Abstract: In this paper,the local errors of displacement and velocity induced by temporal discrete in directintegration are studied.Then,the difference among the system energy,local error of energy as well as the energyfor local errors of displacement and velocity are distinguished,The local error of energy is presented as a posteriorierror estimator which has clear physical meaning. In the estimator not only the local errors of displacement but alsothe local errors of velocity are taken into account.An adaptive time-stepping procedure is presented. The numericalcomputation is also performed,and the presented estimator is proved to be of high accuracy and reliability.