driving range after service hours, with a ball picker collecting the balls from the lawn and dumping them in a water-filled ditch; • a ball elevator, which transports the balls to the washing unit; • a washer, which brushes and cleans the balls with heated water; • a blower unit, which blows the balls into the main hose and dries them; • the main hose, through which the balls are blown to the hopper unit, feeding the balls to the next blower; • a system of hoses and manifolds; which shoots the balls to multiple dispensing units at each floor; • ball dispensers, which supply a selectable quantity of balls to the customer. In addition to these larger systems, connected by hoses and manifolds for distribution to many dispensers on several levels of the driving range, smaller systems are available, which are connected by conveyors and elevators and supply the balls to one dispenser. 489

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Figure 31.6 Components of ball management systems: (a) ball picker; (b) Range Servant ball management system; 1: drive unit; 2: ball picker; 3: ball ditch; 4: elevator; 5: ball washing unit; 6: blower; 7: hose; 8: feed hopper; 9: blower manifold; 10: tee-up; 11: dispenser unit; 12: tee mat Sources: (a) © 2011 Derone Enterprises, reproduced with kind permission; (b) © 2011 Range Servant, reproduced with kind permission

The dispensers are equipped with several payment systems operated with coins, tokens, membership cards, credit cards (for example, Range-Express golf ball dispenser payment system by Easy Picker Golf Products, Lehigh Acres FL, USA), or E-keys and PIN numbers (eTMrange electronic range system by golfrangesystems, Niagara Falls NY, USA). The number of 490

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balls can be preselected and is counted by photo sensors and light gates. ‘Grid and cradle’ systems (Range Servant 2010), located at the dispensers or at the central hopper sort out the damaged balls and ensure that the golfer receives only intact balls. Yet complex systems are prone to failure, as demonstrated in the following case report, especially when management systems and sports equipment are not compatible. A typical incompatibility may occur if the sports equipment is larger than the management system for which it is designed.The size of golf balls, however, is regulated (USGA 2010) with a diameter of not less than 1.680 inches (42.67 mm), as is the mass (‘weight’) of not more than 1.620 ounces avoirdupois (45.93 g); the spherical symmetry; the initial velocity, limiting the speed of the ball off the clubface: maximally 76.2 m/s at a launch angle of 75 degrees; the overall distance standard: a ball hit by the Iron Byron (standardised golf swing robot) cannot fly further than 256 metres. For the coefficient of restitution, USGA and the Royal and Ancient Golf Club of St Andrews agreed to limit the coefficient of restitution of a golf ball bouncing off the club head to a maximum of 0.83. The material of golf balls is not regulated. The following case report, experienced by the author of this chapter, illustrates how factors other than size and mass can seriously disturb the smooth operation of a ball management system. A driving range in country A was supplied with range balls from a sporting goods vendor in country B.The range balls were manufactured in country C and the ball management system in country D. After two months of operation, the customers started to complain that the dispensers did not count the balls properly and that they were supplied with fewer balls than selected. Finally, the system stopped working owing to a ball jam in the main hose. Subsequent investigation revealed the cause of the jam, shown in Figure 31.7.The system failed because of two factors: • The incompatibility of the operating temperature of the ball management system (washing water and/or air of blower system) and the melting point of the ball’s cover material. As shown in Figure 31.7, several balls are inseparably melted together and thus are too heavy to be blown through the main hose into the hopper.The ball manufacturer claimed that the ball’s cover material is composed from two different Surlyn® monomers, which, according to MatWeb (www.matweb.com) have a melting point of 90 degrees C. • The disintegrating surface finish polluting the photo sensors at the dispensers. The range balls were supplied with a thin polyurethane surface finish, which adds a shiny touch to the balls. External weather conditions and repeated ball collisions inside the system (washer, blower, hopper, hoses with 90-degree bends, manifolds) caused the surface cover to crack and flake off.A thick layer of polyurethane flakes and dust accumulated beneath the hopper and was blown into the dispensers. The flakes polluted the entire interior of the dispenser units including the surface of the photo sensors and light gates, resulting in false counts of range balls. The ball management system had to be shut down, which caused a severe financial loss to the owner of the driving range.The local company commissioning the system rejected any responsibility and blamed the ball manufacturer.The latter claimed that their balls were manufactured according to the standard prescribed by the rules. The problem offers two solutions: either the ball management system is not suitable for handling these specific range balls or conversely, the balls are not suited for being managed by the system.The three parties involved are the manufacturers of the balls and the ball management system and the owner of the driving range. The following arguments need to be considered for solving the problem and preventing such problems in the future. 491

