## 1 Introduction  
The two arms of a tuning fork can be regarded as two symmetrical cantilever beams with one end fixed and the other end free, which can achieve decoupled measurement of fluid density and viscosity. Tuning forks are widely employed in industrial production. For example, tuning forks are used to measure the density and viscosity of wine in real time, thereby achieving quality monitoring of the wine fermentation process [1, 2]. Tuning forks are used to measure the density and viscosity of gases, in order to monitor the gas polymerization process in high-temperature and high-pressure containers [3]. The smoke and dust content of engine oil can be monitored by measuring its density and viscosity through tuning forks [4; 5]. In the process of oil and gas exploitation [6; 7], using a tuning fork to measure the density and viscosity of downhole fluid can identify fluid [8; 9], determine fluid composition, and divide the oil-water interface, which is of great significance for optimizing oil and gas reservoir management and intelligent oil-field exploitation [10; 11]. In-depth study of the relationship between tuning-fork resonance characteristics and fluid characteristic parameters is helpful in optimizing the design of the tuning fork sensor, thereby enhancing the accuracy of fluid density and viscosity measurements using tuning forks. In 1998, [PERSON] established an explicit analytical model based on the Euler-Bernoulli cantilever-beam theory, which relates the resonant frequency and quality factor of a cantilever beam vibrating in a fluid to fluid density and viscosity. The model can express the quantitative relationship between the resonant frequency and quality factor of the tuning fork, the size of the tuning fork, the material parameters, and the fluid density and viscosity, but only for fluid with small viscous effects, such as air [12]. In 2011, [PERSON] et al. developed an implicit analytical model using an equivalent circuit approach, relating tuning-fork resonant frequency and quality factor to fluid density and viscosity. The model contained undetermined coefficients related to the material and size of the tuning fork, and the accuracy of the model was affected by the parasitic capacitance of the tuning fork [13]. In 2014, [PERSON] et al. compared the vibration of a tuning fork in a fluid to the damped vibration of a pendulum-spring system. Based on the forces exerted on the tuning fork in the fluid, they derived an implicit analytical model relating tuning-fork resonant frequency and quality factor to fluid density and viscosity, with the effective area of interaction between the tuning fork and fluid being an undetermined coefficient [14]. In 2015, [PERSON] et al. studied the feasibility of using an electromagnetic-driven metal tuning fork to measure fluid density and viscosity, and analyzed the sensitivity of tuning fork characteristic parameters to fluid density and viscosity. However, due to the limitation of experimental fluid samples, only the sensitivity of the tuning fork was studied when density and viscosity changed within a small range [15]. In 2020, [PERSON] et al. used the Sobol global sensitivity analysis method instead of the commonly used finite difference method to analyze the sensitivity of tuning forks to fluid density and viscosity, but did not carry out further research into the relationship between the sensitivity of tuning forks and the large-scale variation of fluid density and viscosity [16]. In order to overcome the limitation of existing explicit analytical models, which are only applicable when the viscous effects of the fluid are small, this paper utilizes finite element analysis to simulate the coupling vibration law between tuning forks and fluids. It compares the results of finite element numerical simulations, analytical model predictions, and experimental measurements. Building upon this comparison, the finite element method is employed to simulate the relationship between tuning-fork resonance characteristics and fluid characteristic parameters.