# On the Estimation of the Moving Mass of a TMD Installed on a Lively Structure
## 2 Theoretical Background  
A tuned mass damper, and any other SDOF system attached to a structure, can be conceptually modeled as shown in Figure 1. Without loss of generality, the SDOF system is oriented in the vertical direction, but it could be placed in any other direction as long as it properly couples to the response of the structure. The movement of this conceptual model, which can be subjected to an external force \(f(t)\) and/or to a base acceleration \(\ddot{x}_{s}\) for being attached to a lively structure, can be described by means of the expression in Equation (1), where \(m_{t}\), \(c_{t}\) and \(k_{t}\) account for the physical properties of the SDOF system (moving mass, damping constant and stiffness constant, respectively). The natural frequency, \(\omega_{t}\) in rad/s, and the damping ratio \(\zeta_{t}\) of the isolated SDOF system can be defined so that \(\omega_{t}^{2}=k_{t}/m_{t}\) and \(2\zeta_{t}\omega_{t}=c_{t}/m_{t}\). Also, the relative coordinate \(\xi(t)=x_{t}(t)-x_{s}(t)\) is defined to simplify the way the base movement is accounted for. After introducing this variable change into Equation (1), it is divided by the moving mass \(m_{t}\) and the definitions of the modal properties are introduced, leading to Equation (2). \[m_{t}\ddot{x}_{t}(t)+c_{t}(\dot{x}_{t}(t)-\dot{x}_{s}(t))+k_{t}(x_{t}(t)-x_{s} (t))=f(t) \tag{1}\]  
\[\dot{\xi}(t)+2\zeta_{t}\omega_{t}\dot{\xi}(t)+\omega_{t}^{2}\xi(t)=\frac{1}{m_{ t}}f(t)-\ddot{x}_{s}(t) \tag{2}\]  
Equation (2) is transformed into the Laplace domain by substituting the time variable (\(t\)) by the complex variable (\(s\)) and taking into account that \(\mathcal{L}\{\dot{\xi}(t)\}=s^{2}\Xi(s)\), \(\mathcal{L}\{\dot{\xi}(t)\}=s\Xi(s)\) and \(\mathcal{L}\{\dot{x}_{s}(t)\}=s^{2}X_{s}(s)\), where \(\mathcal{L}\{\cdot\}\) is the Laplace transform operator and the uppercase functions represent the Laplace transform of the corresponding lowercase functions. Then, by letting \(\omega\) be the excitation frequency in rad/s, the complex variable \(s\) is substituted by \(j\omega\) (so \(s^{2}=-j\omega\)), to obtain Equation (3), which is in the frequency domain. This last equation is the core of the novel methodology presented in the next section. Note that, if no force is applied (\(f(t)=0\) N, \(F(\omega)=0\) N), which is the main assumption made in [16], then the term that explicitly contains the moving mass \(m_{t}\) in  
Figure 1: Conceptualization of a TMD, modeled as an SDOF system, installed on a structure. Equation (3) disappears and the transmissibility expression in Equation (4), one of the two expressions presented in [16], is derived. Also note that the structural properties are not included in this formulation, so it is not necessary for them to be known for the presented methodology to be applied. \[(-\omega^{2}+j2\zeta_{t}\omega_{t}\omega+\omega_{t}^{2})\Xi(\omega)=\frac{1}{m_{ t}}F(\omega)+\omega^{2}X_{s}(\omega) \tag{3}\]  
\[Y_{s_{\varphi}^{*}}(\omega)=\frac{X_{t}(\omega)-X_{s}(\omega)}{X_{s}(\omega)}= \frac{\omega^{2}}{-\omega^{2}+j2\zeta_{t}\omega_{t}\omega+\omega_{t}^{2}} \tag{4}\]  
Finally, in this work, some frequency response functions are computed from experimental data. The experimental data always consist in two time series, recorded at a certain sample rate (\(f_{s}\) in samples per second, or S/s). One will act as the input, \(x(t)\), and the other as the output, \(y(t)\). Once registered, the time domain signals may be filtered to remove any undesirable frequency components or detrended to remove any linear trend. In order to estimate the frequency response function that relates the input and the output, both time domain signals are windowed to divide the whole time series into segments of a certain size (\(N_{t}\)). If required, the window function may also weight the segments to avoid frequency leakage problems.