# Dynamic Behaviour of an Automotive Dual Clutch Transmission during Gear Shift Maneuvers
## 4 The Lubrication Model in the Gear Teeth Contacts  
In the present analysis a tribo-dynamic model has been adopted to account for the gear rattle phenomenon generated during the meshing of the gear teeth of all the unloaded gears. The interaction is determined by a force \(F_{ci}\) exerted between the tooth flanks of the gears. This force can be due to a direct contact, when the teeth mesh under loading \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline \(K_{ij}\) & Torsional stiffness of the shaft between i-th and j-th pinion gear \\ \(K_{5p}\) & Torsional stiffness of the shaft between \(Cl_{1}\) and 5 th pinion gear \\ \(K_{6p}\) & Torsional stiffness of the shaft between \(Cl_{2}\) and 6 th pinion gear \\ \(\sigma_{ij}\) & Torsional damping of the shaft between i-th and j-th pinion gear \\ \(\sigma_{5p}\) & Torsional damping of the shaft between \(Cl_{1}\) and 5 th pinion gear \\ \(\sigma_{6p}\) & Torsional damping of the shaft between \(Cl_{2}\) and 6 th pinion gear \\ \(\omega_{e}\) & Angular velocity of the engine \\ \(\omega_{v}\) & Angular velocity of the vehicle wheels \\ \(I_{ip}\) & Moment of inertia of the i-th pinion gear (kgm\({}^{2}\)) \\ \(I_{is}\) & Moment of inertia of the i-th secondary wheel gear (kgm\({}^{2}\)) \\ \(\theta_{ip}\) & Angular rotation of the i-th pinion gear (rad/s) \\ \(\theta_{is}\) & Angular rotation of the i-th secondary wheel gear (rad/s) \\ \(\theta_{wo}\) & Angular rotation of the odd secondary shaft (rad/s) \\ \(\theta_{ave}\) & Angular rotation of the even secondary shaft (rad/s) \\ \(r_{ip}\) & Pitch radius of the i-th pinion gear \\ \(r_{is}\) & Pitch radius of the i-th secondary wheel gear \\ \(F_{ci}\) & Contact Force exerted between the teeth of the i-th gear \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols in the motion equations. conditions, or can be due to the oil between the teeth when the tooth flanks of the unloaded gears are approaching before the contacts. \(F_{ci}\) forces can be expressed by:  
\[F_{ci}(X_{i},x_{i},\dot{x}_{i})=\left\{ \begin{array}{l l}K(X_{i})x_{i},&if&|x_{i} \geq b|\\ S(x_{i})\dot{x}_{i},&if&|x_{i}<b|\end{array} \right. \tag{21}\]  
As regard to the direct force on the correct side of the loaded teeth, the adopted theory is exposed in [16; 17]. During the direct contact phase, a nonlinear elastic force applied at the pitch point is considered. This force is obtained using a stiffness function which depends on the position of the pinion gear, and is evaluated by the sum of some terms corresponding to the various tooth pairs contemporarily in contact. For the i-th tooth pair the stiffness term is:  
\[K_{i}(X)=k_{p}exp\left(C_{a}\Big{|}\frac{X-\epsilon X_{z}/2}{1.125\epsilon_{x} X_{z}}\Big{|}^{3}\right) \tag{22}\]  
where \(\epsilon\) and \(\epsilon_{a}\) are, respectively, the total and the transverse contact ratio. \(X_{z}\) is the transverse base pitch. The variable \(X=r_{p}\theta_{p}\) represents the absolute motion of the pinion gear. Consequently, for a complete meshing cycle, X starts from 0 and ends at \(\epsilon X_{z}\). Term \(k_{p}\) represents the stiffness value at the pitch point, which depends on the tooth parameters, along with the coefficient \(C_{a}\). The lubricated interaction between the unloaded teeth, due to the oil squeeze effect, is developed and detailed in [10]. The force generated by the oil \"squeezed\" between the teeth of a gear pair, in the instants preceding the impact, has been obtained from the Reynolds equation. The deduced gear rattle model, adopted in this analysis, was experimentally verified in a previous analysis and is based on the following hypothesis that considers even the main properties of the oil lubricant:  
1. Only a single tooth pair engaging at the pitch point is assumed. 2. The teeth are considered as rigid bodies with cylindrical shapes and axes parallel to those of the gear wheels. Cylinder 1 represents the tooth of the driving gear, while cylinder 2 represents the tooth of the driven gear wheel. 3.