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Vibration response of wavy surfaced disc in elastohydrodynamic rolling contact
Wear, 139 (1990)
1
I- 15
VIBRATION RESPONSE OF WAVY SURFACED DISC IN ELASTOHYDRODYNAMIC ROLLING CONTACT H. MEHDIGOLI Tribology Section, Department of Mechanical Engineering, Imperial College of Science and Technology, Exhibition Road, London SW7 2BX {U.K.) H. RAHNEJAT On-line Su~ei~lance, ~onito~ng and diagnostics Unit, School of~echaniea~, Aero~u~cal and ~oduction Engineering, Kingston PoZytechnie, Canbury Park Road, Kingston, Surrey (U.K.) R. GOHAR Tribology Section, Department of Mechanical Engineering, Imperial College of Science and Technology, Exhibition Road, London SW7 2BX (U.K.) (Received July 14, 1988; revised July 20, 1989; accepted December 7, 1989)
Summary
The m~itude of vibration and noise generation in rotating machinery is affected by the value of damping. Damping has a significant influence on the reduction of untoward motions. It can be exerted externally, for example, by the support structure and squeeze films in seals and internally by material hysteresis, dry friction and the lubricant squeeze film action in elastohydrodynamic (EHD) contacts. This paper investigates the,influence of EHD lubricant film damping and preloading in the concentrated non-conforming contact of rolling and normally approaching discs. It is shown that under high preloading the damping effect is slight, but the level of preloading affects the value of the natural frequency.
1. Introduction The significance of fluid film damping in the elastohydrodynamic (EHD) contact of rolling and normally approaching rolling elements has recently been investigated by various authors. One of these investigations was the study of squeeze film variation in the non-conforming EHD contact of a pair of cylinders by Herrebrugh [ 11. In his analysis he concluded that the squeeze film damping contribution is more significant in the region of thick films. Rohde et aI. [2f also studied the dynamic response of normally approaching bodies subjected to fluctuating normal load. They compared the results of rigid, elastic and viscoelastic contacts and concluded that the 0043-1648/90/$3.50
@ Elsevier Sequoia/Printed in The Netherlands
2
different response characteristics observed point to significant fluid film damping that may occur under hydrodynamic and EHD conditions. The effect of squeeze film action in lubricated EHD contact of rolling elements under oscillating conditions was also investigated by Walford and Stone [3] as well as Chandra and Rogers [4] under hydrodynamic conditions for normal approach and rebound of lubricated cylinders. More recently Safa and Gohar [5] obtained experimentally transient pressure distributions for the normal approach of a falling and rebounding ball against a flat race under EHD conditions using a manganin microtransducer. Dareing and Johnson [6] have carried out valuable experimental work with a disc machine where the rolling mating discs were subjected to periodic excitations caused by surface corrugations on one disc. They showed that under certain contact conditions resonance occurs with the lubricant film contributing significantly to the damping. Their theoretical model, however, approximated the lubricant film to a linear spring and damper arrangement which is a very simplified representation of an EHD lubricant film. More representative oil film behaviour in rolling and squeezing EHD elliptical contacts was presented by Mostofi [7]. His formula was used in the contact of mating discs and ball to races contacts in deep-groove ball bearings in works presented by Rahnejat [8] and Rahnejat and Gohar [9]. They showed that the contribution of squeeze film damping is more significant in the region of thick films, pertaining rather to hydrodynamic conditions. This paper presents the results obtained when modelling the elliptical EHD contacting geometry of mating discs used in ref. 6. The numerical method used is the same as that described in refs. 8 and 9, where the surface acceleration of the EHD lubricant film boundaries is assumed to vary linearly over a small time interval.
2. Simulation
model
The numerical model is designed to simulate the experiments of Dareing and Johnson [6] who employed a disc machine. The corrugations on the surface of one of the discs, used in the work of ref. 6, is modelled here by a similar sinusoidal function. The contact oil film thickness uses the EHD lubrication (EHL) oil film regression formula of Mostofi and Gohar [lo] with a squeeze velocity term included, as described by Mostofi [ 71. 2.1. Specifications of the simulated disc machine The disc machine used in ref. 6 employs a pair of mating discs, each with a diameter of 102 mm. The source of excitation is a corrugated surface on one disc of amplitude 1.27 pm. There are 120 corrugations around the disc’s circumference. The other disc is crowned to a radius of 102 mm. Each disc and its supporting shaft are mounted on a hydrostatic journal bearings housing which in turn is supported by soft leaf springs. The stiffnesses of the leaf springs are negligible and the bearing housings have very
3 TABLE 1 Lubricant properties Lubricant
Viscosity (at 30 “C Ns rn-?)
piezoviscosity ( m2 N-l)
Shell Vitrea 79
0.890
25.9 x 10-9
Shell Turbo 33
0.034
21.6 x 1O-9
high stiffness compared with the contact stiffness of the mating discs. The disc contact stiffness constant is found from the experimental data of ref. 6, to be k, = 28 (GN m-3)-2. A constant preload of Q = 810 N acts upon the discs through the leaf-springs. The total mass of one disc with its attachments is 20.4 kg. Two types of lubricant are used as is the case in ref. 6 and are listed in Table 1. 2.2. The elastic contact deformation The crowned disc and its mating corrugated disc form an elliptical point contact footprint where the wavelength of the corrugation is much greater than the footprint minor axis and thus does not influence the hertzian contact conditions. The contact load in an elliptical contact according to Hertz [ll] is given by W = K,A3”
(1)
where K, is the contact stiffness of mating discs. The elliptical contact ratio is [ 121 eP * =
i
0.64
1
RX
1.03 F
Y
where R, and R, are the reduced radii in the xz and yz planes respectively (see Fig. 1) such that 1 1 -=-+R, RI, 1 _=-+RY
1 R2x
1 RI,
1 .R2Y
where RI, = R2x
=
RI, and RZy = 00.
