Skip to main content
SpringerLink
Log in

An efficient method for simulating the dynamic behavior of periodic structures with piecewise linearity

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

An efficient method based on the parametric variational principle (PVP) is proposed for simulating the dynamic behavior of periodic structures with large number of piecewise linearity. The formulation for the gap-activated springs is proposed based on PVP, which can accurately determine the states of the gap-activated springs. Based on the periodicity of the system and the precise integration method, an efficient numerical time-integration method is developed to obtain the dynamic responses of the system. For this method, the matrix exponential of only one unit cell of the system is computed, which greatly improves the computational efficiency. Dynamic responses of a 3 degrees of freedom (DOF) piecewise linear system under harmonic excitations are given to demonstrate the validation of the proposed method. The piecewise linear dynamic system can exhibit very complex vibrational behavior, such as stable periodic motion, multi-periodic motion, quasi-periodic motion and chaotic motion, which can be successfully predicted by using bifurcation theory. Moreover, it is demonstrated that the proposed method can be used to efficiently determine dynamic responses of a periodic piecewise linear system with large number of DOFs and large number of gap-activated springs under harmonic excitations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Stoker, J.J.: Nonlinear Vibrations in Mechanical and Electrical Systems. Interscience Publishers, New York (1950)

    MATH  Google Scholar 

  2. Hogan, S.J., Higham, L., Griffin, T.C.L.: Dynamics of a piecewise linear map with a gap. Proc. Roy. Soc. A 463(2077), 49–65 (2007)

    Article  MathSciNet  Google Scholar 

  3. Kevenaar, T.A.M., Leenaerts, D.M.W.: A comparison of piecewise-linear model descriptions. IEEE Trans. Circuits I 39(12), 996–1004 (1992)

    Article  Google Scholar 

  4. Murali, K., Lakshmanan, M., Chua, L.O.: The simplest dissipative nonautonomous chaotic circuit. IEEE Trans. Circuits I 41(6), 462–463 (1994)

    Article  Google Scholar 

  5. Gouzé, J., Sari, T.: A class of piecewise linear differential equations arising in biological models. Dyn. Syst. 17(4), 299–316 (2002)

    Article  MathSciNet  Google Scholar 

  6. Casey, R., Jong, H.D., Gouzé, J.L.: Piecewise-linear models of genetic regulatory networks: equilibria and their stability. J. Math. Biol. 52(1), 27–56 (2006)

    Article  MathSciNet  Google Scholar 

  7. Hommes, C.H., Nusse, H.E.: “Period three to period two” bifurcation for piecewise linear models. J. Econ. 54(2), 157–169 (1991)

    Article  MathSciNet  Google Scholar 

  8. Hommes, C.H., Nusse, H.E., Simonovits, A.: Cycles and chaos in a socialist economy. J. Econ. Dyn. Control. 19(1–2), 155–179 (1995)

    Article  Google Scholar 

  9. Kalmár-Nagy, T., Csikja, R., Elgohary, T.A.: Nonlinear analysis of a 2-DOF piecewise linear aeroelastic system. Nonlinear Dyn. 1(1), 1–12 (2016)

    MathSciNet  Google Scholar 

  10. Gharesifard, B., Touri, B., Basar, T., Shamma, J.: On the convergence of piecewise linear strategic interaction dynamics on networks. IEEE Trans. Automat. Contr. 4(4), 1–1 (2015)

    MATH  Google Scholar 

  11. Motro, R.: Tensegrity: Structural Systems for the Future. Kogan Page Science, London (2003)

    Book  Google Scholar 

  12. Pun, D., Liu, Y.B.: On the design of the piecewise linear vibration absorber. Nonlinear Dyn. 22(4), 393–413 (2000)

    Article  Google Scholar 

  13. Shen, P.H., Lin, S.W.: Mathematic modeling and characteristic analysis for dynamic system with asymmetrical hysteresis in vibratory compaction. Meccanica 43(5), 505–515 (2008)

    Article  MathSciNet  Google Scholar 

  14. Zhang, C.: Theoretical design approach of four-dimensional piecewise-linear multi-wing hyperchaotic differential dynamic system. Optik 127(11), 4575–4580 (2016)

    Article  Google Scholar 

  15. Choi, Y.S., Noah, S.T.: Forced periodic vibration of unsymmetric piecewise-linear systems. J. Sound Vib. 121(1), 117–126 (1988)

    Article  MathSciNet  Google Scholar 

  16. Hudson, J.L., Rossler, O.E., Killory, H.C.: Chaos in a four-variable piecewise-linear system of differential equations. IEEE Trans. Circuits 35(7), 902–908 (1988)

    Article  MathSciNet  Google Scholar 

  17. Kato, T.: A Short Introduction to Perturbation Theory for Linear Operators. Springer, Berlin (1982)

    Book  Google Scholar 

  18. Heck, B.S., Haddad, A.H.: Singular perturbation in piecewise-linear systems. IEEE Trans. Automat. Contr. 34(1), 1722–1727 (1988)

    MathSciNet  Google Scholar 

  19. Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (2011)

    MATH  Google Scholar 

  20. Cao, Q., Xu, L., Djidjeli, K., Price, W.G., Twizell, E.H.: Analysis of period-doubling and chaos of a non-symmetric oscillator with piecewise-linearity. Chaos Solitons Fract. 12(10), 1917–1927 (2001)

