Dynamics with friction, Part II : modeling, analysis and experiment 9789812811271, 9812811273

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SERIES ON STABILITY, VIBRATION AND CONTROL OF SYSTEMS Volume 7

Series B

Dynamics with Friction Modeling, Analysis and Experiment Part II Editors

Ardeshir Guran, Friedrich Pfeiffer & Karl Popp

onol .Joint with 1

s

'""° ^ ? C

Weo), are introduced to obtain

22 D. P. Hess

Figure 13. Typical Poincare map of chaotic block motion for joint with rough surface stiffness.

y - + 2C«

y+nV^AE^V2°V\e-(h^l)-e'^W}sgn(y) m

= ^^COSQt ma

(21)

A quantitative test based on the sign of the Lyapunov exponents of the joint motion is used to identify steady-state chaos. The criterion for chaotic motion is as follows: if the largest Lyapunov exponent is positive (A,, >0), the motion is chaotic; and if the exponent is zero or negative (A,1

represents the natural frequency of the plate when the mode shape consists

of j nodal diameters and n nodal circles. The constant % • and \\fjn can be determined from the initial conditions. McLeod and Bishop gave the natural frequencies of circular plates with various boundary conditions. 2.2

Vibration of a Spinning Membrane

The transverse vibration of a flexible membrane which is spinning at a constant angular velocity, and therefore under the action of centrifugal forces, can be described by the differential equation (Prescott4), d2w

where crr and o e e

1 d ( dw^ a ee d2w (14) r °rr ~ "302 dt< r dr \ dr denote the radial and circumferential stresses respectively. It may

be demonstrated' (Lamb and Southwell1) that the membrane displacement has the solution, w = /?(r)cos;8 cosco t

(15)

where R(r) is given by, rr\j

R(r)

a,p,y,-

(16)

KbJ

and F(») is the Gauss or hypergeometric series, (

a,P,Y,—- = 1

2^2

r

TY" V

J

a(at-l)P(P+l) (l)(2)y(Y+l)

v b

K

v (17) J

with the terms, a = —(j-m) , (3 = — (j+m+2) Y = 7+1 .

(18 a,b,c)

2. Vibrations and Friction-Induced Instability in Discs 33 The hypergeometric series is convergent when r < b. When r = b the series will diverge unless ( a + P) < y . However, from Eqs. (18a,b,c) it can be seen that ( a + (}) = y and the series is therefore divergent unless it terminates. This can be accomplished by letting m = j+2n, where n is a positive integer. Accordingly, the displacement solution of the spinning membrane is given by r

rr\j

1 -

vb;

.2

(n)(n+/+l)

r'

10+1)

b'
(tojj .

sign

is

for nCl < co7 < cu2 > anc ^

tne

minus

sign

is

for

Alternatively it is found that, w - -A j cos/0 cosjClt + A2 sin/0 sinjClt

when

(35)

(36)

GO 7 < nCl < co-2 .

We take, for instance, the case when nCl < a>7 < u)-2 and the amplitude Ax > A2 The motion of the disc is given by, w = A2 cos/(0 - Clt) -f (A j - A2)cos;'0 cosjClt ,

(37)

where the first term represents a backward travelling wave (relative to the moving disc) of amplitude A 2 and the second component is a fixed vibration term of amplitude (Aj-A 2 ). Numerical calculations by Tobias and Arnold showed that by increasing the imperfection, the backward wave was reduced at the expense of an increase in the fixed wave component. Extensive experimental investigations were reported by Tobias and Arnold . The experimental set-up was quite elaborate and consisted of several major components. A high grade steel disc of 12 inches diameter was used in the investigations. It had a thickness of 0.0936" and a tolerance of ±0.0003" which made it a very thin disc in comparison with its other dimensions. Special efforts were made to check its flatness and alignment so that reliable measurements could be taken. Self-aligning ball race bearings were used to support the spindle on which the disc was mounted. The disc was driven by a D.C. motor though a V-shaped belt and the motor was capable of rotation speeds from 400 to 20,000 rpm. One flywheel was attached to the spindle and another to the motor in order to reduce speed variations. A large electromagnet linked to an

