# Design and Experiment of a Passive Vibration Isolator for Small Unmanned Aerial Vehicles
## 2 Design of the Vibration Isolator and Test of Rubber Stiffness
### FEM Analysis of a Fixed-Fixed Beam including Vibration Isolator  
Rubber, a material that embodies both the damping and stiffness properties, was equivalently modeled using the Kelvin-Voigt theory, which incorporates a viscoelastic model to accurately represent the interaction between the rigid and non-rigid parts of rubber. The spring contact boundary conditions were applied to the top and bottom plates of the vibration isolator. Figure 11b illustrates the method used to input the variables for the spring contact condition. The FEM analysis was conducted with stiffness values (K) of 1.796, 2.902, and 4.652 N/mm for the spring contact condition. These values were derived from the stiffness measurement experiments at hardness levels of 30, 50, and 70, respectively. Damping was applied through the mathematical analysis of a mass-spring-damper model. For the FEM analysis (depicted in Figure 12), both ends of the beam, assumed to represent a drone, were fixed, and the optical equipment weighing 1.7 and 2.0 kg was applied as a point mass. Through FEM analysis, the natural frequencies within the drone's operational frequency band of 40-85 Hz were identified. Figure 13 presents the FEM analysis results depicting the natural frequencies and mode shapes for stiffness levels of 30, 50, and 70. During the evaluation process for the FEM analysis, the natural frequencies due to bending of the vibration isolator mounted on the fixed-fixed beam were confirmed. Figure 13a,d,g represent the first natural frequencies with stiffness levels of 30 presenting 9.44 Hz, 50 presenting 10.72 Hz, and 70 presenting 11.04 Hz. Figure 13b,e,h represent the second natural frequencies with stiffness levels of 30 presenting 50.29 Hz, 50 presenting 58.09 Hz, and 70 presenting 64.09 Hz. Figure 13c,f,i represent the third natural frequencies with stiffness levels of 30 presenting 71.31 Hz, 50 presenting  
Figure 11: FEM analysis contact conditions for vibration isolator simulation: (a) equipment of rubber shape and placement; (b) spring contact conditions. Figure 10: Concentrated mass applied to the fixed-fixed beam (WDH: 30 \(\times\) 3 \(\times\) 540 [mm]) with a natural frequency of 10.89 Hz. 79.91 Hz, and 70 presenting 96.18 Hz. Also, considering the drone's maximum takeoff weight, the mass of the optical equipment was set to 2.0 kg for additional FEM analysis. Figure 14 presents the FEM analysis results depicting the natural frequencies and mode shapes for stiffness levels of 30, 50, and 70. During the evaluation process for the FEM analysis, the natural frequencies due to bending of the vibration isolator mounted on  
Figure 12: FEM analysis boundary conditions for the vibration isolator. Figure 13: FEM analysis results: (**a**) Mode 1: bending frequency 9.44 Hz (hardness 30); **(b)** Mode 2: bending frequency 50.29 Hz (hardness 30); **(c)** Mode 3: bending frequency 71.31 Hz (hardness 30); **(d)** Mode 1: bending frequency 10.72 Hz (hardness 50); **(e)** Mode 2: bending frequency 58.09 Hz (hardness 50); **(f)** Mode 3: bending frequency 79.91 Hz (hardness 50); **(g)** Mode 1: bending frequency 11.04 Hz (hardness 70); **(h)** Mode 2: bending frequency 64.09 Hz (hardness 70); **(i)** Mode 3: bending frequency 96.19 Hz (hardness 70). the fixed-fixed beam were confirmed. Figure 14a,d,g represent the first natural frequencies with stiffness levels of 30 presenting 9.10 Hz, 50 presenting 10.21 Hz, and 70 presenting 11.07 Hz. Figure 14b,e,h represent the second natural frequencies with stiffness levels of 30 presenting 49.47 Hz, 50 presenting 57.54 Hz, and 70 presenting 64.81 Hz. Figure 14c,f,i represent the third natural frequencies with stiffness levels of 30 presenting 71.99 Hz, 50 presenting 79.85 Hz, and 70 presenting 85.24 Hz. The natural frequency of the vibration isolator mounted on a fixed-fixed beam, detected within the drone's operational frequency band of 40-85 Hz using FEM results, can potentially cause instability in the drone's operation. Therefore, it is necessary to avoid these frequencies to ensure stable drone performance.