Pressure Boundary Layer in a Transient or Steady Gas Film and Mass Interaction

1970 ◽  
Vol 37 (4) ◽  
pp. 945-953 ◽  
Author(s):  
F. C. Hsing ◽  
H. S. Cheng
Keyword(s):  
Boundary Layer ◽  
Steady State ◽  
High Speed ◽  
Numerical Scheme ◽  
Reynolds Equation ◽  
Equations Of Motion ◽  
Pressure Profile ◽  
Tilting Pad ◽  
Static Solutions ◽  
Gas Bearing

This paper presents a numerical scheme capable of yielding accurate pressure profile for the transient and steady hydrodynamic gas film generated by high-speed relative motion of two nonparallel surfaces. The numerical difficulties associated with high compressibility numbers for the gas film Reynolds equation were overcome by employing a set of systematically generated irregular grid spacings based on a coordinate transformation. By coupling the fluid-film solution with the equations of motion of a tilting pad, the dynamics of the mass film interaction were treated. Results are presented for both steady-state and dynamical solutions. Static solutions for a 120-deg partial-arc gas bearing have been used for comparison.

Transient Dynamics of a Tilting Pad Gas Bearing System

1967 ◽  
Vol 89 (4) ◽  
pp. 499-507 ◽  
Author(s):  
V. Castelli ◽  
J. T. McCabe
Keyword(s):  
Steady State ◽  
Reynolds Equation ◽  
Journal Bearing ◽  
Degree Of Freedom ◽  
Performance Characteristics ◽  
Experimental Results ◽  
Transient Dynamics ◽  
Tilting Pad ◽  
Gas Bearing ◽  
Bearing System

A method for obtaining the performance characteristics of a rotor-tilting pad gas lubricated journal bearing system by solving the appropriate dynamics equations together with the time-transient Reynolds’ equation is outlined. Results for a 4 degree of freedom and an 18 degree of freedom system are given. Comparison with steady-state and experimental results are also discussed.

An Assessment of the Value of Theory in Predicting Gas-Bearing Performance

1961 ◽  
Vol 83 (2) ◽  
pp. 195-200 ◽  
Author(s):  
S. Cooper
Keyword(s):  
Steady State ◽  
Slip Velocity ◽  
Reynolds Equation ◽  
Energy Equation ◽  
Experimental Methods ◽  
Experimental Results ◽  
Theoretical Investigations ◽  
Gas Bearing ◽  
Theoretical Results

The object of the paper is to indicate the value of theoretical investigations of hydrodynamic finite bearings under steady-state conditions. Methods of solution of Reynolds equation by both desk and digital computing, and methods of stabilizing the processes of solution, are described. The nondimensional data available from the solutions are stated. The outcome of an attempted solution of the energy equation is discussed. A comparison between some theoretical and experimental results is shown. Experimental methods employed and some difficulties encountered are discussed. Some theoretical results are given to indicate the effects of the inclusion of slip velocity, stabilizing slots, and a simple case of whirl.

Dynamic Characteristics of a Hydrostatic Gas Bearing Driven by Oscillating Exhaust Pressure

1984 ◽  
Vol 106 (4) ◽  
pp. 477-483 ◽  
Author(s):  
C. B. Watkins ◽  
H. D. Branch ◽  
I. E. Eronini
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Source Model ◽  
Regular Perturbation ◽  
Response Characteristics ◽  
Finite Difference Approximations ◽  
Bode Plots ◽  
Gas Bearing

Vibration of a statically loaded, inherently compensated hydrostatic journal bearing due to oscillating exhaust pressure is investigated. Both angular and radial vibration modes are analyzed. The time-dependent Reynolds equation governing the pressure distribution between the oscillating journal and sleeve is solved together with the journal equation of motion to obtain the response characteristics of the bearing. The Reynolds equation and the equation of motion are simplified by applying regular perturbation theory for small displacements. The numerical solutions of the perturbation equations are obtained by discretizing the pressure field using finite-difference approximations with a discrete, nonuniform line-source model which excludes effects due to feeding hole volume. An iterative scheme is used to simultaneously satisfy the equations of motion for the journal. The results presented include Bode plots of bearing-oscillation gain and phase for a particular bearing configuration for various combinations of parameters over a range of frequencies, including the resonant frequency.

