# On the Estimation of the Moving Mass of a TMD Installed on a Lively Structure
## 1 Introduction
The authors of this work already presented a direct transmissibility-based methodology to accurately estimate both the natural frequency \(\omega_{t}\) and the damping ratio \(\zeta_{t}\) of a TMD installed on a slender structure [16], which works not only for properly tuned mass dampers, but also for any kind of SDOF system installed on a structure, not necessarily tuned to mitigate the contribution of a specific mode. The procedure consists in simultaneously measuring both the acceleration of the TMD moving mass, \(\dot{x}_{t}\), and the acceleration of the point of the structure to which it is attached, \(\dot{x}_{s}\), and estimating the frequency response function between them, called transmissibility function. Two approaches were presented: one in which the frequency response function was directly estimated between the two measured signals and another in which the frequency response function was estimated between the structure acceleration and the relative acceleration (\(\ddot{x}_{t}-\dot{x}_{s}\)). By fitting the data of each frequency response function to the corresponding transmissibility analytical expression, the natural frequency \(\omega_{t}\) and the damping ratio \(\zeta_{t}\) were estimated. In this work, a novel methodology is presented to simultaneously estimate the natural frequency, \(\omega_{t}\), the damping ratio, \(\zeta_{t}\), and the moving mass, \(m_{t}\), of a TMD installed on a lively structure. The major difference between this and the previous work [16] is that the presented procedure enables an estimate of the moving mass of the TMD, \(m_{t}\), to be made, something that could not be done before due to the adopted approach being completely based on transmissibility functions. Now, a two-step methodology is presented. The first step is still based on the estimation of a transmissibility function, but the procedure is completed with a second step in which a couple of frequency response functions are calculated by applying a force on the moving mass. These functions, together with the transmissibility, lead to a curve that can be fitted to the classical frequency response function of an SDOF system, which is explicitly expressed in terms of the three parameters of the system, including its moving mass, \(m_{t}\). As will be shown, the methodology is entirely based on simple expressions, deduced from the SDOF system theory, so it is within the grasp of every mechanical engineer. In addition, it requires a very reduced set of measurements to be performed, leading to a procedure that is very easy and fast to apply. This constitutes a major advantage with respect to others, such as the mass modification technique. In order to compare the quality of the obtained results against the required workload, that technique is also covered in this article. It is a widespread method also conceived to estimate the moving mass of an SDOF system based on the definition of the undamped natural frequency of an SDOF system. As mentioned above, the presented methodology is aimed at estimating, in an easy and rapid way, the whole set of modal parameters (moving mass, natural frequency and damping ratio) of an installed TMD, which is helpful in a variety of scenarios. It is useful, for example, for the fine tuning of the mass damper in order to overcome modifications it may suffer while assembling or transporting it. The TMD properties can also be periodically checked to assess its long-term durability and to foresee tuning or maintenance works. Finally, although this methodology is mainly focused on installed TMDs, it could also be extended to estimating the properties of any substructure modeled as a single-degree-of-freedom system.