Damping of Vibrations [1 ed.] 9789054106760, 9789054106777, 9781315140742, 9781351456678, 9781351456661, 9781351456685

This monograph seeks to strengthen the contributions of Polish scientists and engineers to the study of problems of mech

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Damping of Vibrations [1 ed.]
 9789054106760, 9789054106777, 9781315140742, 9781351456678, 9781351456661, 9781351456685

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Damping of Vibrations Edited by

ZBIGNIEW OSINSKI Institute of Machine Design Fundamentals, Warsaw University o f Technology Warsaw, Poland


Authors: Jerzy Bajkowski, chapter 3; Zbigniew Di}browski, chapter 15; Jaroslaw Dyk, chapter 10; Jan Freundlich, chapter 4; Krzysztof Gol6s, chapter 5; Stanislaw Karczmarzyk, chapter 4; Grzegorz Klekot, chapter 15; Andrzej Kosior, chapter 7; Wlodzimierz Kurnik, chapters II, 12; Jerzy Osinski, chapters 4, 9, 10; Zbigniew Osinski, chapters I, 2, 4, 6; Jerzy Pokojski, chapter 14; Stanislaw Radkowski, chapter 15; Danuta Sado, chapter 10; Zbigniew Skup, chapter 8; Zbigniew Starczewski, chapter 12; Andrzej Tylikowski, chapter 13; Jerzy Wrobel, chapters 7, 14

The publication is financially supported by Polish State Committee for Scientific Research and reviewed by: Roman Bogacz & J6zef Giergiel Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by A.A. Balkema, Rotterdam, provided that the base fee of US$1.50 per copy, plus US$0.10 per page is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, USA. For those organizations that have been granted a photocopy license by CCC, a separate system of payment has been arranged. The fee code for users of the Transactional Reporting Service is for the hardbound edition: 90 5410 676 X/98 US$1.50 + US$0.1 0, and for the student paper edition: 90 5410 677 8/98 US$1.50 + US$0.1 0. Published by A.A.Balkema, P.O. Box 1675,3000 BR Rotterdam, Netherlands Fax: +31.10.4135947; E-mail: [email protected]; Internet site: http://www.balkema.nl A.A. Balkema Publishers, Old Post Road, Brookfield, VT 05036-9704, USA Fax: 802.276.3837; E-mail: [email protected] ISBN 90 5410 676 X hardbound edition ISBN 90 5410 677 8 student paper edition © 1998 A.A.Balkema, Rotterdam Printed in the Netherlands

Disclaimer The publisher has made every effort to trace copyright holders and welcomes correspondence from those they have been unable to contact.





Nom enclature


1. V ibration damping - basic concepts 1.1 Introduction 1.2 Measures of damping used in vibration damping theory 1.3 Hysteresis loops of vibrating systems References


2. V ibration damping by viscous resistance 2.1 Conditions of alternating motion of a viscously dam ped mechanical system 2.2 C riteria for alternating m otion in systems with non-linear characteristics of damping and elasticity forces 2.3 Free vibration damping by a non-linear resistance 2.4 The effect of non-linear damping on forced vibration 2.5 V ibration damping by elimination of non-harmonic resonances References 3. V ibration damping by dry friction 3.1 Introduction 3.2 Dry friction forces 3.3 Examples of mechanical systems with dry friction 3.4 Shock absorbers as an example of modelling systems with dry friction 3.5 M athem atical description of the shock absorber model 3.6 M otion analysis of the linear absorber model References


15 19 22 30 31 31 34 48 51 58 68 70 70 75 78 85 89 91 95



4. V ibration dam ping by internal friction 4.1 Introduction 4.2 Equations of motion of a vibrating system with internal friction 4.3 Description of vibration damping of composite structures w ithin classical theories 4.4 Description of vibration damping of viscoelastic composite structures within the linear theory of elasticity 4.5 Damping modelling of lam inated structures by FEM 4.6 Modelling of damping properties of a machine foundation References 5. Energy dissipation in metals under cyclic loading 5.1 Introduction 5.2 Description of cyclic deformation in m etals 5.3 Cyclic stress-strain curve. Hysteresis loop 5.4 Fatigue life under the condition of uniaxial cyclic loading 5.5 M ultiaxial fatigue damage energy criterion 5.6 The cumulative fatigue damage hypothesis in the case of stepwise loadings 5.7 Modelling of the fatigue crack growth rate in term s of energy References 6. V ibration dam ping by structural friction 6.1 S tructural friction in tensile and torsional systems 6.2 S tructural friction in a tightening joint under stretching 6.3 S tructural friction at bending 6.4 Free vibration of a system clamped in a tightening joint with structural friction taken into account References 7. V ibration dam ping by structural friction in a system with clamped strip and a set of two beams 7.1 Introduction 7.2 Model of the system with a dam ped strip 7.3 Model of system with a set of two beams 7.4 Exact analytical m ethod of examining free vibration of the system with the beam set 7.5 Approxim ate m ethod of analyzing forced vibration of the system with the beam set 7.6 Numerical simulation of free and forced vibration of the system 7.7 The system with a dam ped strip as a vibration elim inator 7.8 Elasto-frictional contact of the strip/base system 7.9 Experim ental examinations of the set of two beams under bending References

97 97 113 124 143 165 169 172 176 176 176 179 182 191 201 209 215 220 220 220 233 238 244

245 245 245 247 250 254 257 260 262 279 280


8. V ibration damping by structural friction in drive systems with multiple-disc clutches 8.1 Introduction 8.2 S tructural friction in the outer disc of a clutch 8.3 Structural friction in the inner disc of a clutch 8.4 Influence of the system param eters on vibration dam ping by structural friction 8.5 Forced vibration of the system with a friction clutch 8.6 Starting-off stage of the system under harmonic excitation 8.7 Starting-off stage of the system under stochastic excitation 8.8 Torsional vibration of the clutch system under a random torque-type excitation 8.9 Energy dissipation in clutch systems with electrorheological fluids References 9. V ibration damping by structural friction - analysis incorporating the finite element m ethod 9.1 Introduction 9.2 Rem arks on the finite element m ethod analysis 9.3 Ways of applying the load 9.4 Edge growth of the pressure and modification of the joint 9.5 Modelling of structural friction damping in discrete-continuous systems 9.6 Examples of numerical simulations of vibration dam ped by structural friction References


283 283 284 295 300 305 309 312 319 325 328

333 333 336 347 348 351 356 357

10. Damping of param etric and autoparam etric vibration 10.1 Introduction 10.2 Param etric vibration damping in systems w ithout external loading 10.3 Param etric vibration damping in systems subjected to a constant external loading 10.4 Param etric vibration damping in toothed gear systems 10.5 Free vibration damping in a two-degrees-of-freedom system with autoparam etric coupling 10.6 Effect of damping on forced vibration of a two-degrees-of-freedom autoparam etric system References

359 359 360

11. Stability and self-excited vibration of shafts 11.1 Introduction 11.2 Equations of autonomous transverse vibration of shafts 11.3 Supercritical bifurcation of a shaft with Von K arm an’s nonlinearity 11.4 Self-excitation hysteresis of a physically non-linear shaft

392 392 394 396 405

363 365 368 382 389



11.5 B razier’s effect in thin-walled shafts 11.6 Internal damping in multi-layer shafts 11.7 Summ ary References 12. R eduction of vibration of rigid rotors with slide bearings 12.1 Introduction 12.2 Dynamic properties of a slide bearing treated as a viscoelastic support element 12.3 Harmonically forced vibration of symmetric rotors 12.4 Vibroisolating properties of the slide bearing system 12.5 V ibration of an asymmetric rotor under harmonic excitation References 13. Active vibration damping in continuous elements of structures and machines 13.1 Introduction 13.2 A concept of active damping by distributed piezoelectric elements 13.3 Stabilization of beam param etric vibration - the model of concentrated moments 13.4 Stabilization of beam param etric vibration - the model with perfectly attached piezolayers 13.5 Stabilization of beam param etric vibration - the model with shear forces in the glue layer included 13.6 Numerical sim ulation of the stabilization of beam param etric vibration 13.7 Stabilization of plate param etric vibration 13.8 Concluding remarks References 14. O ptim ization of dam ping in machine dynamics 14.1 Introduction 14.2 Damping optimization criteria 14.3 Decomposition of large optim ization problems related to generation of dam ping characteristics 14.4 Example of m ulti-objective optim ization of elastic and dam ping properties of a linear vehicle suspension model 14.5 Example of a large optim ization problem related to generation of dam ping and elastic characteristics References

