MODERN DESIGN OF

FOIL BEARNG AND

ROTOR DYNAMIC SYSTEM

Features:

Detailed Numerical Solution for Reynolds Equation

Advanced Numerical Solutions for 3D Navier-Stokes Equations with Turbulent Flows

Advanced Design of Foil Bearing Components

Probabilistic Robust Design of Bearings and Rotor Dynamics

SBIR Report “IMPROVED ANALYTICAL CAPABILITIES FOR FOIL AIR BEARINGS”

AERO PROPULSION LABORATORY

WRIGHT RESEARCH AND DEVELOPMENT CENTER

January 1990

AUTHOR

Erh-Rong Wu Dr. Eng. Sc. Columbia University, P.E.

www.megaresearch.com

Dr. Erh-Rong Wu is a doctorate graduated from Columbia University, New York.

He joined Shaker Research Corporation leaded by Dr. C.H.T. Pan to develop RSVP (Rotor Substructure Vibration Program) for Wright Aeronautical Laboratories. He worked for Allis-Chalmers Corporation Hydro Turbine Division in charge of rotor bearing system design. He served as an adjunct faculty at Pennsylvania State University, Capitol Campus teaching finite element courses. He joined Garrett AiResearch Corporation Foil Gas bearing group to improve foil bearing design, analysis and manufacturing processes. He founded Mega Research Inc. to develop a multi bearing library and rotor-bearing dynamic codes utilizing probabilistic approaches.

CONTENTS

PREFACE

PART 1. ADVANCED FOIL BEARING DESIGN AND ANALYSIS

1.1 Introduction

1.2 Reynolds Equation Approach

1.3 Foil Thrust Bearing

    1.3.1 Fluid Film Region

    1.3.2 Foil and Underspring Region

    1.3.3 Solution Procedure

1.4 Foil Journal Bearing

    1.4.1 Fluid Film Region

    1.4.2 Fluid Film Profile

    1.4.3 Lift-Off Speed and Minimum Film Thickness

    1.4.4 Numerical Solution of Reynolds Equation

1.5 Column Method

PART 2. THREE DIMENSIONAL NAVIER-STOKES APPROACH

2.1 Introduction

2.2 Fluid Film Region

2.3 Foil Region

2.4 Numerical Solution

2.5 Computations of Bearing Coefficients

PART3. PROBABILISTIC ROTOR-BEARING DYNAMICS

3.1 Introduction

3.2 Rotor-Bearing Dynamic Analysis

3.3 Field Transfer Matrix

3.4 Station Transfer Matrix

3.5 Rotor-Bearing Dynamic Module

3.6 Probabilistic Approach

3.7 Rotor Model for Computations

3.8 Computations and Results

3.9 First Order Reliability Method for Finding Design Point

3.10 Concluding Remarks

Part 4.  ADVANCED DESIGN of Foil Bearing Components

PREFACE
This book is intended to provide an advanced and informative reference for foil air/gas bearing analysts, designers and developers. The book consists of basic foil bearing theory, analysis and practice. Furthermore, it gives improved analytical tools for practical applications in design processes.

In the field of lubrication analysis, Reynolds equation approach has been widely employed for liquid lubricant applications where the fluid is incompressible and the governing equation, Reynolds equation, is linear. The solution process is simple and straight forward even with numerical approximations.  As for the case of air or gas bearings, the fluid is compressible and the governing equation is nonlinear. Linearization and numerical approximations are needed to perform analysis. The significant features of this book focus on these mentioned numerical solution techniques.

The book contains two topics. The first topic (Part 1 and Part 2) provides detailed solution techniques for Reynolds Equation with well explained numerical procedures. Chapter 1 provides derivations of Reynolds equation for a foil air bearing configuration. Detailed numerical solution procedures utilizing the well-known Column method are provided. The lift-off speed and the fluid film thickness which are essential in foil bearing operations are discussed. Thermal elasticity behavior in the supporting spring region is considered and numerical approaches are employed. Chapter 2 taking the Reynolds number effects into account, three dimensional Navier-Stocks approach is presented. Complete turbulent flow formulations including energy equations are used for the numerical solution with SIMPLE algorism. Special coordinate transformations for fluid film thickness are developed to facilitate numerical formulations. The technical report submitted under SBIR program to Wright Research and Development Center entitled, “Improved Analytical Capabilities for Foil Air Bearings,” is included as in Appendix.

The second topic (Part 3) outlines conventional rotor-bearing dynamic approaches. Then it presents a new probabilistic approach to consider the random behaviors of design variables. The first order reliability method (FORM) is utilized to prevent critical speed encounter and avoid synchronous vibration. This technical tool renders concepts to quickly define design variables for better rotor vibration control.

Part4. offers advanced design processes for foil bearing components. Utilization of numerical formulations to facilitate complex geometry design of essential bearing parts.

The original SBIR report has been retyped and revised. All formulations and pictures are clean and clear.

                                                                                                                   Erh-Rong Wu

                                                                                                                   Dr. Eng Sc., PE