AN ABSTRACT OF A THESIS

FINITE ELEMENT ANALYSIS OF HELICAL AND DIFFERENTIAL GEARBOX HOUSINGS

Sarika S. Bhatt

Master of Science in Mechanical Engineering

The finite element analysis of two gearbox housings that constitute the driving mechanism of a double bascule movable bridge was performed. Both the triple reduction helical gearbox and the differential gearbox were made of ASTM A36 steel. The triple reduction helical gearbox was a three-stage gearbox transmitting 112.5 h.p. at 174 rpm with a reduction ratio of 71.05:1. The differential gearbox was a single stage gearbox transmitting 150 h.p. at 870 rpm with a reduction ratio of 5:1.

The load calculations for helical, herringbone, and bevel gears were performed

using the MATHCAD software package. The reactions were used to apply loads to the finite element models of the housings. Geometric models of the two gearboxes were built and meshed using the ANSYS finite element program. Linear structural analysis was performed using a combination of shell and solid elements to determine the deflection and to estimate the stress distribution in the housings. Nonlinear analysis was later performed using shell, solid, beam, and gap elements to determine if the interface between the two halves of the housing separated and contributed to any undesirable misalignments of the shafts or bearings.

In the triple reduction gearbox, the axial forces caused a maximum uz

displacement of 0.022 in. The displacements in the differential gearbox were ten times less than the displacements in the triple reduction gearbox. The location and magnitude of these displacements would not contribute to the undesirable misalignment of the shafts and bearings. The maximum von Mises stress in the triple reduction gearbox was 9000 psi and the maximum von Mises stress in the differential gearbox was 6000 psi. The minimum factor of safety in the triple reduction gearbox was four and the minimum factor of safety in the differential gearbox was six. The nonlinear analysis determined that separation did not occur on the interface between the two halves of the gearbox housings.

FINITE ELEMENT ANALYSIS OF HELICAL AND DIFFERENTIAL GEARBOX HOUSINGS

________________

A Thesis

Presented to

the Faculty of the Graduate School

Tennessee Technological University

by

Sarika ShreeVallabh Bhatt

_______________

In Partial Fulfillment

of the Requirements for the Degree

MASTER OF SCIENCE

Mechanical Engineering

_______________

August 2000

ii

CERTIFICATE OF APPROVAL OF THESIS

FINITE ELEMENT ANALYSIS OF HELICAL AND DIFFERENTIAL GEARBOX HOUSINGS

by

Sarika S. Bhatt

Graduate Advisory Committee: _____________________________ ________ Chairperson Date _____________________________ ________ Member Date _____________________________ ________ Member Date

Approved for the Faculty: ___________________________________ Dean of Graduate Studies ___________________________________ Date

iii

STATEMENT OF PERMISSION TO USE

In presenting this thesis in partial fulfillment of the requirements for a Master of

Science degree at Tennessee Technological University, I agree that the University

Library shall make it available to borrowers under rules of the Library. Brief quotations

from this thesis are allowable without special permission, provided that accurate

acknowledgement of the source is made.

Permission for extensive quotation from or reproduction of this thesis may be

granted by my major professor when the proposed use of the material is for scholarly

purposes. Any copying or use of the material in this thesis for financial gain shall not be

allowed without my written permission.

Signature_____________________ Date ______________________

iv

DEDICATION

This thesis is dedicated to my family.

v

ACKNOWLEDGEMENTS

I wish to acknowledge sincere appreciation and gratitude to my graduate advisor

Dr. Christopher Wilson for his unflagging guidance, encouragement, and teaching during

the entire course of my graduate studies. I would also like to thank my committee

members Dr. Darrell Hoy and Dr. Sally Pardue for their guidance and support.

I sincerely acknowledge the remarkable guidance given by Mr. Joe Saxon of

Meritor Automotive towards the successful completion of this thesis.

I owe special thanks to Mr. James Alison at Steward Machine Company,

Birmingham, AL, for providing the technical support whenever needed. A deep

appreciation and gratitude is given to Mr. Joel Seber for his magnificent cooperation and

assistance in the CAE lab.

I am indebted to all my friends and teachers who directly or indirectly contributed

towards this study. I thank the Mechanical Engineering Department at Tennessee

Technological University for providing a phenomenal learning experience during the

course of my graduate studies.

Above all, my heartfelt gratitude to my parents, family, and my husband for their

encouragement, blessings, support, patience, and love that arouses me each day to

continue my exploration.

vi

TABLE OF CONTENTS

Page

LIST OF TABLES............................................................................................................. ix

LIST OF FIGURES ............................................................................................................ x

LIST OF SYMBOLS AND ACRONYMS....................................................................... xv

1. INTRODUCTION ........................................................................................................ 1

Movable Bridges......................................................................................................... 1

Problem Statement ...................................................................................................... 4

Research Plan and Outline .......................................................................................... 7

2. GEARS AND GEAR FORCE ANALYSIS ................................................................. 9

Helical and Herringbone Gears................................................................................... 9

Bevel Gears............................................................................................................... 17

3. GEARBOX SPECIFICATIONS AND LOAD CALCULATIONS........................... 24

Triple Reduction Gearbox......................................................................................... 34

Differential Gearbox ................................................................................................. 37

4. FINITE ELEMENT MODELING AND ANALYSIS ............................................... 41

Creating and Meshing the FE Models ...................................................................... 44

2-D Structural Element (Plane 42)........................................................................ 45

Elastic Shell Element (Shell 63) ........................................................................... 45

Plastic Shell Element (Shell 43) ........................................................................... 46

vii

Page

3-D Structural Solid Element (Solid 45)...............................................................47

3-D Structural Solid Element (Solid 92)...............................................................47

Merging Solid and Shell Elements ....................................................................... 48

Triple Reduction Gearbox Model Geometry and Loads........................................... 51

Model I.................................................................................................................. 51

Model II ................................................................................................................ 53

Model III ............................................................................................................... 54

Model IV............................................................................................................... 55

Model V ................................................................................................................ 57

Differential Gearbox Model Geometry and Loads ................................................... 59

Model SDI I .......................................................................................................... 59

Model SDI II ......................................................................................................... 60

Nonlinear Analysis.................................................................................................... 61

3-D Elastic Beam Element (Beam 4).................................................................... 63

3-D Point to Point Contact Element (Contac 52).................................................. 63

Block Model Geometry and Loads ........................................................................... 65

Triple Reduction Gearbox Model VI........................................................................ 67

Summary of Assumptions in the Analysis of the Gearboxes ................................... 69

5. FEA RESULTS AND DISCUSSION ........................................................................ 71

Triple Reduction Gearbox Models............................................................................ 71

Differential Gearbox Models .................................................................................... 87

viii

Page

Triple Reduction Gearbox Nonlinear Solution ......................................................... 97

6. CONCLUSIONS AND RECOMMENDATIONS ................................................... 107

BIBLIOGRAPHY........................................................................................................... 110

APPENDICES ................................................................................................................ 113

APPENDIX A……………………………………………………………………. 114

APPENDIX B……………………………………………………………………. 120

APPENDIX C……………………………………………………………………. 122

VITA............................................................................................................................... 125

ix

LIST OF TABLES Page

Table 3.1 Shaft Data for the Triple Reduction Gearbox................................................... 26

Table 3.2 Pinion and Gear Specifications for the Triple Reduction Gearbox .................. 26

Table 3.3 Shaft Data for the Differential Gearbox ........................................................... 28

Table 3.4 Pinion and Gear Specifications for the Differential Gearbox........................... 28

Table 3.5 Direction of Tangential Forces ......................................................................... 32

Table 3.6 Direction of Axial Forces [9]............................................................................ 32

Table 3.7 Pinion and Gear Forces in the Triple Reduction Gearbox................................ 33

Table 3.8 Pinion and Gear Forces in the Differential Gearbox ........................................ 33

Table 3.9 Distance between Pinion or Gear and Ends for Input and Output Shaft .......... 36

Table 3.10 Distance between Pinion or Gear and Ends for Intermediate Shafts.............. 36

Table 3.11 Reaction Loads of the Triple Reduction Gearbox .......................................... 37

Table 3.12 Bearing Details of the Triple Reduction Gearbox .......................................... 37

Table 3.13 Distance between Pinion or Gear and Ends for Input and Output Shafts....... 40

Table 3.14 Reaction Loads of Differential Gearbox......................................................... 40

Table 3.15 Bearing Details for Differential Gearbox....................................................... 40

Table 4.1 Plate Dimensions of the Triple Reduction Gearbox ......................................... 43

Table 4.2 Plate Dimensions of the Differential Gearbox.................................................. 43

Table 4.3 Description of Load Cases in Model V ............................................................ 58

x

LIST OF FIGURES Page Figure 1.1 Single Leaf Bascule Bridge [2] ......................................................................... 2

Figure 1.2 Block Diagram of a Operating Mechanism of a Double Bascule Bridge ......... 3

Figure 1.3 Movable Single Bascule Bridge with the Operating Gearbox Mechanism [3] . 4

Figure 1.4 Gearbox Designed and Manufactured by Steward Machine Company [4]....... 5

Figure 1.5 Model of Fabricated Gearbox [6] ...................................................................... 6

Figure 2.1 Helical Gears [7].............................................................................................. 10

Figure 2.2 Herringbone Gears [7]..................................................................................... 11

Figure 2.3 Section of Helical Gear [7].............................................................................. 12

Figure 2.4 Components of Tooth Force in Helical Gears [7] ........................................... 15

Figure 2.5 Straight Bevel Gear [7].................................................................................... 17

Figure 2.6 Section of Bevel Gear [7] ................................................................................ 18

Figure 2.7 Pair of Bevel Gear [7]...................................................................................... 20

Figure 2.8 Tooth Forces in Bevel Gears [7] ..................................................................... 21

Figure 3.1 Sectional View of Triple Reduction Gearbox .................................................25

Figure 3.2 Sectional View of the Differential Gearbox.................................................... 27

Figure 3.3 Example of Straddle Mounting [9].................................................................. 29

Figure 3.4 Example of Overhung Mounting [9] ............................................................... 29

Figure 3.5 Flowchart of the Helical Gear Reaction Calculations ..................................... 31

Figure 3.6 Gear Arrangement and Forces in the Triple Reduction Gearbox.................... 35

xi

Page

Figure 3.7 Gear Arrangement and Forces in the Differential Gearbox ............................ 38

Figure 4.1 Geometric Model of Triple Reduction Gearbox ............................................. 42

Figure 4.2 Geometric Model of Differential Gearbox ...................................................... 42

Figure 4.3 Two Dimensional Solid Structural Element (Plane 42) [13]........................... 46

Figure 4.4 Elastic and Plastic Shell Element (Shell 63 and Shell 43) [13]....................... 46

Figure 4.5 Three Dimensional Structural Solid Element (Solid 45) [13] ......................... 48

Figure 4.6 Three Dimensional Structural Solid Element (Solid 92) [13] ......................... 48

Figure 4.7 Solid Element Model....................................................................................... 50

Figure 4.8 Combined Solid and Shell Element Model .....................................................50

Figure 4.9 Finite Element Model of the Triple Reduction Gearbox with Representative

Loads......................................................................................................................... 52

Figure 4.10 Expanded Radial Load Distribution from Figure 4.9.................................... 53

Figure 4.11 Finite Element Model of Triple Reduction Gearbox with Inner Ring .......... 54

Figure 4.12 Axial Load on the Inner Ring in Model III ................................................... 55

Figure 4.13 Finite Element Model of Model IV ............................................................... 56

Figure 4.14 Finite Element Model of Model IV with Shell Elements on the Far End ..... 56

Figure 4.15 Finite Element Model of Model V with Radial Loads .................................. 58

Figure 4.16 Finite Element Model of Model SDI I .......................................................... 60

Figure 4.17 Finite Element Model of Model SDI II ......................................................... 61

Figure 4.18 Two parts of a Finite Element Model with Contact [12]............................... 62

Figure 4.19 3-D Elastic Beam Element (Beam 4) [13]..................................................... 64

xii

Page

Figure 4.20 3-D Point to Point Contact Element (Contac 52) [13] .................................. 64

Figure 4.21 Block BMN Geometry and Loads................................................................. 65

Figure 4.22 Triple Reduction Gearbox Model VI with Interface and Bolt Preload ......... 68

Figure 5.1 Model I Displacement ux (in) .......................................................................... 72

Figure 5.2 Model I Displacement uy (in) .......................................................................... 72

Figure 5.3 Model I Displacement uz (in) .......................................................................... 73

Figure 5.4 Model I Total Displacement usum (in).............................................................. 73

Figure 5.5 Model I von Mises Stress σeff (psi) ................................................................. 74

Figure 5.6 Comparison of Displacement ux in Models I, II, and III ................................. 76

Figure 5.7 Comparison of Displacement uy in Models I, II, and III ................................. 76

Figure 5.8 Comparison of Displacement uz in Models I, II, and III ................................. 77

Figure 5.9 Comparison of von Mises Stress in Models I, II, and III ................................ 77

Figure 5.10 Model V (Load Case IV) Displacement ux (in)............................................. 79

