Linearized squeeze-film dynamics : model structure and ... - CiteSeerX

263

Linearized squeeze-film dynamics :model structure and the interpretation of experimentally derived parameters C R Burrows, PhD, DSc, CEng, FIMechE School of Mechanical Engineering, University of Bath

N C Kucuk, BSc, PhD CAE Electronics Limited, Saint Laurent, Quebec, Canada M N Sahinkaya, MSc, DPhil Uzunoglu Shipping and Trading Company, Istanbul, Turkey

R Stanway, MSc, DPhil, CEng, MTMechE, MIEE Department of Mechanical Engineering, University of Liverpool

In this paper the authors address the problem of parameter estimation in linearized models of squeeze-film dampers. After demonstrating how the choice of an unsuitable model structure can lead to the misinterpretation of experimentally derived coeflcients, a technique is described for assessing the significance of individual inertia, damping and stifness effects associated with the squeeze-film. An experimental study involving a squeeze-film vibration isolator is described. I t is shown how the technique allows the model structure to be simplijied in a systematic way. NOTATION

(

axx, ax, , .. . etc. oil-film inertia coefficients

transducer gain coefficients in x and y directions respectively radial clearance matrix of measurements and known parameters ... etc. oil-film damping coefficients static eccentricity position excitation forces in x and y directions respectively amplitudes of applied harmonic forces in x and y directions respectively J-1 . etc. combined stiffness Coefficients for oil film and mechanical structure damper-ring mass (per land) integer total number of harmonics included in input forcing sequence complex number relating harmonic displacements in x and y directions time matrix of observations displacements in x and y directions respectively amplitudes of harmonic displacements in x and y directions respectively coefficients static eccentricity ratio matrix of unknown coefficients static attitude angle angular frequency fundamental frequency The M S was received on 20 November 1989 and was accepted for publication on 21 March 1990. C06389 0 IMechE 1990

0954-4062/90

Y, ( Ii

1 (- 1 ( 1T (*

superscripts denoting real and imaginary components respectively denotes differentiation with respect to time denotes estimated quantity transpose of a matrix

1 INTRODUCTION

The application of modern parameter estimation techniques (1) to obtain dynamical models of oil-film bearings in vibration is now becoming an established practice in the turbomachinery field (2). Parameter estimation involves the processing of experimental vibration records to obtain numerical values of the coefficients in differential or difference equation models of the oil-film dynamics. Experimental programmes of this nature are usually justified under circumstances where oil-film bearings are liable to have a profound influence upon the overall vibrational characteristics of the rotating machine and where models derived from lubrication theory are unable to account for observed behaviour (3). What lubrication theory does offer is a tentative structure for the parametric model. Traditionally, linearized models have been used on the assumption that such a treatment is capable of accounting for the steady running behaviour of a well-balanced rotating machine (4). This leads to differential equation models of the oilfilm dynamics involving inertia, damping and stiffness coefficients. Such models are appealing as the coefficients can be related to physical properties of the oilfilm. In principle, such relationships should assist with the choice of model structure for a given rotor-bearing configuration and also facilitate the interpretation of $2.00 + .05

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Fig. 1 General arrangement of squeeze-film vibration isolator

results. In practice, the choice of a suitable model structure for parameter estimation and the interpretation of estimated parameters is not as straightforward as some earlier analyses might imply. Leaving aside for the moment technical aspects of parameter estimation (these will be discussed later in Sections 2 and 3) the choice of model structure is critically dependent upon the ultimate purpose of the parameter estimation experiments. To illustrate this point, first consider the situation where the purpose of identifying a parametric representation is to obtain a model that accounts for the behaviour of oil-film bearings in a rotating machine. Typically the oil-film model might be required as input data for a computer program used to predict the vibrational behaviour of turbomachinery (5). Here, good modelbuilding practice (6) would dictate that a structure involving the minimum number of unknown parameters be chosen. This does not imply, however, that only a single structure can provide adequate characterization of the measured input/output relationship: identified models of different structure can predict observed responses, as can be demonstrated with reference to two recent experimental studies. In reference (7),Stanway et al. demonstrated how the synchronous response of a squeeze-film damper bearing could be accurately predicted on the basis of an identified oil-film model involving four damping coefficients. In contrast, Roberts et al. (8) showed that the step responses of a similar device could be predicted using a model involving one inertia and one damping coefficient. Taken together, the studies illustrate the danger of attaching physical significance to estimated parameters. A model may be adequate in the sense of accounting for observed behaviour without being unique. Part C: Journal of Mechanical Engineering Science

