# Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling
## 7 Forced Response Computed on Test-Cases
Nevertheless, this choice, which does not affect the validity of the methodology, aims to boost the effect of the analyzed phenomenon of the dynamic coupling so to better understand it. In Figure 10 an example of nodal coupling is reported for the system under analysis. That is the case of a nodal coupling between the nodal diameter ND-5 of G\({}_{1}\) and the nodal diameter ND-10 of G\({}_{2}\). By recalling the vector of the physical coordinates of the system defined in Section 2 (Equation (1)), the mode shape shown in Figure 10 contains both the nodal diameters. It means that an excitation of ND-5 of G\({}_{1}\) affects the vibration of G\({}_{2}\) which will vibrate at the same natural frequency with a ND-10 shape. It is important to remark that the ND-5 of G\({}_{1}\) (considered as stand-alone component) has a natural frequency \(\omega_{5}=1118\) Hz (see Figure 9a). When the G\({}_{1}\) is coupled to G\({}_{2}\) by means of the mesh stiffness, the ND-5 of G\({}_{1}\) still has natural frequency \(\omega_{5}\), but the mode shape of the coupled system associated to that natural frequency shows an ND-5 mode shape coupled to an ND-10 of G\({}_{2}\). In other words, it is numerically demonstrated that a mesh stiffness with such a value of \(k_{t}\) causes the coupling between the nodal diameters of the gears without changing remarkably the natural frequencies with respect to those of the gears (considered as stand-alone components). The choice to keep a value of \(k_{t}\) that does not cause a remarkable change in the natural frequencies is a reasonable assumption that verifies what is experimentally found in the industrial applications. As a matter of fact, real test cases are \begin{table}
\begin{tabular}{l c c} \hline \hline  & **GEAR-1** & \\ \hline
**Z1** & **10** & **Teeth** \\ \hline ND & \multicolumn{2}{c}{[0,...,5]} \\  & Mechanical characteristics & \\ mb & 0.2 & (kg) \\ mc & 1 & (kg) \\ ka & \(10^{7}\) & (N/m) \\ kb & \(10^{7}\) & (N/m) \\ kc & \(10^{8}\) & (N/m) \\ damping ratio & 0.005 & (-) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical characteristics of G1 model. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **GEAR-2** & \\ \hline
**Z2** & **20** & **Teeth** \\ \hline ND & \multicolumn{2}{c}{[0,...,10]} \\  & Mechanical characteristics & \\ mb & 0.2 & (kg) \\ mc & 2 & (kg) \\ ka & \(10^{7}\) & (N/m) \\ kb & \(10^{7}\) & (N/m) \\ kc & \(10^{8}\) & (N/m) \\ damping ratio & 0.005 & (-) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical characteristics of G2 model. characterized by mode shapes showing modal couplings between nodal diameters at given natural frequencies, which are practically the same of those of the stand-alone gears. Thus, the test case under exam aims at simulating a real coupled system where the dynamic characteristics of the mode shapes and natural frequencies remain practically unchanged. As it was described in Section 5, the external force acting on the system is a mesh force which travels from one tooth to another one with the mesh stiffness. In other words, a force of value \(F_{m}\) acts on the teeth nodes (with same value but with opposite direction) of a specific \(n_{th}\) teeth pair when it is in contact. So, as for the mesh stiffness, the Fourier series of the mesh force is studied, as described in Section 5. In Figure 11a,b the harmonic content (or Engine Orders, \(EO\)) of the forces, acting on the two gears respectively, is shown in terms of amplitudes of the various \(EO\). As anticipated in Section 6.2 (Equation (73)), a harmonic index \(EO\) of the travelling force excites mode shapes characterized by a specific ND. Since the ND under analysis for G\({}_{1}\) is 5, the \(EO\) excitation that have been selected are: 5, 15, 25, 35. MMTS allows the computation of the modal response of the mode shape of interest due to the selected \(EO\). The forced response in the physical coordinates is easily derived through Direct Modal Transformation (DMT).