Figure 5.11 Model V (Load Case IV) Displacement uy (in)............................................. 79

Figure 5.12 Model V (Load Case IV) Displacement uz (in) ............................................. 80

Figure 5.13 Model V (Load Case IV) von Mises Stress (psi) .......................................... 80

Figure 5.14 Defined Paths on the Triple Reduction Gearbox........................................... 82

Figure 5.15 Comparison of X-Displacement in Model IV and V..................................... 83

Figure 5.16 Comparison of Y-Displacement in Model IV and V..................................... 83

Figure 5.17 Model V (Load Case III) Displacement uz (in) ............................................. 84

Figure 5.18 Model V (Load Case II) Displacement uz (in) ..............................................85

xiii

Page

Figure 5.19 Model V Load (Case I) Displacements uz (in) ..............................................86

Figure 5.20 SDI II (load case I) Near End 2 Displacement ux (in) ................................... 88

Figure 5.21 SDI II (load case I) Far End 1 Displacement ux (in) ..................................... 88

Figure 5.22 SDI II (Load Case I) Far End 1 Displacement uy (in) ................................... 89

Figure 5.23 SDI II (Load Case I) Near End 2 Displacement uz (in)................................. 89

Figure 5.24 SDI II (Load Case I) Far End 1 Displacement uz (in) ................................... 90

Figure 5.25 SDI II (Load Case I) Far End 1 Total Displacement usum (in)....................... 90

Figure 5.26 SDI II Far End 1 von Mises Stress (psi)........................................................ 91

Figure 5.27 SDI II (Load Case II) Near End 2 Displacement ux (in) ............................... 93

Figure 5.28 SDI II (Load Case II) Far End 1 Displacement ux (in).................................. 93

Figure 5.29 SDI II (Load Case II) Far End 1 Displacement uy (in).................................. 94

Figure 5.30 SDI II (Load Case II) Near End 2 Displacement uz (in) ............................... 94

Figure 5.31 SDI II (Load Case II) Far End 1 Displacement uz (in) .................................. 95

Figure 5.32 Comparison of Total Displacement in SDI I and SDI II ............................... 96

Figure 5.33 Comparison of Von Mises Stress in SDI I and SDI II .................................. 97

Figure 5.34 Model VI Displacement ux (in) ..................................................................... 98

Figure 5.35 Model VI Displacement uy (in) ..................................................................... 98

Figure 5.36 Model VI Displacement uy between the First Intermediate and Second

Intermediate Shaft Hole ........................................................................................... 99

Figure 5.37 Gap on the Interface between the First Intermediate and Second Intermediate

Shaft Holes................................................................................................................ 99

xiv

Page

Figure 5.38 Model VI Displacement uz (in) ................................................................... 100

Figure 5.39 Model VI von Mises Stress (psi) ................................................................. 100

Figure 5.40 Defined Paths on the Interface in the Triple Reduction Gearbox ............... 104

Figure 5.41 Displacement ux in Model VI and VII......................................................... 104

Figure 5.42 Displacement uy in Model VI and VII......................................................... 105

Figure 5.43 Displacement uz in Model VI and VII......................................................... 105

Figure 5.44 Von Mises Stress in Model VI and VII ....................................................... 106

xv

LIST OF SYMBOLS AND ACRONYMS

Symbol Description

a center to center distance

b face width

d pitch circle diameter

ha addendum

hf dedendum

i speed ratio

m transverse module

mn normal module

p transverse circular pitch

pa axial circular pitch

pn normal circular pitch

rb back cone radius

rm mean radius

usum total nodal displacement

ux, uy, uz components of nodal displacement in global X, Y, Z directions

z number of teeth

z′ number of teeth on formative gear

Ao cone distance

At tensile stress area

D pitch circle diameter

xvi

Symbol Description

Dp diameter at midpoint along face width

Fi preload

Fp proof load

P resultant force

Pa axial force

Pr radial force

Ps separating force

Pt tangential force

R transverse diametral pitch

Rb back cone radius of formative gear

Rn normal diametral pitch

Sp proof strength

T torque

X, Y, Z global coordinate axes

α transverse pressure angle

αn normal pressure angle

γ pitch angle, pinion

ω angular velocity

ψ helix angle

Γ pitch angle, gear

h.p. horsepower

xvii

AGMA American Gear Manufacturing Association

Symbol Description

ASTM American Standard of Testing and Materials

CCW Counterclockwise

CW Clockwise

FEA Finite Element Analysis

FEM Finite Element Method

LH Left Handed

RH Right Handed

SAE Society of Automotive Engineers

1

CHAPTER 1

INTRODUCTION

This thesis focuses on the force, deflection, and stress analysis of two gearboxes

designed and manufactured by Steward Machine Company, Birmingham, Alabama.

These gearboxes are designed for high torque and low speed applications for operating

movable bridges, heavy hoisting machinery, or other lifting mechanisms. The helical and

herringbone-bevel combination gearbox housings analyzed in this thesis form the driving

mechanism of a double bascule movable bridge.

Movable Bridges

Movable bridges are generally constructed over waterways where it is difficult to

build a fixed bridge high enough for water traffic to pass under it. The common types of

movable bridges are the lifting, bascule, and swing bridges. The bascule bridge is similar

to the ancient drawbridge both in appearance and operation. It may be in one span or in

two halves meeting at the center. It consists of a rigid structure mounted at the abutment

of a horizontal shaft about which it swings in a vertical arc. A single leaf bascule bridge

is shown in Figure 1.1. The need for large counterweights and the presence of high

stresses in the hoisting machinery limit the span of bascule bridges. The largest

constructed span of a double-leaf bridge is 336 ft [1]. A double bascule bridge has two

2

Figure 1.1 Single Leaf Bascule Bridge [2]

leafs on each side and a total of four leafs that open and close when the bridge is opened

and closed. An AC motor drives the differential gearbox D of a double bascule bridge

shown in the block diagram of Figure 1.2. The AC motors are typically rated for 15-150

h.p. at 870 rpm. The differential gearbox D drives the triple reduction gearboxes T on

both sides, which in turn drive the main pinion P. The main pinion drives the rack R

attached to the leaf of the bridge. The differential gearbox allows equal load distribution

between the two output shafts. It consists of herringbone, helical, and bevel gears. The

triple reduction gearbox consists of helical gears. A photograph showing the operating

gearbox mechanism of a single leaf movable bridge is shown in Figure 1.3.

3

Figure 1.2 Block Diagram of a Operating Mechanism of a Double Bascule Bridge

4

Figure 1.3 Movable Single Bascule Bridge with the Operating Gearbox Mechanism [3]

Problem Statement

A gearbox is a complicated structure where the actual loading is idealized using

statically equivalent loads. The design calculations of most gearboxes are very

complicated. A general practice is to employ experience or test data in the sizing of a

gearbox. After fabrication, the gearboxes are tested to loads in excess of the expected

service loads. Deflections are measured using dial indicators. In some cases, strain gages

are used to measure strains at key locations. Steward Machine Company tests gearboxes

at 150 percent of the design load. A gearbox designed and manufactured by Steward

Machine Company is shown in Figure 1.4. The primary components of a gearbox are

5

Figure 1.4 Gearbox Designed and Manufactured by Steward Machine Company [4]

gears, shafts, and bearings. The housing is constructed to hold the gear-shaft and bearing

subassembly in place. The other components include shims, gaskets, oil seals, breathers,

oil level indicators, bearing retainers, and fasteners.

In recognition of the rapidly changing developments in the field of machine

design, an understanding of the characteristic structural behavior of gearbox housings is

desirable. Classical methods, such as the mechanics of materials method and the theory

of elasticity, are difficult to apply to the gearbox housing geometry to predict the

behavior. The finite element method (FEM) is a versatile numerical method widely used

to solve such engineering problems. In this research, the deflection and stress distribution

in the triple reduction and differential gearbox housings are estimated using FEM.

In the technical literature, very little writing was found on the finite element

analysis of gearbox housings. A good description of finite element analysis of gearbox

6

housings was given by V. Ramamurti, et al. [6]. Both cast and fabricated housings were

analyzed and compared for stress and rigidity levels. The geometric model of the

fabricated gearbox is shown in Figure 1.5. Both the gearboxes were made in two halves

to be bolted at the center. For the purpose of linear analysis, the two halves were

considered integral. The geometric modeling was discretized into a number of triangular

plate elements with six degrees of freedom at each node. The bearing holes were modeled

as octagons. The bottom faces of these gearboxes were fixed by constraining all the

degrees of freedom of those nodes. The direction of the inplane force was determined and

distributed among four or five nodes. The finite element analysis of the fabricated

housing predicted stresses in the range of 20 to 27 MPa (3 to 4 ksi) and deflections in the

range of 0.26 to 1.16 mm (0.010 to 0.046 in).

Figure 1.5 Model of Fabricated Gearbox [6]

7

Ramamurti did not study the interface of the two halves of the gearbox. To fully

understand the interface displacements a more elaborate model is required. This thesis is

focused on a closer examination of the interface in the Steward Machine Company

gearboxes. Both the simpler linear approach and a more complex nonlinear approach will

be used.

Research Plan and Outline

The first step towards the finite element analysis (FEA) of the gearboxes was the

calculation of gear forces on all the shafts of both the gearboxes. The technical

background on the types of gears and gear force analysis is discussed in Chapter 2. An

understanding of gear forces is the basis of the calculation of forces acting on the

gearboxes. The specifications of the two gearboxes and the calculation of gear forces are

described in Chapter 3. Subsequently, the loads transmitted to the housing are calculated

in Chapter 3. Generalized MATHCAD programs were written for calculating the loads

from helical, herringbone, and bevel gears and are included as Appendices. The results of

the programs were the reaction loads transmitted to the housing. The next step involved

modeling, discretizing, and solving the geometry of the gearboxes for the FEA. An

overview of the methodology for the application of FEM to obtain desired results is

outlined in Chapter 4. The modeling guidelines, description of element types used, and

meshing of the gearbox models is detailed in Chapter 4. Modeling, meshing, and solving

the finite element models was an iterative process and is discussed elaborately in Chapter

8

4. The commercial finite element package ANSYS was used for modeling and analyses.

The results and discussion of the linear and nonlinear analyses performed are detailed in

Chapter 5. The conclusions and recommendations derived from the analyses are

discussed in Chapter 6.

9

CHAPTER 2

GEARS AND GEAR FORCE ANALYSIS

The discussion on gears and gear forces is adapted from V. B. Bhandari [7]. Gears

are broadly classified into four types: spur, helical, bevel, and worm. In spur gear, the

teeth are cut parallel to the axis of the shaft. The profile of the gear tooth is an involute

curve and remains identical along the entire width of the gearwheel. Spur gears are used

only when the shafts are parallel because the teeth are parallel to the axis of the shaft.

Spur gears impose radial loads on the shafts. In spur gears, the contact between meshing

teeth occurs along the entire face width of the tooth. Therefore, a sudden load application

occurs, resulting in an impact condition and generating noise. Helical, herringbone, and

bevel gears constitute the driving mechanism of the gearboxes in this thesis.

Helical and Herringbone Gears

Helical gears have an involute profile similar to spur gears. The contact between

meshing teeth of helical gears begins with a point on the leading edge of the tooth and

gradually extends along the diagonal line across the tooth. When helical gears mesh,

there is a gradual application of load. Thus, helical gears have smooth engagement and

quiet operation. The teeth of helical gears are cut at an angle with the axis of the shaft as

shown in Figure 2.1. The involute profile of a helical gear is in a plane perpendicular to

10

Figure 2.1 Helical Gears [7]

the tooth element. Helical gears are used in automobiles, turbines, gearboxes, and high

speed applications up to 3000 m/min [7]. The magnitude of the helix angle of the pinion

and the gear is the same; however, the hand of the helix is opposite. For example, a right-

handed pinion meshes with a left-handed gear. Helical gears impose radial and thrust

loads on shafts.

Herringbone gears are a special type of helical gears. They consist of double

helical teeth with a small groove between the two helixes as shown in Figure 2.2. This

groove is required for hobbing and grinding operations. The construction of herringbone

gears results in equal and opposite thrust reactions. Thus, herringbone gears impose only

radial loads on shafts. Herringbone gears are used for parallel shafts.

11

Figure 2.2 Herringbone Gears [7]

A portion of the top view of a helical gear is shown in Figure 2.3. A1B1 and A2B2

are the centerlines of adjacent teeth on the pitch plane. The helix angle ψ is defined as the

angle A1B2A2 between the axis of shaft and the centerline of the tooth on the pitch plane.

The plane of rotation is labeled XX and the plane perpendicular to the tooth elements is

labeled YY. The distance A1A2 is the transverse circular pitch p, measured in the plane of

rotation. The distance A1C is the normal circular pitch pn, measured in a plane

perpendicular to the tooth elements.

