Periodic response and stability of hysteretic oscillators

Asymptotic response of a two DOF elastoplastic system under harmonic excitation. Internal resonance case

“…When, on the contrary, the nonlinear springs behave hysteretically, relations (2) do not hold tree because the fi's are no longer single-valued functions of ~2i's, but are more loosely related functionally. If only periodic motions are sought, the solutions can be found using the harmonic balance method [10,15]. The use of this method leads to a system of algebraic equations with n × 2N unknowns, where N is the number of harmonic components assumed.…”

“…A more general hysteretic law made it possible to show cases of multi-valued frequency response curves and the existence of other branches besides the principal one [8]. However more refined analyses have shown that the presence of hysteresis reduces some of the effects of nonlinearity [10], as does high damping in nonlinear elastic systems. As far as the methods of solution for periodic motions are concerned, the main difference between elastic and hysteretic models lies in the fact that the latter have a multi-valued restoring force which requires the use of integral methods such as harmonic balance or averaging [8], while methods needing derivatives like perturbation, are not possible.…”

“…In [9], for instance, it is shown that results obtained in [5] for one DOF oscillator, by assuming four branches per cycle, are in some cases not correct. Many difficulties associated with the multivaluedness of hysteretic force can be overcome by adding dimensions to the state variables space where the restoring force can be formulated in incremental form by means of only ordinary functions [10,11].…”

Periodic solutions of non-smooth friction oscillators

“…This phenomenon is not covered by linear elastic or viscoelastic models, but can be reproduced if different stiffness coefficients, depending whether the spring is loaded or unloaded, are used. This means to use −k(1 + µ 0 sgn (xx ))x with µ 0 ∈ (0, 1) as restoring force, which leads to (2) with µ(x) = µ 0 |x|. Although the complex behavior of systems with hysteresis cannot be covered by such a simple model, it nevertheless reproduces the phenomenon mentioned above.…”

“…In fact, this type of model is presently used to study vibration control devices; see [12]. A more sophisticated four-dimensional model of a hysteretic oscillator is studied in [2] by means of "the method of harmonic balance", i.e. Galerkin approximations to periodic solutions are computed.…”