Energy conservation in Newmark based time integration algorithms

Energy balance equations are established for the Newmark time integration algorithm, and for the derived algorithms with algorithmic damping introduced via averaging, the so-called a-methods. The energy balance equations form a sequence applicable to: Newmark integration of the undamped equations of motion, an extended form including structural damping, and finally the generalized form including structural as well as algorithmic damping. In all three cases the expression for energy, appearing in the balance equation, is the mechanical energy plus some additional terms generated by the discretization by the algorithm. The magnitude and character of these terms as well as the associated damping terms are discussed in relation to energy conservation and stability of the algorithms. It is demonstrated that the additional terms in the energy lead to periodic fluctuations of the mechanical energy and are the cause of the phenomenon of response 'overshoot', previously observed empirically in the application of Newmark based algorithms to high-frequency components. It is also demonstrated that the stability limit of the explicit Newmark algorithm is reached, when the stiffness term in the algorithmic energy vanishes, and that energy fluctuations take place for integration intervals close to the stability limit. (c) 2006 Elsevier B.V. All rights reserved.