# Variable Guide Vane Scheduling Method Based on the Kinematic Model and Dual Schedule Curves
## 3 Analytical Model of the VGV Link Mechanism
### Model Validation by ADAMS Simulation  
The derived model is validated by comparing the model in ADAMS (commercial mechanism analysis software, MSC software). In the case of the ADAMS model, the analysis is based on the three-dimensional space, so the axial movements of unison rings (also clevis) and longitudinal axis rotations of the vertical links, which are neglected in the analytical model are also included. The same actuator stroke (\(L_{x}\)) was assigned to both models and verified by comparing the vane angles (\(\theta_{l}\)). The ADAMS model consists of the bell crank, vertical links, and clevis (connector of the vertical link to the unison ring), which moves with the unison ring (shown in Figure 4). A 3-D rendering of the actuators above the bell crank is omitted, and instead, the stroke input (\(L_{x}\)) is numerically applied. The vane angle (\(\theta_{l}\)) is obtained from the rotational angle of the unison ring (\(\theta_{5}\)) using Equation (16) to reduce model complexity. Connections between the bell crank and vertical links are defined by the connections of geometric markers which represent the connection points of the bell crank and vertical links, without the 3-D rendering structure. Thus, invisible connections of markers can define relations of motions between the bell crank and vertical links. Different from the analytical model, the clevis (unison ring) has translational motion to the engine axis due to coupled motion with lever arms that rotate at their stem axis. Axial displacement of the clevis is applied using Equation (17) (refer to Figure 3).  
\[x_{a}\ =\ L\big{(}1-\cos\big{(}\sin^{-1}(L_{d^{\prime}}\tan\theta_{5}/L)\big{)} \big{)} \tag{17}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Variables**} \\ \hline Constants & \(L_{a},L_{b},L_{c},L_{d},L_{e},\ L_{d^{\prime}},\ L,\ \theta_{6}\) \\ State variables & \(\theta_{0},\ \theta_{1},\ \theta_{3},\ \theta_{4},\ \theta_{5}\) \\ Inputs & \(L_{x}\) \\ Outputs & \(\theta_{l}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Variables of the VGV mechanistic model.  
Figure 4: Layout of the Adams Mechanistic Model.  
All the joints in the ADAMS model are listed in Table 3.  
The VGV link mechanism has two identical link connections on the right and left side of the compressor, and it is manipulated by two hydraulic actuators. Thus, the inputs of the ADAMS model are two identical actuator strokes, and the outputs are stage vane angles calculated using the clevis rotational angle.  
As presented in Figure 5, the results of the analytical mechanistic model coincide with those of the ADAMS model, even if the planar motion of the unison ring is assumed in the analytical mechanistic model. Axial movement of the vertical links due to the translational motion of the unison ring cannot affect the vane rotational angle because the vertical links are large enough to neglect the translational motion of the unison ring. In this research, the verified analytical mechanistic model is used instead of the ADAMS model to schedule the VGV link mechanism to reduce calculation time and to perform iterative calculations.