The Pennsylvania State University The Graduate School Intercollege Graduate Program in Materials NONLINEAR FINITE ELEMENT MODELING AND ANALYSIS OF A TRUCK TIRE A Thesis in Materials by Seokyong Chae © 2006 Seokyong Chae Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2006 The thesis of Seokyong Chae was reviewed and approved* by the following: Moustafa El-Gindy Senior Research Associate, Applied Research Laboratory Thesis Co-Advisor Co-Chair of Committee James P. Runt Professor of Materials Science and Engineering Thesis Co-Advisor Co-Chair of Committee Co-Chair of the Intercollege Graduate Program in Materials Charles E. Bakis Professor of Engineering Science and Mechanics Ashok D. Belegundu Professor of Mechanical Engineering *Signatures are on file in the Graduate School. iii ABSTRACT For an efficient full vehicle model simulation, a multi-body system (MBS) simulation is frequently adopted. By conducting the MBS simulations, the dynamic and steady-state responses of the sprung mass can be shortly predicted when the vehicle runs on an irregular road surface such as step curb or pothole. A multi-body vehicle model consists of a sprung mass, simplified tire models, and suspension system to connect them. For the simplified tire model, a rigid ring tire model is mostly used due to its efficiency. The rigid ring tire model consists of a rigid ring representing the tread and the belt, elastic sidewalls, and rigid rim. Several in-plane and out-of-plane parameters need to be determined through tire tests to represent a real pneumatic tire. Physical tire tests are costly and difficult in operations. Thus, the parameters for the rigid ring tire model are alternatively predicted by conducting virtual tire tests using a finite element analysis (FEA) tire model. A nonlinear three-dimensional FEA tire model representing a truck tire, 295/75R22.5, is constructed by implementing three-layered membrane elements, hyperelastic solid elements, and beam elements. Then, the FEA tire model is validated by comparing its in- plane and out-of-plane responses with physical measurements. The virtual and physical responses show good agreements. After successful validations of the FEA tire model, virtual tire tests are conducted to predict the in-plane and out-of-plane parameters for the rigid ring tire models for the first time using an FEA tire model. The predicted parameters are implemented in the rigid ring tire model, and the model undergoes water drainage ditches 90° and 45° to the tire running direction to predict dynamic in-plane and out-of-plane tire responses at various tire loads. Vertical displacement of the tire spindle, tire contact forces, and moments are plotted and compared with those of the FEA tire model. The in-plane tire responses show good iv agreements between the results of the two models. On the other hand, the out-of-plane tire responses are relatively not in good agreements due to the significantly different tire contact area geometries of the two tire models on the 45° ditch. In the simulations of the FEA and rigid ring tire models, only constant vertical tire load is applied to the tire models. Additional tire load due to the vertical acceleration of the sprung mass during tire operations is not considered. Thus, a sprung mass and suspension system is assembled with the tire models to include the effect of the vertical sprung mass motion, which represents a quarter-vehicle model and a closer model to real vehicle applications. Then, the models undergo a 90° ditch at various running speeds. The vertical accelerations of the tire spindles are predicted during the ditch runs and compared with measurements to check whether or not the rigid ring tire model in the quarter-vehicle environment predicts acceptable responses. Modern high computational capability enables to establish a reliable virtual tire and quarter-vehicle model test environments. The developed quarter-vehicle model predicts not only dynamic tire responses but also sprung mass responses to irregular road surface inputs. Thus, a trustworthy quarter-vehicle model can replace the conventional field durability tests, which saves product development cost and time. In addition, virtual tests under similar conditions can be easily repeated. Higher quality products at lower cost are undoubtedly promising. v TABLE OF CONTENTS LIST OF FIGURES ……………………………….…………………………….. x LIST OF TABLES……………………………………………………..………… xiv NOMENCLATURE…………………………………………………..………… xv ACKNOWLEDGEMENT……………………………………………..………… xvii CHAPTER 1: INTRODUCTION ………………………………………………. 1 1.1 History of the Wheel and Tire …………………………………………... 3 1.2 Objectives ……………………………………………………………….. 5 1.3 Methodology …………………………………………………………….. 6 1.4 Statement of the Work …………………………………………………... 