## 2 Materials and Methods
### Finite Element Method
#### 2.2.2 Comparison of Finite Element Method and Analytical Method
The difference between the numerical results and the analytical results of the quality factor is large, the relative difference between the numerical results and the experimental results is less than 4%, and the relative difference between the analytical results and the experimental results can even exceed 110%. This shows that the finite element method includes the influence of fluid viscous effects on tuning-fork resonance parameters, and is closer to the actual situation than the approximate analytical method, and the simulation results are more accurate. Figure 4: (**a**) Relative difference of numerical results and analytical results of resonant frequency corresponding to 19 groups of different fluids. (**b**) Relative difference of numerical results and analytical results of quality factor corresponding to 19 groups of different fluids. Figure 3: (**a**) Numerical, analytical, and experimental results of resonant frequency corresponding to 19 groups of different fluids. (**b**) Numerical, analytical, and experimental results of quality factor corresponding to 19 groups of different fluids. where Y represents f or Q, x represents \(\rho\) or n, and Formula (11) represents the relative change in resonant frequency or quality factor corresponding to a relative change in unit density or viscosity. The higher the sensitivity, the more sensitive the tuning-fork resonance parameters are to the changes of fluid characteristic parameters, resulting in higher measurement accuracy. The tuning fork size is set to be consistent with that used in the laboratory in the finite element method. Fluid density increases from 100 kg/m\({}^{3}\) to 1900 kg/m\({}^{3}\), \(\Delta\rho\) is 50 kg/m\({}^{3}\), viscosity increases from 2 mPa\(\cdot\)s to 212 mPa\(\cdot\)s, and \(\Delta\)n is 6 mPa\(\cdot\)s.Figure 6 shows the sensitivity of resonant frequency to density and viscosity, and Figure 7 shows the sensitivity of quality factor to density and viscosity. It can be seen from Figure 6 that the sensitivity of tuning-fork resonant frequency to fluid density and viscosity increases with the increase in fluid density and viscosity, and the sensitivity of tuning-fork resonant frequency to fluid density is greater than to viscosity. It can be seen from Figure 7b that the sensitivity of tuning fork quality factor to fluid viscosity generally increases with the increase in fluid viscosity. The sensitivity value of tuning fork quality factor to fluid density is comparable to that of viscosity. But the sensitivity of quality factor to fluid density does not vary monotonically. [PERSON] et al. [15] analyzed, through experiments, the sensitivity of the metal tuning fork driven by electromagnetic means. However, this approach required setting up experimental platforms and extensive data-processing work. Moreover, the analysis results were limited by experimental samples and only the sensitivity of the tuning fork within a small range of density and viscosity variations was analyzed. In contrast, the finite element method does not require the construction of experimental platforms and experiments, and can analyze the sensitivity of the tuning fork when the density and viscosity change in a wide range, which can greatly save time and economic costs. Figure 6: (**a**) Sensitivity of resonant frequency to density. (**b**) Sensitivity of resonant frequency to viscosity. Figure 7: (**a**) Sensitivity of quality factor to density. (**b**) Sensitivity of quality factor to viscosity.