Abstract

The problem analyzed here concerns the dynamic response of a bidimensional polygonal rigid body simply supported on a harmonically moving rigid ground. The immediate scientific purpose of the paper is to obtain and analyze the dynamic response by using a variational formulation, recently proposed by one of the authors [1], where the dynamics is described as a differential inclusion. This formulation allows us to determine the instantaneous accelerations of the system by means of a mechanical model with friction and unilateral constraints, which does not reduce the degrees of freedom or impose an “a priori” choice of the mechanism activated during the motion. By treating the friction coefficient as a stability parameter, it has been possible to obtain different kind of responses, ranging from rocking to sliding-rocking, and compare them with those obtained in the literature. Sliding-rocking motions obtained so far have exhibited not only harmonic but also interesting and more complex behaviors with chaotic features.

The search for theoretical and numerical instruments able to identify and classify these more complex motions was first performed in the case of rocking, characterized by a smaller number of degrees of freedom. A technique was then implemented for calculating Lyapunov’s exponents also during the time intervals of the impacts. The introduction and evaluation of these exponents can also permit us to perform the stability analysis with respect to overturning, by limiting the analysis and evaluation to the first impact that the system undergoes by starting at rest: in fact, large values of Lyapunov’s exponents before the first impact are connected with overturning during the motion which follows. This circumstance can make it easier to carry out the stability analysis with respect to overturning, as a function of the amplitude and frequency of the excitation.

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