A study of the dynamic behavior of piecewise nonlinear oscillators with time-varying stiffness

The dynamic behavior of a piecewise-nonlinear mechanical oscillator with parametric and external excitations is investigated. The viscously damped oscillator is subjected to a periodically time-varying, piecewise nonlinear restoring function. 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 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 and available experimental data for a special case. Floquet theory is applied to determine the stability of the steady-state harmonic balance solutions.; This solution method is first applied on a single-degree-of-freedom piecewise nonlinear time-varying system to find steady state period-1 and period-eta (eta > 1) motions. The system is characterized by a symmetric restoring function, which consists of three segments: a clearance (dead-zone) segment and two continuously nonlinear segments defined by a linear component, a quadratic term and a cubic term. 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. As a specific application, an elastic sphere-plane interface is studied by using this solution method. The dynamic model of the sphere-plane system includes both a continuous nonlinearity associated with the Hertzian contact and a clearance-type nonlinearity due to contact loss. The accuracy of the dynamic model and solution method is demonstrated through comparisons with experimental data and numerical solutions. A single-term harmonic balance approximation is used to derive a criterion for contact loss to occur. The influence of harmonic external excitation f(tau) and damping ratio zeta on the steady state response is also demonstrated.; Finally, the solution method is extended to a generalized multi-degree-of-freedom dynamical system with multiple clearances, time-varying coefficients, and piecewise nonlinear characteristics. This generalized formulation is applied to a three-degree-of-freedom gear-bearing system to demonstrate its applicability.