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Figure 31.7 Golf balls inseparably melted together

• The manufacturer of range balls: the manufacturer of range balls cannot expect that his balls are managed by a ball management system or a specific system.The manufacturer does not necessarily have to manufacture extra-rugged range balls such that they can withstand the ‘harsh’ conditions of any ball management system. • The manufacturer of the ball management system: the manufacturer of the ball management system cannot expect any problem with range balls if he did not test any existing ball, nor foresee problems with balls developed in the future.The manufacturer of the ball management system can provide a list of range balls tested including the test results (if any), warn of incompatible brands of range balls and reject warranty if operations fails when using balls other than the suitable ones identified (for example, in the operation manual and/or contract). • The client: the client (driving range owner) can, but does not have to, ask for guarantee that the balls purchased are suitable for a specific ball management system.The client can ask for guarantee that the ball management system purchased is suitable for a specific ball or any range ball (what the manufacturer of the system can hardly guarantee); alternatively, the client can ask the ball management system manufacturer for their recommendations of suitable range balls. The solution to the problem outlined in this case report was quite clear, as the word ‘golf ’ did not appear in a single instance in the then text of the ball management system brochure nor in the text of the associated operation manual. The images of the brochure, however, showed a driving range with white balls of the size of golf balls. The term ‘ball management system’ is, however, too ambiguous and can theoretically apply to any (sports) ball. 492

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Summary and future outlook The design of sports facilities is driven by art and engineering. Architecture creates the shapeliness and exciting experience inherently connected to stadia and other sports facilities. The engineering behind the architecture – in most instances, not directly visible to the eye – provides the solid backbone in terms of logistics, operation, modularity, safety (structural, fire, injuries), economics and sustainability. The design of winter sports freestyle parks is a typical example of how facilities evolved from more spectacular to more safe sporting events. Reinke’s (2006/2007) design method of maximising the length of the table top hinges on the demand that ‘photographers and spectators commonly want to see extreme flight distances’ (Reinke 2006/2007 [translated]).The prevention of falls, however, is simply regarded a matter of ‘continuous and careful training on jumps with gradually increasing dimensions’ (Reinke 2006/2007 [translated]).Accepting that accidents occur, independent of the training level of the athlete, led to a paradigm shift and to the design of safe landing slopes. Even the best trained élite athletes with maximal body control are prone to accidents. Hubbard’s (2009) design method hinges on an acceptable safe fall height and is independent of the jump take-off speed, which makes it also independent of the overall dimensions of the in-run and kicker. A comparable paradigm shift at a larger scale is seen in the sustainable development of the Olympic Games. As suggested by Furrer (2002), ‘sustainable Olympic Games must contribute to the sustainable development of the host city and region through their economic, social and environmental legacy’. This goes beyond sustainable design of sports facilities, as the International Olympic Committee (IOC) considers the ‘environment as an integral dimension of Olympism, alongside sport and culture’ (IOC 2012). According to the Olympic Charter (IOC 2011), the IOC’s role with respect to the environment is ‘to encourage and support a responsible concern for environmental issues, to promote sustainable development in sport and to require that the Olympic Games are held accordingly’.This was achieved by the organisers of Olympic Games in various ways. In the Sydney 2000 Summer Games, the key environmental achievements included public transport access, solar power applications, good building material selection, recycling of construction waste, energy and water conservation and wetland restoration (IOC 2012). The Vancouver 2010 Winter Games targeted six corporate-wide sustainability performance objectives: accountability, environmental stewardship and impact reduction, social inclusion and responsibility, aboriginal participation and collaboration, economic benefits, and sport for sustainable living (IOC 2012). The London 2012 Summer Games set five priority themes, centred on climate change, waste, biodiversity, inclusion and healthy living (IOC 2012). The Sochi 2014 Winter Games will incorporate the elements of green standard in Olympic venue development, use of alternative energy sources, carbon neutrality, zero waste (IOC 2012).This makes the Sochi Games the first ones which specifically focus on the carbon footprint of Olympic Games.The engineering of sports facilities will play a vital role in achieving the goals set for current and future Olympic Games.

