## 3 Results
### Additional Validation: Extended Range of Solicitations for Disturbance Rejection Test  
Further experiments with numerical simulations were made to reveal potential limits of validity of the linear reduced order model. In parallel to the full NL 3D model that was performed in Abaqus, some full NL 3D mechatronic simulations were also performed using OOFELIE::Multiphysics. The model was slightly simplified to reduce computational complexity, but the model stays close to the one that was prepared in Abaqus. For example, the model in OOFELIE::Multiphysics introduces the following simplifications:  
* The spacer and the base plate were considered as clamped. * The Cardan joint was simplified. Each of the two Cardan discs was replaced by a hinge with an internal rotation spring. The stiffness of this spring was updated to achieve the right eigenfrequencies of interest. The final value used in the simulations is 2340 N.m/rad. These simplifications can be done because they do not affect the global behaviour of the system. For the simulations, OOFELIE permits performing a monolithic resolution of the flexible mechanism with the associated controller. The controller is defined as an equivalent continuous one, and we can then choose a time step larger than the sampling rate of the original discrete controller. This last fact can save time during the numerical solution. The beta damping factor (stiffness proportionality factor) for the rotation springs at the level of the Cardan joint was set to \(28\times 10^{-6}\) for the simulation. **The disturbance rejection test was performed for the following levels of disturbance:**  
* Case 1: 70% of maximum actuator torque (reference case from the previous section);
* Case 2: 7% of maximum actuator torque;
* Case 3: 700% of maximum actuator torque (unreachable experimentally). Figure 26 shows the rotational velocities for the three cases of disturbance amplitudes. Some scaling factors for Cases 2 and 3 are applied in order to permit the comparison of solutions for the different disturbance levels:  
* Case 2 was scaled by a factor 10 for comparison with reference case (case 1);
* Case 3 was scaled by a factor 1/10 for comparison with reference case (case 1). \begin{table}
\begin{tabular}{l c c c c} \hline \hline  & \multicolumn{2}{c}{**Reference Following**} & \multicolumn{2}{c}{**Disturbance Rejection**} \\ \cline{2-5}  & \(e_{rms}\) & \(e_{pk-pk}\) & \(e_{rms}\) & \(e_{pk-pk}\) \\ \cline{2-5} _Linear ROM_ & 0.0455 & 0.405 & 5.82 & 40 \\ _Full scale nonlinear FEM_ & 0.0449 & 0.279 & 3.05 & 31.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of average and worst-case prediction error of linear and nonlinear model in the closed-loop reference-following and disturbance rejection experimentsFor the left plot, it is observed that the solutions are very close to the results acquired from the full NL Abaqus model and the linear reduced order model. The general shape is the same for all three disturbance levels. On the right plot, at the beginning of the simulation, the curves for 70% and 7% disturbances fit perfectly. This means that we can consider the model as linear almost up to 70% disturbance. For the 700% disturbance, we can observe some significant differences with the reference case. This is related to the fact that the vibration of the rotating flexible blade cannot be considered as linear anymore in a local rotating frame. On Figure 27, the real computed large deflection of the flexible blade is shown at simulation time t = 1 ms for 700% disturbance case. It is clear that this kind of large displacement and rotation in a local frame attached to the flexible arm can only be accurately taken into account using a full 3D non-linear model of the complete mechatronic system. All these results permit us to validate that the use of linear ROM for the controller design was sufficient for the current application considering a reasonable level of solicitations.