Modelling and identification of nonlinear squeeze film dynamics

Journal of Flu&

and Structures (1992) 6, 223-248

MODELLING AND IDENTIFICATION OF NONLINEAR SQUEEZE FILM DYNAMICS H. ESMONDE, J. A. FITZPATRICK, H. J. RICE

AND

F. AxIsAt

Department of Mechanical Engineering, Trinity College, Dublin 2, Ireland. t DEMT, CEA, Centre d’Etudes Nuclkaires de Saclay, 91191 Gif sur Yvette, France (Received 29 October 1990 and in revised form 1 October 1991)

In this paper, using a theoretical model of the dynamics of a viscous squeeze film for a circular plate, the relationship between the plate motion and the associated forces are restructured to enable identification of the various linear and nonlinear components of the system using spectral analysis techniques. A numerical simulation shows that the results obtained using the procedures are in good agreement with the expected values. A series of experiments were conducted and parameter values were derived for each of the nonlinear force components acting in the system. The various physical phenomena associated with the squeeze film are examined for different film sizes, giving an insight into the Row regime in the film. Finally, responses calculated using the derived values with sinusoidal excitation are compared with experimental results and good agreement is obtained.

1. INTRODUCTION SQUEEZE FILM FORCES, which are highly nonlinear systems where two moving surfaces are separated

in nature,

can arise

in engineering

by a viscous fluid. The analysis of the dynamics of liquid squeeze films is of importance in many engineering applications. For modelling the wear in heat exchangers due to tube/baffle impact and for the seismic analysis of reactor fuel rod assemblies, a realistic assessment of the forces transmitted at the squeeze film interfaces is necessary. These forces arise from both damping and inertia effects and are highly nonlinear. As a consequence, it is extremely difficult to determine the contribution of the various forces or to quantify them from experiments. In many circumstances an accurate assessment of these nonlinear forces is required in order to predict accurately vibration and wear patterns. Since a complete solution is not available for the squeeze film problem using the generalized equations for fluid motion, experiments are required to supplement the theoretical analysis. A number of studies have been conducted in which the nonlinear coefficients were identified by a variety of methods. Most of these techniques, however, use effective linearization of many of the terms. For example, a quasi-linear approach was used in the modelling of bearing dynamics by Tichy & Bourgin (1985) where close tolerances brought squeeze film forces into play. Also, the lift forces created by the eccentric movement of the outer casing of journal bearings in an annular space were determined analytically by Burrows et al. (1987) using a linearized Reynolds equation. However, with increased speeds resulting in high squeeze Reynolds numbers, a full nonlinear approach is required to describe correctly all the effects present. Other techniques which have been used include the Invariant Imbedding and the State Variable Filter methods (e.g., Ellis et al. 1988). Both of these track the data in the time domain and estimate the system parameters using linearized regression models. Mulcahy (1980) and Rogers & Ahn (1986) have reported on detailed research work in the area of tube/baffle dynamics. Rogers & Ahn (1986) and Rogers et 0889-9746/W/020223 + 26 $03.00

@ 1992Academic

Press Limited

224

H. ESMONDE

ET AL

(1988) identified nonlinear hydrodynamic mass and damping terms for a cylindrical squeeze film using nonlinear models with sinusoidal excitation and harmonic analysis. The models established were, however, still amplitude-dependent indicating the presence of additional unaccounted for nonlinearities in the system. In order to ensure correct response predictions, all possible nonlinear effects must be accounted for in the modelling procedures, and methods for identifying the dynamic parameters of the system are required as was shown by Aita et al. (1985). In this paper, a nonlinear model is developed to describe the generalized dynamics associated with the squeeze film of a circular flat plate geometry. The elements of the model are reformulated such that the spectral analysis technique for identifying nonlinear systems proposed by Rice & Fitzpatrick (1988) can be applied. A numerical simulation calculates the response of the system to a random forcing function and the data generated is then used to test the identification procedures. The procedures are implemented using data from an experimental rig and test results with supporting numerical simulations are presented for a range of film thicknesses. In addition, the model is refined to give a more detailed representation of the viscous effects. The measured coefficients and models are independently verified by comparing the response of a numerical simulation with experimental data using sinusoidal excitations. al.

2. THEORETICAL

ANALYSIS

The approach used here is similar to that of Kuzma (1967) for the problem of a hat circular plate vibrating close to a fixed parallel surface. As well as the forces generated within the film, edge effects are also considered in the analysis. Since there is no analytical solution to the governing equations, assumptions based on physical considerations are used to simplify the squeeze film problem. For a circular plate geometry using cylindrical co-ordinates, all terms can be considered to be independent of the angle 0 so that the system can be described in terms of r and z only. If the circular plate oscillates in close proximity to a fixed, flat, parallel surface with the film thickness, h, much smaller than the plate radius, R, then, an order of magnitude analysis can be performed (e.g. Tichy & Winer 1970), so that the equations of fluid motion can be reduced to give

The first term in the equation refers to unsteady inertia and accounts for the acceleration of the liquid within the film. The second and third terms arise due to the confinement of the liquid, forcing the particles to move in an arc and give rise to a convective inertia term. The fourth term allows for the pressure gradient along the film, while the fifth term takes the shearing effects in the film into account with a viscous parameter. Equation (1) can be solved iteratively by applying conservation of mass (e.g., Kuzma 1967) to obtain an expression for the pressure, which is then integrated over the lower surface of the plate to calculate the force exerted by the fluid on the plate as

where zd, vd are the plate displacement and velocity respectively at the edge which can be represented as follows:

P,=$p&

IV&

and POis the pressure (3)

