Principal Inertia Representation And Its Application In Simulation Of Rigid-Body System

The kinematics and dynamics of constrained multi-body systems constitute an important part of what is referred to as CAD(Computer Aided Design)and CAE(Computer Aided Engineering).Due to the constraints imposed,the dynamic equations of constrained system are available in a variety of formulations.These different mathematical formulations present some different features both in theory and numerical computation.During last decades,a large amount of effort has been devoted to various theoretical representations of constrained system as well as the related studies of numerical simulation.The influences of accuracy of different representations,however,have not caused intensive investigations in previous works about numerical simulation.In the paper,various kinds of formulations of mass matrix for constrained multi-body dynamic equations as well as their numerical influences have been investigated detailedly,and the major research of this paper can be summarized as the following three parts:(1)The augmented formulations of mass matrix of constrained multi-body system have been specifically studied,and a special representation of mass matrix termed as the principal representation of inertia,has been proposed by the definition of generalized angular velocity.In the standard form of the principal inertia representation,the mass matrix is split into two parts:a constant matrix and a displacement-dependent matrix,of which the proportion is controlled by a scaling parameter,denoted as oa and the scaling parameter o can be determined arbitrarily.Then the generalization of the principal inertia representation is guaranteed for general constrained systems through establishing the mathematical equivalence between the augmented formulation of mass matrix and the principal representation of inertia.(2)In the framework of the principal representation of inertia,the discretization error of kinetic energy has been derived in detail for constrained dynamic system.Error estimation demonstrates that the discretization error of kinetic energy is linear to the scaling parameter in Lagrange'^ frame;the discretization error of kinetic energy is linear to the reciprocal of the scaling parameter in Hamilton's frame.And while the mass matrix can be considered as a function of generalized displacements,the slope of the error function is of the same order as the discretization error.In contrary,the magnitude of discretization error of kinetic energy is independent with the specific form in the augmented formulation of mass matrix.Hence,the principal representation of inertia is not equivalent to the augmented formulation of mass matrix in numerical significance.On basis of the error estimation,the arithmetic and harmonic means of the principal value of inertia are further recommended as reasonable preconditioning values of the scaling parameter a to receive small discretization errors.(3)On basis of the principal representation of inertia,a new approach by determing the(optimal)preconditioning value of the scaling parameter is developed for constrained systems to improve the numerical accuracy of integrations,and is applied for the numerical simulation of two'-dimensional arnd three dimensional rigid bodies.Numerical results verify the validity of the conclusions of error analysis,and the accuracies of numerical integrations are improved impressively.In the example of three-dimensional rigid body rotation in terms of convected base vectors or unit quaternion,the numerical accuracy of integrations whose scaling parameter is given as the arithmetic mean of principal value of inertia,presents about one order of magnitude higher than that of the original schemes.Under these researches,the principal representation of inertia considered as a specific formulation for the mass matrix of constrained dynamic system,provides an extra dimension to improve the accuracy of numerical integrations,and its theoretical and numerical studies need to be urgently developed in the future.