411 412 420 421 423 423 424 430 437 440 447

449 449 451 453 460 465 468 470 474 474 477 477 477 478 479 482 486




A ttenuation of noise in mechanical systems 15.1 Introduction 15.2 Minimization of noise generated by a complex source with with a toothed gear system as an example 15.3 Identification of the vibroacoustic structure of a machine 15.4 A ttenuation of noise of a heavy machine with a hydraulic excavator as an example 15.5 Comments on noise-absorbing structures References

488 488



499 530 534 552 555


Problem s concerned with vibration damping have been a subject of comprehensive interest of scientists and researchers for a long time. Notw ithstanding the variety and im mensity of work done within this domain of study, and despite all possibly m ost accurate solutions and arduous experiments, m any aspects related to dam p­ ing rem ain poorly examined. Some of the reasons responsible for this situation lie in the irreversibility of damping processes, the subsequent energy dissipation and the difficulties encountered in a true reconstruction of the physical characteristics of the phenom ena. A m athem atical description of non-linear forces corresponding to dam ping resistance forces, usually formulated as hypotheses, does not always truly em body the physical and technical reality. V ibration damping still unfolds as a challenging field for theoretical and experimental studies, though researchers are not provided with ever more sophisticated m ethods of investigation. The authors of the present work intend to present their own research achieve­ m ent in this book: some possibly general contributions to certain problems, and com plem entary or new and more precise investigations of some other problems. In order to understand the phenomenon better some basic knowledge appM ng to vibration dam ping is enclosed. It is yet limited to a minimum, leaving only expla­ nations of the essential entries. This work will neither provide the reader with everything th a t is related to vibration dam ping problems, nor will it supply the reader with a survey of all fun­ dam ental theories, m ethods and experiments. The book is m eant for scientists and researchers from institutes dealing with the problems of vibration damping. Engineers, designers, industrial researchers, and machine operators can find innovative formulations and solutions to new problems or to problems th a t have been treated, up to now, in a different way. The authors are aware of the danger th at none of the readers can find a complete solution to the problem he is now involved in. The main intention is to indicate the course of a procedure toward specific and individual tasks through which new m ethods and approaches related to sophisticated research programs can be developed and improved. The other goal of the book is to popularize and strengthen the consider­ able contribution of Polish scientists and engineers to the study of the problems of m echanical vibration. Their share is supported by the reference lists at the end of each chapter. The references consist mainly of the output of the original authors. The first two chapters generalize definitions of measures employed in vibration 11



dam ping, summarize conditions and criteria for m otion to be described as alter­ n ating m otion when accompanied by viscous damping. In the chapter 3 some problems concerning vibration brought about by dry friction are discussed. A part from a classical description, one can find m athem at­ ical relationships in the form of multiple-valued functions expressing the friction forces, w ith shock absorbers taken as examples for the interpretation of the derived formulas. The next chapter is dedicated to internal damping. The eigenproblems of com­ posite structures, relationships between damping decrements and m aterial loss fac­ tors are presented in this p art of the book. C hapter 5 focuses on energy dissipation phenom ena in m etallic elements under cyclic loading. The following problems are included: damage energy criteria for m ultiaxial cyclic loading, damage hypothesis for stepped loads, modelling of the speed of a fatigue fracture expressed in term s of energy. T he problems of vibration damping as a result of so-called structural dam ping are widely discussed in the next four chapters. V ibration dam ping phenom ena in models reflecting different structures and their m ountings are thoroughly treated. Beam- or shaft-like systems and drive systems are often employed in the research. Various m ethods, such as analytical m ethods, numerical simulations and the finite element m ethod, are used to obtain solutions. C hapter 10 presents the modelling of damping in the case of param etric and autoparam etric vibrations of lam inated machine elements. In chapter 11 the problem of self-excited vibration of rotating lam inated shafts is analyzed. The self-excited vibration manifested by bifurcation of the equilibrium position is conditioned by the presence of internal friction. P articularly interest­ ing are the problems of geometric and m aterial non-linearities as well as relations between the resultant internal damping coefficient of the shaft and the physical properties of the lam inated components, including their arrangem ent. T he m aterial enclosed in chapter 12 reflects damping and vibroisolating proper­ ties of rigid rotors supported by slide bearings. The studies of resonant character­ istics reflecting the behaviour of asymmetric rotors allow to propose new boundary conditions for oil film in slide bearings. In chapter 13 the theoretical rudim ents of so-called intelligent structures are dis­ cussed. Those structures are equipped with control elements, and sometimes also w ith a supply system. The essence of the discussion consists of examining initial disturbances of param etric excitation, which provides the system with energy. The energy has to be dissipated by an active damping subsystem. The dissipation capa­ bility of the active damping is proved on the assum ption th a t the excitation force is constituted by a broad-band Gaussian process. The general analytical condition for param etric vibration stabilization in the form of a variational inequality is derived. T he problem of optim al design of damping characteristics for mechanical sys­ tem s with special emphasis on the decomposition of a optim ization task can be found chapter 14. The last chapter is dedicated to problems related to noise attenuation. The m ain interest focuses on vibroisolating structures, their proper design, and mini­ m ization of noise emission in its source. The necessity of simultaneous analysis of the vibration- and the acoustic-type of energy propagation is pointed out.


A , A, A n , Aij B Bij C ,C,Cn D, D 77* H fj1 fjl Tp ) &p fj1 hj, j%j, -c/o)

— —

F, Fj, F G, Gij H T ,H,H{) T, T0 C T.u.tfy

— — — — — — — -

^ (•) fl) j ^ Mi C, Ccr, Cj d e, e / 9 h k , j kg lylpilk m, m tn*,m

— — — — — — -

m atrix of a system, am plitude, equation coefficients m atrix of a system, am plitude, equation coefficients m atrix, integration constant, contact stiffness m atrix of damping, diameter Young’s modulus, Young’s m odulus of a orthotropic m aterial, initial energy, kinetic energy, potential energy forces, cross-section areas Kirchhoff’s modulus force vector, force, the Heaviside function identity m atrix, mass moment of inertia geometric, polar moment of inertia stiffness m atrix, coefficients inertia m atrix, moment of forces norm al forces, number of loading cycles m atrix of forces, forces, probability elements of stiffness m atrix curvature radius spectral density vibration period, friction force supply voltage, m atrix of displacements, displacement energy density functional, functional of friction force dissipation the Fourier transform am plitude, equation coefficients, acceleration am plitude, equation coefficients damping coefficients diameter eccentricity, element of a probabilistic space function acceleration of gravity damping coefficient stiffness coefficients length mass num ber of cycles 13


Nom enclature

P>Po,Po £t, £o V) r) G m

— — — — — — — — — — — — — —

£ 0, 0(.) A, A*

— — — —

v £,£(£,x) p cr, cr2(.) r < p

— — — — — — — — — — — —

A $ Q, V

coefficient, pressure orthogonal eigenvectors, unitary friction force curvature radius, eigenvalues of a linearized system displacement coordinate time phase variables m atrix, displacement deflection of a shaft deflection of a shaft, velocity displacement coordinates angle, coefficient, m aterial constants angle, coefficient, vibration decrement, m aterial param eters angle, dimensionless damping coefficients logarithmic damping decrement, Kronecker’s delta strain, angular acceleration resistance coefficient of the Kelvin-Voigt model vibration frequency ratio, loss coefficient param eter of a curve oil film boundary, function vibration mode component, Lame conditions friction coefficient, actual oil viscosity, stiffness m odulation coefficient, frequency ratio Poisson’s ratio, excitation frequency displacement of journal center, displacement function density, angle stress vector, norm al stress, variance time, dimensionless friction coefficient angle dynamic param eter of a bearing, hysteresis loop area frequency of damped and undam ped vibration longitudinal displacement angle, allowable area set of allowable values nabla operator