225

NONLINEAR SQUEEZE FILM DYNAMICS

Since I&, the average radial velocity at the edges, can be written as

VdR v, = - 2(&j + h) ’ then p = _ 0

p5R2Vd Iv,1. S(z, +h)2

(4)

’

thus, Vd Vd IVdl - 12y (Zd + h)3 - E (Zd + q2

’

(5)

where A = pnR4/8. The first three terms in equation (5) are identical to those determined by Kuzma (1967), with the fourth term introduced to model edge effects. The coefficients of all four terms in equation (5) depend on the flow profile in the film. In steady, developed conditions the profile will not change, so that the coefficients remain constant. However, the flow underneath the plate in the squeeze tilm is unsteady and undeveloped in which case the flow profile will change continuously throughout the motion. Equation (5) can thus be rewritten with the coefficients before the terms replaced by (Y, /3 and y, while keeping 5 in the fourth term. (It should be noted that each of these coefficients is nondimensional.) Thus,

V”,

Vd

vd

Iv,1

(6) >’ -a(&,+h)+B(zd+h)2-YV(Zd+h)3-E(Zd+h)2 Ffl=A Since it is difficult to account for the fluctuations in flow profile during the oscillations, averaged values of the coefficients are determined experimentally using the spectral techniques outlined in the next section. If, however, a uniform profile was prevalent with constant radial velocity throughout the film at a given radius, then a different set of coefficients to those of equation (5) would have been determined. This regime is close to the case where the fluid is accelerated from rest when the plate begins to move. The other extreme is steady developed flow, which exhibits a parabolic flow profile somewhat similar to that for the iterative solution of equation (5) (see Figure 1). The coefficients for each of the first three terms can be compared in Table 1 for each of the three profiles. Each of the terms in equation (6) represents a force contribution arising from a particular physical phenomenon. The first term, ~~~~i,l(zd + h), describes the unsteady inertia effects as the fluid is accelerated beneath the plate during the vibrations. Despite the fact that the average mass of fluid accelerated is independent of the flow

7

/l///j//////////////// ////////////////////

(At

(B)

(C)

Figure 1. Various flow profiles: A, uniform solution; B, iterative solution; C, parabolic solution.

226

H. ESMONDE

E’l AL

TABLE 1

Coefficients used in the simulations

Test

coefficients

CY

P

1.22

O-S

‘C i

X.85

0.57 I.0

Uniform solution

14

I.5

0.0

Parabolic solution

14

1.5

12.0

I ,O’

Iterative solution

1.2

7. I4

1241

I .(I’

*The head loss was set at 14 unsteady flows

for the simulations

as no data is available

for

profile, the coefficient LY, governing the unsteady inertia contribution to the overall fluid force, does depend on the flow regime. In the iterated solution, the coefficient is 20% larger than either of the other two solutions which both have the same value. The second term, /3AV$/(zd + /I)~, defines the convective inertia effects, arising from the turning motion of the fluid. The squared operator indicates that this component of the fluid force always acts so as to separate the surfaces, irrespective of the phase of movement. The unidirectional nature occurs because the radius of curvature of the fluid motion always has the same sign (see Figure 2). The third term in equation (6), yAvVd/(zd + h)',accounts for viscous forces as fluid layers shear over each other in the film. In Table 1 the coefficient for the uniform profile is zero since there is no shearing between fluid layers and hence no viscous forces. This is the case for a fluid accelerated from rest, when any velocity differential confined in the narrow boundary layer at the edges has little overall effect when averaged over the film. The cubed film thickness in the denominator indicates the importance of this term for small films with low squeeze Reynolds numbers (Re,,) so that in the final stages of compression it is this term which stops the surfaces from contacting each other. For larger values of Re,, inertia forces govern the fluid force equation. The pressure losses as the flow enters and leaves the gap at the singularity at the edge of the plate give rise to additional forces described by the fourth term, m equation (6). The coefficient E depends on the profile at the edge EJ.V, Iv,ll(~d + h)2 . of the film, the geometry at this point and whether the flow is entering or leaving the film. Work has been done in this area by Idel’cik (1969) for steady developed flows but no data is available for the unsteady case. As well as the flow profile changing, so does during the the geometry as the gap size varies, and hence : will not stay constant

Figure

2. Origin

of the convective

term.

NONLINEAR

SQUEEZE

227

FILM DYNAMICS

oscillations. The variation of the coefficients (Y, /3, y and 5_ with flow profile is a complex problem, particularly during random vibrations. Vortex formation at the singularity and turbulent flow renders an analytic description of such variation impossible. In this work the terms in equation (6) are retained to describe the different physical aspects in the film, while assuming (Y, p, y and Lj remain constant and thus the values obtained are averaged. 3. IDENTIFICATION

OF DYNAMIC

PARAMETERS

Consider the spring mass damper system shown in Figure 3, where a plate, coupled to a linear spring and damper, oscillates close to a fixed surface. Two forces are present, the excitation force, F,,, and a fluid force, &, due to squeeze film effects. The equation describing the system motion is given by Fe, = Mli, + cV, + kz, - Ffl .

(7) The first three terms on the right-hand side of equation (7) refer to the linear structural elements, which are included in the analysis when the excitation force is used in the analysis. If the plate force is used instead, then the only linear element in the dynamic equation for the system is the plate inertia force. The equation governing system response in this instance can thus be written as F plate= mplatelid- F, .

