This paper presents a step-by-step time integration algorithm for efficiently solving second-order nonlinear dynamic problems. The method employs the rewriting of motion as two sets of first-order differential equations. The interpolation of the relevant quantities is achieved by a particular quadratic polinomial expression for the velocities and forces and is defined by values at the boundaries of the time step. Then the time definite integrals of both first-order ordinary differential equations define the numerical relations in the step. An accurate extrapolation predictor and an adaptive time stepping procedure are used as the time predictor–corrector method.