Modal Frequency Response of a Four-Pad Tilting Pad Bearing With Spherical Pivots, Finite Pivot Stiffness and Different Pad Preloads

Tilting pad journal bearings (TPJBs) provide radial support for rotors in high-speed machinery. Since the tilting pads cannot support a moment about the pivot, self-excited cross-coupled forces due to fluid-structure interactions are greatly reduced or eliminated. However, the rotation of the tilting pads about the pivots introduces additional degrees of freedom into the system. When the flexibility of the pivot results in pivot stiffness that is comparable to the equivalent stiffness of the oil film, then pad translations as well as pad rotations have to be considered in the overall bearing frequency response. There is significant disagreement in the literature over the nature of the frequency response of TPJBs due to non-synchronous rotor perturbations. In this paper, a bearing model that explicitly considers pad translations and pad rotations is presented. This model is transformed to modal coordinates using state-space analysis to determine the natural frequencies and damping ratios for a four-pad tilting pad bearing. Experimental static and dynamic results were previously reported in the literature for the subject bearing. The bearing characteristics as tested are considered using a thermoelastohydrodynamic (TEHD) model. The subject bearing was reported as having an elliptical bearing bore and varying pad clearances for loaded and unloaded pads during the test. The TEHD analysis assumes a circular bearing bore, so the average bearing clearance was considered. Because of the ellipticity of the bearing bore, each pad has its own effective preload, which was considered in the analysis. The unloaded top pads have a leading edge taper. The loaded bottom pads have finned backs and secondary cooling oil flow. The bearing pad cooling features are considered by modeling equivalent convective coefficients for each pad back. The calculated bearing full stiffness and damping coefficients are also reduced non-synchronously to the eight stiffness and damping coefficients typically used in rotordynamic analyses and are expressed as bearing complex impedances referenced to shaft motion. Results of the modal analysis are compared to a two degree-of-freedom second-order model obtained via a frequency-domain system identification procedure. Theoretical calculations are compared to previously published experimental results for a four-pad tilting pad bearing. Comparisons to the previously published static and dynamic bearing characteristics are considered for model validation. Differences in natural frequencies and damping ratios resulting from the various models are compared, and the implications for rotordynamic analyses are considered.

Evaluation of High-Order Resonance of Blade Under Wake Excitation

Avoiding the low-order resonances of blades is one of the main design goals for a mechanical structure designer of turbo machinery. However, we have to accept that there are resonance frequencies in the operating speed range of the blade, for the following reasons: Firstly, the natural frequencies of the blade are closely spaced sometimes, it is impossible to avoid them all. Secondly, in general, the higher of the resonance frequency, the lower the energy of resonance will be. But in recent 10 years, the high-order blade resonances present more and more frequently in turbo machinery, which induce a lot of HCF problems. As the considerations above, studies on the high-order vibration of blades become necessary and important. In the cascade, the high-order vibration of blades is mainly induced by the wakes from upstream. An obvious difference of the wake excitation from the common excitations resides in its asynchronism, that is, the maximum value of aerodynamic force from wakes at each point doesn’t appear at the same time, because except the frequency, the distribution of the aerodynamic force field depends on two parameters: not only amplitude but also phase angle. Both are functions of coordinates. In this paper, the related position in Euclidean Space between the asynchronous excitation field and the modal displacement of blade were deal with to evaluate the strength of the high-order resonance of blade. The effect of the asynchronous aerodynamic force field on the blade resonance was studied either. Finally a method for evaluation of high-order resonance of blade excited by wake fluid is proposed. A numerical case was studied either, which demonstrates that the proposed evaluation on high-order resonance is practical in engineering problem.

Impact of Frequency Dependence of Gas Labyrinth Seal Rotordynamic Coefficients on Centrifugal Compressor Stability

A numerical model developed by Thorat & Childs [1] has indicated that the conventional frequency independent model for labyrinth seals is invalid for rotor surface velocities reaching a significant fraction of Mach 1. A theoretical one-control-volume (1CV) model based on a leakage equation that yields a reasonably good comparison with experimental results is considered in the present analysis. The numerical model yields frequency-dependent rotordynamic coefficients for the seal. Three real centrifugal compressors are analyzed to compare stability predictions with and without frequency-dependent labyrinth seal model. Three different compressor services are selected to have a comprehensive scenario in terms of pressure and molecular weight (MW). The molecular weight is very important for Mach number calculation and consequently for the frequency dependent nature of the coefficients. A hydrogen recycle application with MW around 8, a natural gas application with MW around 18, and finally a propane application with molecular weight around 44 are selected for this comparison. Useful indications on the applicability range of frequency dependent coefficients are given.

