Abstract: A new step-by-step integral procedure of dynamics equations is presented. The general expression of solution of dynamics equations is obtained on the basis of the homogenous analytical solutions of dynamics equations. The explicit analytical integration algorithm, which is characterized by fourth-order accuracy, self-starting and self-correcting, is employed to discretize the equivalent load terms at the right-hand terms of the equations. The transfer matrix is not limited by time step size. If the group solution method is used, the size of stiffness matrix and mass matrix will be reduced and the method will be more cost-effective. Numerical examples show that the results are highly accurate in comparison with those of Newmark, Wilson-θ Houbolt and central difference method. Moreover, the present results are closer to the exact solution, than those of other methods. In addition, the method is also suitable for treatment of nonlinear dynamics problems since no iterative procedure is needed in the present algorithm.