## 2 Theoretical Model of a Quartz Tuning Fork  
### Mechanical Model  
Each prong of the QTF can be considered as a 2D free-clamped beam, since the crystal width \(w\) is smaller than both the QTF thickness \(T\) and length \(L\). [20]. According to the Euler-Bernoulli beam theory, the dynamics of the vibration of a beam is described by the following differential equation [21]:  
\[EI\frac{\partial^{4}y(x,t)}{\partial^{4}x}+\rho A\ \frac{\partial^{2}y(x,t)}{ \partial^{2}t}=0 \tag{1}\]  
Assuming a homogeneous prong, \(E\) is the Young's modulus in the QTF vibration plane, \(\rho\) is the quartz density, \(A\) is the prong section and \(I\) is the moment of inertia. Equation (1) can be solved using the separation variable method with a free-clamped boundary condition. Thus, the resonance frequencies of the in-plane flexural modes are determined [20]:  
\[f^{\prime}_{\ n}=\frac{1}{2\pi}\sqrt{\frac{EI}{\rho A}}k_{n}^{2} \tag{2}\]  
where \(k_{n}^{2}=v_{n}{}^{2}/L^{2}\) and \(v_{n}\) is a number determined by the specific resonance mode (\(n=0\) identifies the fundamental mode). When the QTF is vibrating in air, the prong loses energy due to interaction with the surrounding medium. This effect can be modelled in the Euler -Bernoulli theory as an additional inertia caused by the motion of the molecules in the air and a viscous drag force acting on the prongs. Defining \(u\) as an additive mass per unit of length and \(C_{d}\) as the damping parameter, Equation (1) becomes [21]:  
\[EI\frac{\partial^{4}y(x,t)}{\partial^{4}x}+(\rho A+u)\frac{\partial^{2}y(x,t)}{ \partial^{2}t}+C_{d}\frac{\partial y(x,t)}{\partial t}=0 \tag{3}\]  
In the realistic case of a negligible damping, if \(u\ll\rho A\), the additive mass causes a normalized shift on the frequency of the fundamental mode given by:  
\[\Delta f=\frac{f^{\prime}{}_{o}-f_{0}}{f^{\prime}{}_{o}}=\frac{u}{2\rho A} \tag{4}\]  
where \(f_{0}\) are the QTF resonance frequencies when prongs vibrate in air. In steady motion, the additive mass is proportional to the density of air \(\rho_{air}\), thus \(u=k\rho_{air}\)[22]. According to the ideal gas law, \(\rho_{air}=MP/R\Theta\), where \(M=28.964\) kg/mol is the molar mass, \(P\) is air pressure, \(R=62.3637\) m\({}^{3}\) -Torr/K- mol is the gas constant, and \(\Theta\) is the QTF temperature. Air damping also affects the \(Q\)-factor of the QTF resonance mode. The contribution due to air damping was derived by [PERSON] et al. [9]. In the viscous region, the dynamics can be described by the Navier-Stokes equation which leads to an expression for the \(Q\)-factor contribution \(Q_{air}\) related to air damping given by:  
\[Q_{air}=\frac{4\pi\rho Tw^{2}f_{0}}{3\pi\mu w+\frac{3}{4}\pi w^{2}\sqrt{4\pi \rho_{air}\mu f_{0}}} \tag{5}\]  
where \(\mu\) is the coefficient of the viscosity of air. Thermoelastic and support losses are intrinsic dissipation mechanisms and can be assumed as pressure-independent. [PERSON] et al. [11] developed a model to describe support losses as the effect of a shear force exerted from the vibrating beam on the QTF support, which excites elastic waves propagating into the support with a wavelength greater than the prong width \(w\). With this assumption, the quality factor contribution \(Q_{sup}\) related to support losses can be simplified as:  
\[Q_{sup}=A_{n}\frac{L^{3}}{T^{3}} \tag{6}\]  
with the coefficients \(A_{n}\) depending on the \(n\)-th QTF resonance mode. Thermoelastic losses are due to the inhomogeneity of the stress field, which, in turn, causes an inhomogeneous, local increase in temperature. This temperature gradient is an additional channel of energy dissipation [12]. The corresponding Q-factor contribution \(Q_{TED}\) depends upon prong geometry and can be expressed as:  
\[Q_{TED}\propto\frac{T^{3}}{L^{2}} \tag{7}\]  
These theoretical models describe each loss mechanism as standalone and their dependence on the main physical parameters is explicated. Each loss contribution is independent from the other, but all occur simultaneously for a vibrating QTF prong.