# Dynamic Behaviour of an Automotive Dual Clutch Transmission during Gear Shift Maneuvers
## 4 The Lubrication Model in the Gear Teeth Contacts
Each cylinder has a radius (\(r_{1}\) and \(r_{2}\)) equal to the involute radius at the pitch point. 4. The slip velocity of the oil in the lubricated contact between the teeth (along the x direction) is neglected. 5. The oil viscosity is assumed to be constant. A scheme of the tooth surfaces is then represented in the Figure 2, where, in accordance with the above mentioned hypotheses, the teeth of two wheels are depicted as two cylindrical bodies with radii \(r_{1}\) and \(r_{2}\), approaching each other along the \(\eta\) direction:  
Figure 2: Scheme of the tooth pair contact. The Reynolds equation, with the above-mentioned hypotheses, can be formulated in the mono-dimensional expression:  
\[\frac{1}{\mu}\frac{\partial}{\partial\xi}\left(h^{3}\frac{\partial p}{\partial \xi}\right)=12\frac{\partial h}{\partial t} \tag{23}\]  
with: \(p\) = oil pressure; \(h\) = distance between the teeth surfaces; \(\mu\) = oil dynamic viscosity. The gap height \(h(\xi)\) between the teeth can be approximated by the following formula:  
\[h(\xi)=h_{0}+\frac{1}{2}\xi^{2}\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right) \tag{24}\]  
with \(h_{0}\) indicating the minimum distance between the two surfaces. Assuming:  
\[R=2\left(\frac{r_{1}r_{2}}{r_{1}+r_{2}}\right) \tag{25}\]  
the gap is:  
\[h(\xi)=h_{0}+\frac{\xi^{2}}{R} \tag{26}\]  
Imposing the boundary condition:  
\[\left\{ \begin{aligned} \xi&=-a\to p=0\\ \xi&=0\rightarrow\frac{\partial p}{\partial\xi}=0 \end{aligned} \right. \tag{27}\]  
Integrating the Reynolds equation, the expression of the pressure due to the oil between the teeth can thus be obtained:  
\[p(\xi)=3R^{3}\mu\left(\frac{1}{\left(a^{2}+Rh_{0}\right)^{2}}-\ \frac{1}{ \left(\xi^{2}+Rh_{0}\right)^{2}}\right) \tag{28}\]  
where \(a\) represents the semi-length of the oil film along \(\xi\) direction, related to the lubrication conditions. By integrating the above relation along the \(\xi\) coordinate in the range \((-\)a,a), the analytical expression for the oil squeeze force (21) is obtained:  
\[F_{c}=\dot{x}S(x,a)=-\dot{x}3\mu ZR^{3/2}\frac{\left[a(a^{2}-Rx)\sqrt{Rx}+(a^{2 }+Rx)^{2}\arctan(\frac{a}{\sqrt{Rx}})\right]}{x^{3/2}(a^{2}+Rx)^{2}} \tag{29}\]  
In which \(Z\) indicates the axial width of the gear pair, \(x=\theta_{r}r_{s}=(\theta_{p}-\theta_{s})r_{s}\) and \(\dot{x}=\dot{\theta}_{r}r_{s}=(\dot{\theta}_{p}-\dot{\theta}_{i})r_{s}\). In the present numerical analysis all the various gear pairs have been subject to full-meatus lubrication conditions, that is, with oil distributed along the entire tooth flanks. Moreover, the oil lubrication characteristics are those of a typical lubricant for mechanical transmissions with a SAE 75W90 gradation, operating at a low temperature (oil viscosity \(\mu\) = 0.02 Pa s). In previous experimental analyses, conducted on a specific test rig at the laboratory of Industrial Engineering Dept. of University \"Federico II\", designed to carry out investigations on single unloaded gear pairs, the benefit effect of lubricant in reducing the gear rattle phenomenon [10] has been outlined. Figure 3 reports an example of the comparisons between the experimental and theoretical time histories of the angular relative displacement of a gear pair, coming from the automotive gear box corresponding to the 1 st gear adopted in the present theoretical analysis, under rattling operative conditions. Geometrical and dimensional parameters of the gear pair are reported in Appendix A (Table A1). The stationary operative conditions of the test correspond to: speed = 600 rpm; speed fluctuation = 100 rpm; frequency of speed fluctuation = 4 Hz. In this example the gear pair operates under boundary lubrication regime. The good agreement observed in the theoretical-experimental analyses, in the time domain, enables the model to carry out numerical simulations of unloaded gear pairs of automotive gearboxes under rattling conditions, taking into account the effects of lubrication.