Abstract

This paper presents the experimental response curves for the solutions of Duffing’s equation for that frequency range where higher-order harmonics (called superharmonics as opposed to lower order or subharmonics) of the fundamental are important components of the solution. The experimental results were obtained by solving Duffing’s equation on an electronic differential analyzer. The paper presents corroboration of analytical results for the third-order superharmonic component of solutions which were calculated by a two-term Ritz approximation. The results of this paper indicate that superharmonics of higher order than the third, that is, fifth and seventh order, and even-order harmonics such as second and fourth order can be found in the frequency range studied. This paper points out a definite relationship which exists between the superharmonic generated and the free-vibration curve (“backbone”) of the system. For example, if we compare the maximum displacement of the solution containing an nth order superharmonic with the same amplitude of the free-vibration curve, it is seen that the frequency of the fundamental (the forcing frequency) is 1/n times this natural frequency or, conversely, the frequency of the superharmonic component is n times that of the fundamental and has in fact the frequency of the free vibration for that amplitude. Thus a superharmonic of order n can be pictured as a sustained free vibration which is generated when the frequency of the forcing function is 1/n of the frequency of the free-vibration response curve. The phenomenon of superharmonic oscillations is pictured here as a nonlinear resonance. Experimental evidence of the superharmonic-jump phenomenon as discovered analytically by John Burgess (1) also is presented in this paper.

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