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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Limit Cycle Prediction for a Multi-Degree-of-Freedom System with a Freeplay Non-Linearity

Journal of Applied Nonlinear Dynamics 12(2) (2023) 379--403 | DOI:10.5890/JAND.2023.06.014

A. P. Lewis

School of Physics, Engineering and Computer Science, University of Hertfordshire, Hatfield, Hertfordshire AL10 9AB, United Kingdom

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Abstract

This paper presents a method for predicting limit cycles of a multi-degree-of-freedom system possessing a freeplay non-linearity. The approach taken is through casting the problem as an integro-differential equation. The method is a development of that previously reported in the literature for the simpler case of such a system with a cubic hardening non-linearity. The system considered is based on aeroelastic applications where structural non-linearities of this kind are encountered. Limit cycles stability is determined using an implementation of Floquet analysis based on extending the Hill's Determinant approach that may be used in analysing the Mathieu equation. The limit cycle predictions and Floquet multipliers are compared against predictions from numerical integration to show the validity of the method. Fast Fourier transform analysis is used to provide comparisons with the predictions of harmonic components from the analytical results. As the Floquet analysis also produces an approximation to the motion of the system in the neighbourhood of a limit cycle, in the case of an unstable limit cycle, it was possible to approximate the limit cycle stable manifold in situations where the limit cycle amplitude is much greater than the amount of freeplay.

References

  1. [1]  Woolston, D.S., Runyan, H.L., and Byrdsong, T.A. (1955), Some Effects of System Non-Linearities in the Problem of Aircraft Flutter, NACA TN 3539.
  2. [2]  Woolston, D.S., Runyan, H.L., and Andrews, R.E. (1957), An investigation into the effects of certain types of system non-linearities on wing and control surface flutter, Journal of the Aeronautical Sciences, 24(1), 57-63, January 1957.
  3. [3]  Shen, S.F. (1959), An approximate analysis of non-linear flutter problems, Journal of the Aeronautical Sciences, 28(1), 25-32.
  4. [4]  Laurenson, R.M. and Trn, R.M. (1980), Flutter analysis of missile control surfaces containing structural non-linearities, AIAA Journal, 18(10), 1245-1251.
  5. [5]  Lee, C.L. (1986), An iterative procedure for non-linear flutter analysis, AIAA Journal, 24(5), 833-839.
  6. [6]  Laurenson, R.M., Hauenstein, A.J., and Gubser, J.L. (1986), Effects of Structural Non-Linearities on Limit Cycle Response of Aerodynamic Surfaces, In 27th Structures, Structural Dynamics and Materials Conference, 86-0899.
  7. [7]  Lau, S.L. and Zhang, W. S. (1992), Nonlinear vibrations of piecewise-linear systems by incremental harmonic balance method, ASME Journal of Applied Mechanics, 59(1), 153-160.
  8. [8]  Chicurel-Uziel, E. (2001), Exact, single equation, closed-form solution of vibrating systems with piecewise linear springs, Journal of Sound and Vibration, 245(2), 285-301.
  9. [9]  Liu, L. and Wong, Y.S. (2002), Non-linear aeroelastic analysis using the point transformation method part 1: freeplay model, Journal of Sound and Vibration, 253(2), 447-469.
  10. [10]  Monfared, Z., Afsharnezhad, Z., and Esfahani, J.A. (2017), Flutter, limit cycle oscillation, bifurcation and stability regions of an airfoil with discontinuous freeplay nonlinearity, Nonlinear Dynamics, 90(3), 1965-1986.
  11. [11]  Liu, L. and Dowell, E.H. (2009), High dimensional harmonic balance analysis for dynamic piecewise aeroelastic systems, ASME International Mechanical Engineering Congress and Exposition, Proceedings, 12, 659-669.
  12. [12]  Verstraelen, E., Dimitriadis, G., Rossetto, G.D.B., and Dowell, E.H. (2017), Two-domain and three-domain limit cycles in a typical aeroelastic system with freeplay in pitch, Journal of Fluids and Structures, 69, pp. 89-107.
  13. [13]  Al-Mashhadani, W.J., Dowell, E.H., Wasmi, H.R., and Al-Asadi, A.A. (2017), Aeroelastic response and limit cycle oscillations for wing-flap-tab section with freeplay in tab, Journal of Fluids and Structures, 68, 403-422.
  14. [14]  Wayhs-Lopes, L.D., Dowell, E.H., and Bueno, D.D. (2020), Influence of friction and asymmetric freeplay on the limit cycle oscillation in aeroelastic system: An extended H{e}non's technique to temporal integration, Journal of Fluids and Structures, 96.
  15. [15]  Castellani, M., Lemmens, Y., and Cooper, J.E. (2016), Parametric reduced-order model approach for simulation and optimization of aeroelastic systems with structural nonlinearities, Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 230(8), 1359-1370.
  16. [16]  Dimitriadis, G., Vio, G.A., Cooper, J.E. (2004), Stability and limit cycle oscillation amplitude prediction for multi-DOF aeroelastic systems with piecewise linear non-linearities, Proceedings of the 2004 International Conference on Noise and Vibration Engineering, ISMA, 2101-2114.
  17. [17]  Yoon, Y. and Johnson, E.N. (2021), Prediction of limit cycle oscillations in piecewise linear systems with multiple piecewise nonlinearities, IET Control Theory and Applications, 15(1), 110-125.
  18. [18]  Llibre, J., Oliveira, R.D., and Rodrigues, C.A.B. (2020), Limit cycles for two classes of control piecewise linear differential systems, Sao Paulo Journal of Mathematical Sciences, 14(1), 49-65.
  19. [19]  Chen, H., Wei, F., Xia, Y.-H., and Xiao, D. (2020), Global dynamics of an asymmetry piecewise linear differential system: Theory and applications, Bulletin des Sciences Mathematiques, 160.
  20. [20]  Zhao, Q. and Yu, J. (2019), Limit cycles of piecewise linear dynamical systems with three zones and lateral systems, Journal of Applied Analysis and Computation, 9(5), 1822-1837.
  21. [21]  Lewis, A.P. (2019), Refined analytical approximations to limit cycles for non-linear multi-degree-of-freedom systems, International Journal of Non-Linear Mechanics, 110, 58-68.
  22. [22]  Schmidt, G. and Tondl, A. (1986), Non-linear Vibrations, Cambridge: Cambridge University Press.
  23. [23]  Schmidt, G. and Schulz, R. (1975), Parametererregte Schwingungen, Berlin: Deutscher Verl. der Wissenschaften.
  24. [24]  Ashley, H. and Zartarian, G. (1956), Piston theory -- a new aerodynamic tool for the aeroelastician, Journal of the Aeronautical Sciences, 23, 1109-1118.
  25. [25]  Seydel, R. (2010), Practical Bifurcation and Stability Analysis, Third Edition, Springer, ISBN 978-1-4419-1739-3.
  26. [26]  Venkatasubramanian, V. and Ji, W. (1997), Numerical Approximation of (n-1)-Dimensional Stable Manifolds in Large Systems such as Power Systems, Automatica, 10, 1877-1883.
  27. [27]  Chung, K.W., Chan, C.L., and Lee, B.H.K. (2007), Bifurcation analysis of a two-degree-of-freedom aeroelastic system with a freeplay structural nonlinearity by a perturbation-incremental method, Journal of Sound and Vibration, 299, 520-539.

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