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Wtc2005-63274 Proceedings of WTC2005 World Tribology Congress III September 12-16, 2005, Washington, D.C., USA WTC2005-63274 FREQUENCY DEPENDENT DYNAMIC PROPERTIES OF TILTING PAD JOURNAL BEARINGS: EXPERIMENTAL RESULTS AND UNCERTAINTY ANALYSIS Waldemar Dmochowski Jacek Dmochowski National Research Council Dept. of Systems and Computer Engineering Institute for Aerospace Research Carleton University Ottawa, Ontario, Canada Ottawa, Ontario, Canada ABSTRACT domain and after introducing auto and cross spectral densities The paper presents experimentally obtained TPJB response can be written as to multifrequency excitation and its comparison with (1a) GF F (ω)− mbGF A (ω) = H xx (ω)(GF X ω)+ H xy (ω)GF Y (ω) theoretically obtained data. Uncertainty considerations for the y x y x y y G (ω)− m G (ω) = H (ω)(G ω)+ H (ω)G (ω) (1b) results obtained using the power spectral density method are Fy Fy b Fy Ay yx Fy X yy FyY also presented. It has been concluded that inertia forces and where pivot flexibility effects are behind the variations of dynamic , (2) Hij = kij + iωcij coefficients with frequency of excitation. power spectral densities G ω (autospectral density) and uu () (cross spectral density) are defined as 1. INTRODUCTION Guv (ω ) It has been expected that the dynamic coefficients of 2 ∗ (* denotes complex conjugate) (3) Guv ()ω = lim E[u ()ω v(ω)] tilting-pad journal bearings are frequency dependent ([1]). T →∞ T However, considering uncertainty of the results, the F ,F , A ,A , X ,Y are Fourier transforms of excitation forces in experimental evidence has been rather inconclusive (e.g. [2]). x y x y This paper presents experimental results of bearing response for the horizontal and vertical directions, accelerations in the a wide range of frequency of excitation. The test bearing was a horizontal and vertical directions, and displacements in the five pad TPJB with load-on-pivot configuration. horizontal and vertical directions, respectively, It is well known that the measured bearing dynamic mb - bearing mass coefficients have relatively significant uncertainties. They are fx, fy - components of the dynamic force estimated based on the statistical analysis of the already x, y - shaft center coordinates in the rectangular system obtained results. This paper presents an approach based on an with the origin at the bearing equilibrium position analysis of error propagation for a frequency domain method. k xx , k xy , k yx , k yy - stiffness coefficients c ,c ,c ,c - damping coefficients. 2. EXPERIMENT xx xy yx yy The test rig has been described elsewhere ([3]). The test bearing parameters are presented in Table 1. 4. UNCERTAINTY OF THE EXPERIMENTAL RESULTS Typically, uncertainty of the measured bearing dynamic Table 1 Test bearing parameters properties is evaluated based on statistical analysis of the Pad length 0.0396 m obtained data, which is usually referred to as Type A analysis ([5]). Such an approach gives an idea about the consistency of Nominal bearing diameter 0.0987 m the results. Here, the uncertainty analysis is aimed at Bearing radial clearance / Preload -3 0.088×10 m / 0.3 estimating the potential effect of elemental uncertainties of Number of pads/Pad angular extension 5 / 55° individual measurements on the uncertainty of the evaluated Advantages of using frequency domain have been known coefficients, which is referred to as Type B analysis ([5]). In for many years ([4]). For the experimental setup with the other words, a relationship has been sought linking time floating test bearing the equation of motion in frequency domain and frequency domain uncertainties. 