Using the radii of the discs and eqns. (2) - (4), ep* = 2.5.
(3)
Fig. 1. Radii of the contacting discs.
2.3. The oil film model The minimum oil film thickness formula obtained by Mostofi [7] for EHL elliptical contacts with flow along the minor axis of the elastostatic contact ellipse is h* = 0 . 112~0~5sG*0~46W*~o~045 exp(-123w,*)
exp(O.O76e,*)
(5)
where
ho& u*=---...Rx
G*
’
a
=-’ &
w*= WE, -,
RX2
wp$
(61
A special feature of this formula is that it includes the relationship between the oil film thickness and the squeeze velocity, w, = dh/dt. 2.4. Geometrical considerations If there were no co~ugations, there would be no excitation and the oil film thickness and the contact elastic deformation would remain constant during rolling. Figure Z(a) depicts this case. Using this figure,
C, = R +,+h,+R+, or C, = 2R - 6, + h,
(7)
With the corrugated surface on one disc introducing an excitation function, x(t) = 2 sin(wt), the film thickness and contact deformation of the discs’ surfaces are functions of time and the disc centres undergo motions x1 and x2 respectively. Thus, referring to Fig. 2(b)
5
Fig. 2. Geometrical conditions of the contacting waviness.
surfaces: (a) with no waviness; fb) with
Fig. 3. The non-linear contact model: I$ for the contact spring and KO for the oil film.
c=c*
+x1+x2
(8)
and c = 2R - s(t) +”x(t) + h(t)
(9)
From eqns. (7) - (9) a relation between the geometric variables can be obtained by eliminating the terms C and C, (6 - 6,) + h, + (x1 i-x,) =x(t) + h(t)
(19)
3. The dynamic model The dynamic model is shown in Fig. 3, where the discs are represented by a pair of equal masses m on a common horizontal axis, the stiffness constant of the non-linear elastic contact is K,, and there is a non-linear stiffness-damper arrangement for the oil film thickness. The two masses are preloaded together with the constant force Q in the horizontal (that is x) direction. The contact reaction W(t) is identical in each elastic contact and the oil film since they are in series. As the oscillations occur in the x direction there is no gravity effect. Therefore, the equations of motion for the disc centres are d2x1 m-= dt2 d2x2
m -=
dt2
W(t)-Q
(11)
W(t)-Q
(12)
6
where the elastic contact W(t) is given by eqn. (1). For the oil film, eqn. (5) can be written in the form W(t) = Kh-O( t) exp (-7
dh/dt)}
where fl= 22.23 and y = 2734.3/U. that .K = (0.112)pV
(13) K is a time-independent constant such
13.9G*lo.s&fl+ 2{ exp(54.85 + 1.6e,*))/E,
There are five unknowns h, 6, x,, x2, W and five equations (1) and (10) (13). Therefore, a solution is possible.
4. Method of solution Equations (1) and (3) are doubly differentiated and equated to yield the equation
(14) For a solution to be obtained, the terms d26/dt2 and d&/dt are eliminated from eqn. (14) using the relations d26 -=dt2
d2x
d2h + -dt* dt2
_ = -26 d6
2-
d2x,
(15)
dt2
{P(dG/dt) + rh(d2h/dt2))
dt
(16)
3h
Substituting the above relations in eqn. (14), a third-order differential equation in terms of oil film roof oscillations results such that d*h
2
+
ILz-
dt2
+
+
$4
(17)
The functions G1, I)*, IJI~and ti4 are given in Appendix A. Step by step integration of eqn. (17) using the Runga-Kutta method yields the values of film thickness h and the squeeze film velocity dh/dt. Equations (13) and (1) are used to obtain the values of W and 6. Since x1 and x2 have appeared symmetrically, eqn. (10) is employed to calculate the motions of the discs’ centres.
7
5. The initial conditions In order to solve the differential equation (17), the initial values of functions $i through G4 should be known. Therefore, the values of variables h, dh/dt, d2h/dt2, 6 and d2X,/dt2 must be obtained at t = 0. Since the steady state solution is desired one can apply any eligible initial conditions instead of an actual set. The following initial steps are taken. The value of K is calculated using the governing EHD parameters. W,=Qattimet=O. 6, is calculated using eqn. (1). h = h, at t = 0. Values of dh,/dt and d2h,/dt2 cannot be evaluated analytically. Therefore, the following procedure is adopted. The source of excitation is assumed to be an exponential function Bet and commences at a point Oi far enough from the point o as to make 3c negligible (see Fig. 4). From the continuity of profile, 3c and dx/dt are the same. Thus at t = t, Beto = Lt sin(wt,)
(18)
Beto = 2 w cos(wt,)
(19)
Therefore 1 t, = - tan-'
Cd
0
and
Equation (17) is solved for the time t,. The function in this region is of the form X = B exp(t - toi) (see Fig. 4). The values at t, are the initial values for the rest of the wavy profile.
Fig. 4. The assumed initial wavy profile.
8
dh oi _ d2h,i _ 0 p--m dt, dt,2 Now the initial value of d2h,/dt3 is calculated at t = t,, using eqn. (17). Step by step iterative integration using the Runga-Kutta method can now commence.