References ARUP (2005) National Swimming Centre Beijing:The Water Cube. ARUP Documents. Sydney: ARUP. Böhm, H. and Senner,V. (2008) ‘Safety in big jumps: relationship between landing shape and impact energy determined by computer simulation’, Journal of ASTM International, 5(8): 1–11. Carfrae,T. (2007) ‘Box of bubbles’, Ingenia, 33: 45–51. Furrer, P. (2002) Giochi olimpici sostenibili: utopia o realtà? [Sustainable Olympic Games: utopia or reality?] Bollettino della Società Geografica Italiana [Bulletin of the Italian Geographical Society] 12(7): 795–830. [Italian]

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Franz Konstantin Fuss Hubbard, M. (2009) ‘Safer ski jump landing surface design limits normal impact velocity’, Journal of ASTM International, 6(1): 1–9. IOC (2011) Olympic Charter in Force as From 8 July 2011. Lausanne: International Olympic Committee. Available online at http://www.olympic.org/Documents/olympic_charter_en.pdf (accessed 12 January 2013). IOC (2012) The Environment and Sustainable Development. Lausanne: IOC (International Olympic Committee). Available: http://www.olympic.org/Documents/Reference_documents_Factsheets/ Environment_and_substainable_developement.pdf John, G., Sheard, R. and Vickery, B. (2007) Stadia: A Design and Development Guide. 4th ed. Amsterdam: Architectural Press/Elsevier. McNeil, J. A. and McNeil, J. B. (2009) ‘Dynamical analysis of winter terrain park jumps’, Sports Engineering, 11: 159–64. Müller,W. (1997) ‘Biomechanics of ski-jumping: scientific jumping hill design’, in E. Müller, H. Schwameder, E. Kornexl, C. Raschner (eds), Science and Skiing. London: E. & F. N. Spon, pp. 36–48. Range Servant (2010) Driving Range Equipment: The right partner for you. Driving range equipment 2012(2). Available online at http://www.rangeservant.net/PDFS/RS_katalog_2010_2_low.pdf (accessed June 2012). Reinke, C. (2006/2007) ‘Vom Springen zum Fliegen [From jumping to flying], SnowSport: das Magazin für Schneesport-Profis, 3: 11–13. [German] Snel, R. (1999) Functional Design Proposal for Grave White water Park, Concept Paper. Nieuwegein: Nederlandse Kanobond. Thomson, W. (1st Baron Kelvin) (1887) ‘On the division of space with minimum partitional area’, Acta Mathematica, 11: 121–34.. USGA (2010) Equipment Rules. Rule 5 – The Golf Ball,Appendix III – The Ball. United States Golf Association. Available online at http://www.usga.org/Rule-Books/Rules-on-Clubs-and-Balls/Appendix-III%e2%80%93-The-Ball/ (accessed 12 January 2013). Weaire, D. and Phelan, R. (1994) ‘A counter-example to Kelvin’s conjecture on minimal surfaces’, Philosophical Magazine Letters, 69: 107–10.

