# Dynamic Identification of Tensile Force in Tie-Rods by Interferometric Radar Measurements
## 5 Estimation of Tensile Force Starting from Vibration Data  
### Modified Lagomarsini and Calderini Approach  
As discussed in Section 2, a suitable tensile force dynamic identification approach suitable for reprocessing interferometric radar data is that by [PERSON] and [PERSON] [14]. Anyway, to have representative results, interferometric radar measurements required installing a corner reflector representing a non-negligible additional mass for the tie-rod, thus affecting the modal parameters of the tie-rod. In particular, the mass of the corner reflector, 1.5 kg, was about 30% of the mass of the tie-rod. Notice that, on the other hand, the mass of the accelerometers is negligible compared to the mass of the tie-rod (see Table 2).  
Figure 5: Comparison between the radar (above) and the accelerometric (below) power spectral density (PSD) in the presence of a corner reflector. Applied tensile force: 50 kN.  
To also consider the additional mass of the corner reflector in the structural model, the dynamic equations in [14] were expressly modified. First, the corner reflector was modeled (Figure 6) as a lumped mass located at the point of the tie-rod where it is installed. Then, the domain representing the beam was divided into two parts at the location of the lumped mass, and two different form functions describing the spatial variation of transverse displacement of the beam were introduced (the time dependence, instead, is described by the same function for the two parts of the beam):  
\[\begin{cases}\phi_{1}(x)=C_{1}\ \sin ax+C_{2}\ \cos ax+C_{3}\ \sinh\beta x+C_{4}\ \cosh\beta x&0\leq x\leq l\\ \phi_{2}(x)=C_{5}\ \sin ax+C_{6}\ \cos ax+C_{7}\sinh\beta x+C_{8}\ \cosh\beta x&l\leq x \leq L,\end{cases} \tag{9}\]  
where \(l\) is the distance of the concentered mass from the left end of the beam and \(C_{1},\ldots,C_{8}\) are unknown constants, to be determined by imposing the boundary conditions as in the original approach, and the continuity conditions in correspondence of the section where the additional mass of the corner reflector is attached:  
\[\left\{\begin{array}{l}\phi_{1}(0)=0\\ \phi_{1}^{{}^{\prime}}(0)-\frac{\gamma}{L}\ \phi_{1}{}^{\prime}(0)=0\\ \phi_{2}^{{}^{\prime}}(L)=0\\ \phi_{2}^{{}^{\prime}}(L)-\frac{\gamma}{L}\ \phi_{2}{}^{\prime}(L)=0,\end{array}\right. \tag{10}\]  
with \(\gamma\) defined in (4), and the continuity conditions:  
\[\left\{\begin{array}{l}\phi_{1}(l)-\phi_{2}(l)=0\\ \phi_{1}{}^{{}^{\prime}}(l)-\phi_{2}{}^{\prime}(l)=0\\ \phi_{1}{}^{{}^{\prime\prime\prime}}(l)-\phi_{2}{}^{{}^{\prime\prime}}(l)=0 \\ \phi_{1}{}^{{}^{\prime\prime\prime}}(l)-\phi_{2}{}^{{}^{\prime\prime\prime}}(l )+\left(\frac{\omega}{2}\right)^{2}\frac{m\gamma^{2}}{\delta pL^{2}}\phi_{1}( l)=0,\end{array}\right. \tag{11}\]  
with \(\gamma\) and \(\delta\) defined in (4) and \(\overline{\omega}\) defined in (7). For writing (10) and (11), the rotational inertia of the corner reflector was neglected, considering it as a concentered mass.  
Now it becomes possible to write the linear algebraic system determining the 8 unknown constants C\({}_{1},\ldots,\) C\({}_{8}\). In this case, as for the original method in Section 2, for determining a non-trivial solution, the coefficient matrix of the system must vanish, yielding to a characteristic equation similar to (2), here omitted for the sake of brevity. Finally, the unknown parameters \(EJ,k\) and \(N\) can be determined by a numerical procedure based on minimizing the same error function \(G\) in (8). Therefore, the problem of the beam supporting a concentrated mass can be handled with the same numerical solution approach proposed in [14] by introducing the same parameters described in Section 2.