Amir Younan

A Review of Tilting Pad Bearing Theory

A theoretical basis for static and dynamic operation of tilting pad journal bearings (TPJBs) has evolved over the last 50 years. Originally demonstrated by Lund using the pad assembly method and a classic Reynolds equation solution, the current state of the art includes full thermoelastohydrodynamic solutions of the generalized Reynolds equation that include fluid convective inertia effects, pad motions; and thermal and mechanical deformations of the pads and shaft. The development of TPJB theory is reviewed, emphasizing dynamic modeling. The paper begins with the early analyses of fixed geometry bearings and continues to modern analyses that include pad motion and stiffness and damping effects. The development of thermohydrodynamic, thermoelastohydrodynamic, and bulk-flow analyses is reviewed. The theories of TPJB dynamics, including synchronous and nonsynchronous models, are reviewed. A discussion of temporal inertia effects in tilting pad bearing is considered. Future trends are discussed, and a path for experimental verification is proposed.

Temporal and Convective Inertia Effects in Plain Journal Bearings With Eccentricity, Velocity and Acceleration

This paper extends the theory originally developed by Tichy (Tichy and Bou-Said, 1991, Hydrodynamic Lubrication and Bearing Behavior With Impulsive Loads,” STLE Tribol. Trans. 34 , pp. 505–512) for impulsive loads to high reduced Reynolds number lubrication. The incompressible continuity equation and Navier-Stokes equations, including inertia terms, are simplified using an averaged velocity approach to obtain an extended form of short bearing Reynolds equation which applies to both laminar and turbulent flows. A full kinematic analysis of the short journal bearing is developed. Pressure profiles and linearized stiffness, damping and mass coefficients are calculated for different operating conditions. A time transient solution is developed. The change in the rotor displacements when subjected to unbalance forces is explored. Several comparisons between conventional Reynolds equation solutions and the extended Reynolds number form with temporal inertia effects are presented and discussed. In the specific cases considered in this paper, the primary conclusion is that the turbulence effects are significantly more important than inertia effects.

Model Reduction Methods for Rotor Dynamic Analysis: A Survey and Review

The focus of this literature survey and review is model reduction methods and their application to rotor dynamic systems. Rotor dynamic systems require careful consideration in their dynamic models as they include unsymmetric stiffness, localized nonproportional damping, and frequency-dependent gyroscopic effects. The literature reviewed originates from both controls and mechanical systems analysis and has been previously applied to rotor systems. This survey discusses the previous literature reviews on model reduction, reduction methods applied to rotor systems, the current state of these reduction methods in rotor dynamics, and the ability of the literature to reduce the complexities of large order rotor dynamic systems but allow accurate solutions.

Nonlinear Analysis of Squeeze Film Damper with Entrained Air in Rotordynamic Systems

Squeeze film dampers are widely used in aircraft engines, land-based gas turbines, and other rotating machines to improve system damping. They often have entrained air within the oil film, which is usually not taken into account in rotordynamics analysis due to lack of good formulations of the effects involved. The effects on the damping force calculation can be significant. This work presents a new formulation of the nonlinear Reynolds equation pressure evaluation within the squeeze film damper, including the effects of entrained air, with resulting changes in effective lubricant density and viscosity. Viscosity and density expressions are developed as a function of the air/oil volume fraction. The density of the bubbly oil is a function of the air bubble diameter, which changes due to surface tension effects during lubricant motion in the bubbly oil film. The lubricant viscosity decreases due to the entrained volume of air but increases due to the surface tension effects taken from experimental tests. Pressure supply and bubbly oil film cavitation effects are included in the analysis and end seal effects are evaluated. The nonlinear time-transient forces in the squeeze film damper are evaluated as functions of (1) lubricant and air properties; (2) damper geometry including diameter, length, clearance; (3) end seal properties; and (4) shaft position and velocity. Example cases of pressure calculations and radial and tangential forces are shown. Example nonlinear transient motions are presented for a rigid, symmetrical rotor and for a nonsymmetrical rotor representing a gas turbine–type fan rotor.

