Skip to main content

Reliability function determination of nonlinear oscillators under evolutionary stochastic excitation via a Galerkin projection technique

Abstract

An approximate semi-analytical technique is developed for determining the response first-passage time probability density function of a class of lightly damped nonlinear oscillators subject to evolutionary stochastic excitation. Specifically, relying on a Markovian approximation of the response energy envelope, and on a stochastic averaging treatment, yields a backward Kolmogorov equation governing the evolution in time of the oscillator reliability function. Next, the backward Kolmogorov equation is solved by employing an appropriate orthogonal basis in conjunction with a Galerkin projection scheme. It is noted that the technique can account for arbitrary evolutionary excitation forms, even of the non-separable type. The special case of an undamped oscillator, for which relevant analytical results exist in the literature, is also included and studied in detail. Further, Markovian approximation of the potential energy envelope is considered as well. In comparison with the conventional amplitude-based energy envelope formulation, the intermediate step of linearizing the nonlinear stiffness element is circumvented, thus reducing the overall approximation degree of the technique. An additional significant advantage of the potential energy envelope formulation relates to the fact that its degree of accuracy appears rather insensitive to the nonlinearity magnitude (at least in the considered examples). Pertinent Monte Carlo simulation data are included in the numerical examples as well for assessing the accuracy of the technique.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

References

  1. Vanvinckenroye, H., Andrianne, T., Denoël, V.: First passage time as an analysis tool in experimental wind engineering. J. Wind Eng. Ind. Aerodyn. 177, 366–375 (2018)

    Article  Google Scholar 

  2. Tominaga, Y., Stathopoulos, T.: CFD simulation of near-field pollutant dispersion in the urban environment: a review of current modeling techniques. Atmos. Environ. 79, 716–730 (2013)

    Article  Google Scholar 

  3. Andrianne, T., de Ville de Goyet, V.: Mitigation of the torsional flutter phenomenon of bridge deck section during a lifting phase. In: 8th International Colloquium on Bluff Body Aerodynamics and Applications, Northeastern University, Boston, Massachusetts, USA (2016)

  4. Schuss, Z.: Theory and Applications of Stochastic Processes, Vol. 170 of Applied Mathematical Sciences. Springer, New York (2010)

    Book  Google Scholar 

  5. Preumont, A.: Random Vibration and Spectral Analysis. Kluwer Academic Publishers, Dordrecht (1994)

    MATH  Book  Google Scholar 

  6. Bergman, L.A., Heinrich, J.C.: On the moments of time to first passage of the linear oscillator. Earthq. Eng. Struct. Dyn. 9(3), 197–204 (1981)

    Article  Google Scholar 

  7. Vanvinckenroye, H., Denoël, V.: Average first-passage time of a quasi-Hamiltonian Mathieu oscillator with parametric and forcing excitations. J. Sound Vib. 406, 328–345 (2017)

    Article  Google Scholar 

  8. Vanvinckenroye, H., Denoël, V.: Second-order moment of the first passage time of a quasi-Hamiltonian oscillator with stochastic parametric and forcing excitations. J. Sound Vib. 427, 178–187 (2018)

    Article  Google Scholar 

  9. Lin, Y.K.Y.-K.: Probabilistic Theory of Structural Dynamics. R.E. Krieger Pub. Co, Malabar (1976)

    Google Scholar 

  10. Crandall, S.: First-crossing probabilities of the linear oscillator. J. Sound Vib. 12(3), 285–299 (1970)

    MathSciNet  MATH  Article  Google Scholar 

  11. Yang, J., Shinozuka, M.: First-passage time problem. J. Acoust. Soc. Am. 47(1B), 393–394 (1970)

    Article  Google Scholar 

  12. Oksendal, B.: Stochastic Differential Equations: An Introduction with Applications-Fifth Edition, pp. 1–5 (2003)

  13. Kovaleva, A.: An exact solution of the first-exit time problem for a class of structural systems. Probab. Eng. Mech. 24(3), 463–466 (2009)

    Article  Google Scholar 

  14. Naess, A., Gaidai, O.: Monte Carlo methods for estimating the extreme response of dynamical systems. J. Eng. Mech. 134(8), 628–636 (2008)

