Dynamical Behaviors of a Mass Submitted to the Action of Perpendicular Spring and Excitations with Phase Shift.

Background: In recent years, interesting results have been obtained on the study of the nonlinear dynamics of translational, rotational and pendulum electromechanical systems. Most of these systems consider one-directional motions or coupling of one-directional motions. However, for some applications, actuators can move over different places on a surface or plane. Purpose: The purpose of this work is to study the nonlinear dynamics of a mass moving over a plane when it is subjected to perpendicular springs and perpendicular forces created two periodically excited electromagnets with phase shift. Methods: After the derivation of the equations of motions using mechanics laws and interaction forces created by electromagnets, the harmonic balance method and the fourth-order Runge-Kutta method are used to obtain mathematically and numerically the dynamical states of the system. Results: It is found that a bidimensional periodic motion takes place along the oblique line with slope depending on the phase shift. The frequency-response curves of the periodic motion show resonance, antiresonance and hysteresis. The bifurcation diagrams versus the amplitude of the excitations and the phase shift show that the dynamical states can be periodic or chaotic. The transition from the periodic states to chaos is either through successive period-doublings or abrupt. Conclusion: In this work, the nonlinear dynamics of a mass subjected to the action of perpendicular springs and perpendicular periodic forces with phase shift has been analyzed. It has been found that the mass can exhibit several dynamical states in a plane including chaos characterized by the random distribution of the positions occupied by the mass in the bidimensional space. [ABSTRACT FROM AUTHOR]