The ratio of pn and p from triangle A1A2C is

ψcos21

1 ==AA

CA

p

pn (2.1)

or ψcosppn = . (2.2)

12

Figure 2.3 Section of Helical Gear [7]

The pitch circle diameter of a helical gear with number of teeth z is

πzp

d = . (2.3)

The transverse diametral pitch R is the ratio of the number of teeth z to the pitch circle

diameter d:

13

d

zR = . (2.4)

Comparing Equations 2.3 and 2.4, the product of the transverse circular pitch p and the

transverse diametral pitch R is

π=pR . (2.5)

Substituting Equation 2.5 in to Equation 2.2 leads to

ψcos

RRn = , (2.6)

where Rn is the normal diametral pitch.

The transverse module m and the normal module mn are defined as the inverse of

the transverse diametral pitch R and the normal diametral pitch Rn, respectively. Hence,

R

m1= , (2.7)

and n

n Rm

1= . (2.8)

Substituting Equations 2.7 and 2.8 in to Equation 2.6 leads to

ψcosmmn = . (2.9)

Combining Equations 2.4, 2.7, and 2.9 leads to

14

ψcosnzm

zmd == . (2.10)

The axial pitch pa of the helical gear is the distance A1B2 shown in Figure 2.3.

From triangle A1A2B2, the axial pitch pa is related to the transverse circular pitch p by

ψtan

ppa = . (2.11)

There are two pressure angles, the transverse pressure angle α and the normal

pressure angle αn, in their respective planes. These angles are related by the following

expression:

ααψ

tan

tancos n= . (2.12)

The normal pressure angle is usually 20o [8].

The center to center distance a between the two helical gears with teeth z1 and z2

is

ψψ cos2cos222

2121 nn mzmzdda +=+= (2.13)

or ψcos2

)( 21 zzma n +

= . (2.14)

The speed ratio i is determined from the ratio of speed or number of teeth in the

pinion and gear and is

15

p

g

g

p

z

zi ==

ωω

, (2.15)

where subscripts p and g refer to the pinion and the gear, respectively.

The resultant force P acting on the helical gear as discussed in Bhandari [7] is

shown in Figure 2.4. This force is resolved into three components the tangential

component Pt, the radial component Pr, and the axial component Pa. From triangle ABC,

the radial component is

nr PP αsin= (2.16)

Figure 2.4 Components of Tooth Force in Helical Gears [7]

16

and the resultant of the axial and tangential force BC is

nPBC αcos= . (2.17)

From triangle BDC in Figure 2.4, the axial and tangential forces are

ψαψ sincossin na PBCP == (2.18)

and ψαψ coscoscos nt PBCP == . (2.19)

Combining Equations 2.18 and 2.19 leads to

ψtanta PP = , (2.20)

and combining Equations 2.16 and 2.19 leads to

=

ψα

cos

tan ntr PP . (2.21)

The tangential component Pt is calculated using

d

TPt

2= , (2.22)

where T is the torque transmitted and d is the pitch circle diameter.

17

Bevel Gears

Bevel gears are used to transmit power between two intersecting shafts. There are

two common types of bevel gears: straight and spiral. The gearboxes analyzed in this

thesis have straight bevel gears. A schematic of straight bevel gear from Bhandari [7] is

shown in Figure 2.5. The elements of the teeth are straight lines that converge to a

common apex. The straight bevel gear teeth have an involute profile. The teeth of the

spiral bevel gears are curved. Straight bevel gears are easy to design and manufacture and

give reasonably long service when properly mounted on shafts. They are noisy during

high-speed operation. Bevel gears are not interchangeable and are always made in pairs.

The angle between the axes of intersecting shafts is 90o in most straight bevel gears.

Figure 2.5 Straight Bevel Gear [7]

18

The dimensions of bevel gears are always specified and measured at the large end

of the tooth. The pitch lines of the teeth lie on the surface of an imaginary cone with the

apex at O shown in Figure 2.6. The distance Ao is the cone distance. The pitch angle γ is

the angle the pitch line makes with the axis of the gear. The addendum ha, the dedendum

hf, and the pitch circle diameter D are specified at the large end of the tooth as shown in

Figure 2.6. The back cone of radius rb is an imaginary cone and its elements are

perpendicular to the elements of the pitch cone.

Figure 2.6 Section of Bevel Gear [7]

19

An imaginary spur gear is considered in a plane perpendicular to the tooth at the

large end to derive the terms associated with bevel gears. The pitch circle radius and

number of teeth on this imaginary spur gear are Rb and z′, respectively. The virtual or

formative teeth on the imaginary spur gear are

m

rz b2

=′ , (2.23)

where m is the module at the large end of the tooth. If z is the actual number of teeth on

the bevel gear, then

m

Dz = . (2.24)

From Equations 2.23 and 2.24, the ratio of z and z′ is

D

r

z

z b2=

′. (2.25)

From ∆ ABC in Figure 2.6,

AC

ABBCA =sin (2.26)

or b

o

r

D )2/()90sin( =− γ . (2.27)

Using Equation 2.27 and ( )γγ −= o90sincos yields an expression for the back cone

radius rb

20

γcos2

Drb = . (2.28)

Substituting Equation 2.28 in Equation 2.25 leads to

γcos

zz =′ . (2.29)

A pair of bevel gear is shown in Figure 2.7. The pitch circle diameters of the

pinion and gear are Dp and Dg, respectively. The pitch angle of the pinion is γ and the

pitch angle of the gear is Γ. From the geometry in Figure 2.7,

g

p

g

p

g

p

z

z

mz

mz

D

D===γtan . (2.30)

It can also be shown that p

g

z

z=Γtan . For the pair of bevel gear shown in Figure 2.7,

2

πγ =Γ+ . (2.31)

Figure 2.7 Pair of Bevel Gear [7]

21

According to Bhandari [7], the resultant tooth force between two meshing teeth of

bevel gears is concentrated at the midpoint along the face width of the tooth. The

resultant force acts at the mean radius rm shown in Figure 2.8. The mean radius rm is

−

=

2

sin

2

γbDr p

m , (2.32)

where Dp is the diameter of the pinion at the midpoint along the face width and b is the

face width of the tooth.

Figure 2.8 Tooth Forces in Bevel Gears [7]

22

The resultant force has two components, Pt and Ps, shown in Figure 2.8. Ps is the

separating force between the two meshing teeth. Pt is the tangential component

perpendicular to the plane of the paper. The tangential component is determined from the

relationship,

m

t r

TP = , (2.33)

where T is the torque transmitted by the gears. This analysis is similar to that of the

helical gears and the resulting separating force is

αtants PP = , (2.34)

where α is the pressure angle. The separating force is further resolved into two

components: the axial and radial forces shown in Figure 2.8. For the pinion,

γcossr PP = (2.35)

and γsinsa PP = . (2.36)

Substituting Equation 2.34 into Equations 2.35 and 2.36, respectively, leads to

γα costantr PP = (2.37)

and γα sintanta PP = . (2.38)

The components of the tooth force on the pinion can be determined using

Equations 2.37 and 2.38. The components of tooth forces acting on the gear are equal to

the components of tooth forces acting on the pinion in magnitude, but act in the opposite

direction. The radial component of the gear is equal to the axial component Pa on the

23

pinion. Similarly, the axial component on the gear is equal to the radial component Pr on

the pinion.

24

CHAPTER 3

GEARBOX SPECIFICATIONS AND LOAD CALCULATIONS

This chapter outlines the specifications of the triple reduction and differential

gearboxes. It also includes the calculations of the gear forces magnitude and directions

and the loads transmitted to the housing.

The triple reduction gearbox is the input to the main drive pinion of one leaf of

the bridge. This gearbox weighs approximately 18,000 lb and is driven by the differential

gearbox. The material of the housing is ASTM A36 steel with a modulus of elasticity E

of 30 × 106 psi and Poisson’s ratio ν of 0.29. The housing is joined together by a

combination of welding and bolted joints. A schematic of the gearbox is shown in Figure

3.1. The triple reduction gearbox shafts are designated using capital S’s and a numeral.

The gearbox has two intermediate shafts S2 and S3 besides the input and output shafts S1

and S4. All shafts have helical gears and anti-friction bearing at shaft ends. The gearbox

is designed to transmit 112.5 h.p. at 174 rpm with a reduction ratio of 71.05:1. The

summary of shaft and gear specifications is shown in Table 3.1 and Table 3.2,

respectively. The dimensions and specifications were provided by Steward Machine

Company.

The special differential gearbox drives the triple reduction gearbox and ensures

equal load distribution between the output shafts. The differential gearbox weighs

approximately 1200 lb. The material of the housing is ASTM A36 steel with a modulus

of elasticity E of 30 × 106 psi and a Poisson’s ratio ν of 0.29. A schematic of the gearbox

25

Figure 3.1 Sectional View of Triple Reduction Gearbox

26

Table 3.1 Shaft Data for the Triple Reduction Gearbox

Shaft Diameter (in) Length between bearing ends (in)

Input Shaft 4.503 First Intermediate Shaft 7.004 Second Intermediate Shaft 11.005 Output Shaft 12.506

40.876

Table 3.2 Pinion and Gear Specifications for the Triple Reduction Gearbox

No. of Teeth Diametral Pitch Helix Angle (degree)

Pressure Angle (degree)

Input Shaft Pinion 16 3 15 20

First Intermediate Shaft Pinion

16 2 20.24 20

First Intermediate Shaft Gear

72 3 15 20

Second Intermediate Shaft Pinion

19 1.5 15 20

Second Intermediate Shaft Gear

60 2 20.24 20

Output Shaft Gear 80 1.5 15 20

is shown in Figure 3.2. The differential gearbox shafts are designated using s’s and a

numeral. The gearbox has a differential setup on intermediate shaft s2 and s3 with a

balanced 3-pinion and bevel gear assembly. The bevel gears B1 and B2 are mounted on

shafts s2 and s3 and the bevel pinion meshes with them on both sides. The bevel pinion A

is one of the three pinions on the differential assembly. Herringbone gears are mounted

on the input and intermediate shafts. Helical gears are mounted on the intermediate and

output shafts. The gearbox is designed to transmit 150 h.p. at 870 rpm with a reduction

ratio of 5:1. The summary of shaft and gear specifications provided by Steward Machine

Company is shown in Table 3.3 and Table 3.4, respectively.

27

Figure 3.2 Sectional View of the Differential Gearbox

28

Table 3.3 Shaft Data for the Differential Gearbox

Shaft Diameter (in) Length between bearing ends (in)

Input Shaft 2.560 Intermediate Shaft (Two) 6.503 Output Shaft (Two) 4.726

19.75

Table 3.4 Pinion and Gear Specifications for the Differential Gearbox

No. of teeth Diametral Pitch

Helix Angle (degree)

Pressure Angle (degree)

Input Herringbone Gear 25 5.774 30 17.5 Intermediate Herringbone Gear

125 5.774 30 17.5

Intermediate Helical Gear

79 4 15.55 20

Intermediate Bevel Pinion

12 3 - 20

Intermediate Bevel Gear

42 3 - 20

Output Helical Gear 79 4 15.55 20

These gearboxes, designed and manufactured by Steward Machine Company, are

rated in accordance with the American Gear Manufacturing Association (AGMA)

standards for helical and herringbone-bevel combination enclosed drives. Through-

hardened, alloy steel shafts with large shaft diameters are used to minimize deflections

and assume maximum stability and support for gears. Through-hardened gears and

pinions manufactured from high quality alloy steel forging, casting, welding, or rolled

alloy steel bars are used. For helical gears, the aspect ratio is kept below two and the

overlap ratio is usually kept above two.

29

The reaction loads from the gears vary depending on the type of gears and the

type of bearing mounting. There are two basic types of mounting: straddle and overhung.

The support points for straddle mounting are on shaft ends and the load is applied

between the support points (see Figure 3.3). In overhung mounting, the load is applied

outside the support points (see Figure 3.4). When the forces act downward, the bearing

support reactions for straddle and overhung mounting are shown in Figure 3.3 and Figure

3.4, respectively.

Figure 3.3 Example of Straddle Mounting [9]

Figure 3.4 Example of Overhung Mounting [9]

30

The load calculations for the gearboxes have been done using a mathematical

analysis program, MATHCAD. The MATHCAD program attached in Appendix A can

be used for straddle mounting and the MATHCAD program in Appendix B can be used

for overhung mounting. The procedure for the program input and calculations is shown in

the flowchart in Figure 3.5.

The first input parameters to the program are power and speed indicated in box 1

of the flowchart. A shaft can have one or two gear mountings. A shaft with one gear

mounted is represented as Case I. A shaft with two gears, one driving pinion and other

the driven gear, is represented as Case II. For Case I, the input is the distance between the

gear and both ends indicated in box 2. For Case II, the input constitutes distance between

one end and the gear, distance between the two gears and the distance between the second

end and the gear on that side as indicated in box 3. The next input for Case I is the gear or

pinion specification as indicated in box 4. For Case II, the number of teeth, diametral

pitch, helix angle and pressure angle for both the gear and pinion must also be given. The

direction of tangential and axial forces in box 9 is determined based on the input of boxes

6, 7 and 8 for the direction of rotation, whether the gear is a driving member of a driven

member, and the hand of the gear, respectively.

