Abstract
Two fundamentally different versions of the method of multiple scales (MMS) are currently in use in the study of nonlinear resonance phenomena. While the first version is the widely used reconstitution method, the second version is proposed by Rahman and Burton [1]. Both versions of the second-order MMS are applied to the differential equation obtained for a parametrically excited cantilever beam with a lumped mass at an arbitrary position. The bifurcation and stability of the obtained response show the difference between the two versions. While the Hopf bifurcation phenomena with no jump is found in the case of second-order MMS version I, both jump-up and jump-down phenomena are observed in second-order MMS version II, which closely agree with the experimental findings. The results are compared with those obtained by numerically integrating the original temporal equation.
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Dwivedy, S.K., Kar, R.C. Nonlinear Response of a Parametrically Excited System Using Higher-Order Method of Multiple Scales. Nonlinear Dynamics 20, 115–130 (1999). https://doi.org/10.1023/A:1008358322080
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DOI: https://doi.org/10.1023/A:1008358322080