# Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling
## 2 Model of the System  
The system under analysis is made of two meshing gears (Gear-1 or G\({}_{1}\), and Gear-2 or G\({}_{2}\), Figure 1). For each gear, a local reference system rotating with the gear itself is defined. Each gear is divided into sectors (Z\({}_{1}\) for Gear-1, and Z\({}_{2}\) for Gear-2), one per each tooth (Figure 2). A gear ratio \(\eta\) can be defined for the system under analysis as the ratio between Z\({}_{1}\) and Z\({}_{2}\). Each sector is modelled as a lumped parameter model with two degrees of freedom (dof), or nodes, one for the tooth and one for the gear sector wheel (Figure 3). The rotation of each gear around its own axis is allowed (no radial or axial displacement are allowed, only tangential displacement is allowed). The latter assumption is reasonable for the case of thin walled spur gears where radial and axial displacements can be assumed as negligible. The sectors are then coupled together. The periodic coupling between the teeth of the two gears is modelled by a time-variant mesh stiffness K\({}_{\text{M}}\)(t), described in more detail in Section 3. As shown in Figure 3, the two gears are constrained to ground by means of the stiffness elements \(k_{a,1}\) and \(k_{a,2}\) respectively. The mechanical characteristics of mass \(m_{b,1}\) and \(m_{b,2}\), and stiffness \(k_{b,1}\) and \(k_{b,2}\) are associated to the teeth of the gears, whose displacement coordinates are \(x_{\text{G}_{1},t}\) and \(x_{\text{G}_{2},t}\) (respectively for the teeth of G\({}_{1}\) and the teeth of G\({}_{2}\)). The mechanical characteristics of mass \(m_{c,1}\) and \(m_{c,2}\), and stiffness \(k_{c,1}\) and \(k_{c,2}\) are associated to the gear sector wheel, whose displacement coordinates  
Figure 1: System of two meshing gears. Figure 3: Lumped parameters model of the sectors of the gears Gear-1 and Gear-2. Figure 2: Representation of the sectors of the gears. are \(x_{G_{1},c}\) and \(x_{G_{2},c}\) (respectively for the sector wheel of \(G_{1}\) and the sector wheel of \(G_{2}\)). In Figure 3, a force acting on the meshing teeth dof (equal but with opposite direction for the tooth of \(G_{1}\) and the tooth of \(G_{2}\)) is shown, which represents the force establishing between them when the meshing couple is in contact. Obviously, when the couple is not meshing, no forces are exchanged. Then, the force travels along the circumference of each gear as the system rotates. The excitation force will be discussed more in detail in Section 5. The physical displacement vector (with the corresponding size) is:  
\[\{x\}=\left\{\left\{x_{G_{1}}\right\}^{T},\left\{x_{G_{2}}\right\}^{T}\right\} ^{T}{}_{1 xN}, \tag{1}\]  
where,  
\[\left\{x_{G_{1}}\right\}^{T}=\left\{\left\{x_{G_{1},c}\right\}^{T},\left\{x_{ G_{1},t}\right\}^{T}\right\}^{T}{}_{1x\;2Z_{1}}, \tag{2}\]  
\[\left\{x_{G_{2}}\right\}^{T}=\left\{\left\{x_{G_{2},c}\right\}^{T},\left\{x_{ G_{2},t}\right\}^{T}\right\}^{T}{}_{1x\;2Z_{2}}, \tag{3}\]  
with \(N=2\;Z_{1}+2\;Z_{2}\). \(\{x\}\) including Gear-1 displacement coordinates, \(x_{G_{1}}\), subdivided into \(x_{G_{1},c}\) which indicates the displacement of the nodes of the gear wheel, and \(x_{G_{1},t}\) indicating the displacement of the teeth. The same holds for \(x_{G_{2}}\) of Gear-2. The equation of motion, in matrix form, can be written in general as:  
\[M\;\ddot{x}+\hat{C}\;\dot{x}+\hat{K}(t)\;x=\hat{F}(t), \tag{4}\]  
where \(M\) is the mass matrix; \(\hat{C}\) is the damping matrix; \(\hat{K}(t)\) is the stiffness matrix which includes time-variant parameters, corresponding to the mesh stiffness \(K_{\text{M}}\)(t) used to couple the two gears; \(\hat{F}(t)\) is the force vector containing non-zero values for the teeth dof. As anticipated before, \(\hat{F}(t)\) is a time-variant vector since the mesh force passes from one tooth to another as the system rotates. Then, each tooth is periodically subjected to a force excitation due to the meshing, where the period is equal to the rotation period of the gear.