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Nonlinear homogeneous dynamical systems

Inaudi, Jose A.; Kelly, James M.

UCB/EERC-93/11, Earthquake Engineering Research Center, University of California, Berkeley, 1993, 106 pages (530/I62/1995)

This report deals with some aspects of the dynamical response of nonlinear mechanical systems in which the response to external excitation is scaled by a constant when the excitation and initial conditions are scaled by that constant. This mathematical property is referred to as homogeneity of order one. Oscillators of this class exhibit characteristics typical of linear systems such as period of vibration and decay ratio independent of the amplitude of vibration. Furthermore, in the case of nonlinear multi-degree-of-freedom structures satisfying this homogeneity property, the response in free vibration can exhibit modes of vibration if certain conditions are satisfied. Passive and active vibration-control systems exhibiting this special property are described and some of the features of their dynamical response are analyzed. The systems presented include a passive friction dissipater known as the Energy Dissipating Restraint, a nonlinear causal model of linear hysteretic damping, semi-active viscous dampers, and semi-active friction dampers with modulated contact forces. Numerical methods for the integration of the equations of motion of this type of system are presented. Linearization methods are applied to this type of nonlinearity leading to linear viscoelastic models that can represent the behavior of nonlinear homogeneous systems with very good accuracy. Suggestions for further analytical and experimental research on concepts presented are given.

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