# On the Estimation of the Moving Mass of a TMD Installed on a Lively Structure
## 4 Validation Example  
To validate the proposed methodology, the SDOF system shown in Figure 2a is placed on the lively structure shown in Figure 2b. The SDOF system represents a TMD, but it is not perfectly tuned because it is not required for the purposes of this work. It is composed of a moving mass (letter A in Figure 2a), which is attached to its frame (B) via four slender plates and a spring (C), conferring the ensemble a certain amount of stiffness. The damper (D) consists of a couple of permanent neodymium magnets, which are rigidly joined to the frame, placed very close to a copper plate, which is rigidly attached to the moving mass. As explained in [21,22], the movement of the copper plate inside the permanent magnetic field develops the so-called Eddy currents, creating an effect similar to an ideal viscous damper, since the overall reaction force can be assumed to be proportional to the relative velocity of both elements within a certain range of relative velocities. By means of a precise weighting scale, the total mass of the moving elements can be estimated. To do so, apart from the moving mass itself (A), half of the mass of the elastic members (C) and the mass of the magnets (D) are also included, leading to 32.52 kg. Note that this value may not be fully representative of the effective moving mass due to the dynamic coupling effects that arise when the SDOF system is installed on the structure.  
The ensemble is simply placed on the structure and the frame is designed so it can be assumed to be rigid enough. The lively structure on which the SDOF system is installed is a timber plate with a main span of 13.5 m, a width of 1 m and a height of 0.14 m. It is made of laminated wood, Glulam 24 h. It is simply supported at both ends and, to alleviate the deflection due to its own weight, the middle section is supported by two elastic members made of three springs each. The first mode of this plate is at 2.33 Hz, with a modal damping ratio of 0.27% and a mode shape similar to the first bending mode shape of a simple-supported beam. The generalized mass of that mode at the center section (the place on which the SDOF system will be installed) is 433 kg. Note that the data given in this paragraph is intended to fully present the structure on which the TMD is placed; however, as mentioned above, it is not required for the presented methodology to be applied.  
The example is divided into three parts. First of all, the dynamic properties of the SDOF system are identified by performing a regular experimental modal analysis in the reference scenario with it placed on a rigid floor. Then, the SDOF system is installed on the middle section of the timber plate and, by means of the mass modification technique (Section 3.1), a first estimation of its properties is carried out. Finally, the procedure presented in Section 3.2 is applied to obtain a second estimation of the same properties. Note that all recorded data, as well as the MATLAB code that processes it, is available at a Mendeley Data repository [17]. As mentioned before, the built-in function tfestimate is used to perform the estimation of the experimental frequency response functions by using the estimators \(H_{1}\) and \(H_{2}\) (see Equation (6)), which are then used to calculate the \(H_{2}\) estimator and perform further computations. This function handles the windowing and averaging processes typical of this sort of time-to-frequency-domain transforms. On the other hand, the different curve fittings are carried out by means of the function fmincon, a built-in optimization function. A function error, which calculates the squared error between the experimental FRFs and the synthesized ones for a certain set of properties. The optimization function, then, finds the set of properties that minimizes that error.