*3.4. Viscous Damping*

Viscous damping originates from the fluid resistance. It is considered to be the most significant damping mechanism in MEMS operating in ambient conditions.

During the beam movement in fluid, an additional force related to the medium appears. The quality factor due to viscous damping can be analytically expressed using a normalized time-independent function called hydrodynamic function Γ*hydro*:

$$Q\_{\text{viscous}} = \frac{\frac{4\rho\_b h}{\pi \rho\_f b} + \Gamma\_{hydro}^R(\omega)}{\Gamma\_{hydro}^I(\omega)}\tag{13}$$

where *ρb*, *ρ<sup>f</sup>* , Γ*<sup>R</sup> hydro*, <sup>Γ</sup>*<sup>I</sup> hydro* are the density of the beam, the density of the fluid, and the real and imaginary parts of the hydrodynamic function, respectively. The total hydrodynamic function originates from the linearized Navier–Stokes equation. Thus, it can be represented as a linear combination of hydrodynamic functions originating from each sidewall of the beam cross-section [33]. Pictorially, it is presented in Figure 5, while mathematically it is expressed as:

$$
\Gamma\_{hydro} = \frac{1}{2}\Gamma\_{tb} + \frac{1}{2}\Gamma\_{tb} + \Gamma\_{sq} + \frac{1}{2}\Gamma\_{lr} + \frac{1}{2}\Gamma\_{lr} \tag{14}
$$

where Γ*tb*, Γ*sq*, Γ*lr* are hydrodynamic functions originating from the top and bottom side of the cantilever, squeeze film, and the left and right side of the cantilever, respectively.

**Figure 5.** Scheme of streamlines acting on the cross-section sidewalls of the cantilever oscillating in its first mode of vibration with corresponding hydrodynamic functions. Γ*tb* is used to describe the forces applied at the top and the bottom of the cantilever, Γ*lr* relates to the left and right sides of the cantilever, while Γ*sq* is a hydrodynamic force originating from squeeze film effect.
