Clemson University TigerPrints All Theses Theses 12-2009 ROBUST PARAMETER DESIGN AND FINITE ELEMENT ANALYSIS FOR A NON- PNEUMATIC TIRE WITH LOW VIBRATION Amarnath Proddaturi Clemson University, aprodda@clemson.edu Follow this and additional works at: http://tigerprints.clemson.edu/all_theses Part of the Engineering Mechanics Commons Recommended Citation Proddaturi, Amarnath, "ROBUST PARAMETER DESIGN AND FINITE ELEMENT ANALYSIS FOR A NON-PNEUMATIC TIRE WITH LOW VIBRATION" (2009). All Theses. Paper 725. This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact awesole@clemson.edu.
ROBUST PARAMETER DESIGN AND FINITE ELEMENT ANALYSIS FOR A NON-PNEUMATIC TIRE WITH LOW VIBRATION A Thesis Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Master of Science Mechanical Engineering by Amarnath Proddaturi December 2009 Accepted by: Dr. Lonny L. Thompson, Committee Chair Dr. John C. Ziegert Dr. Paul F. Joseph
ABSTRACT During rolling of a non-pneumatic tire, vibration may be produced by the interaction of collapsible spokes with a shear deformable ring as they enter the contact region, buckle and then snap back into a state of tension. Other potential sources of vibration include the interaction of tire tread with the ground and ring vibration. In the present work, a systematic study of the effects of key geometric design parameters is presented using Taguchi s Robust Parameter Design Method and Orthogonal Arrays. In the present work, a 2D planar finite element model with geometric nonlinearity and explicit time-stepping is used to simulate rolling of the non-pneumatic tire. Vibration characteristics are measured from the FFT frequency spectrum of the time-signals of perpendicular distance of marker nodes from the virtual plane of the spoke, ground reaction forces, and ring vibration. Both maximum peak amplitudes and RMS measures are considered resulting in a total of five output measures which are to be reduced for evaluated optimal design combinations. In the initial study, two different L 8 orthogonal arrays are considered, one with both spoke and ring parameters, and the other, which focuses on only ring variables and interactions between them. Based on the results from the initial study, an L 27 orthogonal array, which combines all key geometric variables and includes the effects of uncontrollable noise factors of rolling speed and ground pushup, is analyzed for robust parametric design. Since there are more than one set of noise factor combinations, Signal-to-Noise (S/N) ratios with Analysis of Variance (ANOVA) methods are used to 1
determine percent contributions and predict the optimal combination level for each control factor, for all vibration measures. 2
DEDICATION This thesis is dedicated to my family Srinivas and Shakunthala Proddaturi, and Narsing Rao and Vijayalaxmi Proddaturi. 3
ACKNOWLEDGMENTS I offer my deepest gratitude to my research advisor Dr. Lonny L. Thompson who was abundantly helpful and offered invaluable assistance, support and guidance throughout this research project and for my thesis writing. I attribute the level of my Masters degree to his encouragement and effort without which this thesis would not have been completed. I would like to thank Dr. Paul F. Joseph and Dr. John C. Ziegert for serving on my committee and for their valuable suggestions for improving my thesis. I would also like to thank Dr. Timothy Rhyne, and Mr. Steve Cron of Michelin and Dr. Andreas Obieglo of BMW for their personal assistance and for financial support for this research project. 4
TABLE OF CONTENTS Abstract... 1 Dedication... 3 Acknowledgments... 4 Table of Contents... 5 List of Tables... 7 List of Figures... 10 Chapter One : Introduction... 13 1.1 Introduction to Tweel TM... 13 1.2 Orthogonal Arrays... 14 1.3 Thesis Objective... 16 Chapter Two : Finite Element Model for Tweel TM... 19 2.1 2-D Planar Tweel Geometry... 19 2.2 Tweel Material Properties... 23 2.3 Rolling Analysis Procedure... 27 2.4 Mesh properties... 29 Chapter Three : Load-Deflection Curves... 35 3.1 Analysis Procedure for Load-Deflection Curves... 36 3.2 Results... 39 3.3 Conclusions... 46 Chapter Four : Measures of Tweel Vibration... 47 4.1 Spoke Vibration... 49 4.2 Ring Vibration... 57 4.3 Ground Interaction Vibration... 63 Chapter Five : Effect of Spoke Thickness and Number of Spoke Pairs on Vibration... 67 5.1 Results... 68 5
5.2 Conclusions... 71 Chapter Six : Taguchi Design Method... 72 6.1 Taguchi Design Method... 73 6.2 Analysis of Variance (ANOVA)... 83 6.3 Taguchi s Robust Parameter Design Method... 84 Chapter Seven : L8-Orthogonal Array Studies... 86 7.1 Setup of L 8 Orthogonal Arrays... 86 7.2 Results... 88 7.3 Discussion... 94 Chapter Eight : L 27 Orthogonal Array Study with Taguchi Robust Parameter Design Method... 97 8.1 L 27 Orthogonal Array Setup... 98 8.2 Results... 100 Chapter Nine : Conclusion... 113 9.1 Conclusions... 113 9.2 Future Work... 115 Appendices... 116 Appendix A: Additional Results for L 8 Orthogonal Array with Spoke and Ring Parameters:... 117 Appendix B: Additional Results for L 8 Orthogonal Array with Ring Parameters and their Interaction... 122 Appendix C: Additional Results for L 27 Orthogonal Array Study... 126 REFERENCES... 139 6
LIST OF TABLES Table 2.1: Geometric and Material Parameters for Reference Tweel model... 22 Table 2.2: Elastic Moduli for Orthotropic Reinforcement material... 24 Table 4.1: FFT Peak Amplitudes of spoke vibration for all the marker nodes during steady rolling 120 rad/sec (Reference model)... 56 Table 5.1: Spoke Vibration results for the maximum peaks obtained from FFT... 69 Table 5.2: Spoke Vibration results for the RMS amplitude obtained from FFT... 69 Table 5.3: Ground Interaction results for the RMS amplitude obtained from FFT.... 70 Table 5.4: Ring Vibration results for Max peak and RMS amplitude obtained from FFT70 Table 6.1: L 8 Orthogonal array showing the two-level values for each of 7 factors [12] 75 Table 6.2: Available Standard Orthogonal Arrays [13]... 77 Table 6.3: L 8 Orthogonal Array with one output measure... 79 Table 6.4: Deviation from mean values for the L8 array considered above... 79 Table 6.5: An L 8 Orthogonal Array with outer array showing the signal-to-noise ratio column... 82 Table 6.6: Orthogonal Array in the Taguchi Robust Parameter design experiment... 85 Table 7.1: L 8 Orthogonal Array with Spoke and Ring Parameters... 87 Table 7.2: L 8 Orthogonal Array with Ring Parameters and their Interactions... 88 Table 7.3: Five output measure values for L 8 Orthogonal Array with Spoke and Ring Parameters... 89 Table 7.4: Deviation from mean for Max peak amp of spoke vibration measure... 90 Table 7.5: Five output measure values for L 8 Orthogonal Array with Ring Parameters and their Interactions... 91 Table 7.6: Deviation from mean for Max peak amp of spoke vibration measure... 91 Table 7.7: Optimal levels for control factors for five output measures of L 8 Orthogonal Array with Spoke and Ring Parameters... 94 Table 7.8: Ranking for importance for reducing vibration in the five output measures using L 8 Orthogonal Array with Spoke and Ring Parameters... 94 Table 7.9: Optimal levels for control factors for all output measures of L 8 Orthogonal Array with Ring Parameters... 96 Table 7.10: Rankings for control factors for all output measures of L 8 Orthogonal Array with Ring Parameters... 96 7
