Combination resonances in the non-linear response of bowed structures to a harmonic excitation

Abstract

A second-order uniform expansion is obtained for the response of a bowed structure (systems with quadratic and cubic non-linearities) to a combination resonance. The results show that combination resonances of the difference type can never be excited. Moreover, the results show that the steady state amplitudes depend strongly on the second-mode excitation amplitude f 2. Two threshold values ζ 1 and ζ 2 for f 2 are identified. When f 2 < ζ 1, combination resonances are never excited, when f 2 > ζ 2, combination resonances are always excited, and when ζ 2 ⩽ f 2 < ζ 2, combination resonances may or may not be excited, depending on the initial conditions. The results also show that combination resonances are very sensitive to the addition of a first-mode excitation amplitude f 1. For a certain range of values of f 1 f 2 and the phase angle τ between the two excitations, combination resonances are quenched and the response consistes essentially of the linear response. The quenching of combination resonances has been verified by numerically integrating the original differential equations in the presence as well as the absence of internal resonances. The analysis also identifies a range of values of f 1 f 2 and τ for which the response is significantly enhanced; this result has also been verified by numerically integrating the original equations in the presence as well as the absence of internal resonances.