# Frequency Sensitivity and Mode Characteristics of Body Structure for a Piezoelectric-Based UUV
## 4 Results and Discussion
### Vibration Modes Analysis  
Figure 5 gives the normal displacement modes of the UUV body structure, since the normal direction vibration is the main vibration mode, so the normal displacement modes are given. Figures 6 and 7 are the shear modes and torsional moment modes of the UUV body. In Figures 5-7, only modes B(2,1) and B(4,2) with the lowest frequency are investigated. From these figures, the following is known:  
(1) In the figures of the displacement modes, shear modes and torsional moment modes with a certain group of (m, n), \(m\) axial wave numbers are distributed on the outside surface of the cylinder along the axial direction, and \(n\) circumferential wave numbers are distributed on an arbitrary circular cross-section of the body structure. (2) For the same order modes, the shear modes and the torsional moment modes have the same mode shape, and the difference between them is that the amplitude varies from one mode to another. (3) As the mode order grows, the amplitude of the shear mode decreases, whereas the amplitude of the torsional moment modes increases. Hence, for lower order vibrations, the shear modes are the primary vibratory form, while for the higher order vibrations, the torsional moment modes are the main vibratory form. In short, the shear modes and the torsional moment modes are influenced by the normal displacement modes of the UUV body structure. Hence, when the dynamic model is established, the mode characteristics of the UUV body structure are determined. Figure 4: Frequency sensitivity with parameter changes. (**a**) \(R_{m}\) changes; (**b**) \(\delta\) changes; (**c**) \(l\) changes; (**d**) \(\rho\) changes. The influences of the UUV system parameters on the amplitude of its modes are also investigated. Based on this objective, two typical modes B(2,1) and B(4,2) are selected to be investigated for each parameter. The results obtained are presented in Figures 8-10 and indicate the following:  
Figure 5: Displacement modes of the UUV body structure with (**a**) B(2,1); (**b**) B(4,2). Figure 6: Shear modes of the UUV body structure with (**a**) B(2,1); (**b**) B(4,2). Figure 7: Torsional moment modes of the UUV body structure with (**a**) B(2,1); (**b**) B(4,2). Figure 8: Effect of wall thickness \(\delta\) on modes amplitudes with (**a**) B(2,1); (**b**) B(4,2). Figure 9: Effect of middle surface radius \(R_{m}\) on modes amplitudes with (**a**) B(2,1); (**b**) B(4,2). (1) As the wall thickness \(\delta\) increases, the amplitude of the shear modes and the torsional moment modes increase. For the modes B(2,1) and B(4,2), the amplitudes of \(F_{\mathrm{Sx\theta}}\) and \(M_{\mathrm{x\theta}}\) at \(\delta\) = 12 mm are three times and nine times larger, respectively, than the values at \(\delta\) = 4 mm. Hence, the torsional moment modes response is more sensitive than the shear modes response with the change of parameter \(\delta\). (2) With the increase of the middle surface radius \(R_{m}\), the amplitude of the shear modes increases slightly, whereas the amplitude of the torsional moment modes decreases. Meanwhile, the amplitude of \(F_{\mathrm{Sx\theta}}\) increase by approximately 21 percent and 17 percent at modes B(2,1) and B(4,2), respectively, while the amplitude of \(M_{\mathrm{x\theta}}\) decrease by 26.1 percent and 25.7 percent, respectively. These results show that the torsional moment vibration modes are apparent when \(R_{m}\) is quite small, and that the shear vibration modes become increasingly significant as \(R_{m}\) becomes larger. (3) Contrary to the law of \(\delta\) changes, as the body length \(l\) increases, the amplitude of the shear modes and torsional moment modes decrease. The decrements of \(F_{\mathrm{Sx\theta}}\) at modes B(2,1) and B(4,2) are 43.8 percent and 40.7 percent, respectively. In addition, the decrements of \(M_{\mathrm{x\theta}}\) are 29.1 percent and 29.8 percent, respectively. The shear modes are more sensitive to the change of body length.