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

Dolatabad, M.R ; Sharif University of Technology | 2022

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  1. Type of Document: Article
  2. DOI: 10.1007/s11071-021-06969-0
  3. Publisher: Springer Science and Business Media B.V , 2022
  4. Abstract:
  5. 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. © 2021, The Author(s), under exclusive licence to Springer Nature B.V
  6. Keywords:
  7. Analytical models ; Dynamic stability ; Inverted pendulum ; Nonlinear vibration ; Closed loop systems ; Dynamical systems ; Equations of motion ; Nonlinear equations ; Pendulums ; State feedback ; Vibration analysis ; Analytical analysis ; Dynamics stability ; Experimental analysis ; Gaining insights ; Non-linear vibrations ; Numerical solution ; Rotary inverted pendulums ; Stable equilibrium ; Vibratory response ; 3D modeling
  8. Source: Nonlinear Dynamics ; Volume 107, Issue 3 , 2022 , Pages 1887-1902 ; 0924090X (ISSN)
  9. URL: https://link.springer.com/article/10.1007/s11071-021-06969-0