# Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling
## 6 Forced Response Computation with MMTS
### Modal Analysis and Transformation of the Equation of Motion in Modal Coordinates  
Since the stiffness matrix \(\hat{K}(t)\) contains time-variant elements, the computation of modal analysis cannot be performed in general. For this reason, the computation of the modal analysis is performed on the mean part of the stiffness matrix \(K_{C}\) (Equation (14)) (\(K_{C}\) is a symmetric matrix). Thus, let us consider the time-invariant (unforced and undamped) part of the equation of motion Equation (4) (Equation (28)) and perform the modal analysis (Equation (29)). \[M\ddot{x}+K_{C}x=0; \tag{28}\]  
\[\det\Bigl{(}K_{C}-\omega^{2}M\Bigr{)}=0,\;\mbox{\it eigenvalue problem}\;\to\; \omega_{n}^{2},\;\psi_{n},\;n=1\div N; \tag{29}\]  
\[\Psi=\bigl{[}\psi_{1},\;\ldots,\psi_{n},\ldots,\psi_{N}\bigr{]}_{N\times N},\; \mbox{\it modal matrix}\]  
\(\omega_{n},\;\mbox{\it natural frequency}\)  
From modal analysis, the natural frequencies \(\omega_{n}\) and mode shapes \(\psi_{n}\) of the (mean) overall system are obtained. Then, let us apply Direct Modal Transformation (DMT, Equation (30)) to the equation of motion using the modal matrix \(\Psi\) and multiply the latter by the transpose of the modal matrix \(\Psi^{T}\). The resulting equation written in modal coordinates (\(u\), vector of modal coordinates) is reported in Equation (31). \[x=\Psi u\;. \tag{30}\]  
\[M_{mod}\;\ddot{u}+\hat{C}_{mod}\;\dot{u}+K_{c,mod}\;u+\hat{D}(t)\;u=\hat{P}(t), \tag{31}\]  
with: \(M_{mod}\), modal mass matrix; \(\hat{C}_{mod}\), modal damping matrix; \(K_{c,mod}\), modal mean stiffness matrix;  
\[\hat{D}(t)=\Psi^{T}\hat{K}_{V}(t)\Psi; \tag{32}\]  
\[\hat{P}(t)=\Psi^{T}\hat{F}(t) \tag{33}\]  
If the mode shapes are normalized with respect to the unitary modal masses, \(M_{mod}\) is equal to the identity matrix and \(K_{c,mod}\) is a diagonal matrix having on its main diagonal the eigenvalues of the system \(\omega_{n}^{2}\). As anticipated in Section 2, damping is not modelled physically into the model. For the sake of simplicity, damping is introduced by means of the modal damping ratio \(\varsigma_{n}\) to be associated to each \(n_{th}\) mode shape. It is possible to obtain the modal damping \(\hat{c}_{n}\) as:  
\[\hat{c}_{n}=2\frac{\varsigma_{n}}{\sqrt{k_{mod}n\;\times\;m_{mod}n}}=2\frac{ \varsigma_{n}}{\sqrt{\omega_{n}^{2}}},\;=1\div N; \tag{34}\]  
with \(m_{mod}\) modal mass of the \(n_{th}\) mode shape (\(m_{mod}=1\), if the mode shapes are normalized with respect to the unitary modal masses) and \(k_{mod}n\) modal stiffness of the \(n_{th}\) mode shape (\(k_{mod}=\omega_{n}^{2}\), if the mode shapes are normalized with respect to the unitary modal masses). Then, the modal damping matrix \(\hat{C}_{mod}\) is a diagonal matrix having on its main diagonal the modal damping \(\hat{c}_{n}\), satisfying the following relation that allows computation of the damping matrix in physical coordinates. \[C=\Psi^{T-1}\;\hat{C}_{mod}\;\Psi^{-1} \tag{35}\]  
The diagonalization of most of the matrices inside the equation of motion allows us to write singularly the equations of motion in modal coordinates (Equation (36)). The only term that is not diagonalized is \(\hat{D}(t)\) being a non-symmetric matrix that was not involved in the modal analysis. As a consequence, in the \(n_{th}\) equation of motion in modal coordinates \(\hat{D}(t)\) must be expressed as sum of the products of the elements \(\hat{D}_{nr}(t)\) (elements of the matrix \(\hat{D}(t)\) at \(n_{th}\) row and \(r_{th}\) column) times the \(r_{th}\) modal displacement \(u_{r}\). \[\ddot{u_{n}}+\hat{c}_{n}u_{n}+\omega_{n}^{2}\;u_{n}+\sum_{r=1}^{N}\bigl{\{} \hat{D}_{nr}(t)\;u_{r}\bigr{\}}=\hat{P}_{n}(t),\;\;\;n=1\div N \tag{36}\]  
The Equation (36) represents the useful equation for the development of MMTS by means of which it is possible to compute the modal response \(u_{n}\), in the frequency domain, of the generic mode shape subjected to a modal force \(\hat{P}_{n}(t)\).