# On the Estimation of the Moving Mass of a TMD Installed on a Lively Structure
## 3 Methodology
### Novel Procedure  
The procedure presented in this work is based on the expression in Equation (3). Its terms can be rearranged and Equation (9) can be obtained, where \(Y_{s\xi}\), defined in Equation (4), accounts for the transmissibility function between the base and moving mass coordinates. Note that the expression in the right hand side of Equation (9) corresponds to the receptance function of an SDOF system, \(h(\omega)\), i.e., the frequency response function relating an input force and the displacement response of that system. As can be seen, it is explicitly expressed in terms of its natural frequency \(\omega_{t}\), its damping ratio \(\zeta_{t}\) and its moving mass \(m_{t}\). \[\frac{\Xi(\omega)}{F(\omega)}-Y_{st}(\omega)\frac{X_{s}(\omega)}{F(\omega)}= \frac{1/m_{t}}{-\omega^{2}+j2\zeta_{t}\omega_{t}\omega+\omega_{t}^{2}}=h(\omega) \tag{9}\]  
The application of this methodology requires the estimation of two response functions: \(\Xi(\omega)/F(\omega)\) and \(X_{s}(\omega)/F(\omega)\). Since they share the same input reference, the applied force \(F(\omega)\), the response of both degrees of freedom, \(x_{t}(t)\) and \(x_{s}(t)\), can be simultaneously recorded, the relative coordinate \(\xi(t)\) can be calculated, and both response functions can be estimated at the same time. The transmissibility function, \(Y_{s\xi}(\omega)\), is also required for the left hand side of Equation (9) to be fully computed. Since, by definition, the transmissibilityfunction requires no force to be applied on the point belonging to the structure, \(f(t)=0\) N (see Section 3), this function cannot be estimated along with the previous response functions. For this reason, this method must be applied in two stages. However, in contrast to the work presented in [16], the transmissibility function is not now required to estimate the modal properties of the SDOF system (\(\omega_{t}\) and \(\zeta_{t}\)), but rather to compute the left hand side of Equation (9), so only one identification or fitting process to the receptance SDOF response function needs to be carried out. This leads to a simultaneous estimation of the natural frequency \(\omega_{t}\), the damping ratio \(\zeta_{t}\) and the moving mass \(m_{t}\) of the SDOF system. More precisely, the proposed methodology could start by computing the transmissibility function between the moving mass of the installed TMD (or a similar device) and the point of the structure on which it is placed. As explained before, this is done by simultaneously measuring the acceleration of the moving mass and the acceleration of the structure to which the SDOF system is attached, while both points are moving due to an external action different from a force applied on the moving mass. A frequency response function is estimated (by using an estimator such as \(H_{\text{e}}\), defined in Equation (6)) by treating the acceleration of the structure as the input and the relative acceleration as the output. Then, by applying a force on the moving mass, the acceleration response of the same two points is recorded again. Now, the frequency response functions relating both accelerations (outputs) and the applied force (input) need to be estimated. For this task to be properly done, it is advisable to apply the force in a way that does not modify the sought moving mass. For example, an impact hammer (or any device that externally exerts an impact on the moving mass) is suitable for this purpose; moreover, any sort of inertial shaker should be avoided, since it needs to be placed on the moving mass to induce a force on it. Finally, once the three response functions are computed, the left hand side of Equation (9) can be calculated and the receptance expression (right hand side of Equation (9)) can be fitted to the resulting complex series of data in the frequency domain. As a result, an estimation of the three values, \(m_{t}\), \(\omega_{t}\) and \(\zeta_{t}\), are obtained at the same time.