# Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling
## 6 Forced Response Computation with MMTS
### Forced Response Computation Using MMTS
In order to solve the \(N\) equations of motion (Equation (49)) in the modal coordinate \(u\) it is necessary to develop the terms \(D_{nr}(t)\) and \(P_{n}(t)\) in their harmonic series:  
\[D(t)=D_{1}(t)+D_{2}(t)+D_{3}(t), \tag{50}\]  
\[D(t)=\sum\limits_{s_{1}=1}^{\infty}\big{[}D_{1}^{s_{1}}\ e^{i\,s_{1}\Omega_{1} t}+\overline{D_{1}^{s_{1}}}e^{-is_{1}\Omega_{1}t}\big{]}+\sum\limits_{s_{2}=1}^{ \infty}\big{[}D_{2}^{s_{2}}\ e^{i\,s_{2}\,\Omega_{2}\ t}+\overline{D_{2}^{s_{ 2}}}e^{-i\,s_{2}\Omega_{2}t}\big{]} \tag{51}\]  
\[ \begin{array}{c}P(t)=P_{1}(t)+P_{2}(t),\end{array}  \tag{52}\]  
\[ \begin{array}{c}P(t)=P_{1,0}+P_{2,0}+\sum\
olimits_{k_{1}=1}^{\infty}\big{[} P_{1}^{k_{1}}\ e^{ik_{1}\Omega_{1}t}+\overline{P_{1}^{k_{1}}}e^{-i\,k_{1}\Omega_{1}t} \big{]}\\ \hskip 14.226378 pt+\sum\
olimits_{k_{2}=1}^{\infty}\big{[}P_{2}^{k_{2}}\ e^{ik_{2}\Omega_{2}t}+\overline{P_{2}^{k_{2}}}e^{-i\, k_{2}\Omega_{2}t}\big{]}.\end{array}  \tag{53}\]  
It is worth to remind that the fundamental frequency \(\Omega_{1}\) is the speed of \(\mathrm{G_{1}}\), the fundamental frequency \(\Omega_{2}\) is the speed of \(\mathrm{G_{2}}\), linked to \(\Omega_{1}\) through the gear ratio \(\eta=\frac{\mathrm{Z1}}{\mathrm{Z2}}\) and the fundamental frequency \(\Omega_{3}\) is the frequency whereby a generic pair of teeth meshes (see Equations (11)-(13)). Substituting the expressions Equation (51) and Equation (53) into Equation (49), one obtains the extended \(p_{th}\) equation of motion in modal coordinates with all the time-variant parameters developed inside (see Appendix A, Equation (A2)). This equation contains all the harmonics of the excitation force, but they can be treated singularly by computing the forced response to each harmonic component of the force. Finally, a sum of all the response contributions can be performed, thanks to the superimposition effect principle, so to compute the overall multi-harmonic response. Thus, the following discussion analyzes the forced response to the generic \(k_{th}\) harmonic component of \(P_{1}(t)\) acting on the teeth of \(\mathrm{G_{1}}\). The same approach is used to compute the forced response to the second set of harmonics \(P_{2}(t)\) acting on the nodes of \(\mathrm{G_{2}}\), but here they are not treated for sake of clarity. Let us consider the following equation (Equation (54)) of the \(p_{th}\) modal equation of the system, where only the generic \(k_{th}\) harmonic component of the force function \(P_{1}(t)\) is considered:  
\[ \begin{array}{c}\ddot{u_{p1}}+\omega_{p}^{2}u_{p1}=-2i\omega_{p}A_{p}^{ \prime}e^{i\omega_{p}t}\\ -\sum\
olimits_{s_{1}=1}^{N}\sum\
olimits_{s_{2}=1}^{\infty}\big{[}D_{1}^{s_{1 }}\ e^{i\,s_{1}}\ A_{r}\ e^{i(s_{1}\Omega_{1}+\omega_{r})t}+\ D_{1}^{s_{1}}\ \overline{A_{r}}\ e^{i(s_{1}\Omega_{1}- \omega_{r})t}\ \big{]}\\ -\sum\
olimits_{r=1}^{N}\sum\
olimits_{s_{2}=1}^{\infty}\big{[}D_{2}^{s_{2}} \ \overline{A_{r}}\ e^{i(s_{2}\Omega_{1}+\omega_{r})t}+\ D_{2}^{s_{2}}\ \overline{A_{r}}\ e^{i(s_{2}\Omega_{1}- \omega_{r})t}\ \big{]}\\ -\sum\
olimits_{s_{1}=1}^{\infty}\big{[}D_{3}^{s_{2}}\ \overline{A_{r}}\ e^{i(s_{ 2}\Omega_{1}+\omega_{r})t}+\ D_{3}^{s_{3}}\ \overline{A_{r}}\ e^{i(s_{3}\frac{\Omega_{1}}{\Omega_{1}- \omega_{r})t}}\ \big{]}\\ -ic_{p}\omega_{p}A_{p}e^{i\omega_{p}t}+P_{1}^{k_{1}}\ e^{i\,k_{1}\Omega_{1}t}+ CC.\end{array}  \tag{54}\]  
It is now possible to investigate and remove the unwanted secular terms, inside the latter equation. The elimination of secular terms represents a solvability condition for the solution of the problem, because of the additional freedom introduced with the new independent variables. In order to eliminate  
Since \(a_{p}\) is a complex quantity, it is convenient to express \(a_{p}\) according to its real and imaginary parts (Equations (68)-(70)).