31

Figure 3.5 Flowchart of the Helical Gear Reaction Calculations

4

1

2

Computes torque

Case I Case II

Shaft has one gear mounted. Assign N=1

Shaft has two gears mounted. Assign N=2

Computes Tangential, Radial and Axial forces for Gear or Pinion

Computes Tangential, Radial and Axial forces for Gear and Pinion

Determines the direction of tangential radial and axial forces

Computes reaction at end 1 and end 2

Input: Power and speed

Input: Distance between end 1 and gear Distance between end 2 and gear

Input: Distance between end 1 and gear/pinion Distance between two gears Distance between end 2 and gear/pinion

Gear or Pinion specifications

Gear and Pinion specifications

Determine whether the gear is driving pinion or the driven gear

Determine direction of rotation of shaft

Determine the hand of the gear

Assign m Assign M

3

5

6

7

8

9

32

The program calculates the torque on a shaft, the diameter of the gears, and the

tangential, separating, and axial forces on the gears. These forces are used to calculate the

reaction loads on the gearboxes. It is important to understand the gear force directions

prior to the calculation of reaction loads on the gearbox housings. The direction of the

tangential, separating, and axial forces change depending on the direction of rotation and

the hand of gear teeth. The direction of these forces also change depending on whether

the gear is a driving pinion, or a driven gear. The bearing selection does not depend on

the angle at which the reactions act. However, the angle at which these loads act is

calculated for finite element analysis. The direction of tangential and axial forces acting

on the gears can be determined from Table 3.5 and Table 3.6. The separating force

always acts on the tooth surface and points towards the center. A summary of gear forces

for both gearboxes follows in Table 3.7 and Table 3.8.

Table 3.5 Direction of Tangential Forces

Hand of Spiral Direction of rotation Driving member Driven member Left hand Clockwise Towards Left Towards Left Left Hand Counterclockwise Towards Right Towards Right Right Hand Clockwise Towards Left Towards Left Right Hand Counterclockwise Towards Right Towards Right

Table 3.6 Direction of Axial Forces [9]

Hand of Spiral Direction of rotation Driving member Driven member Left hand Clockwise Away from viewer Towards viewer Left Hand Counterclockwise Towards viewer Away from viewer Right Hand Clockwise Towards viewer Away from viewer Right Hand Counterclockwise Away from viewer Towards viewer

33

Table 3.7 Pinion and Gear Forces in the Triple Reduction Gearbox

Tangential Force (lb)

Separating Force (lb)

Axial Force (lb)

Input Shaft Pinion 410476.1 × 310486.6 × 310955.3 × First Intermediate Shaft Pinion 410301.4 × 410946.1 × 410586.1 × First Intermediate Shaft Gear 410476.1 × 310486.6 × 410810.2 × Second Intermediate Shaft Pinion

410301.4 × 410946.1 × 410586.1 ×

Second Intermediate Shaft Gear

410301.4 × 410946.1 × 410586.1 ×

Output Shaft Gear 510048.1 × 410607.4 × 410809.2 ×

Table 3.8 Pinion and Gear Forces in the Differential Gearbox

Tangential Force (lb)

Separating Force (lb)

Axial Force (lb)

Input Herringbone Gear 310347.4 × 310876.1 × 310510.2 × Intermediate Herringbone Gear 310347.4 × 310876.1 × 310510.2 × Intermediate Helical Gear 310301.5 × 310335.2 × 310475.1 × Intermediate Bevel Gear 310500.4 × 455.64 310595.1 × Output Helical Gear 310301.5 × 310335.2 × 310475.1 ×

The gear forces were used to calculate the reaction loads that balance out these

forces. When a load is applied, it is the actual load applied to the housing in the correct

direction. It is not the bearing reaction that opposes the load and is in the opposite

direction. The radial reactions represent vector summation of these actual loads in the

correct direction. These radial reactions are distributed on a 90o arc in the finite element

model. A summary of radial reaction loads for both gearboxes is discussed in the next

section.

The triple reduction and differential gearboxes are both used to operate the leaf on

both sides of the bridge. Therefore, both clockwise and counterclockwise rotations of the

34

gearboxes have been reviewed. The bearing details give bore size on housing of the

gearboxes where reaction loads act. Hence, the bearing details for both the gearboxes

have also been tabulated. The bearing selection was carried out by Steward Machine

Company using the guidelines of the Timken Bearing Catalog for taper roller bearings

[10].

Triple Reduction Gearbox

The representation of tangential, separating, and axial forces acting on gears of all

shafts is shown in Figure 3.6. The rotations are viewed from End 1 on right-hand side of

the input shaft. The direction of shaft rotation, End 1, and End 2 are represented in Figure

3.6. The distances between the pinion or gear and End 1 and End 2 are tabulated in Table

3.9 and Table 3.10. These values were chosen to ensure that the distance between bearing

centerlines was consistent for all shafts. These values are not the exact dimensions. For

the input and output shafts, the distance between pinion or gear and End 1 is designated

as B. The distance between End 2 and pinion or gear is labeled A. For intermediate

shafts, the distance between End 2 and pinion or gear is C, the distance between pinion

and gear is designated by D and, the distance between End 1 and pinion or gear is E.

The input shaft S1 of the triple reduction gearbox has RH driving pinion on the

shaft. It takes a torque of approximately 41000 lb-in and rotates at 174 rpm.

35

Figure 3.6 Gear Arrangement and Forces in the Triple Reduction Gearbox

36

Table 3.9 Distance between Pinion or Gear and Ends for Input and Output Shaft

Shaft Distance A (in)

Distance B (in)

Input 34.001 6.875 Output 27.563 13.313

Table 3.10 Distance between Pinion or Gear and Ends for Intermediate Shafts

Shaft Distance C (in)

Distance D (in)

Distance E (in)

First Intermediate 9.25 24.751 6.875 Second Intermediate 9.25 18.313 13.313

The intermediate shaft S2 has both a LH driving pinion and a LH driven gear. The

gear meshes with pinion of the input shaft. The shaft torque is approximately 180000 lb-

in and the shaft rotates at 38.67 rpm.

The second intermediate shaft S3 has a RH driven gear with meshes with pinion

of the first intermediate shaft and RH driving pinion that drives the output shaft gear.

This shaft rotates at 10.312 rpm and takes a torque of approximately 680000 lb-in.

The output shaft S4 has LH gear that is driven by the pinion on the second

intermediate shaft. The torque on the shaft is approximately 2870000 lb-in and the

rotation is at 2.45 rpm.

A summary of reaction loads and bearing detail from the Timken Bearing Catalog

[10] is in Table 3.11 and Table 3.12, respectively. The angles in the radial reactions are

referenced using 0o as the positive X axis.

37

Table 3.11 Reaction Loads of the Triple Reduction Gearbox

Shaft

Radial reaction at End 1 (lb)

Radial reaction at End 2 (lb)

Axial reaction at End 1 (lb)

Axial reaction at End 2 (lb)

CW rotation Shaft S1

o4 6510352.1 ∠× o3 7210616.2 ∠×

310955.3 × 0

CCW rotation Shaft S1

o4 29310331.1 ∠× o4 6510352.1 ∠×

0 310955.3 ×

CW rotation Shaft S2

o4 10010234.2 ∠× o4 6510952.3 ∠×

0 41019.1 ×

CCW rotation Shaft S2

o4 8510209.2 ∠× o4 7310745.3 ∠×

41019.1 × 0

CW rotation Shaft S3

o4 6510871.8 ∠× o4 9910828.6 ∠×

0 410224.1 ×

CCW rotation Shaft S3

o4 28110201.8 ∠× o4 27910827.6 ∠×

410224.1 × 0

CW rotation Shaft S4

o4 10010171.7 ∠× o4 4510817.4 ∠×

410809.2 × 0

CCW rotation Shaft S4

o4 2351066.8 ∠× o4 26310437.3 ∠×

0 410809.2 ×

Table 3.12 Bearing Details of the Triple Reduction Gearbox

Shaft Bearing Housing Bore Diameter (in)

Shaft S1 938/932 8.377 Shaft S2 H 239640/239610 12.601 Shaft S3 EE 295110/295193 19.254 Shaft S4 HM 259048/259010 17.629

Differential Gearbox

The representation of tangential, separating, and axial forces acting on gears of all

the shafts are shown in Figure 3.7. End 1 and 2 and the direction of shaft rotation are also

38

Figure 3.7 Gear Arrangement and Forces in the Differential Gearbox

39

represented in Figure 3.7. The lengths A, B, and O for all the shafts are presented in

Table 3.13.

The input shaft s1 of the differential gearbox has driving herringbone gear at the

shaft center. The shaft rotates at 870 rpm and takes a torque of approximately 10800 lb-

in.

The differential setup being on intermediate shafts s2 and s3 the calculation of

reaction loads is done by superposition of loads from the herringbone, helical, and bevel

gears mounted on them. The reaction due to helical gears can be calculated from Case I

of the program in Appendix A. The load from the herringbone gear involves an overhung

mounting and the reaction can be calculated using the program in Appendix B. The

contribution of loads from bevel gears on the inner end of the shaft has to be added to the

outer ends. The calculation of loads from bevel gears is given in Appendix C. The shaft

rotates at 150 rpm with a torque of approximately 54000 lb-in.

The output shafts s4 and s5 are both cases of straddle mounting and their reactions

can be calculated using Case I of Appendix A. However on the right end shaft, reaction

load at End 1 is transmitted to the outer side of the gearbox housing. Similarly on left end

shaft, reaction load at End 2 is transmitted to the outer side of the gearbox housing. The

torque on the shaft is approximately 54000 lb-in and the rotation is at 174 rpm. The

summary of reaction loads and bearing detail follow in Table 3.14 and Table 3.15. The

angles in the radial reactions are referenced using 0o as the positive X axis.

40

Table 3.13 Distance between Pinion or Gear and Ends for Input and Output Shafts

Shaft Distance A (in)

Distance B (in)

Distance O (in)

Input s1 9.875 9.875 - Intermediate s2,s3 2.875 2.875 4.125 Output s4,s5 2.875 2.875 -

Table 3.14 Reaction Loads of Differential Gearbox

Shaft

Radial reaction at end 1 (lb)

Radial reaction at end 2 (lb)

Axial reaction at end 1 (lb)

Axial reaction at end 2 (lb)

CW rotation Shaft s1

o3 6710367.2 ∠×

o3 6710367.2 ∠× 0 0

CCW rotation Shaft s1

o3 29310367.2 ∠×

o3 29310367.2 ∠×

0 0

CW rotation Shaft s2,s3

o3 33310450.6 ∠×

o3 33310450.6 ∠× 3070 3070

CCW rotation Shaft s2,s3

o30523.878 ∠ o3 20010200.2 ∠× 120 120

CW rotation Shaft s4,s5

o3 6110026.3 ∠× o3 6110026.3 ∠× 0 0

CCW rotation Shaft s4,s5

o3 21510630.4 ∠×

o3 21510630.4 ∠× 310475.1 × 310475.1 ×

Table 3.15 Bearing Details for Differential Gearbox

Shaft Bearing Housing Bore Diameter (in)

Input Shaft s1 SKF-NJ-313 5.512 Intermediate Shaft s2,s3 46790/46720 10.002 Output Shaft s4,s5 JM 624649/624610 7.087

41

CHAPTER 4

FINITE ELEMENT MODELING AND ANALYSIS

This chapter describes modeling, meshing, loading, and solving the FE models of

the gearbox housings. FEM is a numerical method widely used to solve engineering

problems. In this method of analysis, a complex region defining a continuum is

discretized into simple geometric shapes called finite elements. A displacement function

is associated with each finite element. The finite elements are interconnected at points

called nodes. The behavior of each node can be determined by using the properties of the

material. The total set of equations describing the behavior of each node gives a series of

algebraic equations expressed in matrix notation. Solution of these equations gives the

nodal degrees of freedom in the structure. Stresses are calculated using derivatives of

displacements. The evaluation of stresses requires more refined models. The type and

complexity of a model is dependent on the type of results required. The reader can refer

several texts for the fundamentals and understanding of FEM. Logan [11] and Cook [12]

present a comprehensive background of FEM and its applications.

The geometries of the triple reduction and differential gearboxes are shown in

Figure 4.1 and Figure 4.2. The overall dimensions of all the plates of the triple reduction

and the differential gearboxes are shown in Table 4.1 and Table 4.2. In the initial

analyses, the two halves of these gearboxes were assumed to be integrally connected.

