Comparison of methods for computing equivalent viscous damping ratios of structures with added viscous damping

Three analysis methods are compared for determining the equivalent viscous damping ratios in a simple one-story, one-bay system with three different added damping configurations. The analysis methods are modal strain energy using the undamped mode shapes, free vibration log decrement, and complex eigenvalue-eigenvector analysis. The damping configurations include a diagonal configuration, a horizontal configuration, and a toggle-braced system. For each of these systems, a variety of linkage stiffness and damper capacities are considered. In all cases the structure, exclusive of the dampers, is assumed to remain elastic. A brief discussion is provided for the behavior of systems with nonlinear viscous dampers. The results of the study show that significantly different values may be obtained for the effective viscous damping in the same system when different analysis approaches are used. For the toggle-braced system with relatively flexible linkages, it was shown that the modal strain energy approach consistently produced increasing effective damping with increased damper capacity, while the other two methods indicated the opposite results. Subsequent analysis of the toggle-braced structure using the complex eigenvalue-eigenvector approach showed that the effective system damping was indeed decreasing with an increased device damping constant, and that the behavior could be attributed to the axial flexibility of the toggle brace linkage. The analysis indicated that the greater the flexibility of the linkage relative to the damping constant of the device, the greater the phase difference between the deformational velocity in the device and the relative horizontal velocity of the story containing the device. While the phase shift is evident from response history analysis, the most efficient visualization tool is a plot of the damped mode shapes on a complex plane.