Skip to main content
SpringerLink
Log in

Approximate stochastic response of hysteretic system with fractional element and subjected to combined stochastic and periodic excitation

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

Abstract

A method based on statistical linearization is proposed, for determining the response of single-degree-of-freedom hysteretic systems endowed with fractional derivative element and subjected to combined periodic and white/colored excitation. The method is developed by decomposing the system response into a combination of a periodic and of a zero-mean stochastic components. In this regard, first, the equation of motion is cast into two coupled fractional-order nonlinear differential equations with unknown deterministic and stochastic response components. Next, the harmonic balance method for the fractional-order deterministic equation and the statistical linearization for the stochastic equation are used, to obtain the Fourier coefficients of the deterministic response component and the variance of the stochastic response component, respectively. This yields two sets of coupled nonlinear algebraic equations which can be solved by appropriate standard numerical method. Pertinent numerical examples, including both softening and hardening Bouc–Wen hysteretic system endowed with different fractional-orders, are used to demonstrate the applicability and accuracy of the proposed method.

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
Fig. 13

Similar content being viewed by others

References

  1. Hong, S.R., Choi, S.B., Choi, Y.T., Wereley, N.M.: A hydro-mechanical model for hysteretic damping force prediction of ER damper: experimental verification. J. Sound Vib. 285(4–5), 1180–1188 (2005)

    Article  Google Scholar 

  2. Okuizumi, N., Kimura, K.: Multiple time scale analysis of hysteretic systems subjected to harmonic excitation. J. Sound Vib. 272(3–5), 675–701 (2004)

    Article  Google Scholar 

  3. Iwan, W.D.: Response of the bilinear hysteretic system to stationary random excitation. J. Acoust. Soc. Am. 43(3), 545–552 (1968)

    Article  Google Scholar 

  4. Caughey, T.K.: Sinusoidal excitation of a system with bilinear hysteresis. J. Appl. Mech. 27(4), 640–643 (1960)

    Article  MathSciNet  Google Scholar 

  5. Bouc, R.: Forced vibration of mechanical systems with hysteresis. In: Proceedings of the Fourth Conference on Nonlinear Oscillation, Prague (1967)

  6. Wen, Y.K.: Method for random vibration of hysteretic systems. J .Eng. Mech. Div. 102(2), 249–263 (1976)

    Article  Google Scholar 

  7. Baber, T.T., Wen, Y.K.: Random vibration hysteretic, degrading systems. J. Eng. Mech. Div. 107(6), 1069–1087 (1981)

    Article  Google Scholar 

  8. Ismail, M., Ikhouane, F., Rodellar, J.: The hysteresis Bouc-Wen model, a survey. Arch. Comput. Methods Eng. 16(2), 161–188 (2009)

    Article  Google Scholar 

  9. Ikhouane, F., Rodellar, J.: Systems with Hysteresis: Analysis, Identification and Control Using the Bouc-Wen Model. Wiley, Chichester (2007)

    Book  Google Scholar 

  10. Chassiakos, A.G., Masri, S.F., Smyth, A.W., Caughey, T.K.: On-line identification of hysteretic systems. J. Appl. Mech. 65(1), 194–203 (1998)

    Article  Google Scholar 

  11. Yang, G., Spencer, B.F., Jr., Caelson, J.D., Sain, M.K.: Large-scale mr fluid dampers: modeling and dynamic performance considerations. Eng. Struct. 24(3), 309–323 (2002)

    Article  Google Scholar 

  12. Lu, X., Zhou, Q.: Dynamic analysis method of a combined energy dissipation system and its experimental verification. Earthq. Eng. Struct. Dyn. 31(6), 1251–1265 (2002)

    Article  Google Scholar 

  13. Zhu, X., Lu, X., Xu, C.: Parametric identification of mild steel damper based on Bouc-Wen model. Struct. Eng. 27(5), 124–128 (2011)

    Google Scholar 

  14. Rakotondrabe, M.: Bouc-wen modeling and inverse multiplicative structure to compensate hysteresis nonlinearity in piezoelectric actuators. IEEE Trans. Autom. Sci. Eng. 8(2), 428–431 (2011)

    Article  Google Scholar 

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

    Article  Google Scholar 

  16. Di Paola, M., Pirrotta, A., Valenza, A.: Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results. Mech. Mater. 43(12), 799–806 (2011)

    Article  Google Scholar 

  17. Sasso, M., Palmieri, G., Amodio, D.: Application of fractional derivative models in linear viscoelastic problems. Mech. Time-Depend. Mater. 15(4), 367–387 (2011)

    Article  Google Scholar 

  18. Guo, J.W.W., Daniel, Y., Montgomery, M., Christopoulos, C.: Thermal-mechanical model for predicting the wind and seismic response of viscoelastic dampers. J. Eng. Mech. 142(10), 04016067 (2016)

    Article  Google Scholar 

  19. Lewandowski, R., Łasecka-Plura, M.: Design sensitivity analysis of structures with viscoelastic dampers. Comput. Struct. 164, 95–107 (2016)

    Article  Google Scholar 

  20. 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, 865–896 (2016)

    Article  MathSciNet  Google Scholar 

  21. Fragkoulis, V.C., Kougioumtzoglou, I.A., Pantelous, A.A., Beer, M.: Non-stationary response statistics of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation. Nonlinear Dyn. 97(4), 2291–2303 (2019)

    Article  Google Scholar 

  22. Kang, S., Wu, H., Li, Y., Yu, S., Yang, X., Yao, J.: A fractional-order normalized Bouc-Wen model for piezoelectric hysteresis nonlinearity. arXiv preprint arXiv:2003.04917 (2020)

