Potential well escape in a galloping twin-well oscillator

TLDR: In this paper, three different analytical approaches are implemented: (i) the method of harmonic balance, (ii) a method of multiple scales using harmonic and elliptic basis functions, and (iii) the Melnikov criterion.

Abstract: When a bi-stable oscillator undergoes a supercritical Hopf bifurcation due to a galloping instability, intra-well limit cycle oscillations of small amplitude are born. The amplitude of these oscillations grows as the flow speed is increased to a critical speed at which the dynamic trajectories escape the potential well. The goal of this paper is to obtain a simple yet accurate analytical expression to approximate the escape speed. To this end, three different analytical approaches are implemented: (i) the method of harmonic balance, (ii) the method of multiple scales using harmonic and elliptic basis functions, and (iii) the Melnikov criterion. All methods yielded an identical expression for the escape speed with only one key difference which lies in the value of a constant that changes among the different methods. A comparison between the approximate analytical solutions and a numerical integration of the equation of motion demonstrated that the escape speed obtained via the multiple scales method using the elliptic functions and the Melnikov criterion are in excellent agreement with the numerical simulations. On the other hand, the first-order harmonic balance technique and the multiple scales using harmonic functions provide analytical estimates that significantly underestimate the actual escape speed. Using the Melnikov criterion, the influence of parametric and additive noise on the escape speed was also studied.