(8)

The contribution to the overall force from the linear component in this equation is smaller than that of equation (7) since the mass is much smaller and there are no damping or spring forces present. Either of equations (7) or (8) may be used when analysing the fluid forces, and similar results should be obtained for the nonlinear fluid components. Substituting for Ffl from equation (6), equation (7) becomes

VS v, +P(Zd+h)Z-YY(Zd+h)3-E(Zd+h)2

v, iv,1

Using the spectral-based identification procedures detailed by Rice & Fitzpatrick (1988, 1991); the dynamics of this system can be represented in block diagram form as shown in Figure 4(a), where the excitation force is taken as the output with linear and nonlinear variable parameters forming the inputs. The nonlinear model inputs are calculated directly from the time domain response records. The linear characteristics are determined from the transfer function for the first path and the nonlinear

8

t

_...

Fluid force

..____......._..........~~ Figure 3. Spring/dashpot

supported

plate.

228

H. ESMONDE

ET AL.

-+--$,I / (b) Figure

4. (a) Inverse

squeeze

film model;

(b) uncorrelated

model

NONLINEAR

SQUEEZE

FILM DYNAMICS

229

coefficients, cu& /3& yvA_and & from the four successive transfer functions. Since the inputs are partially correlated, the most efficient procedure for determining the transfer functions is to successively condition each input with respect to all previous inputs in the model, to give the system outlined in Figure 4(b), where the inputs are now uncorrelated. The optimized transfer functions for this model are given by L, = Giy.(i-,)!lGii.(i-l)! 9 where the conditioned

WV

spectra may be evaluated iteratively using Gij.,! = Gij.(,-l,! - LjGi,.(,-l)!,

(II)

, j, and n is the number of inputs. An index of wherei,j=r+l,... ,n+l, r=l,... order n + 1 refers to the output y. To gauge the relevance of the conditioned inputs, a partial coherence is used. This is given as

(12) This is a normalized measure of how much of the remaining output after (i - 1) previous terms have been considered is attributable to the ith conditioned input. The partial coherence does not give a measure of the direct effect of the ith input, since the inputs are partially correlated. For example, the first partial coherence, which is calculated the same way as the ordinary coherence function, does not represent the direct effect of the first input since the correlated parts of the other inputs will also be included. However, if the ith partial coherence is equal to zero, then that term is irrelevant, or, if it is equal to unity, then all the successive terms in the analysis are irrelevant. To assess the system as a whole, the multiple coherence is used and is calculated from the partial coherences as Y: = 1 - [(I - Y$Xl

- v;;>. . . (1 - Y$lP

(13) (14)

where G,,, is the unaccounted part of the output spectrum arising from numerical, electrical or mechanical noise. The multiple coherence should be close or equal to unity indicating that all the necessary inputs have been taken into account and the noise term is not significant. The original transfer functions (relating to the squeeze film dynamics) may now be recovered iteratively from the optimized transfer functions using Hi_”= Liy -

2

LijHjy

i=(q-l),(q-2)...1.

j=i+l

(15)

Starting with the last conditioned transfer function L,, = Hqy and working backwards, all the original functions can be evaluated. The conditioned spectral techniques used in this paper are described in detail by Bendat & Piersol (1986). 4. NUMERICAL

SIMULATION

A numerical simulation of equation (9) using the explicit time integration algorithm of DeVogelaere (1955) with random excitation was implemented. This procedure has been shown to be suitable for solving highly nonlinear problems by Axisa et al. (1988)

230

H. ESMONDE

ETAL

and Antunes et al. (1988). The input force was generated from band-limited noise spectra between 2 and 1OOHz with a random phase, using the procedures detailed by Styles & Dodds (1976). Fifty spectral records of 4096 points were then transformed to provide the forcing function with a time step of 0.5 ms. The parameters describing the linear part of the system were: M = 6.5 kg, The four coefficients defining tally (Esmonde 1989) as: (Y= 1.22, The other system

values

k = 79.5 kN/m.

c = 28.75 N s/m, the nonlinear

p = 0.8,

components

y=8.85,

were determined

experimen-

6 = 0.57.

were

R = 0.05 m,

p = 102 kg/m-‘,

Y = 10mh m2/s,

which gives a value for A of 2.45 x 10-j kg m. The mean static film thickness used was h =0_34mm. The input variables for the linear and nonlinear paths as well as the excitation force were stored prior to the spectral analysis. The results from the analysis are presented in Figures 5 and 6 in terms of partial coherence, gain factor and phase for each of the paths and finally the multiple coherence for the overall system is shown in Figure 7. The gain factor for the linear path was obtained by inverting the modulus of the transfer function of the first input since the system had been inverted, the excitation force being taken as the output during the processing stage. This path refers to the linear structural dynamics, so that Figure 5(a) should show the characteristics of a single degree of freedom system having one narrow peak with a magnitude of 3.85 kg-’ (i.e. 1/25;M) at 17.6Hz. Instead, th ere is a collection of smaller peaks at and slightly above this frequency, suggesting the presence of several modes. The phase is also somewhat indistinct around these frequencies, although it has a steady value of 180” below and 0” above the natural frequency. The linear forces within the system for a film thickness of 0.34 mm are comparatively small and are overshadowed by the nonlinear fluid forces. As a result, the noise introduced from the numerical rounding off and the derivation of the nonlinear terms from the previous half step has had a strong influence on this term. The partial coherence for this path is the same as an ordinary coherence function since the spectra are not conditioned in equation (13) for the first term. The drop in coherence, therefore, indicates the presence of a nonlinearity in the system. The sharp drop in coherence below 10 Hz is, however. due to low signal values because of numerical integration errors. The drop around 25 Hz arises because of nonlinearities, and in particular the convective inertia which is dependent on the square of velocity and, therefore, likely to be poorly correlated with velocity or displacement. Towards the higher frequencies, the coherence gradually approaches unity, since the vibration amplitude decreases so that the system becomes quasi-linear. The results for the second path, which accounts for the unsteady inertia force and has &/(z, + h) as input, are shown in Figure 5(b). The transfer function (aA) was preset at 2.99 x low3 kg m (i.e. CY= 1.22) which is the value obtained by the gain factor above 20 Hz. The phase shift is 0” and steady, particularly above this frequency indicating that the gain factor is a positive value. The partial coherence has a low value in the 0 to 20 Hz range, where the gain factor is incorrect, since inertia forces are less important at lower frequencies. The coherence gradually increases at higher frequencies as inertia begins to dominate the response.

