Stability of Gyroscopic Systems

by Guran, Ardeshir; Bajaj, Anil; Ishida, Yukio; Perkins, Noel; D'Eleuterio, Gabriele; Pierre, Christophe

• ISBN13: 9789810226305
• ISBN10: 9810226306
• Format: Hardcover
• Copyright: 1996-12-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Table of Contents

Preface	vii
A. Guran
A. Bajaj
Y. Ishida
N. Perkins
G. D'Eleuterio
C. Pierre
Contributors	xv
STABILITY OF GYROSCOPIC SYSTEMS
Nonlinear Flow-Induced Vibration of Structures	1	(102)
Christopher Folley
Anil Bajaj
Introduction	1	(14)
Cross-Flow Effects	2	(3)
Axial Flow Effects	5	(2)
The Tube Problem	7	(8)
Equations of Motion	15	(15)
Coordinate Systems	15	(3)
Stress-Strain Relationships	18	(1)
The Equations of Motion	18	(5)
Boundary Conditions	23	(1)
Static Equilibrium Solutions	24	(6)
Scaling and Linear Analysis	30	(20)
Nondimensional Parameters and the Introduction of the Order Parameter	30	(6)
Solution of the Linear Problem	36	(5)
The Adjoint System	41	(3)
The Center Manifold Reduction and Asymptotic Expansion	44	(6)
Nonlinear Analysis	50	(27)
Planar Oscillations	50	(2)
Normal Forms, Spatial Oscillations, and Symmetry	52	(7)
Symmetry Breaking Perturbations	59	(2)
Trivial Solutions and Instabilities	61	(2)
Constant Solutions and Instabilities	63	(6)
Hopf Bifurcation from H0	69	(4)
The Takens-Bogdanov Bifurcation	73	(4)
Numerical Results	77	(12)
Broken Rotational Symmetry	78	(3)
Broken time-Phase Symmetry	81	(8)
Conclusion and Some Comments on Experimental Results	89	(5)
Appendix A: Coefficients of the 2-Frequency Wave Normal Form	92	(1)
Appendix B: Coefficients of the O(2)-Takens-Bogdanov Normal Form	93	(1)
References	94	(9)
Nonlinear Oscillations of Rotors	103	(89)
Yukio Ishida
Introduction	103	(1)
Equations of Motion of Symmetrical Systems with a Concentrated Mass	104	(7)
Introduction	104	(1)
4DOF System Where the Deflection and the Inclination Couple with Each Other	104	(6)
Lateral Oscillations and Inclination Oscillations of an Uncoupled Rotor System	110	(1)
Nonlinear Spring Characteristics (Case of a Weak Nonlinearity)	111	(8)
Introduction	111	(1)
Nonlinear Spring Characteristics of a 2DOF Rotor System	112	(4)
Nonlinear Spring Characteristics of a 4DOF Rotor System	116	(3)
Representation by the Normal Coordinates	119	(2)
Introduction	119	(1)
2DOF Rotor System for Inclination Oscillation	119	(1)
4DOF Rotor System	120	(1)
Nonlinear Resonances in Symmetrical Rotor Systems	121	(23)
Introduction	121	(1)
Harmonic Resonance at the Major Critical Speed	121	(1)
2DOF Rotor System	122	(4)
4DOF Rotor System	126	(2)
Subharmonic Oscillation of Order 1/2 of a Forward Whirling Mode	128	(1)
2DOF Rotor System	128	(3)
4DOF Rotor System	131	(3)
Combination Resonance	134	(1)
2DOF Rotor System	134	(4)
4DOF Rotor System	138	(1)
Summary of the Analyses	138	(1)
Summary of Experimental Results	139	(3)
Previous Research on Nonlinear Resonances in Symmetrical Rotor Systems	142	(2)
Nonlinear Forced Oscillations of Unsymmetrical Shafts and Unsymmetrical Rotors	144	(10)
Introduction	144	(1)
Derivation of the Equations of Motion	145	(1)
Equations of Motion of an Unsymmetrical Shaft System	145	(2)
Equations of Motion of Unsymmetrical Rotor Systems	147	(1)
Harmonic Resonance (Unsymmetrical Rotor System)	147	(1)
Theoretical Analysis (Linear System)	147	(1)
Theoretical Analysis (Nonlinear System)	148	(1)
Experimental Results	149	(1)
