Nonlinear Oscillations in Mechanical Engineering

by Fidlin, A.

• ISBN13: 9783540281153
• ISBN10: 3540281150
• Format: Hardcover
• Copyright: 2005-12-16
• Publisher: Springer Verlag

Summary

"Nonlinear Oscillations in Mechanical Engineering" explores the effects of nonlinearities encountered in applications in that field. Since the nonlinearities are caused, first of all, by contacts between different mechanical parts, the main part of this book is devoted to oscillations in mechanical systems with discontinuities caused by dry friction and collisions. Another important source of nonlinearity which is covered is that caused by rotating unbalanced parts common in various machines as well as variable inertias occurring in all kinds of crank mechanisms. This book is written for advanced undergraduate and postgraduate students, but it may be also helpful and interesting for both theoreticians and practitioners working in the area of mechanical engineering at universities, in research labs or institutes and especially in the R and D departments within industrial firms.

Table of Contents

Preface	vii
Introduction	1	(34)
Usual Sources of Nonlinearity in Mechanical Engineering	1	(7)
Geometrical Nonlinearities	1	(1)
Physical Nonlinearities	2	(1)
Structural or Designed Nonlinearities	3	(1)
Constraints	4	(2)
Nonlinearity of Friction	6	(2)
The Basic Ideas of the Perturbation Analysis	8	(9)
Variation of Free Constants and Systems in the Standard Form	8	(2)
Standard Averaging as an Almost Identical Transformation	10	(3)
Method of Multiple Scales	13	(2)
Direct Separation of Motions	15	(1)
Relationship between These Methods	16	(1)
Examples of Elementary Nonlinear Problems Solved by Standard Averaging	17	(7)
Instability and Self Excited Oscillations in the Van Der Pol's Equation	17	(2)
The Main Resonance in a System with a Small Cubic Nonlinearity	19	(2)
Secondary Resonances in the System with Cubic Nonlinearity and Strong Excitation	21	(3)
Axiomatic Theory of Collisions	24	(11)
Impulsive Motion of the Point Mass	24	(1)
Impulsive Motion of a System of Point Masses	25	(2)
Impulsive Motion of a Rigid Body	27	(2)
Collinear Collision of Two Point Masses	29	(2)
Direct Collisions in Mechanical Systems with Ideal Constraints	31	(2)
Concluding Remarks	33	(2)
Oscillations in Systems with Dry Friction	35	(54)
Self Excited Oscillations of the Mass-on-Moving-Belt	38	(17)
The Problem Description; Equations of Motion	38	(2)
Types of Motion	40	(2)
Pure Slip Oscillations	42	(2)
Stick-slip Oscillations	44	(8)
Discussion of the Results	52	(2)
Concluding Remarks	54	(1)
Friction Induced Flutter	55	(9)
Mathematical Basics of Flutter in a System with Two Degrees of Freedom	55	(1)
Wobbling of an Elastically Supported Friction Disc	56	(5)
On the Unstable Behavior of an Asymmetrically Supported Disc (Brake Squeal)	61	(3)
Conclusions	64	(1)
Vibration Induced Displacement. Averaging in Discontinuous Systems	64	(8)
A Simple Example of the Vibration Induced Displacement	65	(2)
Mathematical Basics for the First Order Averaging of the Constant Order Discontinuous Regimes	67	(2)
The Elementary Example of the Vibration Induced Displacement. The First Order Approximation	69	(1)
Discussion of the Results	70	(1)
Conclusions	71	(1)
Vibration Induced Displacement of a Resonant Friction Slider	72	(17)
Problem Description	72	(1)
Equations of Motion	73	(3)
Illustration to System's Behavior	76	(1)
Transformation to the Principal Form: Amplitude of the Resonator	77	(4)
Motion of the Slider: Preparing for Averaging	81	(4)
Performance in Dependence of Parameters; Comparison between Analytic Prediction and Numerical Simulations	85	(2)
Conclusions	87	(2)
Systems with Almost Elastic Collisions	89	(48)
The Basic Ideas of Discontinuous Averaging. Unfolding Transformations	91	(9)
The Basic Idea of the Unfolding Transformation for the Mass Limited at One Side	91	(1)
The Unfolding Transformation and Averaging in Case of Slightly Inelastic Collisions	92	(4)
Comparison between Analytic and Numeric Predictions for the Oscillator Limited from One Side	96	(1)
Unfolding Transformation and Averaging for the Free Mass in a Clearance	97	(3)
The ``Mass-on-Moving-Belt'' Limited at One Side: First Order Approximation	100	(3)
