Dynamic Coefficients Of Bearings Research Articles

Numerical investigation of influence sensitivity of a gas foil bearing parameters on the dynamic coefficients

Most studies on dynamic coefficients of bearings are focused on evaluation using different analytical methods. Minimal emphasis is given to the level of influence of each geometrical variable, the corresponding range of these variables for optimum stiffness and damping and the measure of performance of the analytical method used. The objective of this paper was to study the influence and sensitivity of length-to-diameter ratio, eccentricity ratio, bearing number, whirl ratio, and bearing compliance on the stiffness and damping of gas foil bearing. A numerical model is developed by utilizing the finite difference method to evaluate the dynamic coefficients. The results reveal that the normalized stiffness increases with the bearing number and decreases with increased bearing compliance whereas the normalized damping shows an opposite nature. Further, the stiffness coefficients tend to increase and the damping coefficients tend to decrease corresponding to increase in speed up to 240 krpm. The characteristic data sets obtained from the analysis is used to train an artificial neural network (ANN). Performance of ANN network is evaluated though computation of root-mean-square error (RSME) and regression coefficient (R2) and Mean Absolute Error (MAE). Utilizing the neural network results, a Sobol’s sensitivity test is carried out to identify most effective parameters which have a significant influence on the dynamic coefficients of gas foil bearing. After that, an adaptive neurofuzzy interface system (ANFIS) is established to determine the optimum range of data for which maximum stiffness and damping can be obtained. The results deduce that the neural network shows high efficacy in predicting the output variables correctly with a regression of more than 95%. It is also observed that the variation of dynamic coefficients is the highest for eccentricity ratio whereas lowest for whirl ratio. The maximum stiffness and damping coefficients are also obtained for a wide range of geometrical variables which can help in designing the gas foil bearing.

Computational evaluation of dynamic coefficients of thrust bearings; effect of artificial texturing on thermohydrodynamic performance

Fluid film thrust bearings are commonly used in industry, providing durable and reliable operation at high values of load carrying capacity, accompanied by low friction losses. A major advantage of hydrodynamic fluid film bearings, over other types of bearings, is their enhanced dynamic behaviour, especially under transient or impact loads. Currently, a systematic approach to identify the dynamic coefficients of thrust bearing geometrical configurations utilising high complexity CFD simulation data has not yet been established. It is therefore imperative to develop a method, capable of evaluating the dynamic characteristics of complex bearing designs and allow the evaluation of bearing response under transient loads. In the present work, a computational approach is proposed to estimate the stiffness and damping coefficients of fluid-film thrust bearings. A CFD-based ThermoHydroDynamic (THD) numerical model of the bearing is developed and utilised for performing an initial steady-state simulation at given rotational speed and thrust load, as well as subsequent transient simulations at increasing or decreasing thrust loads. The former simulation is used to calculate the stiffness coefficient of the bearing at the specified conditions, while the latter are appropriately post-processed to estimate the damping coefficient of the bearing at different values of rotor acceleration. The procedure is repeated at different operating conditions, yielding a map of the dynamic coefficients of the bearing. Finally, a single degree of freedom model is generated, which utilises the calculated values of dynamic coefficients to evaluate transient bearing performance under any given thrust load history. The proposed methodology is applied to compare the dynamic response characteristics of a conventional sector-pad tapered-land thrust bearing and a textured tapered-land thrust bearing of the same principal dimensions.

Nonlinear Response of Tilting Pad Journal Bearings to Harmonic Excitation

In the experimental identification of dynamic bearing coefficients, usually small perturbations around the static equilibrium position are assumed and linear coefficients are considered. In the literature, studies on non-linear effects in plain journal bearings, especially from a numerical point of view, are reported. Few similar studies can be found on tilting pad journal bearings (TPJB). The present work reports some peculiar aspects observed during the experimental identification procedure of TPJB linear dynamic coefficients. The tests are performed on a test bench designed for large size journal bearings operating at high peripheral speeds and static loads. A quasi-static procedure is developed to quickly check the results obtained from the usually adopted dynamic excitation. It consists of applying a slowly rotating force to the floating stator and measuring the relative displacement of the stator from the rotating shaft. Different levels of static and dynamic load are applied to two different TPJBs with four and five pads. Deformed orbits have been observed increasing the ratio between dynamic load and static load, suggesting the presence of non-linearity. Similar results are obtained with simple analytical models assuming suitably tuned non-linear stiffness terms.

Large amplitude dynamic behavior of thrust air bearings: Modeling and experiments

Large dynamic loading of air thrust bearings may result in undesired contact between bearing surfaces. The counteracting nonlinear aerodynamic forces delivered by the thin air film in the bearing, which determine if contact will occur, may be computed either by solving the nonlinear Reynolds equation or by using a grid with position dependent dynamic bearing coefficients. It is shown by a numerical case study that both numerical approaches give similar response results. Moreover, it is demonstrated that the model based on the nonlinear Reynolds equation accurately predicts the experimental large amplitude dynamic behavior of an axial-symmetric air thrust bearing. Hence, both numerical approaches may be used to predict if bearing surfaces will make contact due to large impulsive external forces.