Therefore, the bolted connection was not specifically modeled. This assumption lead to a

linear structural analysis of the gearbox housings. In the later analyses, the interface

42

Figure 4.1 Geometric Model of Triple Reduction Gearbox

Figure 4.2 Geometric Model of Differential Gearbox

43

Table 4.1 Plate Dimensions of the Triple Reduction Gearbox

Plate Dimensions (l x b x h) (in)

l/b h/b

Plate A 41.5 × 0.625 × 30 66 48 Plate A1 41.5 × 1 × 28 41.5 28 Plate B1 107 × 0.625 × 41.5 171 66 Plate B2 107 × 1 × 41.5 107 41.5 Plate C1 107 × 0.625 × 17.5 171 28 Plate C2 107 × 1 × 18.75 107 18.7 Plate D1 107 × 4.625 × 12.5 23 2.7 Plate D2 107 × 4.625 × 12.5 23 2.7 Base Plate 107 × 9.25 × 4 11 0.43 Stiffeners 1, 3 and 5 15.25 × 0.625 × 1 24 1.6 Stiffeners 2 and 4 17.5 × 1 × 1 17.5 1 Stiffeners 6 to 10 15.5 × 0.75× 3.625 21 4.8

Table 4.2 Plate Dimensions of the Differential Gearbox

Plate Dimensions (l x b x h) (in)

l/b h/b

Plate A 13.5 × 0.375 × 18.9375 36 50.5 Plate A1 12.5 × 0.5 × 18.9375 25 38 Plate B1 52 × 0.375 × 18.9375 139 50.5 Plate B2 52 × 0.5 × 18.9375 104 38 Plate C1 52 × 0.375 × 7.5 139 20 Plate C2 52 × 0.5 × 7.75 104 15.5 Plate D1 52 × 3.1235 × 6 16 2 Plate D2 52 × 3.1235 × 6 16 2 Base Plate 52 × 18.9375 × 1.5 2.7 0.08 Stiffeners 1 and 2 9.0625 × 0.75 × 1.1875 12 1.6 Stiffeners 3 to 6 6.5 × 0.5 × 3.1227 13 6.2

between the two halves and the bolted connection was modeled to better understand and

interpret deflection and stresses. The gap elements used to model the interface between

the two halves lead to a nonlinear analysis.

44

Creating and Meshing the FE Models

Modeling is based on a conceptual understanding of the physical system and

judgement of the anticipated behavior of the structure. A model is an assembly of finite

elements, which are pieces of various sizes and shapes. The element aspect ratio, which

represents the ratio of the longest and the shortest dimensions in an element, should

ideally be kept close to unity. The element skewness should also be avoided by keeping

the corner angles in quadrilateral elements close to 90o. A suitable mesh should minimize

the occurrences of high aspect ratio and excessive skewness. In addition, the mesh must

have enough elements to provide accurate results without wasting time in processing and

in interpreting the results.

Geometric modeling and meshing of these gearboxes with suitable elements and

optimum degrees of freedom was an iterative and challenging process. First, a coarse

mesh was made and the overall response of a structure was evaluated. In a 2-D case, a

fine mesh should be used only where stress changes are rapid. In 3-D meshing, abrupt

changes in shape could force the use of finer mesh over the entire structure depending on

the structure geometry. Finer meshes were made for the gearboxes to interpret deflections

and stresses more accurately and to check the convergence of the solutions. Modeling and

meshing was done using the preprocessor in ANSYS. The following subsections describe

the element types used to construct the FE models of the gearboxes. The element

description is taken from the ANSYS Element Manual [13].

45

2-D Structural Element (Plane 42)*

This element type is used for 2-D modeling of solid structures. It has four nodes

having two degrees of freedom at each node, translation in the nodal X and Y directions.

The geometry, node locations, and element coordinate system for this element are shown

in Figure 4.3.

The Plane 42 element was used to model the areas of plates D1 and D2. Then the

elements were extruded to create 3-D elements (Solid 45). The original 2-D elements

were then deleted.

Elastic Shell Element (Shell 63)*

This element type is suited for modeling thin shell structures. Shell 63 has both

membrane and bending capabilities. The element has six degrees of freedom at each

node, translation in the nodal X, Y, and Z directions and rotations about the nodal X, Y,

and Z axes. The geometry, node locations, and the element coordinate system for this

element are shown in Figure 4.4.

Plates A, A1, B1, B2, C1, and C2 shown in Figure 4.3 and Figure 4.4 were

modeled using these thin shell elements. In Table 4.1, the ratio of the smallest inplane

dimension to the plate thickness for plates A, A1, B1, B2, C1, and C2 are in the range of

18 to 171.Therefore, these plates were considered thin.

* The ANSYS element type designation is given in the parentheses.

46

Figure 4.3 Two Dimensional Solid Structural Element (Plane 42) [13]

Figure 4.4 Elastic and Plastic Shell Element (Shell 63 and Shell 43) [13]

Plastic Shell Element (Shell 43)*

This element type is suited to model moderately thick shell structures. It has six

degrees of freedom at each node, translations in the nodal X, Y, and Z directions and

rotations about the nodal X, Y, and Z axes. The geometry, node locations and the

coordinate system for plastic shell and elastic shell are identical.

The stiffeners for both the gearboxes were modeled using these thick shell

elements. In Table 4.1, the ratios of the smallest inplane dimension to the plate thickness

for the stiffeners lie in the range of 24 to 1. Therefore, the stiffeners were modeled using

thick shell elements.

* The ANSYS element type designation is given in the parentheses.

47

3-D Structural Solid Element (Solid 45)*

This element type is used for the 3-D modeling of solid structures. It is defined by

eight nodes each node has three degrees of freedom, translations in the nodal X, Y, and Z

directions. The geometry, node locations, and the element coordinate system for this

element are shown in Figure 4.5.

The thick plates D1 and D2 and the base plate for both the gearboxes were

modeled with these elements. Plates D1 and D2 are 4.625 in thick. In addition, the plates

are located where the shafts enter the gearbox. Excessive deflections could cause

undesirable misalignment of the shafts and bearings. High deflections could also cause

some oil leakage. Therefore, these plates were carefully discretized using Solid 45

elements.

3-D Structural Solid Element (Solid 92)*

This element type has quadratic displacement behavior and is well suited to model

irregular meshes. It is defined by ten nodes each node has three degrees of freedom,

translations in the nodal X, Y, and Z directions. The geometry, node locations, and the

coordinate system for this element are shown in Figure 4.6.

The complex irregular geometry near the intermediate shaft of the differential

gearbox could not be modeled using Solid 45 elements. Therefore, Solid 92 elements

were used to model the irregular geometries.

* The ANSYS element type designation is given in the parentheses.

48

Figure 4.5 Three Dimensional Structural Solid Element (Solid 45) [13]

Figure 4.6 Three Dimensional Structural Solid Element (Solid 92) [13]

Merging Solid and Shell Elements

Solid elements are used for structural components when the thickness is

comparable to the other two dimensions. Shell elements can replace the solid elements

when the thickness is small compared to the other two dimensions. The use of shell

elements significantly reduces the required degrees of freedom and computation time for

a complicated model. According to the Finite Element Handbook [14], a shell-solid

49

interface should be created sufficiently far from the region of interest. The following

example illustrates the issues in shell-solid connections.

A solid block attached to a thick plate is shown in Figure 4.7. This system is

modeled using 3-D solid elements. The same structure can be modeled using a

combination of solid and shell elements as shown in Figure 4.8. The shell elements share

the same nodes as the solids on the interface. In ANSYS, the nodes common to the solid

and shell elements can be merged [15]. The nodes corresponding to the shell elements on

the interface in Figure 4.8 have been merged with coincident nodes of the solid elements.

Thus, the nodes of the solid elements with the translation degrees of freedom are shared

by the shell elements. The nodal forces corresponding to the translation degrees of

freedom will be transmitted from shell elements to the solid elements. However, special

constraint equations are imposed on the common nodes to transmit the nodal moments

corresponding to the rotational degrees of freedom of the shell to the solids. Hence, the

rotation of the shell is coupled with the translation of the solid and the nodal moments are

also transmitted. The triple reduction and differential gearbox geometric models were

discretized using the combination of solid and shell elements for finite element analysis.

A summary of individual model geometry, loads, solution, and results follows in the

sections ahead.

50

Figure 4.7 Solid Element Model

Figure 4.8 Combined Solid and Shell Element Model

51

Triple Reduction Gearbox Model Geometry and Loads

The triple reduction gearbox was first modeled half with a coarse mesh and after

interpretation of results the mesh was refined. After refinements in the region of higher

deflections and stresses, when significant changes in the results did not occur, the mesh

was expanded to model the second half of the gearbox. All the load cases were solved on

the complete model. The following subsections describe the individual models

constructed and loaded.

Model I

Using geometric symmetry, half of the triple reduction gearbox was built in

Model I. The gearbox is geometrically symmetric; however, the loading is not symmetric.

Therefore, Model I was only constructed to identify the acceptable mesh required for half

the gearbox. Thereafter, the geometry was expanded to the other half of the gearbox in

subsequent models. The discretized geometry with 14375 elements and 18102 nodes is

shown in Figure 4.9. Constraint equations were written for the nodes at the solid-shell

interface.

The base plate of the gearbox was completely constrained. Symmetry boundary

conditions were applied at the open end of the gearbox on the plane of geometric

symmetry. The loading represented End 1 of the gearbox when End 1 of the input shaft

rotated clockwise. To simplify the application of loading, the radial loads were applied as

52

Figure 4.9 Finite Element Model of the Triple Reduction Gearbox with Representative Loads

pressure on nodes in a 90o arc. The region on which these loads acted was determined

after the mesh near the bearing surface of the holes was complete. The axial loads were

applied as nodal forces on all nodes on the bearing surface of the holes. The finite

element model with all loads and constraints is shown in Figure 4.9. The radial load

distribution for this load case is shown separately in Figure 4.10. The actual loads used

are summarized in Table 3.12.

53

Figure 4.10 Expanded Radial Load Distribution from Figure 4.9

Model II

Model II was a refinement of Model I. It had 17253 elements and 21395 nodes.

The mesh on plates D1 and D2 and the stiffeners was refined but the connectivity of

plates D1 and D2 with the other plates forced the use of finer mesh on the entire

structure. The refined mesh was a check for the convergence of the solution. The same

loads as in Model I were reapplied at the new node locations. The results from Model I

revealed that the behavior of the housings did not differ when the constraint equations for

the shell and solid connection were not written. For the geometry of the gearbox housings

the loading did not transmit significant moments on the housing. However, the use of

constraint equations was very expensive in terms of computation time without significant

changes in results. Hence in Model II and subsequent models the constraint equations for

the shell-solid interface were not written.

54

Model III

The solid geometry of Model III differed from the geometry of Models I and II. A

small ring along the circumference of the bearing holes was added in the geometry of

Model III. The radial loads were applied in the 90o arc similar to Models I and II. The

application of axial loads was done differently for this model. The axial loads were

applied as pressure on the inner ring along the circumference of the bearing holes. The

discretized model geometry with radial loads is shown in Figure 4.11. The axial loads

applied on the inner ring are shown in Figure 4.12.

Figure 4.11 Finite Element Model of Triple Reduction Gearbox with Inner Ring

55

Figure 4.12 Axial Load on the Inner Ring in Model III

Model IV

Model IV includes the entire geometry of the gearbox. The far end of the gearbox

was modeled using shell elements for all plates. The use of shell elements reduced the

number of degrees of freedom. The near end plates D1 and D2 and the base plate were

modeled using solid elements. The loads on the far end were applied and their

contribution to the overall deflections on the near end was accounted. Model IV had

21890 elements and 26024 nodes. The discretized model geometry is shown in Figure

4.13 and Figure 4.14.

56

Figure 4.13 Finite Element Model of Model IV

Figure 4.14 Finite Element Model of Model IV with Shell Elements on the Far End

57

In the load case solved with Model IV, nodal loads on the far end of the gearbox

were the loads on End 1 when the input shaft rotated counterclockwise. The loads on the

near end were the loads on End 2 when End 1 on the input shaft rotated

counterclockwise. These loads are summarized in Table 3.12.

Model V

Model V was a refinement of Model IV. Model V has a finer mesh on the top and

side plates than Model IV. The model has 23739 elements and 27875 nodes and is shown

in Figure 4.15. Model V was used for further analysis of the triple reduction gearbox

housing. Model V was to be solved for the eight different load cases in Table 4.3. The

loads used are summarized in Table 3.12.

58

Figure 4.15 Finite Element Model of Model V with Radial Loads

Table 4.3 Description of Load Cases in Model V

Load Case Description of applied loads

Case I Loads on End 2 when input shaft rotates CCW Case II Loads on End 1 when input shaft rotates CCW Case III Loads on End 2 when input shaft rotates CW Case IV Loads on End 1 when input shaft rotates CW Case V Only radial loads on End 1 when input shaft rotates CW Case VI Only axial loads on End 1 when input shaft rotates CW Case VII Only radial loads on End 2 when input shaft rotates CW Case VIII Only axial loads on End 2 when input shaft rotates CW

59

Differential Gearbox Model Geometry and Loads

The differential gearbox drives the triple reduction gearboxes on both sides and

ensures equal load distribution to both the triple reduction gearboxes as mentioned in

Chapter 1. The complete gearbox was modeled using the combination of shell and solid

elements. Later, the gearbox model was refined to check the convergence of the solution.

