一种改进的基于Jacobi椭圆函数的随机平均法
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摘要
建立了改进的基于Jacobi椭圆函数的随机平均法,用于预测有界噪声激励作用下硬弹簧和软弹簧系统的随机响应。通过引入基于Jacobi椭圆函数的变换,导出关于响应幅值和激励与响应之间相位差的随机微分方程,应用随机平均原理,将响应幅值近似为一个Markov扩散过程,建立其平均的It随机微分方程。响应幅值的稳态概率密度由相应的简化Fokker-Planck-Kolmogorov方程解出;进而得到系统位移和速度的稳态概率密度。以Duffing-Van der Pol振子为例,研究了硬刚度及软刚度情形下的随机响应,通过与Monte Carlo数值模拟结果比较证实了此方法的可行性及精度。由于广义调和函数是基于线性系统的精确解,Jacobi椭圆函数是基于非线性系统的精确解,研究结果表明基于Jacobi椭圆函数的随机平均法得到的结果与Monte Carlo模拟方法更接近。因此与基于广义调和函数的随机平均相比,基于Jacobi椭圆函数更加精确,因为它是基于保守的非线性系统。
A novel stochastic averaging technique is proposed to evaluate the random responses of nonlinear systems with cubic stiffness to bounded noises.By introducing a transformation based on the Jacobian elliptic functions,the stochastic differential equations with respect to the system amplitude and the phase difference between the imposed excitation and the system response are derived.Applying the stochastic averaging principle yields the associated Itstochastic differential equations.Then,the stationary joint probability density of the amplitude and the phase difference is obtained by solving the corresponding FokkerPlanck-Kolmogorov equation.Numerical results for a representative example with hardening and softening stiffness are given to verify the feasibility and accuracy of the proposed procedure.Compared to the stochastic averaging method based on generalized harmonic functions,the present procedure is of higher accuracy as it is based on the exact solution of the associated conservative nonlinear system.
A novel stochastic averaging technique is proposed to evaluate the random responses of nonlinear systems with cubic stiffness to bounded noises.By introducing a transformation based on the Jacobian elliptic functions,the stochastic differential equations with respect to the system amplitude and the phase difference between the imposed excitation and the system response are derived.Applying the stochastic averaging principle yields the associated Itstochastic differential equations.Then,the stationary joint probability density of the amplitude and the phase difference is obtained by solving the corresponding FokkerPlanck-Kolmogorov equation.Numerical results for a representative example with hardening and softening stiffness are given to verify the feasibility and accuracy of the proposed procedure.Compared to the stochastic averaging method based on generalized harmonic functions,the present procedure is of higher accuracy as it is based on the exact solution of the associated conservative nonlinear system.
引文
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[9] Zhu W Q,Yang Y Q.Stochastic averaging of quasinonintegrable-Hamiltonian systems[J].Journal of Applied Mechanis,ASME,1997,64:157-164.
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[11]Zhu W Q,Huang Z L,Suzuki Y.Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems[J].International Journal of NonLinear Mechanics,2002,37:419-437.
[12]Huang Z L,Zhu W Q,Suzuki Y.Stochastic averaging of strongly non-linear oscillators under combined harmonic and white noise excitations[J].Journal of Sound and Vibration,2000,238:233-256.
[13]Tien W M,Namachchivaya N S,Coppola V T.Stochastic averaging using elliptic functions to study nonlinear stochastic systems[J].Nonlinear Dynamics,1993,4:373-387.
[14]Huang Z L,Zhu W Q,Ni Y Q,et al.Stochastic averaging of strongly nonlinear oscillators under bonded noise excitations[J].Journal of Sound and Vibration,2002,254:245-67.
[15]Coppola V T,Rand R H.Averaging using elliptic functions:Approximation of limit cycles[J].Acta Mechanica,1990,81:125-142.
[16]Coppola V T.Averaging of strongly nonlinear oscillators using elliptic functions[D].Ithace,NY:Cornell University,1989.
[17]Okabe T,Kondou T,Ohnishi J.Elliptic averaging methods using the sum of Jacobian elliptic delta and zeta functions as the generating solution[J].International Journal of Non-Linear Mechanics,2011,46:159-169.
[18]Rakaric Z,Kovacic I.An elliptic averaging method for harmonically excited oscillators with a purely non-linear non-negative real-power restoring force[J].Communications in Nonlinear Science and Numerical Simulation,2013,18:1888-1901.
[19]Okabe T,Kondou T.Improvement to the averaging method using the Jacobian elliptic function[J].Journal of Sound and Vibration,2009,320:339-364.
[20]Holmes P J.A nonlinear oscillator with a strange attractor[J].Philosophical Transactions of the Royal Society London,Series A-Mathematical and Physical Sciences,1979,292(1394):419-448.
[21]Lin Y K,Cai G Q.Probabilistic Structural Dynamics:Advanced Theory and Applications[M].New York:McGraw-Hill,1995.
