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A new fractional equivalent linearization method for nonlinear stochastic dynamic analysis

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Abstract

In this paper, a novel fractional equivalent linearization (NFEL) approach is developed by introducing fractional derivative term into the conventional equivalent linear equation. By appropriately formulating the fractional linearization coefficients with aid of the frequency response function of fractional equivalent linear system, an efficient iteration scheme is developed for determining these coefficients. Thus, the solution of fractional equivalent linear equation can be used to approximate the response of randomly excited nonlinear system. The contribution of the new added fractional derivative term to the classical damping force and the elastic restoring force is then quantitatively illustrated by virtue of the frequency response function of fractional system. The effectiveness of the NFEL is finally demonstrated by a nonlinear and a Duffing oscillator that is subjected to Gaussian white noise. The results are compared with the conventional equivalent linearization and exact solution.

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References

  1. Caughey, T.K.: Equivalent linearization techniques. J. Acoust. Soc. Am. 35(11), 1706–1711 (1963)

    Article  MathSciNet  Google Scholar 

  2. Roberts, J.B., Spanos, P.D.: Random Vibration and Statistical Linearization. Courier Corporation, North Chelmsford (2003)

    MATH  Google Scholar 

  3. Crandall, S.H.: A half-century of stochastic equivalent linearization. Struct. Control Health Monit. 13(1), 27–40 (2006)

    Article  MathSciNet  Google Scholar 

  4. Elishakoff, I.: Stochastic linearization technique: a new interpretation and a selective review. Shock Vib. Dig. 32(3), 179–188 (2000)

    Article  Google Scholar 

  5. Socha, L.: Linearization in analysis of nonlinear stochastic systems: recent results-part i: theory. Appl. Mech. Rev. 58(3), 178–205 (2005)

    Article  Google Scholar 

  6. Wang, Z., Song, J.: Equivalent linearization method using gaussian mixture (gm-elm) for nonlinear random vibration analysis. Struct. Saf. 64, 9–19 (2017)

    Article  Google Scholar 

  7. Cai, G., Lin, Y.: A new approximate solution technique for randomly excited non-linear oscillators. Int. J. Non-Linear Mech. 23(5), 409–420 (1988)

    Article  MATH  Google Scholar 

  8. Spanos, P., Donley, M.: Equivalent statistical quadratization for nonlinear systems. J. Eng. Mech. 117(6), 1289–1310 (1991)

    Article  Google Scholar 

  9. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives theory and applications, vol. 1993. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  10. Dai, H., Zheng, Z., Wang, W.: A new fractional wavelet transform. Commun. Nonlinear Sci. Numer. Simul. 44, 19–36 (2017)

    Article  MathSciNet  Google Scholar 

  11. Bagley, R.L., Torvik, J.: Fractional calculus-a different approach to the analysis of viscoelastically damped structures. AIAA J. 21(5), 741–748 (1983)

    Article  MATH  Google Scholar 

  12. Rossikhin, Y.A., Shitikova, M.V.: Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent results. Appl. Mech. Rev. 63(1), 010801–1–52 (2010)

    Google Scholar 

  13. El-Wakil, S., Abulwafa, E.M.: Formulation and solution of space-tsime fractional boussinesq equation. Nonlinear Dyn. 80(1–2), 167–175 (2015)

    Article  Google Scholar 

  14. Huang, D., Xu, W., Xie, W., Liu, Y.: Dynamical properties of a forced vibration isolation system with real-power nonlinearities in restoring and damping forces. Nonlinear Dyn. 81(1), 641–658 (2015)

    Article  Google Scholar 

  15. Li, C., Ma, Y.: Fractional dynamical system and its linearization theorem. Nonlinear Dyn. 71(4), 621–633 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Machado, J., Galhano, A.: Fractional dynamics: a statistical perspective. J. Comput. Nonlinear Dyn. 3(2), 021201–1 (2008)

    Article  Google Scholar 

  17. Wen, S., Shen, Y., Li, X., Yang, S.: Dynamical analysis of mathieu equation with two kinds of van der pol fractional-order terms. Int. J. Non-Linear Mech. 84, 130–138 (2016)