Table 8.1: Control factors with their level values... 98 Table 8.2: Noise factors with their level values... 98 Table 8.3: L 27 Orthogonal Array with outer array used in the Taguchi Robust Parameter Design Experiments... 99 Table 8.4: Max Peak Amplitude Results for spoke vibration... 101 Table 8.5 Main-effects chart values for Max peak amplitude for spoke Vibration.... 102 Table 8.6: ANOVA for S/N (Max peak Amp of Spoke Vibration)... 103 Table 8.7: Optimum levels for each control factor for all the five output measures... 104 Table 8.8: Optimal Predicted Combination... 106 Table 8.9: Predicted S/N Ratio Values of Optimal combination... 107 Table 8.10: Comparison of predicted S/N Ratio values of optimal combination with all other runs... 108 Table 8.11: Max Peak Amplitude Results for spoke vibration for all optimal combinations including reference model... 109 Table 8.12: Stiffness Estimate between ground pushup limits... 110 Table 8.13: Levels for the optimal combination, Optimal-2... 111 Deviation from mean for RMS amp of spoke vibration measure... 117 Deviation from mean for RMS amp of ground interaction measure... 118 Deviation from mean for Max peak amp of ring vibration measure... 119 Deviation from mean for RMS amp of ring vibration measure... 120 Deviation from mean for RMS amp of spoke vibration measure... 122 Deviation from mean for RMS amp of ground interaction measure... 123 Deviation from mean for Max peak amp of ring vibration measure... 124 Deviation from mean for RMS amp of ring vibration measure... 125 RMS Amplitude Results for the spoke vibration.... 127 Main-effects chart values for RMS peak amplitude for spoke Vibration... 128 ANOVA for S/N ratio of RMS Amp of Spoke Vibration... 129 RMS Amplitude Results for the ground interaction.... 130 Main-effects chart values for RMS amplitude for ground interaction... 131 ANOVA table (only sum of squares) for S/N ratio of RMS Amp of ground interaction 132 MAX Peak Amplitude Results for the Ring Vibration... 133 Main-effects chart values for Max peak amplitude for ring vibration... 134 8
ANOVA table for S/N ratio of Max peak Amp of ring vibration... 135 RMS Amplitude Results for the Ring Vibration... 136 Main-effects chart values for RMS amplitude for ring vibration... 137 ANOVA table for S/N ratio of RMS Amp of ring vibration... 138 9
LIST OF FIGURES Figure 1.1: Tweel: A Non-pneumatic tire... 14 Figure 2.1: 2D-Planar Reference Tweel Model showing all the parts in the assembly... 20 Figure 2.2: Tweel Plug-in Interface showing reference Tweel parameters... 21 Figure 2.3: Stress-strain curve for Marlow material model.... 26 Figure 2.4: Uniform mesh used for dynamic rolling analysis... 31 Figure 2.5: Detailed view of uniform mesh used for dynamic rolling analysis... 32 Figure 2.6: Model with coarse/fine mesh regions for static load-deflection analysis... 33 Figure 2.7: Detailed view of course/fine regions for static load-deflection analysis... 34 Figure 3.1: Deformed Tweel Model with Coarse-Fine mesh regions, after 25 mm ground push-up... 38 Figure 3.2: Coarse-Fine Mesh (zoomed view of elements in both fine and coarse mesh in spoke pairs and ring)... 39 Figure 3.3: Load-deflection curves of two models one with cooling step and other without cooling step... 41 Figure 3.4: Load-deflection curves of with three different shear beam thicknesses... 42 Figure 3.5: Load-deflection curves with three different spoke thicknesses and number of spoke pairs.... 43 Figure 3.6: Load-deflection curves for Tweel model with and without tread.... 45 Figure 4.1: The final deformation in rolling step at 0.405 sec in the reference model... 48 Figure 4.2: Spoke marker nodes shown in Red.... 50 Figure 4.3: Spoke marker nodes shown by red dots: (a) undeformed, (b) deformed.... 51 Figure 4.4: Distance between the top and bottom nodes of spoke for the rolling step; measures spoke length change.... 52 Figure 4.5: Perpendicular distance of each marker node from the plane of the spoke during rolling step... 53 Figure 4.6: FFT spectrum of Perpendicular Distance of the Upper Quarter Node for the Reference Tweel model... 55 Figure 4.7: Ring marker nodes (nodes marked in red color)... 58 Figure 4.8: Detailed view of Ring marker nodes (nodes marked in red color)... 59 Figure 4.9: Change in Ring Thickness over all Steps (for Reference model)... 60 10
Figure 4.10: Ring Thickness over rolling Step (for Reference model)... 61 Figure 4.11: FFT of changes in Ring Thickness (for Reference model)... 62 Figure 4.12: Vertical Reaction Force over All Steps (for Reference model)... 63 Figure 4.13: Vertical Reaction Force for Rolling Step (for Reference model)... 64 Figure 4.14: FFT of Vertical Reaction Force (for the Reference model)... 65 Figure 5.1: (a) 30 spoke pairs with 3.36 mm spoke thickness. (b) 20 spoke pairs with 5.04 mm spoke thickness... 68 Figure 7.1: Interaction plot between spoke thickness and spoke length for Max peak Amplitude spoke vibration measure.... 90 Figure 7.2: Interaction plot between shear beam thickness and inner coverage for Max peak amp of spoke vibration measure... 92 Figure 7.3: Interaction plot between shear beam thickness and outer coverage for Max peak amp of spoke vibration measure... 92 Figure 7.4: Interaction plot between shear beam thickness and spoke thickness for Max peak amp of spoke vibration measure... 93 Figure 8.1: Main-effects Chart for the Max peak amplitude for spoke vibration... 102 Figure 8.2: Combined Result of Percentage Contribution (using ANOVA) for S/N Ratio Values of all the Output Measures... 104 Interaction plot between spoke thickness and spoke length for RMS amp of spoke vibration measure... 118 Interaction plot between spoke thickness and spoke length for RMS amp of ground interaction measure... 119 Interaction plot between spoke thickness and spoke length for Max peak amp of ring vibration measure... 120 Interaction plot between spoke thickness and spoke length for RMS amp of ring vibration measure... 121 Interaction plot between shear beam thickness and other factors for RMS amp of spoke vibration measure... 122 Interaction plot between shear beam thickness and other factors for RMS amp of ground interaction measure... 123 Interaction plot between shear beam thickness and other factors for Max peak amp of ring vibration measure... 124 Interaction plot between shear beam thickness and other factors for Max peak amp of ring vibration measure... 125 Main-effects Chart for the RMS peak amplitude for spoke vibration.... 128 11
Main-effects Chart for the RMS amplitude for ground interaction.... 131 Main-effects Chart for the Max peak amplitude for ring vibration.... 134 Main-effects Chart for the RMS amplitude for ring vibration... 137 12
