FEM for Springs
Bearbeitet von
Masayoshi Shimoseki, Toshio Hamano, Toshiyuki Imaizumi
1. Auflage 2003. Buch. xiv, 234 S. Hardcover ISBN 978 3 540 00046 4
Format (B x L): 15,5 x 23,5 cm Gewicht: 1170 g
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Contents
1 SPRINGS AND ELASTIC COMPONENT 1
1.1 SPRING CONSTANT . . . 1
1.1.1 Definition of the Spring . . . 1
1.1.2 Tangential Gradient . . . 2
1.1.3 System of Multiple Degrees of Freedom . . . 2
1.2 ELASTIC COMPONENT IN A VIBRATION SYSTEM . . . 3
1.2.1 Vibration Equations . . . 3
1.2.2 Chords of a Guitar . . . 4
1.2.3 Wave Equation . . . 6
1.3 FIRST STAGES OF ANALYSIS . . . 7
1.3.1 Orientation . . . 7
1.3.2 Steps of Analysis . . . 8
1.3.3 A Pitfall in the Approximate Solution . . . 9
1.3.4 Element Stiffness of Elastic Component . . . 10
1.4 ONE-DIMENSIONAL COMBINATION OF COMPONENTS . . . 11
1.4.1 Coupling Between Components . . . 11
1.4.2 Generalized Matrix Equation for Coupled Elastic Components 13 1.4.3 Verification of Boundary Condition Type . . . 14
1.4.4 Parallel Coupling of Elastic Components . . . 16
1.4.5 Transverse Stiffness of Elastic Components . . . 17
1.5 PLANE STRUCTURES . . . 19
1.5.1 Transformation of Coordinates . . . 19
1.5.2 Obliquely Connected Components . . . 21
1.5.3 From Components to Finite Elements . . . 25
2 OUTLINE OF FINITE ELEMENT METHOD (FEM) 29 2.1 FUNDAMENTALS OF ELASTO-PLASTICITY DYNAMICS . . . 29
2.1.1 Viewpoint of Continuum Dynamics . . . 29
2.1.2 General Equations . . . 30
2.1.3 Basic Equations for Linear Elastic Body . . . 31
2.1.4 Principle of Virtual Work . . . 33
2.2 EXPANSION TO NONLINEAR PROBLEMS . . . 34
vi CONTENTS
2.2.1 Geometrical Nonlinearity . . . 34
2.2.2 Material Nonlinearity . . . 36
2.3 EXPANSION TO DYNAMIC PROBLEMS . . . 38
2.3.1 Mass and Damping Matrix . . . 38
2.3.2 Natural Frequency . . . 40
2.3.3 Simulation . . . 44
2.4 SPATIAL DISCRETIZATION . . . 45
2.4.1 Derivating Procedure of Element Stiffness . . . 45
2.4.2 Stiffness of Truss Elements . . . 46
2.4.3 Element Stiffness of Plane Stress . . . 48
2.4.4 Element Stiffness of a Three-dimensional Elastic Body . . . 50
3 ROLE OF FEM IN SPRING ANALYSIS 53 3.1 COMPARISON OF FEM WITH CONVENTIONAL DESIGN METH- ODS . . . 54
3.1.1 Assumption in Model Construction . . . 54
3.1.2 From Linear to Nonlinear . . . 56
3.2 THE UTILIZATION OF FEM SOFTWARE . . . 57
3.2.1 Use of Commercial Software . . . 57
3.2.2 Selection of Commercial Software . . . 58
3.2.3 Development of Dedicated Programs . . . 59
3.3 EFFECTIVENESS IN DESIGN PRACTICE . . . 60
3.3.1 Single Spring and Peripheral Parts . . . 60
3.3.2 Simulation of the Manufacturing Process . . . 61
3.4 PROSPECT OF FUTURE APPLICATION . . . 63
3.4.1 Optimum Design . . . 63
3.4.2 Nonlinear Problems in Manufacturing Simulation . . . 64
3.4.3 Necessity of Material Data . . . 66
4 CLASSIFICATION AND APPLICATION OF ELEMENT 69 4.1 INTRODUCTION OF VARIOUS ELEMENTS . . . 69
4.1.1 Beam Elements . . . 69
4.1.2 Plate Elements . . . 72
4.1.3 Axisymmetric Elements . . . 74
4.1.4 Cubic Elements (Solid Elements) . . . 76
4.1.5 Contact Elements . . . 77
4.2 SELECTION OF ELEMENT AND DISCRETIZING PRACTICE . 78 4.2.1 Selection of Elements . . . 78
4.2.2 Tips on Discretization . . . 80
5 ELEMENTARY ANALYSIS 83 5.1 FORMED WIRE SPRINGS . . . 84
5.2 STABILIZER BARS . . . 94
5.3 HELICAL COMPRESSION SPRINGS . . . 100
5.3.1 Static Analysis . . . 101
CONTENTS vii
5.3.2 Analysis of Eigenvalue . . . 119
5.4 HELICAL EXTENSION SPRINGS . . . 121
5.5 HELICAL TORSION SPRINGS . . . 125
5.6 SPIRAL SPRINGS . . . 128
5.7 LEAF SPRINGS . . . 137
5.8 FLAT SPRINGS . . . 151
5.9 STRESS CONCENTRATION . . . 155
5.9.1 Stress Concentration on the Periphery of a Center Bolt Hole for Leaf Springs . . . 155
5.9.2 Stress Concentration at the Slit Bottom of a Disc Spring . . 159
5.9.3 Stress Concentration at the End of a Torsion Bar Spring . . 163
6 EXPANSION OF ANALYTICAL HANDLING 169 6.1 STABILIZER BARS . . . 170
6.1.1 Tubular Stabilizer Bars . . . 171
6.1.2 Effect of Bush . . . 176
6.2 HELICAL COMPRESSION SPRINGS . . . 180
6.2.1 Problem of Contact . . . 181
6.2.2 Non-circular Cross Section . . . 188
6.2.3 Presetting . . . 193
6.2.4 Surging . . . 198
6.3 LEAF SPRINGS . . . 201
6.3.1 RBA Type Leaf Springs . . . 202
6.3.2 Effect of Shackle and Contact Plate . . . 205
6.3.3 Hysteresis Characteristic . . . 209
6.3.4 Wind-up . . . 213
6.3.5 Presetting . . . 215
6.4 DISC SPRINGS, RING SPRINGS . . . 219
6.4.1 Disc Springs . . . 219
6.4.2 Ring Springs . . . 223
INDEX 229
