Vibrations of mechanical systems with regular structure (eBook)

Preface 5
Contents 7
Chapter1 Introduction 12
Chapter2 Mechanical Vibratory Systems with Hierarchical Structure. Simulation and Calculation Methods 18
2.1 Introduction 18
2.2 Models of Mechanical Systems with Lumped Parameters 19
2.2.1 Coefficients of Static Stiffness and Compliance 20
2.2.2 Static Stiffness Coefficients for a Beam 22
2.3 Reduction of Models with Lumped Parameters 24
2.4 Coefficients of Dynamic Stiffness and Compliance 26
2.4.1 Coefficients of Dynamic Stiffness 27
2.4.2 Coefficients of Dynamic Compliance 28
2.5 Dimension Reduction of Dynamic Compliance Matrix 32
2.6 Determining Dynamic Compliance Using Experimental Methods 33
2.7 Fundamentals of Finite-Element Method. Analytical Approaches 36
2.7.1 Stiffness Matrix for Beam Finite Element 37
2.7.2 Stiffness Matrix for Assembled System 39
2.7.3 Boundary Conditions and Various Ways of Subsystems Connecting 40
2.8 Decomposition Methods Taking into Account Weak Interactions Between Subsystems 41
2.8.1 Coefficients of Dynamic Interactions 42
2.8.2 Decomposition by Partition into Independent Substructures 45
2.8.3 Other Decomposition Methods 47
2.8.4 Coefficients of Weak Interaction and Criteria for Ill -- Conditions of Matrices 48
Part I Systems with Lumped Parameters 49
Chapter3 Vibrations of Regular Systems with Periodic Structure 50
3.1 Introduction: Some Specific Features of Mechanical Systems 50
3.2 Wave Approach at Vibrations of Mechanical Systems with Periodic Structure 51
3.3 Vibrations of Frames with Periodic Structure 56
3.3.1 Combining Finite Elements Method and Dispersion Equation 56
3.3.2 Vibrations of Grate Frames 59
3.4 Dynamic Properties of Laminar Systems with Sparsely Positioned Laminar Ribbing 61
3.4.1 Dispersion Equation for Ribbed Laminar Systems: Conditions for Possibility of Continualization 62
3.4.2 Vibrations of a Single-Section Lamina with Laminar Ribbing: Comparison of Discrete and Continuous Models 67
3.5 Finite-Element Models for Beam Systems: Comparison with Distributed Parameters Models 70
3.5.1 Dispersion Equation for FE-Model of Beam 70
3.5.2 Comparing Models with Distributed Parameters and Finite Elements Models at Different FE-Mesh 73
3.5.3 Beam Systems 77
3.6 Hierarchy of Mathematical Models: Superposition of Wave Motions 79
3.7 Vibrations of Self-Similar Structures in Mechanics 82
3.7.1 Self-Similar Structures: Basic Concepts 82
3.7.2 Vibrations in Self-Similar Mechanical Structures: Dispersion Equation 83
Chapter4 Vibrations of Systems with Geometric Symmetry. Quasi-symmetrical Systems 88
4.1 Introduction 88
4.2 Basic Information about Theory of Groups Representation 89
4.2.1 Basic Concepts and Definitions 89
4.2.2 Examples of Applying Groups Representation Theory 92
4.3 Applying Theory of Group Representation to Mechanical Systems: Generalized Projective Operators of Symmetry 95
4.3.1 Features of Mechanical Systems with Symmetric Structure 95
4.3.2 Generalized Projective Operators and Generalized Modes 96
4.4 Vibrations of Frames with Cyclic Symmetry 98
4.4.1 Stiffness and Inertia Matrices 98
4.4.2 Projective Operators for Frame: Generalized Modes 100
4.4.3 Analysis of Forced Vibrations 104
4.5 Effect of FE-Mesh on Matrix Structure 104
4.5.1 The Square Frame: Generalized Modes 105
4.6 Vibroisolation of Body on Symmetrical Frame: Vibrations Interaction 107
4.7 Quasi-symmetrical Systems 109
4.7.1 Vibrations Interaction at Slight Asymmetry 109
4.7.2 Quasi-symmetrical Systems: Free Vibrations 111
4.7.3 Quasi-symmetrical Systems: Forced Vibrations 113
4.8 Hierarchy of Symmetries: Multiplication of Symmetries 114
4.9 Periodic Systems Consisting from Symmetrical Elements 116
4.10 Generalized Modes in Planetary Reduction Gear due to Its Symmetry 117
4.10.1 Dynamic Model of Planetary Reduction Gear 117
4.10.2 Generalized Normal Modes in Planetary Reduction Gear: Decomposition of Stiffness Matrix 118
