Analysis of a Split-Path Gear Train with

Fluid-Film Bearings

by

Andrew V. Wolff

Thesis submitted to the faculty of the

Virginia Polytechnic Institute and State University

In partial fulfillment of the requirements for the degree of

Master of Science

in

Mechanical Engineering

Committee Members: R. Gordon Kirk, Chair

Charles Reinholtz Daniel J. Inman

May 6, 2004 Blacksburg, VA

Keywords: helical, gearbox, split path, split torque

Copyright 2004

Analysis of a Split-Path Gear Train with

Fluid-Film Bearings

Andrew Wolff, M.S.

Virginia Polytechnic Institute and State University, 2004

Advisor: R. Gordon Kirk

(Abstract)

In the current literature, split path gear trains are analyzed for use in helicopter

transmissions and marine gearboxes. The goal in these systems is to equalize the

torque in each path as much as possible. There are other gear trains where the

operator intends to hold the torque split unevenly. This allows for control over the

gearbox bearing loading which in turn has a direct effect on bearing stiffness and

damping characteristics. Having control over these characteristics is a benefit to a

designer or operator concerned with suppressing machine vibration.

This thesis presents an analytical method for analyzing the torque in split path gear

trains. A computer program was developed that computes the bearing loads in

various gearbox arrangements using the torque information gathered by the analytical

method. A case study is presented that demonstrates the significance of the analytical

method in troubleshooting an industrial gearbox that has excessive vibration.

iii

To my father,

Dr. David A. Wolff,

and my mother,

Dr. Linda D. Wolff

iv

Acknowledgements

I would like to thank my advisor, Dr. Gordon Kirk, for his guidance throughout my

graduate work at Virginia Polytechnic Institute and State University. I appreciate the

invitation to conduct rotor dynamics research after attending his class on the topic. I

would also like to extend my thanks to Dr. Charles Reinholtz and Dr. Daniel J.

Inman as members of my advisory committee.

Finally I would like to thank my parents and Jen for their support and love

throughout my academic career. It has been rewarding and exciting sharing the

graduate student experience with Jen.

v

Table of Contents

page

Abstract ii

Dedication iii Acknowledgements iv List of Figures vii List of Tables ix Nomenclature x

Chapter 1 Introduction 1

1.1 Literature Review................................................................... 2 1.2 Research Objectives................................................................ 4 Chapter 2 Bearing Loads in a Gearbox 5

2.1 Introduction............................................................................ 5 2.2 Concepts and Definitions..................................................... 5 2.3 Visual Basic.NET Code ...................................................... 10 Chapter 3 Split Path Gear Trains 16

3.1 Introduction ........................................................................... 16 3.2 Concepts and Definitions .................................................... 16 3.3 Analytical Model .................................................................... 21 3.4 Computer Program Split Path Calculation ....................... 28

Chapter 4 Case Study: CRF Test Stand 33

4.1 Introduction ........................................................................... 33 4.2 Analytical Model Comparison ............................................. 41

Chapter 5 Conclusions and Recommendations 50

5.1 Conclusions ............................................................................ 50 5.2 Recommendations ................................................................. 51

vi

Table of Contents (continued) page References 53

Appendix A -- Gear Layout 1 Code Segment 54 Appendix B -- Gear Layout 2 Code Segment 58

Appendix C -- Bearing Profile Plotting Program 65

Vita 67

vii

List of Figures

page

2.1 Helical Gear Nomenclature ....................................................................... 7

2.2 Helical Gear Mesh Force Components .................................................. 8

2.3 Helical Gear Axial Force Directions ..................................................... 9

2.4 Gear Layout Option Screen ..................................................................... 10

2.5 Force Vector Diagram for Gear Layout 1 ............................................. 11

2.6 Bearing Location Input Screen for Gear Layout 1 ................................ 12

2.7 Shaft Mass Input Screen for Gear Layout 1 .......................................... 13

2.8 Gear Parameter Input Screen for Gear Layout 1 ................................. 14

2.9 Bearing Loads Results Screen for Gear Layout 1 .................................. 15

3.1 Split Path Gear Train – Front View ......................................................... 18

3.2 Split Path Gear Train – Top View .......................................................... 19

3.3 Conceptual Plot of Split Path Torque .................................................... 20

3.4 Split Path Torque – Each torque path has identical stiffness .............. 26

3.5 Split Path Torque – Path B has greater stiffness than path A ............ 27

3.6 Split Path Torque – Path A has greater stiffness than path B ............ 27

3.7 Bearing and Gear Arrangement for Gear Layout 2 ............................... 28

3.8 Force Vector Diagrams for Gear Layout 2 ............................................. 29

3.9 Bearing Location Input Screen for Gear Layout 2 ................................ 30

3.10 Gear Parameter Input Screen for Gear Layout 2 ................................. 31

3.11 Bearing Loads Result Screen for Gear Layout 2 .................................... 32

4.1 CRF Test Stand Gear Train ...................................................................... 34

4.2 High Speed Pinion Shaft DyRoBeS Model ............................................ 36

4.3 Multiple Station Forced Response with estimated bearing loading ...... 36

4.4 High Speed Pinion Shaft Bearing Profile ................................................. 38

4.5 Matlab plot used to match measured bearing profile ........................... 40

4.6 Torque split plot using the CRF gear train parameters ........................ 44

viii

List of Figures (continued)

page

4.7 Torque Split in Path B as a Function of Clocking Angle –

CRF gear train parameters ........................................................................... 45

4.8 Torque in Each Path using as a Function of Clocking Angle –

CRF gear train parameters .......................................................................... 45

4.9 BePerf Bearing Diagram Updated with Correct Loading Vector ........ 46

4.10 Multiple Station Forced Response – 10,000 HP ..................................... 47

4.11 3D Forced Response – 10,000 HP – 0.56 Torque Split ......................... 47

4.12 Sensitivity Plot – Changes in Support Stiffness ....................................... 48

4.13 Sensitivity Plot -- Changes in Torque Splits ............................................ 49

ix

List of Tables

page

4.1 Comparison Between DyRoBeS and VT-FAST Software ..................... 35

4.2 BePerf Approximate and Improved Bearing Properties ........................ 41

4.3 Stiffness Factors in each Path of the CRF test stand .............................. 42

4.4 Bearing Support Stiffness Sensitivity ......................................................... 49

4.5 Bearing Load Sensitivity ............................................................................... 49

x

Nomenclature

ac = pt = transverse circular pitch

ae = pn = normal circular pitch

Cb = Bearing clearance

Cp = Lobe radial clearance

Cxx = Bearing damping in the x-direction

D = Bearing inner diameter

Fa = Helical gear axial force component

Fn = Helical gear transmission force

Fr = Helical gear radial force component

Ft = Helical gear tangential force component

GR = gear reduction ratio of the input pinion and compound shaft gear

kA = torsional stiffness of path A

kB = torsional stiffness of path B

Kxx = Bearing stiffness in the x-direction

L = Bearing length

L/D = Length to Diameter ratio