# On the Estimation of the Moving Mass of a TMD Installed on a Lively Structure
## 4 Validation Example
### Mass Modification Technique  
The device is then placed on the structure. The mass modification technique is applied by considering three different scenarios: the original scenario in which the moving mass equals \(m_{t}\) and no extra mass is added (\(\Delta m_{0}=0\) kg), named scenario S0; a situation in which a mass of \(\Delta m_{1}=2.02\) kg is added to it, (scenario S1); and a scenario in which a total mass of \(\Delta m_{2}=4.04\) kg (2 \(\times\) 2.02 kg) is added, (scenario S2). The added mass represents 6.55% of the original moving mass in the first scenario and 13.10% in the second one. In each case, the structure is excited by walking on it (thus, not applying any direct force on the moving mass of the SDOF system, as explained in Section 3.2) and the acceleration of the moving mass (\(\dot{x}_{t}(t)\)) is synchronously recorded, together with the acceleration of the point of the structure on which the SDOF system is located (\(\ddot{x}_{s}(t)\)). The relative acceleration, \(\ddot{\xi}(t)=\dot{x}_{t}(t)-\dot{x}_{s}(t)\) is then calculated. The transmissibility function between \(\dot{x}_{s}(t)\) (input) and \(\ddot{\xi}(t)\) is estimated via the \(H_{v}\) estimator in a similar way to that explained in Section 4.1. This time, however, the time series data are segmented by means of the Hanning window  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Case** & \(m_{t}\) **(kg)** & \(\epsilon_{m}\) **(\%)** & \(f_{t}\) **(Hz)** & \(\epsilon_{f}\) **(\%)** & \(\zeta_{t}\) **(\%)** & \(\epsilon_{\xi}\) **(\%)** \\ \hline Rigid floor & 30.84 & - & 2.345 & - & 1.87 & - \\ Mass modification & 34.52 & 11.9 & 2.340 & 0.20 & 2.01 & 7.48 \\ Novel method & 31.85 & 3.27 & 2.342 & 0.12 & 1.99 & 6.42 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Estimated properties of the SDOF system in the three presented cases.  
Figure 3: Fitted frequency response function of the isolated SDOF system.  
of \(2^{15}=32,768\) points, instead of the rectangular one, as it is a long continuous recording. An overlapping of \(2^{14}=16,384\) points (50%) between two consecutive segments is used to overcome some problems that may arise due to the low weighting values at both window ends. The time series recording lasts around 425 s (170,000 points at 400 S/s), so a total of 9 averages can be computed by means of the tfestimate function. The resulting transmissibility functions in the three scenarios are shown in Figure 4 (thick lines).  
The three transmissibility functions, one per scenario, are fitted to their corresponding expression (Equation (4)). The fitted curves are also shown in Figure 4 (thin lines). The set of identified natural frequencies and damping ratios are shown in Table 2 together with the mass added in each scenario (\(\Delta m_{i}\), where \(i\) stands for the number of the scenario: 0, 1 or 2) and an estimation of the moving mass by applying Equation (7) (the scenario S0 is taken as the reference for both estimations). As explained Section 3.1, the points \(f_{0}^{2}/f_{t_{i}}^{2}\), which are equal to \(\omega_{10}^{2}/\omega_{t_{i}}^{2}\), plotted against the mass increment \(\Delta m_{i}\), should draw a straight line. This plot is shown in Figure 5. As can be seen, the three points draw an almost perfectly straight line, with a slope of 0.0290 kg\({}^{-1}\), leading to an original moving mass of \(m_{t}=34.52\) kg, and an independent term above 0.999, as expected. This moving mass value, together with the natural frequency and damping ratio of the scenario S0 (with no added mass), is included in Table 1 (row \"Mass modification\") for comparison purposes and the relative errors (where \(\epsilon_{m}\) stands for the mass value error, \(\epsilon_{f}\) for the natural frequency error and \(\epsilon_{\zeta}\) for the damping ratio error) are estimated with respect to the reference case (\"Rigid floor\").  
Figure 4: Experimental transmissibility functions (thick lines) and fitted curves (thin lines) associated to the three scenarios S0 (blue), S1 (red) and S2 (green).