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INDEX

bicycle, power 118 bicycle, racing 111 biofeedback 46, 47, 59, 61 Bird’s Nest 480 blade 146, 147, 150, 152 blade, cleaver 150 blade, macon 150, 151 blade, pencil 150 blade, smoothie 151 blade, spoon 150 Bluesign® standard 22 Bluetooth 86 boat lift 482 boat, speed 146 bounce 420, 433 boundary layer 247, 377, 393 brake 136 brake, disc 137 brake, dynamic 282 brake, figure-of-eight 282 brake, hybrid 284 brake, rappel rack 284 brake, rim 136 brake, semi-automatic 284 brake, static 282 brake, tube 284 bramble 350

accelerometer 49, 65 acid, nitric 17 acid, polylactic 18 acid, sulphuric 17 Adidas 5, 27 aerodynamics 356, 377, 392, 411, 424 air spring 134 airfoil 163 aluminium 15, 16 American football 65, 410 amplifier 54 analogue-to-digital converter 54 anchor 278 angular velocity 64, 66 Ant 86 apparel 233 aramid fibres 12, 336, 341 AstroTurf 455 Atti compression tester 357 Australian football 65, 67, 410 ball management system 489 ball, bounce see bounce ball, rebound 476 ball, roll distance 476 bamboo fibre 19 barefoot shoes 229 baseball 325, 386 baseball, aerodynamics 392 baseball, bat 325 bat performance factor 332 bearing 135 belay devices 281 bicycle, aerodynamics 118, 122, 123 bicycle, derailleur 111 bicycle, frame 112, 114, 115, 131 bicycle, performance model 115

cam profile 288 camming angle 289 camming devices 287 cams 287 carbon fibres 12, 14, 131, 141, 158, 191, 299, 336, 341 carbon footprint 5 carving ski xxv, 60 centre of percussion 307, 310 495

Index drag coefficient xxv, 248, 356, 394, 412, 424, 428, 442 drag crisis 248, 394 drag force 413 drag, aerodynamic 161, 166, 167, 237, 246 drag, form 145 drag, hydrodynamic 145, 237 drag, skin friction 145 drag, wave 145 drive train 138 dynamic stiffness 369, 389

centre of pressure 52, 64, 70 chainring 139 chalk 290 chocks 287 Clegg impact surface tester 476 climbing 277 climbing gyms 480 climbing hold, smart 70 climbing wall, smart 70 climbing, dead pointing 74 CO2 emissions 7 coefficient of friction 75, 117, 162, 285, 290, 318, 417, 433 coefficient of restitution 297, 307, 308, 315, 359, 364, 365–9, 386, 396, 407, 421, 432 coefficient of restitution, apparent 313, 315, 332 coil springs 134 Coliseum 480 collision efficiency 315, 326, 334 composites 3, 12, 14–16, 131, 141, 158, 191, 299, 336, 341 computational fluid dynamics 125, 261, 358, 415, 443 conservation of energy 349 conservation of momentum 350 cork 401, 361 corking 336 cotton 18 court pace rating 434 crankarm 139 crankset 132 crew 141 cricket 89 cricket ball 266, 361 cricket ball, aerodynamics 377, 380 cricket ball, manufacturing 361 cricket ball, smart 64 cricket bats 44 cricket swing 380–3 cricket, bowler 64 cricket, bowling 89 cricket, laws 362 cross-linking 352 cushioning, shoes 216

Eco Circle 20 Ecobambu 18 eco-design 22, 23 Eco-indicator 10, 11 ecological footprint 28 ecowool 18 eco-yarns 18 edge grip 194, 198 efficiency 163 electric charge 53, 54 electrical resistance 48 electrocardiogram 84, 89, 90 electrogoniometers 48 electromotive force 49, 76 energy footprint 5 energy return xxiv, 221, 225 energy return, shoes 221 energy transfer, shoes 226 energy, conservative xxiv, xxvi energy, kinetic 161 energy, loss 161 energy, non-conservative xxiv, xxv, 45 energy, transfer 226 environmental burden 4 environmental impact 5, 10, 11, 29 environmental sustainability 5 ethyltetrafluoroethylene 483 exergaming 94 exertion games 94 fabrics 235 face guard 269 fall factor 278 fall protection 277 fatigue 136 fibreglass 12, 14, 191, 336, 341 fibres, aramid 12, 336, 341 fibres, carbon 131, 141, 158, 191, 299, 336, 341 fibres, glass 191, 336, 341 FIFA (International Federation of Football Associations) 440, 455 FIH (International Hockey Federation) 399 FINA (International Federation of Swimming Associations) xxv finite element model 262, 263, 375

damper 131 damping 131, 133, 134, 198, 310 damping ratio 199, 319 data logger 54 data storage 44 data transfer 44 dead spot 307, 309 deflection 48 design process 112, 113 dimples 356 displacement 48 drag area 117 496

