## 3 Modeling of an All Movable Fin with Rotational Freeplay
### Description of the All Movable Fin  
The numerical experiments of nonlinear system identification were carried out on a supersonic flutter test platform as shown in Figure 3. The test platform was assembled on a rigid base with an all movable fin connecting by an elastic beam. Based on the convenience of performing the ground vibration tests (GVTs) and the compatibility requirements of mounting on the side walls of the wind tunnel, the rigid base was designed with sufficient weight and stiffness. To simulate the installation environment of the fin, a rotating device was also considered inside the rigid base with rotating fixed. The base was made of steel while the all movable fin, specified as a skeleton-cover structure, was manufactured using aluminum-alloy. With several additional weights inside, the whole mass of the all movable fin is 3.20 kg. The size of the elastic beam was designed to meet the stiffness requirements, which was also made of steel.  
The all movable fin was tested and identified by linear system identification techniques to obtain the natural frequency and mode shapes. Figure 4 presents the experimental set-up of GVTs performed using LMS Test.Lab. Burst random excitation was applied on the two selected nodes by a shaker, separately. Furthermore, the displacements of four selected nodes were sampled by four laser sensors. The sampling frequency was set to 2000 Hz to avoid the bias of low-frequency sampling. Table 1 gives the natural frequencies and damping ratios of the first 2 modes of all movable fin up to 150 Hz of interests. The mode shapes of the first two modes are shown in Figure 5.  
Figure 2: The restoring force of freeplay.  
The finite-element (FE) model was established by MSC Nastran [38] and has been modified based on the test results of GVTs using FEMTools [39], an optimization software developed by Dynamic Design Solutions. Figure 6 is the FE model of the all movable fin and elastic beam. The white-colored component is the all movable fin, whereas the cyan-colored component is the elastic beam. To reduce the computational cost, the FE model is used to solve the FE model.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode** & **Natural Frequency (Hz)** & **Damping Ratio (\%)** & **Description** \\ \hline
1 & 90.6 & 0.72 & Rigid fin bending mode \\
2 & 102.1 & 0.56 & Rigid fin rotating mode \\ \hline \hline \end{tabular}
\end{table}
Table 1: Natural frequencies and damping ratios of the first 2 modes of all movable fin up to 150 Hz.  
Figure 4: The experimental set-up of ground vibration tests.  
Figure 5: The first 2 mode shapes of all movable fin.  
Figure 3: The supersonic flutter test platform.  
computational burdens and achieve faster nonlinear calculations, the linear elements of the FE model were condensed using the modal synthesis technique. The whole FE model was reduced in terms of several elastic modes of all movable fin and two assumed modes involve the rigid bending and rotating motion of all movable fin. Freeplay nonlinearity was finally introduced to the rigid rotating DOF. The freeplay regions considered in this paper are listed in Table 2.  
A 4-5 th-order Runge-Kutta time integration was performed with the same sampling frequency of 2000 Hz as GVTs during the numerical investigation. The mechanical and electrical disturbances in measurements were considered by adding a 0-150 Hz white noise to the synthetic signals, and the noise-signal ratio was set to 2%.