Periodic response and stability of hysteretic oscillators

Abstract

Difficulties encountered in the study of the response of hysteretic systems under periodic force lie in the multivalued nature of the constitutive relationship. In this paper, some of these difficulties are circumvented by assuming an incremental formulation which results in an ordinary nonlinear problem with single-valued functions, though with an enlargement of the phase space. Consideration is given only to periodic oscillations that are found through the harmonic-balance method with many components; there thus ensues a system of algebraic equations that is solved numerically. Stability is studied by the linearized Poincar´ map determined via numerical integration. A simple hysteretic oscillator, that presents degrading and non-degrading behaviour, is considered. The results clearly show that the influence of higher harmonics is far from negligible. While non-degrading oscillators reveal stable behaviour over all the frequency range, in the degrading case there is instability that allows either saddle-node or Hopf bifurcation