3.4.1. Viscous Damping on the Top and the Bottom

Γ*tb* describes the viscous damping on the front and the back of the cantilever.

Sader [34] used the exact analytic solution for a circular-cross section cantilever. Then he used a multiplicative correction function Ω*sader* in order to provide a more precise result in case of infinitely thin rectangular beams. Correction function Ω*sader* depends on the Reynolds number and therefore on the width and frequency of the cantilever. The expression of Γ*tb* is given in Equation (15) and the Ω*sader* expression in [34].

$$\Gamma\_{tb}(\omega) = \left(1 + \frac{4iK\_1(-i\sqrt{iR\_\varepsilon})}{\sqrt{iR\_\varepsilon}K\_0(-i\sqrt{iR\_\varepsilon})}\right)\Omega\_{sader}(\omega)\tag{15}$$

*<sup>K</sup>*0, *<sup>K</sup>*<sup>1</sup> are modified Bessel functions of the second kind, *Re* <sup>=</sup> *<sup>ρ</sup> <sup>f</sup> <sup>ω</sup>b*<sup>2</sup> <sup>4</sup>μ*<sup>f</sup>* is the Reynolds number, *ρ<sup>f</sup>* is the density of the fluid, and μ*<sup>f</sup>* is the dynamic viscosity.