Model SDI I

Model SDI I was the first model of the differential gearbox. This gearbox weighs

about eighteen times less than the triple reduction gearbox. Since the differential gearbox

is much smaller than the triple reduction gearbox, the complete model of this gearbox

was made. Unlike the triple reduction gearbox, the far end of the differential gearbox end

was also modeled using solid elements. The finite element model had 14391 elements

and 18048 nodes. As noted before, the irregular geometry on plates D1 and D2 in these

differential gearbox models was meshed using 3-D structural solid elements. The

discretized model is shown in Figure 4.16. The loads listed in Table 3.15 were applied.

The loads on the near end represented the loads on End 2 when End 1 on the input shaft

rotated clockwise. The loads on the far end were the loads on End 1 for the same rotation.

60

Figure 4.16 Finite Element Model of Model SDI I

Model SDI II

Model SDI II had a finer mesh than SDI I. The model had 20314 elements and

25326 nodes and is shown in Figure 4.17. The loads presented in Table 3.15 were

applied.

61

Figure 4.17 Finite Element Model of Model SDI II

Nonlinear Analysis

After an understanding of the overall behavior of the housings it was important to

study the behavior on the interface between plates D1 and D2 which were modeled

integrally in the analysis so far. The results on the interface could indicate whether the

deflections contributed to any undesirable misalignment of the shafts and bearings.

Therefore, the interface between plates D1 and D2 in the gearbox was modeled using gap

elements, which eventually made the analysis nonlinear. A general description of

nonlinearities is given by Cook [12].

62

Gap and contact nonlinearity was used to model the connection between

components of the gearbox housing in this study. In some problems, two structures or

parts of a structure may make contact when a gap closes, may separate after being in

contact, or may slide on one another with friction. Part 1 and 2 in Figure 4.18 can be

modeled using gap elements. Nodes on adjacent surfaces are connected using gap

elements. A very small gap is defined between the two parts to specify the gap direction.

No forces are exerted when there is a gap between parts. Normal and shear forces

proportional to the spring stiffness of the gap element act on the interface if the gap

closes when loads are applied. This gap and contact nonlinearity defines the interface

between plates D1 and D2 in Figure 4.1 and Figure 4.2 in the triple reduction and

differential gearboxes.

Nonlinear problems require iterative solutions and longer run times than linear

problems. As an example, a simple nonlinear block model of two separate blocks with an

interface between them and a bolt simulation was first modeled and solved. Beam

elements were used to simulate the bolts and gap elements were used to model the

interface.

Figure 4.18 Two parts of a Finite Element Model with Contact [12]

63

3-D Elastic Beam Element (Beam 4)*

The 3-D beam element has six degrees of freedom at each node, translation in the

nodal X, Y, and Z directions and rotation about the nodal X, Y, and Z directions. Beam 4

has tension, compression, torsion, and bending capabilities. The geometry, node

locations, and the element coordinate system for this element are shown in Figure 4.19.

Beam elements were used both to simulate a bolt and to model a stiff region at the bolt

locations in the structure.

3-D Point to Point Contact Element (Contac 52)*

The Contac 52 element represents two surfaces which may maintain or break

contact and may slide relative to each other. This element has three degrees of freedom at

each node, translation in the X, Y, and Z directions. A Contac 52 element has two nodes,

and the input includes stiffness and initial gap. The gap direction can be specified by a

very small predefined gap between the two surfaces. The geometry and node locations

are shown in Figure 4.20.

* The ANSYS element description is given in the parentheses

64

Figure 4.19 3-D Elastic Beam Element (Beam 4) [13]

Figure 4.20 3-D Point to Point Contact Element (Contac 52) [13]

65

Block Model Geometry and Loads

Before modeling and solving the complicated gearbox geometry, a simple

nonlinear block model of two separate halves with an interface and bolt simulation was

modeled and solved. The results of this model were reviewed carefully to understand the

bolt simulation on the interface. This analysis with the simple model served as a

guideline to perform nonlinear analysis of the gearbox housing geometry. The guidelines

given by Mr. Joe Saxon, Meritor Light Vehicle Systems, Inc., Gordonsville, TN, were

used to simulate the bolted connection [16].

Blocks M and N shown in Figure 4.21 were modeled using 3-D solid elements

(Solid 45).

Figure 4.21 Block BMN Geometry and Loads

66

The interface was modeled using gap elements (Contac 52). The bolt was modeled using

a beam element (Beam 4). This complete geometry was referred to as BMN.

The gap elements were created between each node on the two adjacent surfaces.

To define the gap direction, an initial gap was given as 1 × 10-5 in. A network of beam

elements was used to transfer forces from the bolt to the adjacent material (see Figure

4.21). The network was created using beam elements (Beam 4) with a cross-sectional

area of 2 in2. This cross section is slightly larger than the cross-sectional area of the bolt.

The modulus of elasticity E for the beams was given as 30 × 107 psi. Points A and B in

Figure 4.21 are the centers of the bolt on the top half and the bottom half, respectively.

Rigid constraint equations were written for the nodes surrounding the center node on both

halves. The translation nodal degrees of freedom ux and uz and the rotational degrees of

freedom rx, ry, and rz were coupled for the center nodes A and B.

The bolt preload was applied as seen in Figure 4.21 on nodes A and B. This

preload was applied to simulate the cone frustum with a compression zone in the bolted

region. The bolt preload Fi, for a given bolt size and material was calculated using the

procedure given in Shigley [17]

pi FF 75.0= , (4.1)

where Fp is the proof load

ptp SAF = . (4.2)

Here, Sp is the proof strength of the given material and At is the tensile-stress area. For the

given bolt material of SAE Grade 5 and a bolt size of 211 in, the proof strength and

67

tensile-stress area from Shigley are 74 kpsi and 1.405 in2, respectively [17]. The bolt

preload Fi, using Equations 4.1 and 4.2 was calculated to be 77977.5 lb.

The results of this model and the calculated preload were then used to define a

constraint equation for the nodal degree of freedom uy between nodes A and B. After

writing the constraint equation for uy, other loads were applied to the block model and the

solution obtained.

Triple Reduction Gearbox Model VI

After experimenting with the small block model, the actual gearbox model was

made. The interface between plates D1 and D2 was modeled using Contac 52. The

presence of the seven bolts, shown in Figure 4.1, was simulated using beam elements

(Beam 4) and constraint equations. The complete model geometry, referred to as Model

VI, is shown in Figure 4.22. The model had 16837 elements and 20360 nodes. The bolt

material was SAE Grade 5 and the bolt size was 211 in. Using Equations 4.1 and 4.2, the

material proof strength and tensile-stress area, the bolt preload for these bolts was

calculated to be 77977.5 lb. Unlike the other models, Model VI was solved once with the

constraint equations for shell-solid interface and the second time without these constraint

equations. As explained for the simple block geometry, the constraint equations for the

bolt simulation were written to form a network of beams. The nodal degrees of freedom

ux, uz, rx, ry, and rz for the bolt center nodes were coupled and the bolt preload was

68

Figure 4.22 Triple Reduction Gearbox Model VI with Interface and Bolt Preload

applied on the bolt center nodes. The results of this load case were used to write the

special constraint equations for uy. Then, all other loads were applied to solve the model.

The load case solved for nonlinear analysis represented the loads on End 1 when input

shaft rotated clockwise. End 1 loads for input shaft rotating clockwise is the worst

loading condition for the triple reduction gearbox. The loads used are presented in Table

3.12.

69

Summary of Assumptions in the Analysis of the Gearboxes

The assumptions made in this thesis are summarized below.

• The actual loading in the gearboxes was idealized using statically equivalent loads.

• The geometry of the gearbox housings was simplified to accommodate the analysis

within the limitations of the ANSYS University High Option (32000 elements and

nodes).

• Fillets, bearing retainers, fasteners, oil level indicators, and other small components

were not modeled.

• The top and bottom halves of the gearboxes were considered integral for the linear

analysis.

• Perfect welding was assumed between the structural components.

• The weight of the gearbox assembly, which was approximately 18000 lb for the triple

reduction gearbox and approximately 1200 lb for the differential gearbox, was

neglected.

• The axial forces were applied as nodal forces around the bearing holes and the radial

forces were applied as pressure on a 90o arc in the bearing holes. These loads are a

simple approximation to the actual contact loads.

• By using gap elements on the interface, the sliding between the two halves of the

housing was ignored in the nonlinear analysis.

• An initial gap of 1 × 10-5 in was given between the two halves to define the gap

direction.

70

• The bolted region of the housing was modeled using beam elements to simulate a bolt

and a network of beams to distribute the forces to adjacent solid elements.

71

CHAPTER 5

FEA RESULTS AND DISCUSSION

The various models discussed in Chapter 4 were solved to obtain deflection and

stresses for different loading conditions. Both the magnitude and location of large

deflection and stresses were important. Steward Machine Company considers a deflection

of 0.020 in or more as significant [20]. A gearbox design with deflections on the order of

0.020 in would require special evaluation. Stresses in Steward Machine Company

gearboxes tend to be small. Steward Machine Company does not have specified factors of

safety for stresses in gearboxes.

Triple Reduction Gearbox Models

Model I was solved for two conditions. First, the moments from the shell

elements were ignored at the shell-solid interface. Second, the constraint equations to

transfer moments across the shell-solid interface were written. The results of Model I

with constraint equations were very similar to the results without constraint equations. In

addition, the use of constraint equations was very time-consuming. Therefore, only the

Model I results without constraint equations are presented. The contour plots for

displacements ux, uy, uz, total displacement usum, and the von Mises stress σeff for Model I

are shown in Figure 5.1 through Figure 5.5. Both displacements and stresses are plotted

using nodal quantities.

72

Figure 5.1 Model I Displacement ux (in)

Figure 5.2 Model I Displacement uy (in)

73

Figure 5.3 Model I Displacement uz (in)

Figure 5.4 Model I Total Displacement usum (in)

74

Figure 5.5 Model I von Mises Stress σeff (psi)

A contour plot of ux is given in Figure 5.1. The maximum value of ux was 0.0024

in. The location for the maximum ux displacement was on the top half of the input shaft

hole (first hole from the right in Figure 5.1). The largest ux displacement on plates A and

A1 was 001.0− in. A contour plot for displacement uy is shown in Figure 5.2. The

maximum value of uy was 0.0035 in. This maximum displacement was on the stiffener on

the second intermediate shaft hole (third hole from the right in Figure 5.2). The

displacement uy on the top plate B1 was 0035.0− in. A contour plot of uz is given in

Figure 5.3. The maximum value of uz was 0.017 in. The location for maximum uz

displacement was on the top of the second intermediate shaft hole. A contour plot of the

total displacement usum is shown in Figure 5.4. The maximum usum and uz displacements

were the same.

75

A contour plot of von Mises stress σeff in psi is given in Figure 5.5. The maximum

value of σeff was 8364 psi. This value is artificially high because the fillets on the sharp

edges of the stiffener were not modeled. The stress values further away from the sharp

edges were approximately 3000 psi. Even for a maximum value of 8364 psi the factor of

safety for the gearbox housing would be four.

Based on the results of Model I, constraint equations were not used for Model II

or any other subsequent gearbox models. The mesh refinement in Model II did not

produce results that were significantly different from Model I. Therefore, the contour

plots for Model II are not presented.

Model III was similar to Model I. Additionally, Model III included a small

circumferential ring at the bearing holes. Axial loads were applied as a pressure on the

inside circumference of the ring. The resulting displacements and stresses were not

significantly different than the results of Model I and II.

The displacements and von Mises stress for Models I-III are compared in Figure

5.6-Figure 5.9 along the top edge of plate D1. The refinement in Model II and the change

in loading on Model III did not change the deflections and stresses. The ux displacements

plotted in Figure 5.6 were all below 0.001 in. The variation in ux was small along the top

edge of plate D1. The uy displacements in Figure 5.7 were larger than the ux

displacements. The maximum uy displacement along the top edge on plate D1 was 0.0022

in. The uz displacements plotted in Figure 5.8 were the largest of all the displacements.

The maximum uz displacement was 0.018 in. The maximum total displacement usum was

also 0.018 in. The location of the maximum uz and usum was on top of the second

76

0.00E+00

4.00E-04

8.00E-04

1.20E-03

1.60E-03

2.00E-03

2.40E-03

0 20 40 60 80 100

Distance along the top edge of plate D1, in

X-d

isp

lace

men

t, in

Model I

Model II

Model III

Figure 5.6 Comparison of Displacement ux in Models I, II, and III

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

0 20 40 60 80 100

Distance along the top edge of plate D1, in

Y-

Dis

pla

cem

nt,

in

Model I

Model II

Model III

Figure 5.7 Comparison of Displacement uy in Models I, II, and III

77

0.00E+00

2.00E-03

4.00E-03

6.00E-03

8.00E-03

1.00E-02

1.20E-02

1.40E-02

1.60E-02

1.80E-02

2.00E-02

0 20 40 60 80 100

Distance along top edge of plate D1, in

Z-D

isp

lace

men

t, in

Model I

Model II

Model III

Figure 5.8 Comparison of Displacement uz in Models I, II, and III

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 20 40 60 80 100

Distance along top edge of plate D1, in

von

Mis

es S

tres

s, p

si

Model I

Model II

Model III

Figure 5.9 Comparison of von Mises Stress in Models I, II, and III

intermediate shaft hole. The von Mises stresses σeff along the top edge of plate D1 are

plotted in Figure 5.9. The maximum stress was 4935 psi around the sharp edge of the

stiffener on top of the second intermediate shaft. These results are consistent with the

78

loading, since for this load case I, the radial loads act upward on the second intermediate

shaft bearing hole and the axial loads pull on the plate D1 and D2.

