[ Log In | Register]
! Skip Navigation Links
Skip Navigation Links
Image of Journal of Applied Nonlinear Dynamics
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

Stability of Dynamic Systems under Parametric Excitations with Multiple Frequencies

Journal of Applied Nonlinear Dynamics 15(1) (2026) 57--81 | DOI:10.5890/JAND.2026.03.004

Jian Deng

Department of Civil Engineering, Lakehead University, Thunder Bay, Ontario P7B 5E1, Canada

Download Full Text PDF

 

Abstract

The stability of dynamic systems under parametrically periodic excitations with a single frequency has been extensively investigated. However, parametrically arbitrary excitations with multiple frequencies are common in various sciences and engineering, including earthquakes and blasting. This paper proposes a new numerical method to study the stability of dynamic systems under parametric excitations with multiple frequencies. The critical step of the numerical method involves approximating the system with multiple frequencies by a system with a single principal frequency (or period) as closely as possible. Subsequently, a numerical algorithm is proposed to calculate both the state transition matrix on one principal period, which determines the dynamic stability, and the responses at any specific time. The efficiency and accuracy of the proposed numerical method are demonstrated. As an application example, the dynamic stability of a column under parametric loads with multiple frequencies is obtained through parametric studies involving the magnitude of incommensurate frequency, the number of frequencies, damping, and semi-rigid connections.

References

  1. [1]  Bolotin, V.V. (1964), Dynamic Stability of Elastic Systems, Holden-Day.
  2. [2]  Waters, T.J. (2010), Stability of a two-dimensional Mathieu-type system with quasi-periodic coefficients, Nonlinear Dynamics, 60(3), 341-356.
  3. [3]  Deng, J. (2023), Numerical simulation of stability and responses of dynamic systems under parametric excitation, Applied Mathematical Modelling, 119, 648-676.
  4. [4]  Hsu, C.S. (1974), On approximating a general linear periodic system, Journal of Mathematical Analysis and Applications, 45, 234-351.
  5. [5]  Sinha, S.C., Chou, C.C., and Denman, H.H. (1979), Stability analysis of systems with periodic coefficients: an approximate approach, Journal of Sound and Vibration, 64(4), 515-527.
  6. [6]  Zhong, Z., Liu, A., Fu, J., Pi, Y.L., Deng, J., and Xie, Z. (2021), Analytical and experimental studies on out-of-plane dynamic parametric instability of a circular arch under a vertical harmonic base excitation, Journal of Sound and Vibration, 500, 116011.
  7. [7]  Sanders, J.A. and Verhulst, F. (1985), Averaging Methods in Nonlinear Dynamical Systems, Springer, New York.
  8. [8]  Nayfeh, A.H. and Mook, D.T. (1979), Nonlinear Oscillations, Wiley, New York.
  9. [9]  Liu, D. (2001), Dynamic stability analysis of frame systems by application of the finite element method, Doctoral thesis, University of Louisville, USA.
  10. [10]  Deng, J., Shahroudi, M., and Liu, K. (2023), Dynamic stability and responses of beams on elastic foundations under a parametric load, International Journal of Structural Stability and Dynamics, 23(2), 2350018.
  11. [11]  Deng, J. (2021), Analytical and numerical investigations on pillar rockbursts induced by triangular blasting waves, International Journal of Rock Mechanics and Mining Sciences, 138, 104518.
  12. [12]  Deng, J., Kanwar, N.S., Pandey, M.D., and Xie, W-C. (2019), Dynamic buckling mechanism of pillar rockbursts induced by stress waves, Journal of Rock Mechanics and Geotechnical Engineering, 11, 944-953.
  13. [13]  Murdock, J. (1978), On the Floquet problem for quasi-periodic systems, Proceedings of the American Mathematical Society, 68(2), 179-184.
  14. [14]  Davis, S.H. and Rosenblat, S. (1980), A quasiperiodic Mathieu–Hill equation, SIAM Journal on Applied Mathematics, 38(1), 139-155.
  15. [15]  Zounes, R.S. and Rand, R.H. (1998), Transition curves for the quasi-periodic Mathieu equation, SIAM Journal on Applied Mathematics, 58(4), 1094-1115.
  16. [16]  Broer, H. and Simó, C. (1998), Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena, Bulletin of the Brazilian Mathematical Society, 29(2), 253-293.
  17. [17]  Puig, J. and Simó, C. (2011), Resonance tongues in quasi-periodic Hill–Schrödinger equation with three frequencies, Regular and Chaotic Dynamics, 16(1), 61-78.
  18. [18]  Sharma, A. and Sinha, S.C. (2018), An approximate analysis of quasi-periodic systems via Floquet theory, Journal of Computational and Nonlinear Dynamics, 13(2), 021008.
  19. [19]  Subramanian, S.C. and Redkar, S. (2021), Lyapunov–Perron transformation for quasi-periodic systems and its applications, Journal of Vibration and Acoustics, 143, 041015.
  20. [20]  Subramanian, S.C., Waswa, P.M., and Redkar, S. (2022), Computation of Lyapunov–Perron transformation for linear quasi-periodic systems, Journal of Vibration and Control, 28(11-12), 1402-1417.
  21. [21]  Nayfeh, A.H. (1983), Response of two-degree-of-freedom systems to multifrequency parametric excitations, Journal of Sound and Vibration, 88(1), 1-10.
  22. [22]  Madabhushi, G., Knappett, J., and Haigh, S. (2010), Design of Pile Foundations in Liquefiable Soils, Imperial College Press.
  23. [23]  Giraldo-Londoño, O. and Aristizábal-Ochoa, J.D. (2014), Dynamic stability of slender columns with semi-rigid connections under periodic axial load: theory, Dyna, 81(185), 56-65.
  24. [24]  Xie, W-C. (2006), Dynamic Stability of Structures, Cambridge University Press.
  25. [25]  Pipes, L.A. (1964), The dynamic stability of a uniform straight column excited by a pulsating load, Journal of the Franklin Institute, 277(6), 534-551.
  26. [26]  Deng, J., Ballesteros Ortega, J., Liu, K., and Gong, Y. (2024), Theoretical and numerical investigations on dynamic stability of viscoelastic columns with semi-rigid connections, Thin-Walled Structures, 198, 111758.
  27. [27]  Humar, J.L. (2002), Dynamics of Structures, 2nd ed., CRC Press.
  28. [28]  Chen, L.Q. and Yang, X.D. (2006), Vibration and stability of an axially moving viscoelastic beam with hybrid supports, European Journal of Mechanics A/Solids, 25, 996-1008.
  29. [29]  Pipes, L.A. and Harvill, L.R. (2014), Applied Mathematics for Engineers and Physicists, 3rd ed., Dover Publications.
  30. [30]  Merkin, D.R. (1997), Introduction to the Theory of Stability, Springer, New York.
  31. [31]  Richards, J.A. (1975), On the choice of steps in the piecewise-constant Hill-equation model of a quadrupole mass filter, International Journal of Mass Spectrometry and Ion Processes, 18, 11-19.
  32. [32]  Xu, C., Wang, Z., and Li, B. (2021), Dynamic stability of simply supported beams with multi-harmonic parametric excitation, International Journal of Structural Stability and Dynamics, 21(2), 2150027.

Copyright © 2011-2026 L & H Scientific Publishing. All rights reserved. Wildcard SSL Certificates