# Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling
## 7 Forced Response Computed on Test-Cases
Here, the forced response (expressed as the \(P2P\) measure of the multi-harmonic response developed in the time domain) of the teeth of the two gears is computed, in a given operational speed range (the reference speed is the speed of G\({}_{1}\)) where the excitation of the mode shape in Figure 8 occurs. What is expected is to see a resonance of the G\({}_{1}\) due to the excitation of the ND-5 by some \(EO\) of the mesh force and an \"induced\" resonance of the G\({}_{2}\) due to the action of the latter \(EO\) exciting the G\({}_{1}\). The fact that the second resonance is induced by the first one is demonstrated by the fact that no excitation of the ND of G\({}_{2}\) should be present for that operational frequency conditions. As a matter of fact, by looking at the Campbell diagrams of the gears, you can note that for G\({}_{1}\) (Figure 12a) the involved \(EO\) crosses the natural frequency line in that operational speed range, while for G\({}_{2}\) (Figure 12b) no crossing of the natural frequency lines occurs by the involved \(EO\). Thus, the conclusion is that the second resonance on G\({}_{2}\) is caused by the first one on G\({}_{1}\). The \(P2P\) measure of the multi-harmonic response of the two gears computed using MMTS is reported in Figure 13. Here, also the \(P2P\) measure computed by means of DTI is shown in order to make a comparison between the two results. It is worthy to note that there is a big difference in terms of computational time for the construction of the forced response using the two methodologies (MMTS and DTI). In more detail, a DTI study can take some hours, and the computational time can increase considerably as the number of dof of the system increases as well as the resolution of the operational speed range and the integration time interval increase. As consequence, the computational efforts can be practically unsustainable for systems with a large number of dof and very low damping ratios  
Figure 10: Mode shape of the system. _Nodal Coupling_ between ND-5 of G\({}_{1}\) and ND-10 of G\({}_{2}\). whereby DTI must integrate for larger time interval in order to reach a steady-state response, where the transient part is completely extinguished (i.e., a real test-case of gear system where damping ratio is lower than 0.1%). On the other hand, MMTS, operating in the frequency domain, results in a very fast calculations of the forced response which can take few minutes and then it allows to select the \(EO\) of interest which have an influence in a given operational speed range, neglecting the minor effects of other \(EO\) and so reducing considerably the computational efforts. In both the analysis, two resonance peaks are visible in a given operational speed range (speed of G\({}_{1}\) from 210 Hz to 240 Hz). The blue curves are the resonances of G\({}_{1}\) computed respectively with MMTS and DTI. The same for the red curves. The resonance of G\({}_{2}\) is induced by the resonance of G\({}_{1}\), as anticipated before. By looking at the figures Figure 14a,b showing respectively the FFT of the time domain responses, computed through DTI, of the gears (respectively shown in the figures Figure 15a,b), it is easy to note that the main harmonic component of the multi-harmonic response in both cases is exactly the natural frequency of the mode shape shown in Figure 10 which couples ND5 of G\({}_{1}\) to ND10 of G\({}_{2}\). This represents an additional proof that the resonance of G\({}_{2}\) is directly induced by the excitation of that single mode shape by the mesh force on the G\({}_{1}\). This is a clear example of dynamic coupling between two meshing gears and MMTS allowed to forecast the resonance of G\({}_{2}\) induced by the excitation of the ND of G\({}_{1}\) and this could not be possible if you have considered the gears as stand-alone components. In that case, no interaction between the ND of the gears can be studied. Figure 11: (**a**) Engine Orders of mesh force acting on G\({}_{1}\). (**b**) Engine Orders of mesh force acting on G\({}_{2}\).