6. Results and discussion The excitation frequency w is a function of the rotational speed of the corrugated disc and the number of corrugations. There are 120 corrugations around the circumference of the corrugated disc. For the rotational speed of N rev mm-’ w=27rN-
120 60
= 41rNrad s-l
The numerical calculations are carried out at different disc speeds to obtain the amplitude-speed response characteristics of one disc with either lubricant and with different values of preload Q. Generally two preload values of 810 N and 2100 N are used as in ref. 6. The results are presented in both time and frequency domains; the latter are then subjected to a comparative study with the findings of ref. 6. 6.1. Time domain analysis Time histories of the lubricant film thickness and the elastic contact deformation are presented in Fig. 5. The results exhibit much smaller amplitudes of oil film oscillations than the corresponding contact deflection, 6. The steady state of oil film thickness is 3.85 pm with a small ripple oscillation superimposed on it of peak-to-valley amplitude of 0.35 pm (i.e. 9% of
P .FJ
3.5.~
h
Tim [msl Fig. 5. Time histories of the oil film thickness and contact elastic deflection: Shell Vitrea 79 lubricant; N = 360 rev min-‘.
Q = 810 N;
9
the steady oil film thickness). The corresponding static contact deflection is 9.19 pm with a dynamic deflection of peak-to-valley amplitude of 6.475 pm superimposed on it. These simulated results are obtained with the Shell Vitrea oil at the disc speed of 360 rev min-’ and Q = 810 N. The time history relative displacement of discs’ centres (i.e. x = x1 - x2) is shown in Fig. 6(a). The initial 90 ms of the response show evidence of its decaying transient characteristics due to fluid film damping action. In Fig. 6(b) the steady state double amplitude of x is 3.9 pm which is of the order of the oil film thickness. The transient portion of the response consists of beats between two frequencies, one due to the free response of the system and the other at the forcing frequency induced by the corrugations, in this case at 720 Hz. The fluid film dampens the free oscillation very slowly indeed. The steady state conditions are eventually attained at the forcing frequency with the vibration maintained by the rotation of corrugated disc. This trend is observed to be even slower from the x oscillation time history of Fig. 7 for
-6.0
(a)
(b)
J
0
20
40 Tiie
Time
80
80
I GO
[%I
[msl
Fig. 6. Time history of relative displacement: Shell Vitrea 79 lubricant; N = 360 rev min-‘.
(a) 0 - 90 ms; (b) 100 - 140 ms. Q = 810 N;
10
the lower viscosity Shell Turbo 33 oil, where steady state conditions are not yet achieved after 10 000 time steps (or 1 s). Figure 8 shows the x oscillation response with Shell Vitrea oil and an increased preload of 2100 N. Comparing this with Fig. 6(b) shows that with higher preloads, the time to reach the steady state condition increases as the oil film thickness and hence oil film damping both decrease. The narrow band record of the response in Fig. 8 is indicative of sharply resonant characteristics due to the low fluid film damping, giving a maximum peak to valley x amplitude of 14.5 pm as opposed to 3.9 pm with the lower preload of 810 N and the same lubricant in Fig. 6. The observed response behaviour is easily explained by the nature of the EHD film which has almost rigid contact characteristics. The higher the preload, the greater its rigidity and the lower the ripple shown in Fig. 5.
1.5
F 2
0.0
;
-1.5
-3.0
950
960
970
Time
980
000
I
1000
[msl
Fig. 7. Time history of relative displacement of disc centres: Q = 810 N; Shell Turbo 33 lubricant; N = 360 rev min-‘.
-8.0 1 100
110
120 Time
130
140
I 150
[msl
Fig. 8. Time history of relative displacement of disc centres: Q = 2100 N; Shell Vitrea 79 lubricant; N = 360 rev min-‘.
11
Thus there is little opportunity for the film surfaces to acquire a squeeze velocity sufficient to cause significant damping. Increasing the lubricant viscosity has only a slight effect. To get some idea of the damping present, the steady state amplitude from Fig. 6(b) for Shell Vitrea 79, was subtracted from Fig. 6(a) and a similar procedure was carried out for the time response for the Shell Turbo 33 oil. The equivalent linear damping factors were 0.007 for Shell Vitrea 79 and 0.0018 for the Shell Turbo 33. These are both very small values, slowing the negligible contribution an EHD film makes to total damping. Had the preloads been less and/or the amplitude of disc centre oscillation been greater, with the film being modelled to come out of the EHD regime during oscillation and then enter a piezo viscous or iso-viscous regime as the load decreased, then more lubricant film damping would have resulted. This was demonstrated in ref. 9 where the lubricant film model varied between EHD and isoviscous to accommodate the rolling elements entering and leaving the low and high loading zones of the ball bearings. Most of the damping contribution was from the unloaded zone. 6.2. The frequency domain analysis To obtain the frequency response of the system, the steady state amplitude of x is plotted against the disc speed under the constant preload of 810 N for both the test lubricants (see Fig. 9). The natural frequency of the system is unchanged with the choice of lubricant as it is governed by the amount of preload Q and the equivalent contact stiffness which is given by k, (as the oil film acts almost as a rigid layer, k, = m). The amplitude of oscillations is slightly greater with the lower viscosity lubricant, owing to its lower damping properties. The natural frequency is at disc speed of about 275 rev min- ‘. When the preload is increased to 2100 N, the natural frequency is moved to a disc speed of 325 rev min-’ for the Shell Vitrea 79 lubricant (see Fig. 10). This increase in natural frequency is caused by the higher preload making the vibrations occur in a stiffer region of the contact springs. Figure 11 shows a comparison of the numerical results of Fig. 9 and the experimental results of ref. 6 for the Shell Vitrea 79 lubricant under
100
200
300
Discspeed,B.P.M.1
Fig. 9. Amplitude frequency
400
500
plot for relative displacement of disc centres: Q = 810 N.
100
200
300
400
500
Discspeed,[R.P.M.] Fig. 10. Amplitude frequency plot for relative displacement of disc centres: Q =) 2100 N; Shell Vitrea 79 lubricant.
100
200 Disc
300 Speed
400
500
[R.P.H.]