Secondly, consider the situation where the purpose of parameter estimation is to gain physical insight into the mechanisms at work within the oil-film. Under these circumstances the uniqueness of estimated parameters must be considered in addition to the adequacy of the identified model. In line with this philosophy, Burrows and Stanway (9) argued the case for adopting an oil-film model involving a total of twelve coefficients -four inertia, four damping and four stiffness terms. The justification for this approach is that if the experimental conditions are designed such that the values of the estimated parameters are unique with respect to the input/ output data then the possibility of misinterpreting coefficient values is eradicated. With reduced-order models (7, 8) the misinterpretation of results is inevitable if assumptions concerning the significance of neglected terms are not justified. Recent evidence has emerged to support the view that inertia effects in squeeze-film dampers are liable to be significant (lC12). The strategy proposed in reference (9) offers a systematic means of elucidating the significance of squeeze-film inertia. In what follows here the authors address the general problem of estimating the inertia, damping and stiffness coefficients associated with a squeeze-film damper bearing. After stating the problem, conditions for ensuring identifiability of oilfilm coefficients are examined. It is demonstrated, by an example, how models which are adequate in accounting for observed behaviour do not automatically lead to increased physical insight by providing unique values for inertia, damping and stiffness coefficients. This leads to a summary (taken from reference (9)) of the conditions, necessary and sufficient, to ensure unique parameter values. A suitable algorithm for parameter estimation is developed and finally an experimental 0 IMechE 1990

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study is described to illustrate the application of this algorithm. The apparatus employed in the experimental study consists of a model squeeze-film bearing (Fig. 1). This test facility was chosen deliberately for the present investigation as it is capable of satisfying the experimental conditions necessary in order to estimate up to twelve squeeze-film coefficients. A previous investigation involving the same facility demonstrated that reducedorder models could account for observed behaviour. In the present investigation it is shown that identification of more general twelve-coefficient models not only accounts for observed behaviour but also provides insight into the physical mechanisms at work. 2 STATEMENT OF PROBLEM

Consider a squeeze-film vibration isolator in which the motion of a damper-ring of mass 2 m is constrained by centering springs. Forces are developed within the squeeze-film which is contained in the annulus between the damper-ring and a bearing housing. Assume that the bearing has two lands separated by a central circumferential groove through which oil is supplied. For small displacements in the x and y directions (vertical and horizontal respectively) the equations of motion per land can be written (9)

(rn

+ uxx)x+ a,, j; + c,, k + L,, x

-

+ C,,L + k,,y

+ uyy)j;+ a,$ +

CY,j

= F,

+ L,,y + c,, k + k-,, x = F ,

3 CONDITIONS FOR IDENTIFIABILITY OF OIL-FILM COEFFICIENTS

A start is made by considering an experiment involving a physical system believed to be described by the squeeze-film bearing model [equations (l)]. Assume that the system is subjected to harmonic forcing of the form F , = F,exp0wt) and F, = P,exp(jot) such that the resulting displacement responses take the form 1990

{ - 0 2 ( m + uxx)+ jwc,,

+ C,,>X

+ (-02u,, + jwc,, + E,,)F = F, { -w2(m + u,,) + jwc,, + Ly,}Y + ( --02uy, + joc, + Cy,)8 = F,

(2a) (2b)

which relate the (complex) displacement responses to the applied forces in terms of the various squeeze-film coefficients. As part of the experiment the displacement responses 8 and Y are measured. These measurements can be related by defining a function T(jo)such that

P = T(jo)R

(3) In general T(jw)will be a complex quantity. However, for purposes of illustrating what information can be extracted from the experiment it is assumed that the displacement orbit is circular, such that Too)= -j. Using equation (3) to eliminate Y from equation (2a), then

C{ - w2(m + a,,) + L,, + oc,J

+ j(ocxx+ 02uxy- EXy)]X = F, [{ -w2(rn

where a,, etc. are squeeze-film inertia coefficients, c, etc. are squeeze-film damping coefficients and k,, etc. represent the combined stiffness of the oil film and the external springs. Applied forcing is represented by the terms F, and F,. The problem addressed here is that of estimating all twelve coefficients in equations (1) from measurements of the applied forces F, and F, and (typically) the resulting displacement responses x and y. If it is assumed that the mass of the damper-ring and the stiffnesses of the external springs can be determined from experiments performed in the absence of oil, then the significance of each inertia, damping and stiffness term due to the oil-film can be readily assessed from numerical values of the coefficients in equations (1). Before describing an approach it is worthwhile to consider the implications of attempting to estimate parameters in a reduced-order model without prior knowledge of the significance of neglected coefficients. This will serve to justify the adoption of the twelvecoefficient model.

