Abstract
The paper presents a method to evaluate optimal viscous damping ratios and cumulative ductility factors of a structure with bilinear hysteretic dampers and dynamic mass dampers, subjected to an arbitrary earthquake ground motion. It is the principle of the proposed method that the total energy of the structure with arbitrarily specified ductility factors is constructed by summing the modal energies when the modal deformations are agreed to the modal ductility factors computed through the corresponding eigenvalue analysis. The numerical examples show the procedure has high accuracy in estimating cumulative ductility factors as well as the maximum response values.