Vibration damping - basic concepts

1.1 INTRODUCTION V ibration damping is one of the m anifestations of the dissipation of mechanical energy related to motion in mechanical systems. Damping processes have been studied for a long time. The development of research on problems related to vi­ bration entails studying its damping. Admittedly, those studies lag behind other aspects of the investigation of vibrating motion. It so happens th a t dam ping forces are small compared to other interactions present in a mechanical system, yet their m athem atical description remains much more complicated. Therefore, as a first approxim ation those forces are neglected. As a next step viscous linear dam ping is introduced which makes the analysis of the vibration process easier. Actually, any process responsible for the occurrence of damping is very intricate and the knowledge of it is insufficient. At the same time the role of dam ping is significant, though not always fully appreciated. Machine elements undergoing vibration can suffer from some harmful phenomena which will be given in the following list: 1. Disturbances in the proper working of a machine. Excessive vibration can lead to the faulty performance of a machine or device. Heavy machinery can serve here as an example of such a situation. We observe severe vibration in crane-like machines, such as building and assembly cranes used for fitting up elements of buildings, ships, etc. Vibration of a crane jib or a line can cause undesirable motion of an element while being m ounted, thus m aking it difficult and letting it take too long to fit it. Similar problems can appear in mine hoists where the vibration of a cage hinders bringing it to a fast standstill on the proper level. In machine tools vibration is responsible for a poor surface roughness. Vibration of plane panels and steering elements can affect the ability to control an airplane. V ibration of cable railway cars can result in failures, seizure of elements or any other sym ptom s of im proper performance. Valve vibration can cause leaking. Vibration contributes to the disconnection of screws or tightening joints. 2. Shortening of the fatigue life of a machine. Vibrating m otion is the m ain cause for the appearance of variable stresses in machine elements. This can lead to damage of fatigue character. Especially sensitive to such a kind of failure are 15


Damping of vibrations

th e elem ents in w hich the stress tends to concentrate. T h e fo llo w in g m ach in e elem ents, fo r exam p le, are often exposed to fa tig u e cracking: shafts, bearings, ro to r blades, welds. D am age o f th in -w a lle d elem ents, especially car bodies and fuselages o f airplanes, also poses an exam p le o f the h a rm fu l influence o f v ib ra tio n . N a tu ra lly , a ll the elem ents used for w o rk in g in v ib ra tin g m o tio n , such as suspension springs, e x h ib it accelerated w ear sym p tom s. V ib ra tio n can in d ire c tly lead to w ear o f other im p o rta n t m ach ine elem ents. For instance, the transverse m o tio n o f ro ta tin g shafts im plies a n o n -u n ifo rm pressure d is trib u tio n betw een the teeth in gears. T h e sam e effect causes the fast w ear o f slide and ro llin g bearings. V ib ra tio n causes w ear o f flexib le el­ em ents m ade o f ru b b er or polym ers. I t fastens the subsidence o f m ach in e fo u n d a tio n leadin g to undesired loading d istrib u tio n s, thus exposing the m a ­ chine to dam age or a t least lim itin g its fatig u e life.

am plitude

3. Harmful influence on the human body. V ib ra tio n o f m achines and vehicles has a h a rm fu l influence on the h u m a n organism . V ib ra tio n o f ro ad , off-ro ad and ra ilw a y vehicles poses a w ell-kno w n and very serious p ro b lem fo r designers. V ib ra tio n is brought ab o u t by surface irreg u larities or by the w o rk in g o f the engine. In airplanes the fuselage constitutes a n a tu ra l w ay o f tra n s m ittin g v ib ra tio n to crew and passengers. Low -frequency v ib ra tio n s , such as those re la te d to p itch in g and sway o f ships, are h a rm fu l as w ell (seasickness). A n a l­ ogous m o tio n can appear in tow er cranes, and, consistently, in drivers cages. V ib ra tio n o f build ings induced by m achines noxiously affects th e people w o rk in g there. A n in d iv id u a l and com plex p ro b lem is posed by percussive tools and th e ir in ju rio u s influence on the o p eratin g personnel. T ir in g and w ea ry in g effects can be caused by v ib ra tio n o f steering elem ents (th e steering w heel in a car). V ib ra tio n o f gases as w ell as o f m echanical elem ents can be a source o f noise. E v ery m achine, vehicle or sim ple device em its noise h a rm fu l and harsh fo r people w orking in its close neighbourhood. T h e in te n s ity o f noise em ission peaks a t resonance. C onsistently, one avoids such a s itu a tio n by designing o p era tin g p aram eters so th a t they cannot coincide w ith n a tu ­ ra l frequencies corresponding to m achine elem ents. V ib ra tio n outside o f the resonant zone is o f negligible significance (sm a ll am p litu d e s ), see F ig . 1.1.

d a m p in g o f negligible influence

d a m p in g o f / sign ificant ^ / influence £

>1__________ hL. frequency

Figure 1.1. Effect of damping on resonant vibration

Vibration damping - basic concepts


The role of damping is so often disregarded for the following reason: it is some­ times stated th a t a proper assortm ent of operating frequencies satisfactorily solves the problem of vibration and noise avoidance. Naturally, the proper design of op­ erating param eters is one of the fundamental principles to be fulfilled, though it does not settle the problem entirely. Almost every machine is characterized by m any natural frequencies. In practice, only the main resonances can be removed by design. Still, there can be some elements which reveal resonant behaviour if excitation goes through a broad spectrum of frequencies. For instance rotor blades can have different natural frequencies, thus the threat of resonance by one of them is not excluded during starting of the rotor. In vehicles we observe many resonant frequencies, and the excitation usually has a very broad spectrum . The presence of dam ping forces is necessary in this case to secure the safe perform ance of a sus­ pension system. Damping decreases the vibration am plitude, thus m aking it less hazardous. This fact often escapes from the attention of designers and engineers. Sometimes a machine element with slight damping properties works near a reso­ nance zone and quickly gets spoiled, which immediately depreciates the reputation of a producer or manufacturer. Another im portant process in which the role of dam ping cannot be neglected is the process of passing through the resonance zone. It should be noted th a t m any machine elements have to work at frequencies higher than those corresponding to a resonance. Poorly dam ped elements then exhibit transitory vibration with very high amplitudes. Artificially increased damping soothes the vibration, m aking it harmless (see Fig. 1.2).

w eak dam p in g



I \






||l^ tim e


st a r t-o ff strong dam p in g .



A 1 \ Aa a - V / V,/ ” tim e

Figure 1.2. Effect of damping on passing through resonance A nother instant of vibration excitation is an impulse. Any element activated to vibrate th a t way undergoes free vibration. In this case strong dam ping stifles the m otion (Fig. 1.3).

