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E-raamat: Strongly Nonlinear Oscillators: Analytical Solutions

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Strongly Nonlinear Oscillators: Analytical SolutionsSuurem pilt
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This book outlines an analytical solution procedure of the pure nonlinear oscillator system, offering a solution for free and forced vibrations of the one-degree-of-freedom strong nonlinear system with constant and time variable parameter. Includes exercises.

This book provides the presentation of the motion of pure nonlinear oscillatory systems and various solution procedures which give the approximate solutions of the strong nonlinear oscillator equations. The book presents the original author’s method for the analytical solution procedure of the pure nonlinear oscillator system. After an introduction, the physical explanation of the pure nonlinearity and of the pure nonlinear oscillator is given. The analytical solution for free and forced vibrations of the one-degree-of-freedom strong nonlinear system with constant and time variable parameter is considered. Special attention is given to the one and two mass oscillatory systems with two-degrees-of-freedom. The criteria for the deterministic chaos in ideal and non-ideal pure nonlinear oscillators are derived analytically. The method for suppressing chaos is developed. Important problems are discussed in didactic exercises. The book is self-consistent and suitable as a textbook for students and also for professionals and engineers who apply these techniques to the field of nonlinear oscillations.

Arvustused

From the book reviews:

This book is devoted to analysis of solutions of the second-order ordinary differential equations (or systems) which describe oscillations of mechanical (and related) systems. the book can be recommended to engineers as a good source of methods and examples. (Henryk odek, Mathematical Reviews, January, 2015)

1 Introduction
1(4)
2 Nonlinear Oscillators
5(12)
2.1 Physical Models
5(2)
2.2 Mathematical Models
7(10)
References
11(6)
3 Pure Nonlinear Oscillator
17(32)
3.1 Qualitative Analysis
18(4)
3.1.1 Exact Period of Vibration
20(2)
3.2 Exact Periodical Solution
22(4)
3.2.1 Linear Case
24(1)
3.2.2 Cubic Nonlinearity
25(1)
3.3 Adopted Lindstedt-Poincare Method
26(4)
3.4 Modified Lindstedt-Poincare Method
30(3)
3.4.1 Comparison of the LP and MLP Methods
31(1)
3.4.2 Conclusion
32(1)
3.5 Exact Amplitude, Period and Velocity Method
33(1)
3.6 Solution in the Form of Jacobi Elliptic Function
34(4)
3.6.1 Example
37(1)
3.7 Solution in the Form of a Trigonometric Function
38(3)
3.7.1 Example
39(1)
3.7.2 Conclusion
40(1)
3.8 Pure Nonlinear Oscillator with Linear Damping
41(8)
3.8.1 Parameter Analysis
43(3)
3.8.2 Conclusion
46(1)
References
47(2)
4 Free Vibrations
49(38)
4.1 Homotopy-Perturbation Technique
51(6)
4.1.1 Duffing Oscillator with a Quadratic Term
54(2)
4.1.2 Conclusion
56(1)
4.2 Averaging Solution Procedure
57(1)
4.3 Solution in the Form of an Ateb Function
58(7)
4.3.1 Small Nonlinear Deflection Functions
59(3)
4.3.2 Differential Equation with a Linear Dominant Term
62(3)
4.4 Solution in the Form of the Jacobi Elliptic Function
65(6)
4.4.1 Oscillator with Nonlinear Elastic Force
67(4)
4.5 Solution in the Form of a Trigonometric Function
71(5)
4.5.1 Oscillator with Small Linear Damping
73(3)
4.6 Conclusion
76(1)
4.7 Oscillator with Linear Damping
77(10)
4.7.1 Van der Pol Oscillator
79(4)
4.7.2 Conclusion
83(1)
References
84(3)
5 Oscillators with Time Variable Parameters
87(40)
5.1 Oscillators with Slow Time Variable Parameters
88(1)
5.2 Solution in the Form of the Ateb Function
88(5)
5.2.1 Oscillator with Linear Time Variable Parameter
91(2)
5.3 Solution in the Form of a Trigonometric Function
93(8)
5.3.1 Linear Oscillator with Time Variable Parameters
95(1)
5.3.2 Non-integer Order Nonlinear Oscillator
96(1)
5.3.3 Levi-Civita Oscillator with a Small Damping
96(4)
5.3.4 Conclusion
100(1)
5.4 Solution in the Form of a Jacobi Elliptic Function
101(10)
5.4.1 Van der Pol Oscillator with Time Variable Mass
103(8)
5.4.2 Conclusion
111(1)
5.5 Parametrically Excited Strong Nonlinear Oscillator
111(16)
5.5.1 Solution Procedure
113(9)
5.5.2 Numerical Simulation
122(1)
5.5.3 Conclusion
123(1)
References
124(3)
6 Forced Vibrations
127(34)
6.1 Oscillator with Constant Excitation Force
128(14)
6.1.1 Solution of the Odd-Integer Order Oscillator
131(3)
6.1.2 The Oscillator with Additional Small Nonlinearity
134(3)
6.1.3 Examples
137(3)
6.1.4 Conclusion
140(2)
6.2 Harmonically Excited Pure Nonlinear Oscillator
142(19)
6.2.1 Pure Odd-Order Nonlinear Oscillator
142(3)
6.2.2 Bifurcation in the Oscillator
145(2)
6.2.3 Harmonically Forced Pure Cubic Oscillator
147(6)
6.2.4 Numerical Simulation and Discussion
153(5)
6.2.5 Conclusion
158(1)
References
159(2)
7 Two-Degree-of-Freedom Oscillator
161(30)
7.1 Two-Mass System
161(12)
7.1.1 Two-Degree-of-Freedom Van der Pol Oscillator
164(8)
7.1.2 Conclusion
172(1)
7.2 Complex-Valued Differential Equation
173(18)
7.2.1 Adopted Krylov-Bogolubov Method
174(2)
7.2.2 Method Based on the First Integrals
176(11)
7.2.3 Conclusion
187(1)
References
187(4)
8 Chaos in Oscillators
191(32)
8.1 Chaos in Ideal Oscillator
192(14)
8.1.1 Homoclinic Orbits in the Unperturbed System
193(2)
8.1.2 Melnikov's Criteria for Chaos
195(3)
8.1.3 Numerical Simulation
198(4)
8.1.4 Lyapunov Exponents and Bifurcation Diagrams
202(1)
8.1.5 Control of Chaos
203(2)
8.1.6 Conclusion
205(1)
8.2 Chaos in Non-ideal Oscillator
206(17)
8.2.1 Modeling of the System
206(2)
8.2.2 Asymptotic Solving Method
208(1)
8.2.3 Stability and Sommerfeld Effect
209(5)
8.2.4 Numerical Simulation and Chaotic Behavior
214(4)
8.2.5 Control of Chaos
218(1)
8.2.6 Conclusion
218(2)
References
220(3)
Appendix A Periodical Ateb Functions 223(4)
Appendix B Averaging of Ateb Functions 227(4)
Appendix C Jacobi Elliptic Functions 231(2)
Appendix D Euler's Integrals of the First and Second Kind 233(4)
Index 237
Prof. Dr. Livija Cveticanin (Mrs.) University of Novi Sad Faculty of Technical Sciences Trg. D. Obradovica 6 21000 Novi Sad Serbia phone: +381-21-485-2237 affiliation: University of Novi Sad, Serbia
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