# Tuning of a Viscous Inerter Damper: How to Achieve Resonant Damping Without a Damper Resonance
## 4 Damping of the Building Model
### Root Locus Analysis  
As shown by the sequence of root locus diagrams in Figure 6, the optimal tuning of the viscous inerter damper is derived from a modal interaction that creates a bifurcation point in the complex frequency plane (in this case the normalized \(\rho\)-plane). For the optimal front-end inertance, obtained by the design procedure in Table 1 and summarized in Table 3, the complex poles \(\rho=\omega/\omega_{1}\) are obtained by solving the eigenvalue problem (48) from \(c\to 0\) to \(c\to\infty\). Figure 8 shows the root locus diagram for the optimally tuned viscous inerter damper. By comparison with the almost optimal loci in Figure 6b for \(\mu=0.1000\), the optimal tuning curve in Figure 8 for \(\mu=0.1051\) almost realizes the desired bifurcation point. The circles represent the free natural frequencies, while the squares are the clamped natural frequencies. The tuning procedure in Table 1 secures that the product of the first two normalized roots is unity, according to the inverse point relation in (37). Figure 9a shows a detailed plot of the semi-circular loci for the mode 1 roots. The circle represents the real-valued root for \(\beta\to 0\). The other free limit root is located at the origin, as depicted by the circle at the origin in Figure 8. For increasing \(\beta\), the root from \(\rho=1\) moves into the complex plane, approximately along the backbone unit circle, toward the bifurcation point. The other root leaves the origin along the imaginary axis, meets  
Figure 8: Root loci for inertance tuning to mode \(s=1\). The front-end mass ratio \(\mu=0.1051\) is obtained by the tuning procedure in Table 1.  
its conjugate pole decending from \(\rho=i\infty\), whereby it follows the backbone unit circle toward the bifurcation point. In the present case, the root from \(\rho=1\) branches off before the ideal bifurcation point and traces the semi-circular path towards the smaller of the two clamped natural frequencies (blue square), while oppositely, the root from the imaginary axis follows the semi-circular path toward the larger clamped frequency. The ideal root locus diagram is obtained by solving the quartic polynomial in (39) with optimal \(\bar{\mu}_{1}\) and plotted in Figure 9b. It is seen that for the idealized sodf model, without actual influence from other modes, the two semi-circular branches do meet exactly at a bifurcation point, where the damper parameter is \(\beta_{bif}=\beta_{opt}/\sqrt{2}=\sqrt{\frac{1}{4}\bar{\mu}_{1}}\,\). Thus, the inability of the root loci for the 20-story shearframe building model in Figure 9b to exactly meet at a bifurcation point indicates the approximations applied in the representation of residual mode effects in Section 3.2. In the root locus diagram in Figure 9, the asterisks represent the roots for optimal \(\beta\) from the design formula (42). The inclined straight line intersects the origin, whereby its inclination approximates the modal damping ratio. It is seen that both optimal roots are placed on the same line, which demonstrates a balanced damper design, where both roots associated with the target mode 1 are equally damped. The two damping ratios \(\zeta_{1-}\) and \(\zeta_{1+}\) at the two asterisks in Figure 9 are presented in Table 4 for \(k_{0}=0\). It is seen that the two damping ratios for mode 1 and \(k_{0}=0\) are identical (0.1126) to the fourth decimal and practically equal to \(\zeta_{*}=0.1132\), which validates the accuracy of the proposed inertance tuning procedure in Table 1.  
As indicated in Figure 1c, the pure viscous inerter damper is in practice implemented with a spring in parallel to the dashpot to avoid drift in the damper stroke. It is important to note that in the present analysis, the stiffness \(k_{0}\) of the spring is not chosen to create an additional resonance with the front-end inertance \(m\) and thus kept sufficiently small to avoid dynamic interaction with mode 1.