Probabilistic Solutions of the Stretched Beam Systems Formulated by Finite Difference Scheme and Excited by Gaussian White Noise

  • Guo-Kang ErEmail author
  • Vai Pan Iu
  • Kun Wang
  • Hai-En Du
Conference paper
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 33)


The probabilistic solutions of elastic stretched beams are studied when the beam is discretized by finite difference scheme and excited by Gaussian white noise which is fully correlated in space. The nonlinear multi-degree-of-freedom system about the random vibration of stretched beam is formulated by finite difference scheme first. Then the relevant Fokker-Planck-Kolmogorov equation is solved for the probabilistic solutions of the system by the state-space-split and exponential polynomial closure method. Monte Carlo simulation method and equivalent linearization method are also adopted to analyze the probabilistic solutions of the system responses, respectively. Numerical results obtained with the three methods are presented and compared to show the computational efficiency and numerical accuracy of solving the Fokker-Planck-Kolmogorov equation by the state-space-split and exponential polynomial closure method in analyzing the probabilistic solutions of the beams discretized by finite difference scheme and excited by Gaussian white noise. The techniques of using the state-space-split procedure for dimension reduction of the beam systems are discussed through the given beam systems with different space distributions of excitations.


Stretched beam Nonlinear random vibration FPK equation MDOF system Finite difference SSS-EPC method 



The results presented in this paper were obtained under the supports of the Science and Technology Development Fund of Macau (Grant No. 042/2017/A1) and the Research Committee of University of Macau (Grant No. MYRG2018-00116-FST).


  1. 1.
    Herbert, R.E.: Random vibrations of a nonlinear elastic beam. J. Acoust. Soc. Am. 36, 2090–2094 (1964)CrossRefGoogle Scholar
  2. 2.
    Herbert, R.E.: On the stresses in a nonlinear beam subject to random excitation. Int. J. Solids Struct. 1, 235–242 (1965)CrossRefGoogle Scholar
  3. 3.
    Fang, J., Elishakoff, I., Caimi, R.: Nonlinear response of a beam under stationary random excitation by improved stochastic linearization method. Appl. Math. Mod. 9, 106–111 (1995)CrossRefGoogle Scholar
  4. 4.
    Elishakoff, I., Lin, Y.K., Zhu, L.P.: Random vibration of uniform beams with varying boundary conditions by the dynamic-edge-effect method. Comput. Methods Appl. Mech. Eng. 121, 59–76 (1995)CrossRefGoogle Scholar
  5. 5.
    Er, G.K.: The probabilistic solutions of some nonlinear stretched beams excited by filtered white noise. In: Proppe, C. (ed) IUTAM Symposium on Multiscale Problems in Stochastic Mechanics 2012. Procedia IUTAM, Vol. 6, pp. 141–150. Elsevier (2013)Google Scholar
  6. 6.
    Masud, A., Bergman, L.A.: Solution of the four dimensional Fokker-Planck equation: still a challenge. In: Augusti, G., Schuëller, G.I., Ciampoli, M. (eds.) Proceedings of ICOSSAR’2005, pp. 1911–1916, Millpress Science Publishers, Rotterdam (2005)Google Scholar
  7. 7.
    Booton, R.C.: Nonlinear control systems with random inputs. IRE Trans Circ Theory. CT-1(1), 9–18 (1954)CrossRefGoogle Scholar
  8. 8.
    Kazakov, I.E.: Generaliztion of the method of stochastical linearization to multi-dimensional systems. Autom Remote Control. 26, 1202–1206 (1965)Google Scholar
  9. 9.
    Socha, L., Soong, T.T.: Linearization in analysis of nonlinear stochastic systems. ASME J. Appl. Mech. Rev. 44, 399–422 (1991)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Proppe, C., Pradlwater, H.J., Schuëller, G.I.: Equivalent linearization and Monte Carlo simulation in stochastic dynamics. Prob. Eng. Mech. 18, 1–15 (2003)CrossRefGoogle Scholar
  11. 11.
    Metropolis, N., Ulam, S.: Monte Carlo method. J. Am Stat Assoc. 14, 335–341 (1949)CrossRefGoogle Scholar
  12. 12.
    Harris, C.J.: Simulation of multivariable nonlinear stochastic systems. Int. J. Num. Meth. Eng. 14, 37–50 (1979)CrossRefGoogle Scholar
  13. 13.
    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)CrossRefGoogle Scholar
  14. 14.
    Wu, W.F., Lin, Y.K.: Cumulant-neglect closure for nonlinear oscillators under random parametric and external excitations. Int. J. Non-Linear Mech. 19, 349–362 (1984)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ibrahim, R.A., Soundararajan, A., Heo, H.: Stochastic response of nonlinear dynamic systems based on a non-Gaussian closure. ASME J. Appl. Mech. 52, 965–970 (1985)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Sun, J.-Q., Hsu, C.S.: Cumulant-neglect closure method for nonlinear systems under random excitations. ASME J. Appl. Mech. 54, 649–655 (1987)CrossRefGoogle Scholar
  17. 17.
    Hasofer, A.M., Grigoriu, M.: A new perspective on the moment closure method. ASME J. Appl. Mech. 62, 527–532 (1995)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Martens, W., von Wagner, U., Mehrmann, V.: Calculation of high-dimensional probability density functions of stochastically excited nonlinear mechanical systems. Nonlinear Dyn. 67, 2089–2099 (2012)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sun, Y., Kumar, M.: Numerical solution of high dimensional stationary Fokker-Planck equations via tensor decomposition and Chebyshev spectral differentiation. Comput Math Appl. 67, 1961–1977 (2014)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Er, G.K.: Methodology for the solutions of some reduced Fokker-Planck equations in high dimensions. Ann. Phys. (Berlin). 523(3), 247–258 (2011)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Er, G.K., Iu, V.P.: A new method for the probabilistic solutions of large-scale nonlinear stochastic dynamic systems. In: Zhu, W.Q., Lin, Y.K., Cai, G.Q. (eds.) Nonlinear Stochastic Dynamics and Control, IUTAM Book Series, Vol. 29, pp. 25–34. Springer, New York (2011)CrossRefGoogle Scholar
  22. 22.
    Er, G.K.: An improved closure method for analysis of nonlinear stochastic systems. Nonlinear Dyn. 17(3), 285–297 (1998)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Er, G.K.: The probabilistic solutions to non-linear random vibrations of multi-degree-of-freedom systems. ASME J. Appl. Mech. 67(2), 355–359 (2000)CrossRefGoogle Scholar
  24. 24.
    Soong, T.T.: Random Differential Equations in Science and Engineering. Academic, New York (1973)zbMATHGoogle Scholar
  25. 25.
    Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics. McGraw-Hill, New York (1995)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University of MacauMacau SARPeople’s Republic of China

Personalised recommendations