Automatic generation and numerical integration of differential-algebraic equations of multibody dynamics

Abstract

Abstract An algorithm for computing the matrix and vector functions in the numerical integration method of constrained equations of motion in multibody mechanical system dynamics is presented. Cartesian coordinates and Euler parameters have been used as the state variables to construct the mathematical model for a spatial multibody mechanical system. Four basic constraints, which describe the orthogonality and the parallelism of pairs of vectors, have also been written and will serve as building blocks of the kinematic joint library. Derivatives of these basic constraints are then derived by applying differential operators to the multivariable and vector-valued constraint functions. Verification is done by comparing the numerical solution with results computed by a symbolic computation software. In multibody dynamics, the acceleration of constrained equations of motion is a function of the state variables. Using the differentials of joint library functions and generalized force functions, the Jacobian of the acceleration with respect to the state variables is obtained. The efficiency and accuracy of computing this analytic Jacobian matrix are demonstrated by comparing execution time and numerical results with those obtained by applying a first order linearization method. The proposed computational technique is also suitable for various formulations of the equations of motion, for instance, the Euler angles instead of the Euler parameters can be used for the orientation variables. The final goal is to construct a standard library of joint constraints and force expressions for automatic generation of constrained equations of motion and to enhance the numerical integration methods for multibody dynamics.