## 3 Case Study
### Modal Test  
As shown in Figure 5, the study object is a bolted joint structure which is composed of two lap plates and four bolts. The size of the lap plate is measured at about 200 mm long \(\times\) 80 mm wide \(\times\) 8 mm deep and the connecting length is 80 mm. The lap plate is made  
Figure 4: The flowchart for the parameter identification method of bolted joints under different pre-tightening torque. of aluminum alloy. Bolts and nuts are made from low carbon steel of which the nominal diameter is M10. The material parameters of the above two components are shown in Table 2.  
The test sample bolted structure was made by machining operation. A torque wrench was used to control the pre-tightening torque of a single bolt in the group. In this way, the modal test is carried out by hammering under different pre-tightening torque as shown in Figure 6. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Material** & \(\rho\)**/(kg/m3)** & **E/(Pa)** & **G/(Pa)** \\ \hline Aluminum alloy & 2750 & 6.9 \(\times\) 10\({}^{10}\) & 2.69 \(\times\) 10\({}^{10}\) \\ Low-carbon steel & 7900 & 2.1 \(\times\) 10\({}^{11}\) & 8.08 \(\times\) 10\({}^{10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The material parameters of components. Figure 5: Bolted lap structure (mm). Figure 6: Modal test of the connection structure. The piezoelectric accelerometers used in this modal test, had a mass of 20 g, charge sensitivity of 6.05 pC/ms\({}^{2}\), and frequency range from 0.5 Hz to 5 kHz. The CL-YD-303 hammer had a reference sensitivity of 3.99 pC/N. CRAS V7.0, a vibration and dynamic signal acquisition and analysis system developed by Nanjing AnZheng Software Engineering Company, was used as the analytical instrument, and MaCras was used as the modal analysis software. To simulate free boundary conditions, a spring rope suspension was applied to counteract gravity; this method is effective in avoiding imported errors which affect the dynamic characteristics of the structure. The suspension plane is orthogonal to the test direction to avoid the effect of suspension conditions on the test results. The accelerometer is arranged at the end of the structure to avoid the vibration type node. Thirteen test points are arranged along the length direction of the lapping structure. The sampling frequency is set to 5000 Hz. The arrangement of measuring points is shown in Table 3.  
The pre-tightening torque \(T_{N}\) = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 25, 28 N\(\cdot\)m are applied to the bolt through a torque wrench. The first four order modal frequencies under different pre-tightening forces are obtained through multiple modal tests, which are all shown in Table 4. The experimental modal data are plotted and the relationship between the test frequency and the pre-tightening torque is obtained, as shown in Figure 7.  
As can be seen from Figure 7, with the increase of pre-tightening torque the first four order modal frequencies also increase at the same time. The modal frequencies of the first, the third and the fourth order gradually increase with the increase of the pre-tightening torque, while the second modal frequencies tend to be stable when the pre-tightening torque comes to 14 N\(\cdot\)m.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Pre-Tightening Torque**} & \multicolumn{4}{c}{**Modal Frequency (Hz)**} \\ \cline{2-5}  & **(N\(\cdot\)m)** & **Mode 1** & **Mode 2** & **Mode 3** & **Mode 4** \\ \hline
2 & 375.04 & 962.47 & 2032.03 & 2844.67 \\
4 & 387.64 & 975.00 & 2068.01 & 2910.88 \\
6 & 387.50  
The damping ratio of the bolted connection structure is related to many factors, such as material performance, medium condition, surface roughness, processing method, pre-tightening force state and loading frequency, the influence of which is mostly reflected in nonlinear relations. In addition, the measurement errors will also have an impact on the damping ratio.