Fig. 11. Comparison of the experimental results of ref. 6.
the same conditions. The resonant frequency is lower in the experimental results owing to the lower value of the system’s effective stiffness. The higher stiffness value in the numerical results arises from the assumption that the discs are held by rigid supports. In practice, however, disc supports have finite stiffnesses. With the numerical results a peak amplitude at resonance is not obtained because the conditions there bring about contact separation and break down the computational process as the value ah/at increases positively and eqn. (13) fails to provide a finite value. Therefore, the resulting band of frequencies for which a finite amplitude solution cannot be obtained, account for the difference in the width of the experiments and theoretical amplitude-frequency plots. If the preload is varied at a constant rotational speed, we get Fig. 12(a) which shows a decrease in displacement amplitude with an increase in preload. In contrast, Fig. 12(b) shows that if speed is increased to 360 rev min-’ , there is an increase in displacement amplitude with preload. Now the lower the preload the lower the resonance speed, so on a family of frequency response curves for different preloads, as for Fig. 10, we would obtain lower preloads with larger amplitudes for a given disc speed on the rising part of the curves with the situation reversing on the falling part. Thus preload
13
1500
1000
500
PIN
(a)
0
-I 500
@I
1000
1500
2000
2500
QN
Fig. 12. Displacement amplitude Shell Vitrea 79 lubricant.
us. preload:
(a) 200
rev min-‘;
(b) 360 rev min-‘.
appears to be an important factor in the design of gears. Its chosen value must be influenced by the operating speed if the contact spring resonant condition is to be avoided. 7. Conclusions The theoretical model for simulat~g the contact spring characteristics of gears has shown the fo~ow~g. (1) Increasing the preload increases the natural frequency. (2) The modelled EHD film offers very low damping, though with a more viscous lubricant it is more effective. (3) There is reasonable correlation with experimental results though these showed higher damping because of contributions from other factors which could not be modelled here. References 1 K. Herrebrugh, El~tohydrodyn~ic squeeze films between two cylinders in normal approach, Trans. ASME., J. Lubr. Technctl., 92 (1970) 292 - 302.
14 D. Whicker and J. F. Booker, El~tohydrodynami~ squeeze films: 2 S. M. Rohde, effects of viscoelasticity and fluctuating load, Trans. ASME., J. Lubr. Technol., 101 (1979) 74 - 80. 3 T. L. H. Walford and B. J. Stone, The sources of damping in rolling element bearings under oscillating conditions, Proc. Inst. Mech. Eng., 197C, (1983) 225 - 232. 4 A. Chandra and R. J. Rogers, The normal approach, contact and rebound of lubricated cylinders, Trans. ASME., J. Lubr. Technol., 105 (1983) 271 - 279. 5 M. M. A. Safa and R. Gohar, Pressure distribution under a ball impacting a thin lubricant layer, Trans. ASME., J. Tribal., 108 (1986) 372 - 376. 6 D. W. Dareing and K. L. Johnson, Fluid film damping of rolling contact vibrations, J. iWe&. Eng. Sk., 17 (4) (1975) 214 - 219. Oil film thickness and pressure distribution in EHL contacts, Ph.D. 7 A. Mostofi, Thesis, Imperial College of Science and Technology, University of London, 1981. 8 II. Rahnejat, Computational modelling of problems in contact dynamics, Eng. Anal., 2 (4) (1985) 192 - 197. 9 H. Rahnejat and R. Gohar, The vibration of radial ball bearings, Proc. ht. Mech. Eng., 199 (G’C3) (1985) 181 - 193. in elastohydro10 A. Mostofi and R. Gohar, Oil film thickness and pressure distribution dynamic point contacts, J. Mech, Eng. Sci., 24 (4) (1982) 173 - 181. 11 H. Hertz, Miscellaneous Papers, Macmillan, New York, 1986. elastohydrodynamic lubrication of point 12 B. J. Hamrock and D. Dowson, Isothermal contacts, Trans. ASME., J. Lubr. Technol,, 99 (1977) 264 - 276. Influence of vibration on the oil film in concentrated contacts, Ph.D. 13 H. Rahnejat, Thesis, Imperial College of Science and Technology, University of London, 1984.
Appendix A The functions 11/1and $4 in eqn. (17) are defined as follows:
t-41)
\i/=-P+2fl
---
2
yh
h
ah
2p63'2hY
-’ ah
3KK,
zew
at
I( 11 ah
Y
at
11/3’ R1 +@) _ /32hP-21j3’2 exp(y(ah/&)) 3PKK, yh2 3hP63’2 exp{y( ah/at)) $4
=
--m
2PKK,rn
$
+
(A3) 2(W(t) - 9)
i
Appendix B: nomenclature
c
*
% E EX G*
(-1
centre-to-centre distance of discs ellipticity ratio modulus of elasticity (1 - v)2f7rE ffl&
(A4)
15
h h* K K, :
Q Rl_w
R2x
R
R2,
R;,Y’R, t u v W, w,* W W* &I x Y
oil film thickness WR, an EHL time-dependent parameter load-deflection constant of proportionality mass of disc disc speed in rev mm-’ constant preload radii of discs in the x direction radii of discs in the y direction reduced radii in the x and y directions time mean discs’ speed in the direction of rolling WJWR, squeeze speed, ah/Cl t %lU integrated oil film pressure distribution WWRx2 relative motion of discs’ centre in the horizontal x direction waviness function wave amplitude transferred x co-ordinate relative motion of discs in the y direction
Greek symbols piezoviscosity 6 = 22.23 ; y = 2734.3/U Y deflection 6 lubricant viscosity at atmospheric pressure and at 30 “C. 170 V Poisson’s ratio wave frequency 0 Subscripts 192 0 Oi
relate to the contacting discs corresponds to the initial conditions indicates adjusted initial conditions
1
I- 15
VIBRATION RESPONSE OF WAVY SURFACED DISC IN ELASTOHYDRODYNAMIC ROLLING CONTACT H. MEHDIGOLI Tribology Section, Department of Mechanical Engineering, Imperial College of Science and Technology, Exhibition Road, London SW7 2BX {U.K.) H. RAHNEJAT On-line Su~ei~lance, ~onito~ng and diagnostics Unit, School of~echaniea~, Aero~u~cal and ~oduction Engineering, Kingston PoZytechnie, Canbury Park Road, Kingston, Surrey (U.K.) R. GOHAR Tribology Section, Department of Mechanical Engineering, Imperial College of Science and Technology, Exhibition Road, London SW7 2BX (U.K.) (Received July 14, 1988; revised July 20, 1989; accepted December 7, 1989)
Summary
The m~itude of vibration and noise generation in rotating machinery is affected by the value of damping. Damping has a significant influence on the reduction of untoward motions. It can be exerted externally, for example, by the support structure and squeeze films in seals and internally by material hysteresis, dry friction and the lubricant squeeze film action in elastohydrodynamic (EHD) contacts. This paper investigates the,influence of EHD lubricant film damping and preloading in the concentrated non-conforming contact of rolling and normally approaching discs. It is shown that under high preloading the damping effect is slight, but the level of preloading affects the value of the natural frequency.