[22]Anderson T J,Nayfeh A H,Balachandran B.Experimental verification of the importance of the nonlinear curvature in the response of a cantilever beam[J].Journal of Vibration Acoustics,ASME,1996,118:21-27.
[23]Feng Z H,Lan X J,Zhu X D.Principal parametric resonances of a slender cantilever beam subject to axial narrow-band random excitation of its base[J].International Journal of Non-Linear Mechanics,2007,42:1170-1185.
[24]方同.工程随机振动[M].北京:国防工业出版社,1995.
[25]金肖玲.多自由度强非线性随机系统的响应与稳定性研究[D].杭州:浙江大学,2009.Jin Xiaoling.Response and Stability of multi-degreeof-freedom strongly nonlinear stochastic systems[D].Hangzhou:Zhejiang University,2009.
[2] Khasminskii R Z.On the averaging principle for stochastic differential equations[J].Kibernetika,1968,4:260-299.
[3] Stratornovich R L.Topics in the Theory of Random Noise[M].New York:Gordon Breach,1967.
[4] Roberts J B.The energy envelope of a randomly excited nonlinear oscillator[J].Journal of Sound and Vib ration,1978,60:177-185.
[5] Roberts J B,Spanos P D.Stochastic averaging:An approximate method of solving random vibration problems[J].Internationd Journal of Non-Linear Mechanics,1986,21:111-134.
[6] Zhu W Q,Lin Y K.Stochastic averaging of energy envelope[J].Journal of Engineering Mechanics,1991,117:1890-1905.
[7] Wang Y,Ying Z G,Zhu W Q.Stochastic averaging of energy envelope of Preisach hysteretic systems[J].Journal of Sound and Vibration,2009,321:976-993.
[8] Zhu W Q.Nonlinear stochastic dynamics and control in Hamiltonian formulation[J].Applied Mechanics Reviews,2006,59:230-248.
[9] Zhu W Q,Yang Y Q.Stochastic averaging of quasinonintegrable-Hamiltonian systems[J].Journal of Applied Mechanis,ASME,1997,64:157-164.
[10]Zhu W Q,Huang Z L,Yang Y Q.Stochastic averaging of quasi-integrable Hamiltonian systems[J].Journal of Applied Mechanics,ASME,1997,64:975-984.
[11]Zhu W Q,Huang Z L,Suzuki Y.Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems[J].International Journal of NonLinear Mechanics,2002,37:419-437.
[12]Huang Z L,Zhu W Q,Suzuki Y.Stochastic averaging of strongly non-linear oscillators under combined harmonic and white noise excitations[J].Journal of Sound and Vibration,2000,238:233-256.
[13]Tien W M,Namachchivaya N S,Coppola V T.Stochastic averaging using elliptic functions to study nonlinear stochastic systems[J].Nonlinear Dynamics,1993,4:373-387.
[14]Huang Z L,Zhu W Q,Ni Y Q,et al.Stochastic averaging of strongly nonlinear oscillators under bonded noise excitations[J].Journal of Sound and Vibration,2002,254:245-67.
[15]Coppola V T,Rand R H.Averaging using elliptic functions:Approximation of limit cycles[J].Acta Mechanica,1990,81:125-142.
[16]Coppola V T.Averaging of strongly nonlinear oscillators using elliptic functions[D].Ithace,NY:Cornell University,1989.
[17]Okabe T,Kondou T,Ohnishi J.Elliptic averaging methods using the sum of Jacobian elliptic delta and zeta functions as the generating solution[J].International Journal of Non-Linear Mechanics,2011,46:159-169.
[18]Rakaric Z,Kovacic I.An elliptic averaging method for harmonically excited oscillators with a purely non-linear non-negative real-power restoring force[J].Communications in Nonlinear Science and Numerical Simulation,2013,18:1888-1901.
[19]Okabe T,Kondou T.Improvement to the averaging method using the Jacobian elliptic function[J].Journal of Sound and Vibration,2009,320:339-364.
[20]Holmes P J.A nonlinear oscillator with a strange attractor[J].Philosophical Transactions of the Royal Society London,Series A-Mathematical and Physical Sciences,1979,292(1394):419-448.
[21]Lin Y K,Cai G Q.Probabilistic Structural Dynamics:Advanced Theory and Applications[M].New York:McGraw-Hill,1995.
[22]Anderson T J,Nayfeh A H,Balachandran B.Experimental verification of the importance of the nonlinear curvature in the response of a cantilever beam[J].Journal of Vibration Acoustics,ASME,1996,118:21-27.
[23]Feng Z H,Lan X J,Zhu X D.Principal parametric resonances of a slender cantilever beam subject to axial narrow-band random excitation of its base[J].International Journal of Non-Linear Mechanics,2007,42:1170-1185.
[24]方同.工程随机振动[M].北京:国防工业出版社,1995.
[25]金肖玲.多自由度强非线性随机系统的响应与稳定性研究[D].杭州:浙江大学,2009.Jin Xiaoling.Response and Stability of multi-degreeof-freedom strongly nonlinear stochastic systems[D].Hangzhou:Zhejiang University,2009.