Contact vibrations

Journal of Sound and Vibration (1972) 22 (3) 297-322

CONTACT VIBRATIONS P. RANGANATHNAYAK Bolt Beranek and Newman Inc. Cambridge, Massachusetts 02138, U.S.A. (Received 26 November 1971, and in revised form 17 March 1972) Elastic bodies in nominally point contact execute vibrations in the contact mode, i.e. with the bodies vibrating as rigid masses on the non-linear contact spring. These vibrations may be caused either by an oscillating external force or, for rolling or sliding bodies, surface waviness. The purpose of this paper is to develop some of the theoretical groundwork necessary for detailed physical explanations of experimentally observed phenomena in vibratory point contact. Analyses are presented for three cases: undamped free vibrations, forced damped vibrations with a sinusoidal input, and vibrations with a broadband random input. The approximate method of first-order harmonic balance is shown to yield inaccurate solutions, and heuristic arguments for the construction of response diagrams are presented. A comparison with experimental data shows these arguments to be reasonably accurate. Three problems of interest in rolling and/or sliding contact are considered in some detail: loss of contact, plastic deformation, and the formation of corrugations. The problem analyzed here occurs in diverse areas: the contact of gear teeth, ball-bearing ball-race contact, and wheel-rail contact, to name a few. 1. INTRODUCTION A recent paper by Carson and Johnson [1] presents qualitative arguments for believing that such phenomena as the inelastic deformation, sometimes leading to corrugations, of railway track, road surfaces, gear teeth and rolling bearing races have their roots in the nature of point (or Hertzian) contact between elastic-plastic solids. Carson and Johnson then present experimental data obtained on a rolling/sliding contact disc machine which support these arguments. They conclude with a series of intriguing observations but defer explanations to a later date. The intent of our paper is to present some of the theoretical groundwork necessary for detailed physical explanations of experimental observations on such systems. However, we also present conjectures (yet to be tested) on the specific mechanisms involved in the formation of corrugations on rolling surfaces. The mode of vibration analyzed corresponds to mainly rigid-body motions of the contacting bodies, with deformations localized near the contact region, and is termed the "contact mode". In this paper, these localized deformations are assumed to be elastic; the resulting analyses are thus applicable to predictions of the likelihood of inelastic deformation, but not to the subsequent motion. The equation of motion for contact vibrations is shown to be essentially non-linear (i.e. with no "small" non-linearity parameter), and quite apart from its useful applications, which are numerous, is of theoretical interest because of the type ofnon-linearity involved. This nonlinearity leads to severe difficulties in the application of approximate techniques for the analysis of non-linear differential equations. Section 2 contains a development of the equations of motion for contact vibrations, and includes a discussion of the conditions under which these vibrations are important. 19 297

298

P. RANGANATH NAYAK

In section 3, an account is given of a detailed investigation of undamped free vibrations in the contact mode. The results are characteristic of systems represented by a mass on a softening--and asymmetric--spring. These investigations arc put to use in section 4, which deals with forced deterministic vibrations. The first half of section 4 deals with the use of an approximate method, the method of harmonic balance. Even a first-order solution leads to serious computational difficulties; moreover, a comparison with the exact solutions obtained in section 2 shows the results of the approximate method to be of questionable accuracy. Comparisons with the experimental data of Carson and Johnson [1] confirm this suspicion. As an alternative (and we believe, better) approach, the remainder of section 4 is devoted to a heuristic analysis based on nonrigorous, intuitive arguments, for the generation of response curves for sinusoidal inputs. The results of such an analysis arc shown to fit Carson and Johnson's data reasonably well. The more useful case (for roiling contact) of random inputs is analyzed in section 5, with attention being restricted to broadband inputs. A solution of the related Fokker-Planck equation for the joint probability-density of response variables (displacement and velocity) is used to examine the probability of plastic deformation and the frequency with which plastic indentations occur. We end with a conjecture that neither of these by itself is important for the formation of corrugations (as opposed to lone plastic indentations), but that their ratio, giving the average length of plastic indentations, is important. The effect of both the rolling velocity and the static normal load on various statistics is examined, and tentative agreement is found with various observations of Carson and Johnson. Finally, section 6 presents both a critique and a discussion of the analyses of the previous sections. Included are such questions as the validity of the model, discrepancies between theory and experiment, the possibility of subharmonic resonances, and the influence of tractive forces on plastic deformation and the formation of corrugations. 2. FORMULATION OF THE PROBLEM 2.1.

PHYSICAL INVESTIGATIONS

For clarity of exposition of the physics of the contact problem, we first consider a specific case: a ball roiling on an elastic half-space, as shown in Figure 1(a). When the surface of the half-space is wavy, dynamic loads are generated at the interface between it and the bail, and it is of interest to know the amplitude of these forces. In most such analyses, the ball is taken to be a lumped mass and the half-space to be rigid, and the surface waviness is then an input to a lumped linear system (see Figure l(b)). We now proceed to examine when such an approximation is justified. Assume that the surface waviness is described by w ( x ) = Wo sin (2,rx/A) -= oJ0 sin k x .

(1)

Denoting the instantaneous contact force by Pc, we have (2)

Pc(t) = Po + P ( t ) ,

where Po is the mean (static) force and P (t) the dynamic force, with zero mean. For the linear system of Figure l(b), if it is assumed that the mass M2 is large, i.e. to 2 M 2 >>Ks,

toM2 >>c,

~o = Vk,

(3)

where V is the rolling velocity, then the dynamic force is P(t)

= Ml 0 2

wo[(l _p2)2 + (2~p)211/2 sin (oJt + 40,

(4)

299

CONTACT VIBRATIONS

where co~ = Ks/M1,

p = w/COo,

~ = c/2ooo M l ]

and q~= tan_l {12_~z} .

)

(5)

In order to determine when this linearized solution for the dynamic force is valid, we examine the elastic deflections of the ball caused by the contact force Pc(t), now assuming both the ball and the half-space to be elastic. Elastic deflections that are comparable in magnitude to the input displacement, equation (1), imply that the center of mass of the ball does

>

Velocity

1."

,%:

(a)

Moss

M2

(b) Figure 1. (a) An elastic ball with a suspensionsystemrolling on a wavyelastichalf-space.(b) Lumped-mass representation of the system of Figure l(a). not move the same amount as the point of contact between the ball and half-space, an assumption that is implicit in the formulation of the rigid-body model of Figure l(b). According to Hertz's theory of elastic point contact [2], the relationship between the contact force Pc and the downward motion of the center-of-mass of the ball, z, is Pc = Cz 3/2,

(6)

where C is a function only of the ball radius R and the elastic constants of the ball and halfspace. It is understood in equation (6) and the following that z 3/z = 0 if z < 0. Equation (6) is valid for time-varying Pc if the frequency of variation is small compared to the first resonance frequency of the elastic ball; approximately, we require that C t / R >>¢0,

where C~ is the propagation speed of longitudinal waves in the ball material.

300

P. RANGANATHNAYAK

Combining equations (2), (4) and (6), we obtain Z = (?o/C +

(Ml o9~w o / C ) [(1 _p2)2 + (2~p)211/2sin (cot + 4)} 2/3.

(7)

When the damping ratio ~ is small, ~ ~ 0, and it may be seen that the variable part of z (z is the downward motion of the center-of-mass of the ball) and the input w(t) (the upward motion of the contact point) are in phase. Defining a static Hertzian deflection

(8)

Z s = ( P o / C ) 2/3,

we obtain, as a first approximation to z, z ~ z~{1 + ( 2 g l co~/3e0)w0[(1 -p2)2 + (2~p)2]1/2 sin (oJt + 4)}.

(9)

The requirement that the amplitude of the variable part of z be small compared to the input w0 (necessary to justify the neglect of contact deflections) then reads (2M, w~zsl3Po)[(1 _pZ)2 + (2~p)Z]11z < 1.

