# Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling
## 6 Forced Response Computation with MMTS
### Forced Response Computation Using MMTS  
MMTS operates by substituting the independent time variable \(t\) with the time scales \(t_{0}\), \(t_{1}\), \(t_{2}\),..., where \(t_{0}=\varepsilon^{0}t\), \(t_{1}=\varepsilon^{1}t\) and \(t_{2}=\varepsilon^{2}t\) and \(\varepsilon\) is the scale factor which describes the time scale. The relation between the original time variable and the new time scales is expressed in Equation (37). Here, the series approximating the old variable \(t\) is truncated at the second order (power \(\varepsilon^{2}\)):  
\[t=t_{0}+t_{1}+o\Big{(}\varepsilon^{2}\Big{)}=t+\varepsilon t+o\Big{(} \varepsilon^{2}\Big{)}=t+\tau+o\Big{(}\varepsilon^{2}\Big{)}. \tag{37}\]  
As a consequence, the dependent variable \(u_{n}(t)\) as well as the derivative operators need to be written (Equations (38)-(40)), in the new time scale variables:  
\[u_{n}=u_{n0}(t,\tau)+\varepsilon\;u_{n1}(t,\tau)+o\Big{(}\varepsilon^{2}\Big{)},\;n=1\div N; \tag{38}\]  
\[\frac{d}{dt}\Rightarrow\frac{\partial}{\partial t}+\varepsilon\frac{\partial }{\partial\tau}+o\Big{(}\varepsilon^{2}\Big{)}, \tag{39}\]  
\[\frac{d^{2}}{dt^{2}}\Rightarrow\frac{\partial^{2}}{\partial t^{2}}+2 \varepsilon\frac{\partial^{2}}{\partial t\partial\tau}+o\Big{(}\varepsilon^ {2}\Big{)}. \tag{40}\]  
The solution to the problem requires the proper manipulation of the equations of motion. Recalling Equation (36), the manipulation consists of associating some terms of the equation to the coefficient \(\varepsilon\) (Equations (41)-(43)). Let us associate the time-variant part of the stiffness matrix, \(\hat{D}(t)\), the damping matrix \(\varepsilon_{n}\) and the modal force \(\hat{P}_{n}(t)\) to the scale factor \(\varepsilon\):  
\[\hat{D}(t)=\varepsilon\;D(t), \tag{41}\]  
\[\varepsilon_{n}=\varepsilon\;c_{n}; \tag{42}\]  
\[\hat{P}_{n}(t)=\varepsilon\;P_{n}(t). \tag{43}\]  
Substituting Equations (41)-(43) into Equation (36), the latter becomes:  
\[\ddot{u_{n}}+\varepsilon\;c_{n}\dot{u_{n}}+\omega_{n}^{2}\;u_{n}+\sum_{r=1}^{ N}\{\varepsilon\;D_{nr}(t)\;u_{r}\}=\varepsilon\;P_{n}(t),\;n=1:N. \tag{44}\]  
Once the new equation of motion Equation (44) is defined, the MMTS operates by substituting the new expression of the modal response \(u_{n}(t)\) (Equation (38)) and the new derivative operators (Equations (39) and (40)) into Equation (44). The resulting equation of motion will be an equation where all the terms are characterized by a multiplying coefficient that is in general a power of the scale factor \(\varepsilon\) (\(\varepsilon^{n}\)). Then, a separation of the terms according to the power of \(\varepsilon\) is performed, by creating \(n\) different equations. The new equations are reported below (Equations (45) and (46)). * Equation corresponding to the power \(\varepsilon^{0}\): \[\ddot{u_{n0}}+\omega_{n}^{2}u_{n0}=0\;.\] (45)
* Equation corresponding to the power \(\varepsilon^{1}\): \[\ddot{u_{n1}}+\omega_{n}^{2}u_{n1}=-2\frac{\partial^{2}u_{n0}}{\partial t \partial\tau}-\sum_{\tau=1}^{N}\{D_{nr}(t)\;u_{r0}\}-c_{n}\frac{\partial u_{n0 }}{\partial t}+P_{n}(t).\] (46)The solution of Equation (45) can be written in the general form  
\[u_{n}=u_{n0}(t,\tau)+\varepsilon\;u_{n1}(t,\tau)+o\Big{(}\varepsilon^{2}\Big{)},\;n=1\div N; \tag{38}\]0  
where \(A_{n}(\tau)\) is a function of the time variable \(\tau\). Substituting Equation (47) into Equation (46) and adopting the notation in Equation (48), one obtains Equation (49):  
\[u_{n}=u_{n0}(t,\tau)+\varepsilon\;u_{n1}(t,\tau)+o\Big{(}\varepsilon^{2}\Big{)},\;n=1\div N; \tag{38}\]1  
\[ \begin{array}{c}u_{n1}^{\sim}+\omega_{n}^{2}u_{n1}=-2\big{[}i\omega_{n}A_{n }^{\prime}e^{i\omega_{n}t}+CC\big{]}-\sum\limits_{\tau=1}^{N}\big{\{}D_{nr}(t )\ \big{[}A_{r}e^{i\omega_{r}t}+CC\big{]}\big{\}}-\\ c_{n}\big{[}i\omega_{n}A_{n}e^{i\omega_{n}t}+CC\big{]}+P_{n}(t),\end{array}  \tag{49}\]  
where the quantity \(CC\) represents the complex conjugate of the previous term.