Subharmonic Resonance of Order 1/2 of a Forward Whiring Mode (Unsymmetrical Shaft System)	150	(1)
Theoretical Analysis	150	(2)
Experimental Results	152	(1)
Previous Researches on Nonlinear Oscillations of Unsymmetrical Shaft Systems and Unsymmetrical Rotor Systems	152	(2)
Nonstationary Oscillations during Passage through Critical Speeds	154	(12)
Introduction	154	(1)
Equations of Motion	154	(1)
2DOF Rotor System for Lateral Oscillations	154	(1)
2DOF Rotor System of Inclination Oscillation	155	(1)
Harmonic Resonance at the Major Critical Speed	156	(1)
2DOF System for Lateral Oscillations (the Direct Integration Method)	156	(1)
2DOF System for Inclination Oscillation (the Asymptotic Method)	157	(2)
2DOF System for Inclination Motion (the Complex-FFT Method)	159	(2)
Experimental Results	161	(1)
Subharmonic Resonance of Order 1/2 of a Forward Whirling Mode	161	(1)
2DOF System for Inclination Oscillations (the Complex-FFT Method)	162	(2)
Experimental Results	164	(1)
Combination Resonance	165	(1)
Previous Research on Nonstationary Oscillations in a Rotor System with Nonlinear Spring Characteristics	166	(1)
Forced Oscillations of a Rotor System with Internal Damping and Nonlinear Spring Characteristics	166	(7)
Introduction	166	(1)
Internal Damping (Structural Damping)	167	(1)
Equations of Motion	167	(1)
System with Internal Damping and Linear Spring Characteristics	168	(2)
System with Internal Damping and Nonlinear Spring Characteristics	170	(1)
Subharmonic Resonance of Order 1/2 of a Forward Whirling Mode	170	(1)
Subharmonic Resonance of Order 1/2 of a Backward Whirling Mode	171	(1)
Experimental Results	171	(2)
Previous Research on Nonlinear Oscillations of Rotor Systems with Internal Damping	173	(1)
Nonlinear Oscillations of a Continuous Rotor	173	(9)
Introduction	173	(1)
Equations of Motion and Natural Frequencies	173	(3)
Derivation of Ordinary Differential Equations	176	(1)
Harmonic Resonances at the Major Critical Speed [pfn]	176	(1)
The Subharmonic Resonance of Order 1/3 of a Forward Whirling Mode	177	(1)
Combination Resonances [2pfm -- pfn]	177	(1)
Case of m = n	178	(2)
Case of m ≠ n	180	(1)
Summary of Theoretical Analysis	180	(1)
Experiments	181	(1)
Previous Research on Nonlinear Oscillations in Continuous Rotors	182	(1)
Nonlinear Oscillations of a Rotor Containing a Transverse Crack	182	(6)
Introduction	182	(1)
Spring Characteristics and Equations of Motion	182	(2)
Numerical Simulations	184	(1)
Harmonic Resonance at the Major Critical Speed	184	(3)
Previous Research on Nonlinear Analysis of Cracked Rotors	187	(1)
References	188	(4)
Dynamics of High-Speed Axially Moving Material Systems	192	(45)
Noel Perkins
Sheng-Jiaw Hwang
Introduction	193	(2)
Linear Theory	193	(1)
Mechanics of Super-critical Response	194	(1)
Super-Critical Cable Response	195	(17)
Overview	195	(1)
Translating Cable Model	196	(2)
Equilibrium and Stability Analyses	198	(3)
Theoretical and Experimental Results	201	(10)
Summary and Conclusions	211	(1)
Super-Critical Beam Response	212	(23)
Overview	212	(1)
Translating Beam Model	212	(3)
Equilibrium and Bifurcation Analyses	215	(5)
Stability Analysis	220	(3)
Application to Band/Wheel Systems	223	(11)
Summary and Conclusions	234	(1)
Acknowledgements	235	(1)
References	235	(2)
Gyroelastic Dynamics	237	(83)
Gabriele D'Eleuterio
Introduction	237	(8)
An Historical Perspective	238	(2)
