Tilted excitation implies odd periodic resonances

TL;DR

This paper investigates how breaking symmetry in a parametric pendulum through tilted excitation leads to the emergence of odd periodic resonances, combining numerical, experimental, and analytical methods.

Contribution

It introduces a novel analysis of odd resonances caused by asymmetric excitation, using Melnikov theory to analytically predict bifurcation loci and explain the phenomenon.

Findings

Odd resonances occur with tilted excitation.

Analytical bifurcation loci are derived using Melnikov theory.

Experimental results confirm the theoretical predictions.

Abstract

Our aim is to unveil how resonances of parametric systems are affected when symmetry is broken. We showed numerically and experimentally that odd resonances indeed come about when the pendulum is excited along a tilted direction. Applying the Melnikov subharmonic function, we not only determined analytically the loci of saddle-node bifurcations delimiting resonance regions in parameter space, but also explained these observations by demonstrating that, under the Melnikov method point of view, odd resonances arise due to an extra torque that appears in the asymmetric case.