(10)

The quantity (3Po12M~ zs) 112defines a frequency we, which we shall call the contact resonance frequency. Thus we require (oJ~/co~) [(1 _p2)2 + (2~p)2ll/Z ~ 1,

co2¢= 3Po/2M, zs.

(11)

Assuming that ~ is small, we then have two limiting inequalities to be satisfied for the lumpedmass model of Figure l(b) to be a valid representation of the problem of Figure l(a). (i) p = O (1). We require ~Oo 2 ~ co~. (12) Defining a contact stiffness kc by

kc = 3Po/2Zs = ~C2/3 p 1/3,

(13)

we find that equation (12) requires Ks ~ k¢.

(14)

In most applications, this inequality is readily satisfied. Thus for forcing frequencies that are of the order of the suspension resonance frequency, the lumped-mass model is generally likely to be valid. (ii) p >> 1. We now require oJz ~ co¢z. (15) Converting inequalities (12) and (15) into words, the forcing frequency w has to be small compared to the contact resonance frequency oJ¢ for the lumped-mass representation to be valid. Thus, for sufficiently short-wavelength waviness of the surface in Figure l(b), elastic deflections near the contact region may no longer be neglected. Specifically, if (2zr V/h) = O[(3Po/2M1 zs)l/2],

(16)

an analysis taking Hertzian deflections into account is required. We now proceed to such an analysis for the case of the rolling contact of two bodies, only one of which has a wavy surface (see Figure 2). It is assumed that the system is precompressed to a static load P0 by bringing the fixed surfaces S~ and $2 together. Moreover, the suspension springs are assumed to be sufficiently soft so that the spring force always remains at its static value P0If y~ and Yz are the inward motions of the two masses, measured from the position where the masses are just touching in the absence of surface waviness, then the Hertzian deflection is z =Yl +Yl + w,

(17)

301

CONTACT VIBRATIONS "//////Z/////Z S

Spring

i mass

Surfacewaviness/w~/,~ /

/

SurfaceelocltyV ,52"

Figure 2. Two elastic bodies in rolling contact, one with a smooth, one with a wavy surface. where w is the surface waviness, measured radially outwards. The contact force is then Pc = C ( y l +Y2 + w) 3/2,

(18)

and the equations of motion for the two masses become M~y~+c~Pl+Pc=Po,

i = 1,2.

(19)

If it is now assumed that Cl/m I = ¢2/M2 ~ c/M,

(20)

where M = M I M z / ( M 1 + ME), then equations (19) may be combined to give M y + e ) = Po - Pc = Po - C ( y + w)3/2,

(21)

Y =- Yt + Y2.

(22)

where 2.2. NORMAUZEDEQUATIONor MOTION The input w ( x ) in equation (21) may be transformed into a function of time w ( t ) where V = x / t is the surface velocity of the wavy surface in the direction of rolling. By using the definition of the contact resonance frequency, equation (11), and of the Hertzian deflection z, equation (17), equation (21) may be transformed into dZ ~7 drl 2 Y dr + 2~ ~ r + ~ [H('r]) r/3/2- 1] = f ( r ) ,

(23)

~7 = Z/Zs,

(24)

where r = to c t,

~ = c/2e% M ,

d2s~ . ~ d s e f ( r ) = ~72r2 + z g ~ ,

~ =-- wlzs,

(25)

and H(~/) is the Heaviside unit step-function, defined by H(7/> 0 ) = 1, H ( r / < 0 ) = 0. Equation (23) will be taken to be the standard form of the equation of motion for contact vibrations. It may be seen that the displacement input ~:(r) is equivalent to an externally applied forcef(r), given by equation (25).

302

P. R A N G A N A T H

NAYAK

A brief discussion of the physical interpretation of equation (23) is worthwhile. The quantity ~/is the normalized Hertzian compression. When r/< 0, loss of contact occurs, and then represents the separation of the contacting surfaces. When loss of contact occurs, the equation of motion is linear; when there is no loss of contact, however, the equation is nonlinear. Moreover, the non-linearity is of a type that does not appear to have been investigated in the literature on non-linear vibrations, and presents considerable difficulties, as we shall see in section 4.1. 3. UNDAMPED FREE VIBRATIONS For undamped free vibrations, the equation of motion becomes + 3 2 - [ n ( r / ) ' q 3/2 - - 1] = 0 ,

(26)

where the dot now indicates differentiation with respect to r = toe t. Though this case may appear to be only of academic interest, it in fact provides considerable insight into the problem of forced, damped vibrations. Equation (26) may be solved exactly by noting that for any solution, the normalized Hamiltonian H must be a constant: H = ½7)2 + ~gVs/2 _ ~-~7= constant = H0.

(27)

= + V~(no + ~ - ~/~)'~.

(28)

Solving for ~/, we get

Next, observe that for a periodic oscillation, ~) = 0 when ~/takes its maximum value, 7/m~x. The constant H0 is then found from equation (27): H0 =

4- ~ 5 / 2 - ~-r/m~" Tgqmax

(29)

If the maximum value occurs at ~-= 0, the time r to reach an arbitrary value ~ is given by ~max

f dr: ~rnax

f

~'d~/

(30)

-q

Thus, once a value oft/max has been chosen, it is possible to obtain r as a function of~/, and by inversion, 7/as a function of T. However, we are more interested in the relation between the period To of the oscillation and ~7,,,~. Let ~Tm~nbe the minimum value of r/. Then from considerations of symmetry of the oscillation, "qnlax

½%= f

d~/

(31)

"qmln

where ~) is a function of r/and ~Tm,~only. From equations (28) and (29), ,~ = ,V/8---~[(~Sm/2 x __ O ( , o ) ,y]5/2) - - 2.5(,~max __ ~ ) ] 1 / 2

(32)

The lower limit of the integral in equation (31), ~min, may be found as follows. We know that when ~7= ~qmin,4/= 0. We first determine whether ~/ml,is positive or negative. If it is negative, then ~) must have a real value for ~/= 0. From equation (32), we see that this is possible if and only if 3/2 ~ 2 5, or ~Tmax>~ 1"84. 7]max (33)

303

CONTACT VIBRATIONS

Thus loss of contact occurs only when "qma,I> 1"84. The equality indicates incipient loss of contact. To determine ~min when inequality (33) holds, we set ~/= "qminin equation (32) and equate ~ to zero, obtaining ~min

=

~max

--

(34)

O "~5/2 ~ --,max"

When condition (33) is not satisfied, ~m~. has to be determined from 0"4. -5/2 --qmin

--

~min

,~.~5/2 -= O ~'--qmax

(35)

T]max,

which is easily done on a digital computer. Finally, we define a normalized dimensionless frequency f2: f2 = 2zr/%. From equation (26), we see that the static value oft/is r/= 1. It is possible to show from equation (31) that as ~/ma~~ 1, f2 ~ 1. Thus £2 = 1 represents the small-amplitude free-vibration 2.0

]3.0

~ 6.0

1.8

2.6

5"0

2'4

4.5

1.6

2" 6

~! 4O

E'6

2"5

1"4

2'0

I-2

1'5

J l.O

1.0

I'0 i ' 0'8

,~, 0.6 A

1,4 G E

~V 04

t'2

0'2

o eo

o e5

, I 0"70

I

0.75

Normalized

I 0.80 vibration

[ 0"85

I 090

14 095

I'0 I0

frequency, J~

Figure 3. Results for free undamped vibration of the system of Figure 2. The curves are characteristic of second-order systems with softening asymmetrical springs. frequency. Alternatively, the dimensional contact resonance frequency of equation (11) is the frequency at which the system executes free vibrations of small amplitude. A straightforward numerical integration of equation (31) yields a relationship between ?~maxand -Q, and is shown in Figure 3. Note that ~max = Zmax/Zs"Also shown in Figure 3 are the following quantities. (i) The normalized "amplitude" of vibration, defined by ~7o = (Zma, -- Zmin)/2Z~.