Damping of vibrations


Low d am p in g prolongs the free v ib ra tio n , so disturbances in the p erfo rm an ce o f a m ach ine (e.g. a g rid e r) last longer. A t the same tim e the n u m b er o f load in g cycles increases, w hich results in quicker appearance o f a fa tig u e fra c tu re (c u m u la tiv e fa tig u e d am a g e). In the case o f repeated im pulses a t a low level o f d a m p in g there is a greater p ro b a b ility th a t the effect o f one im pulse combines w ith th e effect o f a n o th er im pulse (F ig . 1.4 ). b)



.2 *4—•

U I -2 M— TD

t im e stro n g d a m p in g

ifl CL>

t im e


im p u ls e

iml p| u l s e

w e a k d a m Pin S

F ig u re 1.3. D a m p in g o f im pulses a)

tim e 2 (see F ig . 1.6) fo r w hich the v ib ra tio n a m p litu d e is \ / 2 less th a n the resonant one. N o w the loss fa c to r can be calcu lated as: 02 — oi ! ? = — ----------

&0re s


(1 .1 4 )


In th e stu d y o f v ib ra tio n , hysteresis loops as measures o f energy dissip ation are very often a pplied. A hysteresis loop can be obtained by recording th e m a g n itu d e o f a force versus the displacem ent b rought abou t by its actio n. I f there exists any d a m p in g in the system then the graphical rep resentation o f this recording resembles a loop. T h e area enclosed by the loop is p ro p o rtio n a l to the a m o u n t o f energy dissi­ p a te d in the system w ith in one period. T h e re are different possible in te rp re ta tio n s o f hysteresis loops according to the k in d o f force recorded d u rin g m easurem ents.

Vibration damping - basic concepts


Let us introduce the three following types of hysteresis loops: a) External loop — reflecting the relation between externally applied force and dis­ placements measured in the system. b) Internal loop — if displacement is related to internal forces appearing in the systems. c) Damping hysteresis loop — when damping forces are taken into consideration. Let us regard the following equation of motion of a vibrating system:

x = Fi(x , i) + Fe(t)


where F i (x , x) and Fe(t) correspond to internal and external forces, respectively. The external and the internal loops can be then described by: Fe = Fe(x)


Fi = Fi(x) We can write down the following relationship between them:

Fi(x) = x(x) - Fe{x)


Owing to this relationship we can convert the external loop into an internal subtracting the values corresponding to the external loop from acceleration dent on displacement. M ultiplication of (1.17) by dx = i d t enables one to the infinitesimal work done by existing forces. Integration over the entire leads to:


F{[x(t )] x(t) dt = f

x(t) x(t) dt — f




Fe[x(t)] x(t) d£

one by depen­ obtain period


The first integral on the right-hand side of equation (1.18) equals zero for periodic motion. Hence, we can derive the well-known formula:

L i= L e


which says th at dissipated energy is equal to the work done by external force. If it is possible to split the internal force into a restitution (elastic or gravita­ tional) and a damping force, i.e.:

Fi = F,(x) + Fd(x)


then for a given restitution force Fs(x) we can determine the dam ping hysteresis loop:

Fd(x) = Fi(x) - Fs(x) = x(x) - Fe(x) - F,(x)

( 1.21)


Damping of vibrations

For a very slow motion, i.e. at x « 0 and x « 0, we obtain a static hysteresis loop. Consider a linear system with viscous damping and harmonic excitation: x + 2 h i + wjjx = qs i n vt


In this case we have: Fe =

q sin i/t




Fd =

-2 * i

Fi =

—Wga; —2/ii

x =


qC sin(i/l —6)

where: 1

C 6


\Z(^o - I/ 2)2 + (2hi/)2 —2hv arctan (1.24) qCv cos(i/f — 6) = ± v \ J q 2C 2 — x 2




—gCV2sin(i4 — 6) = —i/2x

The external loop can be expressed by: F e = (u;2 — v 2)x ± 2h v \ J q 2C 2 — x 2


For the internal one we write: Fi = —o;2x ± 2h v\ Jq 2C 2 —x 2


while the dam ping loop is given by: F d = T 2 h v s / q 2C 2 - x 2


The dam ping hysteresis loop has the form of an ellipse:

(afe) +fe) =1

(1 28 )

The internal loop can be obtained by adding —to2x to ordinates of the ellipse. The external loop can be found by adding (—a;2 + v 2)x to the ellipse ordinates. All the hysteresis loops are presented in Figs 1.7, 1.8 and 1.9.

Vibration damping - basic concepts





100 -2,8 \







' ) 2,8 *


250‘ \




50 t




- 1,2

Q = 100 Wa= 70



i---------- r - - \

V -*50- \

i--------- i



2,8 *

\ \



C; 700


Fig u re 1.7. Hysteresis loop fo r viscous dam p ing

L e t us now consider a v ib ra tin g system w ith d ry (C o u lo m b ) fric tio n . T h e m o tio n o f the system is ru led by the equation:

x + T s g n x + w 2x =

10 0 -



100 -







\ « \ _________________ < i— i ^

i -0,6

\0 ,6

1,4 x


6 0 -



t -1.4






0,4 v



-40 -

-60-8 0 -10 0-

0) _ _ _ _ ------- --






Io i °>'

1 rV^ T1



- 20 ;


F ig u re 1.8. Hysteresis loop for viscous d am p ing

F ig u re 1.9. Hysteresis loop fo r viscous d am p ing

L e t us now look a t an exam p le o f a system w ith s tru c tu ra l d am p in g . C onsider a rod clam p ed a t one end and equipped w ith a lu m p ed mass a t the oth er. Its v ib ra tio n in the lo n g itu d in a l direction (ro d subjected to compressive or tensile lo a d in g ), to rsio n al or transverse v ib ra tio n , can be expressed by the follo w in g set o f equations:

un+1(Fi) = un(Fin) + mu = —Fi + Fe

+ {F' ~ / m )2 s g n f i

(1 -3 0 ) (1 .3 1 )

where m denotes the mass of the vibrating body, F{ and Fe are the internal and

Vibration damping - basic concepts


e x te rn a l forces, respectively, u the displacem ent (deflection, angle o f to rsio n ) and C , D constants depending on the geom etry o f the system and fric tio n forces ap­ pearin g there. E q u a tio n (1 .3 0 ) defines the relationship betw een th e displacem ent and in te rn a l force. E q u a tio n (1 .3 1 ) describes the m o tio n o f the system subjected to a force Fe and the in te rn a l force o rig in atin g fro m d am p in g and elasticity.


q= 10


T- 6

v=5 --------


q= 10 T=6

Q) i 8


1 0 ---------, „ z z —


-0,10 -0,05

= 0


: —

Q 0 5 0,10


_ S ------------------------------------------------------------------------













co0 = 10 v= 15



) 0




■,0 E -0,04

= -0,02





S 0,04



10 0 -20



-1 0

-0,10 -0,05



0,10 0,15














d>F, d>,

20 10





-10 -20

-4 -0,10




0,10 0,15


e)F 20 10


0 -10

-2 0





0,10 0,15


0 -4 8



F ig u re 1.10. Hysteresis loop for dry fric tio n F ig u re 1.11. Hysteresis loop for dry fric tio n

Damping of vibrations


q - 10


T= 1

co0 = 5 v = 0,4



— -0,3









-s I -





1 ... I

-0,3 -0,2




-0,3 -0,2




-0,3 -0,2




c) 10

0 -10

d> F 10

0 -10

F ig u re 1.12. Hysteresis loop for dry fric tio n

F o r the needs o f v ib ra tio n analysis o f the considered system le t us express equations (1 .3 0 ) and (1 .3 1 ) in term s o f a d iffe ren tial relationship w ith Fi as the variab le:

mFi{A + BFi) + mF?B + F, = Fe

(1 .3 2 )

W e w ill exam ine the behaviour o f the system under h arm o n ic e x c ita tio n o f a con­ s ta n t a m p litu d e :

Fe = Q sin i/t


Vibration damping - basic concepts


For fin d in g the stead y-state v ib ra tio n we can use a n u m erical procedure. For given values o f Fi we com pute the displacem ents u and accelerations u. These d a ta a llow us to draw the hysteresis loop: the extern al Fe = Fe(u) and th e in te rn a l —Fi = —Fi(u). T h e la tte r can be dire ctly ob tain ed by su b tra c tin g in e rtia l forces as shown in Figs 1.13 and 1.14.




