Optimal damping adapted to a set of initial conditions

Vibrating systems can respond to an infinite number of initial conditions and the overall dynamics of the system can be strongly affected by them. Therefore, it is of practical importance to have methods by which we can determine the damping that is in some sense optimal for all initial conditions, or for a given set of initial conditions. For a single and multi degree of freedom systems, we determine the optimal damping coefficients adapted to different sets of initial conditions using the known method of minimizing the (zero to infinity) time integral of the energy of the system, averaged over a set of initial conditions, and using two new methods that we introduce. One method is based on determining the damping for which the energy of the system, averaged over a set of initial conditions, drops the fastest to a given threshold value. The other method is based on determining the damping that gives minimal average settling time of the system, where we take that the system settled when its energy dropped to a given threshold value. We show that the two new methods give results for optimal damping that are in excellent agreement with each other, but are significantly different from the results given by the minimization of the average energy integral. More precisely, for considered multi degree of freedom systems and sets of initial conditions, the two new methods give optimal damping coefficients that converge to the critical damping of the first mode as the target energy threshold decreases. On the other hand, for these same systems and sets of initial conditions, the method of minimizing the average energy integral gives optimal damping coefficients which are deep in the overdamped regime with respect to the first mode.