(36a)

Note that when loss of contact occurs, we have, from equation (34) T]O

-5/z = 0.~ -- --qmax"

(36b)

(ii) The normalized, time-averaged area of contact, (At>/As, where As is the static area of contact. (iii) The normalized space-time-maximum normal stress, which occurs at the center of the contact region, UmMax/Cr~a,,where O'ma x s is the space-maximum static normal stress. The relationship between the area A or the normal stress cr and the contact load is discussed in detail in reference [2]. The general relations are

A~

132/3 --e ,

°'max

~ p~ c 1/3 ,

(37)

304

v. RANGANATHNAYAK

where the constants of proportionality depend on the geometrical configuration and elastic constants of the contacting bodies. Figure 3 contains a wealth of information. For example, no free vibrations can occur for .Q > 1. This is a characteristic of non-linear one-degree-of-freedom systems with softening springs [3]. From equation (26), it may be seen that the returning force acts as a hardening spring for ~] > 1, but as a softening spring for ~] < 1, the net effect over a period of vibration being to have a spring that is softer than the static spring. The characteristic curves shown in Figure 3 are called the free-vibration "splines", and are of importance because the forcedvibration response diagrams (~]m,~or % as functions o f ~ ) have these splines as "backbones". The splines therefore help in estimating the accuracy of approximate solutions of the forced vibration problem. Figure 3 also indicates that loss of contact occurs for ~ < 0"953. For very low values of damping, even very small surface waviness amplitudes are therefore likely to cause loss of contact, if they appear at a frequency ~ < 0.953. Another point of interest is that at low values of Q, the time-averaged area of contact (A} t changes substantially from the static value As. This is of some significance in models of friction wherein the instantaneous friction force is assumed proportional to the instantaneous area of contact. With such a model, a substantial reduction in the mean friction coefficient would be predicted for contacting bodies vibrating in the contact mode. Finally, we note that very substantial dynamic normal stresses can occur during contact vibrations. This is of prime importance in investigations of surface deterioration, such as those of Carson and Johnson [1]. This is an appropriate place to strike a cautionary note, however, by observing that the standard form of the equation of motion applies only when plastic deformation is not occurring. Thus, the analyses of this paper will serve to predict when plastic deformation will occur, but not to describe the nature of subsequent contact vibrations. 4. FORCED VIBRATIONS WITH SINUSOIDAL INPUTS We now proceed to analyze the case where the input ~: (see equation (25)) is given by ~: = ~:ocos (g2~-+ ~),

q~= arbitrary phase.

(38)

Equation (23) then reads •/ + 2~4/+ J[H(r/) ~73/2- 1] = -£2~o(122 + 4~2)'/2 cos (g2~"+ ~ - 0),

(39)

0 = tan-' (2~/g2).

(40)

where Writing f0 -- ~ 0 ( ~

~ + 4~2) '~,

4, = 0 - ¢,

(4l)

we find that equation (39) becomes + 2~) + ~[H0)) n 3/2 - l] = -fo cos (Q~- - ~s),

~ arbitrary.

(42)

Before attempting to solve this equation, let us review briefly the general characteristics the solution might he expected to have. In section 3, we saw that the behavior of the undamped free vibration amplitude, defined by equation (36a), is characteristic of systems with softening springs. For forced vibrations with damping, we thus expect the following behavior [3, 4, 5] as the input frequency ~ is slowly swept upwards. Initially, for sufficiently low ~, there is only one stable solution (see Figure 4(a)). A t some frequency, dependent on the magnitude of the input and on the damping, the solution bifurcates, there now being two

305

CONTACT VIBRATIONS

stable solutions. However, for small perturbations of the input, the system continues to respond at the smaller amplitude. As the frequency is increased further, a third solution, an unstable one, also becomes possible. At some frequency, dependent mainly on the input amplitude, the low-amplitude stable solution coincides with the unstable solution, resulting in an upward jump to the high-amplitude stable solution. For further increases in ~, there is only one possible solution, the continuation of the high-amplitude solution. When sweeping down in frequency, the high-amplitude response remains stable all the way until a downward jump takes place, as shown in Figure 4(a). Hayashi [4] has shown that the locus of the unstable solution divides initial conditions in a certain region of frequencies into two sets. Initial conditions such as point I~ in Figure 4(a), lying above the locus, cause the system to (o)

,...Free vibration spline ~

-'--

"~--

jUnstable

~

~" o

a

o

I

I

solution

jz, "x x

Z2

~'\ \\ I

I

(b)

~

t// /~'k Free vibration spiine

Unstable solution [ .

I I I I

Input frequency, ,/2 Figure 4. Anticipated behavior of the forced response: (a) response amplitude and (b) mean response, as functions of input frequency.

converge to the high-amplitude stable solution. Initial conditions such as point 12, lying below the locus, cause convergence to the low-amplitude stable solution. Finally, one expects the free-vibration spline, shown in Figure 3, to act as a backbone for the bent-over resonance peak of Figure 4(a). Similar comments apply to the time-averaged or mean response, (~7). A comparison of equations (6) and (37) shows that ~ ~ A, where A is the instantaneous area of contact. Thus (~) ~ (A). The time-averaged contact area (A) for free vibration is shown in Figure 3. To some scale, this is also the free-vibration spline for (~7), and we expect it, as before, to act as a backbone for the forced-vibration response curve for (~7)- This is shown qualitatively in Figure 4(b). Comparing Figures 4(a) and 4(b), we see that when the response amplitude jumps upward, the mean response should jump downward, and v i c e versa. There is thus a range of input frequency in which a high response amplitude can be coupled with a low mean value of the

306

P. RANGANATH NAYAK

response. This is a clear indication of the possibility of loss of contact (corresponding to */< 0) occurring during vibrations at these frequencies. 4.1. HARMONICBALANCE The best analytical approach to solving equation (42) appears to be the method of harmonic balance [4]. The general approach is as follows. Assuming that a periodic solution to equation (42) exists, one expands the solution in a Fourier series: */= a0 + ~ (ag'cosjO-r + bg' sinjO~-),

(43)

d=l

with a0, ag' and bg' (j = 1 ... oo) being unknowns. The quantity n(*/)*/3/2 can now be expanded in a Fourier series: H(*/) ,/3/2 = A0 + ~ Ag' cos j O t + Bj sinfl2~-,

(44)

J'=l

where the A0, Ag' and Bj are functions of the a0, ag' and bg'. After introducing equations (43) and (44) into equation (42), one matches coefficients for the harmonic terms, obtaining a series of equations involving the unknowns a0, ag' and b j, which can, in principle, be solved. In practice, however, the determination of the constants Ao, Ag' and Bg' becomes intractable when any terms beyond the first harmonic are considered. As our first and last approximation we therefore assume ~7= a0 + al cos ~Q-r, (45) the phase ~bin equation (42) being adjusted so that */ma~occurs at 7 = 0. From equation (45) we obtain H(*/) */3/2 = Ao(ao, al) + A l (ao, aO cos ~2~-+ higher-order terms, where

27r

Ao = ~ and

1f

(a0 + a l c o s x )

3/2 dx

1

2,

(46)

.4, lf(ao+a,cosx)3/2cosxdxj 0

It is understood in equations (46) that the integrands vanish when the terms in brackets are negative. Introducing equations (45) and (46) into equation (42), and neglecting the higher-order terms, we get 3Z(Ao- 1) + (3ZAl -- al g22) cos g21 - 2~al g2 sin ~

= -fo cos (g2~-- ~b)

(47)

Equating coefficients of the harmonic terms, and setting the constant term equal to zero, we get Ao(ao, al) = 1. f 2 = [Z3Al(ao, al) _ al £22]2 + 4~2 a2 ~2,)

and -

=tan

where f0 is defined in equation (41).

