AN ABSTRACT OF A THESIS FINITE ELEMENT ANALYSIS OF HELICAL AND DIFFERENTIAL GEARBOX HOUSINGS. Sarika S. Bhatt

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1 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 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 u z displacement of 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.

2 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

3 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 ii

4 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 iii

5 DEDICATION This thesis is dedicated to my family. iv

6 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. v

7 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 GEARS AND GEAR FORCE ANALYSIS... 9 Helical and Herringbone Gears... 9 Bevel Gears GEARBOX SPECIFICATIONS AND LOAD CALCULATIONS Triple Reduction Gearbox Differential Gearbox FINITE ELEMENT MODELING AND ANALYSIS Creating and Meshing the FE Models D Structural Element (Plane 42) Elastic Shell Element (Shell 63) Plastic Shell Element (Shell 43) vi

8 vii Page 3-D Structural Solid Element (Solid 45) D Structural Solid Element (Solid 92) Merging Solid and Shell Elements Triple Reduction Gearbox Model Geometry and Loads Model I Model II Model III Model IV Model V Differential Gearbox Model Geometry and Loads Model SDI I Model SDI II Nonlinear Analysis D Elastic Beam Element (Beam 4) D Point to Point Contact Element (Contac 52) Block Model Geometry and Loads Triple Reduction Gearbox Model VI Summary of Assumptions in the Analysis of the Gearboxes FEA RESULTS AND DISCUSSION Triple Reduction Gearbox Models Differential Gearbox Models... 87

9 viii Page Triple Reduction Gearbox Nonlinear Solution CONCLUSIONS AND RECOMMENDATIONS BIBLIOGRAPHY APPENDICES APPENDIX A. 114 APPENDIX B. 120 APPENDIX C. 122 VITA

10 LIST OF TABLES Page Table 3.1 Shaft Data for the Triple Reduction Gearbox Table 3.2 Pinion and Gear Specifications for the Triple Reduction Gearbox Table 3.3 Shaft Data for the Differential Gearbox Table 3.4 Pinion and Gear Specifications for the Differential Gearbox Table 3.5 Direction of Tangential Forces Table 3.6 Direction of Axial Forces [9] Table 3.7 Pinion and Gear Forces in the Triple Reduction Gearbox Table 3.8 Pinion and Gear Forces in the Differential Gearbox Table 3.9 Distance between Pinion or Gear and Ends for Input and Output Shaft Table 3.10 Distance between Pinion or Gear and Ends for Intermediate Shafts Table 3.11 Reaction Loads of the Triple Reduction Gearbox Table 3.12 Bearing Details of the Triple Reduction Gearbox Table 3.13 Distance between Pinion or Gear and Ends for Input and Output Shafts Table 3.14 Reaction Loads of Differential Gearbox Table 3.15 Bearing Details for Differential Gearbox Table 4.1 Plate Dimensions of the Triple Reduction Gearbox Table 4.2 Plate Dimensions of the Differential Gearbox Table 4.3 Description of Load Cases in Model V ix

11 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] Figure 2.2 Herringbone Gears [7] Figure 2.3 Section of Helical Gear [7] Figure 2.4 Components of Tooth Force in Helical Gears [7] Figure 2.5 Straight Bevel Gear [7] Figure 2.6 Section of Bevel Gear [7] Figure 2.7 Pair of Bevel Gear [7] Figure 2.8 Tooth Forces in Bevel Gears [7] Figure 3.1 Sectional View of Triple Reduction Gearbox Figure 3.2 Sectional View of the Differential Gearbox Figure 3.3 Example of Straddle Mounting [9] Figure 3.4 Example of Overhung Mounting [9] Figure 3.5 Flowchart of the Helical Gear Reaction Calculations Figure 3.6 Gear Arrangement and Forces in the Triple Reduction Gearbox x

12 xi Page Figure 3.7 Gear Arrangement and Forces in the Differential Gearbox Figure 4.1 Geometric Model of Triple Reduction Gearbox Figure 4.2 Geometric Model of Differential Gearbox 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] Figure 4.5 Three Dimensional Structural Solid Element (Solid 45) [13] Figure 4.6 Three Dimensional Structural Solid Element (Solid 92) [13] Figure 4.7 Solid Element Model Figure 4.8 Combined Solid and Shell Element Model Figure 4.9 Finite Element Model of the Triple Reduction Gearbox with Representative Loads Figure 4.10 Expanded Radial Load Distribution from Figure Figure 4.11 Finite Element Model of Triple Reduction Gearbox with Inner Ring Figure 4.12 Axial Load on the Inner Ring in Model III 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 Figure 4.15 Finite Element Model of Model V with Radial Loads Figure 4.16 Finite Element Model of Model SDI I Figure 4.17 Finite Element Model of Model SDI II Figure 4.18 Two parts of a Finite Element Model with Contact [12] Figure D Elastic Beam Element (Beam 4) [13]... 64

