# Synthesis of the Energy-Saving Dry Dual Clutch Control Mechanism
## 2 Materials and Methods
### The Mathematical Model  
A design model of the clutch shift mechanism is proposed to analyze the effect of DCT's carriage movement on clutch margins (Figure 4). The maximum engine torque, the clutch margin, and the number of friction pairs determine the required compression force \(Q\) for the clutch friction pairs. The clutch margin for the light vehicle is \(\beta=1.2\), \(1.7\)-\(2.0\) for all-terrain vehicles and \(3.0\) for heavy machinery. The calculation conditionally considers a single rotary lever and pressure spring. However, the number of levers can reach \(3\)-\(4\), and the number of springs can reach \(18\)-\(42\), depending on the transmitted engine torque. The total pressing force \(F_{np}\) is summarized by all the pre-compressed springs. It is conventionally attached to the clutch swing lever \(NP\), which is connected to the clutch pressure plate via trailing lever \(NM\) at point \(M\). When moving the carriage from position \(G_{0}\) to positions \(G_{1}\) and \(G_{2}\), the pressure plate moves from position \(M\) to positions \(M_{1}\) and \(M_{2}\) (Figure 3a). The action line of the total force \(Q\) coincides with the lever \(NM\). It is also located at a distance \(L\) from the pressure spring line of action. The coordinate system \(XOY\) is used to study the clutch operation. The axis \(OX\) is directed along the compression spring action line. The axis \(OY\) is directed along the swivel disc in its neutral position \(ON_{0}\). The full switch-on of the clutch is considered. Vertical lines \(M_{1}\) and \(M_{2}\) determine the extreme positions of the friction pressure plates for the switched-on 1 st and 2 nd clutch, respectively, with coordinates \(x_{M1}\) and \(x_{M2}\). A pre-compressed pressure spring is used to provide the required force \(Q\). The proposed design scheme allows determining dependencies between the force \(Q_{j}\) (\(j=1\), \(2\)) on the pressure plate and force \(F_{npj}\) of the clutch pressure spring when switching on the 1 st (Figure 3b) and the 2 nd (Figure 3c) clutch, so during vertical movement of the carriage from the neutral position \(G_{0}\) to working positions \(G_{j}\) by the distance of \(\Delta_{Kj}\) (Figure 3a). The equilibrium condition of the \(j\)-th lever relative to the reference point \(G_{j}\) is as follows:  
\[\Sigma M_{G_{j}}(F)=0, \tag{1}\]  
or for the cases of switching on the 1 st and the 2 nd clutches, respectively:  
\[F_{np1}\Delta_{K1}=Q_{1}(L+\Delta_{K1});\;F_{np2}\Delta_{K2}=Q_{2}(L-\Delta_{ K2}). \tag{2}\]  
Figure 4: The original DCT: (**a**)—the design scheme; (**b**)—1 st clutch; (**c**)—2 nd clutch. from which the following expressions can be obtained:  
\[Q_{1}=\frac{F_{np1}\Delta_{K1}}{L+\Delta_{K1}}=\frac{F_{np1}}{\gamma_{1}+1};\ Q_{2}= \frac{F_{np2}\Delta_{K2}}{L-\Delta_{K2}}=\frac{F_{np2}}{\gamma_{2}-1};\ (\gamma_{2}>1), \tag{3}\]  
where the following parameter is introduced:  
\[\gamma_{j}=\frac{L}{\Delta_{Kj}}. \tag{4}\]  
Formula (3) shows that the inequality \(F_{np1}>F_{np2}\) ensures equal forces on the pressure disc (\(Q_{1}=Q_{2}=Q\)) in the case of similar carriage movements. To fulfill this condition, the design schemes in Figure 5 are considered. When the support carriage \(G\) is moved to position \(G_{j}\), the support lever rotates under the force \(F_{npj}\) from the pressure spring at point \(P_{j}\). This fact provides the required compression force \(Q_{j}\) of the clutch friction pairs. For this case, pressure spring forces on the platter surface will be as follows:  
\[F_{npj}={}_{np}\big{(}\Delta_{0}-\Delta_{j}\big{)}, \tag{5}\]  
where \(c_{np}\)--compression spring stiffness [kN/mm]; \(\Delta_{0}\)--spring pre-deformation [mm] for the neutral position of the carriage (when clutches are switched off); \(\Delta_{j}\)--additional deformation of the pressure spring when turned on the \(j\)-th clutch (from Figure 4, \(\Delta_{1}=OP_{1}\), \(\Delta_{2}=OP_{2}\)). Particularly, after switching on the 1 st clutch (Figure 4), the rotary lever reaches the position \(N_{j}G_{j}\), where \(x_{N1}=-z_{1}\).