Harmonic solutions for a class of non-autonomous piecewise linear oscillators
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 the West almost half a century ago. See for example [\textit{S. W. Shaw} and \textit{P. J. Holmes}, J. Sound Vib. 90, 129--152 (1983; Zbl 0561.70022)]. However, a numerical example provided to illustrate the theory uses an interesting aeroelastic model studied numerically in late seventies of the last centry by NASA engineers which definitely got my attention.
scientific article; zbMATH DE number 7382110
scientific article