4.10.2.1 Vibration Types in Satellite Subsystems 119
4.10.3 Free Vibrations 120
4.10.4 Forced Vibrations due to Slight Error in Engagement 121
4.10.5 Vibrations Interaction at Violation of Symmetry 122
Part II Systems with Distributed Parameters 124
Foreword to Part II 124
Chapter5 Basic Equations and Numerical Methods 125
5.1 Elementary Cells: Connectedness 125
5.2 Fundamental Matrices for Systems with Regular Structure 126
5.2.1 Matrices of Dynamic Compliance and Dynamic Stiffness 126
5.2.2 Mixed Dynamic Matrix 127
5.2.3 Transition Matrix 129
5.3 Finite Difference Equations 130
5.4 Mixed Dynamic Matrix as Finite Difference Equation 132
5.5 Transmission Matrix 133
Chapter6 Systems with Periodic Structure 134
6.1 Introduction 134
6.2 Dynamic Compliances and Stiffness for Systems with Periodic Structure 136
6.3 Dynamic Compliances of Single-Connectedness System 138
6.4 Transition Matrix 142
6.5 Forced Vibrations 143
6.6 Vibrations of Blades Package 144
6.7 Collective Vibrations of Blades 146
Chapter7 Systems with Cyclic Symmetry 148
7.1 Natural Frequencies and Normal Modes for Systems with Cyclic Symmetry 148
7.1.1 Natural Frequencies 148
7.1.2 Normal Modes 151
7.2 Vibrations of Blades System 151
7.2.1 Different Designs of Blades Connecting 151
7.2.2 Natural Frequencies for Blades System 152
7.2.3 Normal Modes for Blades System 153
7.3 Numerical and Experimental Results for Blades with Shroud 154
7.3.1 Free Ring Connection 154
7.3.2 Blades with Paired-Ring Shroud 156
7.3.3 Blades Shrouded by Shelves 158
Chapter8 Systems with Reflection Symmetry Elements 161
8.1 Reflection Symmetry Element and Its Dynamic Characteristics 161
8.1.1 Dynamic Stiffness and Compliance Matrices for Reflection Symmetry Element 163
8.1.2 Mixed Matrix for Reflection Symmetry Element 164
8.2 Finite Differences Equations 167
8.3 Special Types of Boundary Conditions 170
8.3.1 Nonclosed Systems 170
8.3.2 Closed Systems 173
8.4 Filtering Properties of System with Reflection Symmetry Elements 175
8.5 Numerical Examples 176
8.5.1 Single-Connectedness Systems 176
8.5.2 Two-Connectedness Systems 182
8.5.3 Three-Connectedness System 185
8.6 Systems Consisting of Skew-Symmetric (Antisymmetric) Elements 188
Chapter9 Self-Similar Structures 192
9.1 Introductory Part: Examples of Self-Similar Mechanical Structures 192
9.2 Dynamic Compliances of Self-Similar Systems 193
9.3 Vibrations of Self-Similar Systems: Numerical Examples 195
9.4 Transition Matrix 196
9.5 Self-Similar Systems with Similar Matrix of Dynamic Compliance 198
9.6 Vibrations of Self-Similar Shaft with Disks 199
9.7 Vibrations of Self-Similar Drum-Type Rotor 201
Chapter10 Vibrations of Rotor Systems with Periodic Structure 205
10.1 Rotor Systems with Periodic Structure with Disks 205
10.2 Rotor with Arbitrary Boundary Conditions: Natural Frequencies and Normal Modes 209
Chapter11 Vibrations of Regular Ribbed Cylindrical Shells 212
11.1 General Theory of Shells 212
11.2 Dynamic Stiffness and Transition Matrix for Closed Cylindrical Shells 214
11.3 Dynamic Stiffness and Transition Matrix for Cylindrical Panel 217
11.4 Dynamic Stiffness and Transition Matrices for Circular Ring with Symmetric Profile 218
11.5 Vibrations of Cylindrical Shell with Ring Ribbing under Arbitrary Boundary Conditions 220
11.6 Vibrations of Cylindrical Shell with Longitudinal Ribbing of Nonsymmetric Profile 224
11.6.1 Dynamic Stiffness and Transition Matrices for Longitudinal Stiffening Ribs 225
11.6.2 Numerical Calculation of Shell with Longitudinal Ribbing 226
Appendix A Stiffness and Inertia Matrices for a Ramified System Consisting of Rigid Bodies Connected by Beam Elements 228
Appendix B Stiffness Matrix for Spatial Finite Element 234
Appendix C Stiffness Matrix Formation Algorithm for a Beam System in Analytical Form 235
Appendix D Stiffness Matrices for a Planetary Reduction Gear Subsystems 237
References 238
Index 243