Study On Numerical Methods For Differential Algebraic Equations In Multibody Systems Dynamics

Multibody system dynamics plays an important role in aerospace,marine engineering,automotive,and robotics.Simulations of multibody dynamics are affected by many factors,including the integrators and dynamics modeling.Lie group integrators have great potential on the implementation of the software in future,due to its ability to get rid of singularities without introducing any redundant constraint and its capability to describe the large overall motion.The dynamic eqautions of rigid body and flexible body’s dynamics modeling in multibody systems can be expressed as differential algebraic equations,which can be solved by two appraches: the first one is to solve the monolithic dynamic equations straightforward,and the second one is the co-simulation technology proposed by scholars in recent years,also known as the multi-solver technology,which is to divide the monolithic system into several subsystems that are integrated sperately in each time step.The co-simulation technique has the potential to expore the computational ability of individual integrators in each subsystem.However,as a result of a large delay of data communication between subsystems,the co-simulation has not been widely used in multibody systems,and the co-simulation of coupled rigid-flexible multibody systems remains to be solved.This paper developed several implicit Lie group integrators,extended several explicit algorithms that solve ordinary differential equations to solve differential algebraic equations,explored the saturation phenomenon of angular velocities in the simulation of high-speed rotation dynamics,and proposed a new co-simulation algorithms for the three dimentional rigid-flexible coupling system.The specific research contents and results are as follows:The index-2\index-3 backward difference Lie group integrators are developed.The numerical research on the capability of these Lie group integrators into the non-stiff and stiff multibody systems(including elastic damping system and rigid-flexible coupling system)was performed.Numerical results show that similar to the generalized-α Lie group integrator,there are spikes on the curves of accelerations computed by index-3backward difference Lie group integrators.Moverover,compared to the generalized-α Lie group integrator,backward difference Lie group integrators can achieve a better computational efficiency due to their adaptive step-sizes and adaptive orders.In stiff problems,index-3 backward difference integrator can achieve order from 2 and 3,and index-2 backward difference integrator can achieve order from 3 and 5,which can achieve a high efficiency.To apply explicit integrators to solve differential-algebric equations,two new methods are proposed to convert the differential-algebraic equations into first-order ordinary differential equations.One of the methods is to introduce a stabilized parameter to tranform the constraint equations and hidden velocity constraint equations into the ordinary differential equations,and combine with dynamical equations to get a set of ordinary differential equations,where the constraint drifting can be effectively confined by adjusting the stabilized parameter.The other method is to use a continuation projection method to select a set of independent generalized coordinates,which resolves the discontinuity of independent generalized velocities that are selected by the traditional method of selecting arbitrary independent generalized coordinates.The ordinary differential equations transformed by the first method are suitable for the explicit Runge-Kutta-Chebyshev algorithm and the extrapolated stabilized Runge-Kutta algorithm,which are introduced into multibody system dynamics and further developed into explicit Lie group integrators.Numerical research shows that in the simulation of non-stiff and stiff multibody systems,the Runge-Kutta-Chebyshev algorithm and its corresponding Lie group integrators can achieve better computational efficiency as well as limit the constraint drifting effectively.To solve the angular velocity saturation phenomenon in high-speed rotations,the dynamics is computed by Lie group integrators,and the numerical research found that Lie group integrators can solve the angular velocity saturation phenomenon in unconstrained high-speed rigid-body systems.In constrained systems,this saturation phenomenon can be resolved if the differential-algebric equations are converted to the ordinary differential equations with respect to angular velocities.Meanwhile,a new method of introducing the non-linear geometrical elastic forces of absolute nodal coordinate formulation into the floating frame of reference method is proposed to resolve the saturation problem in the high-speed rotation of flexible dynamics.A co-simulation algorithm based on the waveform relaxation method was proposed,where the differential-algebric equations are transformed into first-order ordinary differential equations,and the monolithic system is divided into two subsystems with fast and slow variables according to the characteristic of the dynamical system,in which each subsystem is solved seperately and information is exchanged on discrete communication time nodes.This process is iterated repeatedly until the convergence condition is achieved.Numerical studies found that in the planar conservative system,this co-simulation method has less energy dissipation compared to the commonly-used co-simulation methods,and it is suitable for the co-simulation of the three dimensional rigid-flexible multibody systems.To implement Lie group algorithms developed in the thesis,the C++ object-oriented programming,distributed-component modeling method,and Marker technology are adopted to design a computational multibody system program.With the file of a given format about the mass,constraints,external forces and other modeling information of multibody systems,all state quantities can be calculated by running and reading the text file.The framework can realize the automatic integration of constraint equations and the automatic generation of differential-algebraic equations,which has some generalities in the program design.