    Article  Google Scholar 

  15. Au, S.-K., Wang, Y.: Engineering risk assessment and design with subset simulation (2014)

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

    MATH  Book  Google Scholar 

  17. Kougioumtzoglou, I.A., Zhang, Y., Beer, M.: Softening Duffing Oscillator Reliability assessment subject to evolutionary stochastic excitation. ASCE-ASME J. Risk Uncertain. Eng. Syst. Part A Civ. Eng. 2(2), C4015001 (2016)

    Article  Google Scholar 

  18. Vanmarcke, E.H.: On the distribution of the first-passage time for normal stationary random processes. J. Appl. Mech. 42(1), 215 (1975)

    MATH  Article  Google Scholar 

  19. Náprstek, J., Král, R.: Evolutionary analysis of Fokker–Planck equation using multi-dimensional Finite Element Method. Procedia Eng. 199, 735–740 (2017)

    Article  Google Scholar 

  20. Canor, T., Denoël, V.: Transient Fokker–Planck–Kolmogorov equation solved with smoothed particle hydrodynamics method. Int. J. Numer. Methods Eng. 94(6), 535–553 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  21. Coleman, J.J.: Reliability of aircraft structures in resisting chance failure. Oper. Res. 7(5), 639–645 (1959)

    MATH  Article  Google Scholar 

  22. Kougioumtzoglou, I.A., Spanos, P.D.: Stochastic response analysis of the softening Duffing oscillator and ship capsizing probability determination via a numerical path integral approach. Probab. Eng. Mech. 35, 67–74 (2014)

    Article  Google Scholar 

  23. 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. Part B Mech. Eng. 1(2), 021005 (2015)

    Article  Google Scholar 

  24. Spanos, P.D., Kougioumtzoglou, I.A.: Galerkin scheme based determination of first-passage probability of nonlinear system response. Struct. Infrastruct. Eng. 10(10), 1285–1294 (2014)

    Article  Google Scholar 

  25. Yurchenko, D., Mo, E., Naess, A.: Reliability of strongly nonlinear single degree of freedom dynamic systems by the path integration method. J. Appl. Mech. 75(6), 061016 (2008)

    Article  Google Scholar 

  26. Li, J., Chen, J.: Stochastic Dynamics of Structures. Wiley, New York (2009)

    MATH  Book  Google Scholar 

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

    MathSciNet  Google Scholar 

  28. Lin, Y., Cai, G.: Some thoughts on averaging techniques in stochastic dynamics. Probab. Eng. Mech. 15(1), 7–14 (2000)

    Article  Google Scholar 

  29. Red-Horse, J., Spanos, P.: A generalization to stochastic averaging in random vibration. Int. J Non-Linear Mech. 27(1), 85–101 (1992)

    MathSciNet  MATH  Article  Google Scholar 

  30. Roberts, J.J., Spanos, P.P.: 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 

  31. Zhu, W.Q.: Stochastic averaging methods in random vibration. Appl. Mech. Rev. 41(5), 189 (1988)

    Article  Google Scholar 

  32. Proppe, C., Pradlwarter, H., Schuëller, G.: Equivalent linearization and Monte Carlo simulation in stochastic dynamics. Probab. Eng. Mech. 18(1), 1–15 (2003)

    Article  Google Scholar 

  33. Socha, L.: Linearization Methods for Stochastic Dynamic Systems, pp. 1–5. Springer, Berlin (2008)

    MATH  Book  Google Scholar 

  34. Roberts, J.B., Spanos, P.D.: Random Vibration and Statistical Linearization. Dover Publications, Mineola (2003)

    MATH  Google Scholar 

  35. Spanos, P.-T.D.: Numerics for common first-passage problem. J. Eng. Mech. Div. 108(5), 864–882 (1982)

    Google Scholar 

  36. 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 (2016)

    Article  Google Scholar 

  37. 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 

  38. Spanos, P., Solomos, G.P.: Barrier crossing due to transient excitation. J. Eng. Mech. 110(1), 20–36 (1984)

    Article  Google Scholar 

  39. Canor, T., Caracoglia, L., Denoël, V.: Perturbation methods in evolutionary spectral analysis for linear dynamics and equivalent statistical linearization. Probab. Eng. Mech. 46, 1–17 (2016)

    Article  Google Scholar 

  40. Luo, A.C.J., Huang, J.: Analytical period-3 motions to chaos in a hardening Duffing oscillator. Nonlinear Dyn. 73(3), 1905–1932 (2013)

    MathSciNet  Article  Google Scholar 

  41. 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)

