# Bond Graph-Based Approach to Modeling Variable-Speed Gearboxes with Multi-Type Clutches
## 8 Modeling and Numerical Verification of the Variable-Speed Gearbox  
### UncBG Model of a Double-Row Double Planetary Gearbox  
According to the power flow path and structural characteristics of the gearbox, different nodes and energy storage elements were set based on the BG theory, and appropriate source elements were selected. Finally, the causal relationship between the nodes and elements was determined. The uncBG model of the gearbox was established, as shown in Figure 14.  
There were three TF elements, corresponding to the three gear meshing points in the gearbox: \(\text{TF}_{1}=r_{\text{p1}}/r_{s}\), \(\text{TF}_{2}=-r_{\text{r2}}/r_{\text{p2}}\), \(\text{TF}_{3}=-r_{\text{r1}}/r_{\text{p1}}\). Different meshing points were set to different fluctuation values, according to whether there was a uniform load mechanism: \(\Delta\text{TF}_{1}=0.005\text{TF}_{1}\times\text{randn}\), \(\Delta\text{TF}_{2}=0.002\text{TF}_{2}\times\text{randn}\), \(\Delta\text{TF}_{3}=0.002\text{TF}_{3}\times\text{randn}\). Figure 14: Double-row double planetary uncBG model. Figure 13: Dynamic design scheme of the transmission system based on the uncBG model. State space equations:  
\[\left\{ \begin{array}{l}\dot{q}_{5}=\omega_{in}-\frac{1}{I_{8}}p_{8}-{\rm TF}_{1} ^{\prime}\frac{1}{I_{14}}p_{14}\\ \dot{p}_{8}=\frac{1}{C_{3}(t)}q_{5}-R_{7}\frac{1}{I_{8}}p_{8}-\frac{1}{C_{2}(t )}q_{17}-\frac{1}{C_{2}(t)}q_{21}-S_{\epsilon}\\ \dot{p}_{14}={\rm TF}_{1}^{\prime}\frac{1}{C_{5}(t)}q_{5}-R_{13}\frac{1}{I_{14 }}p_{14}-\frac{1}{{\rm TF}^{\prime}}\frac{1}{C_{2}(t)}q_{17}-\frac{1}{{\rm TF}^ {\prime}}\frac{1}{C_{2}(t)}q_{17}-\frac{1}{{\rm TF}^{\prime}}\frac{1}{C_{2}(t)} q_{21}\\ \dot{q}_{17}=\frac{1}{I_{8}}p_{8}+\frac{1}{{\rm TF}^{\prime}}\frac{1}{I_{14}}p _{14}-\frac{1}{I_{23}}p_{26}\\ \dot{q}_{21}=\frac{1}{I_{8}}p_{8}+\frac{1}{{\rm TF}^{\prime}_{3}}\frac{1}{I_{14 }}p_{14}-\frac{1}{I_{23}}p_{23}\\ \dot{p}_{23}=\frac{1}{C_{2}(t)}q_{21}-R_{24}\frac{1}{I_{23}}p_{23}+S_{\epsilon o }\\ \dot{p}_{26}=\frac{1}{C_{2}(t)}q_{17}-R_{28}\frac{1}{I_{26}}p_{26}+S_{\epsilon f }(p(t))\end{array} \right. \tag{27}\]  
On expressing the abovementioned differential Equation (27) with state space equations for matrices and vectors, we obtain:  
\[\dot{X}={\rm A}X+{\rm B}U \tag{28}\]  
Here, A is a seven-order matrix containing all the coefficients of the state variable on the right-hand side of Equation (27). B is the matrix comprising the coefficients of the input variable on the right-hand side of Equation (27). The derived mathematical model is a multi-energy domain shift dynamics model with uncertain factors, including mechanical energy and hydraulic energy, involving uncertain factors such as input random signal, driving hydraulic fluctuation, time-varying meshing stiffness, and comprehensive error. A mathematical model based on the bond graph model was established in MATLAB, and specific parameters adopted in the simulation are presented in Tables 2 and 3. Figure 15a shows the shift dynamics simulation scheme of the double-row double planetary gearbox. Based on Figure 14 and Equation (27), the Simulink scheme of the compound planetary train based on the bond graph was obtained, as shown in Figure 15b, and the Runge-Kutta method was used to solve the shift model. and 366 rad/s respectively, with fluctuations of 6 and 2.5%, and the transmission ratio was 2.42 and 1.42 respectively; The rotating speeds of the planetary gears were 1779 and 1264 rad/s respectively, with fluctuations of 2.5 and 1.7%. The average rotating speeds of the ring gear R1 were 0.1 and 182 rad/s respectively, and the average rotating speeds of ring gear R2 were 256 and 0.001 rad/s respectively. During the gear-shifting process, different parts exhibit different dynamic responses of speed increase or deceleration. The steady-state values of output shaft, planetary gear, ring gear R1, and ring gear R2 rotating speed were close to the theoretical values.