13 xii Page Figure D Point to Point Contact Element (Contac 52) [13] Figure 4.21 Block BMN Geometry and Loads Figure 4.22 Triple Reduction Gearbox Model VI with Interface and Bolt Preload Figure 5.1 Model I Displacement u x (in) Figure 5.2 Model I Displacement u y (in) Figure 5.3 Model I Displacement u z (in) Figure 5.4 Model I Total Displacement u sum (in) Figure 5.5 Model I von Mises Stress σ eff (psi) Figure 5.6 Comparison of Displacement u x in Models I, II, and III Figure 5.7 Comparison of Displacement u y in Models I, II, and III Figure 5.8 Comparison of Displacement u z in Models I, II, and III Figure 5.9 Comparison of von Mises Stress in Models I, II, and III Figure 5.10 Model V (Load Case IV) Displacement u x (in) Figure 5.11 Model V (Load Case IV) Displacement u y (in) Figure 5.12 Model V (Load Case IV) Displacement u z (in) Figure 5.13 Model V (Load Case IV) von Mises Stress (psi) Figure 5.14 Defined Paths on the Triple Reduction Gearbox Figure 5.15 Comparison of X-Displacement in Model IV and V Figure 5.16 Comparison of Y-Displacement in Model IV and V Figure 5.17 Model V (Load Case III) Displacement u z (in) Figure 5.18 Model V (Load Case II) Displacement u z (in)... 85

14 xiii Page Figure 5.19 Model V Load (Case I) Displacements u z (in) Figure 5.20 SDI II (load case I) Near End 2 Displacement u x (in) Figure 5.21 SDI II (load case I) Far End 1 Displacement u x (in) Figure 5.22 SDI II (Load Case I) Far End 1 Displacement u y (in) Figure 5.23 SDI II (Load Case I) Near End 2 Displacement u z (in) Figure 5.24 SDI II (Load Case I) Far End 1 Displacement u z (in) Figure 5.25 SDI II (Load Case I) Far End 1 Total Displacement u sum (in) Figure 5.26 SDI II Far End 1 von Mises Stress (psi) Figure 5.27 SDI II (Load Case II) Near End 2 Displacement u x (in) Figure 5.28 SDI II (Load Case II) Far End 1 Displacement u x (in) Figure 5.29 SDI II (Load Case II) Far End 1 Displacement u y (in) Figure 5.30 SDI II (Load Case II) Near End 2 Displacement u z (in) Figure 5.31 SDI II (Load Case II) Far End 1 Displacement u z (in) Figure 5.32 Comparison of Total Displacement in SDI I and SDI II Figure 5.33 Comparison of Von Mises Stress in SDI I and SDI II Figure 5.34 Model VI Displacement u x (in) Figure 5.35 Model VI Displacement u y (in) Figure 5.36 Model VI Displacement u y between the First Intermediate and Second Intermediate Shaft Hole Figure 5.37 Gap on the Interface between the First Intermediate and Second Intermediate Shaft Holes... 99

15 xiv Page Figure 5.38 Model VI Displacement u z (in) Figure 5.39 Model VI von Mises Stress (psi) Figure 5.40 Defined Paths on the Interface in the Triple Reduction Gearbox Figure 5.41 Displacement u x in Model VI and VII Figure 5.42 Displacement u y in Model VI and VII Figure 5.43 Displacement u z in Model VI and VII Figure 5.44 Von Mises Stress in Model VI and VII

16 LIST OF SYMBOLS AND ACRONYMS Symbol a b d h a h f i m m n p p a p n r b r m u sum u x, u y, u z z Description center to center distance face width pitch circle diameter addendum dedendum speed ratio transverse module normal module transverse circular pitch axial circular pitch normal circular pitch back cone radius mean radius total nodal displacement components of nodal displacement in global X, Y, Z directions number of teeth z number of teeth on formative gear A o A t D cone distance tensile stress area pitch circle diameter xv