Models I-III were preliminary models used to determine an acceptable mesh

density for the triple reduction gearbox model. Models I-III do not physically represent

the complete gearbox. Therefore, the complete gearbox geometry was built in Models IV

and V.

Model IV was the first meshed geometry of the complete gearbox. Model V was a

uniformly refined mesh of the complete gearbox. The contour plots of Model IV are not

presented because the displacements and stresses in Models IV and V were almost

identical.

The contour plots for nodal displacements ux, uy, uz, total displacement usum, and

the von Mises stress σeff for Model V on End 1 side (the near end for load case IV) when

input shaft rotates clockwise (load case IV) are shown in Figure 5.10 through Figure

5.13. End 2 (the far end for load case II) of the gearbox was not modeled complete like

the near end. Therefore, the results obtained on End 2 (the far end) are ignored.

A contour plot of ux for Model V is given in Figure 5.10. The location of the

maximum displacement was on the side plates A and A1. The maximum magnitude of ux

was 0.0164 in. By comparison, the maximum ux in the half Model I was only 0.001 in.

The magnitude of displacement ux is larger than in Models I-III because Model V does

not have the symmetry boundary constraints that were applied to Models I-III.

A contour plot of uy is shown in Figure 5.11. The maximum uy was –0.038 in on

the top plate B1. In Model I, uy on the top plate B1 was ten times less than uy in Model V.

79

Figure 5.10 Model V (Load Case IV) Displacement ux (in)

Figure 5.11 Model V (Load Case IV) Displacement uy (in)

80

Figure 5.12 Model V (Load Case IV) Displacement uz (in)

Figure 5.13 Model V (Load Case IV) von Mises Stress (psi)

81

The top and bottom plates in Model V were not subject to the symmetry constraint

applied in Models I-III; therefore, the magnitude of displacement uy also increased.

However, since this maximum displacement was not on or around the plates D1 and D2 it

was not considered significant.

A contour plot of uz is given in Figure 5.12. The magnitude of the maximum uz

displacement was 0.018 in and the location was on top of the second intermediate shaft

hole (the third hole from the right side in Figure 5.12). The uz displacements on the top

plate D1 were higher in comparison to the bottom plate D2. The displacements decreased

gradually on both sides of the second intermediate shaft hole.

The maximum total displacement of 0.038 in was on the top plate. The maximum

displacement and location was the same as the maximum uy displacement in Figure 5.11.

The maximum total displacement on the near end was 0.018 in on top of the second

intermediate shaft hole. This magnitude and location was the same as in the uz

displacement plot in Figure 5.12. The total displacement contours are not plotted.

The contour plot of the von Mises stress σeff is shown in Figure 5.13. The

maximum stress value on the near end was 9000 psi. As with the previous models, Model

V does not include fillets at the base of the stiffeners. Therefore, the actual maximum

stress was probably lower than 9000 psi.

For comparing the results at specific locations in the gearbox model, various paths

were defined on the gearbox housing. The paths defined on the triple reduction gearbox

are shown in Figure 5.14. Path AB is along the top edge of plate D1. The path GH is on

82

Figure 5.14 Defined Paths on the Triple Reduction Gearbox

top of the second intermediate shaft hole along the edge of plate D1. Path CD is along the

side plates A and A1 and path EF is on the top plate B1.

The ux results of Models IV and V were compared along path CD in Figure 5.15.

The values of ux for Models IV and V along path CD were almost identical. The

maximum ux displacement along path CD was 0.013 in. The uy displacement on Models

IV and V were compared along path EF in Figure 5.16. The maximum uy displacement in

both Models IV and V was -0.034 in on the top plate.

83

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

1.0E-02

1.2E-02

1.4E-02

0 10 20 30 40 50 60

Distance along path CD, in

X-

Dis

pla

cem

ent,

in

Model IV

Model V

Figure 5.15 Comparison of X-Displacement in Model IV and V

-4.00E-02

-3.50E-02

-3.00E-02

-2.50E-02

-2.00E-02

-1.50E-02

-1.00E-02

-5.00E-03

50 70 90 110 130 150 170

Distance along path EF, in

Y-d

isp

lace

men

t, in

Model VModel IV

Figure 5.16 Comparison of Y-Displacement in Model IV and V

The results of Models IV and V were not significantly different. Model V had a

more uniformly refined mesh compared to Model IV. Therefore, all further linear

analyses of the triple reduction gearbox was performed using Model V. The load cases

described in Table 4.3 were solved using Model V.

84

After obtaining the solution at End 1 when input shaft rotated clockwise in load

case IV, Model V was solved for load case III representing the other end, End 2 (the near

end for load case III) when input shaft rotated clockwise. The results of ux and uy were

similar to the results on End 1 in load case IV when input shaft rotated clockwise. These

results are not presented. The uz displacement plot in Model V on End 2 side (the near

end) is shown in Figure 5.17. The maximum uz for load case III was 0.022 in. The

location of the maximum uz was on top of the output shaft hole (first hole from left end in

Figure 5.17). The maximum uz on the shaft hole was 0.021 in. The total displacement

usum on the shaft hole was also 0.021 in. This result was consistent with the maximum

axial loads acting on the output shaft for load case III. The displacement on the output

shaft is more than 0.02 in and may be of concern to Steward Machine Company.

Figure 5.17 Model V (Load Case III) Displacement uz (in)

85

After the load cases where the input shaft rotates clockwise were solved, the load

cases when the input shaft rotates counterclockwise were solved. When input shaft

rotated counterclockwise, the loads on End 1 and End 2 of the gearbox housings were

solved in load cases II and I, respectively. For the counterclockwise rotation of the input

shaft, the ux and uy displacement plots were similar to the results for clockwise rotation of

input shaft. Therefore, the ux and uy displacement plots are not presented. The uz

displacement plot for load case II in Figure 5.18 has a maximum value of 0.022 in. The

location of the maximum uz is on top of the output shaft hole where the axial loads act in

the Z-direction for load case II. The maximum total displacement usum for load case II on

the output shaft hole was 0.021 in and may be of concern to Steward Machine Company.

Figure 5.18 Model V (Load Case II) Displacement uz (in)

86

The uz plot for load case I is presented in Figure 5.19. The maximum uz displacement was

0.017 in for load case I. The maximum uz displacement was located on top of the second

intermediate shaft hole where the maximum axial loads were acting for load case I. The

maximum total displacement usum was also 0.017 in at the same location.

The results of Model V showed that the uz displacement was maximum on the

second intermediate or the output shaft depending on the load case. For the input shaft

rotating clockwise, the maximum uz displacement on one end was on top of the second

intermediate shaft hole and the maximum uz displacement on the other end was on top of

the output shaft hole. These results show that as the input shaft rotates clockwise, the

axial loads pull outward on the input, first intermediate, and the second intermediate shaft

holes on one end. On the other end the axial loads pull outward on the output shaft hole.

Figure 5.19 Model V Load (Case I) Displacements uz (in)

87

Thus, the gearbox is being pulled apart from both shaft ends, pulled down on the top plate

B1, and pulled in on the side plates A and A1.

Differential Gearbox Models

The differential gearbox was modeled using two different meshes as explained in

Chapter 4. SDI I was the coarse mesh of the differential gearbox housing. SDI II was a

uniformly refined mesh. The contour plots of SDI I are not presented because the results

did not differ significantly from the results of SDI II. The SDI II contour plots of nodal

displacements ux, uy, uz, total displacement usum, and the von Mises stress σeff for a

clockwise rotation of the input shaft (load case I) are shown in Figure 5.20 through

Figure 5.26.

The contour plots of displacement ux are given in Figure 5.20 and Figure 5.21 for

End 2 (the near end) and End 1 (the far end), respectively. The maximum ux displacement

was 0.0007 in. The location of maximum ux can be seen in Figure 5.21 near the

intermediate shaft hole on End 1 side. Unlike the triple reduction gearbox, the axial loads

in the differential gearbox were not the major cause of the displacements. The maximum

ux displacement was caused by the radial loads acting on the intermediate shaft at that

location.

The contour plot for displacement uy is shown in Figure 5.22. The maximum

positive uy displacement of 0.0009 in occurred on the stiffener between the input and

88

Figure 5.20 SDI II (load case I) Near End 2 Displacement ux (in)

Figure 5.21 SDI II (load case I) Far End 1 Displacement ux (in)

89

Figure 5.22 SDI II (Load Case I) Far End 1 Displacement uy (in)

Figure 5.23 SDI II (Load Case I) Near End 2 Displacement uz (in)

90

Figure 5.24 SDI II (Load Case I) Far End 1 Displacement uz (in)

Figure 5.25 SDI II (Load Case I) Far End 1 Total Displacement usum (in)

91

Figure 5.26 SDI II Far End 1 von Mises Stress (psi)

intermediate shaft on End 1 side (the far end). The maximum negative displacement of

0017.0− in occurred on the top plate B1.

The contour plots of displacement uz are given in Figure 5.23 and Figure 5.24 for

End 2 (the near end) and End 1 (the far end), respectively. The maximum uz on both the

near and far end was 0.0037 in. The location of the maximum uz displacement was on top

of the intermediate shaft hole on both ends. The maximum uz on the intermediate shaft

hole was 0.0034 in.

The contour plot for total displacement usum is shown in Figure 5.25. The

maximum total displacement of 0.0038 in was on top of the intermediate shaft hole. The

maximum usum displacement on the intermediate shaft hole was 0.0035 in.

92

The von Mises stress plot for End 1 (the far end) is shown in Figure 5.26. A

maximum stress of 6102 psi was found near the stiffener between the input and

intermediate shaft. The actual stress value should be lower because the fillets on the sharp

edges of the stiffener were not modeled. Even for a maximum value of 6102 psi, the

factor of safety for this gearbox housing was six.

The contour plots for nodal displacements ux, uy, uz, and the von Mises stress σeff

for SDI II for a counterclockwise rotation of the input shaft (load case II) are shown in

Figure 5.27 through Figure 5.31.

The contour plot of displacement ux on End 2 (the near end) is given in Figure

5.27. The maximum displacement of 0004.0− in occurred on the output shaft where the

radial load on the hole was acting. The plot for ux on End 1 (the far end) is shown in

Figure 5.28. The maximum displacement locations cannot be seen in this plot. However,

the ux displacements around the intermediate shaft shown on the far end side were only

slightly smaller than the maximum value of 0.0004 in on the near end side.

The contour plot for displacement uy is shown in Figure 5.29 for End 1 (the far

end). The maximum positive uy of 0.0005 in was seen on the stiffener on top of the

output shaft hole. The maximum negative uy of 0009.0− in was seen on the top plate B1.

The maximum positive uy was at a location where the radial loads were acting. The

maximum negative uy was caused by the axial loads pulling on the gearbox from both

shaft ends. Therefore, both the radial and axial loads contributed to the displacements in

this gearbox.

93

Figure 5.27 SDI II (Load Case II) Near End 2 Displacement ux (in)

Figure 5.28 SDI II (Load Case II) Far End 1 Displacement ux (in)

94

Figure 5.29 SDI II (Load Case II) Far End 1 Displacement uy (in)

Figure 5.30 SDI II (Load Case II) Near End 2 Displacement uz (in)

95

Figure 5.31 SDI II (Load Case II) Far End 1 Displacement uz (in)

The contour plots for displacement uz are shown in Figure 5.30 and Figure 5.31

for End 2 (the near end) and End 1 (the far end), respectively. The maximum uz on the

near and far end was 0.0017 in. The location of the maximum uz displacement was on top

of the intermediate shaft hole on both ends. The uz displacement was mainly caused by

the axial loads pulling on the shaft ends on both sides.

The total displacement usum had the same magnitude of 0.0017 in and was on top

of the intermediate shaft hole. The contour plot for usum is not shown.

The total displacements in SDI I and II are plotted and compared in Figure 5.32

for load case I. The displacement values are plotted along the top edge of plate D1. The

displacement values did not differ significantly for SDI I and II. The plot shows the

96

0.00E+00

5.00E-04

1.00E-03

1.50E-03

2.00E-03

2.50E-03

3.00E-03

3.50E-03

4.00E-03

4.50E-03

0 10 20 30 40 50 60

Distance along the top edge of plate D1, in

To

tal d

isp

lace

men

t, in

SDI I

SDI II

Figure 5.32 Comparison of Total Displacement in SDI I and SDI II

maximum total displacement usum of approximately 0.0038 in for the refined model SDI

II. The location of the maximum displacement was on top of the intermediate shaft hole.

The von Mises stresses are plotted and compared in Figure 5.33 for load case I in

SDI I and II. The stresses are plotted along the top edge of plate D1. The stresses from

both models were very similar, except at the stiffeners. Along the top edge of plate D1 a

maximum stress of approximately 1600 psi was observed on the sharp edges of the

stiffeners.

