# Design and Experiment of a Passive Vibration Isolator for Small Unmanned Aerial Vehicles
## 2 Design of the Vibration Isolator and Test of Rubber Stiffness
### Comparative Analysis between Fixed-Fixed Beam Numerical Analysis and FEM Analysis  
As drones are complex structures that often experience harmonic vibrations, an intuitive assessment of the damping effects of a vibration isolator may be challenging owing to the natural frequencies of the complex structure. Vibration experiments based on fixed-fixed beams enable the measurement of the vibrational characteristics of the system, aiding the understanding of the behavior of vibration isolators and providing insights into improving the design. Numerical analysis and FEM analysis is one of the methods for analyzing the dynamic characteristics of engineering systems. It enables the identification of a system's natural frequencies and mode shapes, thus allowing for the early detection and prevention of potential issues such as resonance induced at these frequencies. This approach provides an intuitive means for the optimization of design parameters. In this study, a test setup was developed to simulate drone vibrations. According to [PERSON] et al. (2016), the vibrations transmitted to the optical equipment affect the \(Z\)-axis translational motion more than in the X-Y translational movements. Therefore, only the translational motion along the \(Z\)-axis was considered to simulate the drone's vibration environment. Figure 8: Drone ground vibration test accelerometer data FFT result. Figure 7: Drone ground vibration test set-up and sensor specification. Figure 9 represents a scheme for depicting the experimental environment through numerical analysis. The numerical analysis, applying the mass conditions of the actual experimental environment, is as follows in Equation (1):  
\[f_{1}=\frac{1}{2\pi}\left[\frac{3 EI(a+b)^{3}}{a^{3}b^{3}m}\right] \tag{1}\]  
In Equation (1), \(a\) represents the distance from one end to the concentrated mass, and \(b\) is that from the other end to the concentrated mass. The parameters for the fixed-fixed beam used here are as follows: \(E=78.5\) GPa, \(I=6.75\times 10^{-11}\) m\({}^{4}\), \(a=0.16\) m, and \(b=0.39\) m. To ensure the reliability of the numerical analysis results, m was varied across eight cases from 0.5, 1, 1.5, 1.75, 2.0, 2.25, 2.5, and 3 kg comparing numerical and FEM analysis under various mass conditions. Table 5 presents the results of a comparative analysis between the numerical and FEM analyses of the first natural frequency, with each concentrated mass case applied in the range of 0.5-3 kg across eight cases. Due to the limitations in the mass of optical equipment cases that can be mounted on the vibration isolator, reliability was ensured by comparing and analyzing FEM analysis with numerical analysis. An observed trend is the decrease in the first natural frequency as the mass increases. The discrepancy between the numerical analysis and FEM analysis due to mass increase showed a maximum error rate of 3.3%, thus verifying the mathematical modeling's adequacy through the analysis results. The FEM analysis was conducted on a fixed-fixed beam by applying a total mass of 2.17 kg, which includes the actual optical equipment mass of 1.7 kg and vibration isolator plate mass of 0.47 kg intended for attachment to the fixed-fixed beam. The FEM analysis with the experimental mass applied is shown in Figure 10.  
The first natural frequency of the fixed-fixed beam obtained through FEM analysis is 10.89 Hz, showing a 0.9% error rate compared to the numerical analysis result of 10.99 Hz. Through comparative analysis between numerical analysis and FEM analysis, a fixed-fixed beam simulating the drone's first natural frequency was implemented. \begin{table}
\begin{tabular}{c c c c} \hline
**Load (kg)** & **Numerical Analysis (Hz)** & **FEM Analysis (Hz)** & **Error (\%)** \\ \hline
0.5 & 20.62 & 21.50 & 4.0 \\
1.0 & 15.57 & 15.74 & 1.1 \\
1.5 & 13.02 & 13.00 & 0.1 \\
1.75 & 12.14 & 12.08 & 0.5 \\
2.0 & 11.42 & 11.33 & 0.8 \\
2.25 & 10.81 & 10.70 & 1.0 \\
2.5 & 10.29 & 10.17 & 1.2 \\
3.0 & 9.44 & 9.31 & 1.4 \\ \hline \end{tabular}
\end{table}