17 Symbol D p F i F p P P a P r P s P t R R b R n S p T Description diameter at midpoint along face width preload proof load resultant force axial force radial force separating force tangential force transverse diametral pitch back cone radius of formative gear normal diametral pitch proof strength torque xvi X, Y, Z global coordinate axes α α n γ ω ψ Γ h.p. transverse pressure angle normal pressure angle pitch angle, pinion angular velocity helix angle pitch angle, gear horsepower

18 AGMA Symbol ASTM CCW CW FEA FEM LH RH SAE American Gear Manufacturing Association Description American Standard of Testing and Materials Counterclockwise Clockwise Finite Element Analysis Finite Element Method Left Handed Right Handed Society of Automotive Engineers xvii

19 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 1

20 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 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.

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

22 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

23 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

24 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 in). Figure 1.5 Model of Fabricated Gearbox [6]

25 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

26 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.

27 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 9

28 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 righthanded 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.

29 11 Figure 2.2 Herringbone Gears [7] A portion of the top view of a helical gear is shown in Figure 2.3. A 1 B 1 and A 2 B 2 are the centerlines of adjacent teeth on the pitch plane. The helix angle ψ is defined as the angle A 1 B 2 A 2 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 A 1 A 2 is the transverse circular pitch p, measured in the plane of rotation. The distance A 1 C is the normal circular pitch p n, measured in a plane perpendicular to the tooth elements. The ratio of p n and p from triangle A 1 A 2 C is p n A1C = = cosψ (2.1) p A A 1 2 or p n = p cosψ. (2.2)

30 12 Figure 2.3 Section of Helical Gear [7] The pitch circle diameter of a helical gear with number of teeth z is zp d =. (2.3) π The transverse diametral pitch R is the ratio of the number of teeth z to the pitch circle diameter d:

31 13 z R =. (2.4) d Comparing Equations 2.3 and 2.4, the product of the transverse circular pitch p and the transverse diametral pitch R is pr = π. (2.5) Substituting Equation 2.5 in to Equation 2.2 leads to R R n =, (2.6) cosψ where R n is the normal diametral pitch. The transverse module m and the normal module m n are defined as the inverse of the transverse diametral pitch R and the normal diametral pitch R n, respectively. Hence, m = 1, (2.7) R and m n 1 =. (2.8) R n Substituting Equations 2.7 and 2.8 in to Equation 2.6 leads to m n = mcosψ. (2.9) Combining Equations 2.4, 2.7, and 2.9 leads to

32 zm d = zm = n. (2.10) cosψ 14 The axial pitch p a of the helical gear is the distance A 1 B 2 shown in Figure 2.3. From triangle A 1 A 2 B 2, the axial pitch p a is related to the transverse circular pitch p by p p a =. (2.11) tanψ There are two pressure angles, the transverse pressure angle α and the normal pressure angle α n, in their respective planes. These angles are related by the following expression: tanα n cosψ =. (2.12) tanα The normal pressure angle is usually 20 o [8]. The center to center distance a between the two helical gears with teeth z 1 and z 2 is d1 d 2 z1mn z2mn a = + = + (2.13) 2 2 2cosψ 2cosψ or mn ( z 1 + z2 ) a =. (2.14) 2cosψ The speed ratio i is determined from the ratio of speed or number of teeth in the pinion and gear and is

33 ω p i = = ω g z z g p 15, (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 P t, the radial component P r, and the axial component P a. From triangle ABC, the radial component is P r = Psinα (2.16) n Figure 2.4 Components of Tooth Force in Helical Gears [7]

34 and the resultant of the axial and tangential force BC is 16 BC = P cosα. (2.17) n From triangle BDC in Figure 2.4, the axial and tangential forces are P = BC sinψ = P cosα sinψ (2.18) a n and P = BC cosψ = P cosα cosψ. (2.19) t n Combining Equations 2.18 and 2.19 leads to P = tanψ, (2.20) a P t and combining Equations 2.16 and 2.19 leads to tanα P = n r P t. (2.21) cosψ The tangential component P t is calculated using 2T P t =, (2.22) d where T is the torque transmitted and d is the pitch circle diameter.

