Energy based schemes for damping ratio identification in linear vibrating systems are proposed for the cases of both underdamped and overdamped systems, together with an energy based settling criterion. For the underdamped case, the total energy obtained from the free response to initial conditions is used to estimate the damping ratio. The method was validated numerically and compared with the logarithmic decrement and half power bandwidth method. The energy based damping identification method estimates resulted in errors below 1%, which were significantly lower than those of the comparison methods. The robustness of the damping ratio estimation was studied as well. For overdamped systems, two formulas for damping identification were proposed. The overdamped formulation applies to free responses initiated by an initial displacement, and numerical results showed errors under 2%. In addition, three energy based settling criteria that are insensitive to initial conditions were introduced. When compared with the standard 2% settling-time criterion, it showed consistent agreement with the 2% vibration amplitude band. The results indicate that the proposed energy based measures provide accurate and practical tools for damping identification and settling assessment.
Hajiyeh et al. (2026) studied this question.