# Calculation and Verification of the Real-Time Working Characteristics of a Viscous Coupling
## 4 Theoretical Analysis and Calculation of the Hump Mode of the VC  
When the VC operates at a large speed difference for a long time, the dissipation of the generated heat is insufficient. The excess heat will increase the temperature, inflate the volume, and increase the pressure, and the delivered torque will increase sharply. This is the hump phenomenon. There are several explanations for formation mechanism of the hump phenomenon in reference [7]. No final conclusion has been reached on it so far, but the theory in reference [14] is well accepted. When the hump phenomenon occurs, the internal pressure is very high and difficult to measure. This necessitates the theoretical evaluation. Based on the generation mechanism of the hump phenomenon, the following assumptions are made to evaluate the internal pressure:  
1. The solubility of silicone oil to air is very low. The proportion of air is considered to be quite small (ignorable) inside the silicone oil;
2. When the hump phenomenon occurs, the gas inside the VC is considered to be an ideal gas, and thus follows the physical law of ideal gases;
3. During the hump phenomenon, the pressure of gas inside the VC is the same as that of the other parts. Based on the above assumptions, the combined gas law is applied to calculate the internal air pressure, which will be used as the pressure inside the shell of the VC. From the combined gas law:  
\[\frac{PV}{T}=C(Constant) \tag{60}\]  
The initial fill-up rate of the VC is \(C_{L0}\), the volume of the shell is \(V\), the initial air volume is \(V_{1}\), the initial pressure inside the shell is \(P_{1}\), the initial temperature is \(T_{1}\); at time \(t\), the volume of the air is \(V_{2}\), the internal pressure of the shell is \(P_{2}\), and internal temperature is \(T_{2}\). Equation (60) yields:  
\[\frac{P_{1}V_{1}}{T_{1}}=\frac{P_{2}V_{2}}{T_{2}} \tag{61}\]  
Therefore:  
\[V_{1}=(1-C_{L0})V \tag{62}\]  
\[V_{2}=(T_{2}-T_{1})\beta V_{1}+V_{1}=V(1-C_{L0}-\beta C_{L0}(T_{2}-T_{1})) \tag{63}\]  
Substitute the above two equations into Equation (61), yielding  
\[\frac{P_{1}(1-C_{L0})}{T_{1}}=\frac{P_{2}(1-C_{L0}-\beta C_{L0}(T_{2}-T_{1})) }{T_{2}} \tag{64}\]  
Figure 5: The limited slip torque of the VC. Therefore:  
\[P_{2}=P_{1}\frac{T_{2}}{T_{1}}\bigg{(}\frac{1-C_{L0}}{1-C_{L0}-\beta C_{L0}(T_{2}- T_{1})}\bigg{)} \tag{65}\]  
As long as the temperature \(T_{2}\) at time t is known, the internal pressure \(P_{2}\) can be obtained. According to the Coulomb friction law, the maximum torque transmitted during the hump mode is:  
\[T=\frac{2}{3}\pi KnfP(r_{2}{}^{3}-r_{1}{}^{3}) \tag{66}\]  
In the above, \(f\)--the coefficient of friction between plates; and \(K\)--the coefficient of contact distribution between the plates. The hump phenomenon appears when the volume of silicone oil is equal to the volume of the internal shell, so the moment of occurrence can be obtained through the calculation of the instantaneous filling rate of the silicone oil. The instantaneous filling rate is:  
\[C_{L}=\frac{V_{s}}{V}=C_{L0}(1+\beta_{s}\Delta t) \tag{67}\]  
In the above, \(C_{L}\)--the instantaneous filling rate of the oil; \(V_{s}\)--the instantaneous volume of the oil. The filling rate of the oil is almost 100% when the hump phenomenon occurs, namely, \(C_{L}=1\). So, the trigger temperature is:  
\[T_{s}=T_{0}+\frac{1-C_{L0}}{\beta_{S}C_{L0}} \tag{68}\]  
The variation of internal pressure caused by the rising temperature during different speed differences is shown in Figure 6, it can be seen that for a certain speed difference, the increase of the pressure was nonlinear. The bigger the speed difference was, the faster it increased.