In this paper, we present a new explicit procedure using periodic cubic B-spline interpolation polynomials to solve linear and nonlinear dynamic equation of motion governing single degree of freedom (SDOF) systems. In the proposed approach, a straightforward formulation was derived from the approximation of displacement with B-spline basis in a fluent manner. In this way, there is no need to use a special pre-starting procedure to commence solving the problem. Actually, this method lies in the case of conditionally stable methods. A simple step-by-step algorithm is implemented and presented to calculate dynamic response of SDOF systems. The validity and effectiveness of the proposed method is demonstrated with four examples. The results were compared with those from the numerical methods such as Duhamel integration, Linear Acceleration and also Exact method. The comparison shows that the proposed method is a fast and simple procedure with trivial computational effort and acceptable accuracy exactly like the Linear Acceleration method. But its power point is that its time consumption is notably less than the Linear Acceleration method especially in the nonlinear analysis.