  23. Hatchell, B.K., Mauss, F.J., Amaya, I.A., Skorpik, J.R., Silvers, K.L., Marotta, S.A.: Missile captive carry monitoring and helicopter identification using a capacitive microelectromechanical systems accelerometer. Struct. Health Monit. 11(2), 213–224 (2012)

    Article  Google Scholar 

  24. Zhang, Y., Spanos, P.D.: Efficient response determination of a M-D-O-F gear model subject to combined periodic and stochastic excitations. Int. J. Non-Linear Mech. 120, 103378 (2020)

    Article  Google Scholar 

  25. Yang, J.: Vibration analysis on multi-mesh gear-trains under combined deterministic and random excitations. Mech. Mach. Theory 59, 20–33 (2013)

    Article  Google Scholar 

  26. Xu, Y., Liu, Q., Guo, G., Xu, C., Liu, D.: Dynamical responses of airfoil models with harmonic excitation under uncertain disturbance. Nonlinear Dyn. 89(3), 1579–1590 (2017)

    Article  MathSciNet  Google Scholar 

  27. Harne, R.L., Dai, Q.: Characterizing the robustness and susceptibility of steady-state dynamics in post-buckled structures to stochastic perturbations. J. Sound Vib. 395, 258–271 (2017)

    Article  Google Scholar 

  28. Dai, Q., Harne, R.L.: Investigation of direct current power delivery from nonlinear vibration energy harvesters under combined harmonic and stochastic excitations. J. Intell. Mater. Syst. Struct. 29(4), 514–529 (2018)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Fragkoulis, V.C., Kougioumtzoglou, I.A., Pantelous, A.A.: Statistical linearization of nonlinear structural systems with singular matrices. J. Eng. Mech. 142(9), 04016063 (2016)

    Article  Google Scholar 

  31. Pasparakis, G., Fragkoulis, V., Beer, M.: Harmonic wavelets based response evolutionary power spectrum determination of linear and nonlinear structural systems with singular matrices. Mech. Syst. Signal Process. 149, 107203 (2021). https://doi.org/10.1016/j.ymssp.2020.107203

    Article  Google Scholar 

  32. Kong, F., Spanos, P.D.: Stochastic response of hysteresis system under combined periodic and stochastic excitation via the statistical linearization method. ASME J. Appl. Mech. 88(5), 051008 (2021)

    Article  Google Scholar 

  33. Zhang, Y., Spanos, P.D.: A linearization scheme for vibrations due to combined deterministic and stochastic loads. Probab. Eng. Mech. 60, 103028 (2020)

    Article  Google Scholar 

  34. Kong, F., Spanos, P.D.: Response spectral density determination for nonlinear systems endowed with fractional derivatives and subject to colored noise. Probab. Eng. Mech. 59, 103023 (2020)

    Article  Google Scholar 

  35. Roberts, J.B., Spanos, P.D.: Random Vibration and Statistical Linearization. Dover Publications, New York (2003)

    MATH  Google Scholar 

  36. Spanos, P., Zeldin, B.: Random vibration of systems with frequency-dependent parameters or fractional derivatives. J. Eng. Mech. 123(3), 290–292 (1997)

    Article  Google Scholar 

  37. Shinozuka, M., Deodatis, G.: Simulation of stochastic processes by spectral representation. Appl. Mech. Rev. 44(4), 191–204 (1991)

    Article  MathSciNet  Google Scholar 

  38. Liu, Z., Liu, W., Peng, Y.: Random function based spectral representation of stationary and non-stationary stochastic processes. Probab. Eng. Mech. 45, 115–126 (2016)

    Article  Google Scholar 

  39. Singh, M.P., Chang, T.S., Nandan, H.: Algorithms for seismic analysis of mdof systems with fractional derivatives. Eng. Struct. 33(8), 2371–2381 (2011)

    Article  Google Scholar 

  40. Chopra, A.K.: Dynamics of Structures. Prentice Hall, New Jersey (2011)

    Google Scholar 

  41. Kanai, K.: Semi-empirical formula for the seismic characteristics of the ground. Bull. Earthq. Res. Inst. Univ. Tokyo 35(2), 309–325 (1957)

    Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 52078399) and by the Fundamental Research Funds for the Central Universities(WUT:212274016). The first author would like to thank the Chinese Scholarship Council (CSC) for financial support (File No. 201706955030) during his visit to Rice University as a visiting scholar. The first author would like to greatly thank Professor Pol D. Spanos at Rice University for the discussing the statistical linearization method for fractional-order systems.

Funding

All sources of funding for the research reported have been declared.

Author information

Authors and Affiliations

  1. School of Civil Engineering and Architecture, Wuhan University of Technology, 122 Luoshi Road, Wuhan, 430070, Hubei, China

    Fan Kong & Renjie Han

  2. School of Safety Science and Emergency Management, Wuhan University of Technology, 122 Luoshi Road, Wuhan, 430070, China

    Yuanjin Zhang

  3. School of Civil Engineering, Hefei University of Technology, 193 Tunxi Road, Hefei, 230009, China

    Fan Kong

Authors
  1. Fan Kong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Renjie Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yuanjin Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

KF conceived the study, derived the theoretical formulation and finalized the manuscript. HRJ composed the relevant codes, drafted the manuscript. ZYJ participated in coding, and drafting the manuscript.

Corresponding author

Correspondence to Yuanjin Zhang.

Ethics declarations

Conflict of interest

The authors declare that they have no competing interests.

Code availability

All codes relevant to the present work are available upon request.

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

Kong, F., Han, R. & Zhang, Y. Approximate stochastic response of hysteretic system with fractional element and subjected to combined stochastic and periodic excitation. Nonlinear Dyn 107, 375–390 (2022). https://doi.org/10.1007/s11071-021-07014-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-07014-w

Keywords

Search

Navigation