Dynamics of a motorized spindle supported on water-lubricated bearings

The purpose of this paper is to investigate the dynamic performances of a motorized spindle supported on water-lubricated bearings. A modified transfer matrix method considering both of the translational and tilting dynamic coefficients of the bearings is established. The turbulent Reynolds equation is adopted and numerically solved by the perturbation method and the finite difference method, and the dynamic characteristics of the water-lubricated journal bearings are obtained; the effects of the eccentricity ratio, tilting angle, and the rotational speed on the dynamic coefficients of the water-lubricated journal bearings are analyzed. The critical speed, the dynamic stiffness of spindle nose, and unbalance response of the motorized spindle are investigated. Finally, a comparative study of rotor dynamic behaviors between the 32- and the eight-coefficient bearing models is conducted. The numerical predictions obtained by the 32-coefficient bearing models correlate well with the experimental values available in the literature.

Dynamic Analysis of Integrally Geared Compressors with Varying Workloads

Integrally geared compressors are characterized by compact and high efficiency machines, which are widely used in modern processing industries. As an important part of integrally geared compressors, a geared rotor-bearing system exhibits complicated dynamic behaviors. When running at rated speeds, a coupling system likely produces resonance with an adjusted workload, and a critical load phenomenon occurs. The dynamic coefficients of bearings, axial force and torque, and gear meshing stiffness vary with workload because of the interaction between rotors. In this study, a dynamic model of a geared rotor-bearing system influenced by the dynamic coefficients of bearings, axial force and torque, and gear meshing stiffness is developed. The dynamic responses of the coupling system are calculated and analyzed by using a typical five-shaft integrally geared compressor as an example. The effects of different parameters on the dynamic behaviors of the proposed system are also considered in the discussion. The geared rotor-bearing system is further investigated to examine the failure mechanism of the critical load.

Nonlinear dynamic coefficients prediction of journal bearings using partial derivative method

For decades, linear dynamic coefficients of bearings are widely used to evaluate the stability and dynamic response of oil-film bearing-rotor systems. However, there often exist many strong nonlinear excitation sources in rotor systems. Therefore, the linear oil-film forces cannot reflect the nonlinearity characteristics of rotor systems. Some researchers present numerical methods for prediction of bearing nonlinear dynamic coefficients in order to represent nonlinear oil-film forces of bearing, while their methods are closely dependent on the size and shape of journal orbits. This article gives a computational method which is independent on the journal orbit to predict bearing dynamic coefficients using partial derivative method. The reliability of the numerical method is verified by comparing the linear dynamic coefficients of two-axial-groove bearing with the published results. The variation of dynamic coefficients with eccentricity ratios and length-to-diameter ratios of a simple oil-lubricated 360° journal bearing model is discussed. The numerical results indicate that bearing dynamic coefficients are dramatically increasing with the growth of eccentricity ratios and length-to-diameter ratios, especially when eccentricity ratio is larger than 0.8. The numerical model presented in this article will greatly simplify the solution of oil-film forces in nonlinear dynamic analysis of rotor systems.

Multi-orifice active tilting-pad journal bearings—Harnessing of synergetic coupling effects

The dynamic performance of tilting-pad journal bearings with controllable radial oil injection is theoretically investigated, exploring the synergetic effect between passive elastohydrodynamics and active radial lubricant injection. The flexible tilting-pads are modelled as linearly elastic using finite elements. To reduce computational work, a pseudo-modal reduction is applied. Curves of dynamic bearing coefficients as well as the corresponding phase-magnitude plots are presented. Strong synergy between elastohydrodynamics and active control is observed, in particular for bearings with injection orifices far from the pivot line.

Limits of Dynamic Coefficients of Self-Acting Tilting-Pad Gas Bearings

Many previously published articles have studied the effects of perturbation frequency on the dynamic coefficients of gas bearing, but they all do not give the rounded definition of dynamic coefficients in the whole frequency range. In this article, the dynamic stiffness and damping coefficients of tilting-pad bearings are calculated by the partial derivative method and the limits of dynamic coefficients are obtained by using the limit theory. Furthermore, the equivalent dynamic coefficients of tilting-pad bearings are also calculated, which makes it possible to denote the dynamic characteristics of the tilting-pad bearing with eight dynamic coefficients. In the article, the dynamic coefficients of the bearing are investigated in the whole frequency range. The numerical results indicate that the changing trends of dynamic coefficients with perturbation frequency are well in agreement with their limits. With the increase of perturbation frequency, the dynamic stiffness coefficients are all close to their corresponding limits, and the limits of all dynamic damping coefficients are equal to zero. Moreover, the cross-coupling terms of dynamic coefficients of the tilting-pad gas bearing are close to zero, which further reveals the main reason that the performance of the tilting-pad bearings is better than the generic fixed-pad bearings.