CHAPTER ONE: INTRODUCTION 1.1 Introduction to Tweel TM Recently, Michelin proposed a non-pneumatic tire design, which has potential for improved handling and rolling resistance, in addition to other beneficial properties [1]. A picture of a recent non-pneumatic wheel design is shown in Figure 1.1 [23]. The non-pneumatic wheel structure is molded from polyurethane (PU) material and has three main parts: the shear beam, the hub, and a number of thin, deformable spokes. The shear beam is the layer in the outer ring between imbedded cables, which form inextensible reinforcements. Under loading, the shear beam in the region of contact deforms primarily in shear. The shear beam enables the wheel to have a large contact patch with relatively low and uniform contact pressure [1]. Rubber tread is bonded to the outer surface of the ring outer coverage. The spokes connect the hub to the shear beam and support the weight of the vehicle. The spokes underneath the hub in the contact patch buckle and collapse when the wheel is loaded, while the spokes above support the load through tension. While the study of vibration and noise from pneumatic tires has a long history, there have so far been relatively few studies on the vibration and sound characteristics of non-pneumatic tires. In [2, 6], a computational method was described to examine spoke dynamics in a high speed rolling non-pneumatic wheel described in [1]. By varying the geometric parameters, a wide array of ride and handling qualities can be generated. For applications with significant load such as passenger vehicles, early prototypes have 13
exhibited vibration and noise at higher speeds which is undesirable. In order to identify the causes of acoustic noise and optimize design parameters for a rolling non-pneumatic non wheel, a computational method has been developed for modeling spoke dynamics and vibration during high-speed speed rolling rolling[21, 23]. Figure 1.1: Tweel: A Non-pneumatic tire 1.2 Orthogonal Arrays The orthogonal array is an experimental tool invented by Dr. G Taguchi in Japan in the 1950s to reduce greatly the number of experiments when a large number of changing variables are considered [12]. The orthogonal arrays were developed to determine the influence of each control factor being studied on both the mean result and variation from that result. An orthogonal array is a tool of designing experiments that usually requires only a fraction of the full factorial combinations of the experiments. An orthogonal array means the design is balanced. That is all control factors in the design have levels weighted equally [16].. Because of this property, each control factor will be 14
evaluated independently of all the other control factors, so the effect of one control factor does not influence on the estimation of another control factor. In [21, 23] two Orthogonal Array studies were conducted, an L 8 array with six geometric variables at two variable levels and a L 9 array with four geometric variables at three variable levels. Each geometry created was tested for spoke and ground interaction vibration. Spoke length and curvature were predicted to be the two most influential variables on the amplitude of spoke, and ground interaction vibration. A short spoke with large curvature resulted in the smallest amplitude of vibration for both measurements. Results from confirmation experiments show that the L 8 and the L 9 arrays experienced interactions between variables, indicating that the recommended levels for some variables are dependent on the levels of other variables. Further orthogonal array experiments and studies were suggested in order to gain a greater understanding of the magnitude of interactions between variables. In [21, 23], each geometric variable was varied by high and low levels derived from a 25% deviation from a baseline reference model. Since, the spoke length was estimated to have the greatest effect on vibrations, and since the variation in vibration amplitude between the long and short spokes was large, and may skew results for variables shown to have less effect, it is suggested that the percent deviation be reduced. In addition, the L 8 array in the previous work showed unexpected levels of influence for the shear beam, inner coverage, and outer coverage. Because the shear beam is crucial to the performance of the non-pneumatic tire, interactions between it and the inner and outer coverage should be examined further. 15
1.3 Thesis Objective In previous studies [21, 23], spoke length, spoke curvature, spoke thickness, shear beam thickness, inner coverage, and outer coverage were studied together with interaction between spoke lengths and spoke thickness in an L 8 orthogonal array. The levels used for spoke length in this previous study were 54mm and 90mm, which are 25% from the reference length of 72mm. Using these levels, it was shown that the spoke length and spoke curvature are the two most influencing parameters affecting Tweel vibration. Since these two spoke variables dominate the overall effect on vibration, the differences and importance of the other design variables were difficult to characterize. One of the objectives of the present study is to better characterize the effects of the other variables. This will be accomplished by reducing the spoke length levels from 25% change to 13.9% change from the reference length. Another goal of the present work is to study an additional L 8 orthogonal array with variables of shear beam thickness, inner coverage, and outer coverage, leaving appropriate columns open to expose interactions between the ring variables. While the L 8 array only evaluates two levels, this should give a first-order indication of which variables have interactions, and how strong the interactions are. The final objective is to combine all key geometric variables and include the effects of uncontrollable factors of rolling speed and ground pushup in a Robust Parametric Design study. For a complete study, an L 27 orthogonal array is proposed with combinations of 9 geometric design parameters considered as control factors with three levels each. To provide a robust prediction of the importance of each control factor in 16
the inner array for reducing Tweel vibrations, and determine optimal levels, four different combinations of the two noise factors of rolling speed and ground pushup are repeated. Since there will be more than one sets of uncontrollable noise factor combinations, Signal-to-Noise (S/N) ratios with Analysis of Variance (ANOVA) will be used to determine percent contributions and predict the optimal combinations of each control factor. Another goal is to study the effect of geometric variables on vertical stiffness of the Tweel using load-curves. A roadmap of the Thesis is as follows: Chapter 2 describes the characteristics of the two-dimensional Tweel finite element model created using ABAQUS. The characteristics include reference geometry parameters, material properties, analysis procedure, mesh geometry, and element type. Chapter 3 discusses the mesh and analysis procedure for computing static loaddeflection plots. The effects of adding pretension due to cooling, changes in shear beam thickness, changes in combined spoke thickness and number of spoke pairs, and inclusion of tread, on stiffness and load are studied. Chapter 4 describes the method of measurement of spoke, ring, and groundinteraction vibration measures during rolling, and the use of FFT spectrums to determine peak and RMS amplitude values for each output measure. 17
