Skip to main content

Non-stationary response statistics of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation

Abstract

An approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional derivative elements and subjected to evolutionary stochastic excitation. Specifically, resorting to stochastic averaging/linearization leads to a dimension reduction of the governing equation of motion and to a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is a Fokker–Planck partial differential equation governing the evolution in time of the non-stationary response amplitude PDF. Next, assuming an appropriately chosen time-dependent PDF form of the Rayleigh kind for the response amplitude, and substituting into the Fokker–Planck equation, yields a deterministic first-order nonlinear ordinary differential equation for the time-dependent PDF coefficient. This can be readily solved numerically via standard deterministic integration schemes. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. The technique can account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. A hardening Duffing and a bilinear hysteretic nonlinear oscillators with fractional derivative elements are considered in the numerical examples section. To assess the accuracy of the developed technique, the analytical results are compared with pertinent Monte Carlo simulation data.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. Oldham, K., Spanier, J.: The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, New York (1974)

    MATH  Google Scholar 

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

    MATH  Book  Google Scholar 

  3. 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, 010801–52 (2010)

    Article  Google Scholar 

  4. Di Paola, M., Failla, G., Pirrotta, A., Sofi, A., Zingales, M.: The mechanically based non-local elasticity: an overview of main results and future challenges. Philos. Trans. R. Soc. A 371(1993), 20120433 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  5. Tarasov, V.E.: Fractional mechanics of elastic solids: continuum aspects. J. Eng. Mech. 143(5), D4016001–8 (2017)

    Article  Google Scholar 

  6. 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 

  7. Koh, C.G., Kelly, J.M.: Application of fractional derivatives to seismic analysis of base-isolated models. Earthq. Eng. Struct. Dyn. 19(2), 229–241 (1990)

    Article  Google Scholar 

  8. Lee, H.H., Tsai, C.-S.: Analytical model of viscoelastic dampers for seismic mitigation of structures. Comput. Struct. 50(1), 111–121 (1994)

    MathSciNet  Article  Google Scholar 

  9. Shen, K.L., Soong, T.T.: Modeling of viscoelastic dampers for structural applications. J. Eng. Mech. 121(6), 694–701 (1995)

    Article  Google Scholar 

  10. Rüdinger, F.: Tuned mass damper with fractional derivative damping. Eng. Struc. 28(13), 1774–1779 (2006)

    Article  Google Scholar 

  11. Makris, N., Constantinou, M.C.: Fractional-derivative Maxwell model for viscous dampers. J. Struct. Eng. 117(9), 2708–2724 (1991)

    Article  Google Scholar 

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

    Article  Google Scholar 

  13. Shokooh, A., Suárez, L.: A comparison of numerical methods applied to a fractional model of damping materials. J. Vib. Control 5(3), 331–354 (1999)

    Article  Google Scholar 

  14. Agrawal, O.P.: Stochastic analysis of dynamic systems containing fractional derivatives. J. Sound Vib. 5(247), 927–938 (2001)

    Article  Google Scholar 

  15. Agrawal, O.P.: Analytical solution for stochastic response of a fractionally damped beam. J. Vib. Acoust. 126(4), 561–566 (2004)

    Article  Google Scholar 

  16. Huang, Z.L., Jin, X.L.: Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative. J. Sound Vib. 319(3–5), 1121–1135 (2009)

    Article  Google Scholar 

  17. Spanos, P.D., Evangelatos, G.I.: 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(9), 811–821 (2010)

    Article  Google Scholar 

  18. Chen, L., Zhu, W.: Stochastic jump and bifurcation of Duffing oscillator with fractional derivative damping under combined harmonic and white noise excitations. Int. J. Non-Linear Mech. 46(10), 1324–1329 (2011)

    Article  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. Failla, G., Pirrotta, A.: On the stochastic response of a fractionally-damped Duffing oscillator. Commun. Nonlinear Sci. Numer. Simul. 17(12), 5131–5142 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  21. Di Lorenzo, S., Di Paola, M., Pinnola, F.P., Pirrotta, A.: Stochastic response of fractionally damped beams. Probab. Eng. Mech. 35, 37–43 (2014)

