Banakh / Kempner Vibrations of mechanical systems with regular structure

A Group Theoretical Approach with Engineering Applications
1. Auflage 2010
ISBN: 978-3-642-03126-7
Verlag: Springer
Format: PDF
Kopierschutz: Wasserzeichen (»Systemvoraussetzungen)

E-Book, Englisch, 252 Seiten

In this book, regular structures are de ned as periodic structures consisting of repeated elements (translational symmetry) as well as structures with a geom- ric symmetry. Regular structures have for a long time been attracting the attention of scientists by the extraordinary beauty of their forms. They have been studied in many areas of science: chemistry, physics, biology, etc. Systems with geometric symmetry are used widely in many areas of engineering. The various kinds of bases under machines, cyclically repeated forms of stators, reduction gears, rotors with blades mounted on them, etc. represent regular structures. The study of real-life engineering structures faces considerable dif culties because they comprise a great number of working mechanisms that, in turn, consist of many different elastic subsystems and elements. The computational models of such systems represent a hierarchical structure and contain hundreds and thousands of parameters. The main problems in the analysis of such systems are the dim- sion reduction of model and revealing the dominant parameters that determine its dynamics and form its energy nucleus. The two most widely used approaches to the simulation of such systems are as follows: 1. Methods using lumped parameters models, i.e., a discretization of the original system and its representation as a system with lumped parameters [including nite-element method (FEM)]. 2. The use of idealized elements with distributed parameters and known analytical solutions for both the local elements and the subsystems.

Autoren/Hrsg.