35 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 90 o in most straight bevel gears. Figure 2.5 Straight Bevel Gear [7]

36 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 A o is the cone distance. The pitch angle γ is the angle the pitch line makes with the axis of the gear. The addendum h a, the dedendum h f, 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 r b is an imaginary cone and its elements are perpendicular to the elements of the pitch cone. Figure 2.6 Section of Bevel Gear [7]

37 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 R b and z, respectively. The virtual or formative teeth on the imaginary spur gear are 2rb z =, (2.23) m 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 D z =. (2.24) m From Equations 2.23 and 2.24, the ratio of z and z is z = z 2rb D. (2.25) From ABC in Figure 2.6, AB sin BCA = (2.26) AC or o sin( 90 ) = ( D / 2) γ. (2.27) r b o Using Equation 2.27 and γ = sin( 90 γ ) cos yields an expression for the back cone radius r b

38 D r b =. (2.28) 2cosγ 20 Substituting Equation 2.28 in Equation 2.25 leads to z z =. (2.29) cosγ A pair of bevel gear is shown in Figure 2.7. The pitch circle diameters of the pinion and gear are D p and D g, 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, D p mz p z p tan γ = = =. (2.30) D mz z g g g It can also be shown that z g tan Γ =. For the pair of bevel gear shown in Figure 2.7, z p π γ + Γ =. (2.31) 2 Figure 2.7 Pair of Bevel Gear [7]

39 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 r m shown in Figure 2.8. The mean radius r m is D p bsinγ rm =, (2.32) 2 2 where D p 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]

40 22 The resultant force has two components, P t and P s, shown in Figure 2.8. P s is the separating force between the two meshing teeth. P t is the tangential component perpendicular to the plane of the paper. The tangential component is determined from the relationship, T P t =, (2.33) r m 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 P = tanα, (2.34) s P t 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, P = cosγ (2.35) r P s and P = sinγ. (2.36) a P s Substituting Equation 2.34 into Equations 2.35 and 2.36, respectively, leads to P = tanα cosγ (2.37) r P t and P = tanα sinγ. (2.38) a P t The components of the tooth force on the pinion can be determined using Equations 2.37 and 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 P a on the

41 pinion. Similarly, the axial component on the gear is equal to the radial component P r on the pinion. 23

42 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 psi and Poisson s ratio ν of 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 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 psi and a Poisson s ratio ν of A schematic of the gearbox 24

43 Figure 3.1 Sectional View of Triple Reduction Gearbox 25

44 26 Table 3.1 Shaft Data for the Triple Reduction Gearbox Shaft Diameter (in) Input Shaft First Intermediate Shaft Second Intermediate Shaft Output Shaft Length between bearing ends (in) Table 3.2 Pinion and Gear Specifications for the Triple Reduction Gearbox No. of Teeth Diametral Pitch Helix Angle Pressure Angle (degree) (degree) Input Shaft Pinion First Intermediate Shaft Pinion First Intermediate Shaft Gear Second Intermediate Shaft Pinion Second Intermediate Shaft Gear Output Shaft Gear 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.

45 Figure 3.2 Sectional View of the Differential Gearbox 27

46 28 Table 3.3 Shaft Data for the Differential Gearbox Shaft Diameter (in) Length between bearing ends (in) Input Shaft Intermediate Shaft (Two) Output Shaft (Two) Table 3.4 Pinion and Gear Specifications for the Differential Gearbox No. of teeth Diametral Pitch Helix Angle Pressure Angle (degree) (degree) Input Herringbone Gear Intermediate Herringbone Gear Intermediate Helical Gear Intermediate Bevel Pinion Intermediate Bevel Gear Output Helical Gear 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. Throughhardened, 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.

47 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]

48 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.

49 Input: Power and speed 1 31 Case I Computes torque Case II 2 Shaft has one gear mounted. Assign N=1 Shaft has two gears mounted. Assign N=2 3 4 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 5 Gear or Pinion specifications Gear and Pinion specifications Computes Tangential, Radial and Axial forces for Gear or Pinion Computes Tangential, Radial and Axial forces for Gear and Pinion 6 Determine direction of rotation of shaft 7 Determine whether the gear is driving pinion or the driven gear 8 Determine the hand of the gear Assign m Assign M 9 Determines the direction of tangential radial and axial forces Computes reaction at end 1 and end 2 Figure 3.5 Flowchart of the Helical Gear Reaction Calculations

50 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

51 Table 3.7 Pinion and Gear Forces in the Triple Reduction Gearbox Tangential Force (lb) Input Shaft Pinion First Intermediate Shaft Pinion First Intermediate Shaft Gear Second Intermediate Shaft Pinion Second Intermediate Shaft Gear Output Shaft Gear Separating Force (lb) Axial Force (lb) Table 3.8 Pinion and Gear Forces in the Differential Gearbox Tangential Force (lb) Separating Force (lb) Axial Force (lb) Input Herringbone Gear Intermediate Herringbone Gear Intermediate Helical Gear Intermediate Bevel Gear Output Helical Gear 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 90 o 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

52 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 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 lb-in and rotates at 174 rpm.

