## 5 Numerical Examples
### Csd  
To validate the structural model used in the present work, a clamped flexible blade, shown in Figure 13 and taken from Cheng [36], was first simulated. The results computed by the present method were compared with Dymore and with the Geometrically Exact Beam Theory (GEBT) code [37]. The blade was 1 m long and subjected to a transversal periodic force \(f=10\sin(50t)\) N at the free tip. Since the geometric and material properties were not provided in Cheng [36], the cross-sectional properties reported in Cheng [36] were used directly. The properties are reported in Table 1 for completeness. The blade was discretized by means of 20 three-node finite volume beam elements. Table 2 lists the fundamental frequency results, and Figure 14 plots the axis displacement, transversal displacement, and flap rotation history at the blade tip. It can be seen that the current result has a good coincidence with those in Dymore and GEBT. Note that the maximum flap rotation was about 47\({}^{\circ}\) and that the maximum tip deflection reached about 0.5 m, 50% of the beam span. In a nutshell, the beam had definitely large rotations and deformations. Validations were performed for the baseline rotor at an advance ratio of 0.151 and a thrust coefficient equal to 0.00457. The blade was discretized by 40 finite volume beam elements, and 20 elements were placed in the blade inboard region from the hub center to \(r/R=0.22\), where a significant variation in blade cross-sectional properties took place. The region from \(r/R=0.22\) to tip had a constant mass distribution and structural properties and was discretized by means of 20 equally spaced beam elements. In order to model the pitching control system stiffness, as stated in [13], a torsion spring was modeled at the pitching hinge and the spring stiffness was tuned to match a non-dimensional first torsion frequency of 3.845 /rev, which was reported in the HART II document. At first, the fan plot was predicted by the current CSD method for the HART II rotor blade to verify the structure modeling, as shown in Figure 15. The dominant natural mode shapes, i.e., the first three flap shapes, the first two lead-lag shapes, and the first torsion shape, were also compared at 100% rotational speed with those from the HART II documents [39; 41] and with those computed by UMARC [42]. As seen in Figure 16, the present results correlate well with those computed by UMARC; the difference in the torsion mode at the root was due to the different boundary condition applied at the pitching hinge: in the present simulation, a lumped torsion spring was used, and this led to a twist discontinuity at the pitching hinge. Figure 14. Tip displacement of an uncoupled flexible blade under excitation \(f=10\sin(50t)\) at the tip. \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline
**Station, m** & **x-CG, m** & **x-TA, m** & **x-EA, m** & **EA, N** & **EI\({}_{f}\), Nm\({}^{2}\)** & **EI\({}_{l}\), Nm\({}^{2}\)** & **GI, Nm\({}^{2}\)** & **Twist,\({}^{\circ}\)** & **m, kg/m** & **I\({}_{l}\), gm** & **I\({}_{f}\), gm** & **I\({}_{p}\), gm** \\ \hline
0.0 & 0.0 & 0.0 & 0.0 & \(1.26\times 10^{8}\) & 3000 & 14,000 & 380 & 4.24 & 3.67 & 0.4 & 0.4 & 0.8 \\
0.075 & 0.0 & 0.0 & 0.0 & \(1.26\times 10^{8}  
In addition, as seen from Figure 15, the present model is able to predict transitions between modes of vibration. In general, the predicted mode shapes are in good agreement with those from the HART II document data. Figure 16: HART II rotor blade natural mode shapes. Figure 15: HART II rotor blade frequencies. In the simulation, each blade was modeled by 40 \(\times\) 20 panels and an azimuthal angle step of 3\({}^{\circ}\) was used. The rotor was trimmed in order to reach the experiment thrust coefficient and zero longitudinal and lateral moments at an effective shaft tilt of 4.5\({}^{\circ}\) using the Newton-Raphson iterative method [35; 43].