# Dynamic Identification of Tensile Force in Tie-Rods by Interferometric Radar Measurements
## 5 Estimation of Tensile Force Starting from Vibration Data
### Estimation of the Tensile Force  
Now, starting from the experimentally estimated frequency, it is possible to apply the modified version of the approach by [PERSON] and [PERSON] described in Section 5.1 to determine the dynamically identified tensile forces to be compared to the known ones to verify the accuracy of the procedure under investigation. Calculations were performed through an expressly written MATLAB code; in particular, the required constrained minimization of the error function (8) was performed by using the MATLAB function _fmincon_. The minimization domain was constructed by assigning as lower limits a bending stiffness \(EJ_{inf}=0\), corresponding to a taut string behavior, and rotational stiffness of the constraints at the ends of the beam \(k_{inf}=0\); accordingly, in view of (1) an upper bound of the tensile force determined by the first natural frequency \(f_{1}\) is:  
\[N_{sup} =4\rho L^{2}f_{1}^{2}. \tag{12}\]  
For the upper limit of the domain, a flexural stiffness increased by 30% over the assumed reference value of the bending stiffness \(EJ_{s}\), that is, \(EJ_{sup}=1.3\ EJ_{s}\), was considered. Moreover, a rotational stiffness of the constraints at the ends of the beam \(k_{sup}=100\ \frac{EJ_{s}}{L}\), practically corresponding to a perfectly clamped constraint. In this case, a closed-form solution for the relationship between the first natural frequency \(f_{1}\) and the tensile force \(N\) is, after some analytical developments:  
\[f_{1}=\sqrt{\frac{N}{4\rho L^{2}}}\Bigg{[}1+2\sqrt{\frac{EJ_{sup}}{NL^{2}}}+ \left(4+\frac{\pi^{2}}{2}\right)\frac{EJ_{sup}}{NL^{2}}\Bigg{]}. \tag{13}\]  
Solving (13) for determining \(N\), it is possible to determine a lower bound for the tensile force \(N_{inf}\). In Table 7, the range variation considered for the tensile force \(N\) and for the parameters \(\gamma\) and \(\delta\) in (4) is summarized. Notice, in passing, that the column of Table 7 containing the upper bound of the tensile force \(N_{sup}\) allows for appreciating the error in determining the tensile force coming from applying the basic natural frequency-based method (considering the first natural frequency). It is easily seen that this error is quite large, ranging from 12.3% (applied tensile force 150 kN) to 55.8% (applied tensile force 50 kN). In what follows, tensile forces estimated starting from interferometric radar determined frequencies are compared to those estimated starting from accelerometric determined frequencies. Figure 7 (left) shows the three-dimensional plot of the objective function \(G\) in (8) as a function of the variables \(N\) and \(\gamma\), for fixed \(\delta=1\); a representation of the values of \(G\) in false color scale in the plane \(N\)-\(\gamma\), also for \(\delta=1\), is represented in Figure 7 (right). In both cases, the evaluations are based on dynamic data acquired by radar interferometry for applied tensile force 50 kN. Similarly, starting again from the same experimental data, Figure 8 (left) shows the three-dimensional plot of \(G\) as a function of the variables \(\gamma\) and \(\delta\), for fixed \(N=50\) kN; a representation of the values of \(G\) in false color scale in the plane \(\gamma\)-\(\delta\), also for \(N=50\) kN, is given in Figure 8 (right). To minimize the objective function G, the same practical tricks discussed in [14] were adopted. In particular, for fast convergence of the optimization algorithm, a good initial estimate of the unknown variables, as the starting point for the \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Applied Tensile** & \(N_{sup}\) **(kN)** & \(N_{inf}\) **(kN)** & \(\gamma_{sup}\) & \(\gamma_{inf}\) & \(\delta_{sup}\) & \(\delta_{inf}\) \\ \hline
50 & 61.4 & 22.1 & 100 & 0 & 1.3 & 0 \\
100 & 116.3 & 58.7 & 100 & 0 & 1.3 & 0 \\
150 & 168.4 & 97.3 & 100 & 0 & 1.3 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Ranges for the variables of the minimization problem. needed.