## 1 Introduction  
During the last years of the past century, several seismic events (i.e., 1994, Northridge, USA; 1995, Kobe, Japan) took place that substantially shifted the focus of earthquake engineering [1, 2]. Even though many of the buildings involved were designed according to modern seismic codes, non-structural damage resulted in enormous economic losses and traumatic post-event recovery, making the engineering community realize that enhanced seismic performance, beyond the mere life-safety level, was necessary [3, 4]. Recent seismic events (2010, Chile; 2011, Christchurch, New Zealand) have thrust the need for even more demanding objectives, with emphasis on structural separability, damage-protection and resilience [5, 6, 7]. In this regard, declassification of the military viscous damper technology (developed during the cold war) during the 1990s allowed application of these devices to vibration and earthquake protection of civil structures [8]. Addition of viscous dampers to structures is beneficial in several ways: first, seismic energy input is dissipated through viscous mechanisms rather than structural hysteresis, thus providing an effective protection to the main structural elements [9, 10]; second, interstorey drift, peak storey acceleration and peak velocity are reduced, preventing damage to non-structural elements [11, 12]. For these reasons, viscous dampers have been widely used to provide added seismic resilience to structures [13, 14, 15]. The Japanese case is particularly illustrative: The Japan Society of Seismic Isolation (JSSI) was founded in 1993, and, soon after, the 1995 Kobe earthquake boosted the application of passive protective technologies in this country [16]; by 2011, supplemental damping systems had been already validated with full-scale tests [17] and around 1000 buildings in Japan already counted with passive protection systems [18]; during the 2011 Tohoku-Oki Earthquake, 11 buildings equipped with these types of systems were instrumented with sensors; the recorded signals and post-earthquake inspection showed their excellent seismic behavior [19]. The first codified provisions for supplemental damping systems were developed in the US [20]. Revision of the European seismic code currently under work contemplates inclusion of clauses relative to these systems [21]. The behavior of these devices has been thoroughly studied [22; 23; 24; 25; 26; 27; 28; 29]. A fractional derivative model has been proposed as the most accurate mathematical representation. For practical purposes, this model has often been simplified to a pure dashpot. Added viscous damping systems can be modeled by the serial combination of a spring and a dashpot, where the former represents the flexibility of the different elements linking the pure viscous element to the main structure, and the damper flexibility itself; this last term is often neglected. For a very rigid spring, the system can be assimilated to a pure dashpot, and a closed-form solution under harmonic load is possible; this conveniently allows for the use of simplified methods, such as response spectrum analysis (RSA) [30] implemented in ASCE/SEI 7-16 [31]. However, experimental [32; 33] and analytical [34] studies have shown that inclusion of damping system flexibility is essential in achieving an accurate approximation to the dynamic structural response of systems equipped with added viscous damping. Moreover, a minimum level of stiffness is required for adequate performance of the damping system, as it has been shown that excessive flexibility leads to both substantial reduction of dissipated energy and appearance of damper forces in phase with structural restoring forces, the latter being detrimental for the main structure. Rules have been proposed to estimate the minimum amount of stiffness required to achieve a reasonable level of system efficiency [35; 36]. However, they are usually based on the behavior of Single-Degree-Of-Freedom (SDOF) systems.