# A Novel Test Rig for the Basic Nonlinear Characterization of Bolted Joints
## 5 Dynamic Tests and Analysis  
### The Excitation and Measurement Systems  
The test rig setup (Figure 11) aims to simplify the structure as a one degree of freedom system, in order to have a clean and direct measurement of the frictionally damped response. For this reason, an electromagnetic shaker fixed on a test-bench provides the dynamic excitation, which is measured by a force transducer screwed on one side to the bottom surface of the screw and on the other side to the shaker. In this way, the only constraint of the system is represented by the location of the input excitation that is continuously measured and controlled during a force controlled, stepped sine test.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Descriptions** & **Value** & **Unit** \\ \hline \(p\) & the pitch & 1.95–2.05 & mm \\ \(d_{2}\) & pitch diameter of bolt & 14.95–15.05 & mm \\ \(d_{out}\) & nut outer diameter & 22.95–20.55 & mm \\ \(d_{in}\) & nut inner diameter & 17.95–18.05 & mm \\ \(\mu_{th}\) & thread friction coefficient & 0.14–0.25 [33] & \\ \(\mu_{b}\) & screw head friction coefficient & 0.19–0.25 [33] & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Bolt geometrical parameters.  
Figure 10: Test results of the applied torque and corresponding bolt tension, (**a**) typical result and linear fitting, (**b**) repeated tests.  
The system response is measured by an accelerometer glued at the outer surface of the upper beam (i.e., the beam A in Figure 4) in the middle of the beam's length. The strain gauges used to assess the value of the static tension preload are also employed to monitor the preload variation during the dynamical tests. Time-varying forces and accelerations are collected and processed by the data acquisition system LMS TestLab. The force-controlled stepped sine method was used to excite the bolted joint in a frequency range containing the bending vibration mode detected by the numerical modal analysis. The frequency response functions (FRFs) were obtained by exciting the bolted joint for 1000 cycles for each frequency step before control iterations are made, which ensures to obtain the steady-state response and more robust control of the force.  
The bolted joint was tested under different controlled levels of excitation in order to study the nonlinear behavior caused by the contact interfaces. The force control provided by the LMS TestLab can be graphically observed in Figure 12a, where the trend of the controlled force vs. the excitation frequency is shown. The results show that the force can be easily controlled if a tolerance no smaller than 10% (\(\pm\)5%) is set with respect to the nominal values of 0.1, 0.3, 0.5, 1, 1.5 N excitation peak amplitude.  
Figure 11: Experimental system for the dynamical tests of the bolted joint.  
Figure 12: Verification of testing results, (**a**) the controlled force amplitude for the selected frequency range, (**b**) repeatability check of the FRF measurements.  
In order to check the repeatability of the test practice, the FRFs of three different tests were obtained by using a 1 N dynamic excitation and 3 kN as the bolt tension (Figure 12b). Although the test results show a negligible variation of the resonance frequency, the amplitude of the response changes significantly after several test repetitions (\(N\approx 10\)) performed at different excitation amplitudes, due to the settling of the contact status at the interfaces. Nevertheless, two different aspects must be pointed out. First, the dynamic response tends to be stable as the number of repetitions increases, including also a variation of the dynamic excitation. Second, the FRFs evaluated in the third test session of the figure (i.e., the tests Test_M+1, Test_M+2, M \(\approx 20\)) were obtained after disassembly/assembly operations of the test rig, and the repeatability of the test measurement is still verified.