Dynamic Equation of Constrained Mechanical System

This paper modifies an explicit dynamic equation of constrained mechanical system. Kinematic position of the system is defined by generalized coordinates, which are imposed on constraints. The equations of motion in the form of the Lagrange equations with undetermined multipliers are constructed based on d’Alambert-Lagrange’s principle. Dynamic equations are presented to the mind, resolved relative accelerations. Expressions for the undetermined multipliers are defined by considering the possible deviations from the constraints equations. For constraints stabilization additional variables used to estimate the deviations caused by errors in the initial conditions and the use of numerical methods. For approximation of ordinary differential equations solution, in particular, the nonlinear equations of first order, use explicit numerical methods. Linear equations of the constraints perturbation are constructed. The matrix of the coefficients of these equations is selected in the process of the dynamic equations numerical solution. Stability with respect to initial deviations from the constraints equations and stabilization of the numerical solution depend on the values of the elements of this matrix. As a result values for the matrix of coefficients corresponding to the solution of the dynamics equations by the method of Euler and fourth order Runge-Kutta method are defined. Suggested method for solving the problem of stabilization is used for modeling of the disk motion on a plane without slipping.

unconstrained system, holonomic constraints, nonholonomic constraints, stabilization, Taylor series, numerical solution