Chapter 5 presents a dynamic analysis study of Tweel for three approximately equal weight models, with different spoke thicknesses and number of spoke pairs Chapter 6 describes the steps in Taguchi s Robust Parameter Design Method and orthogonal array theory and usage. Procedures for analysis of results including deviations of level means by mean method, and Signal-to-Noise (S/N) ratio method are described. Percent contributions determined from Analysis of Variance (ANOVA) are explained. Chapter 7 presents results for two different L 8 orthogonal arrays, one with spoke and ring control factors, and other focusing only on ring variables. Chapter 8 presents results for a complete L 27 orthogonal array study with 9 geometric design parameters in the inner array and combinations of rolling speed and ground push up variables defined in an outer array. The geometric design variables are considered as controllable factors, while the speed and push up are considered as uncontrollable noise factors. Results of ANOVA and main effects for S/N ratio values are used to predict optimal combinations of levels of control factors. A confirmation experiment for this optimal combination shows all output measures are reduced as predicted. Chapter 9 gives conclusions and provides suggestions for future work. 18
CHAPTER TWO: FINITE ELEMENT MODEL FOR TWEEL TM In this chapter, the finite element model, material properties, and analysis procedures used are described. In the present work, a large number of experiments with different combinations of changing geometric variables, loads, and rolling speeds are studied. For efficient numerical solutions with reduced cost and time, a 2D planar finite element model is used to predict both vertical-load deflection curves for stiffness analysis and dynamic rolling analysis for vibration studies. In the following, only geometric properties for a Reference Tweel model are described. Changes to these geometric parameters for other experiments are discussed in later Chapters. The Reference Tweel in this study has the same geometric and material parameters as used in previous studies [21, 23], except for the inclusion of a tread and mesh refinement in the present study. Procedures for defining initial pre-tension due to cooling, loading, and rolling steps are the same. Constraints used to tie different parts in the Tweel assembly are described in [2, 6]. 2.1 2-D Planar Tweel Geometry As discussed earlier, the different parts present in the assembled Tweel model are: (a) Analytically rigid ground, (b) Spokes, (c) Ring, (d) Reinforcement, and (e) Tread. The geometry for all parts, including the assembled mesh with constraints are automatically generated in ABAQUS/CAE using a plug-in supplied by Michelin. Figure 2.1 shows all these parts in the assembly. 19
Figure 2.1: 2D-Planar Reference Tweel Model showing all the parts in the assembly A plug-in is a piece of software that installs itself into another application to extend the capabilities of that application [8]. This plug-in has two modules. The first module helps in creating a window with forms allowing the user to define most geometric variables. This module of the plug-in utilizes python scripting to create a graphical user interface. The graphical user interface window is shown in Figure 2.2. The second module accepts the variables from the graphical interface module as inputs, and creates the parts, assembly, and finite element model. 20
Figure 2.2: Tweel Plug-in Interface showing reference Tweel parameters The interface is divided into different categories: (a) Tweel dimensions, (b) shear layer dimensions, (c) spoke parameters, (d) materials selections, and (e) analysis parameters, and (f) mesh refinement. The plug-in was modified in [22] to allow for the creation of a tread and uniform meshes. The geometric and material parameters for the reference Tweel are given in Table 2.1. The material parameters in the plug-in define elastic properties for the different parts in the model. 21
Table 2.1: Geometric and Material Parameters for Reference Tweel model Geometric Section Geometric Variables Dimensions assigned Tweel Dimensions Outside diameter (Do) 593 mm Hub diameter (Dh) 410 mm Tread Thickness (Tth) 2.95 mm Shear Layer Dimensions Ring Thickness (Tr) 19.5 mm Outside Coverage (Toc) 7 mm Inside Coverage (Tic) 2.5 mm Spoke Parameters Number of Spoke Pairs 25 Spoke Thickness (Ts) 4.2 mm Spoke Curvature 8 mm Spoke Derad Outer 0.15 Spoke Derad Inner 0.6 Type of Curve: Triple or Single Triple Material Selections Ring Elastomer B836 x 1 Spoke Elastomer B836 x 1 Cable 4x26 1.8 Cable Comp. Factor 4 Marlow Law for Spokes Yes Analysis Parameters Deflection 15 2D type: Plane Strain or Plane Stress Plane Strain Mesh Refinement Coarse - Normal - Extra Fine Yes In the present work, a non-uniform mesh is used for the static load-deflection analysis, while a uniform mesh is used for the dynamic rolling analysis. The nonuniform mesh has a coarse region and a fine region. Both these meshes are discussed in detail in Section 2.4. 22
2.2 Tweel Material Properties The shear beam, inner and outer coverage s, and spokes are molded with the same polyurethane (PU) material. In the finite element model, the PU materials are modeled as isotropic, hyperelastic with nearly incompressible Marlow properties. Polyurethane materials have relatively low modulus that allows for large strain with low stress. Low stress values are important for cyclic loading during dynamic rolling to provide long fatigue life. The two reinforcing layers on the top and bottom of the shear layer are modeled with orthotropic elastic materials and defined with a very stiff modulus in the tangential (circumferential) direction. These two reinforcing layers ensure that the ring length is conserved, an important characteristic of wheel design [21, 23]. The rubber tread is bonded to the ring outer coverage and is modeled as hyperelastic with nearly incompressible Neo-Hooke properties. The same material properties are used for all models. 2.2.1 Reinforcements In the present model, the reinforcements are modeled with thin orthotropic layers of 0.66 mm thickness. The stress-strain relations for elastic orthotropic materials take the form [9]: 23
σ D D D σ D D σ D = σ σ sym σ 0 0 0 11 1111 1122 1133 11 0 0 0 22 2222 2233 22 0 0 0 33 3333 33 0 0 12 1212 12 13 1313 13 23 2323 23 D D D 0 ε ε ε γ γ γ In the above, subscript 11 corresponds to the radial direction, 22 corresponds to the tangential direction, and 33 is the out-of-plane direction. The cylindrical coordinates are defined using a local material orientation assigned to the reinforcement parts. The local material coordinates reorient to follow the deformation of elements in the reinforcement layer [21, 23]. The elastic moduli are defined in Table 2.2. The modulus D 2222, for the tangential direction, is a factor of 100 larger that the shear modulus, D 1212. The radial modulus D 1111 is small indicating negligible bending stiffness. Table 2.2: Elastic Moduli for Orthotropic Reinforcement material Moduli Value (dn/mm 2 ) D1111 0.01 D1122 0 D2222 3217.75757575758 D1133 0 D2233 0 D3333 0.01 D1212 32.1775757575758 D1313 0 D2323 0 Other important material properties for the reinforcement layer are [21, 23], 24
Mass density ρ= 10 2 4 1.54 x10 ( dn sec / mm ) Thermal Expansion Coefficient α11= α33 = 0 and α 22 = 1.2 x10 / 5 0 C Specific heat c= x J Kg K 8 4.75 10 / ( ) Thermal Conductivity k = 20 W / ( m K ) 2.2.2 Ring and Spokes As discussed earlier, the ring and spokes are molded from the same isotropic polyurethane (PU) material. The hyperelastic properties of the spokes, shear beam layer, inner coverage and outer coverage are defined using a strain energy potential based on a Marlow model. The Marlow strain energy model is recommended when a set of test data (uniaxial or biaxial) is available [8]. Michelin has supplied uniaxial stress-strain data, which was utilized in both the current and previous models [21, 23]. The stress-strain curve corresponding to the Marlow material model data is shown in Figure 2.3. Using this model, for same amount of nominal strains, compressive stresses are large when compared to tensile stress. 25
