Investigations On The Galerkin Variational Integrators Of Multibody System Dynamics

The dynamical models of multibody systems are usually differential/algebraic equations, the numerical methods of them are of great importance in the areas of multibody system dynamics. A lot of achievements have been made over the past 30 years by applied mathematicians and dynamists on investigations of stability,efficiency and accuracy of their computational methods, but all these methods were based on concepts of local linear analysis for numerical stabilization, and consistencies with their counterpart continuous models have not considered, thus not applicable to long term simulations with larger time steps. These drawbacks can be overcome by using contemporary geometric integrators.Geometric integrators are the numerical methods which can inherit the intrinsic invariants of the original continuous models, these invariants are energy, momentum,sympleticity, Lie group structure, etc.. But each traditional geometric integrator usually preserves a specific invariant which can be improved in stabilization by preserving multiple invariants simultaneously. The variational integrator is an excellent example of geometric integrators with capacity of multiple invariants such as symplecticity,momentum and energy of conservative systems, based on which Galerkin variational integrator can provide a unified platform for higher order algorithms, but its application was limited to general conservative dynamical systems without constraints,let alone general multibody systems. The investigations of this thesis focus on higher order Galerkin variational integrators of multibody system dynamics, with contributions:1) A systematic survey is conducted on mathematical models of multibody system dynamics, traditional numerical methods of differential/algebraic equations and contemporary geometric integrators for dynamical systems, inspiring the investigationson Galerkin integrators for multibody system dynamics.2) The Galerkin variational integrators for conservative dynamical systems without constraints are designed firstly based on Lagrange interpolation polynomials combining Gauss, Radau and Lobatto quadrature formulas respectively, which are then extended to systems with holonomic and nonholonomic constraints.3) The Galerkin variational integrators are designed for rigid body dynamics using relative coordinates, Euler angles, Euler parameters and director vectors respectively, which are fundamentals of corresponding integrators for multibody systems.4) The mathematical models of rigid multibody system dynamics are derived systematically via introducing concepts of basic constraints for construction of constraint library, along with their Galerkin variational integrators.The main innovations are:1) The higher order Galerkin variational integrators are designed for the general dynamical systems under different constraints and forces, which is different from the previous investigations on systems with holonomic constraints and conservative forces.2) The higher order Galerkin variational integrators for a constrained rigid body dynamics described by different coordinates are of importance for extensions to dynamics of multibody systems in different coordinates. It is different from the previous researches on dynamics of a particle or a free rotational rigid body.3) Based on the properties of director vector coordinate modeling method of multibody system dynamics, we propose a systematic method for construction of constraint library and generalized force library. Then, the higher order Galerkin variational integrators for rigid multibody system dynamics with director vector coordinates are designed. The similar works have not been found from the literatures up till now.The explorations in this thesis on higher order Galerkin variational integrators for multibody system dynamics provide systematic framework for other continuous models of multibody system dynamics, and can be extended to investigations on flexible multibody system dynamics, and design of optimal control, optimization based on multibody systems.