In order to obtain efficient calculation methods for dynamical problems of large and complex structures. A class of explicit numerical integration methods without dissipative property is proposed through adjusting parameters to improve the stability of algorithms. Accelerations are used as the basic variables in the proposed methods which are on the basis of the Taylor‘s formula. The analyses of stability and accuracy are performed for the proposed methods. Finally, some numerical examples are utilized for the comparison of the proposed methods, the leapfrog central difference method (LCDM), the Newmark-β method and the exact solution. The results show that the proposed methods have the same stability condition as the LCDM under undamped condition. The amplitude attenuation of the proposed methods is 0, which means they are non-dissipative. The periodic error is about half of that of the implicit Newmark-β method. The proposed methods provide a unified format of the LCDM and the Zhai method without dissipation, and more explicit integration algorithms can be derived directly.