1 0.5 Nominal Stress (dn/mm 2 ) 0-0.5-1 -1.5-2 -0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 Nominal Strain Figure 2.3: Stress-strain curve for Marlow material model. Other important material properties for the PU material include [21, 23], Mass density ρ= 10 2 4 1.1x10 ( dn sec / mm ) 0 Thermal Expansion Coefficient α = 0.0002 / C Specific heat c= x J Kg K 9 2 10 / ( ) Thermal Conductivity k = 0.3 W / ( m K ) A Poisson s ratio of ν = 0.45 is used to model a nearly incompressible hyperelastic material. 26
2.2.3 Tread As discussed above, one of the differences in model used in previous study [21, 23], and model used in this present study is the inclusion of tread. The hyperelastic properties of the rubber tread are defined using a Neo-Hooke strain energy potential with coefficients C 10 = 0.0833 and D 1 = 1.241384. Michelin supplied the coefficients used to define the Neo-Hooke model. Other important material properties for the tread material include [22], Mass density ρ= 10 2 4 1.1x10 ( dn sec / mm ) 0 Thermal Expansion Coefficient α = 0.00017 / C Thermal Conductivity k = 0.3 W / ( m K ) 2.3 Rolling Analysis Procedure In the previous work [21, 23], a new step procedure has been developed for the finite element simulations. The same procedure is followed in this study. For this model, zero friction is assigned between the contact of the tread and rigid ground. This is a conservative assumption since it allows the ring to respond more freely than if it were constrained by frictional forces at the ground [21, 23]. Initial rolling is defined using a predefined velocity field for all nodes in the model relative to an angular speed of -120 rad/sec at the hub center. Different Tweels are designed for the left side and right side of the vehicle. The negative sign of the angular speed in the model corresponds to clockwise rotation of a right side Tweel in forward motion in the direction of initial spoke curvature. The next step enforces the steady rolling speed of -120 rad/sec at the hub center, which corresponds to 129.6 kph (80.4 27
mph). Cooling and loading are also combined in this step. The loading corresponds to a vertical displacement of 15mm, minus the small amount of thermal shrinkage due to cooling from 125 to 25 degrees Celsius. The load is calculated from the reaction force at the reference point on the rigid ground and corresponds to one-quarter static vehicle weight. After the final cooling and load are reached, the model rolls at steady-state. All analysis steps are performed in Abaqus/Explicit. The procedure therefore requires no restart file, which greatly increases efficiency. By starting from an initial predefined speed, and combining the cooling and loading into a common step, the procedure reduces considerably the computation time for steady-state rolling analysis. The steps used in the analysis procedure in Dynamic Explicit are as follows [21, 23] 2.3.1 Step 0: Initial This is an initial step automatically generated by ABAQUS. This step is used to define the model s starting condition in the load and interactions modules. Using a predefined field, the initial velocity of all nodes in the finite element model is defined with respect to an angular speed of -120 rad/sec at the hub center. Temperature is also defined to be 125 0 C in the predefined field of load module in this step. Starting the Tweel with an initial velocity reduces the numerical noise that would be created if the Tweel were to start at zero and have to quickly ramp up to -120 rad/sec. The analytical rigid ground is fixed in all degrees of freedom (dof) in this step. The translational dof of the hub center are also fixed for all steps. 28
2.3.2 Step 1: Initial Rolling This step is defined for a time of 0.005 sec with constant temperature 125 0 C. In this step, the hub centre angular speed (VR 3 ) is allowed to rotate freely so that it becomes closer to steady state. 2.3.3 Step 2: Rolling-Loading-Cooling This step is defined for a time of 0.1 sec. In this step the hub center is rotated with constant angular velocity VR 3 = -120 rad/sec. Temperature is decreased to 25 0 C during this step. The cooling introduces pre-stress to simulate the manufacturing process. Ground push-up is defined with a displacement of 15 mm in the positive Y-direction (upwards). The displacement is linearly ramped to 15 mm over the step by defining an instantaneous ground velocity of 150 mm/sec over this 0.1 sec step. 2.3.4 Step 3: Steady-State Rolling This step is defined for a time of 0.3 sec. The hub center continuous to rotate with constant angular velocity VR 3 = -120 rad/sec. Temperature is maintained at 25 0 C over the step. Ground is fixed in all degrees of freedom in this step. During this step, the data for output measures is collected from the output database of ABAQUS for further analysis. 2.4 Mesh properties The element type used for the 2D planar models is CPE4R (4-node bilinear plane strain quadrilateral, reduced integration, hourglass control). The simulations in this work are modeled using plane strain elements shown in Figure 2.2, similar to previous 2D 29
models [21, 23]. Previous studies have shown that the simulation results for the models with plane strain elements more closely match results from a 3D model when compared to plane stress elements. A previous study also indicated that the plane strain models required nearly half the computational time compared to plane stress models [21, 23]. Two different meshes are used for static and dynamic analysis. For dynamic rolling analysis, a uniform mesh is used with the same element size for the ring and all spoke pairs. The uniform mesh used in the present study is an optimized mesh obtained from a convergence study [22]. This optimized mesh is different from the mesh used in [21, 23]. Figure 2.4 and Figure 2.5 show the uniform mesh used for all dynamic rolling models. In Chapter 3, vertical load-deflection curves for analysis of Tweel stiffness are studied. For the static load-deflection analysis, since most of deformation occurs in the lower one-third of the model a fine mesh is used in this region, while in the other twothirds, a coarse mesh is used. Figure 2.6 and Figure 2.7 shows the coarse and fine mesh regions used for static load-deflection analysis. 30
Figure 2.4: Uniform mesh used for dynamic rolling analysis 31
Figure 2.5: Detailed view of uniform mesh used for dynamic rolling analysis 32
Figure 2.6: Model with coarse/fine mesh regions for static load-deflection analysis 33
Figure 2.7: Detailed view of course/fine regions for static load-deflection analysis 34