    Article  Google Scholar 

  21. Hu, H.Y.: Primary resonance of a harmonically forced oscillator with a pair of symmetric set-up elastic stops. J. Sound Vib. 207(3), 393–401 (1997)

    Article  Google Scholar 

  22. Narimani, A., Golnaraghi, M.E., Jazar, G.N.: Frequency response of a piecewise linear vibration isolator. J. Vib. Control 10(12), 1775–1794 (2004)

    MATH  Google Scholar 

  23. Deshpande, S., Mehta, S., Jazar, G.N.: Optimization of secondary suspension of piecewise linear vibration isolation systems. Int. J. Mech. Sci. 48(4), 341–377 (2006)

    Article  Google Scholar 

  24. Comparin, R.J., Singh, R.: Frequency response characteristics of a multi-degree-of-freedom system with clearances. J. Sound Vib. 142(1), 101–124 (1990)

    Article  Google Scholar 

  25. Moussi, E.H., Bellizzi, S., Cochelin, B., Nistor, I.: Nonlinear normal modes of a two degrees-of-freedom piecewise linear system. Mech. Syst. Signal Process. 64, 266–281 (2015)

    Article  Google Scholar 

  26. Xu, L., Lu, M.W., Cao, Q.: Nonlinear vibrations of dynamical systems with a general form of piecewise-linear viscous damping by incremental harmonic balance method. Phys. Lett. A 301(1–2), 65–73 (2002)

    Article  MathSciNet  Google Scholar 

  27. Wong, C.W., Zhang, W.S., Lau, S.L.: Periodic forced vibration of unsymmetrical piecewise-linear systems by incremental harmonic balance method. J. Sound Vib. 149(1), 91–105 (1991)

    Article  Google Scholar 

  28. Liu, H.R., Li, J.Z., Hou, D.X., Yin, R.R., Jiang, J.S.: Analysis of dynamical characteristic of piecewise-nonlinear asymmetric hysteretic system based on incremental harmonic balance method. Discrete Dyn. Nat. Soc. 2015(12), 1–8 (2015)

    MathSciNet  Google Scholar 

  29. Kong, X., Sun, W., Wang, B., Wen, B.: Dynamic and stability analysis of the linear guide with time-varying, piecewise-nonlinear stiffness by multi-term incremental harmonic balance method. J. Sound Vib. 346(1), 265–283 (2015)

    Article  Google Scholar 

  30. Zhong, W.X., Zhang, R.L.: Parametric variational principles and their quadratic programming solutions in plasticity. Comput. Struct. 30(4), 887–896 (1988)

    Article  MathSciNet  Google Scholar 

  31. Zhong, W.X., Sun, S.M.: A finite element method for elasto-plastic structure and contact problem by parametric quadratic programming. Int. J. Numer. Methods Eng. 26(12), 2723–2738 (1988)

    Article  Google Scholar 

  32. Zhang, H.W., Zhang, X.W., Chen, J.S.: A new algorithm for numerical solution of dynamic elastic-plastic hardening and softening problems. Comput. Struct. 81(17), 1739–1749 (2003)

    Article  MathSciNet  Google Scholar 

  33. Zhang, H.W., He, S.Y., Li, X.S.: Two aggregate-function-based algorithms for analysis of 3D frictional contact by linear complementarity problem formulation. Comput. Methods Appl. Mech. Eng. 194(50–52), 5139–5158 (2005)

    Article  Google Scholar 

  34. Yu, S.D.: An efficient computational method for vibration analysis of unsymmetric piecewise-linear dynamical systems with multiple degrees of freedom. Nonlinear Dyn. 71(3), 493–504 (2013)

    Article  MathSciNet  Google Scholar 

  35. Acary, V., De Jong, H., Brogliato, B.: Numerical simulation of piecewise-linear models of gene regulatory networks using complementarity systems. Physica D 269(2), 103–119 (2014)

    Article  MathSciNet  Google Scholar 

  36. Zhang, H.W., Zhang, L., Gao, Q.: An efficient computational method for mechanical analysis of bimodular structures based on parametric variational principle. Comput. Struct. 89(23), 2352–2360 (2011)

    Article  Google Scholar 

  37. Zhong, W., Williams, F.: A precise time step integration method. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 208(6), 427–430 (1994)

    Article  Google Scholar 

  38. Sha, D., Sun, H., Zhang, Z., Yin, F.: A variational inequality principle in solid mechanics and application in physically non-linear problems. Commun. Appl. Numer. Methods 6(1), 35–45 (1990)

    Article  Google Scholar 

Download references

Acknowledgements

The authors are grateful for the support of the Natural Science Foundation of China (No. 11572076) and the 973 program (No. 2014CB049000).

Author information

Authors and Affiliations

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian, 116023, People’s Republic of China

    Dongdong He, Qiang Gao & Wanxie Zhong

Authors
  1. Dongdong He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiang Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wanxie Zhong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiang Gao.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, D., Gao, Q. & Zhong, W. An efficient method for simulating the dynamic behavior of periodic structures with piecewise linearity. Nonlinear Dyn 94, 2059–2075 (2018). https://doi.org/10.1007/s11071-018-4475-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-018-4475-8

Keywords

Search

Navigation