# Bond Graph-Based Approach to Modeling Variable-Speed Gearboxes with Multi-Type Clutches
## 3 BG Models of PGTs
### Single-Pinion PGT  
The most common and widely used PGT has a single planetary gear transmission mechanism between the ring gear and the sun gear. The kinematic relationship between the single-pinion PGT (S-PGT) components is shown in Figure 2. The following relationship can be obtained by assuming equal linear velocities at the gear meshing point [13]:  
\[\omega_{p}r_{p}=(\omega_{c}-\omega_{s})r_{s} \tag{1}\]  
\[\omega_{p}r_{p}=(\omega_{r}-\omega_{c})r_{r} \tag{2}\]  
the characteristic equation is  
\[\omega_{s}-(\beta+1)\omega_{c}+\beta\omega_{r}=0 \tag{3}\]  
Here, \(\omega_{p}\), \(\omega_{s}\), and \(\omega_{r}\) are the rotation speeds of the planetary gear, the sun gear, and the ring gear, respectively. The parameters \(r_{p}\), \(r_{s}\), and \(r_{r}\) are the pitch radii of the planetary gear, the sun gear, and the ring gear, respectively. The parameter \(\beta\) is the PGT parameter, defined as \(\beta=z_{r}/z_{s}\) (\(z_{r}\): number of teeth of the ring gear; \(z_{s}\): number of teeth of the sun gear). The kinematic relationship between the parts is fundamental for establishing the BG model. A good dynamic model of a PGT also should consider the effects of the gear moment of inertia, rotational damping, and meshing stiffness.  
As shown in Figure 3, there are four one-nodes in the bonding diagram model of a single-pinion PGT, representing the sun gear, the planet carrier, the ring gear, and the planetary gear. An independent PGT cannot accomplish power transmission, and should be combined with input equipment (e.g., motor and engine), a clutch, and other components. Therefore, the sun gear, the ring gear, and the planetary carrier can be used as the input  
Figure 1: Double-row double planetary gearbox: (**a**) three-dimensional drawing, (**b**) structure diagram.  
components. The first three can be connected to an external system as an input or output power part; the key causality was not considered in the present study. The R and I elements on each one-node represent the rotational damping and the moment of inertia of the corresponding part, respectively; the 0-node represents the meshing point; finally, C on node 0 represents flexibility.  
When the contact between gears is regarded as an elastic contact, the BG model features three energy-storage parameters. The parameter \(\theta_{sp}\) is the relative angular displacement between the planet and the sun, while \(\theta_{rp}\) is the relative angular displacement between the planet and ring. The parameters \(\tau_{s}\), \(\tau_{r}\), and \(\tau_{c}\) are the torques of the sun gear, the ring gear, and the planetary carrier, respectively.  
For node 1, the rotation speeds for all the keys are the same, and the algebraic sum of the torques is 0, as shown in Figure 3.  
\[\dot{\theta}_{sp}=\omega_{c}-\omega_{s}+\omega_{p}\mathrm{TF}_{2} \tag{4}\]  
\[\dot{\theta}_{rp}=\omega_{c}-\omega_{r}-\omega_{p}\mathrm{TF}_{1}{}^{-1} \tag{5}\]  
For node 0, the torques for all the keys are the same, and the algebraic sum of the rotation speeds is 0. The torque balance equation of the S-PGT is as follows:  
\[\tau_{s}=\theta_{sp}C_{sp}^{-1}-I_{s}\dot{\omega}_{s}-R_{s}\omega_{s}\]  
\[\tau_{r}=\theta_{rp}C_{rp}^{-1}-I_{r}\dot{\omega}_{r}-R_{r}\omega_{r}\]  
\[\tau_{c}-I_{c}\dot{\omega}_{c}-R_{c}\omega_{c}=\tau_{sp}+\tau_{rp}\]  
\[\tau_{rp}\mathrm{TF}_{1}=I_{p}\dot{\omega}_{p}+R_{p}\omega_{p}+\tau_{sp} \mathrm{TF}_{2}^{-1}\]  
Figure 3: BG model of the S-PGT.  
Figure 2: Kinematic diagram of the S-PGT.  
According to the motion characteristics of the S-PGT, the torques on the sun gear and ring gear are applied in the same direction. The carrier torque is opposite the sun gear torque.