    Article  Google Scholar 

  22. Spanos, P.D., Malara, G.: Nonlinear random vibrations of beams with fractional derivative elements. J. Eng. Mech. 140(9), 04014069–10 (2014)

    Article  Google Scholar 

  23. Di Matteo, A., Kougioumtzoglou, I.A., Pirrotta, A., Spanos, P.D., Di Paola, M.: Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral. Probab. Eng. Mech. 38, 127–135 (2014)

    Article  Google Scholar 

  24. Spanos, P.D., Kougioumtzoglou, I.A., dos Santos, K.R.M., Beck, A.T.: Stochastic averaging of nonlinear oscillators: Hilbert transform perspective. J. Eng. Mech. 144(2), 04017173 (2017)

    Article  Google Scholar 

  25. Liaskos, K., Pantelous, A.A., Kougioumtzoglou, I.A., Meimaris, A.T.: Implicit analytic solutions for the linear stochastic partial differential beam equation with fractional derivative terms. Syst. Control Lett. 121, 38–49 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  26. Kougioumtzoglou, I.A., Spanos, P.D.: Harmonic wavelets based response evolutionary power spectrum determination of linear and non-linear oscillators with fractional derivative elements. Int J. Non-Linear Mech. 80, 66–75 (2016)

    Article  Google Scholar 

  27. Kougioumtzoglou, I.A., dos Santos, K.R.M., Comerford, L.: Incomplete data based parameter identification of nonlinear and time-variant oscillators with fractional derivative elements. Mech. Syst. Signal Process. 94, 279–296 (2017)

    Article  Google Scholar 

  28. Kougioumtzoglou, I.A., Spanos, P.D.: An approximate approach for nonlinear system response determination under evolutionary stochastic excitation. Curr. Sci. 97, 1203–1211 (2009)

    MathSciNet  Google Scholar 

  29. Roberts, J.B., Spanos, P.D.: Stochastic averaging: an approximate method of solving random vibration problems. Int. J. Non-Linear Mech. 21(2), 111–134 (1986)

    MathSciNet  MATH  Article  Google Scholar 

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

    MATH  Google Scholar 

  31. 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)

    Article  Google Scholar 

  32. Spanos, P.D., Di Matteo, A., Cheng, Y., Pirrotta, A., Li, J.: Galerkin scheme-based determination of survival probability of oscillators with fractional derivative elements. J. Appl. Mech. 83(12), 121003–9 (2016)

    Article  Google Scholar 

  33. 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 

  34. Solomos, G.P., Spanos, P.T.D.: Oscillator response to nonstationary excitation. J. Appl. Mech. 51(4), 907–912 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  35. Kougioumtzoglou, I.A.: Stochastic joint time-frequency response analysis of nonlinear structural systems. J. Sound Vib. 332(26), 7153–7173 (2013)

    Article  Google Scholar 

  36. Tubaldi, E., Kougioumtzoglou, I.A.: Nonstationary stochastic response of structural systems equipped with nonlinear viscous dampers under seismic excitation. Earthq. Eng. Struct. Dyn. 44(1), 121–138 (2015)

    Article  Google Scholar 

  37. dos Santos, K.R.M., Kougioumtzoglou, I.A., Beck, A.T.: Incremental dynamic analysis: a nonlinear stochastic dynamics perspective. J. Eng. Mech. 142(10), 06016007–7 (2016)

    Article  Google Scholar 

  38. Spanos, P.D., Giaralis, A., Politis, N.P., Roesset, J.M.: Numerical treatment of seismic accelerograms and of inelastic seismic structural responses using harmonic wavelets. Comput. Aided Civil Infrastruct. Eng. 22(4), 254–264 (2007)

    Article  Google Scholar 

  39. Beck, J.L., Papadimitriou, C.: Moving resonance in nonlinear response to fully nonstationary stochastic ground motion. Probab. Eng. Mech. 8(3–4), 157–167 (1993)