2. Vibrations and Friction-Induced

Instability

in Discs

39

amplifier and oscillator was used to excite the disc into vibration. Experiments were conducted with steel cylinders of half an inch diameter and height which were glued to the disc surface in order to create preferred modal configurations. Good correlation for the frequency ratio of the preferential frequencies with theoretical values was found in the stationary disc case. When a static force was applied to the rotating disc (with masses attached), the disc was sufficiendy imperfect to cause the two configurations to form two critical speed resonances. While the presence of nonlinearity ruled out a comprehensive explanation using the linear analysis for the resonances that occurred, the experimental results did show that the introduction of a large imperfection would bring about a reduction in the maximum amplitude of the stationary wave. For instance, the reported amplitudes of the critical speeds of the two configurations were only V9 and 2 / 13 of the value for the nearly perfect disc. In a more recent study Strange and MacBain carried out experiments on imperfect discs using holographic interferometry. They reported significant differences in the level of damping present in 'bladed' tuned and mistuned discs in the modes with 3 or 4 nodal diameters. There were unexplained differences in the level of damping in the two preferred configurations of the mode with 2 nodal diameters. 3.

Excitation by a Transverse Mass-Spring-Damper System

A seminal contribution on the dynamics of discs under the action of a transverse massspring-damper system was made in the two articles by Iwan and Stahl and Iwan and Moeller^ . They identified instabilities that were strongly associated with stiffness, damping (at supercritical speeds), modal interactions, and a terminal instability. The work of Mote and his colleagues on the subject of moving loads on discs, spans three decades. This research work is exhaustive and exceptionally diverse in the treatment of many types of moving load-stationary disc problems. For example, Mote9'10 examined the free vibration of initially stressed discs induced by rotation and in-plane thermal gradients, which increased the fundamental (or lowest mode) natural frequency. The applicability of Green functions to plates subjected to an arbitrary circumnavigating moving force of harmonic or other excitation form was suggested by Mote11. Mote considered the problem of the moving load stability of a circular plate sliding freely on an inner floating central collar, with particular application to the design of circular saws. Recently the vibration and parametric excitation of stationary asymmetric circular plates containing small imperfections and subjected to moving loads, was discussed by Yu and Mote . The mechanism of instability of such a disc was addressed by Shen and Mote14 who successively applied classical nonlinear methods like the techniques of slowly varying amplitudes and phase averaging, the method of successive approximation (Valeev1 ) and the Krylov-Bogoliubov-Mitropolsky method to the rotating mass-springdamper load. The parametric resonance of a stationary circular plate with viscoelastic

40

J. E.

Mottershead

inclusion subjected to a rotating spring was described by Shen and Mote16. A new perturbation method was developed by Shen and Mote17 based on Valeev's work and demonstrated the parametric resonance of an asymmetric stationary circular plate under the action of a rotating spring. Shen and Mote used the multiple scales approach for an analysis of instability in stationary discs due to a rotating mass. 3.1

Stationary Disc with a Rotating Mass-Spring-Damper System

The equation of motion for a stationary classical annular plate excited by a rotating massspring-damper system, illustrated in Figure 3, can be written in the form (Iwan and Stahl7),

Figure 3: Transverse m-k-c system

2. Vibrations and Friction-Induced

d2w

Ir A. dr

Eh'

2

12(1-v2)

dt

dr'

Instability

in Discs

41

r2 ae 2 (38)

= - _ L 5(e-£f)S(r-/-0) m — hr dt o

+ Q.

_

—

30

+ Q. —

dt

ae

The transverse load system consists of mass, m, stiffness, k, and damping, c. The mass slides without separation and traces a circle at radius r0 on the surface of the disc. Thus the deflection of the mass is identically the transverse deflection of the plate at the position defined by the two Dirac delta functions. The transverse motion of the mass is a function of both time and angular position. For the solution of equation (38), Iwan and Stahl assumed an eigenfunction expansion solution for the disc of a form similar to that for a free disc (c.f. Eq. (13)) and given by, oo

oo

w(r,6,0 = E

E

Tjn(t) sin(/e-i)yr))/?7>1(r) .