-l

2~al O

~ _ - - ~ ,

t

)

(48)

CONTACT

VIBRATIONS

307

Having specified ~:0, ~:and #2, one can then do a numerical search for values of a0 and a l that satisfy the first two equations of (48). The phase shift ¢ (see equation (38)) is then found from

¢=o-¢, where 0 and ¢ are given by equation (40) and the last of equation (48), respectively. This procedure is obviously time-consuming (and therefore expensive), and points to the futility of attempting higher-order approximations to 7. Before proceeding to the forced, damped vibrations, it is obviously worthwhile to see how good an approximation the method of harmonic balance gives to the undamped free-vibration solution, which was obtained exactly in section 3. This is done by setting sc = ~ = 0 in equations (48). Proceeding with the search for (a0, a~) pairs described above, one obtains 7 ~ and 70 as functions of £2 as shown in Figure 5. Also shown on Figure 5 are the exact solutions for this case.

t

'

'

'

'

'

'

'

'

1

E o_ 5

4-

-.

X

E E

o

o Z

--

0,3

I 0'4

I 05

I 0"6

! 0.7

F 0'8

Normalized

vibration

~ 0.9

I I0

frequency,

bl

1 1'2

; 1.5

d'2

Figure 5. Free vibration results for the system of Figure 2: a comparison of first-order harmonic balance

solutions with exact solutions of Figure 3. The approximate solution is inaccurate for P < 0'95. solution; - - - , harmonic balance solutions.

, "Exact"

It is clear from the comparison in Figure 5 that, for £2 < 0.95, significant errors are made by the first-order harmonic balance solution, particularly in predictions of 7max- The obvious conclusion is that the true solution 7(t) is sufficiently rich in higher harmonics that equation (45) represents a poor approximation to it. However, if one keeps in mind that true values are likely to be higher than those predicted, one might still be able to extract some useful information from a first-order harmonic balance. A case studied experimentally by Carson and Johnson [1] has ~:0 = 0.9. The quantity reported by them, however, is not the compression amplitude 70, but the amplitude of vibration of the center of mass of one disc relative to that of the other, y in equation (22). Defining X = y/z~, we have, from equations (17), (22), (24) and (38), x =

7 -

~ =

n -

~o cos

(~-

+

¢).

The amplitude of X may be defined as Xo = 1/2(Xmax-- Xmin)" Figure 6 shows the measured values of X0 for ~:0 ~ 0.9, as well as the predicted values for different ~. According to the analytical solutions, one would expect to see an upward jump in X0 at #2 ~ 0.67 when sweeping up in frequency and a downward jump when sweeping down in frequency at a value #2 that depends on ~, due to the instability of the response before the jump.

308

P. RANGANATH

NAYAK

From Figure 6, we see that the experimental data do, indeed, have an upward jump at ~2 ~ 0.67. Seeing that, of the values shown on Figure 5, ~ = 0.05 gives the best fit between theory and experiment, we nevertheless find major discrepancies. The first of these is that the measured amplitudes are larger than the predicted ones, particularly for large amplitude. This, however, is what we expected, and causes no serious concern. The second discrepancy is the presence of the peak at f2 ~ 0-5 in the experimental data. Carson and Johnson call this a sub-harmonic resonance, erroneously, we believe. Sub-harmonic resonances are defined by "response frequency = forcing frequency/n, n = 2, 3 ...", and usually occur at forcing frequencies higher than the small-amplitude resonance frequency. There are at least two possible explanations for this peak. One is that the data points were observed when sweeping down in frequency and represent the downward jump. Alternatively, the peak may result from distortion in the forcing function. Distortion would introduce a second harmonic in the input

I

N°

~k,

I

I

I

i

[

I I. I

I

I

~ =0"05

4 -&

I

I

r

I "~,~.\i /

Y

!

i \

"l,,\

t

I

Vi

I'--- S=O'l

3 o~

El ,3

2 N -6 E o Z

-//

..

~'~16

JJ

-. \

" ~~.

]l

\

I

0

0"4

I

I

!

I

I

l

0.5

0.6

0.7

0.8

0.9

1.0

Normalized

forcing

frequency,

I

1'2

I

1'3

1'4

J'2

Figure& A comparison of harmonic balance solutions for the system of Figure 2 for three values of damping with the experimental data of Carson and Johnson [1]. Xois the amplitude of vibration of the centers of mass of the two discs relative to each other. ~ = 0.05 gives the best fit, which is still not very good. ~:o= 0'9; , harmonic balance solution; - - - , smooth curve through Carson and Johnson's data [1].

at ~ = 1, exciting a quasi-linear resonance of the system. For further comments on this disagreement between theory and experiment, see section 6. Figure 7 shows another comparison between theory and experiment, for ~: ~ 0.5. Comments similar to the above apply here. There is now, however, a major discrepancy between the predicted and measured jump frequencies, for which we have not been able to arrive at a satisfactory explanation. See section 4.2.2 for a discussion of the factors influencing the upward jump frequency. Despite the fact that the harmonic balance approach is apparently not accurate enough for a detailed solution to the contact vibration problem, useful information may nevertheless be gleaned from it. For example, we have seen that the solution is good up to just beyond loss of contact (Figure 5). One could, therefore, obtain an estimate of ~:o to just cause loss of contact for given 12. and ~. To do this, one sets a0 = a~ in equation (45), and obtains, from equation (46), Ao = 8a/~2a~/Z/3rr, AI = 16"V'~2a3o/Z/5rr. (49)

CONTACT

VIBRATIONS

309

Substituting these into equations (48) and using the definition offo, equation (41), one finds a0 = (37r/8"~¢/2)2/3 ~ 0"89 0"89 [(0"903 - 02) 2 + (2[~2)2] 1/2

¢o

J

=

'

and tan~

(50)

0.903--Q2"

Equations (50) indicate that when incipient loss of contact occurs, the maximum compression will be 7]max 2a0 = 1.78, a value that agrees reasonably well with that found for free vibrations (see equation (33)). The amplitude of dynamic compression is r/0 = a0 = 0.89. The second equation of (50) then gives the magnitude of the input se sufficient to just cause loss of contact. =

4

I

I

I

I

I

I

l

~" = 0 " 0 5

~ 3

] i

~J

=

g

E z

0-4

I

I

I

I

I

1

I

I

0.5

0'6

0.7

0.8

0.9

1.0

I.I

1"2

I

1.3

1,4

~,5

Normalized forcing frequency, J2

Figure 7. Similar to Figure 6, but with a lower waviness amplitude. Note that both here and in Figure 6, for high response amplitudes the predicted amplitudes are lower than the measured amplitudes: a result to be expected from Figure 5. ~:0= 0"5; , harmonic balance solution; - - - , smooth curve through Carson and Johnson's data [1]. Paradoxically, however, loss of contact may occur for lower values of ~:0. To see this, refer to Figure 8, which shows the sort of behavior one would expect for the amplitude ~70as one swept the input amplitude at constant g2 and ~. The arrows indicate the direction of sweep. The upward jump would correspond to the upward jump in the response diagram of Figure 9(a). The downward jump would correspond to the downward jump in the response diagram of Figure 9(b). Clearly, the value of ~:0in Figure 9(b) is less than the value in Figure 9(a). Now if incipient loss of contact were to correspond to point A in Figure 8, with ~:0 = ~:A, lOSS of contact would certainly be caused by the value ~:B,if one were sweeping down in amplitude. Moreover, ~B < ~:A"However, point B does not correspond to incipient loss of contact, and would not be covered by the preceding analysis. If incipient loss of contact were to correspond to points such as C or D in Figure 8, then the lowest value of ~:o for loss of contact would, indeed, be given by equation (50). It is thus clearly necessary to determine what value of ~0 would give a resonance diagram such as that of Figure 9(b), i.e. with a downward jump in amplitude at the specified value of .(2, and to determine whether or not loss of contact occurs there. This question, and consequently the completion of the discussion of loss of contact, we postpone to the end of section 4.2.2.

