Unique Loss Factor Images for Complex Dynamic Systems

Over the past century, a number of scalar metrics have been proposed to measure the damping of a complex system. The present work explores these metrics in the context of finite element models. Perhaps the most common is the system loss factor, which is proportional to the ratio of energy dissipated over a cycle to the total energy of vibration. However, the total energy of vibration is difficult to define for a damped system because the total energy of vibration may vary considerably over the cycle. The present work addresses this ambiguity by uniquely defining the total energy of vibration as the sum of the kinetic and potential energies averaged over a cycle. Using the proposed definition, the system loss factor is analyzed for the cases of viscous and structural damping. For viscous damping, the system loss factor is found to be equal to twice the modal damping ratio when the system is excited at an undamped natural frequency and responds in the corresponding undamped mode shape. The energy dissipated over a cycle is expressed as a sum over finite elements so that the contribution of each finite element to the system loss factor is quantified. The visual representation of terms in the sum mapped to their spatial locations creates a loss factor image. Moreover, analysis provides an easily computed sensitivity of the loss factor with respect to the damping in one or more finite elements.