B> 20

c ,0 \ £





0 — - X - ----------------


-1,2 -0,8








% -0,4


u : n u 0 ml---




_____ _____ __________ _____




£ 0.4-----X — ^ ^ s . -----------------------------

.s o--------------- X X -------------i











X -4





20 —


. pkl

d o


v -------------------------------------------------—



-1,2------------------------------------_Z^__ -1,2 -0,8



displacement u-u(t)

7,2^— -------------------------------------

f . , -------------------- 1 S 5 HM ------------------------------------


\ s


displacement ^u(t)



displacement u-u(t) F ig u re 1.13. Hysteresis loop for s tru c tu ra l fric tio n

-2Cr— -------6 -4 -2 0 displacement



J~ 6


Fig u re 1.14. Hysteresis loop for stru c tu ra l fric tio n

T h e y reveal a re la tio n betw een the e xtern al and in te rn a l loops. T h is re la tio n p ro ­ vides us w ith some in fo rm a tio n ab o u t the influence o f d am p in g on the v ib ra tin g m o tio n o f the system . P a rtic u la rly , it can in d icate a n o n -lin e a rity in th e system . T h e relationships presented h ith e rto appear to be useful in e x p e rim e n ta l re­ search dedicated to v ib ra tio n d am p ing phenom ena. In the case w hen a description o f the d am p in g forces cannot be found we excite the system and record the exci­ ta tio n along w ith the corresponding displacem ent and acceleration. T h is w ill m ake


Damping of vibrations

it possible to determ ine an external hysteresis loop and an internal one, and thus to describe the dam ping forces present in the examined system. Generally, the character of these forces is described by poorly verified models.

REFEREN CES 1. Giergiel J., Vibrations of mechanical systems (orig. title in Polish: D rgania ukladow mechanicznych), Krakow: Wyd. AGH 1986 2. Giergiel J., Damping of mechanical vibrations (orig. title in Polish: Tlum ienie drgan mechanicznych), Warszawa: PW N 1990 3. Osinski Z., Theory of vibrations (orig. title in Polish: Teoria drgan), Warszawa: PW N 1980 4. Osinski Z., Damping of mechanical vibrations (orig. title in Polish: Tlum ienie drgan mechanicznych), Warszawa: PW N 1986 5. Osinski Z., Hysteresis loops of the vibrating system, Machine Dynamics Prob­ lems, 5(1993) 6. Osinski Z., Dynamic hysteresis loops corresponding to structural friction (orig. title in Polish: Dynamiczne p§tle histerezy przy tarciu konstrukcyjnym ), Materialy VIII Sympozjum Dynamiki Konstrukcji, Rzeszow 1993



Vibration damping by viscous resistance

2.1 C O N D IT IO N S O F A L T E R N A T IN G M O T IO N O F A V IS C O U S L Y D A M P E D M E C H A N IC A L S Y S T E M L e t us consider a one-degree-of-freedom system w ith lin e a r elastic force character­ istics and lin e a r dam p ing. T h e dam p in g characteristics can be w ritte n dow n as: R( x) = cx

(2 .1 )

Such a k in d o f d am p in g force appears in the case o f solids m o vin g a t a re la tiv e ly low speed in viscous fluids. A lin ea r (viscous) d am p in g description can be in c o rp o ra te d to ex am in e d am p in g phenom ena in m a n y m echanical systems. Because an analysis o f a given system becomes easy in the case o f lin e a r d am p in g it is very o ften used as an ap p ro x im a te and local description o f m ore com plex system s. T h e e q u a tio n o f m o tio n has the follo w in g form : m x + cx + kx = 0

(2 .2 )

w here m denotes the mass o f v ib ra tin g body, c is the viscous d a m p in g coefficient, an d k stands fo r the e lastic ity coefficient. E q u a tio n (2 .2 ) can be tra n s fo rm e d in to : x + 2 h i + uj\ x = 0

(2 .3 )

w here 2OL /i — C , CJq2 — * m m T h e character o f the solution to (2 .3 ) is s tric tly related to the q u an tities h and ljq. I f h < cjq, then the solution is:

x = (Ci cos u t -f C2 sin cut) exp (—ht) 31



Damping of vibrations

or x = A exp(—ht) sin(u;t + o the solution takes the following form: x = (Ci* + C2) exp ( —h t)

(2 .8)

V ibration velocity is then given by: x = [C i(l —/it) —/1C2] exp(—/it)


For the initial conditions according to (2.6) we obtain the following values of the integration constants: C i = x p h + vp C


W hen /i >

— Xp


(2 .10)

we have:

x = [Ci exp(rt) -f C2 e x p (-rt)] e x p (-/it)

( 2 . 11)

Vibration damping by viscous resistance


or x = [A\ cosh(rt) 4- A 2 sinh(—rt)] e x p ( - h t )


where r = y/h? —Uq. The integration constants corresponding to the initial condi­ tions (2.6) are: vp + x p ( h - r) ' 2r q _ vp 4- xp(h + r) 2 ^ _




Ai = x0 A2 = v* +r h x < > Through a detailed analysis we learn th at the general character of m otion is similar in both cases. The displacement, after reaching its extrem e value, tends toward zero within an infinite period of time. In some cases, when the initial velocity is negative and sufficiently high, the system passes through its equilibrium position only once. As can be seen, the damping force strongly affects the motion of a system , which can be vibrating or not (aperiodic). The motion is often said to be alternating (oscillating) when the number of zero-displacements the system passes through is infinite, and non-alternating when the number of zero-displacements is finite. In our case a criterion resolving whether the m otion is alternating or not confines to the following condition: h < cj0


When: h>u> 0


then the m otion is not a vibration. The damping coefficient at which h = u>o is called the critical damping: her =- ^0


and correspondingly: ccr = 2h crm —

m — 2V k m


This criterion is not related to the initial conditions but to the properties of the system.


Damping o f vibrations

2.2 C R ITER IA FO R ALTERNATING MOTION IN SYSTEMS W ITH NON-LINEAR CHARACTERISTICS OF DAMPING AND ELASTICITY FORCES Let us consider the equation of motion in the following form: x -f R ( x ) 4- S( x ) = 0


where R ( x ) represents a non-linear damping force, and S (x ) a non-linear elasticity force. Suppose th a t damping and elasticity characteristics can be expressed by the following formulas: R ( x ) = 2h i 4- du

< > is

Figure 2.2. Example of investigating the singular point

Vibration damping by viscous resistance



I f h > uq the characteristic e q ua tion has tw o rea l and d is tin c t roots: tano and a fin ite num ber o f x(t) = 0 fo r h > ljq. I t should be noted th a t in th e case o f lin e a r systems we have either zero or one p osition x ( t ) = 0 fo r h > u>o, depending on the in itia l conditions. L e t us w rite dow n the follo w in g d ifferen tial equation: x + 2 h i + {x) = 2 (h - b)x + 0 is chosen in such a w ay th a t the s tra ig h t lin e b2x is an u p p er lim it o f th e spring characteristics S(x) for x > 0 and the lower lim it fo r x < 0 (see F ig . 2 .3 ). s w (1




Figure 2.3. Non-linear elastic characteristic


Damping of vibrations

It can be clearly seen th at 6 > the function S (x ) we find that:


for any value of S(x). In such a description of

g(x)x < 0


independently of the sign of the product f { x ) x . E quation (2.35) can be transform ed into an integral form:

x = z + [ { ^ [ i( 0 ] + 0[*(O]} exp[—w0(£ - t)]d£ Jo


with z being the solution to the linearized problem. The integrand:

$ (£) = ( £ - * ) exp[-w 0(< - 0 ]


has a fixed sign within the interval (0, 2), namely: $ (0 < 0


Let us assume the following initial conditions: x (0) = z (0) = A > 0 , i(0 ) = i ( 0) = 0


