An analytical approximated solution and numerical simulations of a non-ideal system with saturation phenomenon

Abstract

Nowadays, researches about non-ideal problems have been increased considerably in technical-scientific community. Nonlinear problems have been widely studied due to their particularities in real problems, mainly when these nonlinearities induce interactions between a dynamical system with its excitation source, these kind of systems are called non-ideal systems. One of the excitation sources, which is of non-ideal kind, is an unbalanced DC motor with limited power supply. When it is coupled to a dynamical system, this system is subjected to Sommerfeld effect, where jump phenomena may occur. However, when the same dynamical system is of two-degrees-of-freedom and possesses 2:1 internal resonance, saturation phenomenon occurs. Therefore, in this work, the dynamical behavior of a non-ideal three-degrees-of-freedom weakly coupled system associated with quadratic nonlinearities in the equations of motion is investigated. The full system consists of two nonlinear mechanical oscillators coupled through quadratic nonlinearities and which produces a 2:1 internal resonance between their translational movements. In the bigger oscillator, an unbalanced DC motor is used as an excitation source. Under these conditions, equations of motion of the system were obtained using Lagrange’s method, and the method of multiple scales was applied to find an analytical approximated solution of the equations of motion. In addition, numerical simulations of the equations of motion were carried out to analyze the response of the non-ideal system by varying the torque of the motor. It is shown that when the excitation frequency is near to second natural frequency of the main system, saturation and jump phenomena occur. Furthermore, this work investigates the ranges of some torques of the motor, which causes the phenomena, and explores the possibility to harvest energy from high-amplitudes of vibration in a future work.