310

P. RANGANATH NAYAK

In concluding this discussion o f the h a r m o n i c balance method, t h o u g h one m a y grant that considerable insight has been gained into the problem t h r o u g h it, it has to be admitted that it is nevertheless sufficiently inaccurate so that one must seek alternative methods o f solution.

D/ o

~X3 m O.

o

Loss of contact

C

I

I

I I ~'

I I I

E Z

Normalized surface waviness amplitude~ ~o

Figure 8. The dynamic Hertzian compression amplitude as a function of the waviness amplitude for fixed D and ~. If incipient loss of contact occurs at A, loss of contact certainly occurs at B, where ~:B < ~:A. t2 = I20 constant; ~ = ~0 constant. 1

(a)

E

O

o=_ Eo

I

o

i

;2o I

(b) a:z

.ta E "O

.N

# J2o

I.o

Normalized forcing frequency, J'2

Figure 9. (a) Response diagram corresponding to the upward jump in Figure 8. ~:= ~¥, ~ = ~0. (b) Response diagram corresponding to the downward jump in Figure 8. ~:= ~:B,~ = ~o. 4.2. ALTERNATIVE APPROACHES

4.2.1. Power series expansion of the non-linearity A m e t h o d described by Hayashi [4] is to use a polynomial approximation for the term 73/2 in equation (42): 7"13/2 =

e 0 -}- e I "t~ - ] - . . .

-~-

c m 9~m.

311

CONTACT VIBRATIONS

One could then use the method of harmonic balance with considerably more ease, as it would be possible to obtain closed-form relations between the Ao, Aj and Bj of equation (44) and the a0, aj and bj of equation (43). Higher-order approximations would then be more feasible. However, fairly complex non-linear algebraic equations in the constants a0, aj and b~ would still have to be solved, and may pose considerable difficulty. We have not explored this method further. 4.2.2. Heuristic arguments Having granted the difficulty of straightforward analytical techniques, one might legitimately ask how much may be learned by the judicious use of heuristic arguments. The answer is, a fair amount, if one is interested in broad strokes rather than fine detail. Our intent in the following is to obtain a broad (but reasonably accurate) picture of the response diagram, by concentrating attention on a few critical regions. If we assume that the compression amplitude ~70is small, we may attempt a solution = 1 + e(t),

[El < 1,

(51)

and then determine the conditions under which such a solution would be valid. Substituting this into equation (39), we have + 2ge + ~ m --O¢0(~ 2 + 4gz) 1/2 cos (Or + ~ - 0). The solution to this linear equation is O~0(O 2 + 4~2)~/2 • - [(1 - 02) z + (2~Q)2] 1/2 cosOr,

(52)

where the phase angle q~of equation (52) has the value q~= 0 _ tan_l ( ~ )

= tan_l [O( 1

2~

]

-- O 2 -- 4~)J"

(53)

Imposing the requirement [e I < 1, we find O¢O(O 2 q- 4~2) 1/2 [(1 -- O2) 2 q- (2~O)2] 1/1 <- l ;

(54)

when 2~ < O, we find some interesting limiting cases. The approximation of equation (51) is valid for all O if ~:o < 2~. This, however is not a case of interest. For ~ small but not large compared to ~0, inequality (54) becomes equivalent to

(1-ff22)2>~(2~O)2

and

02~:o< [1 __~2[.

(55)

Again, two limiting cases are found. (i) O >> 1.

~0 < 1.

(56a)

~o < l~Oz.

(56b)

(ii) O ~ 1.

From a comparison of equations (56a) and (56b), we see that the constraint on ~0 for the approximation of equation (51) to be valid is more stringent at high frequencies than at low frequencies. In any case, equation (52) may be used as a guide to the low- and high-frequency behavior of ~/. For reasonably small values of ~, the frequency at which the downward jump occurs in the response diagrams of Figures 8 or 9 is very close to the frequency at which the response

312

P. RANGANATHNAYAK

and the inputf(r) are out of phase by 90 °, which is also the frequency at which maximum power is dissipated. This frequency thus corresponds to the resonance frequency of linear systems. If we multiply the equation of motion, equation (23), by ~ and average over a whole period, we get

The first and third terms, being averages of total derivatives, must vanish, leaving 2~(~2> = (~f(r)),

(58)

which merely states "Average Dissipated Power = Average Input Power". If one now assumes that near the downward jump, where the response diagram intersects the free vibration spline, the velocity ~ in equation (58) is the same as the undamped freevibration velocity at the same frequency, then it is possible to determine the one value of £2 which satisfies equation (58) for a given ~:0and ~ (see equations (25) and (38) for the connection betweenf(r) and ~:0), by assuming further that ~ a n d f ( r ) are in phase. Suppose, for example, that the free vibration solution may be approximated by 7/= ~/m(£2)+ %(£2) cos (Or + ~ + 7r/2).

(59)

Then, with f ( r ) = g2~/~-~ + 4~2 ~:0cos (£2r + ~), equation (58) yields ~ 0 = ½X/~'~ + 4~z ~:0,

(60)

an equation that may be solved for .(2 as a function of ~:oand ~, by using Figure 3. If however 4~z ¢ £22, equation (60) becomes 70 = ½(~:0/0 ~. (61) The solution to equation (61), designated ~, is shown in Figure 10. The value of T0just before the downward jump, ~0, or the maximum compression occuring during oscillations just before the downward jump, ~=a,, may be found for given values of~0 and ~ by first determining from Figure 10, and then determining ~0 and (/maxfrom Figure 3, for £2 = ~. It is clear from equation (61) that increasing ~:0 by some factor or decreasing ~ by the same factor decreases the downward jump frequency by the same amount. Incidentally, the average power dissipated at this frequency is (/'I)ma x = ~7}2~¢~2= ~2 ~4/4~, (62) being given as in Figure 10. Finally, assuming that the forced-vibration resonance curve is wrapped around the freevibration spline, we wish to determine how broad the resonant peak is. We extrapolate a result which is exactly true for linear systems, and approximately true for many non-linear systems: the half-power points on the resonance curve are separated by a frequency band A£2 given approximately by A£2 = 2~. (63) The half-power points are defined as those points at which the dissipated power is half the maximum dissipated power, given by equation (62). Equation (63) is even approximately valid, however, only when the input f ( r ) in equation (23) has a frequency-independent amplitude. In our case, the amplitude varies approximately as £22 (for small 0, and one would expect the dissipated power to vary as £24, as, for example, in equation (62). We therefore apply

CONTACT VIBRATIONS

313

the half-power criterion to the dissipated power divided by/24. For any frequency, we calculate the dissipated power from the free-vibration spline: (/-/(/2)) = ~o~(/2)/2~, and determine the half-power points from (/-/(/2))//24 = ~2(~r~)/ff~2= ½(Hrnax)l~4 = ~2/8 ~.

(64)

Equation (64) may be reduced to r/o(/2) = 1/~/2(seo/2/2~).

(65)

The solution to this equation, designated/2~/2, is also shown in Figure 10. One thus expects the response curve to have a bandwidth of approximately 2~ at [2 =/2J/2. rO

1

I

]

I

I 2

P 4

I 6

I 8

J

i

[

I

I

', 12

I 14

I i6

18

20

09

0"8

07

06

05

0

I IO

22

Figure 10. The downward jump and half-power frequencies ~ and .Qu2,as functions of waviness amplitude ~o and damping ~. These curves were obtained by heuristic arguments.

