# Tension Force Detection of Short Suspender by Using Laser Doppler Vibrometer
## 3 Results  
### Vibration Frequency under Different Tension Force  
Figure 4 shows the relationships between tension force and vibration frequency for steel rod-type short suspenders of samples 1-3 with different lengths. It can be seen that the vibration frequency of specimens increases with the increase of tension force and decreases with the increase of suspender length.  
Figure 4: Test results under different tension forces.  
Figure 3: Experiment operation for testing vibration frequency by using LDV.  
Figure 2: Test procedure of vibration frequency for prestressed steel rod-type short suspenders by using LDV.  
The typical tension string theory used to describe the relationships between vibration frequency and tension force is demonstrated in Equation (1) [15]. Meanwhile, the modified model considering the bending stiffness of cables is displayed in Equation (2) [18]. These two models have been used to calculate the theoretical value of the vibration frequency of short suspenders of specimens 1-3 on the basis of each level tension force, and the differences between the experimental and calculated values of tension force are listed in Table 2.  
\[T=4 ml^{2}f^{2} \tag{1}\]  
where \(T\) is tension force, \(m\) is linear density of suspenders, \(l\) is length of suspenders, and \(f\) is the first order vibration frequency of short suspenders.  
\[T=4 ml^{2}f^{2}-\frac{\pi^{2}EI}{l^{2}} \tag{2}\]  
where \(EI\) is the bending stiffness of short suspenders.  
\[\delta_{f}=\left|\frac{f_{M}-f_{\rm C}}{f_{M}}\right|\times 100\% \tag{3}\]  
where \(\delta_{f}\) is the deviation of the frequency, \(f_{M}\) is the measured frequency, and \(f_{\rm C}\) is the calculated frequency. It can be seen from Equations (1) and (2) that natural frequencies of different orders all can be used to estimate tension force for cables based on vibration theory, but the expression of the formula is different. If the first-order natural frequency can be clearly obtained by LDV, it is not necessary to do a multi-order modal test. Furthermore, the calculated results obtained by using first order and high order vibration frequency have the same accuracy [25]. Hence, the first order vibration frequency has been used to calculate tension force in this study.  
It can be seen from Table 2 that there is a large error between the experimental and theoretical values of vibration frequency, and this is consistent with the conclusion obtained by [PERSON] et al. [30]. The big error mainly comes from the following two aspects [1,12,14]. On one hand, the boundary conditions in models are simplified as either hinged or fixed constraints, reducing the calculation accuracy. On other hand, the bending stiffness and sag of short suspenders are also ignored in the derivation process of these two models. Therefore, it is difficult to obtain the accurate calculation results of vibration frequency for short suspenders by using the existing theoretical models.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Tension/kN**} & \multirow{2}{*}{\(f_{M}\)**Hz**} & \multicolumn{2}{c}{**Typical Model in Equation (1)**} & \multicolumn{2}{c}{**Modified Model in Equation (2)**} \\ \cline{3-6}  & & \(f_{\rm C}\)/Hz & \(\delta_{f}\)/\% & \(f_{\rm C}\)/Hz & \(\delta_{f}\)/\% \\ \hline
2.3 & 959.38 & 151.70 & 84.19 & 154.00 & 83.95 \\
4.4 & 997.81 & 208.91 & 79.06 & 210.59 & 78.89 \\
6.8 & 1105.63 & 259.90 & 76.49 & 261.25 & 76.37 \\
9.1 & 1155.94 & 300.11 & 74.04 & 301.28 & 73.94 \\
11.2 & 1186.56 & 333.30 & 71.91 & 334.36 & 71.82 \\
14.1 & 1214.06 & 373.97 & 69.20 & 374.91 & 69.12 \\
17.3 & 1238.44 & 413.64 & 66.60 & 414.49 & 66.53 \\
23.1 & 1278.75 & 478.98 & 62.54 & 479.72 & 62.49 \\
26.2 & 1293.44 & 509.29 & 60.62 & 509.98 & 60.57 \\
29.3 & 1307.50 & 538.91 & 58.78 & 539.56 & 58.73 \\
32.4 & 1325.0 & 566.46 & 57.25 & 567.08 & 57.20 \\ \hline \hline \end{tabular}
\end{table}
Table 2: First order vibration frequency of sample 1 under different tension force.