# Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance
## 1 Introduction
### Inerter Dampers  
A recent extension of the classic TMD is the introduction of the so-called inverter, a two-terminal inertia element [11] that is most commonly realized by converting translational deflection into rotation [12], whereby gearing systems can be used to realize extraordinary large effective damper masses [13; 14]. Because the absorber theory dictates that damper performance increases with damper mass, the mass gearing property has accelerated research and development of inerter-based devices.  
The basic inerter-type dampers mostly combine the inerter with a spring and dashpot to create a fundamental resonant damper. The inerter and spring create a damper resonance that is commonly targeted to the dominant vibration mode of the structure. The fundamental analysis of tuned inerter dampers (TIDs) is introduced in [15], in which the damper is tuned to optimal response reduction, compared to the performance of the classic TMD, and investigated for mass ratios of up to 50% of the fundamental modal mass. In [16], tuning expressions for inerter dampers are carefully derived for various damper configurations and extended in [17] to maximize the base-isolation performance when the sdof structure is exposed to stochastic loading.  
Analytical tuning expressions for TIDs are presented in [18; 19], where the modal coupling with other modes is explicitly accounted for by additional (artificial) flexibility and inertia terms. This inclusion of residual mode correction enables the derivation of accurate explicit tuning formulae for inerter dampers, an approach also followed in the present paper. An exact numerical optimization of the two fundamental TIDs is conducted in [20; 21] for an sdof structure exposed to stochastic base excitation.  
Tuned inerter dampers have been proposed for several applications in high-rise building technology, such as for seismic retrofit [22; 23] and the optimization of multiple TIDs for the vibration control of an mdof structure model under stochastic base excitation. Furthermore, inerter dampers have been extended by the introduction of negative-stiffness springs [24; 25], which improves the TID's damping capability, and by the installation of hysteretic dampers in [26].  
A particular property of inerter elements is their theoretical ability to connect, and thus transfer force, between floors separated by several stories. An example is provided by [27], in which the inerter bridges four stories to create a larger relative damper deflection for the wind-induced vibration control of a high-rise building. The ultimate realization defines the so-called ground inerter damper, for which one of the inerter terminals is attached to the building support level. In [28], the inerter damper is tuned for optimal vibration damping, while in [29], the grounded concept is applied for seismic isolation.  
The simplest case of grounded dampers are those acting as base-isolation systems. In [30], tuning expressions are derived optimal base-isolation applications of inerter dampers. As in [17], the optimally tuned inerter dampers aim at minimizing the transmissibility from base to structure. Inerter dampers are proposed for base-isolation of storage tanks by [31], while simple inerter systems (with hysteretic damping) are proposed for seismic isolation of building models in [32].