✍ Scribed by A. Hassan
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The order three superharmonic resonance in the harmonically excited Duffing oscillator with hardening non-linearity is considered. The second order perturbation solutions determined by the multiple scales and frequency-damping expansion methods are compared with each other, and with the exact numerical results. By using the example of third superharmonic resonance in the Duffing oscillator, this work establishes the following:
(1) the approximate periodic solutions determined by the frequency-damping expansion procedure are based on the fast time scale only; (2) these solutions provide an approximation for the steady state response only; (3) the second order approximate solutions for the third superharmonic resonance in the Duffing equation determined by these methods can uncover a second type of frequency response curve; (4) both second order approximations predict extraneous solutions; and (5) the extension, achieved by the second order approximation, in the range of the system parameters for which the perturbation solution provides a reasonable approximation to the actual system response, may depend on the problem on hand and, in this regard, performance of the frequency-damping expansion procedure may not be substantially superior to the performance of the multiple scales -reconstitution procedure.
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