1. Introduction The significance of fluid film damping in the elastohydrodynamic (EHD) contact of rolling and normally approaching rolling elements has recently been investigated by various authors. One of these investigations was the study of squeeze film variation in the non-conforming EHD contact of a pair of cylinders by Herrebrugh [ 11. In his analysis he concluded that the squeeze film damping contribution is more significant in the region of thick films. Rohde et aI. [2f also studied the dynamic response of normally approaching bodies subjected to fluctuating normal load. They compared the results of rigid, elastic and viscoelastic contacts and concluded that the 0043-1648/90/$3.50
@ Elsevier Sequoia/Printed in The Netherlands
2
different response characteristics observed point to significant fluid film damping that may occur under hydrodynamic and EHD conditions. The effect of squeeze film action in lubricated EHD contact of rolling elements under oscillating conditions was also investigated by Walford and Stone [3] as well as Chandra and Rogers [4] under hydrodynamic conditions for normal approach and rebound of lubricated cylinders. More recently Safa and Gohar [5] obtained experimentally transient pressure distributions for the normal approach of a falling and rebounding ball against a flat race under EHD conditions using a manganin microtransducer. Dareing and Johnson [6] have carried out valuable experimental work with a disc machine where the rolling mating discs were subjected to periodic excitations caused by surface corrugations on one disc. They showed that under certain contact conditions resonance occurs with the lubricant film contributing significantly to the damping. Their theoretical model, however, approximated the lubricant film to a linear spring and damper arrangement which is a very simplified representation of an EHD lubricant film. More representative oil film behaviour in rolling and squeezing EHD elliptical contacts was presented by Mostofi [7]. His formula was used in the contact of mating discs and ball to races contacts in deep-groove ball bearings in works presented by Rahnejat [8] and Rahnejat and Gohar [9]. They showed that the contribution of squeeze film damping is more significant in the region of thick films, pertaining rather to hydrodynamic conditions. This paper presents the results obtained when modelling the elliptical EHD contacting geometry of mating discs used in ref. 6. The numerical method used is the same as that described in refs. 8 and 9, where the surface acceleration of the EHD lubricant film boundaries is assumed to vary linearly over a small time interval.
2. Simulation
model
The numerical model is designed to simulate the experiments of Dareing and Johnson [6] who employed a disc machine. The corrugations on the surface of one of the discs, used in the work of ref. 6, is modelled here by a similar sinusoidal function. The contact oil film thickness uses the EHD lubrication (EHL) oil film regression formula of Mostofi and Gohar [lo] with a squeeze velocity term included, as described by Mostofi [ 71. 2.1. Specifications of the simulated disc machine The disc machine used in ref. 6 employs a pair of mating discs, each with a diameter of 102 mm. The source of excitation is a corrugated surface on one disc of amplitude 1.27 pm. There are 120 corrugations around the disc’s circumference. The other disc is crowned to a radius of 102 mm. Each disc and its supporting shaft are mounted on a hydrostatic journal bearings housing which in turn is supported by soft leaf springs. The stiffnesses of the leaf springs are negligible and the bearing housings have very
3 TABLE 1 Lubricant properties Lubricant
Viscosity (at 30 “C Ns rn-?)
piezoviscosity ( m2 N-l)
Shell Vitrea 79
0.890
25.9 x 10-9
Shell Turbo 33
0.034
21.6 x 1O-9
high stiffness compared with the contact stiffness of the mating discs. The disc contact stiffness constant is found from the experimental data of ref. 6, to be k, = 28 (GN m-3)-2. A constant preload of Q = 810 N acts upon the discs through the leaf-springs. The total mass of one disc with its attachments is 20.4 kg. Two types of lubricant are used as is the case in ref. 6 and are listed in Table 1. 2.2. The elastic contact deformation The crowned disc and its mating corrugated disc form an elliptical point contact footprint where the wavelength of the corrugation is much greater than the footprint minor axis and thus does not influence the hertzian contact conditions. The contact load in an elliptical contact according to Hertz [ll] is given by W = K,A3”
(1)
where K, is the contact stiffness of mating discs. The elliptical contact ratio is [ 121 eP * =
i
0.64
1
RX
1.03 F
Y
where R, and R, are the reduced radii in the xz and yz planes respectively (see Fig. 1) such that 1 1 -=-+R, RI, 1 _=-+RY
1 R2x
1 RI,
1 .R2Y
where RI, = R2x
=
RI, and RZy = 00.
