## 4 Discussion  
Based on these deductions, the numerical model is revisited to explore further agreement with the proposition than one of the shims has been malfunctioning during the experimental tests. Recalculation of the NES-gear system response in the examined frequency range with Equations (1)-(3) updated with the quintic stiffness torque given in Equation (6) leads to the amplitudes shown in Figure 11. The reduction in the nonlinear stiffness coefficient causes a significant shift in the jump-down frequency of the NES under the same damping conditions with the corresponding variation in the gear amplitudes. In fact, the numerical results are in good agreement with the experimentally recorded amplitudes, confirming that the NES was working with nearly half its effective stiffness. Combining this with the static identification leads to the conclusion that one of the NES shims was losing contact with the NES rocker. Nevertheless, and although the NES prototype did not follow the optimised response, the model results in Figure 11 prove the theoretical predictability of the NES design and, by extension, the expected optimised operation, should further refining modification be allowed.  
Further, Figure 12 compares the gear time history when the NES is active, with the gear response recorded with the NES in locked state for pinion speeds of 700 rpm, 800 rpm and 900 rpm. In Figure 12a,b, the active gear amplitude is found reduced, whereas in Figure 12c, it is found nearly unaffected by the presence of the NES. This expected behaviour is consistent with the modelling results and it is attributed to the NES action, as Figure 13 shows. The NES oscillation amplitude when the pinion runs at 800 rpm is significantly higher, leading to a reduction in the gear speed fluctuations. On the other hand, moving to 900 rpm, where the gear is shown to be almost unaffected, the NES is found to oscillate at very low amplitudes.  
Figure 11: Experimental results of peak-to-peak gear amplitude compared with modelling results.  
Figure 12: Time histories of the gear vibrations for active and locked NES and mean speed of the pinion; (**a**) 700 rpm; (**b**) 800 rpm; (**c**) 900 rpm.  
Last, the system energy is considered to further enhance the validation of the NES action. In Figure 13(a), the NNM of the system lying within the frequency range of interest is plotted according to the initial design target as informed by the model in Section 2; the statically identified restoring torque was seen to closely match the design target, and the dynamically identified restoring torque was inferred to correspond to the partial malfunction of the prototype, with only one shim applying effective torque to the gear shaft. Juxtaposing the vibration energy of the system at different applied speeds shows that the system response is dictated by the influence of the NNM corresponding to \(T_{NL}/2\). Additionally, calculation of the energy damped by the NES close to the observed activation range shows the significant transfer of energy to the NES, such as in the cumulative energy dissipated by the NES when the pinion speed was 700 rpm and 800 rpm, as shown in Figure 13(b). The response of the system at these speeds corresponds to the reduced gear amplitudes that were observed in the frequency plot in Figure 11 and the time histories in Figure 12, with an expected drop in the dissipated energy when the NES moves out of its activation range to 900 rpm and 1000 rpm.  
Figure 13: Time histories of the gear and the NES vibrations for active NES and mean speed of the pinion; (**a**) 800 rpm and (**b**) 900 rpm.  
Considering the results from Figures 11-14, the NES is shown to reduce the gear speed fluctuations in a manner predictable from and agreeing with the model. Despite the non-optimised operation of the physical prototype, the fundamental justification supporting the motivation of this experimental study is confirmed.