0 IMechE

At) = exdot). Substitution of these expressions into equations (1) and cancellation of the exp0ot) terms leads to the frequency domain equations

x(t) = S exp0wt) and

(4a)

and similarly eliminating X from equation (2b), (1)

(rn

265

+ a,,) + L,,

- wc,,}

+ j(ocy, - o Z a y x+ &,JY

=

F, (4b)

Equations (4) are of the form

+ j8,)X (a, + j&)Y

(a,

P, = P, =

(54

(5b) and as a set of linear algebraic equations these can be solved to yield unique values for the four coefficients a,, ..., 8,. To obtain information on the squeeze-film parameters from knowledge of a,, .. ., 8, requires a number of prior assumptions. Using the experimental approach described, it is impossible to distinguish between the effect of squeezefilm inertia, damping and stiffness. For example, the expression a, is composed of terms involving direct inertia, cross-damping and direct stiffness. Even if the stiffness is assumed to be negligible (which is a reasonable assumption for an uncavitated squeeze-film), there is no way of distinguishing between the effects of a,, and c,, . Only if a further assumption is made-that oilfilm inertia is negligible-an the four damping terms be estimated. Such as approach is obviously unsatisfactory where it is the significance of the inertia terms that is being sought. Moreover, the analysis highlights the dangers of misinterpretation which arise through the use of reduced-order models. Using the concept of structural identifiability, Burrows and Stanway (9) showed the minimum test requirements for estimating a unique numerical value for each of the twelve squeeze-film coefficients. To summarize, it is necessary to: (a) excite the oil-film, first in the x direction and then in the y direction using a multiple-frequency test signal which persistently excites all the system modes; Roc Instn Mech Engrs Vol 204

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(b) measure displacement responses in the x and y directions ; (c) measure the corresponding force vector; (d) apply a suitable data analysis package. In a test facility designed specifically for measuring squeeze-film dynamics, it is straightforward to satisfy requirements (a), (b) and (c). If excitation is applied only in a single direction then a maximum of ten unique coefficients can be obtained from the experiments (9). Requirement (d) can, in principle, be satisfied in a multitude of ways. A time-domain approach involving the estimation of parameters in state-space models was outlined in reference (4). However, a frequency-domain algorithm has been found to be of more practical value and will be used in the experiments described here.

mF = 4 PARAMETER ESTIMATION ALGORITHM

A frequency-domain algorithm for estimating oil-film stiffness and damping terms has been described previously by Burrows and Sahinkaya (13).This algorithm can readily be extended to accommodate oil-film inertia terms. Consider a squeeze-film isolator described by equations (1) where the excitation forces F x and F y are derived from Schroeder phased harmonic sequences (SPHS) (14). Such sequences contain multiple frequencies and provide persistent excitation of the oilfilm. Because the sequences are periodic the displacement responses of the damper-ring can be expanded into real and imaginary components: x(t) =

X

+

exMot) = (Xr jXi)exp6wt)

(6) F J t ) and

Similar expressions can be written for fit), F#). The expression for x(t) given in equation (6) [along with the similar expressions for At), Fx(t) and Fy(t)] can be substituted into equations (1) and after collecting real and imaginary terms we can write WF@F = C F

SPHS

(7)

Shaker

F,(tj

+

4 1

for n = 1, ... , N. The subscript n denotes the nth harmonic in the range from 1 to N. Subscript 1 denotes data obtained from experiments where the system is excited only along the x axis and subscript 2 only along the y axis. Equation (7) has been written on the assumption that the only forcing terms acting on the system are the artificially generated SPHS. This assumption is justifiable in a test facility that does not rotate and where, as a consequence, there are no unbalance forces. The parameter estimation algorithm is not dependent upon this simplification as the frequency component corresponding to the running speed can readily be filtered out, Any signal corruption at frequencies above Nu, will also be filtered. The significance of employing a test facility in which there are deliberately no rotating force components is discussed later. x(r)

FFT

Squeeze4 1 m

-bearing

y(,)

471

I B

1\’1

Form wF,

cF

[equation (7jl APP’Y equation (8) to obtain leastsquares estimate of 0,

I

SPHS Schroeder phased harmonic sequence FFT Fast Fourier transform

Fig. 2 Schematic diagram showing parameter estimation procedure Par1 C: Journal of Mechanical Engineering Science

