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Analytical and experimental analyses of nonlinear vibrations in a rotary inverted pendulum

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Abstract

Gaining insight into possible vibratory responses of dynamical systems around their stable equilibria is an essential step, which must be taken before their design and application. The results of such a study can significantly help prevent instability in closed-loop stabilized systems by avoiding the excitation of the system in the neighborhood of its resonance. This paper investigates nonlinear oscillations of a rotary inverted pendulum (RIP) with a full-state feedback controller. Lagrange’s equations are employed to derive an accurate 2-DoF mathematical model, whose parameter values are extracted by both the measurement and 3D modeling of the real system components. Although the governing equations of a 2-DoF nonlinear system are difficult to solve, performing an analytical solution is of great importance, mostly because, compared to the numerical solution, the analytical solution can function as an accurate pattern. Additionally, the analytical solution is generally more appealing to engineers because its computational costs are less than those of the numerical solution. In this study, the perturbative method of multiple scales is used to obtain an analytical solution to the coupled nonlinear motion equations of the closed-loop system. Moreover, the parameters of the controller are determined using the results of this solution. The findings reveal the existence of hardening- and softening-type resonances at the first and second vibrational modes, respectively. This leads to a wide frequency range with moderately large-amplitude vibrations, which must be avoided when adjusting a time-varying set-point for the system. The analytical results of the nonlinear vibration of the RIP are verified by experimental measurements, and a very good agreement is observed between the results of both approaches.

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Funding

No funding was received to assist with the preparation of this manuscript.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Sharif University of Technology-International Campus, Kish, Iran

    Mohammadjavad Rahimi Dolatabad & Amir Ali Akbar Khayyat

  2. Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

    Abdolreza Pasharavesh

Authors
  1. Mohammadjavad Rahimi Dolatabad
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  2. Abdolreza Pasharavesh
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  3. Amir Ali Akbar Khayyat
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Contributions

All authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Mohammadjavad Rahimi Dolatabad and Abdolreza Pasharavesh. The first draft of the manuscript was written by Mohammadjavad Rahimi Dolatabad and all authors commented on previous versions of the manuscript. Dr. A. Khayyat and Dr. A. Pasharavesh read and approved the final manuscript.

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Correspondence to Abdolreza Pasharavesh.

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Dolatabad, M.R., Pasharavesh, A. & Khayyat, A.A.A. Analytical and experimental analyses of nonlinear vibrations in a rotary inverted pendulum. Nonlinear Dyn 107, 1887–1902 (2022). https://doi.org/10.1007/s11071-021-06969-0

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  • DOI: https://doi.org/10.1007/s11071-021-06969-0

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