## 5 Results
### Test-Case 4: Four-Bar Flexible-Link Multibody System  
The method has been applied to the four-bar flexible-link multibody system proposed in [4; 17]. The dynamic of these systems is nonlinear and disturbed by gravity forces. Even if the dynamic of such a system is non-linear, linearized or piecewise-linear models have been proved to be effective in the synthesis of control schemes [4; 17; 30; 31; 32]. To apply the proposed method, the system has been modeled through ODE by exploiting an Equivalent Rigid Link System (ERLS) from which elastic displacements are defined. Euler-Bernoulli beam finite elements are adopted to model the link flexibility. The resulting non-linear model, formulated through the Principle of Virtual Work, has then been linearized about an equilibrium configuration of interest. The model accounts for the mutual coupling between the \"rigid-body motion\" of the ERLS and the small elastic displacements with respect of the ERLS itself. Additionally, it accounts for the effect of gravity on both the \"rigid-body\" and the elastic dynamics. More details are provided \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{Stage 1: \(k_{0}\)} & \multicolumn{2}{c}{Stage 2: \(k\)=\(k_{0}\)+V\(k_{r}\)} \\ \hline \(f_{0}\) & \(g_{0}\) & \(f\) & \(g\) \\ \hline \(1.0479\) & \(-0.0053\) & \(7.0008\) & \(25.6070\) \\ \(1.1222\) & \(3.8335\) & \(1.709\) & \(1.9598\) \\ \(2.8765\) & \(1.3771\) & \(4.5693\) & \(1.8524\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Gains (test-case 3). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Open-Loop**} & \multicolumn{2}{c}{**Desired**} & \multicolumn{2}{c}{**Closed-Loop Poles with \(k_{0}\)**} & \multicolumn{2}{c}{**Closed-Loop Poles with \(k\)**} \\ \multicolumn{2}{c}{**Poles**} & \multicolumn{2}{c}{**Closed-Loop Poles**} & \multicolumn{2}{c}{**Closed-Loop Poles with \(k\)**} \\ \hline \(-0.918\pm 1.7606i\) & \(-1.500\pm 3.000i\) & \(-1.500\pm 3.000i\) & \(-1.500\pm 3.000i\) \\ \(0.0947\pm 2.5229i\) & - & \(0.1306\pm 2.7016i\) & \(-2.6323\pm 0.2707i\) \\ \(-0.8848\pm 8.4415i\) & - & \(-4.6282\pm 5.5858i\) & \(-3.134\pm 0.3095i\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Open-loop and closed-loop poles (test-case 3). Figure 9: Open-loop and closed-loop poles of the aircraft wing under an air stream (test-case 3). in [4; 17; 28]. The finite element model of the mechanism is shown in Figure 10 while the system parameters are reported in Table 8. All the links, except the frame, are flexible. Sixteen DOFs are adopted to model the system: one coordinate is the rotation of the ERLS, to represent the \"rigid-body\" motion. The remaining DOFs are the displacement of the 15 elastic coordinates. The mechanism is moved by a motor driving the crank rotation. Such an actuator ensures controllability of all the modes, although higher frequency modes are less controllable [17]. The open-loop system is unstable due to the presence of gravity, as it can be inferred from Table 9. The system stiffness and damping matrices are both asymmetric and can be decomposed into a symmetric and an asymmetric part, as done in the second test-case: \(\mathbf{K}=\mathbf{K}_{s}+\mathbf{K}_{a}\) and \(\mathbf{C}=\mathbf{C}_{s}+\mathbf{C}_{a}\). Figure 10: Four-bar linkage: system model (test-case 4) (reproduced from [16]).