In this study, we consider a method for investigating the stochastic response of a nonlinear dynamical system affected by a random seismic process. We present the solution of the probability density of a single/multiple-degree of freedom (SDOF/MDOF) system with several statically stable equilibrium states and with possible jumps of the snap-through type. The system is a Hamiltonian system with weak damping excited by a system of non-stationary Gaussian white noise. The solution based on the Gibbs principle of the maximum entropy of probability could potentially be implemented in various branches of engineering. The search for the extreme of the Gibbs entropy functional is formulated as a constrained optimization problem. The secondary constraints follow from the Fokker-Planck equation (FPE) for the system considered or from the system of ordinary differential equations for the stochastic moments of the response derived from the relevant FPE. In terms of the application type, this strategy is most suitable for SDOF/MDOF systems containing polynomial type nonlinearities. Thus, the solution links up with the customary formulation of the finite elements discretization for strongly nonlinear continuous systems.

Institute of Theoretical and Applied Mechanics Prosecká 809 76 190 00 Prague 9 Czech Republic

Zobrazit více v PubMed

Askar A., Cakmak A. Seismic waves in random media. Probab. Eng. Mech. 1988;3:124–129. doi: 10.1016/0266-8920(88)90024-0. DOI

Langley R. Wave transmission through one-dimensional near periodic structures: Optimum and random disorder. J. Sound Vib. 1995;188:717–743. doi: 10.1006/jsvi.1995.0620. DOI

Manolis G. Acoustic and seismic wave propagation in random media. In: Cheng A.D., Yang C.Y., editors. Computational Stochastic Mechanics. Elsevier; Southampton, UK: 1993. pp. 597–622. Chapter 26.

Náprstek J. Wave propagation in semi-infinite bar with random imperfections of density and elasticity module. J. Sound Vib. 2008;310:676–693. doi: 10.1016/j.jsv.2007.03.048. DOI

Náprstek J., Fischer C. Planar compress wave scattering and energy diminution due to random inhomogeneity of material density; Proceedings of the 16th World Conference on Earthquake Engineering; Santiago, Chile. 9–13 January 2017.

Náprstek J. Sound wave scattering at randomly rough surfaces in Fraunhofer domain. In: Tassoulas J., editor. Proceedings of the Engineering Mechanics 2000 ASCE; Austin, TX, USA. 21–24 May 2000; Austin, TX, USA: University of Texas; 2000. 8p paper N-8.

Zembaty Z., Castellani A., Boffi G. Spectral analysis of the rotational component of eartquake motion. Probab. Eng. Mech. 1993;8:5–14. doi: 10.1016/0266-8920(93)90025-Q. DOI

Zembaty Z. Vibrations of bridge structure under kinematic wave excitations. J. Struct. Eng. 1997;123:479–488. doi: 10.1061/(ASCE)0733-9445(1997)123:4(479). DOI

Kanai K. Seismic-Empirical Formula for the Seismic Characteristics of the Ground. Volume 35. Bulletin of the Earthquake Research Institute; Tokyo, Japan: 1957. pp. 309–325.

Tajimi H. A statistical method of determining the maximum response of a building structure during an earthquake; Proceedings of the 2nd World Conference on Earthquake Engineering; Tokyo, Japan. 11–18 July 1960; Tokyo-Kyoto, Japan: Science Council of Japan; 1960. pp. 781–798.

Lin Y., Cai G. Probabilistic Structural Dynamics. McGraw-Hill; New York, NY, USA: 1995.

Bolotin V. Random Vibrations of Elastic Systems (in Russian) Nauka; Moskva, Russia: 1979.

de Boor C. Practical Guide to Splines. Springer; New York, NY, USA: 1987.

Malat S. Multiresolution approximation and wavelets. Trans. Am. Math. Soc. 1989;315:59–88.

Huang N.E., Shen Z., Long S.R., Wu M.C., Shih H.H., Zheng Q., Yen N.C., Tung C.C., Liu H.H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A. 1998;454:903–995. doi: 10.1098/rspa.1998.0193. DOI

Zembaty Z., Krenk S. Response spectra of spatial ground motion. In: Flesch R., editor. Proceedings of the 10th European Conference on Earthquake Engineering; Vienna, Austria. 28 August–2 September 1994; Rotterdam, The Netherlands: Balkema; 1994. pp. 1271–1275.

