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Steady-state periodic solutions of the nonlinear wave propagation problem of a one-dimensional lattice using a new methodology with an incremental harmonic balance method that handles time delays

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Abstract

A new methodology is developed in this work to solve a one-dimensional (1D) nonlinear wave propagation problem. In its response, the space variable is converted to time functions at different locations, and the resultant time functions are merged with the time variable. Since the merged variables are essentially time-delay variables, the main point of the methodology is that the governing equation of the nonlinear wave propagation problem can be converted to a corresponding nonlinear delay differential equation (DDE) with multiple delays. The new methodology is shown how to formulate the converted DDE, and a modified incremental harmonic balance method is used to solve the 1D nonlinear DDE by introducing a delay matrix operator, where a formula of the Jacobian matrix is derived and can be efficiently and automatically used in the Newton method. This new methodology is demonstrated by solving examples of 1D monatomic chains under the nonlinear Hertzian contact law and with cubic springs. Results well match those in previous works, but derivation effort of solutions and calculation time can be significantly reduced here. When the algorithm of the methodology is compiled as a computer program, there is no additional derivation required to solve wave propagation problems associated with different governing equations.

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References

  1. Graff, K.F.: Wave Motion in Elastic Solids. Courier Corporation, Chelmsford (2012)

    MATH  Google Scholar 

  2. Hussein, M.I., Leamy, M.J., Ruzzene, M.: Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook. Appl. Mech. Rev. 66(4), 040802 (2014)

    Article  Google Scholar 

  3. Wang, K., Liu, Y., Yang, Q.: Tuning of band structures in porous phononic crystals by grading design of cells. Ultrasonics 61, 25–32 (2015)

    Article  Google Scholar 

  4. Ganesh, R., Gonella, S.: From modal mixing to tunable functional switches in nonlinear phononic crystals. Phys. Rev. Lett. 114(5), 054302 (2015)

    Article  Google Scholar 

  5. Vakakis, A.F., King, M.E., Pearlstein, A.: Forced localization in a periodic chain of non-linear oscillators. Int. J. Non-Linear Mech. 29(3), 429–447 (1994)

    Article  Google Scholar 

  6. Vakakis, A.F., King, M.E.: Nonlinear wave transmission in a monocoupled elastic periodic system. J. Acoust. Soc. Am. 98(3), 1534–1546 (1995)

    Article  Google Scholar 

  7. Sreelatha, K., Joseph, K.B.: Wave propagation through a 2D lattice. Chaos Solitons Fractals 11(5), 711–719 (2000)

    Article  MathSciNet  Google Scholar 

  8. Narisetti, R.K., Leamy, M.J., Ruzzene, M.: A perturbation approach for predicting wave propagation in one-dimensional nonlinear periodic structures. J. Vib. Acoust. 132(3), 031001 (2010)

    Article  Google Scholar 

  9. Narisetti, R.K., Ruzzene, M., Leamy, M.J.: A perturbation approach for analyzing dispersion and group velocities in two-dimensional nonlinear periodic lattices. J. Vib. Acoust. 133(6), 061020 (2011)

    Article  Google Scholar 

  10. Manktelow, K., Leamy, M.J., Ruzzene, M.: Comparison of asymptotic and transfer matrix approaches for evaluating intensity-dependent dispersion in nonlinear photonic and phononic crystals. Wave Motion 50(3), 494–508 (2013)

    Article  MathSciNet  Google Scholar 

  11. Packo, P., Uhl, T., Staszewski, W.J., Leamy, M.J.: Amplitude-dependent Lamb wave dispersion in nonlinear plates. J. Acoust. Soc. Am. 140(2), 1319–1331 (2016)

    Article  Google Scholar 

  12. Autrusson, T.B., Sabra, K.G., Leamy, M.J.: Reflection of compressional and Rayleigh waves on the edges of an elastic plate with quadratic nonlinearity. J. Acoust. Soc. Am. 131(3), 1928–1937 (2012)

    Article  Google Scholar 

  13. Wang, J., Zhou, W., Huang, Y., Lyu, C., Chen, W., Zhu, W.: Controllable wave propagation in a weakly nonlinear monoatomic lattice chain with nonlocal interaction and active control. Appl. Math. Mech. 39(8), 1059–1070 (2018)

    Article  MathSciNet  Google Scholar 

  14. Narisetti, R.K., Ruzzene, M., Leamy, M.J.: Study of wave propagation in strongly nonlinear periodic lattices using a harmonic balance approach. Wave Motion 49(2), 394–410 (2012)

    Article  MathSciNet  Google Scholar 

  15. Frandsen, N.M., Jensen, J.S.: Modal interaction and higher harmonic generation in a weakly nonlinear, periodic mass-spring chain. Wave Motion 68, 149–161 (2017)

