Primary Resonance in a Weakly Forced Mathieu Equation With Parametric Damping

Abstract

Abstract In this work we study the primary resonance of a parametrically damped Mathieu equation with direct excitation. A potential application is wind-turbine blade vibration, as aeroelastic effects with a cyclic angle of attack may induce parametric damping, while cyclic stiffening and direct excitation are also known to be relevant. The parametric stiffness, parametric damping, and the direct forcing, all have the same excitation frequency, with phase parameters between these excitation sources. The parametric amplification at primary resonance is examined by applying the second-order method of multiple scales. With parametric stiffness and direct excitation, it is known that there is a primary parametric resonance that is an amplifier under most excitation phases, but can be a slight suppressor in a small range of phases. The parametric damping is shown to interact with the parametric stiffness to further amplify, or suppress, the resonance amplitude relative to the resonance under parametric stiffness. However, without the parametric stiffness, the parametric damping does not enhance the resonance. The phase of the parametric damping excitation, relative to the parametric stiffness, has a strong influence on the amplification or suppression characteristics. There are optimal phases of both the direct excitation and the parametric damping for amplifying or suppressing the resonance. The effect of the strength of parametric damping is also studied. Numerical simulations validate the perturbation analysis.