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The least-squares estimate for the matrix of coefficients is then given by

(w: wF)-'w: CF

(8) The estimation procedure is illustrated in Fig. 2. The forcing signals F,(t) and F,(t) and the corresponding displacements x ( t ) and y(t) are sampled and quantized. A discrete-time, fast Fourier transform provides the real and imaginary components of each signal at a predetermined number of frequencies. These components provide the data to form the arrays WF and C, in equation (7). The parameter array QF is then computed digitally using equation (8). In addition to estimating aFthe computer program also calculates the standard errors and confidence bounds for the estimated coefficients. Thus, a measure of the accuracy and reliability of each individual oil-film coefficient is available. If the experimental data are not sensitive to a particular coefficient then this is reflected in a large standard error and wide confidence bound. These statistical measures enable the significance of each coefficient to be quantified and provides the basis for systematic simplification of the model structure. @, =

5 EXPERIMENTAL FACILITY

5.1 Mechanical arrangement

The general arrangement of the squeeze-film vibration isolator is shown in Fig. 1. This facility has been used previously in experiments to measure the four damping coefficients associated with traditional mathematical models of uncavitated squeeze-films (7) and also in experiments to identify the damping laws associated with the squeeze-film under various operating conditions (15). The isolator itself consists of two main elements: (a) a non-rotating damper-ring symmetrically supported by a flexible shaft; (b) a bearing housing containing two plain lands separated by a central circumferential groove. Bearing housing

The flexible shaft provides the static load capacity to support the damper-ring. Figure 3 shows how the support is provided. A film of oil in the annulus between the damper-ring and the bearing housing provides the squeeze-film forces. Lubricant is supplied to the annulus by a pump through holes at the top and bottom of the circumferential groove. No end seals are fitted and thus, prior to recirculation, the oil is free to discharge into an open reservoir. The critical bearing parameters are : Bearing land length Damper-ring radius Radial clearance Lubricant viscosity

12.0 mm 60.0 mm 0.254 mm 0.06 N s/m2

Dry friction was minimized by carefully controlling the shrink-fit of the damper-ring onto the shaft and suitable design of the shaft supports. Referring to Fig. 3, at one extremity the shaft is firmly held in a taper-lock device while the other end is free to slide in a linear bearing. This arrangement produces a structure with extremely low internal damping. The structural damping is quantified in Section 6.1. Adjustment of the bearing housing to achieve the desired position of static equilibrium is achieved in the vertical direction by the mechanism shown in the sectioned view (Fig. 4). Referring to Fig. 4,a hardened steel ball rides on the tapered section of a screw which passes through the underside of the housing. As the screw is rotated the ball presses on the underside of the housing, causing it to rise in a vertical channel. The horizontal position of the housing is adjusted by the upper set of screws shown in Fig. 4. When the desired equilibrium position has been established, the housing is locked in position by four bolts. The equilibrium position is defined in terms of the static eccentricity ratio c0 and the static attitude angle 1,9~.These terms are defined in relation to the bearing clearance circle which is superimposed onto Fig. 4. In a real application, the damper-ring would normally be mounted on the outer ring of a rolling element

\ \\

Damper-ring

Fig. 3 Side elevation of squeeze-film vibration isolator showing arrangement for supporting damper-ring (the electromagnetic shakers have been omitted for clarity) @ IMechE.1990

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Fig. 4

Sectioned view of bearing housing showing lubricant feed arrangements and mechanism for adjusting vertical position of the housing

bearing. Excitation would arise from forces generated by a rotating shaft within the inner race. In the arrangement described here there are no rotating components and excitation is provided by the two electromagnetic shakers shown in Fig. 1. With this configuration any type of forcing can be simulated, but the nature of the oil squeeze-film may differ compared with the situation where forcing is due to out-of-balance components. When a rotating force is applied the squeeze-film rotates around the annulus, but this phenomenon will not necessarily occur when arbitrary forcing is applied along the x and y axes. The same limitation is inherent in the test procedure described by Roberts et al. (8). To ensure direct correspondence between the test conditions and those experienced in normal operation it would be necessary to superimpose a suitable mu1tiplefrequency test signal upon a rotating force component. To achieve this would require some further developments in the design of an experimental facility. The significance of this problem is referred to later.