Fischer C. Decomposition of the seismic excitation. In: Frýba L., Náprstek J., editors. Proceedings of the EURODYN’99 Conference; Prague, Czech Republic. 7–10 June 1999; Rotterdam, The Netherlands: Balkema; 1999. pp. 1111–1116.

Náprstek J., Fischer O. Comparison of classical and stochastic solution to seismic response of structures. In: Bisch P., editor. Proceedings of the 11th European Conference on Earthquake Engineering; Paris, France. 6–11 September 1998; Rotterdam, The Netherlands: Balkema; 1998. p. 13. CD ROM.

Náprstek J. Non-stationary response of structures excited by random seismic processes with time variable frequency content. Soil Dyn. Earthq. Eng. 2002;22:1143–1150. doi: 10.1016/S0267-7261(02)00141-0. DOI

European Committee for Standardization, editor. Eurocode 8. Design Provisions for Earthquake Resistance of Structures. CEN; Brussels, Belgium: 1996.

Tondl A., Ruijgrok T., Verhulst F., Nabergoj R. Autoparametric Resonance in Mechanical Systems. Cambridge University Press; Cambridge, UK: 2000.

Náprstek J. Non-linear auto-parametric vibrations in civil engineering systems. In: Topping B., Neves L.C., Barros R., editors. Trends in Civil and Structural Engineering Computing. Saxe-Coburg Publications; Stirlingshire, UK: 2009. pp. 293–317. Chapter 14.

Pugachev V., Sinitsyn I. Stochastic Differential Systems—Analysis and Filtering. J. Willey; Chichester, UK: 1987.

Arnold L. Random Dynamical Systems. Springer; Berlin/Heidelberg, Germany: 1998.

Gichman I., Skorochod A. Stochasticheskije Differencialnyje Uravnenija (in Russian) Naukova dumka; Kiev, Ukraine: 1968.

Grasman J., Van Herwaarden O. Asymptotic Methods for the Fokker–Planck Equation and the exit Problem in Applications. Springer; Berlin/Heidelberg, Germany: 1999.

Náprstek J. Some properties and applications of eigen functions of the Fokker–Planck operator. In: Fuis V., Krejčí P., Návrat T., editors. Proceedings of the Engineering Mechanics 2005; Svratka, Czech Republic. 9–12 May 2005; Brno, Czech Republic: FME TU; 2005. p. 12.

Weinstein E., Benaroya H. The van Kampen expansion for the Fokker–Planck equation of a Duffing oscillator. J. Stat. Phys. 1994;77:667–679. doi: 10.1007/BF02179455. DOI

Spencer B., Bergman L. On the numerical solution of the Fokker–Planck equation for nonlinear stochastic systems. Nonlinear Dyn. 1993;4:357–372. doi: 10.1007/BF00120671. DOI

Masud A., Bergman L. Application of multi-scale finite element methods to the solution of the Fokker–Planck equation. Comput. Methods Appl. Mech. Eng. 2005;194:1513–1526. doi: 10.1016/j.cma.2004.06.041. DOI

Náprstek J., Král R. Finite element method analysis of Fokker–Planck equation in stationary and evolutionary versions. Adv. Eng. Softw. 2014;72:28–38. doi: 10.1016/j.advengsoft.2013.06.016. DOI

Náprstek J., Král R. Theoretical background and implementation of the finite element method for multi-dimensional Fokker–Planck equation analysis. Adv. Eng. Softw. 2017;113:54–75.

Jaynes E. Information theory and statistical mechanics. Phys. Rev. 1957;106:620–630. doi: 10.1103/PhysRev.106.620. DOI

Kullback S. Information Theory and Statistics. Chapman and Hall; New York, NY, USA: 1959.

Sarlis N.V., Skordas E.S., Varotsos P.A., Nagao T., Kamogawa M., Tanaka H., Uyeda S. Minimum of the order parameter fluctuations of seismicity before major earthquakes in Japan. Proc. Natl. Acad. Sci. USA. 2013;110:13734–13738. doi: 10.1073/pnas.1312740110. PubMed DOI PMC

Varotsos P.A., Sarlis N.V., Skordas E.S., Tanaka H. A plausible explanation of the b-value in the Gutenberg–Richter law from first Principles. Proc. Jpn. Acad. B. 2004;80:429–434. doi: 10.2183/pjab.80.429. DOI

Burg J. New concepts in power spectra estimation; Presented at the 40th Annual International SEG Meeting; New Orleans, LA, USA. 10 November 1970.

