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Reduction of settling time by multi-frequency pulsed parametric excitation

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Abstract

Introducing a time-periodicity into a system parameter leads to parametric excitation, which in general, may cause a parametric resonance with exponentially increased vibration. Applying a parametric excitation but carefully tuning its frequencies to multiple parametric anti-resonance frequencies is investigated here. The parametric excitation here is realized by an open-loop control at the system boundary that allows for an energy flow into or from the system. A parametric anti-resonance successfully triggers an energy transfer between specific vibration modes of the system and occurs in systems with at least two degrees of freedom. Such an energy transfer increases the overall dissipation of kinetic energy of a lightly damped system. This contribution presents an approach to accelerate the mitigation of transient vibrations by applying a multi-frequency parametric excitation with two or more parametric anti-resonance frequencies. The potential application in a MEMS sensor arrangement consisting of two and more coupled flexible beams exemplifies the method. Starting from the minimum system with two degrees of freedom, the averaging method is applied to analyze the transient slow flow, leading to an analytical approximation of the transition time response of a pulsed multi-frequency parametric excitation system. For a specific example, a reduction of 96.7% of the transient vibrations is achievable.

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The authors declare that all data supporting the simulations and examples of this study are available within the article.

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Acknowledgements

The author MRB thanks the IPN-SIP-202315 and the Austrian Academy of Sciences for their support for the development of this project.

Funding

This work was supported by the Joint Excellence in Science and Humanities (JESH) programme of the Austrian Academy of Sciences through the grant given to the first author.

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Authors and Affiliations

  1. Instituto Politécnico Nacional, UPIBI, Av. Luis Enrique Erro S/N, 07738, Mexico City, Mexico

    Miguel Ramírez-Barrios

  2. Research Center for Microtechnology, Vorarlberg University of Applied Sciences, 6850, Dornbirn, Austria

    Fadi Dohnal

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Correspondence to Miguel Ramírez-Barrios.

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Appendix

Appendix

Fig. 13
figure 13

Comparison between the full solution \(x_1(t)\) of the original system and the envelope \(\vert \hat{u}_1 \vert \) of the approximate system

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The solution \(x_1(t)\) of beam 1 of the original systems for configurations with 2DOF, 3DOF and 4DOF defined in Eqs. (16), (23) and (30) is compared directly with the solution \(\vert \hat{u_1} \vert \) of the approximated slow flows in Eqs. (17), (25) and (32). This is possible due to the definition in Eq. (6). The solutions are transformed according to Eq. (4) into the physical time t. The system parameters are taken from the list given in Table 2. The initial condition is chosen as \(x_1(0) = 0\) in all cases. Figure 13 highlights the quality of the approximation between the physical time evolution and the envelope represented by the slow flow.

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Ramírez-Barrios, M., Dohnal, F. Reduction of settling time by multi-frequency pulsed parametric excitation. Nonlinear Dyn 112, 7185–7198 (2024). https://doi.org/10.1007/s11071-024-09281-9

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