Comparison of Tilting-Pad Journal Bearing Dynamic Full Coefficient and Reduced Order Models Using Modal Analysis (GT2009-60269)

There is a significant disagreement in the literature concerning the proper evaluation of the experimental identification and frequency response of tilting-pad journal bearings (TPJBs) due to shaft excitations. Two linear models for the frequency dependence of TPJBs have been proposed. The first model, the full coefficient or stiffness-damping (KC) model, considers N p tilting pads and two rotor radial motions for N p +2 degrees of freedom. The dynamic reduction of the KC model results in eight frequency-dependent stiffness and damping coefficients. The second model, based on bearing system identification experimental results, employs 12 frequency-independent stiffness, damping, and mass (KCM) coefficients; pad degrees of freedom are not considered explicitly. Experimental data have been presented to support both models. There are major differences in the two approaches. The present analysis takes a new approach of considering pad dynamics explicitly in a state-space modal analysis. TPJB shaft and bearing pad stiffness and damping coefficients are calculated using a well known laminar, isothermal analysis and a pad assembly method. The TPJB rotor and pad KC model eigenvalues and eigen-vectors are then evaluated using state-space methods, with rotor and bearing pad inertias included explicitly in the model. The KC model results are also nonsynchronously reduced to the eight stiffness and damping coefficients and are expressed as shaft complex impedances. The system identification method is then applied to these complex impedances, and the state-space modal analysis is applied to the resulting KCM model. The damping ratios, natural frequencies, and mode shapes from the two bearing representations are compared. Two sample TPJB cases are examined in detail. The analysis indicated that four underdamped modes, two forward and two backward, dominate the rotor response over excitation frequencies from 0 to approximately running speed. The KC model predicts additional nearly critically damped modes primarily involving pad degrees of freedom, which do not exist in the identified KCM model. The KCM model results in natural frequencies that are 63-65% higher than the KC model. The difference in modal damping ratio estimates depends on the TPJB considered; the KCM estimate was 7-17% higher than the KC model. The results indicate that the KCM system identification method results in a reduced order model of TPBJ dynamic behavior, which may not capture physically justifiable results. Additionally, the differences in the calculated system natural frequency and modal damping have potential implications for rotordynamic analyses of flexible rotors.

Feasibility of Gas-Expanded Lubricants for Increased Energy Efficiency in Tilting-Pad Journal Bearings

Lubricants are necessary in tilting-pad journal bearings to ensure separation betweensolid surfaces and to dissipate heat. They are also responsible for much of the undesir-able power losses that can occur through a bearing. Here, a novel method to reducepower losses in tilting-pad journal bearings is proposed in which the conventional lubri-cant is substituted by a binary mixture of synthetic lubricant and dissolved CO

Properties and Performance of Gas-Expanded Lubricants in Tilting Pad Journal Bearings

Lubricants enable proper function and reduce friction in rotating machinery, but they can also contribute to power loss and heat buildup. Gas-expanded lubricants (GELs) have been proposed as tunable mixtures of lubricant and CO2 under pressure with properties such as viscosity that can be controlled directly in response to changing environmental or rotordynamic conditions. In this work, experimental results of GEL viscosity, gas diffusivity, and thermal conductivity were combined with high-pressure phase equilibrium data to understand how these mixtures will behave in tilting pad journal bearings under a range of industry-relevant high-speed conditions. Simulations were carried out using the experimental data as inputs to a thermoelastohydrodynamic model of tilting pad journal bearing performance. Viscosity could be easily tuned by controlling the composition of the GEL and the effect on bearing efficiency was appreciable, with 14–46% improvements in power loss. This trend held for a range of lubricant chemistries with polyalkylene glycols, polyalpha olefins, and a polyol ester tested in this work. Diffusivity, which drives how readily CO2 and lubricants form homogenous mixtures, was found to be a function of the viscosity of the synthetic lubricant, with more viscous lubricants having a lower diffusivity than less viscous formulations. Model results for a bearing in a pressurized housing suggested that cavitation would be minimal for a range of speed conditions. Other bearing parameters, such as eccentricity, temperature, and minimum film thickness were relatively unchanged between conventionally lubricated and GEL-lubricated bearings, suggesting that the efficiency improvements could be achieved with few performance tradeoffs.