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

54 Table 3.9 Distance between Pinion or Gear and Ends for Input and Output Shaft Shaft Distance A (in) Distance B (in) Input Output 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 Second Intermediate 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 lbin and the shaft rotates at 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 rpm and takes a torque of approximately 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 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 0 o as the positive X axis.

55 Table 3.11 Reaction Loads of the Triple Reduction Gearbox Shaft CW rotation Shaft S1 CCW rotation Shaft S1 CW rotation Shaft S2 CCW rotation Shaft S2 CW rotation Shaft S3 CCW rotation Shaft S3 CW rotation Shaft S4 CCW rotation Shaft S4 Radial reaction at End 1 (lb) 4 o o o o o o o o Radial reaction at End 2 (lb) o 37 Axial reaction Axial reaction at End 1 (lb) at End 2 (lb) o o o o o o o Table 3.12 Bearing Details of the Triple Reduction Gearbox Shaft Bearing Housing Bore Diameter (in) Shaft S1 938/ Shaft S2 H / Shaft S3 EE / Shaft S4 HM / 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

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

57 39 represented in Figure 3.7. The lengths A, B, and O for all the shafts are presented in Table 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 lbin. 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 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 lb-in and the rotation is at 174 rpm. The summary of reaction loads and bearing detail follow in Table 3.14 and Table The angles in the radial reactions are referenced using 0 o as the positive X axis.

58 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 s Intermediate s2,s Output s4,s Table 3.14 Reaction Loads of Differential Gearbox Shaft CW rotation Shaft s1 CCW rotation Shaft s1 CW rotation Shaft s2,s3 CCW rotation Shaft s2,s3 CW rotation Shaft s4,s5 CCW rotation Shaft s4,s5 Radial reaction at end 1 (lb) 3 o o o Radial reaction Axial reaction Axial reaction at end 2 (lb) at end 1 (lb) at end 2 (lb) 3 o o o 3 o o 3 o o 3 o o Table 3.15 Bearing Details for Differential Gearbox Shaft Bearing Housing Bore Diameter (in) Input Shaft s1 SKF-NJ Intermediate Shaft s2,s / Output Shaft s4,s5 JM /

59 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 41

60 42 Figure 4.1 Geometric Model of Triple Reduction Gearbox Figure 4.2 Geometric Model of Differential Gearbox

61 Table 4.1 Plate Dimensions of the Triple Reduction Gearbox Plate Dimensions (l x b x h) l/b h/b (in) Plate A Plate A Plate B Plate B Plate C Plate C Plate D Plate D Base Plate Stiffeners 1, 3 and Stiffeners 2 and Stiffeners 6 to Table 4.2 Plate Dimensions of the Differential Gearbox Plate Dimensions (l x b x h) l/b h/b (in) Plate A Plate A Plate B Plate B Plate C Plate C Plate D Plate D Base Plate Stiffeners 1 and Stiffeners 3 to 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.

62 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 90 o. 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].

63 2-D Structural Element (Plane 42) * 45 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.

64 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.

65 3-D Structural Solid Element (Solid 45) * 47 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 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.

66 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

67 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.

68 50 Figure 4.7 Solid Element Model Figure 4.8 Combined Solid and Shell Element Model

69 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 elements and 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

70 52 Figure 4.9 Finite Element Model of the Triple Reduction Gearbox with Representative Loads pressure on nodes in a 90 o 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 The actual loads used are summarized in Table 3.12.

71 53 Figure 4.10 Expanded Radial Load Distribution from Figure 4.9 Model II Model II was a refinement of Model I. It had elements and 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.

72 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 90 o 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 The axial loads applied on the inner ring are shown in Figure Figure 4.11 Finite Element Model of Triple Reduction Gearbox with Inner Ring

73 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 elements and nodes. The discretized model geometry is shown in Figure 4.13 and Figure 4.14.

74 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

75 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 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 elements and nodes and is shown in Figure 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.

