# Dynamic Identification of Tensile Force in Tie-Rods by Interferometric Radar Measurements
## 2 Tensile Force Dynamic Identification in Tie-Rods: Theoretical Approaches
This system has a non-trivial solution only if the determinant of the coefficient matrix of the system vanishes, that is, if the following equation is fulfilled:  
\[ \begin{split} 2 kEJ(\alpha^{2}+\beta^{2})[\beta\,\sin(\alpha L)\cosh( \beta L)-\alpha\,\cos(\alpha L)\sinh(\beta L)]+\sin(\alpha L)\sinh(\beta L) \cdot\\ \cdot[E^{2}J^{2}(\alpha^{2}+\beta^{2})^{2}-k^{2}(\alpha^{2}- \beta^{2})]-2\alpha\beta k^{2}[\sin(\alpha L)\cosh(\beta L)-1]=0,\end{split}  \tag{2}\]  
where the tensile force appears inside \(\alpha\) and \(\beta\), being:  
\[\alpha=\sqrt{\frac{N}{2 EJ}\bigg{(}\sqrt{1+\frac{4\omega^{2}\rho EJ}{N^{2}}}-1 \bigg{)}},\quad\beta=\sqrt{\frac{N}{2 EJ}\bigg{(}\sqrt{1+\frac{4\omega^{2}\rho EJ }{N^{2}}}+1\bigg{)}}, \tag{3}\]  
with \(\omega\) the separation variable, representing the circular frequency of the part of the response function depending on the time. The method consists of finding the unknown parameters \(EJ\), \(k\) and \(N\), which fulfill (2). Nevertheless, a general closed-form solution does not exist, unlike simple extreme cases. Figure 1: General tie-rod structural model for tensile force dynamic identification approaches. Therefore, numerical methods should be employed. To this aim, in [14], the following three parameters are introduced:  
\[\varphi=2\pi^{2}\frac{EJ_{S}}{NL^{2}},\quad\gamma=\frac{kL}{EJ_{S}},\quad\delta= \frac{EJ}{EJ_{S}}, \tag{4}\]  
where \(EJ_{s}\) represents an empirical reference value for the bending stiffness \(EJ\), (2) assumes the following a-dimensional form:  
\[ \begin{array}{l}g(\varphi,\gamma,\delta,\frac{\omega_{n}}{\varpi})=\sin\alpha \,\sinh\!\beta\!\left[\left(\alpha^{2}+\beta^{2}\right)^{2}-\frac{\gamma^{2}} {\delta^{2}}(\alpha^{2}-\beta^{2})\right]\\ +2\frac{\gamma}{\delta}(\alpha^{2}+\beta^{2})(\beta\,\sin\alpha\,\cosh\beta- \alpha\,\cos\alpha\sinh\!\beta)-2\frac{\gamma^{2}}{\delta^{2}}\alpha\beta(\cos \alpha\,\,\cosh\beta-1),\end{array}  \tag{5}\]  
(3) becomes:  
\[\alpha=\sqrt{\frac{1}{\delta\varphi}\left(\sqrt{1+\delta\varphi^{2}\Big{(} \frac{\omega_{n}}{\varpi}\Big{)}^{2}}-1\right)},\quad\beta=\sqrt{\frac{1}{ \delta\varphi}\Bigg{(}\sqrt{1+\delta\varphi^{2}\Big{(}\frac{\omega_{n}}{ \varpi}\Big{)}^{2}}+1\Bigg{)}}, \tag{6}\]  
where \(n=1,2,\dots\), and \(\varpi\) can be assumed as the exact solution for the first modal frequency evaluated in the ideal case of a \"pinned-pinned beam\" with bending stiffness \(EJ_{s}\):  
\[\varpi=\pi^{2}\sqrt{\frac{EJ_{S}}{\rho L^{4}}} \tag{7}\]  
(see [14] for details). In general, the knowledge of three natural frequencies \(\omega_{n}\), \(n=1,2,\dots\), is necessary for determining \(\varphi\), \(\gamma\) and \(\delta\), related to the unknown parameters \(EJ\), \(k\) and \(N\). Specifically, a system of three nonlinear equations should be solved through numerical techniques. However, this procedure may be affected by significant instabilities; therefore, in [3], it is suggested to determine the three unknown parameters through the minimization of an error function \(G\) defined as follows:  
\[G(\varphi,\gamma,\delta)=g_{1}^{2}+g_{2}^{2}+g_{3}^{2}, \tag{8}\]  
where \(g_{i}=g(\varphi,\gamma,\delta,\frac{\omega_{i}}{\varpi})\), \(i=1,2,3\). Furthermore, in [14], a suitable procedure for minimizing (8), performed through an international mathematics and statistics library (ISML) routine. In short, the approach in [14] considers as input data for the tie-rod the length \(L\), easily measurable in situ, and the density, and require an empirical assumption on the reference value \(EJ_{S}\) for the bending stiffness, to be estimated through direct measurements of the tie-rod's section and the quality of the steel. Then, it allows for identifying the tensile force \(N\) by experimentally determining three natural frequencies. This can be due to employing even only one accelerometer; the possibility of determining the tensile force without any information about mode shapes makes this approach suitable for reprocessing interferometric radar measurements.