5.2 Instrumentation and measurements The static eccentricity of the damper-ring in the bearing housing is monitored by mechanical dial gauges. Other static measurements are the oil pressure at the inlet to the housing and its temperature as the oil is discharged from the annulus. In the absence of rotation there is no significant temperature change across the squeeze-film. Forces applied to the damper-ring are measured by quartz load cells connected to charge amplifiers. The displacement responses of the damper-ring are measured by two pairs of non-contacting capacitive displacement probes, one pair in the vertical plane and one set in the horizontal plane. The signals from each pair are averaged to provide the mean displacement from equilibrium of the damper-ring. Care was taken to prevent oil from being deposited between the displacement transducer heads and the damper-ring. However, there was evidence of oil ingress and this was accounted for in determining the transducer constant for use in the estimation procedure. Part C: Journal of Mechanical Engineering Science

6 EXPERIMENTAL PROCEDURE

6.1 Estimation of structural parameters Before supplying oil to the squeeze-film isolator, a series of tests was undertaken to determine the structural characteristics of the damper-ring on its supports. These tests involved the application of an SPHS signal with a cut-off frequency of 100 Hz to each of the electromagnetic shakers in turn. The corresponding displacements of the damper-ring in both directions were measured, and along with the input force signal were recorded using a frequency-modulated magnetic tape recorder. Subsequent analysis of the signals could then be performed off-line. The frequency domain estimator [equations (7) and (S)] was applied to the force-displacement data to estimate the effective damper-ring inertia and the equivalent viscous damping and stiffness of the vibration isolator’s structure. Analysis of the data using a dualchannel F F T analyser showed that the frequency range of interest was from 5 to 50 Hz. The estimator was programmed accordingly. The FFT analyser was also used throughout the tests to monitor the system responses and obtain frequency response functions. The estimated values for the structural parameters are shown in Table 1. The estimated stiffness values of approximately 250 kN/m per bcaring land are in close agreement with those obtained independently by Stanway et al. (7). The estimated value of effective damper-ring inertia of 4.9 kg per bearing land compares with the value of 4.5 kg quoted in reference (7). The equivalent viscous damping in the structure is negligible in relation to the damping provided when oil is supplied to the annulus, thus confirming the result in reference (7). The small variation of the damper-ring mass in the two directions is due to slight imperfections in the alignment of the rods that couple the shakers to the damperring. The goodness-of-fit values were determined as part of the estimation procedure. Details of the computations are given in reference (16). A value of unity indicates a @ IMechE 1990

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Table 1 Estimates of structural coefficients obtained in the absence of oil Inertia -

Damping N slm

Stiffness N/m x lo5

ka

Axis

Estimate

Standard error

Estimate

Standard error

Estimate

Standard error

Goodness of fit

X

5.246 5.101

0.157 0.138

111.98 214.71

22.60 22.99

9.833 9.772

0.325 0.294

0.927 0.941

Y

film, values of eccentricity larger than 0.6 were not examined. The initial experiments provided estimates of all twelve squeeze-film coefficients and these are shown in Table 2. Further experiments involving reduced-order models are shown in Tables 3, 4 and 5. These results are discussed in detail in the section that follows. With the arrangement described, the data needed to produce the parameter estimates for the full model (Table 2) were obtained in less than 20 minutes. This represents a significant saving in time compared with tests where only a single frequency is injected at any given time. The approach also reduces the likelihood of errors being introduced due to parameter drift during an extended test procedure.

perfect model. Thus since the values in Table 1 exceed 0.92 for both axes, it can be said that the model structure and parameter estimates provide a good representation of the physical system.

6.2 Estimation of oil-film coefficients Given the numerical values of the structural parameters shown in Table 1 it was possible to proceed with the estimation of the squeeze-film inertia, damping and stiffness coefficients. After calibrating the displacement transducers to allow for the ingress of oil, the damperring was excited first in the x direction and then in the y direction. In this way it was possible to discriminate between the displacements associated with forces F , and F , . Following FFT analysis, both sets of frequency responses (that is direct and cross-axis) were supplied to the estimation algorithm [equations (7) and (8)l. Interest was concentrated on the range of static eccentricity ratios from zero to 0.6. For an oil inlet pressure of 50 kN/m2 this allowed the gains of the vibrator amplifiers to be kept constant whilst providing suitable excitation levels for linear analysis. To examine higher static eccentricity ratios would have required larger force levels and would have involved either a change in amplifier gain or larger perturbations at lower eccentricity ratios. To avoid the possibility of overexcitation and consequent non-linear behaviour of the squeeze-

