When does the method of harmonic balance give a correct prediction for mechanical systems?

Abstract

We investigate the validity of the harmonic balance method for nonlinear, multi-degree-of-freedom mechanical system with time-periodic forcing and linear damping. We provide conditions under which an approximate periodic solution obtained from this method correctly signals the existence of an actual periodic response of the full nonlinear system. These conditions improve classical results from the literature and provide a-priori computable conditions for the validity and accuracy of the harmonic balance method. Our proof is based on Newton's method in Banach spaces for an appropriately chosen functional. We also derive error bounds for the harmonic balance method and illustrate these on mechanical examples.

Keywords:

• Harmonic balance
• Newton's method
• periodic solution
• nonlinear mechanical system

2010 Mathematics Subject Classifications:

• 37N05
• 49M15

Disclosure statement

No potential conflict of interest was reported by the authors.