## 1 Introduction  
Structural control systems have turned into a standard technology to improve the dynamic response of civil engineering structures subjected to dynamic actions, such as wind forces or earthquake loads [1]. These control systems can be classified into four major groups: passive, active, hybrid, and semiactive controllers. Passive systems are widely accepted by the engineering community because of their mechanical simplicity, low power requirements, and controllable force capacity [2]. Among passive systems, one of the most commonly used and tested devices throughout the years has been the tuned mass damper (TMD). This system consists of attaching an additional mass linked to the main structure using a spring and a viscous damper, which is optimally tuned to one of the fundamental vibration frequencies of the system in order to transfer energy among the vibrating modes by making the structure more flexible [3]. The original TMD formulation was first proposed by [PERSON] in [4] as a vibration absorber with no damping to control periodic resonance vibrations. Subsequently, [PERSON] and [PERSON] [5] developed the theory of vibration absorbers, including viscous damping to the system to be effective under different frequencies of random vibrations. Conventional tuning methodologies have been proposed thereafter, considering the harmonic loads, and random stationary and nonstationary white noise processes for single degree of freedom systems [6; 7; 8; 9]. The first full-scale implementations of TMDs were aimed to control dynamic displacements caused by wind-induced vibrations. In that sense, TMDs have been deployed in several structures around the world, including the Centre Point Tower in Sydney, the Citicorp Center in New York, the Chiba Port Tower in Japan, the CN Tower in Canada, and, more recently, the Taipei 101 tower in Taiwan [10; 11]. Several variations based on the conventional TMD formulation have been studied and applied thereafter, including pendulum-tuned mass dampers (PTMD) [12; 13; 14], tuned liquid column dampers (TLCD) [15; 16; 17], bidirectional TMDs [18] and hybrid, semi-active, and active TMDs [19; 20; 21]. However, some early investigations concluded that TMDs were not effective in reducing the response of buildings subjected to seismic excitation [22; 23], which was due to two principal reasons. First, the limitations associated with the amount of mass added to the structural system, and, more importantly, that conventional TMDs can suppress effectively a single vibration mode. Instead, wide-band multi-modal damping can be achieved by complex systems, such as nonlinear TMDs [24; 25], multiple positioned TMDs [26], or the amplification of the inertial mass in conventional TMDs using inerter devices (TMDIs) [27; 28; 29]. In spite of this, since the early 2000s, some researchers have proved the effectiveness of numerical iterative methods in the tuning of linear mass dampers for seismic applications. Classic techniques have been used for that purpose, such as minimax optimization [30], response surface methodology [31], and nonlinear gradient-based optimization [32]. Moreover, conventional population-based metaheuristics and evolutionary algorithms have been used to improve the computational efficiency in the tuning procedure; among them, particle swarm optimization [33; 34]. harmony search [35; 36; 37; 38; 39; 40], ant colony [41; 42], flower pollination algorithm [43; 44; 45], bat algorithm [46], cuckoo search [47], genetic algorithms [48; 49; 50], and, more recently, chaotic optimization [51]. Furthermore, various of these investigations have included the effects of dynamic soil-structure interaction (SSI) [52; 53; 54; 55]. Even though the tuning of linear TMDs on single degree of freedom systems is a very well addressed problem in the literature, most of these works were based on the assumption that closed-form expressions, like those presented in [6; 7; 8; 9], are not valid for multi-degree of freedom systems subjected to actual seismic loads.