# Bond Graph-Based Approach to Modeling Variable-Speed Gearboxes with Multi-Type Clutches
## 6 Coupling the Clutch Models to the PGT Model
The final multi-system coupled transmission system model is shown in Figure 10.  
The above analysis reveals that in addition to the uncertain factors such as the gear meshing and tooth surface contact fluctuations at the tooth level in the gearbox, there is also the effect of external excitation, including the engine speed fluctuations, load torque fluctuations, and the aerodynamic torque variation of the rotor. It is worth noting that the transmission system is realized through the hydraulic clutch during the shifting process; thus, the uncertainty of the loading hydraulic fluctuation will also contribute. Finally, the manufacturing errors in the production process of parts, and assembly errors during the gearbox assembly process will directly affect the dynamic characteristics of the transmission system. Therefore, ignoring the uncertainty in a dynamic model will significantly affect the quality and cost of the corresponding transmission system [34]. The time-varying meshing stiffness is the main source of uncertainty in the gear transmission modeling. The meshing phenomenon is responsible for the gearbox noise. In gear transmission, the meshing stiffness changes with time owing to the transition between single- and double-teeth meshing zones, which is related to the geometric and physical characteristics of gear teeth. The widely used methods for calculating the time-varying meshing stiffness of gears are the Ishikawa deformation formula, the mathematical elasticity method, the ISO standard, and the energy method. In addition, many analytical methods have been developed, especially numerical methods (such as the finite element method), for calculating the meshing stiffness. In this work, the average tooth stiffness was calculated using the formula in the ISO standard. In the process of gear transmission, there is a periodic alternation between the single-tooth and double-tooth meshing states, which leads to regular changes in the gear meshing stiffness. Therefore, the time dependence of the meshing stiffness is expressed by the Fourier series, as follows:  
\[k(t)=k_{av}+\sum_{i=1}^{n}(k_{2i}\sin(i\omega t))=k_{av}+\sum_{i=1}^{n}k_{i} \cos(i\omega t+\varphi_{i}) \tag{21}\]  
here, \(k_{av}\) is the average meshing stiffness; \(k_{i}\) is the \(i\)th harmonic amplitude of the stiffness; \(\omega\) is the meshing frequency; and \(\varphi_{i}\) is the initial phase of \(i\)th harmonic. In addition, \(\varphi_{i}=\arctan(-k_{2i-1}/k_{2i})\), \(k_{a}=\sqrt{k_{2i-1}^{2}+k_{2i}^{2}}\). The first harmonic term in Equation (21) is taken as the meshing stiffness used for the uncertainty analysis, and Figure 12 shows the meshing stiffness of the involute gear. \[k(t)=k_{avg}+k_{a}\cos(\omega t+\varphi_{1}) \tag{22}\]  
Figure 11: Main subsystems of the helicopter transmission model. In this approach, the meshing stiffness values for a single pair of teeth and a double pair of teeth are calculated according to the calculation method of tooth stiffness of the involute gear in the ISO formula draft. The end face coincidence degree is obtained through the geometric parameters and pressure angle of the meshing gear to obtain the average meshing stiffness \(k_{mg}\) and stiffness variation amplitude \(k_{a}\). The mathematical expression of this stiffness form is relatively simple; it can effectively reflect the influence of the actual meshing stiffness on the system to a certain extent, and is widely used in research studies on gear nonlinear dynamics.