*3.3. Support Losses*

The cantilever presented in Figure 1 is held by a support. During the cantilever movement a part of the energy is dissipated into the support. This dissipation is described by the support quality factor. An analytical solution for support losses in case of a clampedfree cantilever was proposed by Hao [32] and takes the following form:

$$Q\_{support} = \left(\frac{0.24(1-\nu)}{(1+\nu)\Psi}\right)\frac{1}{(\frac{a\_n}{\pi}\chi\_n)^2} \left(\frac{L}{h}\right)^3\tag{12}$$

where *ν*, *αn*, *χ<sup>n</sup>* is the Poisson's ratio, a mode constant, and a mode shape factor, respectively. For the clamped-free cantilever fundamental mode *n* = 1 and *α*<sup>1</sup> = 1.875, the mode shape factor *<sup>χ</sup>*<sup>1</sup> <sup>=</sup> sin(*α*1)−sinh(*α*1) cos(*α*1)+cosh(*α*1) and <sup>Ψ</sup> <sup>=</sup> 0.336. It can be seen from Equation (12) that the energy dissipation from the support is inversely proportional to (*L*/*h*)3. If we look at a fixed frequency, without considering the length of the cantilever, then the quality factor of the support is *Qsupport* ∝ √ <sup>1</sup> *ω*3 *nh*3 .