In our heuristic synthesis of the forced-vibration response curve, we have so far obtained the following: (i) high- and low-frequency asymptotes; (ii) the amplitude and frequency at the downward jump; (iii) the "half-power" points; (iv) the bandwidth at the half-power points. To complete the synthesis, we need to know the frequency at which the upward jump occurs. At this point, simple heuristic arguments fail. There is no analogy between the upward jump and any phenomenon in linear systems. We know of no physical explanation that attaches to it, apart from that of instability. A rather more complex argument works, however. In our own studies of the harmonic balance technique (not reported here) and in other studies of non-linear vibrations [5], it has been observed that the frequency at which the jump occurs is practically independent of ~, being a function primarily of the input amplitude. (The amplitude of response just before the jump occurs, however, does depend on ~.) Since the harmonic balance technique is good for small response amplitudes, one may obtain the upward jump frequency, designated ~, by using a high value of damping (thus ensuring low response amplitude). The results of such an exercise are shown in Figure 11. We are now in a position to consider the question of whether the characteristic jumps will or will not occur. Clearly, the jumps will not occur for a given ~:0and ~ i f ~ (downward jump frequency) is greater than ~ (upward jump frequency). Whether or not ~ > ~ can be determined by comparing Figures 10 and 11. For example, for ~:0 = 0.9, we have, from Figure 11, ~ ~ 0.66. Setting .O = ~ = 0.66, we get, from Figure 10, ~:o/~~ 8-6, or ~:o ~ 8.6~ = 0.9. 20

314

P. RANGANATH

NAYAK

Clearly, no jumps will occur for ~ > 0.9/8.6 ~ 0.105. In fact, as ~:o varies from 0.3 to 3, the minimum value of ~ for no jumps varies from about 0.08 to about 0.13. Thus it appears possible by a combination of harmonic balance and the heuristic arguments described above to obtain a reasonably good synthesis of the response diagram. One example <%

~.o

t

1 l

I

]

I

cr ct. 0"8 0-7 Q.

0.6

"6 0 5 E ::" (3.4 0

0.5

1.0 J.5 2.0 2,5 3,o 3.5

Normalizedwavinessamplitude,¢"o

Figure 11. The upward jump frequency ~ as a function of the waviness amplitude ~:o. ~ is practically independent of the damping.

1

I

1

1

1

I

I

I

I

0"7

0"8

0"9

I'0

1'1

1.7'

~spline

~:~o.___~co~o. 45

-~- 2

,'2

0"3

0'4

0'5

0'6

1'3

Normalized forcing frequency, ~ F i g u r e 12. Comparison of Carson and Johnson's experimental data [1] with a curve synthesized from the heuristic arguments of section 4.2.2, and with the help of Figures 10 and 11. ~:0 = 0"9, ~ = 0 . 0 5 ; , synthesized curve; - - - , smooth curve through Carson and Johnson's data [1].

o f such a synthesis is shown in Figure 12, along with Carson and Johnson's data [1]. A comparison with Figure 6 shows that the synthesis technique gives a much better fit to the data than the harmonic balance method by itself. We note that in the synthesized response of Figure 12, the results of the harmonic balance method were used for £2 > 1 and for the lower branch of the response curve for 12 < 0.67. It is useful to summarize some of the heuristic arguments detailed above.

CONTACT

315

VIBRATIONS

(i) For low and high frequencies (compared to #2 = 1) or for very high damping or for very low input amplitude, the equation of motion may be linearized by assuming that the response amplitude is small compared to the mean response (~ 1). (ii) An increase in the input amplitude ¢0 decreases the frequency of the downward jump, ~, while increasing the response amplitude just before the jump. It has a similar effect on the frequency of and amplitude before the upward jump. However, it has relatively little effect on the bandwidth of the resonance peak. (iii) An increase in the damping ~ increases ~, while decreasing the response amplitude just before the jump. It has relatively little effect on ~, but decreases the response amplitude before the upward jump. (iv) The quantity 2~ is a good measure of the "half-power" bandwidth of the resonance peak. b

I

I

[

0.6

0.7

0.8

I

l

I

I

I

I I.O

I I.I

I 1,2

I 1.3

I

¢04

0.5

E

I

0

~ 0"5

i 0.9 '1k Normolized

forcing

I 1,4

1,5

frequency, d'2

Figure 13. The minimum waviness amplitude necessary for loss of contact. To the fight of the vertical arrows, incipient loss of contact occurs,and equation (50)applies; to the left,loss of contact occurs, for a finite length of time during each oscillation. (v) At the downward jump, the velocity of response and the forcing function f0-), equation (23), are approximately in phase. The dissipated power may be estimated by approximating the forced-vibration response waveform by the free-vibration waveform. (vi) For small £2, the response 7/and the input ~: are in phase; for large £2, they are out of phase by 180° (see equation (53)). We now conclude the discussion of loss of contact, left hanging at the end of section 4.1. We had concluded that to find the minimum value of ~:othat would cause loss of a contact for given.(2 and ~, it would be necessary to determine the value of~:0that would cause a downward jump at #2, and also whether loss of contact would occur just before the jump. This value of ~:omay be obtained from the curve for ~ in Figure 10. For given £2 and ~, we set ~ = 12, and obtain ~:0= ~ x (function of£2) from Figure 10. From Figure 3 we see that loss of contact occurs only for #2 < 0.953. Thus, for Q -<<0.953, the value of ~:0 obtained in this manner must be compared with the value given by equation (50), which holds for incipient loss of contact, in order to determine the minimum value of ~:0for loss of contact. For £2 > 0-953, the minimum value is given by equation (50). Curves obtained in this fashion are shown in Figure 13 for a few values of ~. The minimum value of ~:0 for loss of contact is designated ¢~1,.

316

P. RANGANATH NAYAK

We observe the interesting result from Figure 13 that for small ~2, increasing the damping initially increases ~ " , but eventually causes it to turn around and decrease. There thus appears to be an optimal value of damping for prevention of loss of contact. An intuitive explanation of this result is as follows. The surface waviness initially (during a period of oscillation) forces the bodies apart in a quasi-static manner. As the waviness falls away, the bodies try to follow it, but their ability to do so depends on the magnitude of damping. For high damping, the returning force (the static normal load) is low enough so that the bodies approach each other more slowly than the surface waviness falls away, leading to loss of contact. For lower damping, the bodies are able to ride the surface well. For very low values of damping, the high amplitude response on the resonance peak becomes a possibility, and loss of contact once more becomes easy. 5. BROADBAND RANDOM INPUTS 5.1. THE JOINT PROBABILITY DENSITY

We now turn to the more realistic case (for rolling contact) where the waviness of the rolling surface is random rather than sinusoidal. Although approximate analyses for narrowband inputs are possible [6], we postpone their discussion to a future article, as they are sufficiently complex to deserve a separate treatment. Here, we treat the case of broadband random inputs. It must be left to experimental work to determine whether or not engineering surfaces give rise to broadband inputs at the frequencies we are concerned with. For the sake of clarity, we rewrite the equation of motion derived in section 2: d2 ~7+ 2~ d~7 -dr- T d r + ~[H(~) n

3/2 --

1] = f ( r ) ,

n - z/&,

(66)

where d 2 ~:

f(z) = ~

^ ~d~:

+ 2:~ ~ ,

~ =- w/zs,

r - toe t.

(67)

Here z is the Hertzian compression, zs the static value of z, toc the contact resonance frequency of equation (1t), w the surface waviness, and H(r/), the Heaviside unit step-function, H(~ i> 0) = 1, H07 < 0) = 0. We now assume thatf(-r) is broadband. It is not enough for the bandwidth o f f to be large compared to the half-power bandwidth of the non-linear system ( ~ 2~), as with linear systems; it must be large compared to the width (in frequency) of the bent-over resonance spike of Figures 5 or 7, i.e. large compared to 1. Then an application of the Fokker-Planck equation (see, for example, reference [7]) yields the stationary joint probability-density for and ~: P(~7, ~) = Gexp{-2~/rrq~:f[½il 2 + 3z(0"4H(~) ~5/2 _ ~/)]} (68) where q~ff is the (approximately constant) power-spectral-density (PSD) o f f ( r ) , and G is a normalizing constant such that f ; p07,7})dr]d~ = 1. --co

(69)

--CO

Several facts, well known in the theory of non-linear random vibrations, may be seen from equation (68). The displacement ~/and the velocity 9 are uncorrelated. The velocity ~ has a Gaussian probability distribution, with zero mean and a standard deviation ~

=

(rr~zzl2~) u2.

