多体系统动力学Lie群微分-代数方程约束稳定方法
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摘要
针对多体系统动力学微分-代数方程求解问题,研究基于Lie群表达的约束稳定方法.首先引入新的Lagrange乘子,结合位移约束、速度级约束和加速度级约束方程,构造了新的Lie群微分-代数方程.然后使用向后差商隐式方法和CG(Crouch-Grossman)方法,对微分–代数方程进行离散求解,得到精确度较高的动力学仿真结果.该方法在精确保持各级约束方程的同时,保持旋转矩阵的正交性,并且使系统总能量误差较小.
Constraints stabilization method on Lie group is studied for different-algebraic equations( DAEs) of multibody system dynamics. Considering the displacement constraints,the velocity constraints and the acceleration constraints equations,the new Lagrange multipliers are used to construct the new DAEs on Lie groups.Then,using the ward difference method and CG( Crouch-Grossman) method,high accuracy simulation results is obtained by solving the constraints stabilization DAEs. The constraints stabilization method can accurately maintain the constraint equation at all levels,and can maintain the orthogonality of rotation matrix. Meanwhile,the total energy error of the system is smaller.
Constraints stabilization method on Lie group is studied for different-algebraic equations( DAEs) of multibody system dynamics. Considering the displacement constraints,the velocity constraints and the acceleration constraints equations,the new Lagrange multipliers are used to construct the new DAEs on Lie groups.Then,using the ward difference method and CG( Crouch-Grossman) method,high accuracy simulation results is obtained by solving the constraints stabilization DAEs. The constraints stabilization method can accurately maintain the constraint equation at all levels,and can maintain the orthogonality of rotation matrix. Meanwhile,the total energy error of the system is smaller.
引文
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9 Iserles A,Munthe-Kaas H Z,Nrsett S,et al.Lie-group methods.Acta Numerica,2016,9(2):215~365
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13 Munthe-Kaas H.High order Runge-Kutta methods on manifolds.Applied Numerical Mathematics,1999,29(1):115~127
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15 Terze Z,Müller A,Zlatar D.Lie-group integration method for constrained multibody systems in state space.Multibody System Dynamics,2015,34(3):275~305
2 Fon-Llagunes J M.Multibody dynamics-computational methods and applications.Switzerland:Springer,2016
3 Betsch P.Structure-preserving integrators in nonlinear structural dynamics and flexible multibody dynamics.Switzerland:Springer,2016
4 Jay L O.Symplectic partitioned Runge-Kutta methods for constrained Hamiltonian systems.SIAM Journal on Numerical Analysis,1996,33(1):368~387
5 Lens E,Cardona A.An energy preserving/decaying scheme for nonlinearly constrained multibody systems.Multibody System Dynamics,2007,18(3):435~470
6 Betsch P,Uhlar S.Energy-momentum conserving integration of multibody dynamics.Multibody System Dynamics,2007,17(4):243~289
7 Marsden J E,West M.Discrete mechanics and variational integrators.Acta Numerica,2003,10(1):357~514
8 Ding J Y,Pan Z K.Higher order variational integrators for multibody system dynamics with constraints.Advances in Mechanical Engineering,2014,6(1):383680-1-8
9 Iserles A,Munthe-Kaas H Z,Nrsett S,et al.Lie-group methods.Acta Numerica,2016,9(2):215~365
10 Simo J C,Vu-Quoc L.On the dynamics in space of rods undergoing large motions-A geometrically exact approach.Computer Methods in Applied Mechanics and Engineering,1988,66(2):125~161
11 Lewis D,Simo J C.Conserving algorithms for the dynamics of Hamiltonian systems on Lie groups.Journal of Nonlinear Science,1994,4(1):253~299
12 Crouch P E,Grossman R.Numerical integration of ordinary differential equations on manifolds.Journal of Nonlinear Science,1993,3(1):1~33
13 Munthe-Kaas H.High order Runge-Kutta methods on manifolds.Applied Numerical Mathematics,1999,29(1):115~127
14 Arnold M,Brüls O,Cardona A.Error analysis of generalized-αLie group time integration methods for constrained mechanical systems.Numerische Mathematik,2015,129(1):149~179
15 Terze Z,Müller A,Zlatar D.Lie-group integration method for constrained multibody systems in state space.Multibody System Dynamics,2015,34(3):275~305