CHAPTER THREE: LOAD-DEFLECTION CURVES The ground reaction force is the resultant of all contact pressures of the tire tread interacting with the contacting surface. Since the ground is assumed frictionless, only the force in the vertical direction is significant. The resultant force is measured at the reference point for the ground, modeled as an analytical rigid plane. In the present study, vertical load is applied indirectly by enforcing a ground push-up step. The push-up is defined a linear displacement during the step. The force corresponding to different ground push-up values can be calculated by plotting a load-deflection curve. The slope of the load-deflection curve gives the vertical stiffness for the Tweel model. Since the loading for a typical Tweel for automotive applications is relatively large, causing geometric nonlinearity, since the materials are hyperelastic, the load-deflection curves are nonlinear. Since the load-deflection curves are nonlinear, the tangent (slope) and therefore, the stiffness, are not constant over the loading. Load-deflection curves and stiffness for the Tweel are important input parameters for vehicle dynamics models which simulate design functions such as ride comfort, handling, grip, etc. Therefore, it is also important to know the effect of changing the levels of different geometric variables on Tweel vertical stiffness. In the present study, load-deflection curves for the reference spoke geometry are compared with, (a) different ring thicknesses, (b) combined change in spoke thickness with number of spoke pairs, (c) tread and without tread. In addition, a comparison of the load-deflection curve for the reference geometry with and without cooling is studied. 35
3.1 Analysis Procedure for Load-Deflection Curves To generate load-deflection curves, a static analysis using ABAQUS-Standard is used. The procedure to build the finite element model in ABAQUS/CAE uses the same python script plug-in, as explained in the previous chapter. The analysis procedure for calculating load-deflection curves is discussed below. 3.1.1 Steps The static analysis procedure requires only two steps in the analysis instead of the three required for the dynamic analysis described in Chapter 2. For the static analysis, the Cooling-Loading step is divided into two separate steps. The first step is a cooling step followed by a second loading step with constant temperature. Both steps use the coupled temperature-displacement analysis procedure in ABAQUS-Standard. Both steps are defined as steady state with a time period of 1 sec. Initial temperature is defined to be 125 0 C with a predefined field for the whole model. 3.1.2 Boundary Conditions As explained earlier, loads are applied by moving the ground with a vertical displacement. The reaction force is calculated at the ground reference point. Three boundary conditions defined: 1. Hub center is fixed in all steps. 2. Rigid Ground is fixed during the cooling step, and then has vertical displacement of 25 mm in positive Y direction (upwards) in the loading step. 3. Temperature of the whole model decreased from 125 0 C to 25 0 C in cooling step, and then the temperature is maintained at 25 0 C in the loading step. 36
During the cooling step, the ring contracts, causing the tread to lose contact with the ground plane. As result, during loading step, there is some time for the ground to move in contact with the tread and cause Tweel deflection. To create a Tweel loaddeflection curve, the amount of ground displacement prior to contact must be subtracted to start the load-deflection curve for the deformed Tweel at zero. 3.1.3 Mesh For the coupled temperature-displacement analysis in ABAQUS-Standard, the type used for all elements in the mesh is CPE4RT (4-node plane strain thermally coupled quadrilateral, bilinear displacement and temperature, reduced integration, hourglass control). Reduced integration elements are used in order to be consistent with the elements used for the rolling models performed with dynamic ABAQUS-Explicit analysis. Reduced integration shortens computation time in the explicit rolling models. For the static load analysis, as the ground pushes into the Tweel, only the lower region is in contact with the ground, and most of the deformation occurs in the ring and collapsing spokes in this lower-portion. For efficient static analysis, a fine element mesh is used only in the lower one-third of the Tweel model. In the other two-thirds, a coarse mesh is used. Therefore, the mesh used in these studies has both, coarse and fine regions. Figure 3.1 shows the coarse and fine mesh regions. The seven spoke pairs and ring at the bottom region has a fine mesh. The other regions have a coarse mesh. 37
Figure 3.1: Deformed Tweel Model with Coarse-Fine mesh regions, after 25 mm ground push-up The fine part of the mesh has 3 elements along the thickness in each spoke, and 45 elements along the length, where as the coarse part has only 2 elements along the thickness and 20 elements along the length. In the fine mesh region, the ring variables: shear beam thickness, inner and outer coverage, reinforcements and tread has more 38
elements along the circumference, but the same number in the radial direction. Figure 3.2 shows the difference between the fine and coarse regions of the mesh. Figure 3.2: Coarse-Fine Mesh (zoomed view of elements in both fine and coarse mesh in spoke pairs and ring) 3.2 Results Results for all models are reported in increments of 0.25 mm, which corresponds to 100 increments over 25 mm push-up for 1 sec during the loading step. As discussed earlier, in order to plot the Tweel load-deflection, the gap formed by the contraction of 39
the ring due to cooling prior to loading contact is subtracted from the ground push-up displacement. For all models, the initial gap during the cooling step approximately 1.25 mm. 3.2.1 Effect of Spoke Pre-Tension In this study, load-deflection curves for two different models; one with a cooling step, and one without a cooling step. Figure 3.3 compares the load-deflection curves for both these cases. The load-deflection plot clearly shows a large difference between the reaction force values for the case with cooling, to the case without cooling, for the same displacement. During the initial small deflection, the slope with cooling is larger, indicating increased stiffness initially. However, as the deflection increases, the slopes are nearly parallel indicating approximately same stiffness. 40
6 5 with cooling without cooling Displacement Vs Load 4 Force(dN) 3 2 1 0 0 5 10 15 20 25 Displacement(mm) Figure 3.3: Load-deflection curves of two models one with cooling step and other without cooling step 3.2.2 Effect of Shear Beam Thickness In this study, the load-deflection curves for different shear beam thicknesses, with 8 mm, 10 mm and 12 mm are compared. Figure 3.4 shows the load-deflection curves for the three models. 41
7 6 Displacement Vs Load 8 mm shear beam thickness 10 mm shear beam thickness 12 mm shear beam thickness 5 Force(dN) 4 3 2 1 0 0 5 10 15 20 25 Displacement(mm) Figure 3.4: Load-deflection curves of with three different shear beam thicknesses The load-deflection plot shows a large difference between the reaction force and slopes (stiffness) for values of 8 mm, 10 mm and 12 mm shear beam thickness cases, for the same Tweel deflection. The difference is increasing proportionally with the increase in displacement. In particular, for the reference Tweel geometry (10 mm shear beam thickness), at 15 mm displacement corresponding to a one-quarter vehicle load of approximately 4 dn, the three cases have significantly different values for load. In summary, the reaction force values increase proportionally for the corresponding displacements with increase in the shear ring thickness. 42