Index FIS (International Ski Federation) 173 FISA (International Federation of Rowing Associations) 141, 147 flow regimes 377–8 flow separation 382 flow transition 248, 377 foam 218, 223 force plates 52 force strengthening 417 force vector 52 force vector diagram 63, 71, 72, 74 force weakening, 417 Formula ‘1’ 43 fractal dimension 45, 68 free-body diagram 55, 312 frequency 131 frequency mode 309, 327 frequency response function 199 frequency, natural 131 friction coefficient see coefficient of friction friction, belt 282, 285 friction, directional 419 friction, kinetic 174, 175, 184 friction, rolling 161, 162, 166 friction, skin 249 friction, static 174, 184 friction, xxv 417 frictional anchors 287 Froude number 145

helmet 252, 263 helmet, bicycle 253–62 helmet, cricket 263–5, 268 hockey 399 hockey balls 399, 401 honeycomb structures 191 hot-spot analysis 30 hotspot xxv hull 145, 146 hydrogen peroxide 17 hydrophobicity 175, 183 hyperelasticity 359

gamma function 364 glide ratio 163 global economy xxiv global market 3 global sports industry xxiv GoingStick 475 golf ball 349 golf ball, aerodynamics 356 golf ball, core 352 golf ball, cover 354 golf club 295 golf, driver head 297 golf, driving range 489 golf, putter 303 golf, range balls 489 golf, shaft 296 golf, swing 296 GPS 48, 60, 84 greenhouse effect 483 greenhouse gas 17, 34 gross domestic product xxiv gunwale 141, 148 gutty 350 gyroscopes 49, 64, 66

Jabulani 441 jogging 102–6

IAAF (International Association of Athletics Federations) xxvi ice hockey, puck 339 ice hockey, skate 202, 203 ice hockey, stick 339 impact 262 inclinometers 48 inertial measurement unit 50, 76, 84 instrumentation 43, 59 interferometer 51 intrinsic power 315 inventory tree 8 IOC (Olympic Committee) 493 IPC (International Paralympic Committee) 156, 163 ITF (International Tennis Federation) 423

karabiner 279 keronite 139 kevlar 191 knuckling effect 448 laser vibrometer 198 lay-up 149, 150 lead alloy 11 life cycle 7 lift coefficient 356, 395, 424, 442 linkage, four-bar 132 linkage, shock 133 load cells 52 lubrication 175, 176 magnesium carbonate 290 magnetometers 48 Magnus effect 395, 427 Magnus force 395, 427 maple 326 materials inventory 9 measurement chains 54 MEMS 48 microcontroller 44

Hausdorff dimension 69, 70, 79 Hawk Eye system 49 497

Index Riehle tester 357 RMIT Superbike xxvi, 111–28 rock climbing 70 rock protection 287 rope 277 rope brakes 281 rowing 141 rowing boat 67 rowing, shell 141, 142, 149 rowlocks 141 rugby 410 rugby ball 410, 417

microelectromechanical systems 48 microsensors 83 moment of inertia 44, 312, 326, 327 moment vector diagram 63 monocoque 142, 148 mountain bike 130 mountaineering 277 nanoparticles 12, 13 natural gut 315 Nike 5, 97 Nintendo 95 nylon 316

