# Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling
## 5 Excitation Force  
A mesh force applied to two meshing teeth excites the system. The mesh force \(F(t)\) of Figure 3 corresponds to a torque (for example applied on G\({}_{1}\)) and a resistant torque (applied on G\({}_{2}\)) acting on the system. Since the mesh force rotates along the gears as the system rotates, each tooth of a gear undergoes periodically the same mesh force, with a period that is equal to the rotation period of the gear itself (\(T_{1}\) for teeth of G\({}_{1}\) and \(T_{2}\) for the teeth of G\({}_{2}\)) each tooth is subjected to the same force with a time delay. During meshing, the value of the mesh force is assumed to be constant. Then, considering the generic \(i_{th}\) tooth of a gear, the mesh force \(F_{Gi}(t)\) acting on it will have the same trend of the mesh stiffness in the time domain (rectangular waveform, Figure 4). As a consequence, the force vector \(\hat{F}(t)\) in the equation of motion Equation (4) contains, in correspondence to the teeth dof, the mesh force trends \(F_{Gi}(t)\), properly phased in the time domain according to the position of the tooth which is considered (e.g., if tooth-1 of G\({}_{1}\) undergoes \(F_{1_{1}}(t)\), tooth-2 of G\({}_{1}\) undergoes \(F_{1_{2}}(t)=F_{1_{1}}(t+\Delta T)\), being \(\Delta T\) the meshing time interval). Finally, the force vector \(\hat{F}(t)\) can be represented as  
\[\hat{F}(t)=\left\{ \begin{array}{c}\{0\}_{Z_{1}\times 1}\\ \left\{\begin{array}{c}F_{1_{1}}\\ \vdots\\ F_{1_{2}Z_{1}}\\ \{0\}_{Z_{2}\times 1}\end{array}\right\}_{Z_{1}\times 1}\\ \left\{\begin{array}{c}F_{2_{1}}\\ \vdots\\ F_{2_{2}Z}\end{array}\right\}_{Z_{2}\times 1}\end{array} \right\}_{N\times 1} \tag{18}\]  
As for the mesh stiffness, it is convenient to express the mesh force as a Fourier series, since it is periodic in the time domain. This will be advantageous for the computation of the forced response (Section 6), since each harmonic will be considered separately, computing the overall response as a superimposition of the effects due to each harmonic. As a matter of fact, the system is a linear time-variant system whereby the superimposition principle is valid. Then, let us write the expression of the force function, in the Fourier series. It is convenient to distinguish between the force acting on the teeth nodes of Gear-1 from the force acting on the teeth nodes of Gear-2. The two force sets, expressed in Fourier Series, have fundamental frequencies that are the speed of the Gear-1 (\(\Omega_{1}\)) for the excitation force terms acting on the teeth nodes of Gear-1 and the speed of Gear-2 (\(\Omega_{2}\)) for the excitation force terms acting on the teeth nodes of Gear-2. As a consequence, let us write the general expression of the force acting on the \(i_{th}\) tooth of G\({}_{1}\) and the force acting on the \(j_{th}\) tooth of G\({}_{2}\) respectively in Equations (19) and (20):  
\[F_{1_{i}}(t)=F_{1_{i},0}+\sum\
olimits_{k_{1}=1}^{\infty}[F_{1_{i}}^{k_{1}} \cos(k_{1}\Omega_{1}t)+F_{1_{i}b}^{k_{1}}\sin(k_{1}\Omega_{1}t)]; \tag{19}\]  
\[F_{2_{j}}(t)=F_{2_{i},0}+\sum\
olimits_{k_{2}=1}^{\infty}[F_{2_{j}a}^{k_{2}} \cos(k_{2}\,\Omega_{2}t)+F_{2_{j}b}^{k_{2}}\sin(k_{2}\Omega_{2}t)]. \tag{20}\]  
As for the mathematical expression of the mesh stiffness (Section 3), The Fourier Series of the mesh force can be written by means of the exponential notation.