Stochastic averaging of quasi-integrable and non-resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations
详细信息 查看全文
- 作者：Wantao Jia (1) (2)
Weiqiu Zhu (1) (3) - 关键词：Quasi ; integrable and non ; resonant Hamiltonian system ; Combined Gaussian and Poisson white noise excitations ; Stochastic averaging method ; Perturbation method
- 刊名：Nonlinear Dynamics
- 出版年：2014
- 出版时间：April 2014
- 年：2014
- 卷：76
- 期：2
- 页码：1271-1289
- 全文大小：1,754 KB
- 参考文献：1. Roberts, J.B., Spanos, P.D.: Stochastic averaging: an approximate method of solving random vibration problems. Int. J. Non-linear Mech. 21(2), 111-34 (1986)
2. Zhu, W.Q.: Stochastic averaging methods in random vibration. ASME Appl. Mech. Rev. 41(5), 189-99 (1988)
3. Zhu, W.Q.: Recent developments and applications of the stochastic averaging method in random vibration. ASME Appl. Mech. Rev. 49(10S), S72–S80 (1996)
4. Stratonovich, R.L.: Topics in the Theory of Random Noise, vol. 1. Gordon and Breach, New York (1963)
5. Khasminskii, R.Z.: A limit theorem for the solutions of differential equations with random right-hand sides. Theory Probab. Appl. 11(3), 390-06 (1966)
6. Landa, P., Stratonovich, R.L.: Theory of stochastic transitions of various systems between different states. In: Proceedings of Moscow University, Moscow, pp. 33-5 (in Russian) (1962).
7. Khasminskii, R.Z.: On the behavior of a conservative system with friction and small random noise. Appl. Math. Mech. 28, 1126-130 (1964)
8. Zhu, W.Q.: Stochastic averaging of the energy envelope of nearly Lyapunov systems. In: Random vibrations and reliability. Proceedings of IUTAM Symposium, pp. 347-57. Akademie, Verlag (1983)
9. Zhu, W.Q., Lin, Y.K.: Stochastic averaging of energy envelope. ASCE J. Eng. Mech. 117(8), 1890-905 (1991)
10. Khasminskii, R.Z.: On the averaging principle for stochastic differential It? equation. Kibernetika 4, 260-79 (1968). (in Russian)
11. Roberts, J.B.: The energy envolope of a randomly excited non-linear oscillator. J. Sound Vib. 60(2), 177-85 (1978)
12. Zhu, W.Q., Huang, Z.L., Suzuki, Y.: Response and stability of strongly non-linear oscillators under wide-band random excitation. In. J. Non-linear Mech. 36(8), 1235-250 (2001)
13. Roberts, J.B., Vasta, M.: Response of non-linear oscillators to non-white random excitation using an energy based method. In: IUTAM Symposium on nonlinearity and stochastic structural, dynamics, pp. 221-31. Springer, Berlin (2001)
14. Cai, G.Q., Lin, Y.K.: Random vibration of strongly nonlinear systems. Nonlinear Dyn. 24(1), 3-5 (2001)
15. Zhu, W.Q.: Nonlinear stochastic dynamics and control in Hamiltonian formulation. ASME Appl. Mech. Rev. 59, 230 (2006)
16. Zhu, W.Q., Huang, Z.L., Yang, Y.Q.: Stochastic averaging of quasi-integrable Hamiltonian systems. ASME J. Appl. Mech. 64(4), 975-84 (1997)
17. Zhu, W.Q., Huang, Z.L., Suzuki, Y.: Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems. Int. J. Non-linear Mech. 37(3), 419-37 (2002)
18. Zhu, W.Q., Yang, Y.Q.: Stochastic averaging of quasi-nonintegrable-Hamiltonian systems. ASME J. Appl. Mech. 64(1), 157-64 (1997)
19. Huang, Z.L., Zhu, W.Q., Suzuki, Y.: Stochastic averaging of strongly non-linear oscillators under combined harmonic and white-noise excitations. J. Sound Vib. 238(2), 233-56 (2000)
20. Huang, Z.L., Zhu, W.Q., Ni, Y.Q., Ko, J.M.: Stochastic averaging of strongly non-linear oscillators under bounded noise excitation. J. Sound Vib. 254(2), 245-67 (2002)
21. Zeng, Y., Zhu, W.Q.: Stochastic averaging of / n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations. Int. J. Non-linear Mech. 45(5), 572-86 (2010)
22. Zeng, Y., Zhu, W.Q.: Stochastic averaging of quasi-linear systems driven by Poisson white noise. Probab. Eng. Mech. 25(1), 99-07 (2010)
23. Zeng, Y., Zhu, W.Q.: Stochastic averaging of quasi-nonintegrable-Hamiltonian systems under Poisson white noise excitation. ASME J. Appl. Mech. 78(2), 021002 (2011). doi:10.1115/1.4002528
