Issue 29

V. Zega et alii, Frattura ed Integrità Strutturale, 29 (2014) 334-342; DOI: 10.3221/IGF-ESIS.29.29 336 Bending and torsional resonators We first focus on the modelling of the two types of resonator which constitute the sensing elements of the integrated structure. The flexural resonators are very thin beams attached to the substrate at one end and to the springs at the other end, at a certain distance H from the anchor point (see Fig. 2(a)). The optimal length of H is selected to maximize the sensitivity of the device. The electrostatic driving of the resonator is done by means of an electrode attached to the substrate with initial gap d 0 ; another parallel electrode allows for the sensing. During functioning a polarization voltage V p is applied to the resonator while a small oscillating signal v a ( t ) applied to the driving plate drives the beam to resonance. In the absence of external input, the resonator has the nominal frequency 0 f 0 1 2 m e k k f m    (1) where m k , e k and m are the equivalent mechanical and electrostatic stiffness and the equivalent mass, respectively. Figure 2 : Schematic views of the resonators: (a) in-plane view of bending resonator, (b) in-plane and side view of torsional resonator Their expressions can be obtained as in [13] making use of the Hamiltonian’s principle and searching for the solution, in terms of transverse displacement of the beam, in the form ( , ) ( ) ( ) w y t y W t   . Using as ( ) y  the eigenfunction of a doubly clamped beam these equivalent properties read:   2 3 0 2 2 0 0 2 3 3 0 0 0 2 0 198.492 2 2 0.397 0.397 h m h p p e h EI k EI dy h sV sV k dy h d d m A dy Ah                  (2) where E is the Young’s modulus, A and I the area and the momentum of inertia of the cross section, 0  is the dielectric permittivity, s is the out of plane thickness and  the mass density.