Periodic Motions of a Two-Body System Subjected to Repetitive Impact

1969 ◽  
Vol 91 (4) ◽  
pp. 931-938 ◽  
Author(s):  
David L. Sikarskie ◽  
Burton Paul
Keyword(s):  
Steady State ◽  
High Speed ◽  
Equations Of Motion ◽  
Point Of View ◽  
State Behavior ◽  
State Response ◽  
Stability Of Solution ◽  
Straightforward Method ◽  
Parametric Studies ◽  
Repetitive Impact

The dynamics of a widely used class of hammer impact machines are investigated on the basis of a two-degree-of-freedom idealization. The difficulty in the problem is due to the repetitive impact which introduces a nonlinearity in the system. It is the purpose of the analysis to develop a solution for the steady-state behavior of the system. There are several ways this can be done. One of the most efficient ways, from the point of view of ease of parametric studies of the system, is to convert the problem to a “boundary” value problem. With this technique, the system is governed by the equations of motion between impacts, and further satisfies additional conditions at the beginning and end of each impact cycle. Since the solution is obtained in only one cycle, it thus represents a straightforward method of studying the effect of various system parameters. A fundamental assumption in the analysis is that the steady-state response of the system has a period equal to the forcing period. This is verified for one set of parameters through the use of high-speed movies of an actual machine. There are several other interesting features in the analysis, including multivaluedness of the solution, allowable solution domain, and stability of solution, which have not been completely resolved to date.

scholarly journals Numerical Investigation of Pressure Profile in Hydrodynamic Lubrication Thrust Bearing

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Farooq Ahmad Najar ◽  
G. A. Harmain
Keyword(s):  
Reynolds Equation ◽  
Hydrodynamic Lubrication ◽  
Pressure Profile ◽  
Grid Refinement ◽  
Average Value ◽  
Lubricant Oil ◽  
Pressure Profiles ◽  
Tilting Pad ◽  
Minimum Film Thickness ◽  
Grid Independence

Reynolds equation is solved using finite difference method (FDM) on the surface of the tilting pad to find the pressure distribution in the lubricant oil film. Different pressure profiles with grid independence are described. The present work evaluates pressure at various locations after performing a thorough grid refinement. In recent similar works, this aspect has not been addressed. However, present study shows that it can have significant effect on the pressure profile. Results of a sector shaped pad are presented and it is shown that the maximum average value of pressure is 12% (approximately) greater than the previous results. Grid independence occurs after 24 × 24 grids. A parameter “ψ” has been proposed to provide convenient indicator of obtaining grid independent results. ψ=|(Prefinedgrid-PRefrence-grid)/Prefinedgrid|, ψ≤ε, where “ε” can be fixed to a convenient value and a constant minimum film thickness value of 75 μm is used in present study. This important parameter is highlighted in the present work; the location of the peak pressure zone in terms of (r,θ) coordinates is getting shifted by changing the grid size which will help the designer and experimentalist to conveniently determine the position of pressure measurement probe.

On The Steady State and Dynamic Performance Characteristics of Floating Ring Bearings

1981 ◽  
Vol 103 (3) ◽  
pp. 389-397 ◽  
Author(s):  
Chin-Hsiu Li ◽  
S. M. Rohde
Keyword(s):  
Steady State ◽  
Linear Theory ◽  
High Speed ◽  
Dynamic Performance ◽  
Equations Of Motion ◽  
Nonlinear Behavior ◽  
Journal Bearings ◽  
Transient Problem ◽  
The Stability ◽  
Bearing System

An analysis of the steady state and dynamic characteristics of floating ring journal bearings has been performed. The stability characteristics of the bearing, based on linear theory, are given. The transient problem, in which the equations of motion for the bearing system are integrated in real time was studied. The effect of using finite bearing theory rather than the short bearing assumption was examined. Among the significant findings of this study is the existence of limit cycles in the regions of instability predicted by linear theory. Such results explain the superior stability characteristics of the floating ring bearing in high speed applications. An understanding of this nonlinear behavior, serves as the basis for new and rational criteria for the design of floating ring bearings.