    Article  Google Scholar 

  40. Mitseas, I.P., Kougioumtzoglou, I.A., Beer, M.: An approximate stochastic dynamics approach for nonlinear structural system performance-based multi-objective optimum design. Struct. Saf. 60, 67–76 (2016)

    Article  Google Scholar 

  41. Spanos, P.-T.D., Lutes, L.D.: Probability of response to evolutionary process. J. Eng. Mech. Div. 106(2), 213–224 (1980)

    Google Scholar 

  42. Grigoriu, M.: Stochastic Calculus: Applications in Science and Engineering. Springer, Birkhäuser (2002)

    MATH  Book  Google Scholar 

  43. Grigoriu, M.: Stochastic Systems: Uncertainty Quantification and Propagation. Springer, London (2012)

    MATH  Book  Google Scholar 

  44. Spanos, P.-T.D., Solomos, G.P.: Markov approximation to transient vibration. J. Eng. Mech. 109(4), 1134–1150 (1983)

    Article  Google Scholar 

  45. Liang, J., Chaudhuri, S.R., Shinozuka, M.: Simulation of nonstationary stochastic processes by spectral representation. J. Eng. Mech. 133(6), 616–627 (2007)

    Article  Google Scholar 

  46. Caughey, T.K.: Random excitation of a system with bilinear hysteresis. J. Appl. Mech. 27(4), 649–652 (1960)

    MathSciNet  Article  Google Scholar 

  47. Zhu, W.Q., Cai, G.Q., Hu, R.C.: Stochastic analysis of dynamical system with double-well potential. Int. J. Dyn. Control 1(1), 12–19 (2013)

    Article  Google Scholar 

  48. Bellizzi, S., Bouc, R.: Analysis of multi-degree of freedom strongly non-linear mechanical systems with random input: part I: non-linear modes and stochastic averaging. Probab. Eng. Mech. 14(3), 229–244 (1999)

    Article  Google Scholar 

  49. Di Paola, M., Pinnola, F.P., Spanos, P.D.: Analysis of multi-degree-of-freedom systems with fractional derivative elements of rational order. In: ICFDA’14 International Conference on Fractional Differentiation and its Applications, pp. 1–6, IEEE (2014)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. C. Fragkoulis.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Appendix

Appendix

In this appendix, more details on the derivation of Eqs. (12) and (13) are included for completeness, while the interested reader is also directed to Refs [32, 33]. In this regard, Eqs. (10) and (11) are rewritten, equivalently, in the form

$$\begin{aligned} \beta (A) = \frac{S(A)}{A\omega (A)} - \beta \frac{S_\alpha (A)}{\pi A\omega (A)} - \beta _0 \end{aligned}$$
(52)

and

$$\begin{aligned} \omega ^2(A) = \frac{F(A)}{A} + \beta \frac{F_\alpha (A)}{\pi A} \end{aligned}$$
(53)

where S(A) and F(A) are given by Eqs. (14) and (15), respectively. Further,

$$\begin{aligned} S_\alpha (A) = \int _{0}^{2\pi } \mathscr {D}^{\alpha }_{0,t}(A\cos \phi )\sin \phi \mathrm{d}\phi \end{aligned}$$
(54)

and

$$\begin{aligned} F_\alpha (A) = \int _{0}^{2\pi } \mathscr {D}^{\alpha }_{0,t}(A\cos \phi )\cos \phi \mathrm{d}\phi \end{aligned}$$
(55)

In general, the fractional derivative of order \(\alpha \) for the cosine function is given by

$$\begin{aligned} \mathscr {D}^{\alpha }_{0,t}\mathrm{cos}(bx) = b^\alpha \mathrm{cos}\left( \frac{\alpha \pi }{2} + bx\right) \end{aligned}$$
(56)

where \(0<\alpha <1\). Determining analytically the integrals defined in Eqs. (54) and (55) by employing Eq. (56) is not straightforward, as it involves the evaluation of rather complex integral forms. Nevertheless, appropriately approximating Eqs. (54) and (55) facilitates significantly the related computations. In particular, assuming that the time parameter \(\tau \) takes small values, Eq. (4) becomes [32, 33]

$$\begin{aligned} \dot{x}(t-\tau ) \approx \dot{x}(t)\cos (\omega (A)\tau )+x(t)\omega (A)\sin (\omega (A)\tau )\nonumber \\ \end{aligned}$$
(57)

