Nonlinear oscillations : exact solutions and their approximations
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書誌事項
Nonlinear oscillations : exact solutions and their approximations
Springer, c2020
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内容説明・目次
内容説明
This book presents exact, closed-form solutions for the response of a variety of nonlinear oscillators (free, damped, forced). The solutions presented are expressed in terms of special functions. To help the reader understand these `non-standard' functions, detailed explanations and rich illustrations of their meanings and contents are provided. In addition, it is shown that these exact solutions in certain cases comprise the well-known approximate solutions for some nonlinear oscillations.
目次
1. Nonlinear oscillators in theoretical and practical systems
2. Free conservative oscillators
2.1.Overview: simple harmonic oscillators
2.2.Duffing-type oscillators
2.2.1. Hardening Duffing oscillators
2.2.2. Softening Duffing oscillators
2.2.3. Bistable Duffing oscillators
2.2.4. Pure cubic oscillators
2.3.Quadratic oscillators
2.4.Purely nonlinear oscillators
2.5.Oscillators with a constant restoring force
3. Free damped oscillators
3.1.Lagrangians and conservation laws for damped oscillators
3.2.Quadratic damping
3.2.1. In linear oscillators
3.2.2. In purely nonlinear oscillators
4. Forced oscillators: exacts solutions for specially designed external excitation
4.1.Forced response of Duffing-type oscillators
4.1.1. Exact solutions
4.1.2. Simplification to the case of harmonic excitation: related approximations
4.2.Forced response of purely nonlinear oscillators
4.2.1. Exact solutions
4.2.2. On some simplifications and approximations
4.3.Turning Duffing-type oscillators into simple harmonic oscillators
4.4.Turning Duffing-type oscillators into quadratic oscillators
4.5.Turning purely nonlinear oscillators into other oscillators
4.6.Hsu's approach with multi-term excitations
5. Isochronous oscillators
5.1.Making non-isochronous oscillators isochronous
5.2.Theorems on isochronous response of nonlinear oscillators
5.3.About Huygens' pendulum and its links with simple harmonic oscillators
5.4.Isochronicity in forced nonlinear oscillators
6. Chains of oscillators
6.1.Chains with purely nonlinear springs
6.6.1. Pure cubic case
6.6.2. General case: positive real-power nonlinearity
6.2.Generalization to nonlinear continuous systems
6.3.Equivalent stiffness in systems of parallel nonlinear springs
Appendix 1: On Beta and Gamma functions (definitions, expressions, series expansions)
Appendix 2: On Jacobi elliptic functions (definitions, relationships, Fourier series)
Appendix 3: On Ateb functions (definitions, relationships, Fourier series)
Appendix 4: On Wave functions (definitions, relationships, Fourier series)
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