In summary, the differential gearbox has smaller displacements than the triple

reduction gearbox. The differential gearbox had smaller magnitude of balanced axial

loads, while the triple reduction gearbox had large unbalanced axial loads. Both axial and

radial loads had approximately equal contributions to the displacements in the differential

97

0

200

400

600

800

1000

1200

1400

1600

1800

0 10 20 30 40 50 60

Distance along the top edge of plate D1, in

Str

ess,

psi

SDI I

SDI II

Figure 5.33 Comparison of Von Mises Stress in SDI I and SDI II

gearbox. By contrast, the axial loads caused most of the displacements in the triple

reduction gearbox. The same observations were true for the stresses in the two gearboxes.

Triple Reduction Gearbox Nonlinear Solution

Model VI was built for the nonlinear analysis as discussed in Chapter 4 and

shown in Figure 4.22. The moments from the shell elements were ignored at the shell-

solid interface. The contour plots for displacements ux, uy, uz, and the von Mises stress

σeff for Model VI are shown in Figure 5.34 through Figure 5.39. Model VI was solved for

End 1 side (the near end) when the input shaft rotated clockwise.

98

Figure 5.34 Model VI Displacement ux (in)

Figure 5.35 Model VI Displacement uy (in)

99

Figure 5.36 Model VI Displacement uy between the First Intermediate and Second Intermediate Shaft Hole (1000 × Magnification)

Figure 5.37 Gap (in) on the Interface between the First Intermediate and Second Intermediate Shaft Holes

100

Figure 5.38 Model VI Displacement uz (in)

Figure 5.39 Model VI von Mises Stress (psi)

101

A contour plot of ux is given in Figure 5.34. The maximum ux displacement was

0.0020 in. The location of the maximum ux displacement was on the top half of the input

shaft hole (first hole from the right in Figure 5.34). The largest ux displacement on plates

A and A1 was 0015.0− in. The maximum ux displacement on the interface was

0007.0− in on the left end of the output shaft in Figure 5.34.

A contour plot for displacement uy is shown in Figure 5.35. The maximum uy

displacement was 0.0038 in. This maximum displacement was on the stiffener on the

second intermediate shaft hole (third hole from the right in Figure 5.35). The

displacement uy on the top plate B1 was 009.0− in.

The uy displacement on the interface between plates D1 and D2 was examined

closely to check for any gap between the two plates. The maximum uy displacement that

caused partial gap on the interface was between the first intermediate and second

intermediate shaft holes. A plot showing the uneven gap between the first intermediate

and second intermediate shaft is shown in Figure 5.36. The line CGD is on the front side

of the gearbox and the line AHF is on the interior side. The uy gap displacement at points

A, B, C, D, E, F, G, and H is shown in Figure 5.37. At points A, B, and C on the side of

the second intermediate shaft, the interfaces separated with a gap of 0.0001 in at A,

0.0007 in at B, and 0.0015 in at C. On the first intermediate shaft hole side at points D

and E, the gap was 0.0013 in and 0.0005 in, respectively. There was no gap on the

interior side at points F and H. In the center the bolt preload kept the two halves from

separating. The displacement on the front side at C, G, and D was more than the

102

displacement on the interior side of the plates. Although the gap was much smaller than

0.02 in, it may be of concern to Steward Machine Company.

A contour plot of uz is given in Figure 5.38. The maximum uz displacement was

0.016 in. The location of maximum uz displacement was on the top of the second

intermediate shaft hole.

The total displacement plot was similar to the uz displacement plot and is not

shown. The maximum total displacement of 0.016 in was on the same location as the uz

displacement in Figure 5.38. On the interface both the halves were being pulled out

because of the outward acting axial loads on the shaft holes.

A contour plot of von Mises stress σeff in psi is given in Figure 5.39. In Model VI,

the explicit modeling of the bolted connection locally increased the stresses. The

maximum stress was 8908 psi. The location of the maximum stress was on the interface

between the first intermediate and the second intermediate shaft. This maximum stress

location was different from the maximum stress location in the linear Model I. By

comparison, the magnitude of stress on the interface between the first intermediate shaft

and the second intermediate shaft in Model I was approximately 5200 psi.

Model VII was made by removing the interface and the bolt simulation from

Model VI to provide an appropriate comparison between the linear and nonlinear models.

Removing the interface and its gap elements eliminated the nonlinearity. The results of

Model VI and VII did not differ significantly in the overall behavior.

For comparing the results of these nonlinear and linear models, various paths

were defined on the interface in the gearbox. The paths defined on the interface are

103

shown in Figure 5.40. Path MN was along the interface between plates D1 and D2. This

path is discontinuous because of the presence of the four shaft holes on the interface.

Paths pq, rs, tu, wx, and yz were transverse lines on the interface to closely examine the

displacements on the interface. A comparison of the linear and nonlinear results along

path MN is shown in Figure 5.41 through Figure 5.44.

The ux displacements, shown in Figure 5.41, are very small. The maximum ux

displacement of 0.0007 in was found near the input shaft hole. The uy displacements

given in Figure 5.42 were larger than the ux values. The maximum uy displacement of

0.0014 in was found on the interface between the first intermediate and second

intermediate shaft holes. The uz displacement shown in Figure 5.43 had the largest

magnitudes, 0.013 in. The location of this displacement was on the interface between the

first intermediate and second intermediate shaft holes. The total displacement of 0.013 in

was also on the interface between the first intermediate and the second intermediate shaft

holes. Though less than 0.02 in, this displacement value and location on the interface

could cause some misalignment on the interface between the two shaft holes.

The von Mises stresses on the interface in Model VI and VII are plotted in Figure

5.44. The maximum stress on the interface was approximately 4000 psi. The stresses in

the linear model were more uniform than the stresses in the nonlinear model. The

discontinuity in stress was attributed to the fact that the nonlinear model included the bolt

and a network of stiffer beam elements on the interface.

104

Figure 5.40 Defined Paths on the Interface in the Triple Reduction Gearbox

-1.0E-03

-8.0E-04

-6.0E-04

-4.0E-04

-2.0E-04

0.0E+00

2.0E-04

4.0E-04

6.0E-04

8.0E-04

0 20 40 60 80 100 120

Distance along path MN, in

X-d

isp

lace

men

t, in

Model VII

Model VI

Figure 5.41 Displacement ux in Model VI and VII

105

1.10E-03

1.45E-03

-1.5E-03

-1.0E-03

-5.0E-04

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

0 20 40 60 80 100 120

Distance along path MN, in

Y-d

isp

lace

men

t, in

Model VII

Model VI

Figure 5.42 Displacement uy in Model VI and VII

1.23E-02

1.34E-02

0.0E+00

2.0E-03

4.0E-03

6.0E-03

8.0E-03

1.0E-02

1.2E-02

1.4E-02

1.6E-02

0 20 40 60 80 100 120

Distance along path MN, in

Z-d

isp

lace

men

t, in

Model VII

Model VI

Figure 5.43 Displacement uz in Model VI and VII

106

0

500

1000

1500

2000

2500

3000

3500

4000

0 20 40 60 80 100 120

Distance along path MN, in

Str

ess,

psi

Model VII

Model VI

Figure 5.44 Von Mises Stress in Model VI and VII

The observations along path MN are true for paths pq, rs, tu, wx, and yz.

Therefore, the plots along paths pq, rs, tu, wx, and yz are not presented.

The overall displacements in the linear and nonlinear analyses are approximately

equal. However, the uneven gap between the first intermediate and second intermediate

shaft holes could not have been detected without the nonlinear analysis.

107

•

CONCLUSIONS AND RECOMMENDATIONS

A successful understanding of the characteristic structural behavior of the gearbox

housing under different loading conditions is attained by conducting the finite element

analysis. The analysis serves as the groundwork to improve the design of the gearboxes

and incorporate desirable changes based on the conclusions derived.

The following four conclusions can be made. The first two conclusions are

focussed on the displacement and stress results of the two gearboxes studied in this

thesis. The last two conclusions are focussed on modeling issues that may be important

for the future modeling of similar designs.

1. In the triple reduction gearbox, the axial loads dominate the overall behavior of the

housing and the axial loads cause approximately ninety percent of the overall

displacements. In the triple reduction gearbox, the maximum total displacement of

0.022 in on the output shaft could cause misalignment of the shaft or bearings. In the

differential gearbox, the axial and radial loads equally contribute to the overall

displacements. The differential gearbox is a balanced configuration and the

magnitude of displacements is at least ten times less than the displacements in the

triple reduction gearbox.

2. The stresses in the triple reduction gearbox range from 0 to 9000 psi. The stresses in

the differential gearbox range from 0 to 6000 psi. With a yield strength of 36000 psi,

108

it is concluded that these gearboxes are strong enough for the application.

3. For both gearboxes, the moments transmitted across the shell-solid interface are small

enough that constraint equations are unnecessary. This observation will save both

modeling effort and computational time in future analyses of similar geometries and

loading.

4. The nonlinear analysis of the triple reduction gearbox was necessary to determine

whether a gap will exist at the interface between the two halves of the housing. An

uneven gap, larger on the outside and nearly zero on the inside occurred between the

first and second intermediate shaft holes. A maximum separation of 0.0015 in was

found in the Y-direction between the first and second intermediate shaft holes. This

gap could not be determined using linear analysis. Therefore, a better understanding

of these gearboxes and similar designs in the future can be obtained only through the

nonlinear analysis.

The following three recommendations are suggested based on the results of this

thesis. The first recommendation focuses on the need for experimental validation, while

the other two recommendations focus on further modeling considerations.

1. The first recommendation is to validate the results of the numerical simulation with

experimental measurements. The most important location for measuring the

displacements is the interface between the first intermediate and second intermediate

shafts where the gap was found. Other locations for measuring the displacements are

on the top edge of the second intermediate and the output shaft holes.

109

2. The relationship between the bolt torque and the interface stresses should be further

analyzed to determine the optimal torque required to prevent separation of the two

halves of the gearbox.

3. To minimize the uy and uz displacements in the triple reduction gearbox, the thickness

of the third stiffener could be increased or an additional stiffener could be added

nearby.

110

BIBLIOGRAPHY

111

1. Steinman, David B., Watson, Sara Ruth, Bridges and Their Builders, Dover

Publications Inc., 1957.

2. Earle Gear and Machine Company, Catalog, Philadelphia, not dated.

3. Steward Machine Company, Birmingham, AL, Photograph, courtesy of Alison, James

1999.

4. Steward Machine Company, Catalog, Birmingham, AL, 1998.

5. ANSYS, Inc., Theory Reference, Release 5.4, Eighth Ed., 1997.

6. Ramamurti, V., Arul Kumar, P.S., Jayaraman, K., “Performance Comparison of Cast

and Fabricated Gearbox Casings” Computers and Structures, Vol. 37, 1990, pp. 353-

359.

7. Bhandari, V. B., Design of Machine Elements, Tata McGraw-Hill Publishing

Company, New Delhi, 1994.

8. Dudley, Darle W., Handbook of Practical Gear Design, Technomic Publishing

Company, Inc., Lancaster, PA, 1994.

9. Dudley, Darle W., ed., Gear Handbook, McGraw-Hill Book Company, Inc., 1962.

10. The Timken Company, Timken Bearing Selection Handbook, 1986.

11. Logan, Daryl L., A First Course in the Finite Element Method Using Algor, PWS

Publishing Company, Boston, 1997.

12. Cook, Robert D., Finite Element Modeling for Stress Analysis, John Wiley, New

Jersey, 1995.

13. ANSYS, Inc., Elements Reference, Release 5.4, Eighth Ed., 1997.

112

14. Kardestuncer, H., ed., Finite Element Handbook, McGraw-Hill Book Company, Inc.,

1987.

15. ANSYS Inc., Node Merge, Internet, http://ANSYS56/docu/catalog/english/ansyshelp/

Hlp_C_NUMMRG.html, March 2000.

16. Saxon, Joe, Meritor Light Vehicle Systems, Inc., Gordonsville, TN, May 2000.

17. Shigley, Joseph. E., Mechanical Engineering Design, Fifth Ed., McGraw-Hill Book

Company, Inc., 1989.

18. ANSYS, Inc., Structural Analysis Guide, Release 5.4, Eighth Ed., 1997.

19. Machinery’s Handbook, Twenty-fifth Ed., Industrial Press, Inc., New York, 1996.

20. Alison, James, Steward Machine Company, Birmingham, AL, telephone conversation

with the author, May 2000.

113

APPENDICES

125

VITA

Sarika ShreeVallabh Bhatt was born in Baroda, Gujarat, India, on September 14,

1976. She graduated from St. Xaviers School, Surat, in 1993. The following August she

entered L.D College of Engineering, Gujarat University, Ahmedabad, and received a

Bachelor of Engineering in Mechanical Engineering in August 1997. She worked with

Birla Cellulosic, Aditya Birla Group till August 1998.

She entered graduate school of Tennessee Technological University in fall 1998,

and is a candidate for Master of Science Degree in Mechanical Engineering.