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Vibration reduction of primary structure using optimum grounded inerter-based dynamic vibration absorber

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Abstract

This paper investigates on the \(H_{\infty }\) and \(H_2\) control performances for a novel grounded inerter-based dynamic vibration absorber configuration (GIDVA) by entirely replacing both damper and stiffness of the classic dynamic vibration absorber with grounded inerter-based mechanical network. First, the equations of motion of the coupled system are derived and the analytical solution is established in terms of primary system displacement under harmonic excitation. Then, the optimum tuning parameters are found by first using the extended fixed points theory (EFPT) to get analytical design. It is found that this EFPT yields the approximate but accurate analytical solutions, so this method can be commonly used to design inerter-based DVAs with four fixed points in the frequency response curves of the primary system. Further, to evaluate the control performance of the proposed GIDVA, the numerical \(H_{\infty }\) and \(H_{2}\) optimizations are performed in MATLAB for the case of harmonic and random excitation, respectively, using the Newton–Raphson algorithm with the starting points, the analytical approximate optimal parameters based on the EFPT. Based on the results comparison of the control performance, one can remark that with the same small mass ratio, the proposed GIDVA can decrease the peak vibration amplitude of primary system more than 64% and 50% and enlarge the suppression bandwidth more than 64% and 57% when compared with the classic DVA and the recent published non-traditional inerter-based DVAs (NIDVA-\(C_i\), i=3, 4, 6), respectively, for both optimal designs \(H_{\infty }\) and \(H_2\). However, one can also notice that this improvement decreases with the increase in the mass ratio. Moreover, under random excitation, the GIDVA also shows good control performance with 56% and 44% reduction in terms of mean square and time history responses of primary system. This result reveals the potential of a novel grounded inerter-based dynamic vibration absorber, which provides a reference for the design index of the novel vibration absorber in engineering practice.

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Acknowledgements

The authors would like to thank the associate editor and the anonymous referees for their valuable comments and suggestions, which helped us to improve the manuscript.

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The author(s) received no financial support for the research, authorship and/or publication of this article.

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Authors and Affiliations

  1. Laboratory of Mechanics, Engineering Science Doctoral Training Unit, University of Douala, P.O. Box 2071, Douala, Cameroun

    Berline Kendo-Nouja

  2. Laboratory of Mechanics, Materials and Structures, Department of Physics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroun

    Marcial Baduidana & Aurelien Kenfack-Jiotsa

  3. Department of physics, Higher Teacher Training College, University of Yaounde I, P.O. Box 47, Yaounde, Cameroon

    Aurelien Kenfack-Jiotsa

  4. National Advanced School of Engineering, University of Yaounde I, P.O. Box 812, Yaounde, Cameroun

    Robert Nzengwa

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  1. Berline Kendo-Nouja
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  3. Aurelien Kenfack-Jiotsa
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Correspondence to Marcial Baduidana.

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Appendix

Appendix

Parameters values of the optimum DVAs used to plot the FRCs of the primary system in Figs. 6, 8 and 10.

Table 7 Optimal design parameters for DVAs presented in Fig. 5 (\(\mu =0.02\))
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Kendo-Nouja, B., Baduidana, M., Kenfack-Jiotsa, A. et al. Vibration reduction of primary structure using optimum grounded inerter-based dynamic vibration absorber. Arch Appl Mech 94, 137–156 (2024). https://doi.org/10.1007/s00419-023-02513-1

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