Abstract:Here, the optimal control problem for first-passage reliability of a class of randomly excited multi-DOF nonlinear quasi-integrable Hamiltonian systems with internal resonance was investigated.Based on the stochastic averaging method and dynamic programming principle, the It stochastic differential equations of the optimally controlled system were obtained.Then the backward Kolmogorov equation governing conditional reliability function and the Pontryagin equation governing the mean first-passage time were established.An illustrative numerical example was given.The validity of the theoretical method was verified by Monte Carlo digital simulation.
齐玉明,吴勇军. 一类内共振非线性随机振动系统的可靠性控制[J]. 振动与冲击, 2019, 38(3): 102-107.
QI Yuming, WU Yongjun. Reliability control for a class of nonlinear random vibration systems with internal resonance. JOURNAL OF VIBRATION AND SHOCK, 2019, 38(3): 102-107.
[1] Cox D R, Miller H D. The Theory of Stochastic Processes [M]. Chapman and Hall, New York, 1965.
[2] ZHU Wei-qiu. Nonlinear stochastic dynamics and control in Hamiltonian formulation [J]. Applied Mechanics Reviews, 2006, 59: 230-248.
[3] 朱位秋. 非线性随机动力学与控制-Hamiltonian理论体系框架[M],科学出版社,北京,2003.
ZHU Wei-qiu. Nonlinear Stochastic Dynamics and Control-Hamiltonian Theoretical Framework [M]. Science Press, Beijing, 2003.
[4] Bellman R. Dynamic Programming [M]. Princeton University Press, Princeton, 1957.
[5] Huang C S, Wang S, Teo K L. Solving Hamilton-Jacobi-Bellman equations by a modified method of characteristics [J]. Nonlinear Analysis, Theory, Methods and Applications, 2000, 40(1): 279–293.
[6] Besselink B, Tabak U, Lutowska A, et al. A comparison of model reduction techniques from structural dynamics, numerical mathematics and systems and control [J]. Journal of Sound and Vibration, 2013, 332: 4403-4422.
[7] Roberts J B, Spanos P D. Stochastic averaging: an approximate method of solving random vibration problems [J]. International Journal of Non-Linear Mechanics, 1986, 21: 111-134.
[8] Deng M L, Zhu W Q. Feedback minimization of first-passage failure of quasi integrable Hamiltonian systems [J]. Acta Mechanica Sinica, 2007, 23(4): 437-444.
[9] Zhu W Q, Huang Z L, Deng M L. First-passage failure and its feedback minimization of quasi-partially integrable Hamiltonian systems [J]. International Journal of Non-Linear Mechanics, 2003, 38(8): 1133-1148.
[10] Wu Y J, Huan R H. First-passage failure minimization of stochastic Duffing-Rayleigh-Mathieu system [J], Mechanics Research Communications, 2008, 35(7): 447-453.
[11] Cheung Y K, Xu Z. Internal resonance of strongly non-linear autonomous vibrating systems with many degrees of freedom [J]. Journal of Sound and Vibration, 1995, 180(2): 229-238.