Efficiency and tuning of viscous dampers on discrete systems

An approximate solution is developed to the complex eigenproblem associated with free vibrations of a discrete system with several viscous dampers, in order to facilitate optimal placement and sizing of added dampers in structures. The approximate solution is obtained as an interpolation between the solutions of the two limiting eigenproblems: the undamped eigenproblem and the constrained eigenproblem in which each damper is replaced with a rigid link. An explicit form of the approximate solution is developed for cases in which the difference between these limiting eigensolutions is sufficiently small, and an iterative solution scheme is presented for cases in which the difference is larger. These results allow the efficiency and tuning of viscous dampers to be investigated by solving only the two limiting real-valued eigenproblems. The application of the approximate formulation is illustrated for a 10-story building model with added dampers.