    MathSciNet  MATH  Article  Google Scholar 

  42. Spanos, P.D., Red-Horse, J.R.: Nonstationary solution in nonlinear random vibration. J. Eng. Mech. 114(11), 1929–1943 (1988)

    MATH  Article  Google Scholar 

  43. 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 (2018)

    Article  Google Scholar 

  44. Primožič, T.: Estimating expected first passage times using multilevel Monte Carlo algorithm, M.Sc. in Mathematical and Computational Finance University

  45. Chunbiao, G., Bohou, X.: First-passage time of quasi-non-integrable-Hamiltonian system. Acta Mechanica Sinica 16(2), 183–192 (2000)

    Article  Google Scholar 

  46. 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 

Download references

Acknowledgements

H. Vanvinckenroye was supported by the National Fund for Scientific Research of Belgium. This research has been conducted during a visit of H. Vanvinckenroye at Columbia University

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. Denoël.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Comparison between Galerkin scheme and analytical expression for the average and mean square first-passage times

Comparison between Galerkin scheme and analytical expression for the average and mean square first-passage times

Considering a linear undamped oscillator under white noise excitation (\(S_{w}(\omega ,t)=S_{0}\)), since \({\dot{T}}_{i}(t)=-\pi \lambda _{i}S_{0}T_{i}(t)\) and \({\dot{c}}_{i}(t)=0\), Eq. (44) becomes

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {E}}\left\{ \tau _{H_{c}}\right\} = &{} \sum _{i}A_{i}\varPhi _{i}(H)\ \text {with}\ A_{i}=\frac{1}{\pi \lambda _{i}S_{0}}\frac{F_{1,i}}{F_{2,i}}\\ {\mathcal {E}}\left\{ \tau _{H_{c}}^{2}\right\} = &{} \sum _{i}B_{i}\varPhi _{i}(H)\ \text {with}\ B_{i}=\frac{2}{\pi ^{2}\lambda _{i}^{2}S_{0}^{2}}\frac{F_{1,i}}{F_{2,i}} \end{array}\right. }\nonumber \\ \end{aligned}$$
(55)

Analytical results are derived for this particular case in [7, 8] for the mean and mean square first-passage times \({\mathcal {E}}\left\{ \tau _{H_{c}}\right\} \) and \({\mathcal {E}}\left\{ \tau _{H_{c}}^{2}\right\} \) based on a stochastic averaging treatment and are given by

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {E}}\left\{ \tau _{H_{c}}\right\} = &{} \frac{1}{\pi S_{0}}\varDelta H\\ {\mathcal {E}}\left\{ \tau _{H_{c}}^{2}\right\} = &{} \frac{1}{\pi ^{2}S_{0}^{2}}\varDelta H\left( H+\frac{3}{2}\varDelta H\right) \end{array}\right. } \end{aligned}$$
(56)

where \(\varDelta H=H_{c}-H\) is the energy increase. The equalities to be proven are

$$\begin{aligned} \frac{1}{\pi S_{0}}\sum _{i}\frac{1}{\lambda _{i}}\frac{F_{1,i}}{F_{2,i}}\varPhi _{i}(H)=\frac{1}{\pi S_{0}}\varDelta H \end{aligned}$$
(57)

for the average first-passage time, and

$$\begin{aligned}&\frac{2}{\pi ^{2}S_{0}^{2}}\sum _{i}\frac{1}{\lambda _{i}^{2}}\frac{F_{1,i}}{F_{2,i}}\varPhi _{i}(H)\nonumber \\&\quad =\frac{1}{\pi ^{2}S_{0}^{2}}\varDelta H\left( H+\frac{3}{2}\varDelta H\right) \end{aligned}$$
(58)

for the mean square first-passage time. The lemmas (57) and (58) are demonstrated by projection of the two sides of the equations on the basis of eigenfunctions \(\varPhi _{j}(H)\). Equality is proven by showing that these projections are identical for all \(\varPhi _{j}(H)\).

First, Eq. (57) is proven. Then, the second demonstration for Eq. (58) results in a similar way from the first one.

