# A Novel Test Rig for the Basic Nonlinear Characterization of Bolted Joints
## 5 Dynamic Tests and Analysis
### Comparisons with the Monolithic Beam  
The two-beam bolted joint, as well as the monolithic beam, are tested under the same loading condition for comparison purposes. Figure 13 shows the two FRFs averaged on three repeated tests (bolt tension equal to 5 kN, excitation force amplitude equal to 1 N). As predicted by the numerical simulations, the resonance frequencies of the two test cases differ for about 400 Hz (449.6 Hz vs. 837.2 Hz for the two-beam bolted joint and the monolithic beam, respectively). The experimental resonance frequencies are slightly lower than the numerical ones (i.e., 449.6 Hz vs. 472 Hz and 837.2 Hz vs. 854 Hz, for the two-beams and single beam test case, respectively), since the real constraints at the contact interfaces are not as ideal as happens for the fully bonded constraints used in the simulation. Both FRFs appear to be slightly asymmetrical with respect to the resonance peak, which suggests the presence of the nonlinear behavior of the bolted structures. It is worth pointing out that such a nonlinear trend is also clear for the single beam test case, where just the contact interfaces between head/nut and beams are present, as well as that at the threads. In order to test the nonlinear behavior of the two structures, a set of FRFs was measured by varying the amplitude of the excitation, while keeping the bolt tension constant and equal to 5 kN (Figure 14). In both cases, the resonance frequency and the response amplitude decrease as the excitation force increases from 0.1 to 2 N, a typical effect that is justified by the changing stiffness and damping due to the nonlinear behavior of the structures during vibration. From a detailed look into the plots of Figure 14, it can be noted a different ratio between the FRFs peak amplitudes corresponding to the highest and lowest excitation force (i.e., 0.1 and 2 N, respectively). In particular, the FRF amplitudes for the monolithic beam vary from 749.6 to 355.6 m/s\({}^{2}\)/N, corresponding to a reduction of the peak response equal to 52%, which is smaller than the 67% obtained for the two-beam bolted joint, where the acceleration amplitude decreases from 430.1 to 142.6 m/s\({}^{2}\)/N. Figure 13: Comparisons of the FRF responses between two beams joint and one monolithic beam. The damping induced at the contact interfaces seems to be much more sensitive to the variation of the excitation force rather than the stiffness, since the frequency only decreases from 453 to 449.2 Hz for the two-beams joint, corresponding to a 0.83% difference. This result is probably due to the slight variation of the joint stiffness that mainly depends on the sticking area, which slightly decreases when the force amplitude is increased, while the thin sliding area varies proportionally more than the central sticking area and produces an important variation in the peak response. For the experimental identification of the modal damping ratio \(\zeta\), the rational fraction polynomials method reduced to SDOF, was used in this study [34]. The variation of the parameters \(\zeta\) for a set of measurements obtained for a fixed value of the bolt tension (5 kN) and different dynamic excitation amplitudes are shown in Figure 15. It can be noted that the damping identified for the two test cases (\(\zeta\) = 1 \(\times\) 10\({}^{-3}\)-\(\star\) 10\({}^{-3}\)) is much larger than that of the metal (about 1 \(\times\) 10\({}^{-4}\)-5 \(\times\) 10\({}^{-4}\)) [35], which indicates that the frictional damping induced at the contact interfaces is the main source of the structural damping. The FRF for the two-beams setup shows a modal damping ratio \(\zeta\) larger than that obtained for the monolithic beam. This proves how the contact interface between two beams represents a source of the friction damping. However, the influence of the contact interfaces of the head/nut-beam and threads contacts cannot be ignored. In fact, the FRFs obtained for the monolithic beam are characterized by a modal damping ratio that is several times larger than that of the material.