    Article  Google Scholar 

  18. Chen, L., Zhu, W.: Stochastic averaging of strongly nonlinear oscillators with small fractional derivative damping under combined harmonic and white noise excitations. Nonlinear Dyn. 56(3), 231–241 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Di Paola, M., Failla, G., Pirrotta, A.: Stationary and non-stationary stochastic response of linear fractional viscoelastic systems. Probab. Eng. Mech. 28, 85–90 (2012)

    Article  Google Scholar 

  20. Huang, Z., Jin, X.: Response and stability of a sdof strongly nonlinear stochastic system with light damping modeled by a fractional derivative. J. Sound Vib. 319(3), 1121–1135 (2009)

    Article  Google Scholar 

  21. Li, W., Chen, L., Trisovic, N., Cvetkovic, A., Zhao, J.: First passage of stochastic fractional derivative systems with power-form restoring force. Int. J. Non-Linear Mech. 71, 83–88 (2015)

    Article  Google Scholar 

  22. Dai, H., Cao, Z.: A wavelet support vector machine-based neural network metamodel for structural reliability assessment. Comput. Aided Civ. Infrastruct. Eng. 32(4), 344–357 (2017)

    Article  Google Scholar 

  23. Xu, J.: A PDEM based new methodology for stochastic dynamic stability control of nonlinear structures with fractional-type viscoelastic dampers. J. Sound Vib. 362, 16–38 (2016)

    Article  Google Scholar 

  24. Xu, J., Li, J.: Stochastic dynamic response and reliability assessment of controlled structures with fractional derivative model of viscoelastic dampers. Mech. Syst. Signal Process. 72–73, 865–896 (2016)

    Article  Google Scholar 

  25. Xu, Y., Li, Y., Liu, D.: Response of fractional oscillators with viscoelastic term under random excitation. J. Comput. Nonlinear Dyn. 9(3), 031015–1 (2014)

    Article  Google Scholar 

  26. Xu, Y., Li, Y., Liu, D., Jia, W., Huang, H.: Responses of duffing oscillator with fractional damping and random phase. Nonlinear Dyn. 74(3), 745–753 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Chen, L., Wang, W., Li, Z., Zhu, W.: Stationary response of duffing oscillator with hardening stiffness and fractional derivative. Int. J. Non-Linear Mech. 48, 44–50 (2013)

    Article  Google Scholar 

  28. Shen, Y., Yang, S., Xing, H., Gao, G.: Primary resonance of duffing oscillator with fractional-order derivative. Commun. Nonlinear Sci. Numer. Simul. 17(7), 3092–3100 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shen, Y.-J., Wei, P., Yang, S.-P.: Primary resonance of fractional-order van der pol oscillator. Nonlinear Dyn. 77(4), 1629–1642 (2014)

    Article  MathSciNet  Google Scholar 

  30. Wang, Z., Hu, H.: Stability of a linear oscillator with damping force of the fractional-order derivative. Sci. China Phys. Mech. Astron. 53(2), 345–352 (2010)

    Article  MathSciNet  Google Scholar 

  31. Wang, Z., Wang, X.: General solution of the Bagley–Torvik equation with fractional-order derivative. Commun. Nonlinear Sci. Numer. Simul. 15(5), 1279–1285 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Dai, H., Zheng, Z., Wang, W.: On generalized fractional vibration equation. Chaos Solitons Fractals 95, 48–51 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  33. Dai, H., Zheng, Z., Wang, W.: Nonlinear system stochastic response determination via fractional equivalent linearization and Karhunen–Loeve expansion. Commun. Nonlinear Sci. Numer. Simul. 49, 145–158 (2017)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Civil Engineering, Harbin Institute of Technology, Harbin, 150090, China

    Zhibao Zheng & Hongzhe Dai

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  1. Zhibao Zheng
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  2. Hongzhe Dai
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Correspondence to Hongzhe Dai.

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Zheng, Z., Dai, H. A new fractional equivalent linearization method for nonlinear stochastic dynamic analysis. Nonlinear Dyn 91, 1075–1084 (2018). https://doi.org/10.1007/s11071-017-3929-8

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