Introduction to Perturbation Techniques

by Nayfeh, Ali H.

• ISBN13: 9780471310136
• ISBN10: 0471310131
• Edition: 1st
• Format: Paperback
• Copyright: 1993-08-20
• Publisher: Wiley-VCH

Summary

Similarities, differences, advantages and limitations of perturbation techniques are pointed out concisely. The techniques are described by means of examples that consist mainly of algebraic and ordinary differential equations. Each chapter contains a number of exercises.

Author Biography

Ali H. Nayfeh received his BS in engineering science and his MS and PhD in aeronautics and astronautics from Stanford University. He holds honorary doctorates from Marine Technical University, Russia, Technical University of Munich, Germany, and Politechnika Szczecinska, Poland. He is currently University Distinguished Professor of Engineering at Virginia Tech. He is the Editor of the Wiley Series in Nonlinear Science and Editor in Chief of Nonlinear Dynamics and the Journal of Vibration and Control.

Table of Contents

Introduction	1	(27)
Dimensional Analysis	1	(9)
Expansions	10	(2)
Gauge Functions	12	(5)
Order Symbols	17	(1)
Asymptotic Series	18	(4)
Asymptotic Expansions and Sequences	22	(1)
Convergent Versus Asymptotic Series	23	(1)
Elementary Operations on Asymptotic Expansions	24	(4)
Exercises	24	(4)
Algebraic Equations	28	(23)
Quadratic Equations	28	(11)
Cubic Equations	39	(4)
Higher-Order Equations	43	(2)
Transcendental Equations	45	(6)
Exercises	48	(3)
Integrals	51	(56)
Expansion of Integrands	52	(4)
Integration by Parts	56	(9)
Laplace's Method	65	(14)
The Method of Stationary Phase	79	(9)
The Method of Steepest Descent	88	(19)
Exercises	101	(6)
The Duffing Equation	107	(27)
The Straightforward Expansion	109	(4)
Exact Solution	113	(5)
The Lindstedt-Poincare Technique	118	(3)
The Method of Renormalization	121	(1)
The Method of Multiple Scales	122	(5)
Variation of Parameters	127	(2)
The Method of Averaging	129	(5)
Exercises	131	(3)
The Linear Damped Oscillator	134	(13)
The Straightforward Expansion	135	(1)
Exact Solution	136	(3)
The Lindstedt-Poincare Technique	139	(3)
The Method of Multiple Scales	142	(2)
The Method of Averaging	144	(3)
Exercises	146	(1)
Self-Excited Oscillators	147	(12)
The Straightforward Expansion	148	(3)
The Method of Renormalization	151	(1)
The Method of Multiple Scales	152	(3)
The Method of Averaging	155	(4)
Exercises	157	(2)
Systems with Quadratic and Cubic Nonlinearities	159	(18)
The Straightforward Expansion	160	(2)
The Method of Renormalization	162	(2)
The Lindstedt-Poincare Technique	164	(2)
The Method of Multiple Scales	166	(2)
The Method of Averaging	168	(1)
The Generalized Method of Averaging	169	(4)
The Krylov-Bogoliubov-Mitropolsky Technique	173	(4)
Exercises	175	(2)
General Weakly Nonlinear Systems	177	(13)
The Straightforward Expansion	177	(2)
The Method of Renormalization	179	(2)
The Method of Multiple Scales	181	(1)
The Method of Averaging	182	(2)
Applications	184	(6)
Exercises	188	(2)
Forced Oscillations of the Duffing Equation	190	(26)
The Straightforward Expansion	191	(2)
The Method of Multiple Scales	193	(16)
Secondary Resonances	193	(12)
Primary Resonance	205	(4)
The Method of Averaging	209	(7)
Secondary Resonances	209	(3)
Primary Resonance	212	(1)
Exercises	213	(3)
Multifrequency Excitations	216	(18)
The Straightforward Expansion	216	(3)
The Method of Multiple Scales	219	(7)
The Case ω2 + ω1 ≈ 1	220	(2)
The Case ω2 - ω1 ≈ 1 and ω1 ≈ 2	222	(4)
The Method of Averaging	226	(8)
The Case ω1 + ω2 ≈ 1	230	(1)
The Case ω2 - ω1 ≈ 1 and ω1 ω2	230	(1)
Exercises	230	(4)
The Mathieu Equation	234	(23)
The Straightforward Expansion	235	(1)
The Floquet Theory	236	(7)
The Method of Strained Parameters	243	(4)
Whittaker's Method	247	(2)
The Method of Multiple Scales	249	(4)
The Method of Averaging	253	(4)
Exercises	254	(3)
Boundary-Layer Problems	257	(68)
A Simple Example	257	(11)
The Method of Multiple Scales	268	(2)
The Method of Matched Asymptotic Expansions	270	(9)
Higher Approximations	279	(5)
Equations with Variable Coefficients	284	(12)
Problems with Two Boundary Layers	296	(8)
Multiple Decks	304	(3)
Nonlinear Problems	307	(18)
Exercises	320	(5)
Linear Equations with Variable Coefficients	325	(35)
First-Order Scalar Equations	326	(3)
Second-Order Equations	329	(2)
Solutions Near Regular Singular Points	331	(11)
Singularity at Infinity	342	(2)
Solutions Near an Irregular Singular Point	344	(16)
Exercises	355	(5)
Differential Equations with a Large Parameter	360	(28)
The WKB Approximation	361	(3)
The Liouville-Green Transformation	364	(2)
Eigenvalue Problems	366	(3)
Equations with Slowly Varying Coefficients	369	(1)
Turning-Point Problems	370	(5)
The Langer Transformation	375	(4)
Eigenvalue Problems with Turning Points	379	(9)
Exercises	383	(5)
Solvability Conditions	388	(84)
Algebraic Equations	389	(5)
Nonlinear Vibrations of Two-Degree-of-Freedom Gyroscopic Systems	394	(3)
Parametrically Excited Gyroscopic Systems	397	(4)
Second-Order Differential Systems	401	(5)
General Boundary Conditions	406	(6)
A Simple Eigenvalue Problem	412	(2)
A Degenerate Eigenvalue Problem	414	(4)
Acoustic Waves in a Duct with Sinusoidal Walls	418	(8)
Vibrations of Nearly Circular Membranes	426	(6)
A Fourth-Order Differential System	432	(6)
General Fourth-Order Differential Systems	438	(3)
A Fourth-Order Eigenvalue Problem	441	(4)
A Differential System of Equations	445	(2)
General Differential Systems of First-Order Equations	447	(5)
Differential Systems with Interfacial Boundary Conditions	452	(2)
Integral Equations	454	(4)
Partial-Differential Equations	458	(14)
Exercises	462	(10)
Appendix A Trigonometric Identities	472	(8)
Appendix B Linear Ordinary-Differential Equations	480	(21)
Bibliography	501	(6)
Index	507