Using the radii of the discs and eqns. (2) - (4), ep* = 2.5.
(3)
Fig. 1. Radii of the contacting discs.
2.3. The oil film model The minimum oil film thickness formula obtained by Mostofi [7] for EHL elliptical contacts with flow along the minor axis of the elastostatic contact ellipse is h* = 0 . 112~0~5sG*0~46W*~o~045 exp(-123w,*)
exp(O.O76e,*)
(5)
where
ho& u*=---...Rx
G*
’
a
=-’ &
w*= WE, -,
RX2
wp$
(61
A special feature of this formula is that it includes the relationship between the oil film thickness and the squeeze velocity, w, = dh/dt. 2.4. Geometrical considerations If there were no co~ugations, there would be no excitation and the oil film thickness and the contact elastic deformation would remain constant during rolling. Figure Z(a) depicts this case. Using this figure,
C, = R +,+h,+R+, or C, = 2R - 6, + h,
(7)
With the corrugated surface on one disc introducing an excitation function, x(t) = 2 sin(wt), the film thickness and contact deformation of the discs’ surfaces are functions of time and the disc centres undergo motions x1 and x2 respectively. Thus, referring to Fig. 2(b)
5
Fig. 2. Geometrical conditions of the contacting waviness.
surfaces: (a) with no waviness; fb) with
Fig. 3. The non-linear contact model: I$ for the contact spring and KO for the oil film.
c=c*
+x1+x2
(8)
and c = 2R - s(t) +”x(t) + h(t)
(9)
From eqns. (7) - (9) a relation between the geometric variables can be obtained by eliminating the terms C and C, (6 - 6,) + h, + (x1 i-x,) =x(t) + h(t)
(19)
3. The dynamic model The dynamic model is shown in Fig. 3, where the discs are represented by a pair of equal masses m on a common horizontal axis, the stiffness constant of the non-linear elastic contact is K,, and there is a non-linear stiffness-damper arrangement for the oil film thickness. The two masses are preloaded together with the constant force Q in the horizontal (that is x) direction. The contact reaction W(t) is identical in each elastic contact and the oil film since they are in series. As the oscillations occur in the x direction there is no gravity effect. Therefore, the equations of motion for the disc centres are d2x1 m-= dt2 d2x2
m -=
dt2
W(t)-Q
(11)
W(t)-Q
(12)
6
where the elastic contact W(t) is given by eqn. (1). For the oil film, eqn. (5) can be written in the form W(t) = Kh-O( t) exp (-7
dh/dt)}
where fl= 22.23 and y = 2734.3/U. that .K = (0.112)pV
(13) K is a time-independent constant such
13.9G*lo.s&fl+ 2{ exp(54.85 + 1.6e,*))/E,
There are five unknowns h, 6, x,, x2, W and five equations (1) and (10) (13). Therefore, a solution is possible.
4. Method of solution Equations (1) and (3) are doubly differentiated and equated to yield the equation
(14) For a solution to be obtained, the terms d26/dt2 and d&/dt are eliminated from eqn. (14) using the relations d26 -=dt2
d2x
d2h + -dt* dt2
_ = -26 d6
2-
d2x,
(15)
dt2
{P(dG/dt) + rh(d2h/dt2))
dt
(16)
3h
Substituting the above relations in eqn. (14), a third-order differential equation in terms of oil film roof oscillations results such that d*h
2
+
ILz-
dt2
+
+
$4
(17)
The functions G1, I)*, IJI~and ti4 are given in Appendix A. Step by step integration of eqn. (17) using the Runga-Kutta method yields the values of film thickness h and the squeeze film velocity dh/dt. Equations (13) and (1) are used to obtain the values of W and 6. Since x1 and x2 have appeared symmetrically, eqn. (10) is employed to calculate the motions of the discs’ centres.
7
5. The initial conditions In order to solve the differential equation (17), the initial values of functions $i through G4 should be known. Therefore, the values of variables h, dh/dt, d2h/dt2, 6 and d2X,/dt2 must be obtained at t = 0. Since the steady state solution is desired one can apply any eligible initial conditions instead of an actual set. The following initial steps are taken. The value of K is calculated using the governing EHD parameters. W,=Qattimet=O. 6, is calculated using eqn. (1). h = h, at t = 0. Values of dh,/dt and d2h,/dt2 cannot be evaluated analytically. Therefore, the following procedure is adopted. The source of excitation is assumed to be an exponential function Bet and commences at a point Oi far enough from the point o as to make 3c negligible (see Fig. 4). From the continuity of profile, 3c and dx/dt are the same. Thus at t = t, Beto = Lt sin(wt,)
(18)
Beto = 2 w cos(wt,)
(19)
Therefore 1 t, = - tan-'
Cd
0
and
Equation (17) is solved for the time t,. The function in this region is of the form X = B exp(t - toi) (see Fig. 4). The values at t, are the initial values for the rest of the wavy profile.
Fig. 4. The assumed initial wavy profile.
8
dh oi _ d2h,i _ 0 p--m dt, dt,2 Now the initial value of d2h,/dt3 is calculated at t = t,, using eqn. (17). Step by step iterative integration using the Runga-Kutta method can now commence.