(70)

CONTACTVIBRATIONS

317

On the other hand, the displacement 7 has a probability distribution that is skewed to the left (negative third moment about the mean) [8]. If the quantity (2~/~-~bfs) is large (see the following section for a discussion of ~bss), equation (68) may be approximated by expanding the function (0.4H(7)75/2- 7) in a Taylor series about 7 = 1, where its maximum occurs, to obtain P(7, 4) ~ G exp {--2~/Trqbef[½#2 -- 0"4 + ½(7 -

1)2]}

•

(71)

For this case, we see that 7 is also approximately Gaussian, with a mean value 7 = 1 and a standard deviation equal to that of the velocity: cr~ = or,)= (Tr~bfs/2~)t/2.

(7.2)

Substituting equation (71) into equation (69), we have an approximation for G, valid when (2~/~r~,y) is large: G m (~/~.2 ~ss) exp (-0-8 ~/~-¢bs~,). (73) Combining equations (71) and (73), we obtain P(7, 4) ~ (~/~.2 qbs~,)exp {-2~/zrtDy.r[½~) 2 + ½(7 - 1)2]}•

(74)

5.2. THE PSD (/)yy Suppose the dimensional waviness has a spatial PSD ~ ( k ) , where the wavenumber k = 2rr/A, A being the spatial wavelength. If the surface is being traversed at a velocity V, the frequency PSD is given by

~w~(o~) = ~ ( k )

~_k, fl¢o

t75)

where oJ = Vk, leading to q~ww(co)= (I/V) Cbww(k--- oJ/V).

(76)

Proceeding with a series of steps analogous to this, one finally obtains the PSD o f f ( z ) in O-space: qbss(/2) = oJc q)ss(OJ) = (w d Vz~) (/24 + 4~2/22) q~ww(k= w/V). (77) It is this PSD that we assume is broad compared to 1. When 2~
(78)

As an example, consider a PSD that is frequently encountered on surfaces: qOww(k) = 1ILk'*,

L = constant.

(79)

Substituting this into equation (78), we obtain

q'ss(/2) ~

(V/°'o)3/Lz~

•

(80)

For this case, we have, from equations (70) and (72), cruz,,~ a~ ,,~ rr/Z~ . ( VloJ¢) 3 • 1/ LzZ~.

(81)

Several interesting facts may be learned from this example, which one would expect to hold broadly for other PSD's ~bw,~(k). (i) Decreases in the damping ~ cause en to increase, thereby increasing the probability of large Hertzian compressions. (ii) Increases in the rolling velocity V have a similar effect, except that the effect is much stronger. (c% ~ V 3/2 ~-1/2.) This might explain the "critical" velocity for the formation of corrugation on rolling discs observed by Carson and Johnson [1].

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P. RANGANATH NAYAK

(iii) A decrease in the normal load P0 (see Figure 2) results in a decrease in both oJc(wc p~/6) and in zs(zs ~ P 2o/3),thereby resulting in a large increase in %. At the same time, however, the mean value of the dimensional compression z(z = zs~) is decreased. Thus it is a moot point as to what effect changes in P0 have on the probability of obtaining large values of z. We now examine this question in greater detail. 5.3. PROBABILITY OF PLASTIC DEFORMATION

We continue to assume that the PSD ~//(12) is broad. Specifically, we assume that, in equation (78), (O4 ~ww(k = o.)/V) = N ( V ) ,

(82)

i.e. that the quantity on the left depends on V but is independent of w. Thus equation (78) becomes ~ s s ( ~ ) = constant = N ( V) / Vz 2 w 3. (83) From equation (72), we have 02 = N ( V)/2~ Vz 2 off,

(84)

p(~) ~ ( l / % V ' ~ ) exp [--Jz(~ - 1/%)21 •

(85)

and from equation (74), Suppose z = z p is the Hertzian deformation necessary to cause plastic deformation. We may then evaluate the probability ~ that plastic deformation will occur: ~'(z > z , ) = ~ ( ~ > ~ ) = f p ( ~ ) d ~ 7 z ½ e r f c [ ( 1 / % V ' 2 ) ( z p / z , - 1 ) ] .

(86)

zp/z.

Assuming that ~ does not vary with the static load P0, we may now study the effect of variations in Po on ~ , all other variables being held constant. First, let the contact load corresponding to zp be Pp. Next, note from equation (84) that % = S p ~ l 1/12,

S = constant.

(87)

Equation (86) may then be written as ~ ½erfc{(P~l/12/SV/2) [(Pp/Po) 2/3 -- 1]}.

(88)

If we define rp = Po/er,,

(89)

the argument of the complementary error function is

(Pp11/12 /S~2)(rpt/4 - r~/~).

(90)

This function has a maximum at rp = 0.141, i.e. Po = 0.141 Pp. Noting that erfc(x) decreases as x increases, we see that this load would then give a minimum probability of plastic deformation. Loads higher or lower than 0.141 Pp would give an increased probability, a conclusion that is in striking qualitative agreement with one of the experimental observations of Carson and Johnson [1 ]. A detailed evaluation of the effect of load, however, cannot be based on the approximate P07) we have used; the expression in equation (68) must be used instead. This involves the evaluation of a fairly difficult integral, which we have not as yet attempted. Some further insight into the development of corrugations on the rolling surface may be gained by examining the frequency with which the plastic load Pp is exceeded. We now proceed to do this.

CONTACT VIBRATIONS 5.4. LEVEL-CROSSINGSTATISTICS Rice [9] has shown that the frequency of upcrossings of the level ~

319

=

zp/z s is given by

co

~+ =

/~p(zp/z~, "/1)d~.

(91)

0

Substituting from equation (68) into this, we obtain v+ = (~/V27)p(zdz,).

(92)

Here, p(~/) is the probability density for ~: p(r/) = G exp [-(4~/3zr@//)(0"4-q 5/2 - ~/)],

(93)

where G is again a normalizing constant. If we again make the approximation of equation (71), we obtain 1 f 1 [z,/zs- l\q v+ ~ ~ e x p [ - ~ ~ j j, (94) where % is given by equation (72). To see the effect of the static load Po on v+, we proceed as before and find v+ z (1/2zr)exp [-(P~1/6/2S2)(rl/4 -rp 11112) 2],

(95)

where r, = Po/Pp and S is defined in equation (89). The exponent has a minimum at rp = 1, corresponding to the largest frequency of upcrossings, and a maximum at rp=0-141, corresponding to the smallest frequency of upcrossings. It is interesting to note that for rp ---> 0% the frequency of upcrossings decreases. The interpretation of this is that the contact load spends an increasing length of time above the value for plastic deformation, Pp, seldom coming below it, and therefore, seldom giving the opportunity for upcrossings. Such a situation, one could argue, would tend to give rise to large stretches of continuous plastic deformation rather than corrugations, a result again comparable qualitatively to an observation of Carson and Johnson [1]. This argument gives insight into the development of corrugations as opposed to the occurrence of plastic deformation: it is not the probability of plastic deformation ~ , or the frequency with which the limit load Pp is exceeded, v÷, that is important by itself, but the mean duration of each exceedance. This duration, r ÷, is clearly given by (96)

r + = ~ / v +.