According to conditions (2.23) and (2.24) we have R ( 0) = 0 and 5(A ) > 0, thus £(0) < 0 and there m ust exist an interval (0, r) within which x < 0. Besides, there also exists an interval (0, t{) where x(t) > 0. For the initial conditions given by (2.42) we have perm anently z(t) > 0 within (0,oo). The displacement x(t) = 0 can be determined from the formula: x ( t ) = z(t) + A(^)



a (t) = f m m ] + g i m i M m Jo The sign of A(£) depends, in the light of (2.40), on the sign of the following function: Q ( x , i ) = i/>(x) + g(x)


Vibration damping by viscous resistance


L e t us fin d the upper and low er lim its o f x. For the assum ed in itia l cond itions x rem ains lim ite d : |#| < A. T h u s the fu n c tio n f ( x ) is also lim ite d (see F ig . 2 .4 ) in accordance w ith the expression:

| / ( x ) | < mu>$A


w ith in the in te rv a l (0, A) w here m = su p {/(x )/w g v l}. S(x)k


F ig u re 2.4. E la s tic energy T h e a m o u n t o f energy stored in the system is:

E(t) < E 0
o A \/l + 2m

(2 .4 8 )

N o te th a t w hen fo r any x E ( X i , X 2 ) we have:

|S (x )| < |u ^x|

(2 .4 9 )

and m is zero. A ssum e th a t w ith in the lim its o f x the follow ing in e q u a lity is satisfied:

|f?(i)| = \2bx + ^ ( i ) \ > 2bx



Damping o f vibrations

which implies th a t the damping characteristics R(x) is placed in (0,u>oA\/l + 2m) above the straight line y = 2bx and below this line within (—u>oAy/l -f- 2m, 0). By assum ption, the function ip(x) satisfies the condition:

rp(x)x > 0

which m eans th a t the signs of tp(x) and x are the same. Now let us assume th a t there exists an instant t at which the displacement x is zero:

x(t) = 0


Since the zero-displacements appear in turns we can state th a t within the interval (0, t), x(t) is always negative and so is the function ifr(x). Because x(t) < 0 for t E (0,£), then, according to (2.38), g(x) < 0. Hence:

f2(z, i ) < 0

and by making use of expression (2.41) we obtain:

A(t) > 0

and consistently:

x(t) = z(t) + A(t) > 0

A ssum ption (2.51) is false. There are no zero-displacements within (0,oo). If the dam ping characteristics R( x) disobey condition (2.50) - see the broken line in Fig. 2.5 - then ^ ( x ) is negative and the integral A(t) can also be negative, which allows the displacement x to be zero at a certain time instant t .

Vibration damping by viscous resistance


R(x) a










Fig u re 2.5. N o n -lin e a r d am p in g characteristic

C onsistently, x has its extrem e value A \ a t t = t \ and x ( t i ) = 0. C onsider th e in te rv a l ( —u>oAiy/l + 2 m i , a ; o ^ i \ / l + 2 m i) w hich is sm aller th a n ( —u o A y / l + 2 m , + 2 m ) because ^4i < ^4 due to energy dissipation. I f th e characteristics R ( x ) obeys (2 .5 0 ) then there w ill be no zero-displacem ents s ta rtin g fro m t\ . I f c o n d itio n (2 .5 0 ) is n o t m et then x(J ) = 0 can occur. T h e character o f th e sin g u lar p o in t fo r h > u>o forces us to determ ine such a value o f A n so th a t th e displacem ent x w ill never reach zero w ith in (tn, oo). T h erefo re the num ber o f zero-displacem ents m u st be fin ite . T h u s condition (2 .5 0 ) constitutes the c riterio n en ab lin g one to s tate w h eth e r the m o tio n , s ta rtin g fro m x = A has no zero-displacem ents or passes th ro u g h zero o nly once. A ll th e above considerations prove th a t the character o f m o tio n , i.e. w h e th e r it is a lte rn a tin g or n o t, depends upon the re la tio n betw een lin e a r term s o f e la s tic ity and d am p in g characteristics only. If:

h < hcr = L)q

(2 .5 2 )

th en we observe an a lte rn a tin g m o tio n disregarding the shape o f th e characteristics. If:

h > h cr~ uo

then the motion is not alternating (see Fig. 2.6a).


Damping of vibrations



X ,i

■V \ A A A


b) X 11


0 x



Fig u re 2.6. A lte rn a tin g and n o n -a lte rn a tin g m o tio n

T h is case is characterized by the appearance o f a fin ite n u m b er o f in stan ts a t w hich x(t) = 0. T h e tw o follo w in g situa tions belong to this case: a) m o tio n w ith one and only one zero-displacem ent in d ep en d en tly o f th e in itia l conditions (see F ig . 2 .6 b ), b) m o tio n w ith a fin ite num ber o f zero-displacem ents, depending on th e in itia l conditions (see F ig . 2.6c ), S itu a tio n a) takes place w hen x and x satisfy co nd ition (2 .5 0 ). I f n o t, we observe s itu a tio n fe). T h e s itu a tio n w hen the num ber o f zero-displacem ents depends on the in itia l conditions is different fro m w h a t is observed in lin e a r system s. C o n d itio n (2 .5 0 ) is sufficient fo r the system to pass th ro u g h x — 0 o nly once. T h e inverse co n d itio n , i.e.:

R ( x) < 2 bx

(2 .5 4 )

w ith in ( —V i , V 2) is n o t sufficient to ensure the n u m b er o f zero-displacem ents to be g rea te r th a n one. Y e t the appearance o f a fin ite n um ber o f zero-displacem ents (g re a te r th a n one) is proved in m an y exam ples, thou gh it is h a rd to in d ic a te any general c rite rio n e nab ling precise d e te rm in a tio n o f this n um ber. L e t us proceed to a grap h ical in te rp re ta tio n relevant to the above discussion. L e t the d am p in g characteristics be o f an a rb itra ry shape w ith in ( —V i , V 2 ) and let it satisfy con d itio n (2 .2 3 ), see Fig . 2.7.

Vibration damping by viscous resistance


m i y


n o n - a l t e r n a t i n g ------m o t io n *

HlfiV W m

r =

i Q E —



jJ^ rE E

a lt e r n a t in g m o t i o n \



li=E r “ I A— --■/ F ig u re 2.7. C ritic a l line

T h e s tra ig h t line acr:

r : y = 2 v 0x

(2 .5 5 )

called the c ritic a l line, separates the d iag ram in to tw o areas, I and I I . T h e slope o f th e tan g en t to R(x) a t x = 0:

a: y; = 2hx

(2 .5 6 )

determ ines the character o f the m o tio n . I f this tan g en t line is placed in the area 7, the m o tio n is a lte rn a tin g (in fin ite num ber o f zero-displacem ents). T h e slope o f th e ta n g en t lin e satisfies: h < u>o. I f the c ritical lin e appears in th e area 77, the m o tio n does n o t have an a lte rn a tin g character and the solution has a fin ite n u m b er o f zero-displacem ents. T h e slope o f the tang ent lin e is: h > u>o- In th is case we can d eterm in e w hethe r the num ber o f zero-displacem ents is one or m o re. For this purpose it is necessary to find values o f the coefficients m and 6, and th en to draw th e lin e ap (see F ig . 2.8):

Damping of vibrations


Re­ n u m b e r o f z e ro d is p la c e m e n t not m ore th an one ~(o0A\/l+2rn

J & rm lfT S n fn il.ll,l.lllm iJlA t

+0)qA\/ 1+2m




n u m b e r o f z e r o - id is p l a c e m e n t a f in it e n u m b e r re late d to



in itial a m p l it u d e

Fig u re 2.8. C ritic a l lin e ap: y = 2bx

(2 .5 7 )

T h e s tra ig h t lin e divides the d iag ram in to tw o fields, denoted as A and B , re­ spectively. I f th e d am p in g characteristics R( x) is placed w ith in ( —u o A y / l + 2 m , ujoA^/1 + 2 m ), and e n tirely w ith in the field A , then we have one and o n ly one zero-displacem ent. In this case cond ition (2 .5 0 ) is n o t satisfied. W h e n th e spring characteristics fulfills: \S(x)\ < |u>ol, i-e- w hen f ( x ) < 0, th en th e procedure is sim ­ ple since b can be replaced w ith 2u>o and the s traig h t lin e ap reduces to th e c ritic a l lin e a cr. T h e lin e ap (see F ig . 2 .9 ) enables d e te rm in a tio n o f the velo city vj w hich is ch aracterized by the follo w in g p ro p erty: for any t > tj w hen v < vj th e m o tio n c a n p a s s t h ro u g h it s e q u ilib r iu m p o s it io n x = 0 o n ly o n ce.


li 1/



./ ^ r . V


F ig u re 2.9. C ritic a l velocity

T h e velo city Vj m akes it possible to fin d the in itia l a m p litu d e A fo r w hich the system w ill never reach x = 0 (fo r zero in itia l velo city) or reach it o n ly once (fo r any non-zero in itia l v elo c ity ).

