# On the Estimation of the Moving Mass of a TMD Installed on a Lively Structure
## 1 Introduction  
In the field of structural dynamics, tuned mass dampers (TMDs) are passive devices installed on lively structures to reduce their response level [1, 2]. They are extensively used not only to mitigate vertical vibrations in, for example, light floors [3] or footbridge decks [4, 5], but also horizontal vibration of tall buildings, such as wind turbines [6], chimneys [7], bridge piles or skyscrapers [4, 8]. One of the most widespread ways of modeling TMDs is as single-degree-of-freedom (SDOF) systems composed of a moving mass \(m_{t}\), an elastic member \(k_{t}\) which attaches it to the structure and a damping element \(c_{t}\) which accounts for the energy dissipation, conferring it a natural frequency \(\omega_{t}=\sqrt{k_{t}/m_{t}}\) and a damping ratio \(\zeta_{t}=c_{t}/(2\omega_{t}m_{t})\) as an isolated system [9]. After its installation, and if the device is properly tuned, the movement of both systems is coupled so the kinetic energy flows from one device to another and, during this energy exchange, a part of it is dissipated in the damping element. For the coupling to be effective, the mass damper must be properly tuned, i.e., its natural frequency \(\omega_{t}\) must be close to and below one of the natural frequencies of the structure [9, 10]. In this sense, the TMD only reduces the response of the structure in a relatively narrow frequency band in the vicinity of the affected mode. If the contribution of several modes needs to be mitigated, then several TMDs should be installed or other more sophisticated approaches (active mass dampers, etc.) should be considered. The overall effectiveness of a TMD, as well as the width of the frequency band it affects, can be controlled by means of the amount of moving mass \(m_{t}\): the greater the moving mass, the more effective it is and the wider the affected frequency band. However, since larger moving masses usually involve higher costs, the minimum one fulfilling certain criteria is commonly installed. As mentioned above, after the installation of a TMD, its movement is coupled to the structural response, making it difficult to estimate its properties as an isolated device (\(m_{t},\omega_{t},\zeta_{t}\)). Even if these properties have been precisely estimated before its installation, it is not possible knowledge of them later, for several reasons. In the short term, the properties of a TMD may be affected by the installation process, which usually involves disassembling and reassembling some of its parts. In the long term, the properties may be modified due to the unavoidable wearing that affects the different elements over time. One common way to cope with these issues is to install multiple TMDs with close tuning frequencies [11; 12]. Together, these TMDs cover a wider frequency range in comparison to a single device, so the potential structural modifications or detuning issues are mitigated to some extent. However, installing more than one TMD may not be possible in some structures (for its cost, location issues, etc. ), and a well-tuned mass damper is more suitable. The most common, but indirect, strategy to evaluate the goodness of the tuning consists in estimating the properties [13; 14] or the frequency response of the ensemble (structure with TMD) [15]. By assessing the modal properties (natural frequency and damping ratio) or the amplitude of the frequency response of the structure in the vicinity of the mitigated mode, the tuning of the TMD can be evaluated and, if required, an iterative re-tuning process may be carried out. However, although useful, none of these procedures provide the sought TMD properties by themselves.