Estimation Of Linearised Fluid Film Coefficients In A Rotor Bearing System Subjected To Random Excitation

Transcription

1 Estimation Of Linearised Fluid Film Coefficients In A Rotor Bearing Sstem Subjected To Random Ecitation Arshad. Khan and Ahmad A. Khan Department of Mechanical Engineering Z.. College of Engineering & Technolog A. M. U. Aligarh, India ABSTRACT An identification procedure for estimating sstem parameters of rotating machines speciall the bearing fluid film coefficients is described. A rigid rotor running on fluid film bearings is ecited b random force. An approach for parameter estimation is developed through a first order Volterra kernel representation of the sstem response. A two-degree-of-freedom sstem with cross coupling effects is considered in the present stud. In addition to direct FRFs (frequenc response functions or first order direct kernels, such a sstem requires definition of cross FRFs or cross-kernels and their estimation. The epressions for direct and cross-kernels are constructed in frequenc domain. The parameter estimation procedure involves etraction of first order Volterra kernels from the response of the sstem to a Gaussian white noise ecitation. Eight linearised fluid film coefficients are estimated b fitting the measured comple frequenc response function to those theoreticall calculated. The procedure is illustrated through numerical simulation. The assumptions involved and the approimations are discussed. The influence of ecitation force and probable measurement noise on estimates is illustrated through numerical simulation. INTRODUCTION Rotor bearing sstems pla a vital role in various fields of engineering. In rotating machiner, the dnamic stiffness of the structure that supports the rotating shaft has considerable effect on the machine vibration. It not onl affects the critical speed of the machine, the vibration in between the critical speeds but also affects the forces transmitted from the machine to the ground and noise emanating from the machine. Since the elements of support structure are assembled in series so the most fleible element has major effect on machine vibration characteristic. In large rotating machines bearing is the most fleible element and this is frequentl a fluid film bearing. Bearing characteristic has a major influence on the performance of the rotor bearing sstem. It has been found that rotor-journal-bearing sstem is a tribo-dnamic sstem and the dnamic characteristic of fluid film and other tribo elements are time varing []. The impedance of most fluid film bearings can be represented with reasonable accurac with eight linear springs and damping coefficients as described b Lund [6]. Over the past few ears the principal of structural identification has been applied in different time domain

2 and frequenc-domain techniques of fluid film coefficients estimation [,3,4]. Each method has some relative merits and demerits.the work presented in reference [] showed that a frequenc domain approach was less susceptible to noise and offer more convenience for being implemented on line. An identification procedure for estimating bearing film coefficients is suggested in order to reduce the influence of phase measurements errors on estimated results []. An eperimental procedure for measuring bearing impedance using multi frequenc test signals give more reliable data and agree with the theoretical prediction within 0% [5]. When the ecitation is due to mass unbalance it is single frequenc ecitation and the response is large in the vicinit of critical speed. owever, when the ecitation is from outside source such as in case of support ecitation can be a single frequenc tpe or random involving several frequencies. Lund [6] carried out response spectral densit analsis of rotor sstems due to stationar random ecitation of the base considering ecitation in vertical direction onl. Bhat [7] used modal analsis to analze the response of an undamped single mass rotor mounted on fluid film bearings. Tessarzik et al. [8] analzed eperimentall the response of turbo rotor to random ecitation. The aim of the work described in this paper is to develop a method for estimation of the parameters. Two degree of freedom sstem with direct as well as cross-coupled stiffness and damping coefficients is considered. An approach for parameter estimation is developed through a first order Volterra kernel representation of the sstem response [9]. Using frequenc domain analsis the first order kernel is etracted from the measurement of applied force and response. The influence of ecitation force, measurement noise and errors on the estimated results is analzed. Comple curve fitting technique is emploed to improve the accurac of estimated coefficients. The procedure is illustrated through numerical simulation. In most conditions of laminar flow the inertia coefficients are neglected and therefore we are considering onl the eight coefficients of the fluid film bearing. SYSTEM MODEL AND GOVERNING EQUATIONS OF MOTION A rigid rotor supported on fluid film bearing can be represented as shown in Fig., where the fluid film is represented b eight spring & damping coefficients as discussed earlier. The equation of motion in the vertical and horizontal direction relating the displacement to the forces applied to it is given b m & + c & + c & + k + k f ( t ( = & c k k f ( t = m, c c, c m & + c & ( where is the mass, c are direct linear damping terms; are cross-coupled linear damping coefficients and k, are the direct linear stiffness terms, while k, k are the cross-coupled linear stiffness term. f t, k ( f represents the ecitation given to the sstem in & direction. (t, in the above equation

3 In order to write the equation of motion in non-dimensional form. Let us define k τ = ω n t, ciiω n ω n =, ii =, η =, m mω n X st kij fi ( τ F η =, ij =, f i ( τ =, and X st = X st k Fma k (3 where, i =, ; and j =, ma Therefore the equations of motion can be written in a non-dimensional form as η + η + η + η η = f( τ (4 η + η + η η η = f ( τ (5 where ( denotes differentiation with respect to τ. The solution of equations (4 and (5 is represented in terms of Volterra operators as η τ = f ( τ, f ( (6 [ τ ] [ f ( τ, f ( ] ( η( τ = τ (7 Considering the first order Volterra kernel representation of response, we can write κ ( i κ ( i η ( τ = f ( τ (8 where [ f ( ] [ ] κ ( i κ ( i i τ = h f i τ τ with κ denotes or for i=, i ( dτ Taking Laplace transform of equation (4 & (5 we get (9

