Periodically Forced Nonlinear Oscillators With Hysteretic Damping

Abstract

We perform a detailed study of the dynamics of a nonlinear, one-dimensional oscillator driven by a periodic force under hysteretic damping, whose linear version was originally proposed and analyzed by Bishop in [1]. We first add a small quadratic stiffness term in the constitutive equation and construct the periodic solution of the problem by a systematic perturbation method, neglecting transient terms as $t\rightarrow \infty$. We then repeat the analysis replacing the quadratic by a cubic term, which does not allow the solutions to escape to infinity. In both cases, we examine the dependence of the amplitude of the periodic solution on the different parameters of the model and discuss the differences with the linear model. We point out certain undesirable features of the solutions, which have also been alluded to in the literature for the linear Bishop's model, but persist in the nonlinear case as well. Finally, we discuss an alternative hysteretic damping oscillator model first proposed by Reid [2], which appears to be free from these difficulties and exhibits remarkably rich dynamical properties when extended in the nonlinear regime.

Energy Transport in 1-Dimensional Oscillator Arrays With Hysteretic Damping

“…In [15], a detailed study was performed on the dynamics of a hysteretic damping model introduced by Reid [16], for a linear as well as nonlinear oscillator driven by a T -periodic sinusoidal force. Reid's model was shown to be free from numerical limitations suffered by a hysteretic damping model proposed earlier by Bishop [17], since Reid's oscillator is expressed by a differential equation whose real and imaginary parts have the same solutions.…”

“…Thus, numerical errors occurring during the integration of Reid's model are efficiently controlled, and it is shown to possess T -periodic solutions, which are true attractors. Moreover, in the weakly dissipative case, it exhibits the coexistence of periodic attractors of period nT, n = 2, 3, ..., with very interesting basins of attraction in the space of initial conditions [15]. Reid's model [16] describes the evolution of a one degree-of-freedom (dof) oscillator with mass M under periodic forcing of amplitude f , satisfying the differential equation:…”

“…Model I is a nonlinear extension of the well-known quasi-linear Reid's model [16,15]. In Section 2, we numerically integrate its equations of motion and study the occurrence of supratransmission, analyzing the effect of the system's parameters on the corresponding critical amplitudes.…”