Dinâmica não-linear em vibrações flexurais de um ressonador mecânico cantiléver.

Abstract

In this work we build an inertial table on which we mount a cantilever mechanical resonator made of a thin steel rod in order to obtain high values of the quality factor and thus study non-linear vibrations. We use the Euler-Bernouli model to obtain normal modes and frequencies. From measurements of the fundamental mode frequencies at different lengths we obtain the Young's modulus of the cantilever steel. We also obtain the mass and the effective spring constant in fundamental mode near the tip of the cantilever. We approximate the dynamics of vibrations in the fundamental mode by a one-degree-of-freedom model. With the aim of building a Duffing oscillator, we coupled a neodymium magnet to the free end of the resonator and a fixed support a little above the rod another magnet identical to the one that was coupled to the cantilever. When far apart, the magnets are aligned on the same vertical axis and oriented in opposite directions, but when we reduce the distance between the buckling rod sufficiently, forming a double potential well. We characterize the parameters of this oscillator by measuring the two equilibrium positions of each well and their respective small amplitude oscillation frequencies and corresponding damping rate. In addition to damped oscillations, we also study forced oscillations. We found bistability curves for oscillations in each well. We fit the experimental data to the bistability curves using the harmonic balance method. Finally, we excite vibrations with chaotic characteristics such as a continuously distributed Fourier spectrum, mainly at low frequencies. Finally, we report that the theoretical models we used fit the experimental data of this dissertation very well.