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

Abstract

The study of a two DOF elastoplastic system is formulated in a suitable phase space, velocity and force, in which an originally multi-valued restoring force is represented by a proper function. The asymptotic response can thus be studied using the Poincar6 map concept and avoiding approximate analytical techniques. On account of the peculiarity of this hysteretic system, which has a well-defined yielding point, its dynamics is studied in a reduced dimension phase space using an efficient numerical algorithm. It is shown that the asymptotic response is always periodic with the period of the driven frequency and is always stable. Thus the response of the oscillator is described by its frequency response curves at various intensities of the excitation. The results presented refer to a system with two linear frequencies in a ratio of 1 : 3. The response is highly complex with numerous peaks corresponding to higher harmonics. The effect of coupling in conditions of internal resonance is a strong modification of the frequency response curves and of the oscillation shape of the structure.

Chua Systems With Discontinuities

“…systems with mathematical difficulties such as discontinuities and, or multivalued differential equations. Such models are interesting from the point of view of applications: Impacts, friction and constitutive laws provide unsmooth models of that type [Paoli, 1993;Paoli et al, 1992;Monteiro Marques, 1994;Deimling, 1992;Popp & Stelter, 1990;Pfeiffer & Prestl, 1994;Whiston, 1987;Shaw, 1986;Shaw & Shaw, 1989;Dowell & Schwartz, 1983;Ferri & Bindemann, 1995;Moreau, 1988;Awrejcewicz & Delfs, 1990a;Awrejcewicz & Delfs, 1990b;Capecchi & Vestroni, 1995;Mahla & Badan Palhares, 1993] etc.…”

Periodic and chaotic dynamics in an asymmetric elastoplastic oscillator

“…Capecchi [4] has studied the elastoplastic oscillator using a nonsmooth mapping approach. Capecchi and others [5,6,7] have extended this approach to two degree of freedom hysteretic oscillators. This work has relevance to structural engineering, particularly earthquake engineering, because most structural systems have a hysteretic response to large amplitude excitations.…”