# Bond Graph-Based Approach to Modeling Variable-Speed Gearboxes with Multi-Type Clutches
## 6 Coupling the Clutch Models to the PGT Model  
Taking the two common PGT models (namely, the friction clutch model and the one-way clutch model) as the basic models, the dynamic model of the transmission can be established by coupling, as shown in Figure 9. However, in actual modeling, it is not easy to obtain the dynamic model directly from the structural diagram (e.g., Figure 1). Therefore, based on the structure and relationships in the diagram, a method for determining the relationships between the basic models is proposed here. Step 1. Model of compound PGT  
For transmissions with composite PGTs, there are many structural interconnections between PGT elements or members, which can yield different transmission ratios. Taking two individual PGTs as an example, there are six possible structural connections, as listed below: ring gear carrier, sun gear ring gear, sun gear sun gear, carrier-carrier, ring gearing-gear, and sun gear carrier. In addition, there are some special structures, such as connecting the planetary gears of two PGTs, or sharing a ring gear. Features: The kinematic equation of a composite PGT must be based on the connection mode between the constituting PGTs. The two affixed parts are considered as a whole, and the speed is the same. Each part is represented by node 1, and each meshing point is represented by node 0, considering factors such as the meshing stiffness, rotational damping, and flexibilities of parts. In this way, a dynamic model of a composite PGT is established. Step 2. Determine the external excitation and output of the gear train. Figure 8: Block diagram of the angular displacement difference model. Figure 9: Power transmission relationships among the three basic models. As shown in Figure 9, the external excitation received by a composite PGT model includes the speed of the input shaft \(\omega_{in}\), the load torque of the output shaft \(\tau_{out}\), the torque of the friction clutch \(\tau_{f}\), and the torque of the one-way clutch \(\tau_{o}\); \(T_{e}\), \(\omega_{g}\), and \(\omega_{d}\) are transmitted to the wet friction clutch dynamic model; \(\omega_{ir}\) and \(\omega_{or}\) are transmitted to the one-way clutch dynamic model. Features: In addition to the connection ports with the friction clutch and the one-way clutch, each composite PGT has two ports (\(\omega_{in}\) and \(\tau_{out}\)). Step 3. Analyze the input and output of the hydraulic friction clutch. As shown in Figures 7 and 9, under the hydraulic pressure \(p(t)\), the friction clutch starts to work, the friction coefficient \(\mu(t)\) is related to \(\omega_{g}\) and \(\omega_{d}\) in the gear train model, and the sliding friction torque \(T_{fslip}\) is transmitted to the gear train. However, when \(\left|\omega_{g}-\omega_{d}\right|\leq\varepsilon\), the friction clutch receives an external torque \(T_{e}\) from the gear train. Features: There are three connection ports (\(T_{e}\), \(\omega_{g}\), and \(\omega_{d}\)) between each friction clutch and the gear train, as well as an external port (\(p(t)\)). Step 4. Connection between the one-way clutch and the transmission system. It can be seen from Figures 8 and 9 that the one-way clutch is a passively controlled mechanism. Driven by \(\omega_{ir}\) and \(\omega_{or}\) in the gear train, the working mode is determined according to the angle difference between the inner and outer rings. When \(\theta_{ir}>\theta_{or}\), The torque is transmitted to the gear train. Otherwise, no torque is transmitted. Features: There are two connection ports (\(\omega_{ir}\), \(\omega_{or}\)) between each one-way clutch and the gear train, and no external ports exist. The design scenario consists of subjecting the gearbox to an upstream torque that models the torque from the rotor, and a downstream flow source that models the rotational speed imposed by the engine. The hydraulic pressure is applied by the control system.