3.2.3 Effect of Spoke Thickness and Number of Spoke Pairs Increasing the number of spoke pairs, while keeping all other variables fixed, increases Tweel stiffness. Likewise, with all other variables fixed, increasing the spoke thickness increases stiffness. The effect of changing the spoke thickness and number of spoke pairs combined in the model is studied. In order for each model considered to have approximately the same total weight, models with larger numbers of spoke pairs have thinner spokes. Recall the reference Tweel has 25 spoke pairs with 4.2 mm thickness. For comparison, a model with 20 spoke pairs and spoke thickness (5.04 mm) is considered, together with a model with 30 spoke pairs and spoke thickness (3.36 mm). Figure 3.5 shows the load-deflection curves for all the three models. 6 5 20 pairs and 5.04 mm thick 25 pairs and 4.2 mm thick 30 pairs and 3.36 mm thick 4 Force (dn) 3 2 1 0 0 5 10 15 20 25 Displacement (mm) Figure 3.5: Load-deflection curves with three different spoke thicknesses and number of spoke pairs. 43
From the load-deflection plot, the model with fewer spoke pairs combined with thicker spokes has increased load and stiffness. That is, for a corresponding displacement value, the three models result in three different reaction force values, and, the trend is as the spoke thickness is increasing, the reaction force is also increasing. This shows that the Tweel stiffness is dominated by the change in spoke thickness compared to the change in number of spoke pairs. Therefore, even at 15 mm push-up, the three cases have a significantly different value of reaction force. 3.2.4 Effect of Tread In this study, the effect of a model with tread and a model without tread are compared. All other geometric variables are set to the reference model values. Figure 3.6 shows the load-deflection curves for these two models. 44
6 5 Without Tread With Tread 4 Force (dn) 3 2 1 0 0 5 10 15 20 25 Displacement (mm) Figure 3.6: Load-deflection curves for Tweel model with and without tread. From the load-deflection plot, it is clear that the reaction force values for the corresponding displacement are almost same in both the models; one with thread, and other, without tread. That is, the inclusion of tread in the Tweel model does not significantly change the stiffness in the model. As discussed earlier, the rubber in the tread is modeled with a hyperelastic material. With this material model, on unloading, the same force-displacement curve would result. In real rubber materials there will be some viscoelastic effects which would result in hysteresis upon unloading. Characterization of the viscoelastic behavior of rubbers requires extensive testing and increases computation time during finite element analysis. In the present work, to reduce 45
the computation time and to eliminate uncertainties in characterizing the experimental material model, hyperelastic materials are used. 3.3 Conclusions Results from the load-deflection curves show that the vertical reaction force values and stiffness increase proportionally with increase in the shear ring thickness. Increasing the number of spoke pairs, while keeping all other variables fixed, increases Tweel stiffness. The combined effect of changing the spoke thickness and number of spoke pairs such that total masses are equal shows the model with fewer spoke pairs combined with thicker spokes has increased load and stiffness. Since we are increasing the number of spoke pairs and stiffness is not increasing, this shows that the Tweel stiffness is dominated by the change in spoke thickness compared to the change in number of spoke pairs 46
CHAPTER FOUR: MEASURES OF TWEEL VIBRATION Potential sources of noise include spoke vibration, ring vibration, and vibration of ground contact. During rolling, the spokes buckle and collapse as they pass through the contact region at Tweel bottom and then snap back into tension as they rotate around the top of the wheel. The relatively sudden change from compression to tension can create vibrations in the spokes, which could be a source of sound radiation and noise. Due to the discrete spacing of the spokes around the ring, spoke length changes can interact and cause ring vibration modes. Another possible contributor to noise may be the bending and shear in the ring as passes through the transition region from round to flat in the contact region can excite elastic waves and vibration. Vibrations of the ring and tread can interact and cause fluctuations in contact forces, which may create noise. In previous studies, [2, 6, 7, 10], only spoke vibrations were investigated as a possible source of noise from the Tweel during high-speed rolling. In [21, 23], vibrations in ring and ground force were also measured. Similar to the previous work, in this study also, the vibration was measured during steady state rolling for: (a) perpendicular distances of spoke marker nodes from the plane of the spoke, (b) ring thickness changes, and (c) changes in ground vertical reactions. Changes in ground reactions forces are intended to measure spoke-to-spoke interactions and spoke passing frequencies during rolling. Similar to [21, 23], time signals are processed using FFT for analysis of frequency response. FFT results are reported based on mean signals and a zero padded FFT with Hamming window for improved resolution of harmonics [11]. 47
The mean signal removes the static response at zero frequency in the FFT. Magnitudes of amplitude peaks and RMS of the FFT results will be used to quantify the level of vibration. These results are used as output measures in the design studies discussed in later chapters. Using the rolling analysis procedure discussed in Section 2.3, FFT magnitudes are computed for zero mean signals for the steady-state rolling step (total time from 0.105 sec. to 0.405 seconds). For an angular speed of 120 rad/sec (approximately 129 km/hr), the Tweel makes approximately five revolutions during the rolling step. Following [21], all displacement data is collected at increments of 0.0002 sec, for a sample rate of 5000 Hz during the rolling step. With this resolution, the Nyquist cutoff frequency is 2500 Hz. Figure 4.1: The final deformation in rolling step at 0.405 sec in the reference model 48
Figure 4.1 shows the deformed geometry of the Reference Tweel with uniform mesh at the end of the Rolling Step. The geometric parameters for the Reference Tweel were discussed earlier in Chapter 2. The Reference model will serve as the benchmark for comparisons with geometric design variable studies. In the following the output measures of FFT amplitudes for spoke, ring, and ground force vibration are explained using the Reference model. These same output measures will be used for the design studies in later chapters. 4.1 Spoke Vibration As discussed earlier, it has been hypothesized that noise may be generated during Tweel operation produced by vibration caused by the spokes buckling and snapping back into tension at a high rate of speed [2]. To measure this vibration, the magnitude of the lateral movement of spokes from a virtual plane passing through the spoke ends is calculated. This perpendicular distance from the spoke plane is computed from displacements of marker nodes along the length of the spoke [2,6,21]. The marker nodes are located at the top, upper quarter point, midpoint, lower point and bottom of the spoke. These five marker nodes can be seen in Figure 4.2. Figure 4.3 shows a detailed view of these marker nodes for both undeformed and deformed geometry. The marker nodes are located on the inside edge of the right spoke in a spoke pair. It is assumed that the vibration on the other spoke in the shared spoke pair is similar. 49
