Skip to main content
SpringerLink
Log in

A new fractional moment equation method for the response prediction of nonlinear stochastic systems

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

Abstract

Moment equation method is commonly used in the analysis of nonlinear stochastic systems. However, in some cases, this method may lead to poor accuracy, or even invalid results, because only limited number of moments of the response is available. In order to overcome this limitation, this paper develops a new fractional moment equation methodology. The new method involves the derivation of fractional moment equation of nonlinear systems and the approximation of probability density function (PDF) of response by means of a novel fractional moment closure scheme. Benefitting from the valuable features of fractional moments, i.e., few number of fractional moments contain large amount of statistical information about the variable, the developed method achieves more accurate PDF estimation of the response when compared with the present methods, especially for nonlinear systems with multiple equilibria. The effectiveness of the new method is finally demonstrated by a Duffing and a bistable Duffing oscillator that is subjected to Gaussian white noise. The results are compared with the Gaussian closure approximation and exact solution.

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

Similar content being viewed by others

Abbreviations

c :

The normalization coefficient of PDFs

\(E{[}\cdot {]}\) :

The expectation operator

\(f( \cdot )\) :

The probabilistic density function (PDF)

\(f( { \cdot , \cdot } )\) :

The joint density function

\(F\left( t \right) \) :

The Gaussian white noise with zero mean

\({\hat{f}}( \cdot )\) :

The estimated PDF

\({\hat{f}}( { \cdot , \cdot })\) :

The estimated joint density function

g(x):

The nonlinear restoring force

h(t):

A continuous function

m :

The total number of \(\lambda _{i}\) or \(\alpha _i\) in the estimated PDF

M :

A positive integer

m(t):

A continuous function

n :

Number of fractional moment equations used in \(\varGamma ( {{p_j},{\lambda _i},{\alpha _i}})\)

p(t):

A continuous function

\({S_0}\) :

The spectral density of F(t)

v(x):

Mean up-crossing rate

x(t):

The response of displacement

\({\dot{x}} (t)\) :

The response of velocity

\(\alpha _i\) :

The parameter in estimated PDF

\(\beta \) :

Damping coefficient

\(\varGamma ( {{p_j},{\lambda _i},{\alpha _i}})\) :

The cost function of fractional moment closure scheme

\(\varepsilon \) :

The nonlinear intensity of nonlinear systems

\(\eta {[} {{f_X}( x)} {]}\) :

The entropy of a continuous random variable X

\(\lambda ( x )\) :

An arbitrary differentiable function

\(\lambda _{i}\) :

The parameter in estimated PDF

\(\tau \) :

Time lag of stationary response

\({1_\mathrm{event}}\) :

Indicator function, it equals to 1 if event is true and 0 otherwise

\(\mathbb {N}\) :

The set of integer

\(\mathbb {R}\) :

The set of real number

References

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  3. Dai, H., Zheng, Z., Ma, H.: An explicit method for simulating non-gaussian and non-stationary stochastic processes by karhunen-loeve and polynomial chaos expansion. Mech. Syst. Signal Process. 115, 1–13 (2019)

    Article  Google Scholar 

  4. Qiu, J., Sun, K., Wang, T., Gao, H.: Observer-based fuzzy adaptive event-triggered control for pure-feedback nonlinear systems with prescribed performance. IEEE Trans. Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2019.2895560.

  5. Sun, K., Mou, S., Qiu, J., Wang, T., Gao, H.: Adaptive fuzzy control for non-triangular structural stochastic switched nonlinear systems with full state constraints. IEEE Trans. Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2018.2883374.

  6. Chen, J., Rui, Z.: Dimension-reduced FPK equation for additive white-noise excited nonlinear structures. Probab. Eng. Mech. 53, 1–13 (2018)

    Article  Google Scholar 

  7. Li, J., Chen, J.: Stochastic Dynamics of Structures. Wiley, Hoboken (2009)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Noori, M., Davoodi, H.: Comparison between equivalent linearization and gaussian closure for random vibration analysis of several nonlinear systems. Int. J. Eng. Sci. 28(9), 897–905 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Wu, W.-F.: Comparison of gaussian closure technique and equivalent linearization method. Probab. Eng. Mech. 2(1), 2–8 (1987)

    Article  Google Scholar 

  12. Ricciardi, G., Elishakoff, I.: A novel local stochastic linearization method via two extremum entropy principles. Int. J. Non-linear Mech. 37(4), 785–800 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Er, G.-K.: An improved closure method for analysis of nonlinear stochastic systems. Nonlinear Dyn. 17(3), 285–297 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Er, G.-K.: Crossing rate analysis with a non-gaussian closure method for nonlinear stochastic systems. Nonlinear Dyn. 14(3), 279–291 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Crandall, S.H.: Non-gaussian closure for random vibration of non-linear oscillators. Int. J. Non-linear Mech. 15(4), 303–313 (1980)

    Article  MATH  Google Scholar 

  16. Joo, H.K., Sapsis, T.P.: A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise. Probab. Eng. Mech. 46, 120–132 (2016)