Comparison of Tilting-Pad Journal Bearing Dynamic Full Coefficient and Reduced Order Models Using Modal Analysis

There is significant disagreement concerning the frequency response of tilting pad journal bearings (TPJBs) due to non-synchronous excitations. Two linear models for the frequency dependence of TPJBs have been proposed. The first model, the full-coefficient or KC model, considers Np tilting pads and rotor motions for Np + 2 degrees of freedom. Dynamic reduction of the KC model results in eight frequency-dependent stiffness and damping coefficients. The second model, based on results from bearing system identification experiments, yields twelve frequency-independent stiffness, damping, and mass (KCM) coefficients. Experimental data has been presented to support both models. There are major differences in the two approaches. The analysis in this paper takes a new approach of considering the pad dynamics explicitly in a state-space modal analysis. TPJB shaft and bearing pad stiffness and damping coefficients are calculated using a well known laminar, isothermal analysis and a pad assembly method. The TPJB rotor and pad full system eigenvalues and eigenvectors are then evaluated using state-space methods, with rotor and bearing pad inertias included explicitly in the model. The full bearing coefficient results are also non-synchronously reduced to the 8 stiffness and damping coefficients are and expressed as shaft complex impedances. The system identification method is then applied to these complex impedances, and the state space modal analysis is applied to the resulting KCM model. The damping ratios, natural frequencies, and mode shapes from the two bearing representations are compared. Two example TPJBs are examined in detail. The analysis indicated that four underdamped modes, two forward and two backward, dominate the rotor response over excitation frequencies from 0 to running speed. The full coefficient, non-synchronously reduced model predicts additional critically damped or overdamped modes due to the additional degrees of freedom as compared to the identified KCM model. The KCM model results in natural frequencies that are 63–65 percent higher than the full coefficient model. The difference in modal damping ratio estimates depend on the TPJB considered, with KCM being 7–17 percent higher than the full coefficient model. The full coefficient model also indicates that the bearing pads contribute significantly to the underdamped modes. The results indicate that the system identification method results in a reduced order model of TPBJ dynamic behavior. Additionally, the differences in the modal calculated system natural frequency and modal damping have potential implications for rotordynamic analyses of flexible rotors, such as critical speed and stability analyses.Copyright

A System Identification Method for Excitation-Frequency Dependent Rotordynamic Coefficients

Experimental identification of rotordynamic systems presents unique challenges. Gyroscopics, generally damped systems, and non-self-adjoint systems due to fluid structure interaction forces mean that symmetry cannot be used to reduce the number of parameters to be identified. Rotordynamic system experimental measurements are often noisy, which complicates comparisons with theory. When linearized, the resulting dynamic coefficients are also often a function of excitation frequency, as distinct from operating speed.In this paper, a frequency domain system identification technique is presented that addresses these issues for rigid-rotor test rigs. The method employs power spectral density functions and forward and backward whirl orbits to obtain the excitation frequency dependent effective stiffness and damping. The method is highly suited for use with experiments that employ active magnetic exciters that can perturb the rotor in the forward and backward whirl directions. Simulation examples are provided for excitation-frequency reduced tilting pad bearing dynamic coefficients. In the simulations, 20 and 50 percent Gaussian output noise was considered. Based on ensemble averages of the coefficient estimates, the 95 percent confidence intervals due to noise effects were within 1.2% of the identified value.The method is suitable for identification of linear dynamic coefficients for rotordynamic system components referenced to shaft motion. The method can be used to reduce the effect of noise on measurement uncertainty. The statistical framework can also be used to make decisions about experimental run times and acceptable levels of measurement uncertainty. The data obtained from such an experimental design can be used to verify component models and give rotordynamicists greater confidence in the design of turbomachinery.Copyright

Generalized Stiffness Gear-Mesh Matrix Including EHD Stiffness

This paper presents a method to compute gear mesh stiffness based on the EHD behavior by combined finite element solution of the Reynolds Equation with the elastic contact model. It is shown that this solution requires iterative procedure to balance the computed pressure profile with the external nominal transmission load. This mesh stiffness is load dependent and therefore is a nonlinear phenomenon. The nominal stiffness value is utilized to model a full (12×12) gear mesh matrix for a linear dynamic model of rotor bearing systems including gears to evaluate system dynamics and coupling between lateral/torsional vibrations.Copyright