Parametric Studies on Dynamic Performance of Hydrostatic Flexure Pivot Tilting Pad Gas Bearing With Radial Compliance

2007 ◽  
Author(s):  
Robert N. Petro ◽  
Daejong Kim
Keyword(s):  
High Speed ◽  
Heat Dissipation ◽  
Optimal Choice ◽  
Hollow Shaft ◽  
Bearing Clearance ◽  
Radial Stiffness ◽  
Hybrid Operation ◽  
Tilting Pad ◽  
Rotor Bearing ◽  
Gas Bearing

Flexure pivot tilting pad gas bearings are recognized as an alternative to foil gas bearing [1, 2] for high speed turbomachinery, due to their capability to provide high rotor-bearing stability and simple structure. The flexure pivot design eliminates wear problem of axial pins or sockets at the pivots which are common in traditional tilting pad bearings. Added features such as a pivot offset and pad preloads can also be optimized to further improve the stability. Hybrid flexure pivot tilting pad gas bearing have also been reported [3]. The hybrid bearing has a direct air supply to the bearing clearance through a tiny orifice. It has shown that the hybrid operation of the tilting pad gas bearing can also increase the rotor-bearing stability [3]. In many microturbomachinery applications, hollow shafts are adopted to reduce the rotor weight and increase the bending critical speeds. However, the hollow shaft has a large centrifugal growth at high speeds requiring the gas bearing to have radial compliances. However, the radial compliance within the tilting pads can compromise the rotor-bearing stability because large displacement of the pads along the radial direction can cause hydrodynamic rotor-bearing instability associated with the increased bearing clearance (i.e. decreased effective preload) if the radial stiffness is not designed properly. Analytical studies show that optimal choice of pad radial stiffness could extend operating envelope of flexure pivot tilting pad gas bearing without deteriorating rotor-bearing stability [4]. High speed operation can generate significant amount of heat and adequate heat dissipation mechanism should also be developed. Hybrid operation is considered to have added benefit of effective cooling capability. This paper presents design studies on hybrid flexure tilting pad gas bearing with radial compliance which can accommodate large rotor centrifugal growth and also provide effective cooling mechanism.

Nonlinear Modeling of Tilting-Pad Bearings With Application to a Flexible Rotor Analysis

2013 ◽  
Author(s):  
Jianming Cao ◽  
Tim Dimond ◽  
Paul Allaire
Keyword(s):  
High Speed ◽  
Dynamic Stiffness ◽  
Nonlinear Modeling ◽  
Transient Behavior ◽  
Pressure Profile ◽  
Flexible Rotor ◽  
Time Step ◽  
Assembly Method ◽  
Tilting Pad ◽  
Tilting Pad Bearings

Tilting-pad bearings are widely used in high-speed rotating machines to improve the system’s stability. Linearized static or dynamic stiffness and damping coefficients are often applied to rotordynamic analyses. This method has limits due to the nonlinear effects of tilting-pad bearing under severe unbalance conditions or large shaft vibration. This work presents a new modeling and assembly method of a linear flexible rotor with nonlinear tilting-pad bearings. The pressure profile on each pad is calculated using an approximate finite element method by solving Reynolds equation derived from a nonlinear tilting-pad bearing model. Nonlinear bearing forces are calculated based upon the shaft instantaneous position and velocity with an update at each time step. Effects of the bearing pad&pivot are evaluated first by applying a rigid rotor on tilting-pad bearings first. The nonlinear transient behavior of a flexible eight-stage compressor supported on two tilting-pad bearings is investigated. The nonlinear numerical transient response of the system under severe unbalance conditions, including coupled motions of bearing pads, bearing pivots and the shaft, and nonlinear bearing forces, is solved using a 4th order Runge-Kutta integration after assembling the system together. Under severe unbalance conditions. Sub and super harmonic response is shown to exist from both rotor and bearing components.

Modal Analysis of Ballooning Strings With Small Sag

1999 ◽  
Author(s):  
Rongjun Fan ◽  
Sushil K. Singh ◽  
Christopher D. Rahn
Keyword(s):  
Steady State ◽  
High Speed ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mode Shapes ◽  
Theoretical Predictions ◽  
Nonlinear Equations Of Motion

Abstract During the manufacture and transport of textile products, yarns are rotated at high speed and form balloons. The dynamic response of the balloon to varying rotation speed, boundary excitation, and disturbance forces governs the quality of the associated process. Resonance, in particular, can cause large tension variations that reduce product quality and may cause yarn breakage. In this paper, the natural frequencies and mode shapes of a single loop balloon are calculated to predict resonance. The three dimensional nonlinear equations of motion are simplified via small steady state displacement (sag) and vibration assumptions. Axial vibration is assumed to propagate instantaneously or in a quasistatic manner. Galerkin’s method is used to calculate the mode shapes and natural frequencies of the linearized equations. Experimental measurements of the steady state balloon shape and the first two natural frequencies and mode shapes are compared with theoretical predictions.

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