Harmonic solutions for a class of non-autonomous piecewise linear oscillators
type: Item
instance of: scholarly article
title: Harmonic solutions for a class of non-autonomous piecewise linear oscillators
zbMATH Open document ID: 1484.34107
author: Biliu Zhou
author: Hebai Chen
author: Huidong Xu
author: Jianwen Zhang
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2021-08-16T00:00:00Z
review text: Let \(k(x)\) be a real-valued piecewise linear function defined as \[ k(x) = \begin{cases} k\cdot(x-\delta) & x\geq \delta\\ 0 & -\delta < x < \delta\\ k\cdot(x+\delta) & x\leq -\delta\\ \end{cases} \] and \(q(t)\) a periodic function (excitation) with period \(T>0\). The notation is taken from the paper and \(\delta\) has nothing to do with Dirac delta function. The authors are considering a single degree of freedom piecewise linear oscillator \[ m\ddot{x}+k(x) = q(t) \] with periodic forcing. Denote \(\omega_0 =\sqrt{k/m}\) and \(\lambda=\omega_0/\omega\). The above equation can be rewritten as the system \begin{align*} x' & = y \\ y' & = -g(x)+p(\tau) \end{align*} for \(\tau = \omega t\), where \[ g(x) = \begin{cases} \lambda^2\cdot(x-\delta) & x\geq \delta\\ 0 & -\delta < x < \delta\\ \lambda^2\cdot(x+\delta) & x\leq -\delta\\ \end{cases} \] and \(p(\tau)=q(wt)/m\omega^2\). Note that \('\) in the above equations stand for the differentiation with respect to time \(\tau\). Let two switching surfaces be defined as follows: \[ \Sigma_1 =\{(x,y)\in \mathbb{R}^2| x = \delta\},\ \ \Sigma_2 =\{(x,y)\in \mathbb{R}^2| x = -\delta\}, \] which partition the phase space of the above system into the following three ''zones'' \[ S_r = \{ (x,y)\in \mathbb{R}^2| x>\delta\},\ \ S_l = \{ (x,y)\in \mathbb{R}^2| x<-\delta\},\ \ S_c = \{ (x,y)\in \mathbb{R}^2| -\delta < x < \delta\}. \] The study deals with the existence and uniqueness conditions of harmonic solutions of the above system. These harmonic solutions are referred in the paper as small (confined to a single zone) or large (multi-zone). Based on the relationship between natural and driving frequency, the study is divided into the four cases: \(\lambda > 1\); \(0<\lambda<1\); \(\lambda = m\); \(m<\lambda<m+1\), where \(m\in\mathbb{N}^{+}\). The results obtained are not very deep and they are perhaps more interesting to the engineering community than mathematicians. The authors seem to be unaware of deep studes published on…
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37C60
zbMATH DE Number: 7382110
zbMATH Keywords: harmonic solution
zbMATH Keywords: resonant/non-resonant case
zbMATH Keywords: piecewise linear oscillator
zbMATH Keywords: non-autonomous system
reviewed by: Predrag Punoševac
MaRDI profile type: Publication
full work available at URL: j.cnsns.2021.105912
OpenAlex ID: W3173031266
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DOI: 10.1016/J.CNSNS.2021.105912
description: scientific article; zbMATH DE number 7382110