Stocks N. Information transmission in parallel threshold arrays: Suprathreshold stochastic resonance. Phys. Rev. E. 2001;63:041114. doi: 10.1103/PhysRevE.63.041114. PubMed DOI

McDonnell M., Stock N., Pearce C., Abbott D. Stochastic Resonance: From Suprathreshold Stochastic Resonance to Stochastic Signal Quantization. Cambridge University Press; Cambridge, NY, USA: 2008.

Cieslikiewicz W. Determination of the Probability Distribution of a Sea Surface Elevation via Maximum Entropy Method (in Polish) Rep. Institute of Hydro-Engineering, Polish Acad. Sci.; Gdansk, Poland: 1988.

Sargent M., III, Scully M., Lamb W., Jr. Laser Physics. Addison-Wesley; Reading, UK: 1974.

Dong W., Bao A., Shah H. Use of maximum entropy principle in earthquake recurrence relationship. Bull. Seismol. Soc. Am. 1984;74:725–737.

Herzel H., Ebeling W., Schmitt A. Entropies of biosequencies: The role of repeats. Phys. Rev. E. 1994;6:5061. doi: 10.1103/PhysRevE.50.5061. PubMed DOI

Rosenblueth E., Karmeshu, Hong H.P. Maximum entropy and discretization of probability distributions. Probab. Eng. Mech. 1987;2:58–63. doi: 10.1016/0266-8920(87)90016-6. DOI

Tagliani A. Principle of Maximum Entropy and probability distributions: definition of applicability field. Probab. Eng. Mech. 1989;4:99–104. doi: 10.1016/0266-8920(89)90014-3. DOI

Sobczyk K., Trebicki J. Maximum entropy principle in stochastic dynamics. Probab. Eng. Mech. 1990;5:102–110. doi: 10.1016/0266-8920(90)90001-Z. DOI

Sobczyk K., Trebicki J. Maximum entropy principle and nonlinear stochastic oscillators. Phys. A Stat. Mech. Its Appl. 1993;193:448–468. doi: 10.1016/0378-4371(93)90487-O. DOI

Náprstek J. Principle of maximum entropy of probability density in stochastic mechanics of elastic systems. In: Dobiáš I., editor. Proceedings of the Dynamics of Machines ’95; Prague, Czech Republic. 1–2 February 1995; Prague, Czech Republic: IT ASCR; 1995. pp. 115–122.

Náprstek J. Application of the maximum entropy principle to the analysis of non-stationary response of SDOF/MDOF systems. In: Půst L., Peterka F., editors. Proceedings of the 2nd European Non-Linear Oscillations Conference—EUROMECH; Prague, Czech Republic. 9–13 September 1996; Prague, Czech Republic: IT ASCR; 1996. pp. 305–308.

Gibbs J. Elementary Principles in Statistical Mechanics. Dover Publications; New York, NY, USA: 1960.

Reif F. Statistical Physics—Berkeley Physics Course-Vol.5. Mc Graw Hill; New York, NY, USA: 1965.

Michlin S. Variational Methods in Mathematical Physics (in Russian) Nauka; Moskva, Russia: 1970.

Náprstek J., Fischer C. Semi-analytical stochastic analysis of the generalized van der Pol system. App. Comput. Mech. 2018;12:45–58. doi: 10.24132/acm.2017.407. DOI

Zubarev D. Neravnovesnaja Statističeskaja Termodinamika (in Russian) Nauka; Moskva, Russia: 1971.

Kullback S., Leibler R. On information and sufficiency. Ann. Math. Stat. 1951;22:79–86. doi: 10.1214/aoms/1177729694. DOI

Kloeden P., Platen E. Numerical Solution of Stochastic Differential Equations. Springer; Berlin/Heidelberg, Germany: 1992.

Rosenberg R. Normal modes of non-linear dual-mode systems. J. Appl. Mech. 1960;27:263–268. doi: 10.1115/1.3643948. DOI

Vakakis A., Manevitch L., Mikhlin Y., Pilipchuk V., Zevin A. Normal Modes and Localization in Non-Linear Systems. Wiley; New York, NY, USA: 1996.

Kerschen G., Peeters M., Golinval J., Vakakis A. Non-linear normal modes, Part I: A useful framework for the structural dynamicist. Mech. Syst. Signal Process. 2009;23:170–194. doi: 10.1016/j.ymssp.2008.04.002. DOI