Stability and bifurcations of hysteretic systems subjected to principal parametric excitation

Dynamic behavior of smooth hysteretic systems subjected to harmonic parametric excitation is investigated. Wen's differential equation model for hysteresis, which can describe a large class of hysteretic systems, is used. Employing the piecewise power series expression for hysteresis proposed in the previous paper, the method of multiple scales is applied for the case of principal parametric resonance to obtain the second-order approximate solutions and their stability. It is shown that the hysteretic systems are unstable inside the principal resonance region but the responses are bounded having symmetric periodic solutions. It is also shown that the systems are stable outside the region even if viscous damping does not exist because of Hopf bifurcation. Numerical integrations are also performed to illustrate the nonlinear resonant characteristics of the systems and confirmed that trivial stationary solutions and stable symmetric periodic solutions in principal resonance region bifurcate to produce a pair of non-symmetric periodic solutions. Stability regions with regard to excitation parameters are also illustrated. The theoretical and numerical results are compared to examine the validity of the present analysis.