Dynamic Simulation Of Multi-body System Based On Lie Group

With the increasing complexity of mechanical engineering development,it is necessary to design more efficient and stable numerical algorithms to meet the numerical simulation requirements of multibody system dynamics.The Lie group method characterizes the rotational degrees of freedom for each body by a special orthogonal group to avoid these singularities caused by parameterized expression with inherent Lie group structure.In this paper,the Lie group method is used to establish the model of multibody system dynamics,and expressed as the index 1,2,and 3 differential-algebraic equations.On this basis,a constrained stability method is proposed,which introduces the new Lagrange multipliers to construct a stable index 1 differential-algebraic equations by reducing the number of index.It consists of a modified equation of velocity and acceleration,a differential equation of motion,and a constraint equation of all levels.All constraints are kept at the same time to ensure the stability of the numerical results.The Lie group Runge-Kutta method and the Lie group generalized-? method are constructed to solve the Lie group DAEs,and then the Lie group multistep block method and the Lie group discrete variation method are designed to solve these DAEs.The Lie group multistep block format is proposed for the Lie group model.First,the last item of the Taylor expansion of each variable retained is changed to an unknown parameter.Basing on the definition of Runge-Kutta stability,the Lie group multistep block format can be obtained.As a new general linear method,the Lie group multistep block method,which has simple format and easy to understand,has Runge-Kutta stability,which is A-stable and L-stable.Based on the variational principle in discrete mechanics,the Lie group discrete variational method with constraints can be constructed,adding the Jacobian matrix of the constraint equation and Lagrange multiplier to the minimum of action integral,discretizing the minimum of action integral and then making variation of each variable to obtain the discrete Euler-Lagrange equations with constraints.Because the Lie group discrete variation method is used to solve the equations of each time interval,the errors of the previous step will not affect the solutions of next step,thus avoiding the error accumulation problem of other numerical methods.Therefore,in theory,the Lie group discrete variation method can preserves the energy and the Lie group structure.