7 DISCUSSION OF RESULTS

7.1 The twelve-coefficient model The results in Table 2 show that it is the six direct-axis coefficients-inertia, damping and stiffness-that govern the behaviour of the model with zero attitude angle. = 0.2, the standard Starting with the results for errors on the cross-inertia terms a,, and aYx and the cross-damping terms c,, and cy, indicate that these estimates are not significant. Also their magnitudes are

Table 2 Estimates from twelve-coefficient model, I/I~ = 0" 0.2

60

Coefficient

Estimate

0.4

Standard error

Estimate

0.6

Standard error

Estimate

Standard error

Stiffness N/m x lo5 2kX 2kXY 2kX 2%

5.401 0.373 0.409 4.813

0.220 0.159 0.152 0.110

6.698 0.242 0.310 4.9 13

0.442 0.255 0.225 0.1 29

6.928 0.600 0.467 5.788

2.361 0.653 0.562 0.154

0.447 0.171 0.227 0.086

19.452 2.097 - 1.546 8.582

2.294 0.596 0.546 0.142

2.755 1.185

154.601 26.981 15.292 26.791

12.720 3.840 3.020 0.910

Damping N s/m x lo3 2Cxx 2cxy 2% 2%

6.937 0.204 -0.027 4.345

0.159 0.100 0.1 10 0.069

12.888 1.002 0.067 5.002 Inertia

ke 2% L X Y

2% 2aYY

23.019 0.972 0.730 18.438

1.038 0.723 0.719 0.501

38.9 18 3.107 3.526 19.356

1.400

0.602

Goodness of fit axis y axis x

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0.855 0.966

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Table 3 Estimates from twelve coeficient model, 9, 0.2

80

Coefficient

Estimate

Table 5 Estimates from reduced-order

= 30"

model involving eight oil-film coefficients and two transducer constants

0.4

Standard error

Estimate

Standard error

Coefficient

Stiffness N/m x to5 2gx 2kJ.Y 2%. 2kY?

0.208 0.174 0.157 0.1 32

5.344 -0.286 -0.254 4.926

2Cxx 2% 2% 2%

Damping Ns/m x lo3 2Cxx 2CX? 25, 2c ....

0.149 0.115 0.112 0.087

6.422 0.863 0.647 4.670

Inertia 2%

2% 2ayx 2a,

0.962 0.811 0.727 0.631

21.450 1.806 2.462 19.287

67.840 20.870 17.179 28.570

4.371 0.335 0.362 4.062

0.128 0.098 0.152 0.117

Inertia kg

0.744 0.429 0.407 0.234

14.547 3.991 2.436 7.953

Standard error

Damping N s/m x lo3

0.595 0.443 0.326 0.242

7.430 0.536 0.097 5.726

Estimate

4.90 1 2.886 2.682 1.580

2%, 2% 2% 2%

19.900 1.608 2.358 20.374

0.4 19 0.342 0.498 0.460

h,, h,,

0.692 0.746

0.016 0.017

Goodness of fit x axis y axis

0.945 0.95 I

Goodness of fit x axis y axis

0.782 0.927

0.935 0.960

small in relation to the direct inertia and damping terms. Although all four combined stiffness coefficients are significant, it is obvious that the direct terms ixx and kyyare dominated by the contributions from the shaft that supports the damper-ring. Also, the cross-stiffness terms are negligbly small and probably arise due to asymmetry in the apparatus rather than from the squeeze-film itself. These results are consistent with the well-known theoretical result that the oil-film stiffness terms are zero in an uncavitated squeeze-film. As the eccentricity ratio increases, so the goodness of fit in the x direction decreases. However, the goodness of fit in the y direction remains around 0.97. This can be explained by the fact that increasing the eccentricity

ratio caused a significant reduction in the amplitude of the displacement response in the x direction while the y direction displacement remained at roughly the same level. However, the results display a similar pattern to those for E~ = 0.2, that is the oil-film stiffness effects are negligible and the direct inertia and damping coefficients are much larger than the cross-damping and cross-inertia Coefficients. Further work is required to relate this result to the extent of cavitation and the location of minimum film thickness in the annulus. The goodness of fit is poor in the x direction and this reinforces the earlier observation about the magnitude of the x displacement. This becomes smaller as c0 increases and the signal-to-noise ratio can only be improved by increasing the magnitude of the disturbing force and/or introducing more sensitive transducers.

