# Bond Graph-Based Approach to Modeling Variable-Speed Gearboxes with Multi-Type Clutches
## 4 The Friction Clutch Model  
### The Karnopp Friction Model  
The wet friction clutch is an important part of many mechanical transmission systems, such as vehicles, ships, and helicopters. The engagement characteristics of the friction clutch importantly affect the stability of the transmission system [19]. In a more detailed modeling of the clutch, it is usually necessary to consider the oil film hydrodynamic lubrication and the micro-convex elastic contact model. Transmission system studies usually consider the influence of the clutch control mode on the dynamic response.  
When a clutch is used in a transmission system, the structural changes of the clutch under the two working conditions of disengagement and engagement directly determine the discontinuity of the structure of the transmission system dynamic model, which makes computer simulations quite challenging [20]. When the clutch is disengaged (that is, \(\tau_{f}=0\)), the driving and driven parts do not affect each other. When the clutch is engaged, under the action of hydraulic pressure \(p(t)\), the friction plate and the mating steel plate exhibit the sliding friction, and there is a nonlinear relationship between \(\tau_{f}\), \(\omega_{rel}\) (relative angular velocities of the driving \(\omega_{g}\) and driven \(\omega_{d}\) parts) and \(p(t)\). When the clutch is engaged, the driving and driven parts are regarded as a fixed connection; \(\tau_{f}\) is determined by the external torque \(T_{e}\), and does not exceed the torque capacity of the clutch \(T_{s}\). In this case, the relationship between \(\tau_{f}\) and \(\omega_{rel}\) cannot be determined, and a classical method to solve this problem is to replace the friction curve near the zero relative speed with an inclined straight line passing through the origin, which leads to the classical friction model, shown in Figure 6a [7]. However, this method may exhibit numerical stiffness problems, and makes it difficult to solve the system step-by-step using the conventional explicit method; on the other hand, the implicit method can easily deal with this issue.  
The Karnopp model, shown in Figure 6b, can effectively address the above numerical stiffness problem. A small range of \(\omega_{rel}=0\) accessories is defined as \(|e|\). In this range, \(\tau_{f}\) does not depend on \(\omega_{rel}\), but only on \(T_{e}\) and \(T_{s}\). Hence, the Karnopp model can be described as:  
\[\tau_{f}=\begin{cases}T_{fslip}=f(\omega_{rel},p(t))&|\omega_{rel}|>\varepsilon \\ T_{fslitk}=\left\{\begin{array}{ll}T_{\rm s}{\rm sgn}(T_{e})&|\omega_{rel}| \leq\varepsilon,\,|T_{e}|>T_{s}\\ T_{e}&|\omega_{rel}|\leq\varepsilon,\,|T_{e}|\leq T_{s}\end{array}\right.\end{cases} \tag{13}\]  
where the slipping friction function \(T_{fslip}(\omega_{rel},p(t))\). is defined as [21]:  
\[T_{fslip}=F_{n}\mu(t){\rm sgn}(\omega_{rel}) \tag{14}\]  
here, \(F_{n}\) represents the positive pressure loaded on the friction plate, while \(\mu(t)\) is the dynamic friction coefficient.  
\[F_{n}=N_{f}{\rm A}_{p}r_{e}p(t) \tag{15}\]  
\[\mu(t)=\mu_{c}+(\mu_{s}-\mu_{c})\exp\Bigl{[}-(\omega_{rel}/\omega_{s})^{ \delta}\Bigr{]} \tag{16}\]  
In the above, \(N_{f}\) is the number of active friction surfaces, \(A_{p}\) is the piston area, \(r_{e}\) is the equivalent clutch radius (\(r_{e}=2(r_{o}^{3}-r_{i}^{2})/3(r_{o}^{2}-r_{i}^{2})\)), \(\mu_{c}\) is the Coulomb friction coefficient, \(\mu_{s}\) is the static friction coefficient, \(\omega_{s}\) is the Stribeck velocity, and \(\delta\) is the Stribeck shape factor. For \(\delta=1\), the Tustin model [22] is obtained; for \(\delta=2\), the Gauss exponential model [23] is obtained.