Single‐Degree‐of‐Freedom Nonlinear Homogeneous Systems

Abstract

A class of nonlinear oscillator that scales input‐output relations is identified and studied in this paper. This type of system shows variable stiffness, variable damping, or both, and is piecewise linear in conical regions of the state space. The following passive and semiactive mechanical systems, which exhibit this variable structure, are used to motivate the discussion: structures containing energy‐dissipating devices with different loading and unloading stiffnesses, structures with variable dampers, arid structures with active variable stiffness. The free‐vibration response of single‐degree‐of‐freedom systems with variable stiffness and variable damping is computed. It is demonstrated that the period of oscillation and the decay ratio between consecutive peaks of this type of nonlinear system are independent of the amplitude of oscillation. The statistical‐linearization method is used to estimate the mean‐square response of structures containing nonlinear homogeneous devices and subjected to random excitation. Excellent accuracy is achieved in the estimation of the mean‐square response of these oscillators using the statistical‐linearization method.