Table 4 Estimates from reduced-order model involving six coeficients. 9, 0.2

60

Coefficient

Estimate

0.4

Standard error

Estimate

= 0"

0.6

Standard error

Estimate

Standard error

Stiffness N/m x lo5 ~~~

2kX 2kYY

5.414 4.812

0.294 0.151

6.811 4.937

~

~

0.567 0.178

8.384 5.824

3.158 0.199

0.574 0.116

25.350 8.91 1

3.163 0.159

3.515 0.8 13

174.137 24.469

17.157 1.056

Damping N s/m x lo3 2Cxx 2%

7.021 4.375

0.213

0.095

13.379 5.034

Inertia kg 2% 20,

23.050 18.380

1.393 0.685

38.923 19.041 Goodness of fit

x axis y axis

0.942 0.972

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0.881 0.967

0.650 0.978

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Taken alone, the results for E* = 0.6 as presented in Table 2 would not justify the use of a reducedcoefficient model. However, based upon the results for E* = 0.2 and 0.4 it is reasonable to propose a model employing only six direct coefficients to model the squeeze-film with zero attitude angle. If an attitude angle of other than Go = 0, 90, 180 or 2 7 0 is investigated, then the six-coefficient model is not necessarily adequate. This is shown by the results in Table 3 which were obtained for an attitude angle of = 30”. In particular, the cross-inertia terms are significant in relation to the direct terms. Thus the results support the arguments presented in references (10) to (12) concerning the significance of effective squeeze-film inertia terms. However, if additional forces were to cause the squeeze-film to rotate around the annulus then this may well modify the significance of the various coefficients.

7.2 Reduced-order models For eccentricity ratios of = 0.2, 0.4 and 0.6, Table 4 shows the results obtained by estimating only the six direct terms in the squeeze-film model. Despite the fact that the contribution of the oil-film to the overall stiffness of the system was previously shown to be negligible, the ‘combined’ stiffnesses lxx and g,,, were retained as a precautionary measure. From Table 4 it can be seen, however, that the numerical values obtained represent the direct mechanical stiffness values. The goodness-of-fit data for the reduced-order model indicates that, for Go = 0, direct inertia and damping terms provide as good a representation of the physical as a model involving twelve coefficients. It may be wise also to estimate lxx and R,,, to detect oil-film stiffness effects that would arise in a cavitated squeeze-film. If cavitation is not likely to be a concern, then equations (7) and (8) can be rearranged to estimate transducer gains rather than stiffness values. An example of this is shown in Table 5 where the combined stiffness coefficients Ex, and kyywere assumed to be equal to the stiffnesses of the supporting shaft. The values shown in Table 5 for a concentric orbit are consistent with those discussed earlier. The goodness of fit is above 0.94 in both directions and indicates an especially good representation of the squeeze-film dynamics. 8 CONCLUSIONS

In the development of experimental techniques to identify the dynamics of oil-film bearings, it is essential to establish a suitable model structure. Such a structure involving a total of twelve coefficients-four inertia, four damping and four stiffness terms-was proposed in an earlier paper (9).However, at the time that proposal was made, signal-processing algorithms were not sufficiently well developed to estimate all twelve coefficients and assess the significance of each individual term. In the present paper, the authors have underlined the need to begin with a twelve-coefficient model. This has been achieved by demonstrating how the influence of individual coefficients can be misinterpreted if reducedorder models are used as the starting point for an experimental study. For example, a model involving 0 IMechE

