# Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling
## 7 Forced Response Computed on Test-Cases  
In this Section, a study of the forced response of a test case is presented. The aim is to show when MMTS is applied for such applications and why it is convenient to use it. Such a problem can be studied, on the other hand, by Direct Time Integration (DTI) of the equations of motion but this is a very time-consuming method which makes difficult a detailed study of the forced response over a wide range of operational frequency. Anyway, here DTI is used to validate the methodology that is developed in this paper. More in detail, given a certain system model, characterized by given mass, stiffness and damping matrices, two parallel studies are developed on the system, applying MMTS and DTI respectively. The validation of the MMTS methodology is made by comparing the peak-to-peak (_P2P_) measures of the multi-harmonic response developed in the time domain, calculated with the two methods. Here, the _P2P_ result is plotted against a reference frequency, which is decided at the beginning of the calculation and defines uniquely all the excitations of the system (both parametric and external excitation). The reference frequency chosen for the _P2P_ plots is the speed of Gear-1, \(\Omega_{1}\). As a matter of fact, the speed of Gear-2, \(\Omega_{2}\), is directly connected to \(\Omega_{1}\) through the gear ratio \(\eta\). Then, all the parametric and external excitations are directly defined. Through the test-case analysis, the dynamic coupling phenomenon is investigated, by remarking its causes and consequences. It is demonstrated that the dynamic coupling, caused by the presence of a time-variant mesh stiffness, leads to a nodal coupling of certain nodal diameters of the meshing gears. To clearly note the phenomenon, a specific test case is built. In more detail, the example which has been used several times in this paper is considered. That is the case of a couple of gears (G\({}_{1}\) and G\({}_{2}\)) having respectively Z\({}_{1}\) = 10 teeth and Z\({}_{2}\) = 20 teeth. The system model of each gear, as it was introduced at the beginning of the paper (Section 2) is constituted by two nodes per sector of the gear (the number of sectors is equal to the number of teeth). As consequence, each gear has number of dof equal to twice the number of its teeth. Being the dimension of the gear model twice the number of the sectors, two modal families of natural frequencies can be derived by performing modal analysis of both the gears considering them as stand-alone components. In Figure 9,b the frequency vs. nodal diameter diagram of the two gears (considered as stand-alone components) is reported, given the mechanical characteristics of the gears shown in table Tables 1 and 2. Figure 9: **(a)**_Frequency vs. ND_ diagram of Gear-1. **(b)**_Frequency vs. ND_ diagram of Gear-2. The gears are coupled by means of a mesh stiffness of the same type described in Section 3. So, it assumes for the \(n_{th}\) pair of meshing teeth a constant value equal to \(k_{t}\) when the pair is in contact, it assumes a null value when it is not. In this test-case \(k_{t}\) is equal to \(10^{6}\) N/m. The mesh stiffness causes a modal coupling between some modes characterized by specific nodal diameters of the two gears respectively. It is good to remember that the mesh stiffness does not have a remarkable influence on the natural frequencies, which remains practically equal to those of the two gears considered as stand-alone components, even though it may have an important influence on the dynamic response of the overall system. In such a case the numbers of teeth \(Z_{1}\) and \(Z_{2}\) has a strong relation with each other. This condition emphasizes the nodal coupling between specific nodal diameters of the gears. It is worth to remind that such a system represents an unusual system because, in practice, gear systems are never designed with such numbers of teeth to avoid the same couple of teeth meshes too often.