Theoretical Prediction of the Lift-Off Speed in Aerodynamic Compliant Foil Journal Bearings

In aerodynamic bearings, since the supporting air film is generated by rotor motion, there is no support at the start of motion. As in all such bearings, there is dry rubbing until the rotor achieves sufficient speed to lift-off. Thus, the lower the lift-off speed, the less will be the rubbing and so the greater will be the life of the bearing. This paper focuses on the theoretical prediction of lift-off speed in aerodynamic compliant foil journal bearings based on a generalized solution of elasto-aerodynamically coupled lubrication for compliant foil bearings. A computational method is presented which is used to predict the lift-off speed in aerodynamic foil journal bearings with eccentricity ratio greater than or equal to 1.0. Special emphasis is placed on investigating the effects of the load imposed on the bearing, the nominal radial clearance and the bearing radius on the lift-off speed. The numerical results obtained indicate that lift-off speed decreases with the decrease of load and nominal radial clearance, but with an increase in bearing radius. The eccentricity ratios are all greater than 1.0 at the lift-off speed for the aerodynamic compliant foil journal bearings used in this study.

Experimental and Analytical Investigations of a Low Pressure Model Turbine During Forced Response Excitation

To guarantee a faultless operation of a turbine it is necessary to know the dynamic performance of the machine especially during start-up and shut-down. In this paper the vibration behaviour of a low pressure model steam turbine which has been intentionally mistuned is investigated at the resonance point of an eigenfrequency crossing an engine order. Strain gauge measurements as well as tip timing analysis have been used, whereby a very good agreement is found between the methods. To enhance the interpretation of the data measured, an analytical mass-spring-model, which incorporates degrees of freedom for the blades as well as for the rotor shaft, is presented. The vibration amplitude varies strongly from blade to blade. This is caused by the mistuning parameters and the coupling through the rotor shaft. This circumferential blade amplitude distribution is investigated at different operating conditions. The results show an increasing aerodynamic coupling with increasing fluid density, which becomes visible in a changing circumferential blade amplitude distribution. Furthermore the blade amplitudes rise non-linearly with increasing flow velocity, while the amplitude distribution is almost independent. Additionally, the mechanical and aerodynamic damping parameters are calculated by means of a non-linear regression method. Based on measurements at different density conditions, it is possible to extrapolate the damping parameters down to vacuum conditions, where aerodynamic damping is absent. Hence the material damping parameter can be determined.

On the Frequency Response of a Spinning Disk With Large Deformations

In this paper, the effect of geometrical nonlinear terms, caused by a space fixed point force, on the frequencies of oscillations of a rotating disk with clamped-free boundary conditions is investigated. The nonlinear geometrical equations of motion are based on Von Karman plate theory. Using the eigenfunctions of a stationary disk as approximating functions in Galerkin’s method, the equations of motion are transformed into a set of coupled nonlinear Ordinary Differential Equations (ODEs). These equations are then used to find the equilibrium positions of the disk at different discrete blade speeds. At any given speed, the governing equations are linearized about the equilibrium solution of the disk under the application of a space fixed external force. These linearized equations are then used to find the oscillation frequencies of the disk considering the effect of large deformation. Using multi mode approximation and different levels of nonlinearity, the frequency response of the disk considering the effect of geometrical nonlinear terms are studied. It is found that at the linear critical speed, the nonlinear frequency of the corresponding mode is not zero. Results are presented that illustrate the effect of the magnitude of disk displacement upon the frequency response characteristics. It is also found that for each mode, including the effect of the geometrical nonlinear terms due to the applied load causes a separation in the frequency responses of its backward and forward traveling waves when the disk is stationary. This effect is similar to the effect of a space fixed constraint in the linear problem. In order to verify the numerical results, experiments are conducted and the results are presented.