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only squeeze-film damping terms can account for observed behaviour in response to a simulated rotating force (7). However, using such a model it is not possible to discriminate between inertia, damping and stiffness effects. Basing the approach upon a frequency-domain estimation algorithm it has been shown how the parameters associated with a squeeze-film vibration isolator can be estimated. First, the inherent mass, equivalent viscous damping and stiffness of the damper-ring on its supporting structure were estimated. Then, using this information, a series of models of the squeeze-film dynamics was estimated. For zero static attitude angle it was shown how the cross-axis coefficients were negligible and how a model involving only six coefficients-the direct inertia, damping and stiffness-provided an accurate representation of the squeeze-film’s behaviour. However, for non-zero static attitude angle the cross-axis coupling needs to be accounted for. In general, oil-film stiffness is negligible in relation to the stiffness of the shaft supporting the damper-ring, thus indicating an uncavitated squeeze-film. However, the squeeze-film inertia does need to be included-only direct terms need be used to model the characteristics when the static attitude angle is zero, otherwise cross-axis inertia also needs to be included. The results confirm previous experimental studies which have suggested the need to include effective squeeze-film inertia. The advantage of the approach presented here is that it enables a systematic examination of the influence of all terms in the linear model. Provided that it is possible to excite the oil-film with a suitable forcing signal the approach could also be applied to examine the influence of inertia in journal bearings. Experimental results would be valuable as a means of checking the validity of theoretical predictions concerning the significance of inertia effects (17). The experiments described also emphasize the need to improve still further the techniques for parameter estimation and also to develop experimental apparatus. When testing any type of oil-film bearing under a variety of operating conditions it is not possible to guarantee generous signal-to-noise ratios in both the x and y directions. Consequently, signal-processing algorithms must be effective at rejecting random noise which contaminates measured responses. In this respect, a new parameter estimation algorithm, based upon frequency-domain filtering, has recently been developed. The algorithm has now been applied to squeeze-film data and preliminary results are encouraging (18). Work is also in hand to design a new experimental facility in which wideband forcing can be superimposed upon a rotating force vector. This will enable a more direct comparison with the results reported in references (10) to (12). REFERENCES 1 Eykhofl, P. System identification: state and parameter estimation, 1974 (Wiley-Interscience, London). 2 Ellis, J, Roberts, J. B. and Ramli, M. D. The experimental determination of squeeze-film dynamic coeficients using the state variable filter method of parametric identification. Trans. A S M E , J . Tribology, 1989,111,252-259. 3 Dowson, D. and Taylor, C. M. The state of knowledge in the field of bearing-influenced rotor dynamics. Tribology Int., 1980, 13, 1761 78. Proc Insto Mech Engra Vol 204

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4 Morton, P. G. The derivation of bearing characteristics by means of transient excitation applied directly to a rotating shaft. GEC J . Sci. Technol., 1975,42, 3 1 4 7 . 5 Firoozian, R. and Stanway, R. Modelling and control of turbomachinery vibrations. Trans. ASME, J . Vibr., Acoustics, Stress and Reliability in Des., 1988, 110, 521-527. 6 Box, G. E. P. and Jenkins, G. M. Time-series analysis,forecasting and control, 1976 (Holden Day, San Francisco). 7 Stanway, R., Firoozian, R. and Mottershead, J. E. Estimation of the linearised damping coeficients of a squeeze-film vibration isolator. Proc. Instn Mech. Engrs., Part C, 1987,2Ol(C3), 181-191. 8 Roberts, J. B., Holmes, R. and Mason, P. J. Estimation of squeezefilm damping and inertial coefficients from experimental free-decay data. Proc. Instn Mech. Engrs, Part C, 1986,2OO(CZ), 123-133. 9 Burrows, C. R. and Stanway, R. A coherent strategy for estimating linearised oil-film coefficients. Proc. R . Soc., 1980, 370(A), 89-105. 10 Tichy, J. A. The effect of fluid inertia in squeeze-film damper bearings: a heuristic and physical description. ASME paper 83-GT177,1983. 11 Tichy, J. A. Measurement of squeeze-film bearing forces to demonstrate the effect of fluid inertia. ASME paper 84-GT-11, 1984. 12 San Andre, L A. and Vance, J. M. Effects of fluid inertia and

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turbulence on the force coefficients for squeeze-film dampers. Trans. ASME, J . Engng for Gas Turbines and Power, 1986, 108, 332-339. Burrows, C. R. and Sahinkaya, M. N. Frequency-domain estimation of linearised oil-film coefficients. Trans. ASME, J . Lubric. Technol., 1982,104,210-215. Schroeder, M. R. Synthesis of low peak-factor signals and binary sequences of low auto-correlation. I E E E Trans. on lnformation Theory, 1970, IT-16,85-89. Stanway, R, Mottershead, J. E. and Firoozian, R. Non-linear identification of a squeeze-film damper. 7rans. ASME, J . Trihology, 1988,110,48&491. Burrows, C. R., Sahinkaya, M. N. and Kucuk, N. C. Modelling of oil-film forces in squeeze-film bearings. Trans. ASME, J . Tribology, 1986,108, 262-269. Smith, D. M. Journal bearings in turbomachinery, 1969 (Chapman and Hall, London). Stanway, R., Tee, T. K. and Mottershead, J. E. Identification of squeeze-film bearing dynamics using a recursive, frequencydomain filter. In Machinery dynamics-applications and vibration control problems, Proceedings of ASME Twelfth Biennial Conference on Mechanical Vibration and Noise, Montreal, 1989, pp. 161-176.

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