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

77 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 elements and 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 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.

78 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 elements and nodes and is shown in Figure The loads presented in Table 3.15 were applied.

79 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].

80 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]

81 3-D Elastic Beam Element (Beam 4) * 63 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 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 * The ANSYS element description is given in the parentheses

82 64 Figure D Elastic Beam Element (Beam 4) [13] Figure D Point to Point Contact Element (Contac 52) [13]

83 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

84 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. 66 The gap elements were created between each node on the two adjacent surfaces. To define the gap direction, an initial gap was given as 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 in 2. 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 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 u x and u z and the rotational degrees of freedom r x, r y, and r z 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 F i, for a given bolt size and material was calculated using the procedure given in Shigley [17] F = 0. 75, (4.1) i F p where F p is the proof load F = A S. (4.2) p t p Here, S p is the proof strength of the given material and A t is the tensile-stress area. For the 1 given bolt material of SAE Grade 5 and a bolt size of 1 2 in, the proof strength and

85 tensile-stress area from Shigley are 74 kpsi and in 2, respectively [17]. The bolt preload F i, using Equations 4.1 and 4.2 was calculated to be lb. 67 The results of this model and the calculated preload were then used to define a constraint equation for the nodal degree of freedom u y between nodes A and B. After writing the constraint equation for u y, 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 The model had elements and nodes. The bolt 1 material was SAE Grade 5 and the bolt size was 1 2 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 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 u x, u z, r x, r y, and r z for the bolt center nodes were coupled and the bolt preload was

86 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 u y. 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.

87 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 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 90 o 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 in was given between the two halves to define the gap direction.

88 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. 70

89 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 in or more as significant [20]. A gearbox design with deflections on the order of 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 u x, u y, u z, total displacement u sum, and the von Mises stress σ eff for Model I are shown in Figure 5.1 through Figure 5.5. Both displacements and stresses are plotted using nodal quantities. 71

90 72 Figure 5.1 Model I Displacement u x (in) Figure 5.2 Model I Displacement u y (in)

91 73 Figure 5.3 Model I Displacement u z (in) Figure 5.4 Model I Total Displacement u sum (in)

92 74 Figure 5.5 Model I von Mises Stress σ eff (psi) A contour plot of u x is given in Figure 5.1. The maximum value of u x was in. The location for the maximum u x displacement was on the top half of the input shaft hole (first hole from the right in Figure 5.1). The largest u x displacement on plates A and A1 was in. A contour plot for displacement u y is shown in Figure 5.2. The maximum value of u y was 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 u y on the top plate B1 was in. A contour plot of u z is given in Figure 5.3. The maximum value of u z was in. The location for maximum u z displacement was on the top of the second intermediate shaft hole. A contour plot of the total displacement u sum is shown in Figure 5.4. The maximum u sum and u z displacements were the same.

93 75 A contour plot of von Mises stress σ eff in psi is given in Figure 5.5. The maximum value of σ eff 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 u x displacements plotted in Figure 5.6 were all below in. The variation in u x was small along the top edge of plate D1. The u y displacements in Figure 5.7 were larger than the u x displacements. The maximum u y displacement along the top edge on plate D1 was in. The u z displacements plotted in Figure 5.8 were the largest of all the displacements. The maximum u z displacement was in. The maximum total displacement u sum was also in. The location of the maximum u z and u sum was on top of the second

94 76 X-displacement, in 2.40E E E E E E-04 Model I Model II Model III 0.00E Distance along the top edge of plate D1, in Figure 5.6 Comparison of Displacement u x in Models I, II, and III 2.50E-03 Y- Displacemnt, in 2.00E E E E-04 Model I Model II Model III 0.00E Distance along the top edge of plate D1, in Figure 5.7 Comparison of Displacement u y in Models I, II, and III

95 77 Z-Displacement, in 2.00E E E E E E E E E E E+00 Model I Model II Model III Distance along top edge of plate D1, in Figure 5.8 Comparison of Displacement u z in Models I, II, and III von Mises Stress, psi 5000 Model I 4500 Model II 4000 Model III Distance along top edge of plate D1, in Figure 5.9 Comparison of von Mises Stress in Models I, II, and III intermediate shaft hole. The von Mises stresses σ eff 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

96 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 u x, u y, u z, total displacement u sum, and the von Mises stress σ eff 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 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 u x for Model V is given in Figure The location of the maximum displacement was on the side plates A and A1. The maximum magnitude of u x was in. By comparison, the maximum u x in the half Model I was only in. The magnitude of displacement u x 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 u y is shown in Figure The maximum u y was in on the top plate B1. In Model I, u y on the top plate B1 was ten times less than u y in Model V.