(39)

;=0 n=0

The coordinates (r,9) fixed to the plate were transformed to the system, (r, cp) fixed to the rotating mass-spring-damper arrangement. This transformation allows the equations to be solved more easily, as shall be shown later. The transverse displacement of the plate in the new coordinate system can be expressed in the form of a truncated eigenfunction expansion, w(r,ee 2 1Q

so that the equation of motion may be written in the coordinates (r,6) attached to the rotating disc as, L(w) = P(w),

w = w(r,9,f) ,

(53 a,b)

where dw dw , ~H(r'Q) m •+• c + lew 2 , *< * ,

P(r,Q,t) =

The function H(r,Q)

(54)

is defined as 1 if (r,9) e

DL

H(r,Q) = i

(55) 0 if (r,6) «E D L

where D L represents the loaded surface of net area A L . An approximate solution to Eq. (53a) is sought, for a vanishingly small load area, in the form,

/ w(r,Q,t) « £ ( A < 0 C O S / 8 -

fi/r)siiye)

A_> R-L

(56)

KbJ

y'=l

In this case an eigenfunction expansion without nodal circles was evaluated, and the radial function was of the polynomial type,

b

KJ

where the constant

~Ci

r \*

H vrb;

(57)

p1- is chosen to minimise the potential energy of the disc and C is

a normalisation constant.

46

J. E.

Mottershead

By the application of Galerkin's method, j[ ft j[ 2 * {L(w) - P(w))Vjn rdrdQ-0

(58)

and using the orthogonality property of the eigenfunctions, a set of equations similar in form to Eqs. (43) and (44) are found. It follows that a similar state-space eigenproblem can be formed as in the preceding section. Indeed the results reached by Iwan and Moeller8 for numerical simulations are similar to the previous discussion on the instabilities for a stationary plate subject to a rotating mass-spring-dashpot load. They reported that the principal effect of the inclusion of the centrifugal stresses was to stiffen the disc. Hutton et al. discussed the number and configuration of stationary spring guides for a rotating clamped-free annular disc (while subjected to stationary lateral, concentrated, cutting forces, to improve the design of high-speed circular saws. Specifically they wanted to increase the lowest critical speed by introducing the spring guides. They used a Galerkin approach to obtain approximate solutions to the dynamical equation and found it possible to distribute and locate the springs that can significantly alter the frequency speed characteristics. The dynamic response of read-write floppy discs subjected to axial sinusoidal movements, as experienced by computers on board ships, aircraft and other moving platforms, was considered by Jiang et al. A floppy disc drive system with a spring stabiliser and including the effects of air film surrounding the disc (modelled as a stiffness, and involving use of the Navier-Stokes equation) was investigated by Chonan et al?x in their study of high-speed disc systems suitable for video camera operation. 3.3

Instability Mechanisms

The response ? f a stationary disc due to the separate effects of a rotating spring, mass or damper was studied by Shen and Mote1 In the case of a rotating spring Eq.(38) can be reduced to, +

K

£

flW

e jn

' '

0,

(59)

which is of the same form as the equation considered by Valeev15. To obtain Eq. (59) it is necessary to transform the differential equation of motion into the modal coordinates of the disc. The transformation is explained in Section 5.1. By following Valeev's method, Shen and Mote were able to show that the instability regions occurred at, Q = (co;n - co/p -r o)/g,

g=\j ± l\ * 0,

j,n,lp = 0,1,2,...,

(60 a,b)

2. Vibrations and Friction-Induced

Instability

in Discs

47

where o is a detuning parameter. Whenever (j,n)=(/,p) the instability is a superharmonic parametric resonance, otherwise it is a combination parametric resonance of the summation type. Neither can exist below the first critical speed. The two cases are the stiffness and modal interaction instabilities referred to by Iwan and Stahl . To investigate the effect of a rotating mass Shen and Mote used the method of slowly varying phase and amplitude. For a single mode approximation they used the modal coordinates equation, 2 ajn

£A + q = - p | ( l + cos 2x) £.1 dx2 j2Cl2 [ dt2

-2 sin 2x £l

dt

- (1 + cos 2x)