## 2 Experimental Investigation
### Modal Test of the Threaded Structure  
#### 2.3.1 Nonlinear Behavior of the Threaded Specim  
There are three states associated with a mechanical joint subjected to increased loading: sticking, micro-slip, and macro-slip. The transition from one state to another alters the interface stiffness and frictional dissipation, which corresponds to the hysteresis mechanism [21]. The thread contact surface appears to stick under a low excitation force. However, with the gradual increase in the excitation, the contact status transforms from sticking to micro-slip or even macro-slip. The effect of nonlinear factors on the vibration behavior will gradually increase. The contact characteristics of threaded connections are generally affected by several factors, such as the roughness, lubrication, and thread preload. In this experiment, the thread preload was determined as a factor in the investigation of the two connection states (tight and loose). Because the thread preload cannot be accurately quantified, the loose thread state was obtained by turning the tight thread back one turn. The excitation levels were 0.8 and 0.1 N. To improve the measurement precision, the segmented swept method was applied to the threaded specimen (40_2.0). The excitation frequencies were divided into three sections: 100-1500, 2000-3500, and 3500-5000 Hz. The swept speed was 5 Hz/s. Table 3 presents the measurement results. In Table 3, each natural frequency of the threaded tube decreases by varying degrees compared with the threadless specimen (L40), and the natural frequency of the loosely connected thread decreases more significantly. In addition, when the excitation level increases from 0.1 to 0.8 N, the natural frequencies of the tightly connected threads change slightly, whereas the natural frequencies of the loosely connected threads change significantly. This indicates that the increase in the excitation level does not change the contact state of the tight thread; however, the contact state of the loose thread changes from sticking to micro-slip or macro-slip. Figure 4 shows the FRF curves of the thread \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **1 st (Hz)** & **2 nd (Hz)** & **3 rd (Hz)** \\ \hline L30 & 903.06 & 2595.30 & 4398.10 \\ L40 & 948.50 & 2704.40 & 4572.90 \\ L50 & 998.47 & 2816.30 & 4772.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: First three order frequencies of the threadless specimen from the FEM (source elaborated by authors). Figure 3: Measured FRF at sensor location A\({}_{2}\) of the threadless specimen (source elaborated by authors). threaded specimen (40\(2\)0) obtained in the experiments and FEM. It can be seen that the FRF of the threadless specimen obtained with the FEM appears similar to that obtained in the experiment. The first natural frequency is 948.5 Hz, with an error of 1.1% compared with the experiment. In addition, compared with the threadless structure, the measured natural frequency of the threaded specimen decreases from 938.26 to 815.84 and 753.92 Hz. The first natural frequency of the loosely connected thread decreases more obviously, indicating that the loose connection state is more significantly affected by nonlinear factors than the tight connection state. The FRF of the loosely connected thread is compared with that of the tightly connected thread. It can be seen that variation in the excitation level will result in the transition of the thread contact state. With increasing excitation force, the first natural frequency of the loosely connected thread decreases from 753.92 to 748.84 Hz, and the amplitude of the FRF peak decreases from 1.32 to 1.18. This results from the transition from sticking to the micro-slip or macro-slip state of the thread contact with the increase in the excitation level. Accordingly, the frictional damping increases, causing the amplitude and natural frequency to decrease simultaneously.