Mixed Mode Oscillations and Bifurcation Mechanism in a Nonlinear Beam-Elastic Foundation Under Parametric and External Excitations

TL;DR

This study investigates the nonlinear dynamics and bifurcation mechanisms in a beam-elastic foundation system under parametric and external low-frequency excitations, revealing routes to chaos and bi-stability through bifurcation analysis.

Contribution

It introduces a detailed bifurcation analysis of a nonlinear vibratory system under combined excitations, highlighting the roles of foundation parameters and excitation types in system stability.

Findings

Folding, cusp, and Bogdanov-Takens bifurcations identified.

Slow excitation induces fold bifurcation.

Elastic foundation stiffness affects stability regions.

Abstract

This paper aims to study existence condition of possible bursting oscillations generated by low frequency excitation of a nonlinear vibratory system in the presence of parametric excitation. Slow-fast dissection technique and numerical bifurcation analysis are employed to extract qualitative changes in system response originated from its nonlinear dynamics. Role of all parameters of elastic foundation and excitation model are studies and it is shown that the system exhibits the phenomena of folding, cusp and Bogdanov-Takens bifurcations which are potentially routes to bi-stability and chaos. It can be found that slow excitation of the nonlinear foundation is the main generating factor of fold bifurcation and stiffness of elastic foundation has a remarkable effect on stability region of the beam. In addition, the base excitation of an elastic foundation in form of a traveling wave, adds multi-frequency excitation and parametric resonances necessarily to the system. This study showed investigating nonlinear oscillator under low frequency excitations in framework of slow-fast plays an invaluable role in understanding instabilities in systems that are not captured by standard methods.