Dynamics Design of the Aero-Engine Rotor Joint Structures Based on Experimental and Numerical Study

This study investigated the rigidity and contact state of joint structures that influenced the rotor dynamic characteristics and imbalance response, and the curve for variable structure parameters and the external load. The consideration of rotor joint structures dynamics design was also discussed. The finite-element models were established by using 3D solid elements and surface-to-surface nonlinear contact elements between the interfaces for numerical analysis. The rotor dynamic characteristics were affected by the rigidity of joint structures, and the rotor imbalance response was affected by the contact state of the interfaces. The experimentation for measuring the static rigidity and dynamic contact state of bolted joints with different experimental cycles were performed. Both numerical simulation and experimental results showed that: Firstly, the stiffness of joint structures was not constant. There was a critical load Fcr, when the external load was less than Fcr, the stiffness of joint structures was K1; when the external load was more than Fcr, the bend stiffness of joint structures would drop to K2. The critical load Fcr was influenced by the length of interfaces and preload. Secondly, the contact state of joint structure interfaces varied after a long time of operating with alternating loads. The rotor imbalance was increased by fatigue damage accumulation and irreversible deformation. The study results show that the rigidity and contact state of joint structures vary with external loads and geometry structures, and would affect the rotor system operating. It is advisable to consider the influence of the position, structural parameter and external load of the rotor joint structures on aero-engine structure dynamics design.

Minimizing Dynamic Response of Counter-Rotating Engines Through Optimized Node Placement

It has been previously proposed that a low-speed rotor balancing procedure can be suitable for supercritical shafting (GT2008-50077). That paper documented the necessity of taking into account nodal locations in the bending mode shapes of a supercritical rotor when designing an optimum balance process for such a rotor. This is due to the fact that balance correction forces (or for that matter, any forces) have the least impact when applied near the nodes of a particular mode. This result led to consideration that node location optimization could help with another issue, i.e. the excitation of backward excited whirl modes in a counter-rotating system. When designing a two rotor gas turbine, there are distinct advantages to having the two rotors turn in opposite directions. Among these are the ability to shorten and lighten the engine by reducing the length of the engine since a row of static turning vanes can be eliminated. The engine can be further lightened by inclusion of an inter-shaft bearing which eliminates static bearing support structure. Additional reduction in gyroscopic maneuver loads and deflections can also be achieved, thus resulting in multiple benefits to a counter-rotating system with an inter-shaft bearing. Unfortunately, the excitation of backward whirl modes of one rotor, which would normally not be a major concern in a co-rotating engine, can be a significant issue when excited in such a counter-rotating engine through the inter-shaft bearing, which serves as a conduit for forces from the other rotor. However, the logic of the earlier statement regarding the effectiveness of forces applied at, or near, a nodal point led to the hypothesis that optimizing the nodal locations relative to the interface points between the rotors could minimize the responsiveness of the system. This led to the hypothesis that by optimizing the node placement relative to the inter-shaft bearing, it should be possible to minimize the excitation of the backward modes. This paper examines that proposition and demonstrates that considering this aspect during the design of such an engine could lead to significant benefit in terms of minimized dynamic responses.

A Methodology for Predicting the Response of Blades With Non-Linear Coatings

Ceramic coatings applied by air plasma spray or electron beam techniques as thermal barrier coatings or to improve the erosion or corrosion resistance of blades in gas turbine engines are found to add damping to the system. However, such coatings display non-linear mechanical properties in that the Young’s modulus and the measure of damping are dependent on the amplitude of cyclic strain. To account for the coating nonlinearity, a new methodology for predicting blade response was developed and applied to an actual component coated with a titania-alumina blend ceramic infiltrated with a viscoelastic material. Resonant frequencies, mode shapes, and the forced response of a one blade segment of an integrated disk from a fan stage rotor were computed and compared with results from bench tests. Predicted frequencies agreed satisfactorily with measured values; predicted and observed values of system damping agreed to within 10%. The results of these comparisons are taken to indicate that it is possible to use laboratory-determined material properties together with an iterative finite element analysis to obtain satisfactory predictions of the response of an actual blade with a nonlinear coating.

Maximum Amplification of Blade Response in Mistuned Multi Stage Assemblies

This paper focuses on the determination of the maximum amplification of blade response due to mistuning in multi stage assemblies. The modal optimization strategy developed earlier in connection with single stage models is extended here to multi stage configurations. Theoretical developments are carried out first and lead to the new upper limit of (1+N1+N2(g2/g1)+…)/2, where Ni are the number of blades on the stages and gi = FiTMi−1Fi with Fi the force vector applied on a sector of stage i and Mi its mass matrix. For identical stages, this maximum equals the Whitehead limit observed with single stages but with the number of blades equal to sum of the numbers of blades of the coupled stages. A computational validation of the theoretical results is achieved next on both a single degree of freedom per blade model and a reduced order model of a blisk. These numerical optimization efforts confirm the theoretical developments and demonstrate that such high amplification factors can indeed be achieved with small levels of mistuning. The effects of the number of blades on the different stages, damping in the system, stage coupling strength, etc are discussed in details.