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Asymptotic analysis of circular motions of base- and length-parametrically excited pendula

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Abstract

The regular circular motions of parametrically excited pendula are analyzed via the Multiple Scale Perturbation Method. Two different cases, object of discussions in the literature, are considered, namely the base-excited pendulum (BEP) and the variable-length pendulum (VLP), which exhibit different behaviors, hitherto unexplained. Under the hypothesis that the parametric excitation is of large frequency, the problem is recast in the form of a perturbation of a singular linear differential operator, admitting a double-zero eigenvalue, where the perturbation parameter is identified in the amplitude of the parametric excitation. Similarly to the linear algebraic eigenvalue problem around such a degenerate point, the asymptotic method calls for using fractional power expansions of the perturbation parameter, for both the state variables and the time-derivative operator. However, while the perturbation appears to be regular in the BEP, so that fractional series apply, it is singular in the VLP, so that such expansions degenerate in integer powers. As a consequence, the BEP is more prone to execute circular motions, in the sense that a lower excitation amplitude is required, in comparison with the VLP. Such a result is confirmed by common experiences in the real world, examples of which are given. On the other hand, the VLP admits motions with different angular velocities, while the BEP just admits one velocity, at least to within the order of the asymptotic solution carried out here. The approach followed is believed to give a satisfactory mathematical explanation of the different behaviors of the two systems.

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Authors and Affiliations

  1. Department of Civil, Construction-Architectural and Environmental Engineering, University of L’ Aquila, 67100, L’Aquila, Italy

    A. Luongo

  2. Department of Architecture, University of Roma Tre, 00185, Roma, Italy

    A. Casalotti

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  1. A. Luongo
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Correspondence to A. Luongo.

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Luongo, A., Casalotti, A. Asymptotic analysis of circular motions of base- and length-parametrically excited pendula. Nonlinear Dyn 112, 757–773 (2024). https://doi.org/10.1007/s11071-023-09112-3

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