Second Order Approximation in Systems with Almost Elastic Collisions	103	(6)
General Mathematical Approach	103	(3)
The Second Order Approximation for the Amplitude of the Mass on Moving Belt Limited from One Side	106	(2)
Discussion of the Results and Comparisons with Numeric Experiments	108	(1)
The ``Mass on Moving Belt'' in a clearance	109	(7)
The Governing Equations and the Unfolding Transformation	110	(1)
Analyzing the Unperturbed System and Introducing Energy as the Slow Variable	111	(3)
Discussion of the Results	114	(2)
Resonance of the Impact Oscillator Limited at One Side under External Excitation	116	(8)
Equations of Motion and the Unfolding Transformation	116	(2)
Resonances in the Almost Linear System	118	(1)
Averaging in the Vicinities of the Almost Linear Resonances	119	(2)
Stability of the Stationary Solutions	121	(1)
Discussion of the Results, Comparison Between Analytic and Numeric Predictions	122	(2)
Nonlinear Resonance of the Externally Excited Oscillator in a Clearance	124	(10)
Equations of Motion and the Unfolding Transformation	125	(1)
Analyzing the Unperturbed System and Introducing Slow and Fast Variables	126	(2)
Resonances in the Significantly Nonlinear System	128	(1)
Averaging in the Vicinity of the Main Nonlinear Resonance	129	(2)
Equations Governing the Slow Motions; Discussion of the Results	131	(2)
Comparison between Analytic and Numeric predictions	133	(1)
Conclusions	134	(3)
Systems with Strong Dissipation Due to High Damping or Inelastic Collisions	137	(44)
Averaging in Systems with Strong Linear Damping	138	(5)
The Basic Idea	138	(3)
Averaging in Systems with Strong Damping System with Respect to one or Several Variables	141	(2)
Linear Resonance in a Strongly Damped System with Two Degrees of Freedom	143	(9)
Equations of Motion	143	(2)
Perturbation Analysis. Transformation to the Form suitable for Averaging	145	(3)
Equations of the First Order Approximation. Discussion of the Approach	148	(2)
Comparison with the Numeric Experiment. Discussion of the Results	150	(2)
Averaging in Systems with Inelastic Collisions: Basic Ideas and General Approach	152	(13)
Basic Types of Motion in Systems with Inelastic Collisions: Elementary Examples	152	(6)
On the Practical Importance of Regimes with Long Contact	158	(2)
Regimes with Long Contacts as an Example of the Variable Order Discontinuous Systems	160	(3)
Variable Order Discontinuous Systems in the Standard Form	163	(2)
Basic Regime with Long Contacts for the Mass in a Resonantly Excited Frame	165	(6)
Equations of Motion	165	(1)
Perturbation Analysis. Transformation to the Form Suitable for Averaging	166	(2)
Equations of the First Order Approximation. Discussion of the Results	168	(3)
The Basic Regime with Long Contacts for the Mass over the Resonantly Excited Base	171	(7)
Equations of Motion	171	(2)
The Master and the Slave Variables; the Unperturbed Solution	173	(2)
Equations of the First Order Approximation. Discussion of the Results	175	(3)
Conclusions	178	(3)
Short Notes on the Significantly Nonlinear Resonance	181	(32)
The Basic Example of the Nonlinear Resonance	184	(13)
Elementary Analysis and Natural Scale for the Resonance Domain	184	(3)
The Basic Regimes of the Equivalent Pendulum	187	(2)
Stability of the Stationary Resonance	189	(3)
Resonant Motions: Averaging with Respect to the Oscillations of the Equivalent Pendulum	192	(5)
Nonlinear Resonant Crusher with Almost Elastic Collisions	197	(6)
Problem Description. Equations of Motion	197	(3)
The Unfolding Transformation. The Main Resonance	200	(1)
Averaging with Respect to the Fast Rotating Phase. Stationary Regimes	201	(2)
Nonlinear Resonant Crusher with Inelastic Collisions	203	(7)
Problem Description. Equations of Motion	203	(3)
The Regularizing Transformation. The Main Resonance	206	(2)
Averaging with Respect to the Fast Rotating Phase. Stationary Regimes	208	(2)
Conclusions	210	(3)
High Frequency Excitation: Basic Ideas and Elementary Effects	213	(52)
Classification of Systems with HF Excitation. Weakly Excited Systems	214	(4)
Classification of Systems with HF Excitation	214	(1)
Systems with Weak HF Excitation	215	(1)
The Weakly Excited Pendulum	216	(2)
A Strongly Excited Pendulum with the Oscillating Suspension Point. Stiffening, Softening and Biasing	218	(8)
