# Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling
## 6 Forced Response Computation with MMTS
### Forced Response Computation Using MMTS
The analytical expression of the real and imaginary parts is derived as a function of \(\sigma\):  
\[a_{p}=a_{R}+i\,a_{I}; \tag{68}\]  
\[a_{R}=\frac{\mathcal{P}_{I}\big{(}\mathcal{D}_{I}-c_{p}\omega_{p}\big{)}+ \mathcal{P}_{R}\big{(}2\sigma\omega_{p}+\mathcal{D}_{R}\big{)}}{\omega_{p}^{2} \Big{(}c_{p}^{2}+4\sigma^{2}\Big{)}-\mathcal{D}_{R}^{2}-\mathcal{D}_{I}^{2}}, \tag{69}\]  
\[a_{I}=\frac{a_{R}\big{(}\mathcal{D}_{I}+c_{p}\omega_{p}\big{)}-\mathcal{P}_{I} }{2\sigma\omega_{p}+\mathcal{D}_{R}}, \tag{70}\]  
where:  
\[P_{1}{}^{k_{1}}=\mathcal{P}_{R}+i\,\mathcal{P}_{I}, \tag{71}\]  
\[\mathcal{D}=\mathcal{D}_{R}+i\,\mathcal{D}_{I}, \tag{72}\]  
The latter expressions (Equations (68)-(72)) allow the computation of the modal response of the \(p_{th}\) mode shape in the frequency domain. Each mode shape, connected to a specific nodal diameter of the system, is excited in resonance by some \(EO\) of the mesh force, according to the law reported in the following equation Equation (73), from gear dynamics theory [9]:  
\[EO=mZ\pm ND\,\ m=1,2,\dots. \tag{73}\]  
Thus, the construction of the multi-harmonic forced response is computed by considering, one by one, each \(EO\) of the mesh force, associating it to the mode shapes, which are described by the relative nodal diameter \(ND\), excited by the selected \(EO\) and computing the modal responses. Once the modal responses in the frequency domain are computed for all the \(EO\), they are transformed through DMT (Equation (30)), passing from modal coordinates to physical coordinates, and the forced response of a given node is developed in the time domain (in our case the nodes of the teeth of the gears) as the sum of all the mode shapes contributions (i.e., DMT). The validity of the superimposition effect principle (as explained in the Introduction) allows the summation of all the responses due to the different \(EO\) in the time domain. The result will be a multi-harmonic response, developed in the time domain. Since the response is not described by a single harmonic it is not possible to acquire the amplitude of the time domain response. Thus, the latter will be expressed by acquiring the peak-to-peak measure of the time domain trend as function of a reference frequency (which can be the speed of G\({}_{1}\) or the speed of G\({}_{2}\) as well) defining uniquely the excitations (both parametric and external) of the overall system. As a matter of fact, setting a certain speed for G\({}_{1}\) means also setting the speed of G\({}_{2}\) since the speed of the gears are linked by the gear ratio. As consequence, the mesh stiffness and mesh force, representing the parametric and the force excitations respectively, directly depend on the speed of the two gears. Finally, the reference frequency describes uniquely the operational conditions of the overall system. In the next section an example of forced response is computed on a dedicated test case and a comparison with Direct Time Integration (DTI) method is made to validate the MMTS methodology, developed into this paper.