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While the kinematic and dynamic modeling of mechanical systems is well developed, the numerical solution of the governing equations of many of these models is still a large field under investigation. Since in any constrained mechanical system, the connections of the connected bodies restrict their relative movement, therefore the governing equations of the model usually form a system of differential-algebraic equations that includes both differential and algebraic equations. It should be noted that, unlike the analytical and numerical solution of differential equations, the analytical and numerical behavior of differential-algebraic equations is more complex and completely different from differential equations. In this paper, a high accuracy method for solving the system of differential-algebraic equations arising in mechanical systems is presented. The presented method is based on the use of a pseudospectral method that will transform the solution of the system of differential-algebraic equations into the solution of a system of algebraic equations. Then, an optimization technique is utilized to facilitate solving the obtained system of algebraic equations. At the end, some numerical experiments are performed on several benchmark problems to show the accuracy and applicability of the method. [ABSTRACT FROM AUTHOR]