6. Results and discussion The excitation frequency w is a function of the rotational speed of the corrugated disc and the number of corrugations. There are 120 corrugations around the circumference of the corrugated disc. For the rotational speed of N rev mm-’ w=27rN-
120 60
= 41rNrad s-l
The numerical calculations are carried out at different disc speeds to obtain the amplitude-speed response characteristics of one disc with either lubricant and with different values of preload Q. Generally two preload values of 810 N and 2100 N are used as in ref. 6. The results are presented in both time and frequency domains; the latter are then subjected to a comparative study with the findings of ref. 6. 6.1. Time domain analysis Time histories of the lubricant film thickness and the elastic contact deformation are presented in Fig. 5. The results exhibit much smaller amplitudes of oil film oscillations than the corresponding contact deflection, 6. The steady state of oil film thickness is 3.85 pm with a small ripple oscillation superimposed on it of peak-to-valley amplitude of 0.35 pm (i.e. 9% of
P .FJ
3.5.~
h
Tim [msl Fig. 5. Time histories of the oil film thickness and contact elastic deflection: Shell Vitrea 79 lubricant; N = 360 rev min-‘.
Q = 810 N;
9
the steady oil film thickness). The corresponding static contact deflection is 9.19 pm with a dynamic deflection of peak-to-valley amplitude of 6.475 pm superimposed on it. These simulated results are obtained with the Shell Vitrea oil at the disc speed of 360 rev min-’ and Q = 810 N. The time history relative displacement of discs’ centres (i.e. x = x1 - x2) is shown in Fig. 6(a). The initial 90 ms of the response show evidence of its decaying transient characteristics due to fluid film damping action. In Fig. 6(b) the steady state double amplitude of x is 3.9 pm which is of the order of the oil film thickness. The transient portion of the response consists of beats between two frequencies, one due to the free response of the system and the other at the forcing frequency induced by the corrugations, in this case at 720 Hz. The fluid film dampens the free oscillation very slowly indeed. The steady state conditions are eventually attained at the forcing frequency with the vibration maintained by the rotation of corrugated disc. This trend is observed to be even slower from the x oscillation time history of Fig. 7 for
-6.0
(a)
(b)
J
0
20
40 Tiie
Time
80
80
I GO
[%I
[msl
Fig. 6. Time history of relative displacement: Shell Vitrea 79 lubricant; N = 360 rev min-‘.
(a) 0 - 90 ms; (b) 100 - 140 ms. Q = 810 N;
10
the lower viscosity Shell Turbo 33 oil, where steady state conditions are not yet achieved after 10 000 time steps (or 1 s). Figure 8 shows the x oscillation response with Shell Vitrea oil and an increased preload of 2100 N. Comparing this with Fig. 6(b) shows that with higher preloads, the time to reach the steady state condition increases as the oil film thickness and hence oil film damping both decrease. The narrow band record of the response in Fig. 8 is indicative of sharply resonant characteristics due to the low fluid film damping, giving a maximum peak to valley x amplitude of 14.5 pm as opposed to 3.9 pm with the lower preload of 810 N and the same lubricant in Fig. 6. The observed response behaviour is easily explained by the nature of the EHD film which has almost rigid contact characteristics. The higher the preload, the greater its rigidity and the lower the ripple shown in Fig. 5.
1.5
F 2
0.0
;
-1.5
-3.0
950
960
970
Time
980
000
I
1000
[msl
Fig. 7. Time history of relative displacement of disc centres: Q = 810 N; Shell Turbo 33 lubricant; N = 360 rev min-‘.
-8.0 1 100
110
120 Time
130
140
I 150
[msl
Fig. 8. Time history of relative displacement of disc centres: Q = 2100 N; Shell Vitrea 79 lubricant; N = 360 rev min-‘.
11
Thus there is little opportunity for the film surfaces to acquire a squeeze velocity sufficient to cause significant damping. Increasing the lubricant viscosity has only a slight effect. To get some idea of the damping present, the steady state amplitude from Fig. 6(b) for Shell Vitrea 79, was subtracted from Fig. 6(a) and a similar procedure was carried out for the time response for the Shell Turbo 33 oil. The equivalent linear damping factors were 0.007 for Shell Vitrea 79 and 0.0018 for the Shell Turbo 33. These are both very small values, slowing the negligible contribution an EHD film makes to total damping. Had the preloads been less and/or the amplitude of disc centre oscillation been greater, with the film being modelled to come out of the EHD regime during oscillation and then enter a piezo viscous or iso-viscous regime as the load decreased, then more lubricant film damping would have resulted. This was demonstrated in ref. 9 where the lubricant film model varied between EHD and isoviscous to accommodate the rolling elements entering and leaving the low and high loading zones of the ball bearings. Most of the damping contribution was from the unloaded zone. 6.2. The frequency domain analysis To obtain the frequency response of the system, the steady state amplitude of x is plotted against the disc speed under the constant preload of 810 N for both the test lubricants (see Fig. 9). The natural frequency of the system is unchanged with the choice of lubricant as it is governed by the amount of preload Q and the equivalent contact stiffness which is given by k, (as the oil film acts almost as a rigid layer, k, = m). The amplitude of oscillations is slightly greater with the lower viscosity lubricant, owing to its lower damping properties. The natural frequency is at disc speed of about 275 rev min- ‘. When the preload is increased to 2100 N, the natural frequency is moved to a disc speed of 325 rev min-’ for the Shell Vitrea 79 lubricant (see Fig. 10). This increase in natural frequency is caused by the higher preload making the vibrations occur in a stiffer region of the contact springs. Figure 11 shows a comparison of the numerical results of Fig. 9 and the experimental results of ref. 6 for the Shell Vitrea 79 lubricant under
100
200
300
Discspeed,B.P.M.1
Fig. 9. Amplitude frequency
400
500
plot for relative displacement of disc centres: Q = 810 N.
100
200
300
400
500
Discspeed,[R.P.M.] Fig. 10. Amplitude frequency plot for relative displacement of disc centres: Q =) 2100 N; Shell Vitrea 79 lubricant.
100
200 Disc
300 Speed
400
500
[R.P.H.]
