Numerical integration is considered for second order differential equations on the form where Ais significantly more expensive to evaluate than B; and B is stiff (highly oscillatory) in comparison with A. Examples of such problem are multibody problem with contact forces acting between bodies; and constraints formulated as penalty forces. Based on the splitting A+B of the acceleration field; a numerical integration algorithm; which is explicit in the Aâ€“part and implicit in the Bâ€“part; is suggested. Consistency and linear stability analysis of the proposed method is carried out. Numerical examples with the proposed method is carried out for two simple test problems; and for a complex multibody model of a rotating ball bearing. Comparison with conventional implicit methods is given for each example. The results indicate that the proposed method is more efficient; in terms of number of evaluations of A; at the same accuracy level.