多体系统动力学微分-代数方程L-稳定方法
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摘要
针对多体系统动力学微分-代数方程形式,在时间区间上构造L-稳定方法,分别基于等距节点、Chebyshev节点和Legendre节点等非等距节点建立求解格式,依据Ehle定理及猜想,与Padé逼近式对比得到待定矩阵和向量,从而获得L-稳定求解公式,循环求解过程采用Newton迭代法计算.以平面双连杆机械臂系统为例,使用L-稳定方法进行数值仿真,通过改变时间区间节点数和步长对各个指标结果进行比较,并与经典Runge-Kutta法对比.结果表明,该方法具有稳定性好、精度高等优点,适用于长时间情况下的多体系统动力学仿真.
An L-stable method over time intervals for differential-algebraic equations of multibody system dynamics was presented. The solution scheme was established based on equidistant nodes and non-equidistant nodes such as Chebyshev and Legendre nodes. According to Ehle's theorem and conjecture, the unknown matrix and vector in the L-stable solution formula were obtained through comparison with the Padé approximation. The Newtonian iteration method was used during the solution process. The planar 2-link manipulator system was taken as an example, and the results from the L-stable method were compared for different node numbers in the time interval and different steps in the simulation, with those from the classic Runge-Kutta method. The comparison shows that, the proposed method has the advantages of good stability and high precision, and is suitable for multibody system dynamics simulation under long-term conditions.
An L-stable method over time intervals for differential-algebraic equations of multibody system dynamics was presented. The solution scheme was established based on equidistant nodes and non-equidistant nodes such as Chebyshev and Legendre nodes. According to Ehle's theorem and conjecture, the unknown matrix and vector in the L-stable solution formula were obtained through comparison with the Padé approximation. The Newtonian iteration method was used during the solution process. The planar 2-link manipulator system was taken as an example, and the results from the L-stable method were compared for different node numbers in the time interval and different steps in the simulation, with those from the classic Runge-Kutta method. The comparison shows that, the proposed method has the advantages of good stability and high precision, and is suitable for multibody system dynamics simulation under long-term conditions.
引文
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[14] 徐方暖,王博,邓子辰,等.基于四元数方法的绳系机器人姿态控制[J].应用数学和力学,2017,38(12):1309-1318.(XU Fangnuan,WANG Bo,DENG Zichen,et al.Attitude control of targets captured by tethered space robots based on the quaternion theory[J].Applied Mathematics and Mechanics,2017,38(12):1309-1318.(in Chinese))
[15] 文立平,杨春花,文海洋.非线性泛函积分微分方程多步Runge-Kutta方法的稳定性和渐近稳定性[J].湘潭大学自然科学学报,2018,40(1):1-5.(WEN Liping,YANG Chunhua,WEN Haiyang.Stability and asymptotic stability of multistep Runge-Kutta methods for nonlinear functional-integro-differential equations[J].Natural Science Journal of Xiangtan University,2018,40(1):1-5.(in Chinese))
[16] SHAMPINE L F,WATTS H A.A-stable block implicit one-step methods[J].BIT Numerical Mathematics,1972,12(2):252-266.
[17] 袁兆鼎,费景高,刘德贵.刚性常微分方程初值问题的数值解法[M].北京:科学出版社,2016.(YUAN Zhaoding,FEI Jinggao,LIU Degui.Numerical Solution of Initial Value Problems for Stiff Ordinary Differential Equations[M].Beijing:Science Press,2016.(in Chinese))
[2] WIDLUND O B.A note on unconditionally stable linear multistep methods[J].BIT Numerical Mathematics,1967,7(1):65-70.
[3] GEAR C W.The Automatic Integration of Stiff Ordinary Differential Equations[M].Amsterdam:North Holland Publishing Company,1963.
[4] DAHLQUIST G.Error Analysis for a Class of Methods for Stiff Non-Linear Initial Value Problems[M].Berlin:Springer-Verlag,1975.
[5] BUTCHER J C.A stability property of implicity Runge-Kutta methods[J].BIT Numerical Mathematics,1975,15(4):358-361.
[6] BURRAGE K,BUTCHER J C.Stability criteria for implicit Runge-Kutta methods[J].SIAM Journal on Numerical Analysis,1979,16(1):46-57.
[7] LI S F.Nonlinear stability of general linear methods[J].Journal of Computational Mathematics,1991,9(2):97-104.
[8] EHLE B L.A-stable methods and Pade approximations to the exponential[J].SIAM Journal on Mathematical Analysis,1973,4(4):671-680.
[9] HAIRER E,WANNER G.Solving Ordinary Differential Equations Ⅱ:Stiff and Differential-Algebraic Problems[M].2nd ed.Beijing:Science Press,2006.
[10] 邓子辰,李庆军.精细指数积分法在卫星编队飞行动力学中的应用[J].北京大学学报(自然科学版),2016,52(4):669-675.(DENG Zichen,LI Qingjun.Precise exponential integrator and its application in dynamics of spacecraft formation flying[J].Acta Scientiarum Naturalium Universitatis Pekinensis,2016,52(4):669-675.(in Chinese))
[11] 彭海军,李飞,高强,等.多体系统轨迹跟踪的瞬时最优控制保辛方法[J].力学学报,2016,48(4):784-791.(PENG Haijun,LI Fei,GAO Qiang,et al.Symplectic method for instantaneous optimal control of multibody system trajectory tracking[J].Chinese Journal of Theoretical and Applied Mechanics,2016,48(4):784-791.(in Chinese))
[12] 阚子云,彭海军,陈飙松,等.开放式多体系统动力学仿真算法软件研发(Ⅱ):DAEs求解算法对比[J].计算力学学报,2015,32(6):707-715.(KAN Ziyun,PENG Haijun,CHEN Biaosong,et al.Study of open simulation algorithm software for multibody system dynamics (Ⅱ):comparison of algorithms for solving DAEs[J].Chinese Journal of Computational Mechanics,2015,32(6):707-715.(in Chinese))
[13] 丁洁玉,潘振宽.多体系统动力学微分-代数方程广义-α投影法[J].工程力学,2013,30(4):380-384.(DING Jieyu,PAN Zhenkuan.Generalized-α projection method for differential-algebraic equations of multibody dynamics[J].Engineering Mechanics,2013,30(4):380-384.(in Chinese))
[14] 徐方暖,王博,邓子辰,等.基于四元数方法的绳系机器人姿态控制[J].应用数学和力学,2017,38(12):1309-1318.(XU Fangnuan,WANG Bo,DENG Zichen,et al.Attitude control of targets captured by tethered space robots based on the quaternion theory[J].Applied Mathematics and Mechanics,2017,38(12):1309-1318.(in Chinese))
[15] 文立平,杨春花,文海洋.非线性泛函积分微分方程多步Runge-Kutta方法的稳定性和渐近稳定性[J].湘潭大学自然科学学报,2018,40(1):1-5.(WEN Liping,YANG Chunhua,WEN Haiyang.Stability and asymptotic stability of multistep Runge-Kutta methods for nonlinear functional-integro-differential equations[J].Natural Science Journal of Xiangtan University,2018,40(1):1-5.(in Chinese))
[16] SHAMPINE L F,WATTS H A.A-stable block implicit one-step methods[J].BIT Numerical Mathematics,1972,12(2):252-266.
[17] 袁兆鼎,费景高,刘德贵.刚性常微分方程初值问题的数值解法[M].北京:科学出版社,2016.(YUAN Zhaoding,FEI Jinggao,LIU Degui.Numerical Solution of Initial Value Problems for Stiff Ordinary Differential Equations[M].Beijing:Science Press,2016.(in Chinese))
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