Next, combining Eq. (57) with Eq. (2), the Caputo derivative defined in Eq. (2) takes the form

$$\begin{aligned} \mathscr {D}^{\alpha }_{0,t}x(t)= & {} \frac{1}{\varGamma (1-\alpha )} \left\{ \dot{x}(t) \int _{0}^{t} \frac{\cos (\omega (A)\tau )}{\tau ^\alpha } \mathrm{d}\tau \right. \nonumber \\&+\, \left. x(t)\omega (A) \int _{0}^{t} \frac{\sin (\omega (A)\tau )}{\tau ^\alpha } \mathrm{d}\tau \right\} \end{aligned}$$
(58)

Further, utilizing the integrals [33]

$$\begin{aligned} \int _{0}^{t} \frac{\cos (\omega (A)t)\mathrm{d}\tau }{\tau ^\alpha }= & {} \omega ^{\alpha -1}(A) \left[ \varGamma (1-\alpha )\sin \left( \frac{\alpha \pi }{2}\right) \right. \nonumber \\&+\, \frac{\sin (\omega (A) t)}{(\omega (A) t)^\alpha } + \left. O(\omega (A) t)^{-\alpha } \right] \nonumber \\ \end{aligned}$$
(59)

and

$$\begin{aligned} \int _{0}^{t} \frac{\sin (\omega (A)t)\mathrm{d}\tau }{\tau ^\alpha }= & {} \omega ^{\alpha -1}(A)\left[ \varGamma (1-\alpha )\cos \left( \frac{\alpha \pi }{2}\right) \right. \nonumber \\&-\, \frac{\cos (\omega (A) t)}{(\omega (A) t)^\alpha } + \left. O\left( \omega (A) t\right) ^{-\alpha } \right] \nonumber \\ \end{aligned}$$
(60)

Equation (58) becomes

$$\begin{aligned} \mathscr {D}^{\alpha }_{0,t}x(t)= & {} \omega ^{\alpha -1}(A) \left[ \dot{x}(t) \sin \left( \frac{\alpha \pi }{2}\right) \right. \nonumber \\&+\, \left. x(t) \omega (A) \cos \left( \frac{\alpha \pi }{2}\right) \right] + \frac{\omega ^{\alpha -1}(A)}{\varGamma (1-\alpha )} \nonumber \\&\times \, \frac{\dot{x}(t)\sin (\omega (A) t) - x(t) \omega (A)\cos (\omega (A) t)}{(\omega (A) t)^\alpha } \nonumber \\&+\, O\left( \omega (A) t\right) ^{-\alpha -1} \end{aligned}$$
(61)

Equation (61) constitutes an approximate expression that facilitates the determination of fractional derivatives of order \(0<\alpha <1\), and thus, it is utilized in simplifying the integrands of Eqs. (54) and (55), which become

$$\begin{aligned} S_\alpha (A) = - \pi A\omega ^\alpha (A)\sin \left( \frac{\alpha \pi }{2}\right) \end{aligned}$$
(62)

and

$$\begin{aligned} F_\alpha (A) = \pi A\omega ^\alpha (A)\cos \left( \frac{\alpha \pi }{2}\right) \end{aligned}$$
(63)

respectively. Finally, Eqs. (12) and (13) are derived by considering Eqs. (62) and (63).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fragkoulis, V.C., Kougioumtzoglou, I.A., Pantelous, A.A. et al. Non-stationary response statistics of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitation. Nonlinear Dyn 97, 2291–2303 (2019). https://doi.org/10.1007/s11071-019-05124-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-05124-0

Keywords

  • Fractional derivative
  • Nonlinear system
  • Stochastic dynamics
  • Non-stationary stochastic process
  • Evolutionary power spectrum