-200

c

24 m

z

9

L””

n

Figure 5. Spectra from simulation

for (a) linear path, (b) unsteady h = 0.34 mm.

inertia,

(c) convective

inertia;

(c) (b)

0

(a)

100

Frequency (Hz)

0 Frequency (Hz)

-200

Frequency (Hz)

100

100

232

H. ESMONDE

-2lW

ET AL.

XHJ

___,.

I I)(I

0

Frequency

0

(Hz)

f-requenq

(a)

Figure

6. Spectra

(Hz) (hl

from simulation

for (a) viscous

term,

(b) pressure

loss term for h = 0.34 mm

100

0 Frequency Figure

7. Multiple

coherence

for simulated

(Hz) spectral

estimates

for h = 0.34 mm

NONLINEAR

SQUEEZE

233

FILM DYNAMICS

To describe the convective inertia effects, the third path (PA) has Vi/(z, + h)2 as input. From Figure S(c), above 20 Hz the gain factor approaches its set value of 1-96 X 10e3 kg m (i.e. p = 0.8). The 5180” phase implies that the transfer function is negative (i.e. -/In). The partial coherence has a high value above 20 Hz but decreases slightly at the higher frequencies, since this velocity-dependent term is relatively more important in the intermediate frequency range. Viscous effects were examined in the transfer function for the fourth path which should be a positive constant for (yAv) of 2.17 X lo-’ kg m3/s (i.e. y = 835). Although viscous effects are important at low frequencies, due to the partial coherence between this and other terms, the low frequency noise is present and contaminates the results

od

1

0

Frequency

(Hz)

Figure 8. Spectra of linear path for simulation;

h = 0.95 mm.

234

H. ESMONDE

E7’AL

below 20 Hz, as can be observed from Figure 6(a). The transfer function maintains the correct value above this frequency, although there is a reasonably high coherence level across the entire range. The results for the analysis of the fifth and final term in the model, which accounts for the head loss effects at the edge of the plate, are shown in Figure 6(b). Although there is a low coherence, the correct value for the transfer function (&t) of 1.4 x 10e3 kg m (i.e. I; = O-57) is obtained at the higher frequencies. The low coherence is due to the fact that this term is calculated after four others have been considered, hence at this stage the amount of information left is small. The success of the identification of the overall system is assessed by examining the multiple coherence function. This is plotted in Figure 7 and, for this case, shows a drop below 20 Hz. Above this value the data can be relied upon to describe the system. whereas below either noise is present or some nonlinearities have not been accounted for. Since the features were preset beforehand, the drop in coherence must be due to noise, causing signal deterioration here. This low frequency degradation of signal quality was also noted in equivalent experiments where the results were not preset or known beforehand. When the analysis was performed for a film thickness of O-95 mm the results obtained in Figure 8 show more clearly than those for 0.34 mm the characteristics of a single degree of freedom system defined by the linear structural elements of the model. In this case the motion amplitude was larger, thus increasing the contribution from the linear term and improving its definition. At present, it is not possible to predict the effects of noise or other errors on the results obtained by applying these procedures to nonlinear systems. As a consequence, the numerical simulations provide a noise-free environment in which the intrinsic errors in the identification procedures for a particular model can be assessed. The simulations can therefore be used to distinguish between such intrinsic errors and errors arising from experimental sources. For example, the low frequency effects noted in the experimental results were ignored since they were shown to be unreliable in the simulations. 5. EXPERIMENTAL 5.1. EXPERIMENTAL

TESTS

RIG

The test rig, which is shown schematically in Figure 9, was designed to permit motion of the central section in the vertical direction only and the circular plate was positioned at the bottom of the central tubular section of the rig. The film thickness formed between the base and the plate was varied by inserting annuli of different sizes in the lower third section of the central tube. The tube was excited at the top by a 200 N vibrator which generated a random or sinusoidal input. The base plate supported perspex walls which formed a water-tight enclosure allowing tests to be carried out with liquid films. A series of preliminary tests carried out in air indicated that the system dynamics were linear with a natural frequency of 17.6 Hz and a damping ratio of 2%. These characteristics were determined using swept sine-type excitation. 5.2.

INSTRUMENTATION

The response and force signals were measured using transducers positioned as shown in Figure 9. The force was measured at two points: between the shaker and the tube so

NONLINEAR

SOUEEZE

FILM DYNAMICS

235

Excitation force transducer

r

1 Displacement transducer

I

I

Accelerometer for velocity

I

I

Accelerometer

Plate force transducer

Figure 9. Schematic of rig and instrumentation.

that the excitation force was recorded; and between the plate and the tube, so that the fluid forces plus plate inertia force could be measured. The tube itself was assumed rigid so that response transducers placed along its length also recorded the plate motion. Two accelerometers were used, and the output from one was integrated to give velocity. Values of displacement were recorded using a noncontacting electromagnetic transducer. These time domain values were used to form the inputs and outputs of Figure 4(a), which were then Fourier transformed prior to the spectral conditioning. The output from the tests was recorded on a pen chart for the sinusoidal excitations or on magnetic tape for the random tests. Prior to data logging, the tests were tuned using a two channel HP spectrum analyser. The time domain values for excitation force, displacement, velocity, acceleration and plate force were digitized using a 12 bit a/d converter and stored on a Hewlett-Packard 1000 mini-computer with record lengths of 1024 points sampled at 200 Hz. Spectral estimates were determined using 100 ensemble averages.