When r + = O[Dr], the average spatial length of the plastic indentation gives rise to an input whose frequency is comparable to the contact resonance frequency; subsequent passes over the indentation would then cause a large transient response, leading to the formation of corrugations. A detailed study of this line of reasoning must be postponed to a time when adequate information is available on the PSD of the input, the detailed nature of p(~, ~)), and the characteristics of the transient response. Some insight may be gained, however, by noting that the asymptotic form of erfc(x) is [10] erfc(X)~x~e-X2,

x --~ ~.

(97)

Combining this with equations (88), (95) and (96), we obtain

sV~-;

q-+ ~'~

Pp11/12 (r,1/4 - r p 1 1 / 1 2)"

(98)

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Qualitatively, we find that the smallest mean exceedance time is found for rp ~ 0.141 (Po ~0.141 Pp). As rp--~ I (Po---~ Pp), the mean exceedance time becomes very large. As rp ~ 0, the mean exceedance time again becomes large, There is clearly some value of rp giving rise to a mean exceedance time which is best for the formation of corrugations. It is left to future work to investigate this idea in detail. 6. CRITIQUE AND DISCUSSION In the formulation of the equation of motion for contact vibrations, in the analysis of this equation, and in comparisons of the results with experimental data, many points have been slurred over, which we now wish to discuss briefly. The first of these is our assumption of viscous damping. Whereas there is no doubt that viscous damping will be present to some extent in most engineering applications, it is equally clear that hysteresis losses in the contacting bodies will be present. Furthermore, for a given stress cycle, the energy loss in a unit volume of material is likely to be dependent on the frequency of stressing. Thus, it is likely that a very complicated damping model will emerge. We will remain content to observe that when the magnitude of external damping is high (say > 0.1), the hysteresis losses are unlikely to affect the response. For low values of external damping, the hysteretic losses introduce a significant non-linearity into the damping. Presumably the effects of such a non-linearity would be (i) to increase the downward jump frequency ~, and thereby decrease the maximum response and (ii) to increase the bandwidth of the resonant peaks somewhat. The upward jump frequency ~ as well as the low- and highfrequency asymptotes of the response curve would be relatively unaffected. The net result would therefore be a somewhat shortened, broadened resonant peak. The interested reader is referred to Hayashi [4] for a discussion of deterministic vibrations with non-linear damping and to Crandall et al. [11] for random vibrations; the latter also contains a discussion of hysteretic damping. Next is the substantial discrepancy found between the theoretical and experimental points near ~ = 0.5 in Figure 5. In comments on this discrepancy in section 4, we suggested that the discrepancy might be due to experimental error, implying more confidence than would be justified in our (approximate) analysis. Another possibility remains, however: Hayashi [4] shows for a particular non-linear system that for large values of input, superharmonics may occur and dominate the response, without distortion of the input. This possible explanation of the discrepancy in Figure 5 has not been explored analytically. We might mention, however, that in an analogue computer study of contact vibrations (which we hope to report later), no peak was observed at J2 ~ 0.5. Third, in our discussion of plastic deformations and corrugations, no mention was made of tractive forces in the contact region, which, as has been shown by Johnson and Jefferis [12], may significantly influence the geometry of plastic indentations, as well as the normal load at which they occur. One obvious effect is that they introduce an asymmetry about the center of the contact, making the plastic indentation itself asymmetric (when it occurs). If one now introduces the variable of direction of rolling, it appears possible that differences may occur between the case where the tractive force is in the direction of rolling and where it is in the opposite direction. (The two cases occur, for instance, in a rolling wheel being accelerated and braked, respectively.) For example, if the plastic indentation were to have a ridge on one side, then depending on the direction of rolling, one would in a subsequent pass, see first either the indentation or the prow of the ridge. The response of the system to these two inputs would be expected to be grossly different. These conjectures are, in fact, based upon differences in the speed of development of corrugations observed by Carson and Johnson [1] between what they term "positive" and "negative" sliding, corresponding, respectively, to the tractive force being aligned with and opposed to the direction of rolling.

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Finally, we have omitted all mention of subharmonic response, which is to be expected for £2 > 1 for certain ranges of ~:0 and ~. There are reasons for this omission; the harmonic balance method could be used, but would be subject to the same inaccuracies and computational difficulties as for harmonic vibrations. The alternative of heuristic arguments we have not so far explored, and expect will present considerable difficulties. Analogue computer studies do present a viable third alternative. For the case shown in Figure 12, for example, the analogue results show large subharmonic response at ~2 g 1.2, with both upward and downward jumps. The response amplitude is in fact larger than for the harmonic resonance. Similarly, no mention has been made of the spectrum of the contact force for the case of random inputs. This may be of some interest in analyses of the generation of acoustic noise during rolling contact, as also in the analysis of surface deterioration. Approximate techniques, based either on linearization [13] or heuristic arguments [14] are available for the generation of the response spectra of non-linear systems, but have not been explored for the contact problem. Despite these reservations, one may fairly grant that a substantial beginning has been made in the analysis of the vibratory contact of elastic bodies. It is hoped that in the future, many of the shortcomings of the analysis will be removed, and that it will be placed on a surer footing by a detailed comparison with experiment.

ACKNOWLEDGMENTS I would like to thank my colleagues, Dr Preston W. Smith, Jr for many illuminating discussions on non-linear vibrations, and Mr Robert B. Tanner for generating the analogue computer data briefly mentioned in section 6. This work was supported in its entirety by the United States Department of Transportation, under Contract No. DOT-FR-10031; this support is gratefully acknowledged.

REFERENCES 1. R. M. CARSONand K. L. JOHNSON1971 Wear 17, 59-72. Surface corrugations spontaneously generated in a rolling contact disc machine. 2. S. TIMOSHENKO1951 Theory of Elasticity. New York: McGraw-Hill Book Company, Inc. Second edition. 3. J. P. DEN HARTOG1956 Mechanical Vibrations. New York: McGraw-Hill Book Company, Inc. Fourth edition. 4. C. HAYASHI1964 Nonlinear Oscillations in Physical Systems. New York: McGraw-Hill Book Company, Inc. 5. W. J. CUNN1NGHAM1958 Introduction to Nonlinear Analysis. New York: McGraw-HiU Book Company, Inc. 6. R. H. LYON, M. HECKLand C. B. HAZELGROVE1961 Journal of the Acoustical Society of America 33, 1404-1411. Response of hard-spring oscillator to narrow-band excitation. 7. T. K. CAUGIaEY1963 Journal of the Acoustical Society of America 35, 1683-1692. Derivation and application of the Fokker-Planck equation to discrete nonlinear dynamic systems subjected to white random excitation. 8. A. M. MOOD and F. A. GRAYBILL1963 Introduction to the Theory of Statistics. New York: McGraw-Hill Book Company, Inc. Second edition. 9. S. O. RICE 1944, 1945 Bell System Technical Journal 23, 282-332 and 24, 46-156. Mathematical analysis of random noise. Also reprinted in N. Wax (Ed.) 1954 Selected Papers on Noise and Stochastic Processes. New York: Dover Publications, Inc. 10. N. GAUTSCHI1965 Error Function and Fresnel Integrals, in M. Abramowitz and I. A. Stegun (Eds.) Handbook of Mathematical Functions. New York: Dover Publications, Inc. 11. S. H. CRANDALL,G. R. KHABBAZand J. E. MANNING1964 Journal of the Acoustical Society of America 36, 1330-1334. Random vibration of an oscillator with nonlinear damping.

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12. K. L. JOHNSONand J. A. Jr.rrr.ais 1963 Proceedings of the Institution of Mechanical Engineers Symposium on Fatigue in Rolling Contact. Plastic flow and residual stresses in rolling and sliding contact. 13. T. K. CAUGrlEY 1963 Journal of the Acoustical Society of America 35, 1706--1711. Equivalent linearization techniques. 14. J. E. MANNING1965 The Spectrum of Random Vibration of Mechanical Oscillators with Nonlinear Springs. Sc.D. Dissertation, Massachusetts Institute of Technology.