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Steady-state dynamic response of various hysteretic systems endowed with fractional derivative elements

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Abstract

In this paper, the steady-state dynamic response of hysteretic oscillators comprising fractional derivative elements and subjected to harmonic excitation is examined. Notably, this problem may arise in several circumstances, as for instance, when structures which inherently exhibit hysteretic behavior are supplemented with dampers or isolators often modeled by employing fractional terms. The amplitude of the steady-state response is determined analytically by using an equivalent linearization approach. The procedure yields an equivalent linear system with stiffness and damping coefficients which are related to the amplitude of the response, but also, to the order of the fractional derivative. Various models of hysteresis, well established in the literature, are considered. Specifically, applications to oscillators with bilinear, Bouc–Wen, and Preisach hysteretic models are reported, and related parameter studies are presented. The derived results are juxtaposed with pertinent numerical data obtained by integrating the original nonlinear fractional order equation of motion. The analytical and the numerical results are found in good agreement, establishing, thus, the accuracy and efficiency of the considered approach.

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References

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

    MathSciNet  Google Scholar 

  2. Caughey, T.K.: Random excitation of a system with bilinear hysteresis. J. Appl. Mech. 27, 649–652 (1960)

    MathSciNet  Google Scholar 

  3. Iwan, W.D.: A distributed-element model for hysteresis and its steady state dynamic response. J. Appl. Mech. 33, 893–900 (1966)

    Google Scholar 

  4. Bouc, R.: Forced vibration of mechanical systems with hysteresis. In: Proceedings of the 4th Conference on Non-linear Oscillation, p. 315 (1967)

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

    Google Scholar 

  6. Hughes, D., Wen, J.T.: Preisach modeling of piezoceramic and shape memory alloy hysteresis. Smart Mater. Struct. 6, 287–300 (1997)

    Google Scholar 

  7. Majima, S., Kodama, K., Hasegawa, T.: Modeling of shape memory alloy actuator and tracking control system with the model. IEEE Trans. Control Syst. Technol. 9, 54–59 (2001)

    Google Scholar 

  8. Yu, Y.H., Naganathan, N., Dukkipati, R.: Preisach modeling of hysteresis for piezoceramic actuator system. Mech. Mach. Theory 37, 49–59 (2002)

    MathSciNet  MATH  Google Scholar 

  9. Sauter, D., Hagedorn, P.: On the hysteresis of wire cables in Stockbridge dampers. Int. J. Non-Linear Mech. 37, 1453–1459 (2002)

    MATH  Google Scholar 

  10. Dyke, S.J., Spencer, B.F., Sain, M.K., Carlson, J.D.: An experimental study of MR dampers for seismic protection. Smart Mater. Struct. 7, 693–703 (1998)

    Google Scholar 

  11. Saadat, S., Salichs, J., Noori, M., Hou, Z., Davoodi, H., Bar-on, I., Suzuki, Y., Masuda, A.: An overview of vibration and seismic applications of NiTi shape memory alloy. Smart Mater. Struct. 11, 218–229 (2002)

    Google Scholar 

  12. Carpineto, N., Lacarbonara, W., Vestroni, F.: Hysteretic tuned mass dampers for structural vibration mitigation. J. Sound Vib. 333, 1302–1318 (2014)

    Google Scholar 

  13. Brokate, M., Visintin, A.: Properties of the Preisach model for hysteresis. Zeitschrift für reine und angewandte Mathematik 402, 1–40 (1989)

    MathSciNet  MATH  Google Scholar 

  14. Lubarda, V., Sumarac, D., Krajcinovic, D.: Hysteretic response of ductile materials. Eur. J. Mech. A-Solids 12, 445–470 (1993)

    MATH  Google Scholar 

  15. Ktena, A., Fotiadis, D.I., Spanos, P.D., Massalas, C.V.: A Preisach model identification procedure and simulation of hysteresis in ferromagnets and shape-memory alloys. Phys. B 306, 84–90 (2001)

    Google Scholar 

  16. Spanos, P.D., Cacciola, P., Red-Horse, P.J.: Random vibration of SMA systems via Preisach formalism. Nonlinear Dyn. 36, 405–419 (2004)

    MATH  Google Scholar 

  17. Mayergoyz, I.D.: Mathematical Models of Hysteresis and Their Applications. Elsevier, New York (2003)

    Google Scholar 

  18. Macki, J.W., Nistri, P., Zecca, P.: Mathematical models for hysteresis. SIAM Rev. 35, 94–123 (1993)

    MathSciNet  MATH  Google Scholar 

  19. Spanos, P.D., Kontos, A., Cacciola, P.: Steady-state dynamic response of preisach hysteretic systems. J. Vib. Acoust. 128, 244–250 (2004)

    Google Scholar 

  20. Capecchi, D., Vestroni, F.: Steady-state dynamic analysis of hysteretic systems. J. Eng. Mech. 111, 1515–1531 (1985)

    Google Scholar 

  21. Capecchi, D., Vestroni, F.: Periodic response of a class of hysteretic oscillators. Int. J. Non-Linear Mech. 25, 309–317 (1990)

    MATH  Google Scholar 

  22. Wong, C.W., Ni, Y.Q., Lau, S.L.: Steady-state oscillation of hysteretic differential model. I: response analysis. J. Eng. Mech. 120, 2271–2298 (1994)

    Google Scholar 

  23. Wong, C.W., Ni, Y.Q., Ko, J.: Steady-state oscillation of hysteretic differential model. II: performance analysis. J. Eng. Mech. 120, 2299–2324 (1994)

