Nonlinear Oscillations: Exact Solutions and Their Approximations

ISBN-13: 9783030531744 / Angielski / Miękka / 2021 / 273 str.

Ivana Kovacic

Recenzje

"The book is well written and very easy to read. From the first pages, where the reader finds the motivation, it is an engaging and informatively very beneficial reading. The connection of mechanics and mathematics is described as well. The text is complemented by illustrative pictures." (Petr Hasil, Mathematical Reviews, October, 2022)

Spis treści

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)

 

O autorze

Ivana Kovacic is a Full Professor of Mechanics at the Faculty of Technical Sciences, University of Novi Sad, Serbia, and the Head of the Centre for Vibro-Acoustic Systems and Signal Processing CEVAS at the same faculty. Her research involves the use of quantitative and qualitative methods to study governing equations arising from nonlinear dynamics problems. She is the editor/author of the books ‘The Duffing Equation: Nonlinear Oscillators and their Behaviour’ and ‘Mechanical Vibrations: Fundamentals with Solved Examples’ (both published by Wiley) as well as two textbooks in Serbian. She is a Subject Editor in three academic journals: Journal of Sound and Vibration (Elsevier), Mechanics Research Communications (Elsevier) and Meccanica (Springer). 

 

Blurb

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.