# Design and Experiment of a Passive Vibration Isolator for Small Unmanned Aerial Vehicles
## 4 Vibration Shaker Experiment: Analysis of Damping Characteristic
### Analysis of Damping Characteristic of Vibration Isolators  
The stiffness of the vibration isolator rubber was determined by comparing the extent of damping amount and damping performance from the vibration data at the bending frequency from the random vibration test. Table 7 presents the accelerance data at each resonance point from the random vibration test. The test results indicate that the vibration isolators exhibit a damping amount ranging from a minimum of 13.50 dB to a maximum of 13.61 dB in Section 1, and from a minimum of 22.98 dB to a maximum of 26.57 dB in Section 2, based on the stiffness. Table 8 calculates the damping amount for each test using the damping values from Table 7 and Equation (2). \(L_{B}\) is damping performance, \(A\) is amplitude of undamped data, and \(B\) is amplitude of damped data. \[L_{B}=10 log_{10}\frac{B}{A} \tag{2}\]  
The results of the vibration tests indicated that the vibration isolator with a hardness of 70 exhibited a high damping performance of 453.94 in Section 2, demonstrating relatively higher damping capabilities when compared to the vibration isolators with hardness levels of 30 and 50. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Mass**} & \multirow{2}{*}{**Category**} & \multicolumn{3}{c}{**Hardness of 30**} & \multicolumn{3}{c}{**Hardness of 50**} & \multicolumn{3}{c}{**Hardness of 70**} \\ \cline{3-10}  & &
\begin{tabular}{c} **First** \\ **Natural** \\ **Frequency** \\ \end{tabular} & **Section 1** & **Section 2** & **Natural** & **Section 1** & **Section 2** & **Natural** & **Section 1** & **Section 2** \\ \hline \multirow{4}{*}{1.7 kg} & FEM & 9.4 Hz & 50.2 Hz & 71.3 Hz & 10.7 Hz & 58.1 Hz & 79.9 Hz & 11.0 Hz & 64.1 Hz & 96.2 Hz \\  & Analysis & & & & & & & & \\  & Random & 9.2 Hz & 49.6 Hz & 70.8 Hz & 10.4 Hz & 57.6 Hz & 80.4 Hz & 11.2 Hz & 65.6 Hz & 91.6 Hz \\  & Vibration test & & & & & & & & \\  & Error Rate & 2.6\% & 1.2\% & 0.7\% & 3.0\% & 0.8\% & 0.6\% & 1.4\% & 2.3\% & 5.0\% \\  & FEM & 9.1 Hz & 49.5 Hz & 71.9 Hz & 10.21 Hz & 57.5 Hz & 79.9 Hz & 11.0 Hz & 64.8 Hz & 85.2 Hz \\  & Random & 9.2 Hz & 49.2 Hz & 70.0 Hz & 10.4 Hz & 57.2 Hz & 80.0 Hz & 11.2 Hz & 65.2 Hz & 90.8 Hz \\  & Vibration test & & & & & & & & \\  & Error Rate & 1.0\% & 0.6\% & 2.6\% & 1.8\% & 0.5\% & 0.1\% & 1.1\% & 0.6\% & 6.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison between natural frequency in FEM analysis and resonance frequency in random vibration test. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multirow{2}{*}{\begin{tabular}{c} **Sensor** \\ **Location** \\ \end{tabular} } & \multicolumn{3}{c}{**Magnitude**} \\ \cline{3-8}  & & \multicolumn{3}{c}{**Hardness of 30**} & \multicolumn{3}{c}{**Hardness of 50**} & \multicolumn{3}{c}{**Hardness of 70**} \\ \cline{3-8}  & & **Section 1** & **Section 2** & **Section 1** & **Section 2** & **Section 1** & **Section 2** \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} Random \\ vibration \\ \end{tabular} } & S1 & 7.02 & 14.36 & 8.84 & 12.90 & 7.07 & 10.47 \\ \cline{2-8}  & S2 & \(-\)6.48 & \(-\)8.62 & \(-\)4.77 & \(-\)11.20 & \(-\)6.48 & \(-\)16.10 \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Decibel \\ reduction \\ \end{tabular} } & \multirow{2}{*}{13.50} & \multirow{2}{*}{22.98} & \multirow{2}{*}{13.61} & \multirow{2}{*}{24.1} & \multirow{2}{*}{13.55} & \multirow{2}{*}{26.57} \\ \cline{3-8}  & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Random vibration test results according to hardness (amplitude unit = dB). The performance indicator of a vibration isolator is the damping ratio, which is a measure of how efficiently the isolator material dissipates the energy generated from vibrations and shocks. The damping ratio is calculated using the \(Q\) factor and is given as follows:  
\[Q=\frac{1}{2*\zeta} \tag{3}\]  
Table 9 shows the damping ratios calculated using the FRF graphs from the random vibration test and Equation (3).