7 CHAPTER 2: LITERATURE SURVEY ………………………………………. 10 2.1 Tire Axis System, Tire Forces, and Moments Definitions ……………… 10 2.1.1 Tire axis system ………………………………………………... 10 2.1.2 Tire forces and moments ……………………………………….. 11 2.1.2.1 Tire longitudinal force ………………………………. 12 2.1.2.2 Tire lateral force ……………………………………… 15 2.1.2.3 Tire vertical force …………………………………….18 2.1.2.4 Tire overturning moment …………………………….. 20 2.1.2.5 Tire rolling resistance moment ……………………… 21 2.1.2.6 Tire vertical moment ………………………………… 21 2.2 Tire Models ……………………………………………………………… 21 2.2.1 Early Tire Models ……………………………………………... 23 2.2.2 Finite Element Analysis Tire models …………………………... 31 2.3 Contact and Sliding Interfaces ………………………………………….. 44 2.3.1 Computational Contact Solution Algorithms ………………….. 44 2.3.1.1 Contact search algorithm ……………………………. 46 2.3.1.2 Contact interface algorithm …………………………. 49 2.3.2 Lagrange Multiplier Method …………………………………... 50 2.3.3 Penalty Method ………………………………………………... 51 2.4 Experimental Tire Testing ………………………………………………. 55 vi 2.5 Tire Rubber Material Modeling …………………………………………. 60 2.6 Summary ………………………………………………………………… 66 CHAPTER 3: NONLINEAR FINITE ELEMENT ANALYSIS TRUCK TIRE MODELING …………………………………………….... 67 3.1 Tire Structure, Its Components, and Materials ………………………….. 69 3.1.1 Carcass ………………………………………………………… 70 3.1.2 Belts …………………………………………………………… 71 3.1.3 Tread and Treadbase …………………………………………... 72 3.1.4 Beads ………………………………………………………….. 73 3.1.5 Aspect Ratio …………………………………………………… 73 3.2 Tire Structure and Material Modeling …………………………………... 75 3.2.1 FEA Modeling on Carcass and Belts Using Membrane Elements ……………………………………………………... 76 3.2.2 FEA Modeling on Tread, Treadbase, Tread Shoulders, Bead Fillers Using Solid Elements ………………………………… 81 3.2.3 FEA Modeling of Beads ……………………………………… 83 3.2.4 FEA Rim Model ……………………………………………… 85 3.3 FEA Tire-Rim Assembly Model ………………………………………… 87 3.4 Various Virtual Tire Test Environments …………………………………88 3.4.1 Combined FEA Truck Tire and a Rigid Flat Road Model ……. 89 3.4.2 Combined FEA Truck Tire and a Rigid Cleat-Drum Model ….. 90 3.4.3 Combined FEA Truck Tire and a Rigid Smooth Drum Model ... 91 3.4.4 Combined FEA Truck Tire and a Rigid Road with Water Drainage Ditches Model ………………………………….…… 92 3.5 Summary ………………………………………………………………… 92 CHAPTER 4: FEA TRUCK TIRE MODEL VALIDATION …………………. 94 4.1 Static Validation Tests ………………………………………………….. 94 4.1.1 Tire Vertical Load-Deflection Relationship …………………... 94 4.1.2 Footprint Area …………………………………………………. 96 4.2 Dynamic Validation Tests ………………………………………………. 98 4.2.1 First Mode of Free Vertical Vibration Test ……………….…... 99 4.2.2 Cornering Test ………………………………………………… 101 4.2.3 Yaw Oscillation Test …………………………………..……… 103 4.2.3.1 Amplitude Ratio and Phase Angle …………….…... 104 vii 4.2.3.2 Virtual Yaw Oscillation Test Using FEA Truck Tire Model ………………………………….. 106 4.3 Summary ………………………………………………………………… 109 CHAPTER 5: DETERMINATION OF THE IN-PLANE AND OUT-OF- PLANE PARAMETERS OF A RIGID RING TIRE MODEL USING THE FEA TRUCK TIRE MODEL ……………………. 111 5.1 In-Plane Parameters Determination ……………………………………... 112 5.1.1 Effective Rolling Radius ………………………………………. 114 5.1.2 In-Plane Translational Stiffness of the Sidewall and Residual Vertical Stiffness at Contact Area ……………………………... 116 5.1.3 In-Plane Longitudinal and Vertical Damping Constants of Sidewall and Residual Damping Constant at Contact Area …… 117 5.1.4 In-Plane Rotational Stiffness and Damping Constant of the Sidewall ……………………………………………………….. 118 5.1.4.1 In-Planed Rotational Stiffness of the Sidewall ……… 119 5.1.4.2 In-Planed Rotational Damping Constant of the Sidewall ……………………………………………... 119 5.1.5 Longitudinal Tread Stiffness and Longitudinal Slip Stiffness … 121 5.1.5.1 Longitudinal Slip Stiffness ………………………….. 123 5.1.5.2 Longitudinal Tread Stiffness …………………………123 5.1.6 Comparison of In-plane Tire Responses between the FEA and Rigid Ring Tire Models ……………………………………….. 124 5.1.6.1 90° Water Drainage Ditch Profile ………………….... 125 5.1.6.2 Comparison of the Predicted Vertical Displacements . 125 5.1.6.3 Comparison of the Predicted Longitudinal Contact Forces ……………………………………………….. 127 5.1.6.4 Comparison of the Predicted Vertical Contact Forces . 129 5.2 Out-of-Plane Parameters Determination ………………………………… 131 5.2.1 Out-of-Plane Translational Stiffness and Damping Constant of the Sidewall ……………………………………………….... 132 5.2.2 Out-of-Plane Rotational Stiffness and Damping Constant of the Sidewall ………………………………..………….……. 134 5.2.3 Lateral Free Vibration Tests …………………………………... 137 5.2.4 Cornering Stiffness ……………………………………………. 139 viii 5.2.5. Self-Aligning Stiffness ………………………………………... 139 5.2.6 Relaxation Length ……………………………………………... 140 5.2.7 Comparison of Out-of-Plane Tire Responses between the FEA and Rigid