A Pendulum with the Vertically Vibrating Suspension Point: Equations Governing the Slow Motions	218	(1)
Discussion of the Results for the Vertically Excited Pendulum	219	(2)
The Pendulum with the Horizontally Vibrating Suspension Point: Equations of Slow Motions and System's Behavior	221	(2)
The Pendulum Excited both Vertically and Horizontally	223	(3)
Shifted Resonances of the Pendulum. The Overlapped Slow Excitation and the Slowly Modulated HF Excitation	226	(8)
Two Types of Bi-harmonic Excitation	226	(1)
Obtaining Equations Governing the Slow Motions of the Pendulum	227	(1)
The Effect of the Overlapped slow Excitation	228	(3)
The Effect of the Slowly Modulated HF Excitation	231	(1)
Using the Slowly Modulated HF Excitation in Order to Quench the Slow Excitation	232	(2)
The First Generalization and the Exceptional Role of the Terms Depending on the Velocity in Systems with HF Excitation	234	(5)
The Basic Equation of the Vibrational Mechanics	234	(4)
A Remark on the Exceptional Role of the Terms Depending on the Velocities	238	(1)
Smoothening of Dry Friction in Presence of HF Excitation. Quenching of the Friction Induced Oscillations	239	(10)
Smoothening of Dry Friction: A Simple Example	239	(4)
Slow Translation of a Particle on the Elliptically Vibrating Plane	243	(3)
Quenching of the Self Excited Oscillations Caused by the Negative Friction Gradient	246	(3)
On the Misbehavior of the ``Optimally'' Controlled Pendulum under the Influence of the HF Excitation	249	(16)
Description of the Problem, Equations Governing the Mechanical Subsystem	249	(2)
The Optimal State-Feedback Control	251	(1)
System's Behavior in Presence of the Strong HF Excitation: Numeric Results	252	(1)
Transformation of the System to the Form Suitable for Averaging	253	(3)
The First Order Approximation; the Stationary Pendulum's Tilt	256	(1)
The Second Order Approximation; the Stationary Position of the Cart	257	(3)
Discussion of the Results	260	(2)
A Robust Control with Averaging Observer	262	(3)
Systems with High-Frequency Excitation: Advanced Analysis and Generalizations	265	(34)
Systems with Strong Excitation. General Analysis	266	(6)
Two Mathematical Examples of Systems with Strong Excitation	272	(5)
A System with One Degree of Freedom and Strong HF Excitation Depending on the Velocity	272	(3)
A System with Two Degrees of Freedom and a Skew Symmetric HF Excitation Depending on the Velocities	275	(2)
The Lowest Natural Frequencies of an Elastic Rod with Periodic Structure	277	(3)
Response of a One Degree of Freedom Nonlinear System to a Strong HF External and Parametric Excitation Due to Oscillating Inertia	280	(4)
The Governing Equations and Their Transformation to the Basic Mathematical Form	280	(1)
Obtaining the Equations Governing Slow Motion	281	(1)
Discussion of the Results	282	(2)
Systems with Very strong Excitation in the Special Case of Fast Oscillating Inertial Coefficients	284	(2)
Response of a One Degree of Freedom Nonlinear System to Very Strong HF External and Strong Parametric Excitation due to Oscillating Inertia	286	(4)
Obtaining the Equations Governing the Slow Motion	287	(1)
Discussion of the Results	288	(1)
Large Solutions	289	(1)
Dynamics of a Two Link Pendulum with a Fast Rotating Second Link	290	(8)
Equations of Motion and Their Transformation to the Basic Form for Systems with Very Strong Excitation	291	(1)
Obtaining Equations Governing the Slow Motion	292	(3)
Discussion of the Results	295	(2)
A Short Remark on the Practical Importance of the Considered Solutions	297	(1)
Conclusions	298	(1)
Appendixes	299	(46)
Appendix I: The first Bogoliubov's Theorem for Standard Averaging	299	(4)
Appendix II: On the Attractive Properties of the Asymptotically Stable Equilibrium of the Averaged System	303	(4)
Appendix III: Averaging of Systems with Short Strong Perturbations	307	(10)
Appendix IV: Averaging of Systems with Small Discontinuities of the Right Hand Sides	317	(4)
Appendix V: Averaging of Systems with Small Discontinuities of the Unknown Function	321	(7)
Appendix VI: Averaging of Variable Order Discontinuous Systems	328	(8)
Appendix VII: Hierarchic Averaging in Systems with a Semi Slow Rotating Phase	336	(4)
Appendix VIII: Averaging in Systems with Strong High Frequency Excitation	340	(5)
References	345	(10)
Index	355