6. EXPERIMENTAL

RESULTS

The results for h = 0.34 mm in water for each of the five paths are presented in Figures 10 and 11 in terms of partial coherence and transfer function (gain and phase), with the multiple coherence for the overall system given in Figure 12. For these tests, the force level was such that the r.m.s. amplitude of vibration was of the order of 20% of the gap size. From the linear path results shown in Figure 10(a), it is not possible to distinguish the features expected of a single degree of freedom system with a mass of 6.5 kg, a natural frequency of 17.6 Hz and 2% damping. The plots are, however, similar to those of the numerical simulation. The transfer function gain factor has several peaks at and slightly above 17.6 Hz with a magnitude close to the expected value of 3.85 (determined from the system properties) at a higher frequency than expected. The

700

I

3

3’0

NO

I

(1

_

0

1011

700 t

I

NONLINEAR

SQUEEZE

FILM

0

237

DYNAMICS

!I

0

0

101)

-2nol1

I Frequency

100

0

l(H)

-2d

IO1)

0

0

i

(Hz)

Frequency

(a) Figure

11. Spectra

(b)

from experiments

for (a) viscous

term,

(b) pressure

loss term,

0-A Frequency 12. Multiple

for h = 0.34 mm.

.IUO ,>

0

Figure

(Hr)

coherence

for experimental

(Hz) spectral

estimates

for h = 0.34 mm.

238

H. ESMONDE

ET AL

phase difference holds a steady value of 0” above the resonant frequency and below this value it swings between f180”. This term shows how much influence noise will have on a weak component within a system. The partial coherence exhibits the same trends as found in the simulations with a sharp drop at the lower frequencies. in this case occurring below 6 Hz. There is also a drop in coherence in the middle to low frequency range which gradually disappears as the coherence rises again in the upper frequency range. The second path relating to unsteady inertia effects is shown in Figure 10(b), and the similarity between this and the simulated results is immediately obvious. The gain factor has a large value below 10 Hz which can be attributed to poor signal-to-noise ratios. The CYvalue was obtained by taking the mean of the gain factor between 30 and 60 Hz to avoid the effects of noise at the lower and upper frequency ranges. The low frequency noise can be attributed to low levels of acceleration as in the numerical simulation, whereas the high frequency noise was a specific feature of the experimental results. In the test apparatus, the amplitude of vibration at such frequencies was small, so that extraneous modes in the rig contaminated the response. There is thus a drop in the partial coherence above 85 Hz. The LYvalue was estimated as 1.22. The convective inertia effects are included in the third path shown in Figure IO(c). This negative term has a phase of f180” gradually shifting towards 0” at the higher frequencies. This fact could be due to confusion between this term and that of the head loss which had a 0” phase. At high frequencies, neither of these terms had a very large contribution to the overall force and since, for positive velocities, both parameters have the same form resulting in local cross contamination. The low frequency noise effects are not as marked in the gain factor and a steady value of j3 = 0.8 is maintained in the frequency range from 20 to 80 Hz. The partial coherence is similar to that obtained in the simulations except for the drop off above 80 Hz due to noise. The analysis of the fourth path, which accounts for viscosity forces, is shown in Figure 11(a). The value for the coefficient y was determined in the range 20 to 60 Hz as 8.85 with a 0” phase shift indicating that this is a positive value. Below 20 Hz, the partial coherence is low, indicating little overall contribution from this term in this frequency range. From the experimental results for coherence and gain factor. it would appear that there is a change in viscous effects above 80 Hz. However. noise contamination makes any speculation on possible transition Reynolds number effects impossible. The final term describes head loss effects and it has a rather low coherence since there is very little information left at this stage. From Figure 1 l(b), the mean value for the gain factor between 20 and 80 Hz is E = 0.57 with 0” phase and all the plots are quite similar to those obtained numerically. The multiple coherence obtained from the analysis is shown in Figure 12 and approaches unity above 20 Hz. The linear path for a test at 0.95 mm is shown in Figure 13 where the expected characteristics of a single degree of freedom system are more apparent than the 0.34 mm case. The gain factor is overestimated, but this could have been overcome by windowing the data. As found with the simulations. the results are better because of the increased importance of this term, which enhanced the signal-to-noise ratio. Similar characteristics were evident in the numerical simulation, where the drop below 20 Hz was attributed to poor signal-to-noise ratios due to integration and truncation errors. For the experimental case, the same problem arises since the analysis is performed using velocity terms derived from integration of accelerometer outputs. Thus, the parameters identified can be relied upon to provide adequate representation of the dynamics of the system under study for frequencies in the range 20-80 Hz. By

NONLINEAR

SQUEEZE

239

FILM DYNAMICS

.E

I

6

0-J. (I

100

I 100 Frequency

Figure 13. Experimental

(Hz)

linear path spectra; h = O-95 mm.

changing the amplitude of the force, small variations were found in the results, especially for the viscous force coefficients. It would seem, therefore, that a more sophisticated model is required to account for the viscous force effects.

