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Analysis of some large-scale nonlinear stochastic dynamic systems with subspace-EPC method

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Abstract

The probabilistic solutions to some nonlinear stochastic dynamic (NSD) systems with various polynomial types of nonlinearities in displacements are analyzed with the subspace-exponential polynomial closure (subspace-EPC) method. The space of the state variables of the large-scale nonlinear stochastic dynamic system excited by Gaussian white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system are then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in the other subspace is formulated. Therefore, the FPK equations in low dimensions are obtained from the original FPK equation in high dimensions and the FPK equations in low dimensions are solvable with the exponential polynomial closure method. Examples about multi-degree-offreedom NSD systems with various polynomial types of nonlinearities in displacements are given to show the effectiveness of the subspace-EPC method in these cases.

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References

  1. Soong T T. Random Differential Equations in Science and Engineering. New York: Academic Press, 1973

    MATH  Google Scholar 

  2. Sobczyk K. Differential Equations with Application to Physics and Engineering. Boston: Kluwer Academic Publishers, 1991

    Google Scholar 

  3. Dimentberg M F. An exact solution to a certain nonlinear random vibration problem. Int J Non-Linear Mech, 1982, 17: 231–236

    Article  MathSciNet  MATH  Google Scholar 

  4. Lin Y K, Cai G Q. Exact stationary-response solution for second order nonlinear systems under parametric and external excitations, part II. ASME J Appl Mech, 1988, 55: 702–705

    Article  MathSciNet  MATH  Google Scholar 

  5. Scheurkogel A, Elishakoff I. Non-linear random vibration of a twodegree-of-freedom system. Ziegler F, Schuëller G I, eds. Non-Linear Stochastic Engineering Systems. Berlin: Springer-Verlag, 1988. 285–299

    Chapter  Google Scholar 

  6. Zhu WQ. Nonlinear stochastic dynamics and control in Hamiltonian formulation. Appl Mech Rev, 2006, 59: 230–247

    Article  ADS  Google Scholar 

  7. Booton R C. Nonlinear control systems with random inputs. IRE Trans Circuit Theor, 1954, 1: 9–19

    Google Scholar 

  8. Caughey T K. Response of a nonlinear string to random loading. ASME J Appl Mech, 1958, 26: 341–344

    MathSciNet  Google Scholar 

  9. Lin Y K. Probabilistic Theory of Structural Dynamics. New York: McGraw-Hill, 1967

    Google Scholar 

  10. Socha L. Linearization Methods for Stochastic Dynamic Systems. Berlin: Springer-Verlag, 2008

    MATH  Google Scholar 

  11. Assaf S A, Zirkie L D. Approximate analysis of non-linear stochastic systems. Int J Control, 1976, 23: 477–492

    Article  MATH  Google Scholar 

  12. Soize C. Steady-state solution of Fokker-Planck equation in higher dimension. Probab Eng Mech, 1988, 3: 196–206

    Article  Google Scholar 

  13. Tagliani A. Principle of maximum entropy and probability distributions: Definition and applicability field. Probab Eng Mech, 1989, 4: 99–104

    Article  Google Scholar 

  14. Sobczyk K, Trebicki J. Maximum entropy principle in stochastic dynamics. Probab Eng Mech, 1990, 5: 102–110

    Article  Google Scholar 

  15. Stratonovich R L. Topics in the Theory of Random Noise, vol. 1. New York: Gordon and Breach, 1963

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Crandall S H. Perturbation techniques for random vibration of nonlinear systems. J Acoust Soc Am, 1963, 35: 1700–1705

    Article  MathSciNet  ADS  Google Scholar 

  18. Barcilon V. Singular perturbation analysis of the Fokker-Planck equation: Kramers’ underdamped problem. SIAM J Appl Math, 1996, 56: 446–497

    Article  MathSciNet  MATH  Google Scholar 

  19. Khasminskii R Z, Yin G. Asymptotic series for singularly perturbed Kolmogorov-Fokker-Planck equations. SIAM J Appl Math, 1996, 56: 1766–1793

    Article  MathSciNet  MATH  Google Scholar 

  20. Lutes L D. Approximate technique for treating random vibrations of hysteretic systems. J Acoust Soc Am, 1970, 48: 299–306

    Article  ADS  Google Scholar 

  21. Cai G Q, Lin Y K. A new approximation solution technique for randomly excited non-linear oscillators. Int J Non-Linear Mech, 1988, 23: 409–420

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhu W Q, Soong T T, Lei Y. Equivalent nonlinear system method for stochastically excited Hamiltonian systems. ASME J Appl Mech, 1994, 61: 618–623

    Article  MathSciNet  MATH  Google Scholar 

  23. Spencer Jr B F, Bergman L A. On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems. Int J Non-Linear Mech, 1993, 4: 357–372

    Google Scholar 

  24. Ujevic M, Letelier P S. Solving procedure for a 25-diagonal coefficient matrix: Direct numerical solutions of the three-dimensional linear Fokker-Planck equation. J Comput Phys, 2006, 215: 485–505

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. Er G K. A consistent method for the solutions to reduced FPK equations in statistical mechanics. Physica A, 1999, 262: 118–128

    Article  Google Scholar 

  26. Er G K. The probabilistic solution to non-linear random vibrations of multi-degree-of-freedom systems. ASME J Appl Mech, 2000, 67: 355–359

    Article  MATH  Google Scholar 

  27. Er G K, Zhu H T, Iu V P, et al. PDF solution of nonlinear oscillators under external and parametric Poisson impulses. AIAA J, 2008, 46: 2839–2847

    Article  ADS  Google Scholar 

  28. Harris C J. Simulation of multivariate nonlinear stochastic system. Int J Numer Methods Eng, 1979, 14: 37–50

    Article  MATH  Google Scholar 

  29. Kloeden P E, Platen E. Numerical Solution of Stochastic Differential Equations. Berlin: Springer-Verlag, 1995

    Google Scholar 

  30. Er G K. Methodology for the solutions of some reduced Fokker-Planck equations in high dimensions. Ann Phys (Berlin), 2011, 523: 247–258

    Article  MathSciNet  ADS  Google Scholar 

  31. Er G K, Iu V P. A new method for the probabilistic solutions of largescale nonlinear stochastic dynamic systems. In: Proceedings of the IUTAM Symposium on Nonlinear Stochastic Dynamics and Control-2010. Berlin: Springer-Verlag, 2011

    Google Scholar 

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  1. Faculty of Science and Technology, University of Macau, Macau SAR, China

    GuoKang Er & VaiPan Iu

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  1. GuoKang Er
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  2. VaiPan Iu
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Correspondence to GuoKang Er.

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Er, G., Iu, V. Analysis of some large-scale nonlinear stochastic dynamic systems with subspace-EPC method. Sci. China Phys. Mech. Astron. 54, 1631 (2011). https://doi.org/10.1007/s11433-011-4443-5

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