## 2 Materials and Methods
Figure 3 shows the modelled reduction in the gear speed amplitude when the NES is included (\"active\" result) with respect to the corresponding amplitude without the NES (\"locked\" result) across the speed range that is achievable in the experimental gear pair setup. Note that the NES is attached to the gear in both scenarios, but it is restrained from oscillating in the locked case in order to eliminate an inertia bias in the comparison of the results. The NES assembly has been designed in order to physically realise the predicted force-deflection profile. Different views of the NES assembly are shown in Figure 4. The mechanism to achieve the desired stiffness nonlinearity is by the deflection of steel  
Figure 3: Numerical frequency response curve for \(k_{N}=430\) Nm/rad\({}^{3}\). Figure 2: (**a**) NNMs without the rigid body mode and using the average meshing stiffness; (**b**) zoom-in for the first mode in the experimentally observed frequency range. shims as the rocker oscillates about its mean position, producing a reactionary torque. The latter is rigidly mounted on the rotating shaft that is connected to the primary (host) system. As the rocker deflects the shims, the NES inertia disc (made of aluminium) is engaged, accepting energy from the primary system. The NES disc hosts the shim mounts and itself is mounted on the rotating shaft (of the primary system) through a bearing connection so that it can freely rotate with respect to the shaft. Deep groove ball bearings have been used to reduce friction between the shaft and the NES disc. Each shim is fixed at its end mounts and assembled so that it is in contact with a brass roller. The rocker rotation moves the rollers, deflecting the shims. The force that is deflecting the shims is calculated using beam bending theory, considering that the shims are fixed at both ends and eccentrically loaded. The key assumptions are as follows:  
* The beam is straight (i.e., it has no curvature);
* The beam material elasticity is the same in both tension and compression;
* The stress in the beam is below the elastic limit;
* The deflections are small;
* The beam undergoes pure bending;
* The friction between the rocker and shim is small (the NES damping will be identified experimentally in a later section);
* It is assumed that the rocker touches the shim at a single point of contact. Figure 4: (**a**) Front, (**b**) inclined and (**c**) exploded CAD views of the NES assembly. The configuration of the shim is shown in Figure 5 for unloaded and loaded scenarios. The total length of the rocker arm is 2 h (see Figure 5a), the length of the shim between the mounts is L and, initially, the roller is situated at the shim's midpoint (unloaded scenario). The shims have a rectangular cross-section with width (w) and thickness (t). Following some counter-clockwise rotation of the rocker by angle \(\theta\), the new rocker position is as shown in Figure 5b as A'B'. Using beam bending theory, the force applied at each shim is given by:  
\[F=\left[\frac{3\,L^{3}E\,I\,I}{a^{\prime 3}\,b^{\prime 3}}\right]\times dy \tag{4}\]  
where \(a^{\prime}=\frac{L}{2}+dx,b^{\prime}=\frac{L}{2}-dx,dy=h(1-\,\cos(\theta)),dx=h \,\sin(\theta)\) and \(E\) is the is modulus of elasticity for steel, while \(I\) is the second moment of area \(\left(w\,t^{3}/12\right)\). Values for the abovementioned parameters are provided in Table 1.  
The centrifugal force generated on the shim during the assembly rotation is negligible; hence, it will be ignored. The direction of the force is parallel to the y-axis at point A\({}^{\prime}\). As a result, the torque applied on the rocker due to deflection angle \(\theta\) on both shims is given by:  
\[I_{G}\ddot{\theta}_{G}-R_{p}C_{M}\dot{x}_{t}-R_{G}K_{M}x_{t}+K_{N}(\theta_{G}- \theta_{N})^{3}+C_{N}\Big{(}\dot{\theta}_{G}-\dot{\theta}_{N}\Big{)}=-T_{G}, \tag{2}\]0  
The above expression describes the torque as a function of the deflection angle \(\theta\) and will be validated both statically and dynamically.