Fig. 11. Comparison of the experimental results of ref. 6.
the same conditions. The resonant frequency is lower in the experimental results owing to the lower value of the system’s effective stiffness. The higher stiffness value in the numerical results arises from the assumption that the discs are held by rigid supports. In practice, however, disc supports have finite stiffnesses. With the numerical results a peak amplitude at resonance is not obtained because the conditions there bring about contact separation and break down the computational process as the value ah/at increases positively and eqn. (13) fails to provide a finite value. Therefore, the resulting band of frequencies for which a finite amplitude solution cannot be obtained, account for the difference in the width of the experiments and theoretical amplitude-frequency plots. If the preload is varied at a constant rotational speed, we get Fig. 12(a) which shows a decrease in displacement amplitude with an increase in preload. In contrast, Fig. 12(b) shows that if speed is increased to 360 rev min-’ , there is an increase in displacement amplitude with preload. Now the lower the preload the lower the resonance speed, so on a family of frequency response curves for different preloads, as for Fig. 10, we would obtain lower preloads with larger amplitudes for a given disc speed on the rising part of the curves with the situation reversing on the falling part. Thus preload
13
1500
1000
500
PIN
(a)
0
-I 500
@I
1000
1500
2000
2500
QN
Fig. 12. Displacement amplitude Shell Vitrea 79 lubricant.
us. preload:
(a) 200
rev min-‘;
(b) 360 rev min-‘.
appears to be an important factor in the design of gears. Its chosen value must be influenced by the operating speed if the contact spring resonant condition is to be avoided. 7. Conclusions The theoretical model for simulat~g the contact spring characteristics of gears has shown the fo~ow~g. (1) Increasing the preload increases the natural frequency. (2) The modelled EHD film offers very low damping, though with a more viscous lubricant it is more effective. (3) There is reasonable correlation with experimental results though these showed higher damping because of contributions from other factors which could not be modelled here. References 1 K. Herrebrugh, El~tohydrodyn~ic squeeze films between two cylinders in normal approach, Trans. ASME., J. Lubr. Technctl., 92 (1970) 292 - 302.
14 D. Whicker and J. F. Booker, El~tohydrodynami~ squeeze films: 2 S. M. Rohde, effects of viscoelasticity and fluctuating load, Trans. ASME., J. Lubr. Technol., 101 (1979) 74 - 80. 3 T. L. H. Walford and B. J. Stone, The sources of damping in rolling element bearings under oscillating conditions, Proc. Inst. Mech. Eng., 197C, (1983) 225 - 232. 4 A. Chandra and R. J. Rogers, The normal approach, contact and rebound of lubricated cylinders, Trans. ASME., J. Lubr. Technol., 105 (1983) 271 - 279. 5 M. M. A. Safa and R. Gohar, Pressure distribution under a ball impacting a thin lubricant layer, Trans. ASME., J. Tribal., 108 (1986) 372 - 376. 6 D. W. Dareing and K. L. Johnson, Fluid film damping of rolling contact vibrations, J. iWe&. Eng. Sk., 17 (4) (1975) 214 - 219. Oil film thickness and pressure distribution in EHL contacts, Ph.D. 7 A. Mostofi, Thesis, Imperial College of Science and Technology, University of London, 1981. 8 II. Rahnejat, Computational modelling of problems in contact dynamics, Eng. Anal., 2 (4) (1985) 192 - 197. 9 H. Rahnejat and R. Gohar, The vibration of radial ball bearings, Proc. ht. Mech. Eng., 199 (G’C3) (1985) 181 - 193. in elastohydro10 A. Mostofi and R. Gohar, Oil film thickness and pressure distribution dynamic point contacts, J. Mech, Eng. Sci., 24 (4) (1982) 173 - 181. 11 H. Hertz, Miscellaneous Papers, Macmillan, New York, 1986. elastohydrodynamic lubrication of point 12 B. J. Hamrock and D. Dowson, Isothermal contacts, Trans. ASME., J. Lubr. Technol,, 99 (1977) 264 - 276. Influence of vibration on the oil film in concentrated contacts, Ph.D. 13 H. Rahnejat, Thesis, Imperial College of Science and Technology, University of London, 1984.
Appendix A The functions 11/1and $4 in eqn. (17) are defined as follows:
t-41)
\i/=-P+2fl
---
2
yh
h
ah
2p63'2hY
-’ ah
3KK,
zew
at
I( 11 ah
Y
at
11/3’ R1 +@) _ /32hP-21j3’2 exp(y(ah/&)) 3PKK, yh2 3hP63’2 exp{y( ah/at)) $4
=
--m
2PKK,rn
$
+
(A3) 2(W(t) - 9)
i
Appendix B: nomenclature
c
*
% E EX G*
(-1
centre-to-centre distance of discs ellipticity ratio modulus of elasticity (1 - v)2f7rE ffl&
(A4)
15
h h* K K, :
Q Rl_w
R2x
R
R2,
R;,Y’R, t u v W, w,* W W* &I x Y
oil film thickness WR, an EHL time-dependent parameter load-deflection constant of proportionality mass of disc disc speed in rev mm-’ constant preload radii of discs in the x direction radii of discs in the y direction reduced radii in the x and y directions time mean discs’ speed in the direction of rolling WJWR, squeeze speed, ah/Cl t %lU integrated oil film pressure distribution WWRx2 relative motion of discs’ centre in the horizontal x direction waviness function wave amplitude transferred x co-ordinate relative motion of discs in the y direction
Greek symbols piezoviscosity 6 = 22.23 ; y = 2734.3/U Y deflection 6 lubricant viscosity at atmospheric pressure and at 30 “C. 170 V Poisson’s ratio wave frequency 0 Subscripts 192 0 Oi
relate to the contacting discs corresponds to the initial conditions indicates adjusted initial conditions