Estimating Fundamental Dynamic Properties of Structures with Supplemental Dampers by Means of Generalized Single Degree of Freedom Systems

The dynamic response of complex structures can be estimated by means of Generalized Single Degree of Freedom (G-SDOF) systems. The original concept of the G-SDOF system is revisited using an alternative viewpoint based on the equilibrium of the three resultant dynamic forces associated with the stiffness, mass and damping components, and applied to frame structures equipped with viscous dampers. In particular, it is shown that a generic structure with supplemental dampers can be reduced into two different G-SDOF idealizations, based on the global translational or rotational equilibrium. The mechanical analogies of the two G-SDOF systems provide physical insight into the dynamic behavior of structures with viscous dampers, since their energy dissipation capacities can be graphically assessed in terms of the resultant dynamic forces and corresponding lever arms. The approach is applied to shear-type structures with specific dampers configurations resulting in proportionally damped systems. Novel analytical estimations of their first circular frequencies and modal damping ratios are obtained, providing upper- and lower-bounds of the exact values. Limitations in the use of the approach for structures with different dampers configurations resulting in highly non-proportionally damped systems are also discussed.