7. REMODELLING

THE VISCOUS

TERM

In order to examine the possibility of remodelling the viscous term, a series of tests were conducted in which the film size was varied and the results for thicknesses of O-34, 0.68 and O-95 mm were compared. The plate force measurement was used in this analysis in order to bypass the linear dynamical element of the support system. The results obtained for each of the four nonlinear coefficients at the three heights are presented in Table 2, along with those for O-34 mm using the excitation force. The unsteady inertia force coefficient can be seen to vary slightly between different heights, and similar results were found when both the excitation and plate forces were analysed. The consistency of results between heights confirms the suitability of this

240

H. ESMONDE ET AL

TABLE2 Measured coefficients for different film thicknesses

h (mm)

B

(Y

0.34* 0.34 0.68 0.95

1.25 1.22 1.25 1.27

0.83 O-8 0.97 0.98

Y

6

8.8 8.85 IS.2 18.1

0.37 0.59 0.52 0.44

* Values derived using the excitation force term for describing unsteady inertia effects. It is also close to the value obtained using the iterative solution which gives LY= l-2. The convective coefficients vary, but not in a systematic fashion. The lowest value of 0.8is returned for a film thickness of 0.34mm, whereas values of O-97and O-98are recorded at higher thicknesses. These values are considerably lower than the analytically derived quantities listed in Table 1 for the smallest film thickness. It is possible that this is a consequence of the formation of boundary layers in the film, as shown in Figure 14, which would induce greater movement in the intermediate layers in the film, thus reducing turning motion of the flow. This suggests the presence of relatively larger boundary layers for the smallest film (lower Reynolds squeeze number). The viscous coefficient y increases with film thickness, indicating that viscous forces are more important for larger films. However, as the film thickness increases, the overall force contribution decreases on account of the cubic term in the denominator. Since y varies with h, it must also vary during the oscillations, because the function V,/(z, + /z)~does not fully account for viscous effects. Below 30 Hz the plate oscillations were relatively large, and thus y fluctuated and led to noisy results. This affected all the paths, so the gain factors were evaluated between 30 and 60 Hz where the graphs were better behaved. The upper frequency bound results from the relatively small displacements which effectively linearize the unsteady and viscous terms. This renders it impossible to separate the overall force into its constituent parts. The variation in y over the oscillation cycles was examined further by separating the viscous effects into those above and below the static position of the plate. To achieve this, equation (6) is rewritten as

r’d

v, IVdl

V”,

“d [w$?]

Ffl=n 1-a(z,+h)+p(Zd+h)2-~(Zd+h)2-y1Y(Zd+h)3 -

v, [*I},

(16)

y2y(Zd + q3

III

Plate motion

Figure 14. Boundary

layer formation

in film

NONLINEAR

SQUEEZE

241

FILM DYNAMICS

The viscous term has now been split up so that y, is the coefficient when the plate is below the mean position and y2 is the coefficient above the mean position. This assumes a step change in the viscous action when the plate crosses the mean position as is illustrated in Figure 15(a). The more abruptly the shift occurs from y, to y2 at the mean position h, the more appropriate the mathematical model becomes. In reality the shift would not occur suddenly and the value of y would vary continuously over the cycle, as shown in Figure 15(b). This analysis procedure was used to compare the viscous terms for the three heights. The results for 0.34 mm are shown in Figure 16, and it can be seen that the partial coherence for y, is relatively large, while that for y2 is much smaller. The viscous effect was thus much more important at the lower part of the cycle of vibration. This is to be expected since the flow beneath the plate will be more developed in the lower phase, resulting in an increase in shearing between the layers because of the reducing denominator in the viscous term. Velocity profiles which might be expected at different points in the oscillation cycle are outlined in Figure 17. In position A, the flow is accelerating and the profile is roughly uniform. At the lowest position B, the profile has developed so that the mid-stream velocity is greater than that at the edges. Once the plate begins to rise again due to the momentum of the mid-stream flow, the profile is like that shown in C where initially the edge velocities are greater than that at the centre. As the plate approaches the highest position, as shown in D, the centre flow will again be reasonably uniform, so that the cycle may restart as in A. The flow profile in positions A and D when the plate is above the mean height is relatively uniform with consequently small viscous effects. At position B below the mean height when the plate is descending, the profile is well developed and, hence, there is a viscous force present. When the plate begins to rise, as in C, the low velocity flow near the plates is able to reverse flow direction more easily than at mid

t

Y

J ,-=h

(b) Figure

15. Effect of height

on viscous coefficient:

(a) idealized;

(b) actual.

H.

242

ESMONDE El- Al.

-311r 100

I

1 I I nI

I/

~requcnc~(Hz)

~rcqu~n~~ ct I/ I

t3 I

( t>1

Figure

16. Results

for remodelled

viscous

term.

section. There is thus a variable velocity distribution through the him and, therefore. a viscous shearing effect. The change in coherence became more marked between the y, and yZ components as the mean film thickness increased. For the larger films, the amplitude of vibration is increased and the difference between the local viscous coefficients is more appreciable as the position of the plate changed. The values obtained for y, and yz are shown in Table 3 for the different film thicknesses. From the five-path analysis, the viscous coefficient seemed to increase with mean film height whereas, with the six path analysis, the coefficient increased at the bottom of the cycle. This may be attributable to increasing flow development allowed by the greater vibration amplitudes. Rather than being a function of mean film thickness, the viscous coefficient would thus seem to be dependent on some function of the amplitude of vibration. The values of y1 from the six-path analysis varied from those found in five-path analysis for the 0.34 and 0.68 mm sizes. The value for y1 was unchanged for O-95 mm and the y2 term was very noisy with an almost random phase. This indicates

NONLINEAR

SQUEEZE

c Figure

243

FlLM DYNAMICS

D

17. Possible

velocity

profiles.