4 s s η s + s η( s + s η( s + η( s η( s = F (0 ( η + s η( s + s ( s ( s η η η( s F ( s ( = Solving the above two simultaneous equations (0 and (, the solutions for the operators η (s and η(s are ( s + s F ( s F η = ( ( s + s + ( s + s ( s ( s ( s + s + Fs ( s F η = (3 ( s + s + ( s + s ( s ( s which give ( s + ( s = (4 ( s + s + ( s + s ( s ( s ( s S = (5 ( s + s + ( s + s ( s ( s ( ( and similarl ( ( = (6 ( s + s + ( s + s ( s ( s Also ( s + s + S = (7 ( s + s + ( s + s ( s ( s ( ( ( ( ( ( ( s + s = (8 ( s ( s = (9 ( s + s + COMPUTER SIMULATION The procedure is illustrated through numerical simulation of the response for the nondimensional coupled equations (4 and (5. The forcing functions in the equations are normalized zero mean random forces, f ( and f (. The ecitation forces are τ simulated through random number generating subroutines and are normalized with respect to their maimum values. A tpical sample of such ecitation is shown in Fig.. Owing to the statistical nature of the problem, the procedure is illustrated for various sets of direct and coupled stiffness and damping parameters. The following case studies (Table- have been designed to stud the influence of direct and coupled parameters and errors involved. τ

5 Table : Case Studies Case-I Case-II Case-III Case-IV The response is computationall simulated, for various sets of numerical values of the parameters in equations (4 and (5. The governing equations are then numericall solved using a fourth order Runge-Kutta method, to obtain the responses in and directions. These responses are fed as inputs to the parameter estimation algorithm. The various direct and cross-coupled first order FRFs are etracted from the response and consequentl parameter estimation is carried out. The estimated parameters are compared with those originall used for the simulation of the response and accurac of the estimates and errors involved are studied. RESULTS AND DISCUSSION The power spectrum of the random force averaged over the ensemble of 000 samples is shown in Fig. 3. The non-dimensional responses η(τ and η(τ have been numericall generated in the non-dimensional time range The ensemble is constructed from 000 different samples of the simulated random force. The response spectra are fed as input to the parameter estimation algorithm. The first order direct and cross Volterra kernels i.e. and, Fig 4 and Fig. 5 respectivel are computed using equations (7 and (6 for case-i. These kernels ehibit peak responses corresponding to two critical frequencies. The eight coefficients i.e.,,,,,,, and are obtained using comple curve fit technique [0]. The parameters thus estimated are = , = , = , = = , = , = , = The eact values of the above coefficients are those used as input for numerical simulation of the response. While, the direct as well as cross stiffness terms are seen to be estimated with a ver good degree of accurac, the error in the damping estimates is higher. The curves of Figs. 6 and Fig. 7 shows the errors incurred in the estimate of and due to statistical nature of the Fast Fourier Transform computational procedure and the finite length of samples. The eact values of these

6 kernels are those obtained from epressions, after direct substitution of numerical values of the coefficients emploed for simulation of the responses. The normalized random error as known, can be seen to be maimum in the vicinit of peak responses at critical frequencies. Since the numerical error is higher in the vicinit of the peaks, the error in damping estimates ma be reduced b taking care of these zones. In cases II and III also the estimates are found to be satisfactor. Case IV has been studied to consider the effect of damping where the peaks are found to be no sharper as in Case-I, whereas the parameters are found to be within the acceptable limits. CONCLUSIONS The Volterra kernels etraction is proposed for the estimation of eight-linearised fluid film parameter in frequenc domain. Comple curve fitting technique is emploed in order to improve the accurac of estimated coefficients. The effect of various tpes of disturbances have been analzed using simulated data. The procedure developed gives ver good engineering estimates of fluid film parameters. It has been found that the normalized random error is maimum in the vicinit of peak responses at critical frequencies; the error in damping estimates ma be reduced b taking care of these zones. The simulated results also show the effect of ensemble size on estimated results. It has been found that large sample size is preferable for obtaining sharp peaks that gives more accurate results. It has been also found that the procedure developed is robust to the measurement noise, which can be usefull emploed for design of eperiments. NOMENCLATURE m mass τ nondimensional time c,, direct and cross damping parameters c c ( t, f ( t ( t, f ( t f input force in and directions f normalized input force in and directions F maimum value of the input force ma ( (, first order cross Volterra kernel transform ( (, first order direct Volterra kernel transform k,, k, direct and cross stiffness parameters k k,,, nondimensional direct and cross stiffness parameters,,, nondimensional direct and cross damping parameters η ω n, η nondimensional response in and direction natural frequenc

7 Fig. : Tpical sample of normalised input force Fig. 3: Power spectrum of input force Fig. 4: First order direct Volterra kernel for Case-I ( Fig. 5: First order cross Volterra kernel ( for Case-I Fig. 6: Normalised Error in the estimate of ( for Case-I Fig. 7: Normalised Error in the estimate of ( for Case-I

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