Validating the dynamic coefficients of bearing pedestals in a multi-mode rotor—bearing system

Over the years, methods have been developed to estimate the dynamic coefficients of bearings based on parameter identification techniques. The stage where the estimated coefficients obtained from these methods need to be validated on actual operating machines, especially those operating at hypercritical speeds, has been reached. This article extended the frequency domain parameter estimation technique normally used on bearings to extract the dynamic coefficients of the bearing pedestals in isolation. It is demonstrated here that by using only a single forcing mechanism located at an angle to the bearing pedestal, the eight possible stiffness and damping coefficients can be estimated accurately. An on-line frequency domain algorithm for estimating these coefficients was developed and tested in the laboratory for this purpose. Its ability to reduce testing time and to produce reliable coefficient estimates from structural spurious resonance is observed. Validations of the dynamic coefficients of the bearing pedestals were carried out by comparing the experimental and computer simulated responses and resonance frequencies of the arrangement of two multi-mode rotor—bearing-system's configurations. By using ANSYS Inc. finite element software as a comparison tool, the results indicated that a nine stations rotor model formulated by the stiffness coefficient method is highly accurate at predicting the rotor—bearing system's responses and resonance frequencies. The results also confirmed that rotor bearing's synchronous response is insensitive with respect to individual estimated parameters that have been determined by statistical hypothesis test to be insignificant.

Surface roughness effect on the dynamic stiffness and damping characteristics of compensated hydrostatic thrust bearings

On the basis of Christensen's stochastic model of rough surfaces, this study theoretically analyzes the effect of surface roughness on the dynamic stiffness and damping characteristics of compensated hydrostatic circular step thrust bearings. Both the stochastic Reynolds equation with rotational inertia effect and the recess flow continuity equation with recess volume fluid compressibility effect are derived to take into account the presence of roughness on the bearing surfaces. Using a small perturbation method, the dynamic coefficients of bearings are evaluated. According to the results, the mean stiffness and damping behaviours are significantly affected by the roughness pattern and the height of roughness. The effect of circumferential roughness provides an increase in the mean bearing stiffness and damping characteristics as compared to the smooth-bearing case. However, these trends are reversed for the bearing having radial roughness structure. An illustration of a design example is also included for industrial practice.

Evaluation of Dynamic Coefficients of a Two-Lobe Journal Bearing Using an Electrical Analogy Approach

Abstract A theoretical approach to evaluating capacitance, resistance, capacitive reactance, and impedance of the lower and upper lobes of a two-lobe elliptical hydrodynamic journal bearing under various operating conditions is developed. It is established that the change in capacitance and resistance with the change in eccentricity ratios is nonlinear. The capacitance and resistance, thus determined, are correlated with the dynamic coefficients of bearings using an electrical analogy. It is found that the stiffness and damping coefficients are higher for two-lobe bearings as compared to those of cylindrical bearings having identical dimensional parameters and operating under similar conditions. The analysis may have the potential to diagnose the stability regime of a bearing through the bearing’s electrical parameters. The electrical analogy may be a useful alternative to conventional techniques.

On the Adjusting of the Dynamic Coefficients of Tilting-Pad Journal Bearings

This paper gives a theoretical and experimental contribution to the problem of active modification of the dynamic coefficients of tilting-pad journal bearings, aiming to increase the damping and stability of rotating systems. The theoretical studies for the calculation of the bearing coefficients are based on the fluid dynamics, specifically on the Reynolds equation, on the dynamics of multibody systems and on some concepts of the hydraulics. The experiments are carried out by means of a test rig specially designed for this investigation. The four pads of such a bearing are mounted on four flexible hydraulic chambers which are connected to a proportional valve. The chamber pressures are changed by means of the proportional valve, resulting in a displacement of the pads and a modification of the bearing gap. By changing the gap, one can adjust the dynamic coefficients of the bearing. With help of an experimental procedure for identifying the bearing coefficients, theoretical and experimental results are co...

Calculation of the Dynamic Coefficients of a Journal Bearing, Using a Variational Approach

The dynamic bearing coefficients are obtained from a solution to the variational equivalent of Reynolds equation. A perturbation method is applied to find the individual dynamic coefficients. The Finite Element Method is used in the numerical evaluation of the equations. The flow is assumed to be laminar, the lubricant is Newtonian. Allowance is made for viscosity-temperature dependency in circumferential and axial directions.

The cross-coupling dynamic coefficients of tilting-pad bearings

There is some misconception in the literature concerning the crosscoupling stiffness and damping coefficients of tilting-pad bearings; these have been reported as equal to zero. Analytical expressions of the dynamic coefficients of the bearing in terms of the dynamic coefficients of the individual pads have been derived to show that the cross-coupling coefficients of the bearing cannot in general be zero.