Eigenfrequency Shift in the Newmark Time Integrator

This technical note aims to clarify the influence of numerical time integration schemes (such as the Newmark method) on the eigenfrequency of the system. With a straightforward analysis of three consecutive time instants it is shown that the eigenfrequency of the time-discretized system is different from the eigenfrequency of the original (continuous) system, and that this frequency shift depends on the magnitude of the applied time step. As such, it affects both free and forced vibrations. In particular, for the analysis of forced vibrations at resonance the excitation frequency must be matched with the eigenfrequency of the time-discretized system rather than the eigenfrequency of the system prior to time discretization.