Invoking the orthogonality property (20) with \(f(H)=1\) , the projection of Eq. (57) on the eigenfunctions \(\varPhi _{j}(H)\) yields

$$\begin{aligned}&\frac{1}{\pi S_{0}}\int _{0}^{H_{c}}\left( \sum _{i}\frac{1}{\lambda _{i}}\frac{F_{1,i}}{F_{2,i}}\varPhi _{i}(H)\right) \varPhi _{j}(H)\text {d}H \nonumber \\&\quad =\frac{F_{1,j}}{\pi \lambda _{j}S_{0}}=\frac{1}{\pi S_{0}}\int _{0}^{H_{c}}\left( H_{c}-H\right) \varPhi _{j}(H)\text {d}H \end{aligned}$$
(59)

that can be rewritten, according to (23), as

$$\begin{aligned} \int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H=\int _{0}^{H_{c}}\lambda _{j}\varPhi _{j}(H)\left( H_{c}-H\right) \text {d}H. \end{aligned}$$
(60)

Considering the definition of the Bessel function (38), one finds

$$\begin{aligned}&\int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H\nonumber \\&\quad =-\int _{0}^{H_{c}}\left( H\varPhi _{j}''(H)+\varPhi _{j}'(H)\right) \left( H_{c}-H\right) \text {d}H\nonumber \\&\quad =-\int _{0}^{H_{c}}\left( H\varPhi _{j}'(H)\right) '\left( H_{c}-H\right) \text {d}H. \end{aligned}$$
(61)

Integration by parts of (61) leads to

$$\begin{aligned} \int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H= & {} \left[ \left( H\varPhi _{j}'(H)\right) \left( H_{c}-H\right) \right] _{0}^{H_{c}}\nonumber \\&-\int _{0}^{H_{c}}H\varPhi '(H)\text {d}H. \end{aligned}$$
(62)

The first term in the right-hand side is trivially equal to zero while a second integration by parts of the second term leads to

$$\begin{aligned} \int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H= & {} -\left[ H\varPhi _{j}(H)\right] _{0}^{H_{c}}\nonumber \\&+\int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H. \end{aligned}$$
(63)

Accounting for the boundary condition \(\text {BesselJ}(0,\sqrt{4\lambda _{i}H_{c}})=0\) (21), equality is proven.

The demonstration is similarly done for the mean square first-passage time. After projection on the eigenfunction \(\varPhi _{j}(H)\), Eq. (58) yields

$$\begin{aligned}&\int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H\nonumber \\&\quad =\frac{1}{4}\int _{0}^{H_{c}}\left( H_{c}-H\right) \left( 3H_{c}-H\right) \lambda _{j}^{2}\varPhi _{j}(H)\text {d}H.\nonumber \\ \end{aligned}$$
(64)

Considering the definition of the Bessel function (38), one finds

$$\begin{aligned}&\int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H\nonumber \\&\quad =-\frac{1}{4}\int _{0}^{H_{c}}\left( H_{c}-H\right) \left( 3H_{c}-H\right) \lambda _{j}\nonumber \\&\qquad \times \left( H\varPhi _{j}''(H)+\varPhi _{j}'(H)\right) \text {d}H. \end{aligned}$$
(65)

A first integration by parts leads to

$$\begin{aligned}&\int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H\nonumber \\&\quad =-\frac{1}{2}\lambda _{j}\left[ H\varPhi '(H)\left( H_{c}-H\right) \left( 3H_{c}-H\right) \right] _{0}^{H_{c}}\nonumber \\&\qquad -\frac{\lambda _{j}}{2}\int _{0}^{H_{c}}\left( H-2H_{c}\right) H\varPhi _{j}'(H)\text {d}H. \end{aligned}$$
(66)

The first term in the right-hand side is equal to zero, while the second one can be developed using (38) and a second integration by parts and yields

$$\begin{aligned} \int _{0}^{H_{c}}\varPhi _{j}(H)\text {d}H= & {} \left[ H\varPhi '(H)(H-H_{c})\right] _{0}^{H_{c}}\nonumber \\&-\int _{0}^{H_{c}}H\varPhi _{j}'(H)\text {d}H, \end{aligned}$$
(67)

which proves equality.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Vanvinckenroye, H., Kougioumtzoglou, I.A. & Denoël, V. Reliability function determination of nonlinear oscillators under evolutionary stochastic excitation via a Galerkin projection technique . Nonlinear Dyn 95, 293–308 (2019). https://doi.org/10.1007/s11071-018-4564-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-018-4564-8

Keywords

  • Galerkin scheme
  • Reliability assessment
  • First-passage time
  • Nonlinear oscillator
  • Evolutionary excitation
  • Backward Kolmogorov equation