Figure 4.2: Spoke marker nodes shown in Red. 50
(a) Figure 4.3: Spoke marker nodes shown by red dots: (a) undeformed, (b) deformed. (b) Figure 4.4 shows a plot of the distance between the top and bottom marker node during the rolling step, which indicates change in spoke length. This plot is obtained from the displacement values of the top and bottom marker nodes during the rolling step. The profile of spoke length shows a transition from compression during buckling in the contact region to tension as it rotates around the sides and top. 51
74 72 70 Length (mm) 68 66 64 62 60 0.15 0.2 0.25 0.3 0.35 0.4 time (sec) Figure 4.4: Distance between the top and bottom nodes of spoke for the rolling step; measures spoke length change. An imaginary line (in red color), shown in Figure 4.3, is created by connecting the top node and the bottom node of the spoke. The perpendicular distance of each of the three marker nodes (as top and bottom are fixed to the imaginary line) from the line is recorded with respect to time. This procedure gives three sets of vibration data, one for each of the three marker nodes. These three sets of data can be seen in Figure 4.5 for the rolling step. 52
Perpendicular Distance of Vertical Marker Nodes from Plane of Spoke 0-2 Perpendicular Distance (mm) -4-6 -8-10 -12-14 -16-18 Upper Quarter Node Middle Node Lower Quarter Node 0.15 0.2 0.25 0.3 0.35 0.4 time (sec) Figure 4.5: Perpendicular distance of each marker node from the plane of the spoke during rolling step In the plot, it is clear that the perpendicular distance of upper quarter node and lower quarter node is almost same, at all times of the rolling step, and, the distance of the middle node is more than these two nodes all the times. The difference can be explained because of the curvature of the spoke in the model. The middle node is already far from the plane of the spoke compared to the upper quarter node and lower quarter node. For the same reason, the middle node also has a larger peak displacement during spoke collapse than the other two nodes. The large negative displacements are seen every 0.052 53
seconds corresponding to the spoke collapsing in the contact patch, with frequency f = (120 rad/sec)/(2π) = 19.1 Hz. 4.1.1 Maximum peak amplitude of spoke vibration Similar to the previous work [21, 23], in this work also, the frequencies at the dominate peaks and amplitudes of spoke vibration are found using a MATLAB program that performs a Fast Fourier Transform (FFT) on the spoke displacement data [21, 23]. As discussed earlier, the FFT results in this work are reported using a Hamming Window. The Hamming window should give a clearer representation of the frequencies and amplitudes of vibration. An FFT spectrum of the upper quarter node displacement for the rolling step corresponding to the perpendicular distance time signal in Figure 4.5 is shown in Figure 4.6. 54
700 Magnitude of Hamming Window with Zero Padding 600 X: 38.45 Y: 615.3 500 400 X 300 200 100 X: 286.6 Y: 113.3 X: 611.3 Y: 198.4 0 0 500 1000 1500 Frequency f (Hz) Figure 4.6: FFT spectrum of Perpendicular Distance of the Upper Quarter Node for the Reference Tweel model The figure shows three amplitude peaks occurring at, 38 Hz, 286 Hz, and 611 Hz. The 38 Hz peak shows overtones of the multiples of 19 Hz, until they diminish around 200 Hz. The 19 Hz peaks are a result of the rotational speed of the Tweel, and correspond to the spoke collapses every 0.05 seconds shown in Figure 4.5. Because this 19 Hz peak is a result of the spoke collapse profile when transitioning from tension to collapse in the contact region, its amplitude will not be considered to generate any significant noise [21, 23]. As considered in the previous work [21, 23], sound pressure levels (SPL) at frequencies below 100 Hz do not have significant impact on human perception of noise. It is common practice to significantly attenuate decibel (db) levels of 55
SPL at low frequencies as measured by ISO A-weighted db curves. The two vibration amplitude peaks of interest, which may result in significant noise perception, are the 286 Hz peak and the 611 Hz peak. The FFT plots are capable of locating peaks up to the Nyquist Frequency. As discussed earlier, since the sampling rate of the data is 5000 Hz, the Nyquist Frequency is 2500 Hz. Figure 4.6 shows the FFT up to 1500 Hz, because for this signal there are no significant amplitude peaks for frequencies greater than 1500 Hz. Table 4.1: FFT Peak Amplitudes of spoke vibration for all the marker nodes during steady rolling 120 rad/sec (Reference model) Marker Nodes Peak Amplitudes RMS Amplitude 1st Peak (286 Hz) 2nd Peak (611 Hz) Upper Quarter Node 113 198 26.1 Middle Node 203 25.9 Lower Quarter Node 153 206 28.4 For all the three marker nodes, the peak amplitudes and corresponding frequency are noted. Table 4.1 shows all these values. The second peak does not appear for the middle marker node suggesting that the vibration mode associated with the second peak is excited with zero amplitude at the middle node. When measuring peak amplitudes, the maximum amplitude value over the three marker nodes is selected as the maximum peak amplitude of spoke vibration. In this case, the value is 206. 56
4.1.2 RMS amplitude of spoke vibration Another measure used to quantify the amount of vibration is the RMS amplitude over the frequency range from 200 Hz to 1000 Hz. The calculation of RMS values uses the formula [21], N 1 RMS = ( xi ) N i = 1 2 where N is the total number of intervals in the step, and x i is the data at the i th interval. This RMS amplitude output measure is also used to compare the magnitude of vibration between different design experiments. The RMS value gives a measure of overall vibration magnitude over the entire frequency range whereas the max peak amplitude measures the intensity of vibration at localized frequencies. Both measures are important in characterizing the amplitude of spoke vibration. The RMS values are given in Table 4.1. The maximum amplitude value in all these three cases is selected as the RMS amplitude of spoke vibration. In this case, the value is 28.4. 4.2 Ring Vibration Similar to the spoke vibration, ring vibration also has two measures of vibration, maximum peak amplitude and RMS amplitude. The magnitude of ring vibration will be determined using the two marker nodes shown in Figure 4.7. These nodes are located inside a spoke pair on top surface of the inner coverage and the bottom surface of the outer coverage. The lower node is not selected on the tread and selected on the outer coverage. 57
Figure 4.7: Ring marker nodes (nodes marked in red color) 58
Figure 4.8: Detailed view of Ring marker nodes (nodes marked in red color) The distance between the top and bottom marker node of the ring are used to measure changes in the ring thickness and will be measured over time. The sampling rate and Nyquist Frequency for the ring vibration data is once again 5000 Hz and 2500 Hz, respectively. A set of ring thickness data inside a spoke pair defined by the distance between the top and bottom ring marker nodes for the reference Tweel model are shown in Figure 4.7 and Figure 4.8. 59