## 2 Materials and Methods
### Finite Element Method
#### 2.2.2 Comparison of Finite Element Method and Analytical Method
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**No.**} & \multirow{3}{*}{\begin{tabular}{c} **Density** \\ **kg/m\({}^{3}\)** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} **Viscosity** \\ **mPa-s** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} **Experimental** \\ **Results** \\ \end{tabular} } & \multicolumn{3}{c}{\begin{tabular}{c} **Numerical** \\ **Results** \\ \end{tabular} } & \multicolumn{3}{c}{\begin{tabular}{c} **Relative Difference of** \\ **Numerical Results** \\ \end{tabular} } & \multicolumn{3}{c}{\
\begin{tabular}{c} **Analytical** \\ **Results** \\ \end{tabular} } & \multicolumn{3}{c}{
\begin{tabular}{c} **Relative Difference of** \\ **Analytical Results** \\ \end{tabular} } \\ \cline{3-13}  & & & & \(f\) & & & & & & & & & \\  & & & & & & & & & & & & \\  & & & & & & & & & & & & \\  & & & & & & & & & & & & \\ \hline \(1\) & 828.6 & 2.930 & 29,089 & 58.53 & 29,485 & 57.97 & 1.36 & 0.96 & 29,540 & 122.61 & 1.55 & 109.48 \\ \(2\) & 815.1 & 4.658 & 29,077 & 46.75 & 29,474 & 46.88 & 1.37 & 0.28 & 29,544 & 97.88 & 1.61 & 109.49 \\ \(3\) & 848.4 & 6.096 & 28,925 & 41.98 & 29,302 & 40.54 & 1.29 & 3.43 & 29,923 & 84.04 & 1.38 & 100.19 \\ \(4\) & 812.4 & 7.08 & 29,027 & 37.07 & 29,411 & 38.21 & 1.31 & 3.08 & 29,483 & 79.38 & 1.57 & 114.14 \\ \(5\) & 854.9 & 8.344 & 28,845 & 34.18 & 29,216 & 34.61 & 1.24 & 1.24 & 29,226 & 71.49 & 1.32 & 109.16 \\ \(6\) & 834.3 & 10.64 & 28,877 & 30.40 & 29,240 & 30.92 & 1.26 & 1.71 & 29,277 & 63.86 & 1.39 & 110.07 \\ \(7\) & 849.4 & 11.78 & 28,778 & 27.73 & 29,157 & 29.23 & 1.32 & 5.41 & 29,173 & 60.18 & 1.37 & 117.02 \\ \(8\) & 864.8 & 14.68 & 28,699 & 26.55 & 29,040 & 26.06 & 1.19 & 1.85 & 29,034 & 53.40 & 1.17 & 101.13 \\ \(9\) & 840.7 & 15.18 & 28,776 & 26.38 & 29,124 & 25.86 & 1.21 & 1.97 & 29,150 & 53.12 & 1.30 & 101.29 \\ \(10\) & 866.1 & 18.31 & 28,630 & 23.80 & 28,961 & 23.3 & 1.16 & 2.10 & 28,950 & 47.36 & 1.12 & 98.99 \\ \(11\) & 831.1 & 19,60 & 28,757 & 23.77 & 29,085 & 22.84 & 1.44 & 3.91 & 29,122 & 46.84 & 1.27 & 97.06 \\ \(12\) & 850.3 & 28.62 & 28,542 & 19.38 & 28,878 & 18.69 & 1.18 & 3.56 & 28,885 & 38.20 & 1.20 & 97.11 \\ \(13\) & 853.5 & 36.52 & 28,426 & 16.39 & 28,700 & 16.45 & 1.21 & 0.37 & 28,767 & 33.62 & 1.20 & 105.13 \\ \(14\) & 915.9 & 40.29 & 28,138 & 14.97 & 28,483 & 15.31 & 1.23 & 2.27 & 28,401 & 31.05 & 0.93 & 107.41 \\ \(15\) & 867.9 & 40.70 & 28,329 & 15.56 & 28,665 & 15.47 & 1.19 & 0.57 & 28,643 & 31.57 & 1.11 & 102.89 \\ \(16\) & 849.7 & 69.22 & 28,115 & 12.31 & 28,457 & 11.78 & 1.22 & 4.31 & 28,460 & 24.14 & 1.23 & 96.10 \\ \(17\) & 941.8 & 71.36 & 27,847 & 12.13 & 28,077 & 11.27 & 0.83 & 7.09 & 27,966 & 22.80 & 0.43 & 87.96 \\ \(18\) & 968.3 & 119.78 & 27,380 & 9.01 & 27,628 & 8.66 & 0.91 & 3.88 & 27,480 & 17.17 & 0.37 & 90.57 \\ \(19\) & 886.3 & 210.58 & 27,151 & 6.69 & 27,450 & 6.47 & 1.10 & 3.29 & 27,411 & 13.12 & 0.96 & 96.11 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of numerical results, analytical results, and experimental results. Figure 2: (a) Conductance diagram of the tuning fork in fluid of group 1. (b) Conductance diagram of the tuning fork in fluid of group 1. Figure 3 shows the comparison of numerical, analytical, and experimental results of resonant frequency and quality factor. Figure 4 shows the relative differences between numerical results and experimental results, and the relative differences between analytical results and experimental results, respectively. The relative difference of numerical results is numerical result - experimental result \(\times\) 100%, and the relative difference of analytical results is analytical result - experimental result \(\times\) 100%. From Figures 3 and 4, it can be seen that the numerical results and analytical results of the resonant frequency are highly consistent, with a relative difference of less than 2% from the experimental measurement results.