Large Flexible Space Structures	240	(1)
A Note on Nomenclature	241	(1)
A Continuum Approach	241	(2)
A Simple Example	243	(1)
Presentation	244	(1)
Gyroelastic Continua	245	(5)
An Elastic Continua	245	(3)
The Gyricity Distribution	248	(2)
Constrained Dynamics	250	(7)
Equation of Motion	250	(2)
Derivation Using Variational Principles	252	(1)
Derivation Using ``Balance Equations''	252	(2)
First-Order Form	254	(1)
Discretization	255	(2)
Constrained Modal Analysis	257	(15)
Gyroelastic Modes of B	257	(3)
General Motion of B	260	(1)
Modal Momentum Coefficients for B	261	(2)
Modal Expansion for ε-1	263	(3)
Modal Identities for B	266	(4)
Parallel-Axis Theorems	270	(1)
Sum of Squares of the Modal Periods of B	271	(1)
Numerical Example: A Cantilevered Gyroelastic Rod	272	(11)
Gyroelastic Modes	274	(5)
Modal Coefficients and Identities	279	(4)
A Model for Gyroelastic Vehicles Using Constrained Modes	283	(8)
A Gyroelastic Vehicle V	283	(1)
Equations of Motion for V	283	(2)
First-Order Form	285	(2)
Pseudorigid Modes	287	(2)
Precessional Modes	289	(1)
Numerical Example	290	(1)
Unconstrained Dynamics	291	(3)
Equations of Motion for V	291	(3)
First-Order Form	294	(1)
Unconstrained Modal Analysis	294	(12)
Gyroelastic Modes for V	294	(2)
General Motion for V	296	(1)
Modal Momentum Coefficients for V	297	(2)
Modal Expansion for ε-1	299	(1)
Modal Identities for V	300	(2)
Identities Involving Constrained and Unconstrained Modal Parameters	302	(3)
Sum of Squares of Modal Periods of V	305	(1)
A Numerical Example: The Free-Free Gyroelastic Rod	306	(7)
Gyroelastic Mode Shapes	307	(1)
Modal Coefficients and Identities	308	(4)
Gyroelastic Rod With a Rigid Body	312	(1)
A Few Closing Words	313	(2)
Acknowledgments	315	(1)
References	315	(5)
Fluid Conveying Pipe With Shear and Compressibility	320	(19)
Ardeshir Guran
Teodor Atanackovic
Introduction	320	(1)
Model and Differential Equations	321	(10)
Stability Boundaries	331	(4)
Divergence Instability Boundary	331	(2)
Flutter Instability Boundary	333	(2)
Conclusions	335	(1)
References	336	(3)
Localization Phenomena in Structural Dynamics	339	(64)
Christophe Pierre
Introduction	339	(10)
An Intuitive Illustration	341	(2)
A Brief Review of the Literature	343	(2)
Engineering Significance of the Localization Phenomenon	345	(3)
Scope and Outline of Chapter	348	(1)
Dynamic Wave Formulation for Disordered Periodic Structures	349	(25)
Background on Wave and Modal Properties of Periodic Structures	349	(3)
Transfer Matrices	352	(3)
Characteristic Free Waves in Periodic Structures	355	(2)
Wave Transfer Matrix for a Periodic Bay	357	(1)
Wave Transfer Formulation for Disordered Multi-Coupled Structures	358	(2)
Normal Modes of Finite Periodic and Disordered Structures	360	(3)
Asymptotic Behavior of the Global Wave Transfer Matrix	363	(5)
Calculation of Localization Factors	368	(6)
Weak and Strong Localization in Chains of Single-Degree of Freedom Oscillators	374	(18)
Equations of Motion and Wave Transfer Matrix Formulation	374	(2)
Ordered Assembly Dynamics	376	(2)
Perturbation Approximation of the Localization Factor	378	(4)
Weak and Strong Localization Results	382	(7)
Equivalent Damping Factors	389	(3)
Wave Localization and Conversion Phenomena in a Multi-Coupled Structure	392	(8)
Concluding Remarks	400	(1)
Acknowledgement	400	(1)
References	400	(3)
Author Index	403	(6)
Subject Index	409