Primary and Secondary Resonances of a Weakly Nonlinear Oscillator
Agus Yodi Gunawan (a*), Farrell Theodore Adriano (b)

a) Industrial and Financial Mathematics Research Group, Bandung Institute of Technology Jalan Ganesha 10, Bandung 40132, Indonesia
*ayodi[at]math.itb.ac.id

b) Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology Jalan Ganesha 10, Bandung 40132, Indonesia
*ftheodoreadriano[at]gmail.com

Abstract

In this paper, a weakly nonlinear forced oscillator having a nonlinear term proportional to a multiplication of displacement and its acceleration is studied. Furthermore, this term is assumed to be small such that the perturbation methods can be applied. Using the method of multiple scales, we investigate the steady-state response of the system in the case of primary and secondary resonances (subharmonic and superharmonic resonances). We show that under resonance condition, the steady-state response undergoes jump phenomena, which depends on how close the frequency of the external force to the natural frequency of the system. We also present approximate analytical solution using the multiple scales method and its numerical solution. Both are in good agreement.

Keywords: Weakly Nonlinear Oscillators, Multiple Scales, Perturbation Methods

Topic: MATHEMATICS AND STATISTICS