## 1 Introduction  
In the last decades there has been increasing interest and research on damping devices. When properly added to a structure, these devices improve considerably its dynamic performance [1; 2; 3]. Among the different types of dampers, pure viscous dampers present the unique feature that their behavior is solely dependent on velocity, rather than displacement: under forced harmonic excitation their damping forces are out of phase with the elastic restoring forces, improving the energy dissipation capacity of the structure without increasing its stiffness [4; 5]. If the damping matrix of the system fulfils certain conditions [6], the natural vibration modes of the damped system are real-valued and identical to those of the associated undamped structure. This type of system is referred to as classically (or proportionally) damped, and its solution by modal superposition is known as 'classical modal method'. To estimate the maximum response of a classically damped structure subjected to earthquake loading, the classical modal method is used in conjunction with response spectra, and this is the basis of the well-known Response Spectrum Analysis (RSA), introduced in FEMA-273 [7] and now included [8] or considered for inclusion [9] in several seismic codes around the world for analysis of structures with added viscous damping systems. However, placement of viscous dampers generally results in a non-proportional damping matrix, i.e., one that does not result in real-valued natural damped vibration modes. The modal method can also be applied to this type of systems in a generalized form due to [PERSON] [10]. This method, referred to as 'generalized modal method', has found limited application in structural engineering practice, possibly because (i) it is more prolix than classical modal analysis; (ii) when used with the response spectrum has to rely on approximations of questionable accuracy; and (iii) the physical meaning of the elements involved in the solution is not as well identified as in the classical modal method [11]. To avoid the complexity involved in the generalized modal method, seismic codes impose conditions on the design of structures with viscous dampers, so that effects of non-proportional damping can be ignored, and the classical modal method can be applied. For example, placement of dampers following a complete vertical distribution throughout the building is assumed to reduce non-proportionality effects [12], whereupon the seismic response can be reasonably estimated with the undamped vibration modes, applying the classical modal method [4; 13]. For systems with non-proportional damping, this involves ignoring the off-diagonal coefficients of the exact modal damping matrix. The errors resulting from this approximation have been pointed out in previous studies [14], yet there are, to these authors' knowledge, no systematic investigations that quantify these errors; in particular, their relationship with properties of the system, degree of non-proportionality of the damping matrix, or amount of damping. The primary objective of the current study is to quantify this error for linear structures with viscous dampers having different number of stories, fundamental periods, amounts of damping, distributions of dampers and distributions of lateral stiffness. A second objective is to correlate the error with the level of non-proportionality of the damping matrix measured by different indices proposed in the literature. Finally, recommendations are made regarding the value of existing indices that measure the degree of non-proportionality of the damping matrix, so that errors resulting from application of classical modal analysis to non-proportionally damped systems are kept within acceptable bounds.