97 79 Figure 5.10 Model V (Load Case IV) Displacement u x (in) Figure 5.11 Model V (Load Case IV) Displacement u y (in)

98 80 Figure 5.12 Model V (Load Case IV) Displacement u z (in) Figure 5.13 Model V (Load Case IV) von Mises Stress (psi)

99 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 u y 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 u z is given in Figure The magnitude of the maximum u z displacement was 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 u z 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 in was on the top plate. The maximum displacement and location was the same as the maximum u y displacement in Figure The maximum total displacement on the near end was in on top of the second intermediate shaft hole. This magnitude and location was the same as in the u z displacement plot in Figure The total displacement contours are not plotted. The contour plot of the von Mises stress σ eff is shown in Figure 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 Path AB is along the top edge of plate D1. The path GH is on

100 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 u x results of Models IV and V were compared along path CD in Figure The values of u x for Models IV and V along path CD were almost identical. The maximum u x displacement along path CD was in. The u y displacement on Models IV and V were compared along path EF in Figure The maximum u y displacement in both Models IV and V was in on the top plate.

101 83 X- Displacement, in Model IV Model V 1.4E E E E E E E E Distance along path CD, in Figure 5.15 Comparison of X-Displacement in Model IV and V -5.00E E-02 Model V Model IV Y-displacement, in -1.50E E E E E E Distance along path EF, in 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.

102 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 u x and u y were similar to the results on End 1 in load case IV when input shaft rotated clockwise. These results are not presented. The u z displacement plot in Model V on End 2 side (the near end) is shown in Figure The maximum u z for load case III was in. The location of the maximum u z was on top of the output shaft hole (first hole from left end in Figure 5.17). The maximum u z on the shaft hole was in. The total displacement u sum on the shaft hole was also 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 u z (in)

103 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 u x and u y displacement plots were similar to the results for clockwise rotation of input shaft. Therefore, the u x and u y displacement plots are not presented. The u z displacement plot for load case II in Figure 5.18 has a maximum value of in. The location of the maximum u z 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 u sum for load case II on the output shaft hole was in and may be of concern to Steward Machine Company. Figure 5.18 Model V (Load Case II) Displacement u z (in)

104 86 The u z plot for load case I is presented in Figure The maximum u z displacement was in for load case I. The maximum u z 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 u sum was also in at the same location. The results of Model V showed that the u z displacement was maximum on the second intermediate or the output shaft depending on the load case. For the input shaft rotating clockwise, the maximum u z displacement on one end was on top of the second intermediate shaft hole and the maximum u z 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 u z (in)

105 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. 87 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 u x, u y, u z, total displacement u sum, and the von Mises stress σ eff for a clockwise rotation of the input shaft (load case I) are shown in Figure 5.20 through Figure The contour plots of displacement u x 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 u x displacement was in. The location of maximum u x 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 u x displacement was caused by the radial loads acting on the intermediate shaft at that location. The contour plot for displacement u y is shown in Figure The maximum positive u y displacement of in occurred on the stiffener between the input and

106 88 Figure 5.20 SDI II (load case I) Near End 2 Displacement u x (in) Figure 5.21 SDI II (load case I) Far End 1 Displacement u x (in)

107 89 Figure 5.22 SDI II (Load Case I) Far End 1 Displacement u y (in) Figure 5.23 SDI II (Load Case I) Near End 2 Displacement u z (in)

108 90 Figure 5.24 SDI II (Load Case I) Far End 1 Displacement u z (in) Figure 5.25 SDI II (Load Case I) Far End 1 Total Displacement u sum (in)

109 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 in occurred on the top plate B1. The contour plots of displacement u z 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 u z on both the near and far end was in. The location of the maximum u z displacement was on top of the intermediate shaft hole on both ends. The maximum u z on the intermediate shaft hole was in. The contour plot for total displacement u sum is shown in Figure The maximum total displacement of in was on top of the intermediate shaft hole. The maximum u sum displacement on the intermediate shaft hole was in.