1 Copyright © #### by ASME 1 Copyright © 2005 by National Research Council of Canada The equation for uncertainty of a result y, which is number of records uc (y) nd often referred to as the law of propagation of uncertainty, is as ∆t time increment for sampling data. follows: The elemental uncertainties u were 2.5 µm for I 1/ 2 2 2 (4) displacement and 0.01g for acceleration. uc = ()θ i u (xi ∑ i=1 where 5. CALCULATIONS The calculations have been based on a θ sensitivity coefficients, θ = ∂y ∂x i i i thermoelastohydrodynamic model, which was described in [3]. standard uncertainty associated with the input u()xi xi The calculated dynamic coefficients include the effect of pivot 3E+8 stiffness. m / N 2E+8 , ) y 6. RESULTS AND DISCUSSION y H ( 1E+8 e As an example, the frequency response functions H R i, j 0E+0 0 100 200 300 representing the bearing direct coefficients in the load direction a. Frequency, Hz are illustrated in Figure 1. For a very lightly loaded bearing 3E+8 (2.02 kN, 14876 rpm), the direct bearing stiffness and damping m / N coefficients show very limited variations with frequency of 2E+8 , ) y y excitation (Figure 1a,b). With increased load these variations H 1E+8 ( m I become more evident, which is illustrated in Figure 1c. 0E+0 Although, for this bearing, the damping coefficients b. 0 100 200 300 Frequency, Hz (represented by the slope of the line) have not varied with 3E+8 frequency of excitation, pad inertia effects became evident in m / the bearing stiffness at high frequencies. In the test conditions N 2E+8 , ) y y considered in this study, these effects have been significant at H ( 1E+8 e frequencies above that corresponding to the shaft speed. R 0E+0 The examples of the results’ uncertainties show yet another 0 50 100 150 200 250 300 c. Frequency, Hz advantage of the PSD method. When compared to the time 3E+8 domain method [7], the frequency domain leads to lower m / uncertainty of the results. The uncertainties shown in N 2E+8 , ) y Figures 1 c,d do not exceed 10%. y H 1E+8 ( m I 0E+0 ACKNOWLEDGEMENTS d. 0 100 200 300 Frequency, Hz The authors wish to thank Waukesha Bearing Corporation for permission to publish the experimental results. experiment calculation Figure 1. Frequency response functions representing direct REFERENCES coefficients in the load direction. 1. Parsell, J.K., Allaire, P.E., and Barret, L.E., 1983, a. real part: load 2.02 kN, shaft speed 14876 rpm. “Frequency Effects in Tilting-Pad Journal Bearing Dynamic b. imaginary part: load 2.02 kN, shaft speed 14876 rpm. Coefficients,” ASLE Trans., Vol. 26, pp. 222-227. c. real part: load 4.03 kN, shaft speed 8872 rpm 2. Ha, H. C. and Yang, S.H., 1999, “Excitation Frequency d. imaginary part: load 4.03 kN, shaft speed 8872 rpm Effects on the Stiffness and Damping Coefficients of a Five- Pad Journal Bearing,” ASME Journal of Tribology, Vol. 121, Sensitivity coefficients have been found by introducing θ i pp. 517-522. a small increment of each cross spectral density , 3. Dmochowski, W., Brockwell, K., DeCamillo, S., and ∆Gi, j Gi, j solving the equation (1), and calculating the change to the Mikula, A., 1993, “A Study of the Thermal Characteristics of the Leading Edge Groove and Conventional Tilting Pad Journal frequency response function H . An approach similar to that ij Bearings,” ASME Journal of Tribology, 115 (1993), pp. 219- presented in [6] has been adapted to the PSD method for 226. evaluation of the bearing dynamic coefficients. The following 4. Rouvas, C., Murphy, B.T., Hale, R.K., 1992, “Bearing assumptions: 1. the measurement of the dynamic excitation is parameter identification using power spectral density methods,” error free, which is consistent with the assumption made for the Proceedings of the Fifth International Conference on Vibration PSD method, 2. the noise present in the original time-dependent in Rotating Machinery, ImechE, 1992, pp. 297-303. process is uncorrelated between samples, 3. the variance of the 5. Guide to the Expression of Uncertainty in Measurement, noise, denoted as u 2 (y), is the same for each sample, lead to the 1993, International Organization of Standardization. relationship for the uncertainty of the evaluated cross spectral 6. Betta, G., Liguori, C., Pietrosanto, A., 2000, “Propagation densities: of uncertainty in a discrete Fourier transform algorithm,” 2 Measurement, 27 (2000), pp. 231-239. 2 2 ∆t ⋅Gxx ⋅u (y) (6) u ()Re()Gx, y = u ()Im(Gx, y )= 7. Dmochowski, W and Brockwell K., 1995, “Dynamic n d Testing of the Tilting Pad Journal Bearing,” STLE Tribology where Transactions, 38 (1995), pp. 261-268. 2 Copyright © #### by ASME 2 Copyright © 2005 by National Research Council of Canada.
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