that the two-regime supposition is less applicable at the larger height since the y2 term has little or no effect. There was a fairly large disparity between the head loss term, 5, when determined from the plate force system and when evaluated with the exciter force system. This term is, however, the least important of the nonlinear forces, particularly for the smallest film size. For 0.34 mm there is a variation of 37% while the difference was found to be less than 10% for the larger films. The plate force was considered to give more accurate results for the fluid forces since the linear contribution was smaller. The head loss term is relatively more important for the larger films because larger amplitude vibrations occur, and a greater volume of fluid enters and leaves the gap to produce greater pressure losses. The coefficient increases as the film thickness decreases, as seen from the plate force results shown in Table 2. This would be expected, since the relative change in section is greater at the singularity at the edge of the plate for the smaller gap sizes. The values obtained lie within the range determined for steady developed flow by Idel’cik (1969) at similar Reynolds numbers, even though the flow regime is quite different. The experimentally determined value, 6, is an average quantity, but in fact varies during the oscillations, as did all the terms to a greater or lesser extent. This was as a result of the formulation of the governing input functions to describe the nonlinear effects from a simplistic analytical model. These functions depend on the flow profile which varies during the oscillations, so that it would be necessary to account for this TABLE 3 Results for ‘two-viscous’ term model Ah (mm)

YI

Yz

Y2IYI

o-34 0.68 0.95

6-5 13.7 18.1

0.94 2.2 1.4

0.14 0.16 O-08

244

H. ESMONDE

ET AL.

variation in equation (6) to fully describe the problem. This has not yet been attempted, although the averaged nonlinear coefficients will approximate the problem more closely than a full quasi-linear approach. 8. SINUSOIDAL

SIMULATIONS

AND MODEL

VALIDATION

The results from sinusoidal tests for 0.34 mm and IO Hz are shown in Figure IX. This frequency was chosen so that the vibration responses would be large enough to illustrate the nonlinear aspects of the system. Although the forcing function was subject to a small amount of feedback, this was not included in the simulations, as the overall effects were insignificant. The simulated response for the same conditions as those of the experimental case are presented in Figure 19 using the experimental coefficients found from the spectral analysis for the nonlinear terms. The signals show the same nonlinear form and, from Table 4, the peak-to-peak magnitudes match reasonably well. Simulations were also undertaken using the coefficients given in Table 1 which were determined theoretically, where the values are dependent on the assumed tlow profile. From the comparison of magnitudes in Table 4, it is clear that the coefficients obtained from the spectral tests are the more accurate. The iterative solution is quite close. but, from the analysis of other system configurations reported

Excttatton force ( N )

Uisplacemcnt (mm)

Velocit! (m/a)

Acceleration

(m/a’)

Figure

18. Experimental

sine wave results

for h = 0.34 mm

NONLINEAR

SQUEEZE

245

FILM DYNAMICS

3 x lo-” (a) 2 z i

c

0.0

m s ._ P -3 x lo0.

1

0.1

0.2

0.3

0.2

0.3

Time (s)

Figure 19. Simulated sine wave results for h = 0.34 mm TABLE

Comparison

of simulations

with experimental Experimental results, R,

Displacement (mm) Simulated/experimental

(%)

0.62 100

Velocity (m/s) Simulated/experimental

(%)

5.9 x lo-’ 100

Acceleration (m/s’) Simulated/experimental

(%)

6.1 100

Plate force (N) Simulated/experimental

(%)

61 .O 100

4

values;

Spectral simulation 0.615 99 5.8 x lo--’ 98

Ah = 0.34 mm, frequency Uniform simulation 1.2 194

Parabolic simulation 0.79 127

= 10 HZ Iterative simulation 0.69 111

1.06 x 10-l 180

7.4 x lo-? 125

6.4 x IO-’ 108

6.3 103

10.2 167

7.6 125

6.4 105

49.5 81

41.0 67

51.5 84

49.5 81

H. ESMONDE ET AL

246

~.,.,.,.‘,,.,.,.,.,.,.~

;.,.,

0.2

0. I

0.1)

Il.1

I-IlllC (\I Figure

20. Simulated

force

components for h = 0.34 mm: inertia; - - --, viscous; -CX.

-O-O-. unsteady head loss.

Inertia:

-.

convectiw

by Esmonde (1989), this solution was not shown to be as reliable as the parameter values obtained experimentally. There is. however. some variation in magnitude for the plate force in the test simulation with the coefficients derived from the spectral analysis. This discrepancy arises because the viscous forces which become important for the small films have not been fully described. and the effect of shearing between layers is greater than that forecast in the model. To gain an insight into the individual force components within the Mm. a plot of the various fluid forces was made during the numerical simulation as shown in Figure 20. The viscous force at 0.34 mm is sharply peaked just before the plate minimum position. This force therefore stops plate-to-surface contact. During the compressive phase the convective and head-loss terms, which were less important for this small film, both peak in the same place, leading the viscous force slightly and acting in the same direction. The unsteady inertia term opposes the other forces and acts so as to bring the plate in contact with the surface. Since the viscous term has the instantaneous film thickness cubed as the denominator, it is this force which prevails, and the motion ceases before collision occurs. During the initial part of the expansion phase the unsteady inertia, viscous and head-loss forces act in the same direction, opposing the motion. The convective inertia being a squared term assists the motion. acrmg so as to separate the surfaces, irrespective of the phase of movement. At the end of the expansion phase and at the start of compression when the plate is at the uppermost position and the velocity reduced to zero, it is only the unsteady inertia which has any effect, acting so as to separate the surfaces. 9. CONCLUSIONS The experimental investigation film dynamics. The analytically

