Bifurcation dynamics of 1DOF parametric oscillator with stiffness-hardening characteristic and dry friction

Abstract

In this work, we have mathematically modeled a 1DOF parametric oscillator with stiffness-hardening characteristics and dry friction and investigated both experimentally and numerically the bifurcation dynamics. The system consists of a cart moving along a linear rolling guide widely used in the industry. It has a stiffness comprised of two components: a linear time-variable part generated by a rotating rod of a rectangular cross-section and a nonlinear hardening stiffness caused by magnetic springs. In the case of nonlinear resistance of motion in a rolling bearing, regardless of the true nature of this phenomenon, it was modeled as the sum of viscous damping and the second component mathematically equivalent to dry friction. The trivial solution was observed to be stable in the whole range of parametric excitation frequencies. But there is a frequency range where the system is bistable, a periodic attractor coexists with the stable equilibrium position, and the branch of the periodic orbit is isolated, i.e., not connected with the equilibrium position, and forms an Isola. This distinguishes the analyzed system from the commonly investigated parametric oscillators, including the classical Mathieu equation and its various versions. Our work perfectly agreed between the numerical simulations and the experimental data. Moreover, the mathematical model for different friction values is investigated, showing the transition between the system without dry friction and the actual rig. The stability of the equilibrium position is tested, and the bifurcation dynamics of periodic orbits are presented using numerical continuation methods, obtaining complete agreement between the results obtained using different ways.