sandwich composites 13 Sankey diagram 36 scull 150 sculler 146 seatstay 132 sensors 44 sensors, capacitative 51 Shannon entropy 68 shock absorption 462 shock spike 216, 219 shoes, barefoot 229 shoes, cushioning 216 shoes, energy return 221 shoes, rearfoot control 221 shoes, running 215 silica sand 14 ski 55 ski jump 485 ski, binding 173 ski, camber 172 ski, carving 60 ski, cross country 171, 172 ski, jump 90, 91 ski, running surface 172, 175 ski, smart 59 ski, speedometer 60 ski, wax 174, 178 skiing style 171 smart equipment 45, 59 smart phones 51, 77 SmartWheel 52, 75 snickometer xxv snow cities 480 snow, dry 180, 181 snowboard, camber 190, 194 snowboard, performance 194 snowboard, rocker 190 snowboard, xxv 189 soccer ball, 439 soccer ball, aerodynamics 442 softball 325, 386 softball, bat 325 soil 471, 472 specific energy consumption 37

oar 67, 143, 147, 150, 152 Oeko-Tex® Standard 22 optical encoder 48 optical navigation technology 48, 60 Oscar Pistorius xxvi Össur Cheetah blades xxvi oval ball, smart 65 oval balls 410 pedal 133 PEEK 315 performance 44 piezo shunts 310 piezoelectric effect 52, 53 piezoelectric material 199 polybutadiene 352 polyester 18, 316 polytetrafluoroethylene 177 polyurethane 401 polyvinyl chloride 191, 401 potentiometer 48 power meter 52 power spot 307, 308 precession 66, 67 pressure angle 288 probability distribution 68, 79 product design 46 product life cycle 28 Prony series 359 Puma 6 punt 66 quality function deployment 165 R&A (Royal and Ancient Golf Club) 350 radius of gyration 312 rearfoot control, shoes 221 rebound power 315 recoil factor 326 Reebok 5 regatta 141 Rényi entropy 68, 69 Reynolds number 248, 377, 393, 414, 432, 447 498

Index transmitters 44 tribology 173 turf, artificial 455, 456, 461 turf, backing 458 turf, grass 467 turf, natural 467 turf, soil 470 turf, tufting 459 turf, yarn 456

speed climbing 73 Speedo swim suit xxv spider 139 spin axis 64, 66, 67 spin factor 395 spin rate 64, 66 spoke 135 sport ball, smart 62 sporting goods market xxiv sports apparel 18 sports facilities 480 sportswear 22 stadia 480 stanchion, 134 stiffness, bending 197, 296, 341, 343 stiffness, torsional 197, 296 strain gauges 48, 50 Stribeck curve 175 string dampers 310 suit 243 supply chain 5, 20 supply loop 21 surface pace rating 434 surface roughness 237, 248, 378, 379 suspension 130, 133, 134 sustainable design 3, 4, 7 sustainable manufacturing 14, 27, 29, 30 sustainable recycling 16 sweet spot 307 swimming 89 swing weight 313, 329 SwingerPro 436

UCI (International Cycling Union) 111, 161 USGA (United States Golf Association) 350 vector diagram 63, 71, 72, 74 velocity strengthening 417 velocity weakening 417 venting 253, 256 vibration damping 318 vibration node 307, 309 vibrations 198 viscoelasticity 352, 364 viscosity parameter 364 waste reduction 5 Water Cube 483 Weaire-Phelan foam 483 wettability 177 wheel 135 wheel hop 131 wheelchair 156 wheelchair, ball sports 164 wheelchair, caster 160 wheelchair, chassis 157 wheelchair, ergometer 166 wheelchair, frame 157 wheelchair, handlebar 160 wheelchair, push rim 160 wheelchair, racing 161 wheelchair, seat 157 wheelchair, smart 75 wheelchair, tyre 158 whitewater parks 481 wind tunnel xxvi, 111, 117, 122, 124, 166, 238, 248, 258, 273, 358, 379, 393, 412, 425, 442 winning time 45, 161 wireless network 86

Teamgeist 441 tennis ball 423 tennis balls, aerodynamics 424 tennis court 432 tennis racquet 7, 306 tennis strings 315 tenpin bowling ball 44 tenpin bowling ball, smart 62 textile, fibres 18, 234 textiles 17, 234 thermal comfort 257 thermoplastics 12, 15 thermoset resin 12 titanium 15, 297 torque 64 trampoline effect 329 transducer 44, 51

Xbox Kinect 95 Zigbee 86

499