Banakh, Ludmilla

Kempner, Mark

Inhaltsverzeichnis

1;Preface;5
2;Contents;7
3;Chapter1 Introduction;12
4;Chapter2 Mechanical Vibratory Systems with Hierarchical Structure. Simulation and Calculation Methods;18
4.1;2.1 Introduction;18
4.2;2.2 Models of Mechanical Systems with Lumped Parameters;19
4.2.1;2.2.1 Coefficients of Static Stiffness and Compliance;20
4.2.2;2.2.2 Static Stiffness Coefficients for a Beam;22
4.3;2.3 Reduction of Models with Lumped Parameters;24
4.4;2.4 Coefficients of Dynamic Stiffness and Compliance;26
4.4.1;2.4.1 Coefficients of Dynamic Stiffness;27
4.4.2;2.4.2 Coefficients of Dynamic Compliance;28
4.5;2.5 Dimension Reduction of Dynamic Compliance Matrix;32
4.6;2.6 Determining Dynamic Compliance Using Experimental Methods;33
4.7;2.7 Fundamentals of Finite-Element Method. Analytical Approaches;36
4.7.1;2.7.1 Stiffness Matrix for Beam Finite Element;37
4.7.2;2.7.2 Stiffness Matrix for Assembled System;39
4.7.3;2.7.3 Boundary Conditions and Various Ways of Subsystems Connecting;40
4.8;2.8 Decomposition Methods Taking into Account Weak Interactions Between Subsystems;41
4.8.1;2.8.1 Coefficients of Dynamic Interactions;42
4.8.2;2.8.2 Decomposition by Partition into Independent Substructures;45
4.8.3;2.8.3 Other Decomposition Methods;47
4.8.4;2.8.4 Coefficients of Weak Interaction and Criteria for Ill -- Conditions of Matrices;48
5;Part I Systems with Lumped Parameters;49
5.1;Chapter3 Vibrations of Regular Systems with Periodic Structure;50
5.1.1;3.1 Introduction: Some Specific Features of Mechanical Systems;50
5.1.2;3.2 Wave Approach at Vibrations of Mechanical Systems with Periodic Structure;51
5.1.3;3.3 Vibrations of Frames with Periodic Structure;56
5.1.3.1;3.3.1 Combining Finite Elements Method and Dispersion Equation;56
5.1.3.2;3.3.2 Vibrations of Grate Frames;59
5.1.4;3.4 Dynamic Properties of Laminar Systems with Sparsely Positioned Laminar Ribbing;61
5.1.4.1;3.4.1 Dispersion Equation for Ribbed Laminar Systems: Conditions for Possibility of Continualization;62
5.1.4.2;3.4.2 Vibrations of a Single-Section Lamina with Laminar Ribbing: Comparison of Discrete and Continuous Models;67
5.1.5;3.5 Finite-Element Models for Beam Systems: Comparison with Distributed Parameters Models;70
5.1.5.1;3.5.1 Dispersion Equation for FE-Model of Beam;70
5.1.5.2;3.5.2 Comparing Models with Distributed Parameters and Finite Elements Models at Different FE-Mesh;73
5.1.5.3;3.5.3 Beam Systems;77
5.1.6;3.6 Hierarchy of Mathematical Models: Superposition of Wave Motions;79
5.1.7;3.7 Vibrations of Self-Similar Structures in Mechanics;82
5.1.7.1;3.7.1 Self-Similar Structures: Basic Concepts;82
5.1.7.2;3.7.2 Vibrations in Self-Similar Mechanical Structures: Dispersion Equation;83
5.2;Chapter4 Vibrations of Systems with Geometric Symmetry. Quasi-symmetrical Systems;88
5.2.1;4.1 Introduction;88
5.2.2;4.2 Basic Information about Theory of Groups Representation;89
5.2.2.1;4.2.1 Basic Concepts and Definitions;89
5.2.2.2;4.2.2 Examples of Applying Groups Representation Theory;92
5.2.3;4.3 Applying Theory of Group Representation to Mechanical Systems: Generalized Projective Operators of Symmetry;95
5.2.3.1;4.3.1 Features of Mechanical Systems with Symmetric Structure;95
5.2.3.2;4.3.2 Generalized Projective Operators and Generalized Modes;96
5.2.4;4.4 Vibrations of Frames with Cyclic Symmetry;98
5.2.4.1;4.4.1 Stiffness and Inertia Matrices;98
5.2.4.2;4.4.2 Projective Operators for Frame: Generalized Modes;100
5.2.4.3;4.4.3 Analysis of Forced Vibrations;104
5.2.5;4.5 Effect of FE-Mesh on Matrix Structure;104
5.2.5.1;4.5.1 The Square Frame: Generalized Modes;105
5.2.6;4.6 Vibroisolation of Body on Symmetrical Frame: Vibrations Interaction;107
5.2.7;4.7 Quasi-symmetrical Systems;109
5.2.7.1;4.7.1 Vibrations Interaction at Slight Asymmetry;109
5.2.7.2;4.7.2 Quasi-symmetrical Systems: Free Vibrations;111
5.2.7.3;4.7.3 Quasi-symmetrical Systems: Forced Vibrations;113
5.2.8;4.8 Hierarchy of Symmetries: Multiplication of Symmetries;114
5.2.9;4.9 Periodic Systems Consisting from Symmetrical Elements;116
5.2.10;4.10 Generalized Modes in Planetary Reduction Gear due to Its Symmetry;117
5.2.10.1;4.10.1 Dynamic Model of Planetary Reduction Gear;117
5.2.10.2;4.10.2 Generalized Normal Modes in Planetary Reduction Gear: Decomposition of Stiffness Matrix;118
