3.4.3. Viscous Damping Due to the Squeeze Film Effect

On the cantilever sidewall, where the gas is trapped between substrate and cantilever, there exists an additional counter reactive force originating from squeeze film action. A mathematical description of squeeze film was given by Bao et al. [36]:

$$\Gamma\_{sq}(\omega) = \frac{-4P\_d}{\pi d b \rho\_f \omega^2} (f\_\ell(\sigma) - if\_d(\sigma)) \tag{17}$$

where *Pa*, *d* are the surrounding pressure and air gap shown in Figure 5, *σ* is a squeeze number [36] given by the following equation:

$$
\sigma = \frac{12\mu\_f \omega L^2}{P\_d d^2} \tag{18}
$$

and *fe*(*σ*) and *fd*(*σ*) are functions introduced by Langlois [37] with the following form:

$$\begin{pmatrix} f\_{\varepsilon}(\sigma) & 1 - \sqrt{\frac{2}{\sigma}} \frac{\sinh\left(\sqrt{\frac{\kappa}{2}}\right) + \sin\left(\sqrt{\frac{\kappa}{2}}\right)}{\cosh\left(\sqrt{\frac{\kappa}{2}}\right) + \cos\left(\sqrt{\frac{\kappa}{2}}\right)}\\ f\_{d}(\sigma) & \quad -\sqrt{\frac{2}{\sigma}} \frac{\sinh\left(\sqrt{\frac{\kappa}{2}}\right) - \sin\left(\sqrt{\frac{\kappa}{2}}\right)}{\cosh\left(\sqrt{\frac{\kappa}{2}}\right) - \cos\left(\sqrt{\frac{\kappa}{2}}\right)} \end{pmatrix} \tag{19}$$
