# Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance
## 1 Introduction
### Outline of Paper  
The aim of the present paper is to design a simple viscous inerter damper, for which the inertance is chosen so that the first two structural vibration modes interact, creating an apparent resonant absorber characteristic, with a damper without an internal resonance. Section 2 introduces the mdof building model and the theory for attaching a general inerter damper. The system reduction from a full mdof model to a single-dof model for the targeted vibration mode includes the influence from the other modes by the residual stiffness and inertance terms, of the same form as in [18].  
In Section 3, the viscous inerter damper is introduced, with a viscous dashpot placed in series with the inerter. This model is referred to as the C2 configuration in [17]. This viscous inerter damper is without a spring and thus is in itself non-resonant. However, because of the residual mode flexibility term, the characteristic equation for the reduced-order single-mode model is of quartic order in the complex natural frequency, whereby the equal modal damping from [9; 18; 43] can be applied to create resonant vibration damping. The residual mode correction coefficients are determined from the two limiting natural frequencies for vanishing (free) and infinite (damped) viscous damper coefficients, following the general approach for vibration absorbers in [44]. However, in the present case, the damper is not resonant in itself, which is opposite to the absorbers in [44], which implies that the inertance of the present viscous inerter damper is calibrated by a simple iterative procedure to create the desired modal interaction between the first two modes of the building model. Section 3 concludes by tuning the viscous dashpot so that optimal frequency response curves are obtained for the response velocity amplitude of the structure.  
The performance of the resonantly tuned viscous inerter damper is illustrated in Section 4 for the 20-floor building model in [14; 39]. The damper is placed between groundand bottom floor, similar to a base-isolation system. The iterative tuning procedure gives a mass ratio of 10.5% of mode 1 modal mass, which is safely within what is considered applicable for inerter dampers in the literature. A root locus analysis, in which the complex-valued natural frequencies are traced in the complex plane for an increasing viscous coefficient, shows that the mass ratio tuning almost creates the desired bifurcation point condition. The subsequent frequency response analysis demonstrates that the novel resonant tuning of the simple viscous inerter damper creates a flat plateau for the top-floor velocity amplitude. The numerical example also investigates the influence of placing a spring in parallel to the dashpot, with a stiffness that is sufficiently small to not interfere too much with the sensitive resonant tuning but large enough to re-center the viscous inerter damper in practice (avoid drift).  
The final conclusions in Section 5 highlight the conceptual novelty of the proposed damper configuration, which is simple and robust and creates the desired resonant type vibration mitigation, although the damper is without an internal resonance (no spring included).