Parametric resonance with linear damping: a general formula for the excitation threshold for high orders

Abstract We derive a general formula for the excitation threshold of parametric resonances of an oscillator with linear damping from consideration of the asymptotic properties of the Mathieu equation. This provides a good approximation for resonances of order m ≥ 2, and it is especially useful for high-order resonances in systems with light damping for which other approaches are cumbersome. Parametric resonance is ubiquitous in mechanical and electrical systems and its threshold is an important consideration, e.g., for systems that would be damaged by a high amplitude of resonantly excited motion. We present the expressions in a form useful for understanding systems with high quality factors such as trapped atomic ions, micro-mechanical devices and other oscillators, especially those with low dissipation in vacuum. High-order parametric resonances are extremely narrow making direct numerical simulation computationally intensive as well as less insightful.