24. Zeng, Y., Zhu, W.Q.: Stochastic averaging of strongly nonlinear oscillators under Poisson white noise excitation. In: IUTAM symposium on nonlinear stochastic, dynamics and control, pp. 147-55. Springer, Berlin (2011)
25. Wojtkiewicz, S.F., Johnson, E.A., Bergman, L.A., Grigoriu, M., Spencer, B.F.: Response of stochastic dynamical systems driven by additive Gaussian and Poisson white noise: solution of a forward generalized Kolmogorov equation by a spectral finite difference method. Comput. Methods Appl. Mech. Eng. 168(1), 73-9 (1999)
26. Zhu, H., Er, G., Iu, V., Kou, K.: Probabilistic solution of nonlinear oscillators excited by combined Gaussian and Poisson white noises. J. Sound Vib. 330(12), 2900-909 (2011)
27. Hanson, F.B.: Applied Stochastic Processes and Control for Jump–Diffusions: Modeling, Analysis, and Computation. SIAM, Philadelphia (2007)
28. ?ksendal, B., Sulem, A.: Applied Stochastic Control of Jump Diffusions. Springer-Verlag, Berlin (2005)
29. Jia, W.T., Zhu, W.Q., Xu, Y.: Stochastic averaging of quasi-non-integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations. Int. J. Non-linear Mech. 51, 45-3 (2013). doi:10.1016/j.ijnonlinmec.2012.12.003
30. Liu, W.Y., Zhu, W.Q., Xu, W.: Stochastic stability of quasi non-integrable Hamiltonian systems under parametric excitations of Gaussian and Poisson white noises. Probab. Eng. Mech. 32, 39-7 (2013)
31. Lin, Y.K.: Probabilistic Theory of Structural Dynamics. McGraw-Hill, New York (1967)
32. Di Paola, M., Vasta, M.: Stochastic integro-differential and differential equations of non-linear systems excited by parametric Poisson pulses. Int. J. Non-linear Mech. 32(5), 855-62 (1997)
33. Di Paola, M., Falsone, G.: It? and Stratonovich integrals for delta-correlated processes. Probab. Eng. Mech. 8(3), 197-08 (1993)
34. Dipaola, M., Falsone, G.: Stochastic dynamics of nonlinear-systems driven by non-normal delta-correlated processes. J. Appl. Mech. 60(1), 141-48 (1993). doi:10.1115/1.2900736
35. Xu, Y., Duan, J., Xu, W.: An averaging principle for stochastic dynamical systems with Lévy noise. Physica D 240(17), 1395-401 (2011)
36. Cai, G.Q., Lin, Y.K.: Response distribution of non-linear systems excited by non-Gaussian impulsive noise. Int. J. Non-linear Mech. 27(6), 955-67 (1992)
37. Wu, Y., Zhu, W.Q.: Stationary response of MDOF dissipated Hamiltonian systems to Poisson white noises. J. Appl. Mech. 75(4), 044502 (2008). doi:10.1115/1.2912987
38. Wu, Y., Zhu, W.Q.: Stationary response of multi-degree-of-freedom vibro-impact systems to Poisson white noises. Phys. Lett. A 372(5), 623-30 (2008)
39. Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics: Advanced Theory and Applications. McGraw-Hill, New York (1995) - 作者单位：Wantao Jia (1) (2)
Weiqiu Zhu (1) (3)
1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710129, China
2. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, China
3. Department of Mechanics, State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou, 310027, China - ISSN：1573-269X
文摘
A stochastic averaging method for predicting the response of quasi-integrable and non-resonant Hamiltonian systems to combined Gaussian and Poisson white noise excitations is proposed. First, the motion equations of a quasi-integrable and non-resonant Hamiltonian system subject to combined Gaussian and Poisson white noise excitations is transformed into stochastic integro-differential equations (SIDEs). Then $n$ -dimensional averaged SIDEs and generalized Fokker–Plank–Kolmogrov (GFPK) equations for the transition probability densities of $n$ action variables and $n$ - independent integrals of motion are derived by using stochastic jump–diffusion chain rule and stochastic averaging principle. The probability density of the stationary response is obtained by solving the averaged GFPK equation using the perturbation method. Finally, as an example, two coupled non-linear damping oscillators under both external and parametric excitations of combined Gaussian and Poisson white noises are worked out in detail to illustrate the application and validity of the proposed stochastic averaging method.