    Google Scholar 

  24. Lacarbonara, W., Vestroni, F.: Nonclassical responses of oscillators with hysteresis. Nonlinear Dyn. 32, 235–258 (2003)

    MATH  Google Scholar 

  25. Nutting, P.G.: A new general law deformation. J. Frankl. Inst. 191, 678–685 (1921)

    Google Scholar 

  26. Bagley, R.L., Torvik, P.J.: On the fractional calculus model of viscoelastic behavior. J. Rheol. 30, 133–155 (1986)

    MATH  Google Scholar 

  27. Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  28. Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A.: Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht (2007)

    MATH  Google Scholar 

  29. Atanacković, T.M., Pilipović, S., Stanković, B., Zorica, D.: Fractional Calculus with Applications in Mechanics. Wiley, London (2014)

    MATH  Google Scholar 

  30. 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, 1–52 (2009)

    Google Scholar 

  31. Hwang, J.S., Wang, J.C.: Seismic response prediction of high damping rubber bearings fractional derivative Maxwell model. Eng. Struct. 20, 849–856 (1998)

    Google Scholar 

  32. Papoulia, K.D., Kelly, J.M.: Visco-hyperelastic model for filled rubbers used in vibration isolation. J. Eng. Mater. Technol. 119, 292–297 (1997)

    Google Scholar 

  33. Makris, N., Dargush, G.F., Constantinou, M.: Dynamic analysis of generalized viscoelastic fluids. J. Eng. Mech. 119, 1663–1679 (1993)

    Google Scholar 

  34. Di Matteo, A., Lo Iacono, F., Navarra, G., Pirrotta, A.: Innovative modeling of tuned liquid column damper motion. Commun. Nonlinear Sci. Numer. Simul. 23, 229–244 (2015)

    Google Scholar 

  35. Spanos, P.D., Evangelatos, G.: Response of a non-linear system with restoring forces governed by fractional derivatives: time domain simulation and statistical linearization solution. Soil Dyn. Earthq. Eng. 30, 811–821 (2010)

    Google Scholar 

  36. Duan, J.S., Huang, C., Liu, L.L.: Response of a fractional nonlinear system to harmonic excitation by the averaging method. Open Phys. 13, 177–182 (2015)

    Google Scholar 

  37. Chen, Y.M., Liu, Q.X., Liu, J.K.: Steady state response analysis for fractional dynamic systems based on memory-free principle and harmonic balancing. Int. J. Non-Linear Mech. 81, 154–164 (2016)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  39. Shen, Y.J., Wen, S.F., Li, X.H., Yang, S.P., Xing, H.J.: Dynamical analysis of fractional-order nonlinear oscillator by incremental harmonic balance method. Nonlinear Dyn. 85, 1457–1467 (2016)

    MathSciNet  Google Scholar 

  40. Di Matteo, A., Spanos, P.D., Pirrotta, A.: Approximate survival probability determination of hysteretic systems with fractional derivative elements. Probab. Eng. Mech. 54, 138–146 (2018)

    Google Scholar 

  41. Chen, B., Li, C., Wilson, B., Huang, Y.: Fractional modeling and analysis of coupled MR damping system. IEEE/CAA J. Autom. Sin. 3, 288–294 (2016)

    MathSciNet  Google Scholar 

  42. Liu, X., Huang, Y., Li, H., Zheng, Q., Shi, Y.: Analysis of fractional derivative model for MR damping systems. Appl. Mech. Mater. 29–32, 2102–2107 (2010)

    Google Scholar 

  43. 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)

    Google Scholar 

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

    MATH  Google Scholar 

  45. Zhang, Y., Kougioumtzoglou, I.A.: Nonlinear oscillator stochastic response and survival probability determination via the wiener path integral. ASCE–ASME J. Risk Uncertain. Eng. Syst. B Mech. Eng. 1, 1–15 (2015)

    Google Scholar 

  46. Yar, M., Hammond, J.K.: Stochastic response of an exponentially hysteretic system through stochastic averaging. Probab. Eng. Mech. 2, 147–155 (1987)

    Google Scholar 

  47. Abramowitz, M., Stegun, J.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover, New York (1963)

    MATH  Google Scholar 

  48. Spanos, P.D., Cacciola, P., Muscolino, G.: Stochastic averaging of Preisach hysteretic systems. J. Eng. Mech. 130, 1257–1267 (2004)

    Google Scholar 

  49. Kougioumtzoglou, I.A., Spanos, P.D.: Response and first-passage statistics of nonlinear oscillators via a numerical path integral approach. J. Eng. Mech. 139, 1207–1217 (2013)

    Google Scholar 

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Acknowledgements

A. Di Matteo and A. Pirrotta gratefully acknowledge the support received from the Italian Ministry of University and Research, through the PRIN 2015 funding scheme (Project 2015JW9NJT—Advanced mechanical modeling of new materials and structures for the solution of 2020 Horizon challenges).

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Authors and Affiliations

  1. L.B. Ryon Chair in Engineering, Department of Mechanical Engineering and Materials Science, Rice University, 6100 Main Street, Houston, TX, USA

    P. D. Spanos

  2. Dipartimento di Ingegneria, Università degli Studi di Palermo, Viale delle Scienze, 90128, Palermo, Italy

    A. Di Matteo & A. Pirrotta

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  1. P. D. Spanos
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Spanos, P.D., Di Matteo, A. & Pirrotta, A. Steady-state dynamic response of various hysteretic systems endowed with fractional derivative elements. Nonlinear Dyn 98, 3113–3124 (2019). https://doi.org/10.1007/s11071-019-05102-6

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