110 92 The von Mises stress plot for End 1 (the far end) is shown in Figure 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 u x, u y, u z, and the von Mises stress σ eff for SDI II for a counterclockwise rotation of the input shaft (load case II) are shown in Figure 5.27 through Figure The contour plot of displacement u x on End 2 (the near end) is given in Figure The maximum displacement of in occurred on the output shaft where the radial load on the hole was acting. The plot for u x on End 1 (the far end) is shown in Figure The maximum displacement locations cannot be seen in this plot. However, the u x displacements around the intermediate shaft shown on the far end side were only slightly smaller than the maximum value of in on the near end side. The contour plot for displacement u y is shown in Figure 5.29 for End 1 (the far end). The maximum positive u y of in was seen on the stiffener on top of the output shaft hole. The maximum negative u y of in was seen on the top plate B1. The maximum positive u y was at a location where the radial loads were acting. The maximum negative u y 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.

111 93 Figure 5.27 SDI II (Load Case II) Near End 2 Displacement u x (in) Figure 5.28 SDI II (Load Case II) Far End 1 Displacement u x (in)

112 94 Figure 5.29 SDI II (Load Case II) Far End 1 Displacement u y (in) Figure 5.30 SDI II (Load Case II) Near End 2 Displacement u z (in)

113 95 Figure 5.31 SDI II (Load Case II) Far End 1 Displacement u z (in) The contour plots for displacement u z 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 u z on the near and far end was in. The location of the maximum u z displacement was on top of the intermediate shaft hole on both ends. The u z displacement was mainly caused by the axial loads pulling on the shaft ends on both sides. The total displacement u sum had the same magnitude of in and was on top of the intermediate shaft hole. The contour plot for u sum 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

114 E E-03 SDI I SDI II Total displacement, in 3.50E E E E E E E E Distance along the top edge of plate D1, in Figure 5.32 Comparison of Total Displacement in SDI I and SDI II maximum total displacement u sum of approximately 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

115 97 Stress, psi 1800 SDI I 1600 SDI II Distance along the top edge of plate D1, in 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 The moments from the shell elements were ignored at the shellsolid interface. The contour plots for displacements u x, u y, u z, and the von Mises stress σ eff for Model VI are shown in Figure 5.34 through Figure Model VI was solved for End 1 side (the near end) when the input shaft rotated clockwise.

116 98 Figure 5.34 Model VI Displacement u x (in) Figure 5.35 Model VI Displacement u y (in)

117 99 Figure 5.36 Model VI Displacement u y 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

118 100 Figure 5.38 Model VI Displacement u z (in) Figure 5.39 Model VI von Mises Stress (psi)

119 101 A contour plot of u x is given in Figure The maximum u x displacement was in. The location of the maximum u x displacement was on the top half of the input shaft hole (first hole from the right in Figure 5.34). The largest u x displacement on plates A and A1 was in. The maximum u x displacement on the interface was in on the left end of the output shaft in Figure A contour plot for displacement u y is shown in Figure The maximum u y displacement was 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 u y on the top plate B1 was in. The u y displacement on the interface between plates D1 and D2 was examined closely to check for any gap between the two plates. The maximum u y 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 The line CGD is on the front side of the gearbox and the line AHF is on the interior side. The u y gap displacement at points A, B, C, D, E, F, G, and H is shown in Figure At points A, B, and C on the side of the second intermediate shaft, the interfaces separated with a gap of in at A, in at B, and in at C. On the first intermediate shaft hole side at points D and E, the gap was in and 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

120 102 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 u z is given in Figure The maximum u z displacement was in. The location of maximum u z displacement was on the top of the second intermediate shaft hole. The total displacement plot was similar to the u z displacement plot and is not shown. The maximum total displacement of in was on the same location as the u z displacement in Figure 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 σ eff in psi is given in Figure 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

121 103 shown in Figure 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 The u x displacements, shown in Figure 5.41, are very small. The maximum u x displacement of in was found near the input shaft hole. The u y displacements given in Figure 5.42 were larger than the u x values. The maximum u y displacement of in was found on the interface between the first intermediate and second intermediate shaft holes. The u z displacement shown in Figure 5.43 had the largest magnitudes, in. The location of this displacement was on the interface between the first intermediate and second intermediate shaft holes. The total displacement of 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 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.

122 104 Figure 5.40 Defined Paths on the Interface in the Triple Reduction Gearbox Model VII Model VI X-displacement, in 8.0E E E E E E E E E E-03 Distance along path MN, in Figure 5.41 Displacement u x in Model VI and VII