The present paper is concerned with the numerical solution of stiff differential-algebraic index 3 multibody dynamical systems resulting in the semi-explicit forms employing explicit time integration operators. A novel explicit primal-dual technique is proposed here to overcome the problems such as instabilities, order reduction in convergence and constraint preserving associated with the DAE index 3 system. Instead of simultaneously solving the generalized coordinates and the Lagrange multipliers, the differential equations and algebraic equations are specially treated thus solving the generalized coordinates separately from the Lagrange multipliers. The claims of the proposed technique are illustrated via numerical examples by employing explicit time integration operators in the conservation form. The overall developments demonstrate effective handling of the primal issues that fundamentally preserve the underlying properties under the umbrella of Linear Multistep methods.