Approximate analysis of self-oscillating system (Q2759539)

A nonlinear dynamical system representing a bunch of \(n\) oscillators of the form \(\ddot{x} =A(v)\cdot\dot{x} +B\cdot x +f(x_1)\) is considered. Here \(x\in \mathbb R^n\), \(A(v)\) and \(B\) are constant matrices of dimension \(n\times n\); \(v\in R_+\) is a parameter; \(f^T =(f_1, \ldots, f_n)\) are nonlinear terms and \(f_i:\mathbb R\to \mathbb R\) are odd functions of single generalized coordinate \((f_i(-x_1) =-f_i(x_1))\). Loss of stability of zero solution of the system is related to passing a pair of complex conjugate roots of characteristic equation of the linear approximating system through the imaginary axis. In the paper, an analytic method for construction of periodic solution of the system is considered. An expression for amplitude of self-oscillations of a wheelset in rectilinear motion is obtained.