5.2.10.2.1;4.10.2.1 Vibration Types in Satellite Subsystems;119
5.2.10.3;4.10.3 Free Vibrations;120
5.2.10.4;4.10.4 Forced Vibrations due to Slight Error in Engagement;121
5.2.10.5;4.10.5 Vibrations Interaction at Violation of Symmetry;122
6;Part II Systems with Distributed Parameters;124
6.1;Foreword to Part II;124
6.2;Chapter5 Basic Equations and Numerical Methods;125
6.2.1;5.1 Elementary Cells: Connectedness;125
6.2.2;5.2 Fundamental Matrices for Systems with Regular Structure;126
6.2.2.1;5.2.1 Matrices of Dynamic Compliance and Dynamic Stiffness;126
6.2.2.2;5.2.2 Mixed Dynamic Matrix;127
6.2.2.3;5.2.3 Transition Matrix;129
6.2.3;5.3 Finite Difference Equations;130
6.2.4;5.4 Mixed Dynamic Matrix as Finite Difference Equation;132
6.2.5;5.5 Transmission Matrix;133
6.3;Chapter6 Systems with Periodic Structure;134
6.3.1;6.1 Introduction;134
6.3.2;6.2 Dynamic Compliances and Stiffness for Systems with Periodic Structure;136
6.3.3;6.3 Dynamic Compliances of Single-Connectedness System;138
6.3.4;6.4 Transition Matrix;142
6.3.5;6.5 Forced Vibrations;143
6.3.6;6.6 Vibrations of Blades Package;144
6.3.7;6.7 Collective Vibrations of Blades;146
6.4;Chapter7 Systems with Cyclic Symmetry;148
6.4.1;7.1 Natural Frequencies and Normal Modes for Systems with Cyclic Symmetry;148
6.4.1.1;7.1.1 Natural Frequencies;148
6.4.1.2;7.1.2 Normal Modes;151
6.4.2;7.2 Vibrations of Blades System;151
6.4.2.1;7.2.1 Different Designs of Blades Connecting;151
6.4.2.2;7.2.2 Natural Frequencies for Blades System;152
6.4.2.3;7.2.3 Normal Modes for Blades System;153
6.4.3;7.3 Numerical and Experimental Results for Blades with Shroud;154
6.4.3.1;7.3.1 Free Ring Connection;154
6.4.3.2;7.3.2 Blades with Paired-Ring Shroud;156
6.4.3.3;7.3.3 Blades Shrouded by Shelves;158
6.5;Chapter8 Systems with Reflection Symmetry Elements;161
6.5.1;8.1 Reflection Symmetry Element and Its Dynamic Characteristics;161
6.5.1.1;8.1.1 Dynamic Stiffness and Compliance Matrices for Reflection Symmetry Element;163
6.5.1.2;8.1.2 Mixed Matrix for Reflection Symmetry Element;164
6.5.2;8.2 Finite Differences Equations;167
6.5.3;8.3 Special Types of Boundary Conditions;170
6.5.3.1;8.3.1 Nonclosed Systems;170
6.5.3.2;8.3.2 Closed Systems;173
6.5.4;8.4 Filtering Properties of System with Reflection Symmetry Elements;175
6.5.5;8.5 Numerical Examples;176
6.5.5.1;8.5.1 Single-Connectedness Systems;176
6.5.5.2;8.5.2 Two-Connectedness Systems;182
6.5.5.3;8.5.3 Three-Connectedness System;185
6.5.6;8.6 Systems Consisting of Skew-Symmetric (Antisymmetric) Elements;188
6.6;Chapter9 Self-Similar Structures;192
6.6.1;9.1 Introductory Part: Examples of Self-Similar Mechanical Structures;192
6.6.2;9.2 Dynamic Compliances of Self-Similar Systems;193
6.6.3;9.3 Vibrations of Self-Similar Systems: Numerical Examples;195
6.6.4;9.4 Transition Matrix;196
6.6.5;9.5 Self-Similar Systems with Similar Matrix of Dynamic Compliance;198
6.6.6;9.6 Vibrations of Self-Similar Shaft with Disks;199
6.6.7;9.7 Vibrations of Self-Similar Drum-Type Rotor;201
6.7;Chapter10 Vibrations of Rotor Systems with Periodic Structure;205
6.7.1;10.1 Rotor Systems with Periodic Structure with Disks;205
6.7.2;10.2 Rotor with Arbitrary Boundary Conditions: Natural Frequencies and Normal Modes;209
6.8;Chapter11 Vibrations of Regular Ribbed Cylindrical Shells;212
6.8.1;11.1 General Theory of Shells;212
6.8.2;11.2 Dynamic Stiffness and Transition Matrix for Closed Cylindrical Shells;214
6.8.3;11.3 Dynamic Stiffness and Transition Matrix for Cylindrical Panel;217
6.8.4;11.4 Dynamic Stiffness and Transition Matrices for Circular Ring with Symmetric Profile;218
6.8.5;11.5 Vibrations of Cylindrical Shell with Ring Ribbing under Arbitrary Boundary Conditions;220
6.8.6;11.6 Vibrations of Cylindrical Shell with Longitudinal Ribbing of Nonsymmetric Profile;224
6.8.6.1;11.6.1 Dynamic Stiffness and Transition Matrices for Longitudinal Stiffening Ribs;225
6.8.6.2;11.6.2 Numerical Calculation of Shell with Longitudinal Ribbing;226
7;Appendix A Stiffness and Inertia Matrices for a Ramified System Consisting of Rigid Bodies Connected by Beam Elements;228
8;Appendix B Stiffness Matrix for Spatial Finite Element;234
9;Appendix C Stiffness Matrix Formation Algorithm for a Beam System in Analytical Form;235
10;Appendix D Stiffness Matrices for a Planetary Reduction Gear Subsystems;237
11;References;238
12;Index;243