Vibration damping by viscous resistance


L e t us consider different associations o f elastic and d am p in g characteristics. T h e characteristics are called s tiff characteristics i f they satisfy the fo llo w in g conditions:

\R(x)\ >

(2 .5 8 )

\S(x)\ >

(2 .5 9 )


O bviou sly, th ey are soft if:

. ( d f ? ( i) l* (* )l



j/ ^ > / y


j% y ^








acOrAx 1+-^-? A2


2dm0A . n + 1 A2



2* 2q


F ig u re 2.11. In v e s tig a tio n o f a lte rn a tin g m o tio n

2h - f a l A 2 ( l + ^

2) > 2 ^ 1 + ^

Vibration damping by viscous resistance


T h e in itia l a m p litu d e A p can be estim ated fro m the above in e q u a lity . T h is can be done by draw ing lines corresponding to the rig h t- and le ft-h a n d sides o f the in e q u a lity (see F ig . 2 .12 ) and fin d in g the intersection.

F(A) h

: _ l _ o _________________


F ig u re 2.12. In v e s tig a tio n o f a lte rn a tin g m o tio n For instance, w hen h = 5, u = 1, j = 1, /? = 1, we o b ta in A p = 1.63, w hen h = 5, uj = 1 , 7 = 0.1, /? = 0.1, then = 5.15, and w hen h = 5, a; = 1, 7 = 0 .01, fd = 0.01, we have = 16.3. For any x (0 ) < the system w ill never pass th ro u g h its e q u ilib riu m position again. I t is clearly seen th a t a certain co m b in atio n o f the d am p in g and elastic characteristics can force the system to pass th ro u g h its e q u ilib riu m position a specified num ber o f tim es. L e t us now consider a m echanical system w ith d ry fric tio n . T h e fo rc e -v e lo c ity d ia g ra m is shown in F ig . 2.13a.


R (X) »


R(x) (,

F ig u re 2.13. In v e s tig a tio n o f a lte rn a tin g m o tio n T h e fu n c tio n shows a disco ntinuity a t x = 0. T h e fu n c tio n can be replaced w ith an a p p ro x im a te and continuous fu n c tio n as is shown in F ig . 2 .1 3 b , w here the shape o f the segm ent M N differs in fin ite sim ally fro m the vertical d irectio n . L e t us draw the c ritic al line acr separating the d ia g ra m in to th e areas I and II.


Damping of vibrations

N o te th a t, regardless o f th e value o f uj) the straig h t lin e a is alw ays placed w ith in area I I . T h e m o tio n is then n o n -a lte rn a tin g and it disappears a fte r a fin ite n u m b e r o f oscillations. T h e num ber o f oscillations obviously depends on th e d a m p in g and elastic characteristics. T h e straig h t line ap provides us w ith the in fo rm a tio n w h e th e r th is n u m b e r is one or m ore. In th e case o f a system dam p ed by an aerodynam ic force R ( x ) = /? i2 s g n ( i), we observe th a t h and the stra ig h t line a belong to area 7, see F ig . 2.14. R(x),



/ y




F ig u re 2.14. In v e s tig a tio n o f a lte rn a tin g m o tio n T h e system passes thro u g h its e q u ilib riu m position an in fin ite n u m b e r o f tim es, in d ep e n d en t o f the elastic characteristics. D ra w in g the s tra ig h t lin e ap is needless in th is case.



L e t us look a t a system w ith lin e a r spring characteristics and n o n -lin e a r d a m p in g characteristics, described by the follo w in g function:

R ( x ) = cx + bx3

(2 .6 3 )

w here c and b are the constant coefficients. T h e eq u atio n o f m o tio n has th e fo rm :

m x + c i + bx3 4- kx = 0

(2 .6 4 )

I t can be tran sfo rm ed into:

x + UqX = //(—2hx —/?i3)


Vibration damping by viscous resistance


where cjq = k / m , h = c / 2m, /? = b / m and fi is a formally introduced sm all param eter (after the required transform ations it will be taken as equal to one). The solution can be found through the sm all-param eter Krylov-Bogolyubov m ethod. The first approxim ation has the following form:

x = a co s^

where a and


can be determined from:

= fiAi(a)


dxb — =

ujo +


, x fiB\{a)


The functions A \ and B\ satisfy the following equalities: 1 f 2w A\ (a ) = — ----- / [ —2/i(—acjosin^) —/?3(—aw o sin ^)3] s in ^ d ^ = L'KUq J q = - ha^- Pug a3




f 2'

^ i ( a) = o I znawo Jo = 0

[ —2/i(—aw0 sin V>) —/?3(—aw0 sin V>)3] cos V’dV’ = (2.70)

Hence: da ,3 — = -ha-pula3


JT = "»


dt d^>

By incorporating the initial conditions:

a(0) = a 0 , V’(O) = V’o


we obtain: a0e hi



\/> ~ I £ - « ( e - “ ' - 1) = W0< + V>0


Formula (2.74) describes the vibration amplitude. The effect of dam ping is shown in Fig. 2.15.

Damping of vibrations


a jt

0 Fig u re 2.15. A m p litu d e drop for n o n -lin ear v ib ra tio n W h e n P = 0 the a m p litu d e decays as in the lin ear case. For (3 > 0 the a m p litu d e decays faster, fo r )3 < 0 slower. T h e solution corresponding to /3 < 0 is v a lid o n ly fo r lim ite d values o f x. O utside these lim its the resistance force described by (2 .6 3 ) changes its sign w hich corresponds n ot to energy dissipation b u t to its supply. L e t us d ete rm in e th e ra tio betw een tw o successive am p litu d es at a distance o f h a lf the v ib ra tio n p eriod fro m each other:


aoe-^ + 'M jl -

|f«gag[e-»*T -


A fte r a tra n s fo rm a tio n we o b tain:

= e~h$ 1-1

l - j f c g a 2 [ e - ( " + 1) M ’ - l ]

(2 .7 7 )

1 — § f 'a'oa o[e-n h T ~ 1]

W h e n P = 0, the expression under the square ro o t is one. T h e d a m p in g decrem ent is th a n constant and equal to the decrem ent corresponding to th e lin e a r case. W h e n P 0, th e decrem ent depends on the am p litu d e a. For p > 0 it is g reater th a n in th e lin e a r case b u t tends to it w ith tim e . For P < 0 the value o f the decrem ent is p e rm a n e n tly sm aller b u t it grows as the a m p litu d e decays (see F ig . 2 .1 6 ). f a i l ji a*

p< 0 -

° * F ig u re 2.16. D ecrem en t o f n o n -lin ea r v ib ra tio n

Vibration damping by viscous resistance


The vibration frequency in the first approximation is equal to th a t in the linear and undam ped case. In the second approximation the frequency ensues from the equation describing the phase shift \j>\


The first two term s on the right-hand side of (2.77) express the frequency corre­ sponding to the linear, viscously damped vibration. The third term appears due to non-linearity of the damping characteristics.



