On the Non-Linear Oscillations of a Parallelepiped-Pendulum system with dry-friction

This work studies the dynamic behavior of a simple mechanical model, constituted of a parallelepiped coupled with a pendulum standing on a horizontal plane. When the plane performs vibrational displacements the parallelepiped begins to evolve its positions and, this "parallelepiped-pendulum movements'' constitutes the subject of our concern. Notwithstanding the fact that this study belongs to "mathematical investigations'', we point out that its motivations belong to the "technical world of research''. These studies originate from the the technical scenario of earthquakes. Interaction between simplified buildings model (the "rocking block model'') and earthquakes are studied since a long time and there is a wide literature about them: [1-5]. When you consider masonry buildings or, in general, no-tension structures, like ancient columns, masonry towers, old chimneys, is not possible to think to them like linear elastic systems. Hence, the methods for reducing and controling vibrations like the application of some "tuned mass and energy absorber'' has to be completely revised. As a first step in the long journey to eventually reduce oscillation caused by the base-movement for these no-tension structure, a simple model for the masonry tower is analyzed, with the application on it of a coupled oscillator (simple pendulum), and the adding of some simple cause for dissipation, i.e. the dry-friction. The mathematical investigation on this simple system gave the following results: 1. it is possible to give a numerical relation which ties the angle of oscillation of the parallelepiped with its frequency (case without pendulum and with pendulum); the same simulations can be done after adding damping in the form of dry-friction. 2. subjected to the external excitation due to the movement of the plane on which the parallelepiped is laid, the presence of the pendulum diminishes the oscillation amplitude. This last situation simulates earthquakes, and from these results we can already see some interesting way for reducing damages to structures. The model here studied is extremely simplified. In-plane movement is assumed so, notwithstanding we name it as a parallelepiped, the structure is actually schematized as a "physical rectangle''. Further steps should consider the introduction of other kinds of dampings and the possibility of laying a "physical parallelepiped" on an elastic bed.