## 5 Conclusions  
Addition of linear viscous dampers to frame structures generally results in non-proportional damping, characterized by a populated modal damping matrix and complex-valued damped modes. A well-established procedure to obtain an approximate response of such systems is to apply classical modal analysis neglecting the off-diagonal terms of the modal damping matrix. This study focuses on measurement of the errors incurred with this approximation, their relationship with parameters describing the structural system, and their possible correlation with non-proportionality indices, originally proposed in the literature to measure the extent of damping non-proportionality. A numerical study is performed, including a wide variety of representative frame-type and shear-type planar structures, added damping systems, and amounts of added and inherent damping, subjected to collections of scaled accelerograms. The errors between the 'exact' numerical response and the approximate responses obtained by two procedures based in modal analysis are considered. The two procedures are Time History Modal Analysis (THMA) and Response Spectrum Method (RSA), and in the latter two different combination rules, square-root-of-sum-of-squares (RSA-SRSS) and complete quadratic combination (RSA-CQC) are included. Negative errors indicate that the approximate method under-estimates the results; conversely, positive errors indicate result over-estimation. Based on results of this study, the following conclusions are drawn:  
1. The error in THMA is only mildly dependent on inherent damping and ground type; its median value is close to 0 and generally 80% of the results are within a range of [\(-\)0.15; +0.25]; conservative (max) errors are larger than un-conservative (min) errors. Large error dispersion occurs at the bottom and top of the frames, with a negative correlation between both stories. Shear-type systems present, in general, larger extreme errors than frame-type systems. 2. RSA present similar trends to THMA, but with larger dispersion and a tendency to un-conservative errors at first story and over-conservative errors at the roof. RSA-CQC offers a smaller dispersion than RSA-SRSS. 3. To keep the error under a certain threshold in THMA and RSA, simple criteria based on frame and damping system properties cannot be proposed from the data available, as parameters that are significant for the response averaged through stories, central story, first story and roof are generally different. Moreover, values of parameters that reduce the response error in one story might trigger a large error in another. 4. A correlation of the max and min error in THMA and RSA with four existing non-proportionality indices has been empirically proved and can be used as an alternative method to keep the error under control, especially for THMA. Tabulated data for different error thresholds are provided. The tabulated data can be used to assess the adequacy of modal analysis in buildings equipped with supplemental damping. The underlying concepts can be extended to other engineering problems such as buildings with base isolation, or placement of viscous dampers for vibration control in high-speed railway bridges, etc. These conclusions are limited to linear systems with added linear viscous dampers within the range of structural parameters defined in the parametric study, where only height-wise decreasing stiffness and damping configurations have been considered. However, the range of parameters is wide enough to include most realistic systems. The study included only far-field motions recorded on firm soil conditions. For site classes B and C no significant differences in error results were found. The flexibility of auxiliary elements in the damping systems was not included. The analysis included all modes with exact spectral response, so the influence of these two sources of error (mode truncation and approximate spectral response) was not considered.