# Performance of Numerically Optimized Tuned Mass Damper with Inerter (TMDI)
## 1 Introduction  
The Tuned Mass Damper (TMD), first described by [PERSON] [1], is a widely used passive device used to control resonant vibrations when the disturbing frequency is unknown. It is a damped one-degree-of-freedom oscillator that is connected to the primary structure, whereby the resulting system is a highly damped two-degrees-of-freedom system. The two optimization parameters of the TMD are its natural frequency and damping ratio. For the selected mass ratio between the TMD oscillating mass and the target modal mass of the vibrating structure, these parameters can be optimized for minimum modal displacement, velocity and acceleration for harmonic and broad band excitations [2]. TMDs are widely used in footbridges [3, 4] and high-rise buildings [5, 6] to guarantee the required acceleration limits [7, 8, 9], in stret bridges to provide the required level of damping [10, 11], in railway bridges to avoid the loss of contact between wheels and track [12], and in flag masts and wind power stations to reduce bending vibrations [13, 14]. To enhance the mitigation efficiency and broadband characteristics of TMDs, several concepts have been developed. Active and semi-active TMDs are based on the real-time control of TMD natural frequency and damping according to the actual frequency of vibration [15, 16, 17, 18, 19]. The resulting enhanced vibration suppression can be used to guarantee lower vibration limits or to ensure vibration limits with reduced TMD mass [6]. Another approach to make TMDs more efficient is simply to increase their oscillating mass. However, limited TMD installation space and costs make this approach unfavorable. Given this, theTMD with inerter, abbreviated as TMDI, has received much attention during the past decade. The ideal inerter is defined to exert a force in proportion to the relative acceleration of its two grounds or terminals, respectively. That the development of real inerters with minimum or even zero friction is a challenging task is seen from the works [20,21]. Originally, the inerter was used in vehicle suspension systems to improve ride comfort [22]. The use of inerters in the field of vibration isolation in a more general way is described in [23]. When the inerter is used in combination with the classical TMD, there are many possible TMDI topologies. Some topologies aim at replacing the TMD mass, some intend to increase TMD inertial force and some target producing an additional degree of freedom whereby the primary structure with TMDI behaves as a three-degrees-of-freedom system. A profound overview of possible TMDI topologies together with closed-form solutions of the related parameters can be found in [24, 25, 26, 27]. TMDI efficiency has been investigated for various TMDI configurations for earthquake excitation [28, 29, 30], wind excitation [31, 32, 33, 34], and harmonic and random excitations of the primary structure [35, 36, 37]. The studies in [35, 36] were made for various TMDI parallel and serial arrangements of inerters, and stiffness and damping elements and optimum TMDI parameters in table format are provided. The authors of [37] show that even very complex inerter configurations are thinkable. Besides the investigations of various TMDI topologies, studies have also been performed to investigate the influence of the inerter ground on the primary structure [38, 39]. This literature survey gives a brief overview of the topics related to inerters that are being investigated. The present work aims at analyzing the mitigation efficiency of the TMDI not with analytical parameters but with numerically optimized parameters to ensure optimality. The optimization criterion is minimum primary structure displacement response. The TMDI is assumed to be located at the anti-node position of the first bending mode of the primary structure (Figure 1a), and the inerter is grounded to the antinode position as well because this is the most realistic scenario for real applications. Two typical TMDI configurations are considered.