Rotordynamic Coefficients Measurements Versus Predictions for a High-Speed Flexure-Pivot Tilting-Pad Bearing (Load-Between-Pad Configuration)

Abstract

Experimental dynamic force coefficients are presented for a four pad flexure-pivot tilting-pad bearing in load-between-pad configuration for a range of rotor speeds and bearing unit loadings. Measured dynamic coefficients have been compared to theoretical predictions using an isothermal analysis for a bulk-flow Navier-Stokes (NS) model. Predictions from two models—the Reynolds equation and a bulk-flow NS equation models are compared to experimental, complex dynamic stiffness coefficients (direct and cross-coupled) and show the following results: (i) The real part of the direct dynamic-stiffness coefficients is strongly frequency dependent because of pad inertia, support flexibility, and the effect of fluid inertia. This frequency dependency can be accurately modeled for by adding a direct added-mass term to the conventional stiffness/damping matrix model. (ii) Both models underpredict the identified added-mass coefficient (∼32kg), but the bulk-flow NS equation predictions are modestly closer. (iii) The imaginary part of the direct dynamic-stiffness coefficient (leading to direct damping) is a largely linear function of excitation frequency, leading to a constant (frequency-independent) direct damping model. (iv) The real part of the cross-coupled dynamic-stiffness coefficients shows larger destabilizing forces than predicted by either model. The frequency dependency that is accounted for by the added mass coefficient is predicted by the models and arises (in the models) primarily because of the reduction in degrees of freedom from the initial 12 degrees (four pads times three degrees of freedom) to the two-rotor degrees of freedom. For the bearing and condition tested, pad and fluid inertia are secondary considerations out to running speed. The direct stiffness and damping coefficients increase with load, while increasing and decreasing with rotor speed, respectively. As expected, a small whirl frequency ratio (WFR) was found of about 0.15, and it decreases with increasing load and increases with increasing speed. The two model predictions for WFR are comparable and both underpredict the measured WFR values. Rotors supported by either conventional tilting-pad bearings or flexure-pivot tilting-pad (FPTP) bearings are customarily modeled by frequency-dependent stiffness and damping matrices, necessitating an iterative calculation for rotordynamic stability. For the bearing tested and the load conditions examined, the present results show that adding a constant mass matrix to the FPTP bearing model produces an accurate frequency-independent model that eliminates the need for iterative rotordynamic stability calculations. Different results may be obtained for conventional tilting-pad bearings (or this bearing at higher load conditions).

A Review of Tilting Pad Bearing Theory

“…The quadratic behavior observed in the real part of the impedance occurs with both the Reynolds equation and the bulk flow models. The data generally have better agreement with the bulk-flow model than with the Reynolds equation model, for example, [90], so the improved agreement is attributed to both temporal and convective fluid inertia effects.…”

“…The bearing housing is perturbed and displacements of the bearing relative to the shaft are measured. The method, originally applied to fixed pad hydrostatic bearings [88,89], has been applied recently to flexible pivot bearings [90,91] and fourpad and five-pad tilting pad journal bearings [92][93][94][95]. A brief description of the experimental identification method follows.…”

“…The system identification method employed in [92][93][94][95] assumes that the bearing dynamic properties can be modeled as a two degree-of-freedom system based on tilting pad bearing housing (x, y) translations. The identification procedure described in [88][89][90][91][92][93][94][95] is accomplished in the frequency domain by applying a sinusoidal excitation of the form x = Xe jΩt , y = Ye jΩt , f x = F x e jΩt , f y = F y e jΩt , where the excitation frequency Ω is in general nonsynchronous. Excitation is accomplished by simultaneously applying several sinusoidal forces to the bearing housing with mass M, resulting in a pseudorandom perturbation [88,89].…”

“…When the frequency response data for flexible pivot bearings reported in [90,91] and tilting pad bearings [92][93][94] is interpreted using (48) by the respective authors, several common themes emerge. The reported data are compared to two models: a tilting pad bearing model based solely on the classic Reynolds equation, (1), and a thermohydrodynamic bulk-flow analysis developed by San Andrés and presented in [74][75][76].…”

“…The inclusion of temporal inertia effects in the bulk flow model is justified in [90][91][92][93][94] using the work of Reinhardt and Lund [97]. Reinhardt and Lund retained the temporal and convective inertia terms in their nondimensional formulation of the lubrication problem and obtained a solution that indicated that temporal inertia effects were important in lubricating flows with Re ≥ 100.…”