## 2 Materials and Methods  
The numerical model that is used in this paper has been presented in detail in [PERSON] et al. [27]. For completeness, we reiterate some of the key aspects and findings from this previous work. Figure 1 shows a sketch of the meshed gears with an NES attached on the gear. The equations of motion of the system shown in Figure 1 are given in Equations (1)-(3) below. The reader is referred to previous work [27] for a comprehensive description and detailed documentation of the model parameters. Note that the model parameters are informed by the experimental setup that is used in the current work, and therefore the interested reader is referred to [27] for the gear pair setup details. \[I_{p}\ddot{\theta}_{p}+R_{p}C_{M}\dot{x}_{t}+R_{p}K_{M}x_{t}=T_{p}+T_{f}\cos \omega t, \tag{1}\]  
\[I_{G}\ddot{\theta}_{G}-R_{p}C_{M}\dot{x}_{t}-R_{G}K_{M}x_{t}+K_{N}(\theta_{G}- \theta_{N})^{3}+C_{N}\Big{(}\dot{\theta}_{G}-\dot{\theta}_{N}\Big{)}=-T_{G}, \tag{2}\]  
\[I_{N}\ddot{\theta}_{N}-K_{N}(\theta_{G}-\theta_{N})^{3}-C_{N}\Big{(}\dot{ \theta}_{G}-\dot{\theta}_{N}\Big{)}=0 \tag{3}\]  
where \(\ddot{\theta}_{G,P,N}\), \(\dot{\theta}_{G,P,N}\), \(\theta_{G,P,N}\) denote the angular acceleration, velocity and position of the gear (subscript G), the pinion (subscript P) and the NES (subscript N), respectively. \(R_{P}\) is the pinion radius, \(R_{G}\) is the gear radius, \(T_{P}\) is the pinion average torque, \(T_{f}\) is the pinion fluctuating torque of angular frequency \(\omega\) and \(T_{G}\) is the gear torque. The time-varying meshing stiffness is a function of the angular position of the pinion and the applied torque and it is denoted by \(K_{M}\). The mesh displacement \(x_{t}\) is described by a piecewise linear function with backlash [27]. The damping coefficient in the pinion-gear contact is \(C_{M}\), and the NES damping coefficient is denoted by \(C_{N}\), whereas the NES nonlinear stiffness coefficient is \(K_{N}=k_{3}\), where an essentially cubic nonlinearity is assumed. Rich information for establishing targeted energy transfer in a primary system with an NES is contained in the NNMs of the coupled nonlinear system of oscillators. However, the piecewise linear profile of the gear teeth deflection induced by the presence of backlash introduces difficulties in attaining the NNMs. In order to mitigate computational difficulties, the backlash is ignored in the calculation of the NNMs, leading to a constant average meshing stiffness, without, as will be found in the experimental results, affecting the conclusions drawn. Figure 2 shows the NNMs of the gear pair with an NES possessing a cubic stiffness of \(k_{3}=430\) Nm/m\({}^{3}\), as informed by optimising the NES properties using the presented model [27]. Figure 1: Sketch of the general layout of a gear pair equipped with an NES; \(I_{P}\) represents the pinion, \(I_{G}\) represents the gear; \(I_{N}\) represents the NES; \(k_{M}\) denotes the gear pair meshing stiffness and \(k_{N}\) the stiffness of the NES attachment. The NNM corresponding to the primary system (gear pair) is the one resulting from the meshing stiffness. Due to the very high value of this stiffness, the resulting NNM exists at very high frequencies (solid red line in Figure 2a), far away from the frequency ranges typically observed in the speed fluctuations of automotive driveshafts. On the other hand, the essential nonlinearity of the NES stiffness sweeps through the entire frequency range depending on the vibration energy. Closer observation of this NNM in the frequency range of interest (up to 200 Hz) in Figure 2b reveals the NES frequency coverage that motivates this experimental study to reduce gear speed fluctuations by the action of an NES attachment. Previous modelling work [27] has shown that the inclusion of the NES in the gear pair results in a promising reduction in the gear fluctuations that is linked to the gear rattling root cause.