    Article  Google Scholar 

  17. Makarem, H., Pishkenari, H.N., Vossoughi, G.R.: A modified gaussian moment closure method for nonlinear stochastic differential equations. Nonlinear Dyn. 89(4), 2609–2620 (2017)

    Article  MathSciNet  Google Scholar 

  18. Xiao, Y., Xu, W., Yang, Y.: Response of strongly nonlinear vibro-impact system with fractional derivative damping under gaussian white noise excitation. Nonlinear Dyn. 85(3), 1955–1964 (2016)

    Article  Google Scholar 

  19. Di Paola, M.: Fokker planck equation solved in terms of complex fractional moments. Probab. Eng. Mech. 38, 70–76 (2014)

    Article  Google Scholar 

  20. Yang, Y., Xu, W., Jia, W., Han, Q.: Stationary response of nonlinear system with caputo-type fractional derivative damping under gaussian white noise excitation. Nonlinear Dyn. 79(1), 139–146 (2015)

    Article  MathSciNet  Google Scholar 

  21. 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), 010801-1–010801-52

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

    Article  MathSciNet  Google Scholar 

  23. Zheng, Z., Zhao, W., Dai, H.: A new definition of fractional derivative. Int. J. Non-linear Mech. 108, 1–6 (2019)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. 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 

  26. Zheng, Z., Dai, H.: A new fractional equivalent linearization method for nonlinear stochastic dynamic analysis. Nonlinear Dyn. 91(2), 1075–1084 (2018)

    Article  Google Scholar 

  27. Dai, H., Ma, Z., Li, L.: An improved complex fractional moment-based approach for the probabilistic characterization of random variables. Probab. Eng. Mech. 53, 52–58 (2018)

    Article  Google Scholar 

  28. Cottone, G., Di Paola, M., Metzler, R.: Fractional calculus approach to the statistical characterization of random variables and vectors. Phys. A Stat. Mech. Its Appl. 389(5), 909–920 (2010)

    Article  MathSciNet  Google Scholar 

  29. Inverardi, P Novi, Petri, A., Pontuale, G., Tagliani, A.: Stieltjes moment problem via fractional moments. Appl. Math. Comput. 166(3), 664–677 (2005)

    MathSciNet  MATH  Google Scholar 

  30. Xu, J., Kong, F.: Adaptive scaled unscented transformation for highly efficient structural reliability analysis by maximum entropy method. Struct. Saf. 76, 123–134 (2019)

    Article  Google Scholar 

  31. Xu, J., Wang, D., Dang, C.: A marginal fractional moments based strategy for points selection in seismic response analysis of nonlinear structures with uncertain parameters. J. Sound Vib. 387, 226–238 (2017)

    Article  Google Scholar 

  32. Xu, J., Zhu, S.: An efficient approach for high-dimensional structural reliability analysis. Mech. Syst. Signal Process. 122, 152–170 (2019)

    Article  Google Scholar 

  33. Zhang, X., He, W., Zhang, Y., Pandey, M.D.: An effective approach for probabilistic lifetime modelling based on the principle of maximum entropy with fractional moments. Appl. Math. Model. 51, 626–642 (2017)

    Article  MathSciNet  Google Scholar 

  34. Zhang, X., Pandey, M.D.: Structural reliability analysis based on the concepts of entropy, fractional moment and dimensional reduction method. Struct. Saf. 43, 28–40 (2013)

    Article  Google Scholar 

  35. Tagliani, A.: Maximum entropy density estimation from fractional moments Novi Inverardi PL. Commun. Stat. Theor. Methods 32(2), 327–345 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Dai, H., Zhang, H., Wang, W.: A new maximum entropy-based importance sampling for reliability analysis. Struct. Saf. 63, 71–80 (2016)

    Article  Google Scholar 

  37. Stone, M.H.: The generalized weierstrass approximation theorem. Math. Mag. 21(5), 237–254 (1948)

    Article  MathSciNet  Google Scholar 

  38. Pradlwarter, H., Schueller, G.: Excursion probabilities of non-linear systems. Int. J. Non-linear Mech. 39(9), 1447–1452 (2004)

    Article  MATH  Google Scholar 

  39. He, J., Zhao, Y.-G.: First passage times of stationary non-gaussian structural responses. Comput. Struct. 85(7), 431–436 (2007)

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by Grant from the National Natural Science Foundation of China (Project 11672091). This support is gratefully acknowledged.

Author information

Authors and Affiliations

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

    Hongzhe Dai & Ruijing Zhang

  2. School of Civil Engineering, The University of Sydney, Sydney, NSW, 2006, Australia

    Hao Zhang

Authors
  1. Hongzhe Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruijing Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hongzhe Dai.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dai, H., Zhang, R. & Zhang, H. A new fractional moment equation method for the response prediction of nonlinear stochastic systems. Nonlinear Dyn 97, 2219–2230 (2019). https://doi.org/10.1007/s11071-019-05119-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-05119-x

Keywords

Search

Navigation