Nonlinear vibrational behavior of multi-body dynamical systems with bi-directional piecewise linear spring constraints

Abstract

The nonlinear vibrational behavior of multi-body dynamical systems with piecewise linear spring characteristics under harmonic excitations is investigated in this paper using the Bozzak-Newmark-LCP numerical scheme. Each body is subjected to bi-directional piecewise linear spring constraints, which are activated when the initial gaps between the body and the secondary springs are consumed. The system equations of motion are discretized in the time domain using the Bozzak relaxation scheme. At each time step, a set of governing equations in terms of displacements are obtained using the Newmark integration scheme. Two auxiliary displacement vectors, complementary to the contact force vectors, are introduced to detect the engagement of gap activated springs with a resolution of the time step size. With the aid of a simple transformation, the nonlinear vibration problem of a multiple degrees-of-freedom (d.f.) piecewise-linear dynamical system is reduced to a mathematical programming problem for which an accurate and efficient solution satisfying the piecewise linear constraints everywhere can be obtained. The numerical scheme used in this paper was first validated for a single d.f. system with two-sided gap-activated springs under base excitation using the results available in the literature. Numerical results are obtained and presented for the dynamical behavior of a two-body oscillator with two primary springs and four gap-activated secondary springs under harmonic excitations over a wide range of values of system and excitation parameters.

Keywords

Bifurcation jump multiple degrees-of-freedom piecewise linear constraint stiffness hardening vibration

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