presented derived

here has given a useful insight expression for the fluid force

into squeeze was used to

NONLINEAR

SQUEEZE

FILM DYNAMICS

247

establish the form of the relevant parameters but could not be relied upon to accurately describe the response of the system. It was necessary to determine the significance of the terms previously derived and to evaluate their coefficients experimentally. This was performed efficiently using spectral techniques which conserved the nonlinear form of the parameters. However, variance in magnitude of the coefficients for the nonlinear terms occurs due to oversimplification in the basic model (established for steady developed flow regimes). The model was thus extended, by incorporating an extra path, allowing a more in-depth examination of viscous effects. The results from the revised model indicated more clearly the reason for fluctuations of the coefficients and hence a model for the flow regime was suggested. Good agreement was obtained between simulated and theoretical results for both random and sinusoidal tests indicating the suitability of the numerical algorithm for modelling nonlinear squeeze forces. The random simulations were useful when compared to experimental results, as spurious values due to noise contamination, noted in the simulations, could be discounted from the experimental results. The parameters evaluated in the random analysis procedures were used in sine tests, where it was possible to examine the individual nonlinear force components of the overall force. ACKNOWLEDGEMENT This work was performed in co-operation at C.E.N. Saclay, Paris, France.

with the Commisariat a I’Energie Atomique

REFERENCES A,TA, S., AXISA, F. & CHRIGUI, H. 1985 On the effect of dense fluid on structures during impact. In Proceedings 8th International Conference on Structural Mechanics in Reactor Technology, Brussels, Paper B4/4. ANTUNES, J., AXISA, F., BEAUFILS, B. & GUILBAUD, D. 1988 Coulomb friction modelling in numerical simulation of vibration and wear work rate of multispan tube bundles. In Proceedings ASMElCSMElIMechElIAHR International Symposium on Flow Induced Vibrations and Noise, Vol. 5 (eds Paidoussis, M. P., Chenowith, J. M., Chen, S. S., Stenner, J. R. & Bryan, W. J.), pp. 157-176. New York: ASME. AXISA, F., ANTUNES, J. & VILLARD, B. 1988 Overview of numerical methods for predicting flow-induced vibration. ASME Journal of Pressure Vessel Technology 110,6-14. BENDAT, J. S. & PIERSOL, A. G. 1986 Random Data: Analysis and Measurement Procedures. New York: Wiley Interscience. BURROWS, C. R., SAHINKAYA, M. N. & KUCUK, N. C. 1987 A new model to predict the behaviour of cavitated squeeze-film bearings. Proceedings of the Royal Society, A 411,

445-466. DEVOGELAERE, R. 1955 A method for the numerical integration of differential equations of second order without explicit first derivatives. Journal of Research of the National Bureau of Standards 54, 119-125. ELLIS, J., ROBERTS, J. B. & HOSSEINI, A. H. 1988 A comparison of identification methods for estimating squeeze film damper coefficients. ASME Journal of Tribology 110,119-127. ESMONDE, H. 1989 Spectra1 analysis of non-linear squeeze film dynamics. Ph.D. Thesis, Dublin University. IDEL’CIK, I. E. 1969 Memento des pertes de charge. Collection du Centre de Recherches et d’Essais d’ElectricitC de France. Paris: Eyrolles. KUZMA, D. C. 1967 Fluid inertia effects in squeeze films. Applied Scientific Research 18,15-20. MULCAHY, T. M. 1980 Fluid forces on rods vibrating in finite length annular regions. Journal of Applied Mechanics 47, 234-240. RICE, H. J. & FITZPATRICK,J. A. 1988 A generalised technique for spectra1 analysis of nonlinear systems. Mechanical Systems and Signal Processing 2, 195-207.

248

H.

ESMONDE

ET AL.

RICE, H. J. & FITZPATRICK, J. A. 1991 The measurement of nonlinear damping in single degree of freedom systems. ASME Journal of Vibration and Acoustics 113,132-140. ROGERS, R. J. & AHN, K. J. 1986 Fluid damping and hydrodynamic mass in finite length cylindrical squeeze films with rectilinear motion. In Flow-Induced Vibration-1986 ASME, PVP Vol. 104 (eds Chen, S. S., Simonis, J. C. & Shin, Y. S.), pp, 99-106. New York: ASME. ROGERS, R. J., BERG, M. M. & RAMPEN, W. H. S. 1988 Measurement of non-linear hydrodynamic mass and damping in finite length, cylindrical squeeze films. In Proceedings ASMEICSMEIIMechEIIAHR International Symposium on Flow Induced Vibration and Noise, Vol. 4 (eds M. P. Paidoussis, Au-Yang, M. K. & Chen, S. S.), pp. 55-73. New York:

ASME. STYLES, D. D. & DODDS, C. J. 1976 Simulation of random environments for structural dynamic testing. Experimental Mechanics 16, 416-424. TICHY, J. A. & WINER, W. D. 1970 Inertial considerations in parallel circular squeeze film bearings. ASME Journal of Lubrication Technology 92, 588-592. TICHY, J. A. & BOURGIN, D. 1985 The effect of inertia in lubrication flow including entrance and initial conditions. Journal of Applied Mechanics 52, 759-765. APPENDIX:

NOMENCLATURE

excitation force (N) fluid force (N) plate force (N) autospectrum of i(t) cross spectrum of i(t) and j(t) conditioned auto spectrum conditioned cross spectrum height of squeeze film (m) mass of plate (kg) pressure (Pa) pressure at plate edge (Pa) radial distance (m) radius of plate (m) Reynolds squeeze number w(zd + h)?/v velocity of plate (m/s) acceleration of plate (m/s’) radial velocity (m/s) displacement of plate (m) nondimensional coefficients for nonlinear terms ordinary coherence function of i(t) and j(t) partial coherence function of i(t) and j(t) multiple coherence function pnR4/8

fluid density (kg/m3) absolute viscosity of fluid (Pa s) kinematic viscosity of fluid (m*/s) complex conjugate