Let the dam ping force be described by the following function:

R = cx + e x 3


Let us examine the vibration of a system with linear elastic characteristics harmonic excitation. The equation of motion assumes the form:

m x + c i + e x 3 + k x = H sin vt



where m denotes the system mass, k its stiffness, H the excitation am plitude, and v its frequency. Equation (2.79) can be transform ed into:

x + ax +

= q sin vt —f i x3


where c e a = — , /? = — , m m

o k H = — , q- — m m

The coefficient (3 will be taken as the small param eter presented in the solution expressed in term s of the power series:

x(t) = X0(t) + 0Xi(t) + 0 2x 2(t) +



Damping of vibrations

After substituting this into (2.80) and equating the term s of the same order we obtain the following set of differential equations: xo 4- a x o 4- WqXo = q s i n v t +




The equations have to be solved successively taking into account only periodic solutions and neglecting the term s which describe decaying vibration (transition). The solution corresponding to the second approxim ation is: x = coi sin vt 4- eoi cos vt 4- fi{c\i sin vt 4- e n cos vt 44- C13 sin 3vt 4- ei3 cos 3vt) 4- fi2{c2 i sin vt + e2i cos vt 4-


4- C23 sin 3vt 4- ^23 cos 3vt 4- C25 sin 5vt 4- 625 cos 5vt) The coefficients appearing in (2.83) are given below: coi =

q(u>l - v 2) Ai —qav

e°l =


cn = 7 -T —[(cgieoi + eoi)(wo “ 4 A i1

1/2) ~ (coi +


3 vz

en = - ^ ^ ■ [(c o ie o i + e o i) a ^ + ( c o i + c0ie^1) ( w ^ - i / 2)] vz C13 = ^ ^ [ (3 c o ie o i - CoiHwo - 9u2) + (-C& + 3c0ie ^ )3 a i/] 613 = C21 =

v3 4 A s 3coie°i


eoi)3al/ + (—coi + ScoiejjjXwo —9i/2)]

3!/3 r / 1 2

3 2 2A~ 11 2 C°iei1 2 C°lCl3

o 3 2 Coie°lCn ~ ^ o ie o ie n 4- - e o i^ n 4-

3 \ / 3 3 - 2 eoiei3J (wo - ^ 2) + ( - 2 Coicn “ 2 C«iCl3 ~ coieo ien + 1 2 3 2 - 3c0ie n e i3 — ^ en cn + 2 eu ci3 ) a i / 3" 3 17 1 2 3 2 , o 3 2 621 = 2A i 11 ~ 2 C°lCn _ 2 C°lCl3 ~ c°ie°lCn ’’’ 3coieoiCn — 2 eoieu +






+ 2 eoiei3) a , / + ( “ 2 C°lCn ~ 2 °°iC i3 _ c°ie °i e n +




- 3c0i e 0ie i3 - - e j ^ c n + - e 21c13J(a;g - i / 2)

3i/3 C23 = 2a ;

2 Coien 4- 3coiei3 4- coi^oi^i! — 2 eoien 4"


Vibration damping by viscous resistance


+ 3 e jX e i3) ( o ^ - 9 j^2) + ( - ^ c ^ c u - 3 c^ c i 3 + c0ie 0i e n


^23 =


2 e 0 1 C l1

3^3 2A 3


3 e 0 ! c 13

) Q l/

C01C11 —300-^13 —coieoicn 4- 7:eoien 4-

+ 3e21e i3) 3 a v + ( — jjc g jC n — 3 c o ,c n — 3cq1c i3 + c o ie o ie n +


C25 = ~

2 e o ic u

3 e 0 1 c 13

31/3 \ ( dc o i e i 3 +

2A 5 [ \ 2

) (wg - 9 i/2) ) ( " o ~ 25i/2) +

3 c o ie o iC i3 —

C01 C13 4- 3coie0iei3 4*

^ 5 a i/

+ H

3^3 17

625 = 2A^ I V

3 2 0 2 C°l613 ~~ oie°lCl3

3 2 \ c 2 e°l€ l3J V '

+ ( - | co ici3 + 3c 0ie 0i e 13 + 2 ”‘ “ 2

5 i/ 2^) (wg - 20K-

In the above definitions the follow ing relations were assumed: A i = ( u l - v 2)2 4- a 2v 2 A 3 = (u >0 — 9i /2) 2 4- 9 a 2v 2

(2 .86 )

A 5 = (a ;2 — 2 5 i/2) 2 4- 2 5 a 2i/2 L e t us investigate the p a rtic u la r case o f a resonance o f the first h arm o n ic, i. e. w hen u>o = v. T h e coefficients in (2 .8 3 ) then take the follo w in g values: coi — 0 •“ = Cn = 0


- ® j L 11

4 a 4 t/

C13 = 2 v 2^-s - Ja3 A 3 3 q3 1 ei3 = — 4 a2A3 . i/ 2 —j « 5 —1 c 2i = —43 .5 a A3 (p e2i = 2.25—3-(—e n + ei3) a6 v 2 q2 c23 = “ l-3 3 e i 3 i/ - 4 .5 c i 3a )


Damping of vibrations 2


623 = 3 - ^ - f — ( 0 . 7 5 e i i « - 4 .5 e i3o; + 1 2 c i3i/)

or A 3 2


C25 = 2.25-^--^—(24i/ei3-|-5aci3)

a 2A5 2


e 25 = 2 .2 5 - ^ r - 4 - ( 5 a e x 3 - 2 4 i / c 13) or A s A i = a 2i/2

A 3 = 641/4 + 9a2u2 A 5 = 576i/4 4- 25a V


Because the analysis of the solution is a very tedious, time-consuming process we confine ourselves to the case when:

- > 10

(2 .88)


This condition is generally satisfied as far as mechanical vibration is concerned. The am plitude corresponding to the Xi function is:

Let us examine the m otion within the first resonance zone. For this purpose we have to check the ratios between the am plitudes of the successive harmonics. Com parison of, e.g. the a n and a i3 amplitudes, proves th at their m utual ratio is equal to the ratio of the C13 and c n coefficients:

M Ic i i l

= 8


3 64£ + 9 f

I 240

The th ird harmonic can be neglected in the solution corresponding to the more so as the product (3x\ is of an order less than xq due to the small param eter (3. Analogous considerations pertaining to all the other harmonics prove th a t only the first harm onic is necessary to be taken into account, provided th a t assum ption (2.87) is still operative. The solution can be then w ritten down in the following form:

X = (coi + /?cn + /?2ci2) sin ut + (e0i + /?en + /?2e12) cos v t


x = D s m { v t + 6)



Vibration damping by viscous resistance



D = \ / ( coi + /? c n + /?2c i 2)2 + (e0i + /?en + /?2e i2)2



tan 6 =

coi 4- /?cn 4- /?2ci2 coi


+ /? en + / ^ 2 ei2

Since coi = 0, c\\ = 0 and C21 is small compared to e2i (near the first resonance) the am plitude can be expressed in a simpler form:


D = e0i 4- /?en + /?2e i2

By making use of the relationship between eoi, e n and e2i we can transform (2.92) into:

D — e0i ! W

f i Y a \a j

+ 1 .6 9 (4 ( V ‘ \aJ \a


Let us determ ine the relation between the vibration am plitude D and the excitation am plitude q. In the case of a linear system we observe a direct proportionality between D and q. The non-linearity disturbs such a relation. For j3 = 0 we have:

D e01 1 — = — = — = const q q olv

For (3 ^ 0 the above ratio is a function of q:

7 = f { q)

A graphical representation of D and D/ q versus q is shown in Figs 2.17 and 2.18.


Damping of vibrations

D ii R' q


p=o 1 e01

(co=const) 0




F ig u re 2.17. A m p litu d e o f n o n -lin ea r v ib ra tio n




L 0