    Article  MathSciNet  Google Scholar 

  16. Lazarov, B.S., Jensen, J.S.: Low-frequency band gaps in chains with attached non-linear oscillators. Int. J. Non-Linear Mech. 42(10), 1186–1193 (2007)

    Article  Google Scholar 

  17. Duan, W.S., Shi, Y.R., Zhang, L., Lin, M.M., Lv, K.P.: Coupled nonlinear waves in two-dimensional lattice. Chaos Solitons Fractals 23(3), 957–962 (2005)

    Article  MathSciNet  Google Scholar 

  18. Wang, X., Zhu, W., Zhao, X.: An incremental harmonic balance method with a general formula of Jacobian matrix and a direct construction method in stability analysis of periodic responses of general nonlinear delay differential equations. J. Appl. Mech. 86(6), 061011 (2019)

    Article  Google Scholar 

  19. Zang, H., Zhang, T., Zhang, Y.: Stability and bifurcation analysis of delay coupled Van der Pol–Duffing oscillators. Nonlinear Dyn. 75(1–2), 35–47 (2014)

    Article  MathSciNet  Google Scholar 

  20. Molnar, T.G., Insperger, T., Stepan, G.: Analytical estimations of limit cycle amplitude for delay-differential equations. Electron. J. Qual. Theory Differ. Equ. 2016(77), 1–10 (2016)

    Article  MathSciNet  Google Scholar 

  21. Gilsinn, D.E.: Estimating critical Hopf bifurcation parameters for a second-order delay differential equation with application to machine tool chatter. Nonlinear Dyn. 30(2), 103–154 (2002)

    Article  MathSciNet  Google Scholar 

  22. Dadi, Z., Afsharnezhad, Z., Pariz, N.: Stability and bifurcation analysis in the delay-coupled nonlinear oscillators. Nonlinear Dyn. 70(1), 155–169 (2012)

    Article  MathSciNet  Google Scholar 

  23. Deshmukh, V., Butcher, E.A., Bueler, E.: Dimensional reduction of nonlinear delay differential equations with periodic coefficients using Chebyshev spectral collocation. Nonlinear Dyn. 52(1), 137–149 (2008)

    Article  MathSciNet  Google Scholar 

  24. Butcher, E.A., Bobrenkov, O.A.: On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1541–1554 (2011)

    Article  MathSciNet  Google Scholar 

  25. Insperger, T., Stepan, G.: Semi-discretization method for delayed systems. Int. J. Numer. Methods Eng. 55(5), 503–518 (2002)

    Article  MathSciNet  Google Scholar 

  26. Bayly, P., Halley, J., Mann, B.P., Davies, M.: Stability of interrupted cutting by temporal finite element analysis. J. Manuf. Sci. Eng. 125(2), 220–225 (2003)

    Article  Google Scholar 

  27. Mitra, R., Banik, A., Chatterjee, S.: Dynamic stability of time-delayed feedback control system by FFT based IHB method. WSEAS Trans. Appl. Theor. Mech 4(8), 292–303 (2013)

    Google Scholar 

  28. Wang, X., Zhu, W.: A modified incremental harmonic balance method based on the fast Fourier transform and Broyden’s method. Nonlinear Dyn. 81(1–2), 981–989 (2015)

    Article  MathSciNet  Google Scholar 

  29. Wang, X., Zhu, W.: Dynamic analysis of an automotive belt-drive system with a noncircular sprocket by a modified incremental harmonic balance method. J. Vib. Acoust. 139(1), 011009 (2017)

    Article  Google Scholar 

  30. Wang, X., Zhu, W.: A new spatial and temporal harmonic balance method for obtaining periodic steady-state responses of a one-dimensional second-order continuous system. J. Appl. Mech. 84(1), 014501 (2017)

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank the support from the National Science Foundation of China under Grant No. 11772100.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Alabama, Tuscaloosa, AL, 35405, USA

    Xuefeng Wang

  2. Department of Mechanical Engineering, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD, 21250, USA

    Weidong Zhu & Mao Liu

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  1. Xuefeng Wang
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  2. Weidong Zhu
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  3. Mao Liu
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Correspondence to Weidong Zhu.

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Wang, X., Zhu, W. & Liu, M. Steady-state periodic solutions of the nonlinear wave propagation problem of a one-dimensional lattice using a new methodology with an incremental harmonic balance method that handles time delays. Nonlinear Dyn 100, 1457–1467 (2020). https://doi.org/10.1007/s11071-020-05535-4

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  • DOI: https://doi.org/10.1007/s11071-020-05535-4

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