Quenching of a primary resonance by a combination resonance of the additive or difference type

Abstract An investigation is presented of the interaction of primary resonances and combination resonances of the additive and difference types in single-degree-of-freedom systems with quadratic and cubic non-linearities. The method of multiple scales is used to derive two coupled first order ordinary differential equations that describe the evolution of the amplitude and phase with damping, non-linearity and both primary and combination resonances. These equations are used to determine the conditions for steady state solutions and their corresponding frequency response equation. The latter is used to determine the conditions for quenching or enhancing the response of the system to a primary resonance by the addition of a combination resonance. The analytical results are verified by numerically integrating the original equation.