Period-one motions of a mechanical oscillator with periodically time-varying, piecewise-nonlinear stiffness

The dynamic behavior of a piecewise-nonlinear mechanical oscillator with parametric and external excitations is investigated. The viscously damped single-degree-of-freedom oscillator is subjected to a periodically time-varying, piecewise-nonlinear restoring function defined by a clearance surrounded by continuously nonlinear (quadratic and cubic) regions. Typical applications represented by this oscillator are highlighted. A multi-term harmonic balance formulation is used in conjunction with discrete Fourier transforms and a parametric continuation scheme to determine steady-state period-1 motions of the system due to both parametric and external excitations. The accuracy of the analytical solutions is demonstrated by comparing them to direct numerical integration solutions. Floquet theory is applied to determine the stability of the steady-state harmonic balance solutions. At the end, detailed parametric studies are presented to quantify the combined influence of clearance, quadratic and cubic nonlinearities within reasonable ranges of all other system parameters. A comparison between time-varying and time-invariant systems is also provided to demonstrate the influence of the parametric and external excitations on a piecewise-nonlinear system.