Issue 29

V. Zega et alii, Frattura ed Integrità Strutturale, 29 (2014) 334-342; DOI: 10.3221/IGF-ESIS.29.29 339 x m F y c  Ω2  (12) As a result, the two inertial masses rotate about their respective axis of rotation a-a in the same direction (see Fig. 4(a)) of the angle: Ω 3 3 l tsG Rx m β G y   (13) where R G is the distance between the a-a axis of rotation of a proof mass and the center of gravity of the same mass, is the total length of one folded torsional spring attaching the mass to the substrate and t its in-plane thickness . Figure 4 : Schematic plan view of the structure: (a) detection of the roll angular velocity Ω y , (b) detection of the linear acceleration a z In particular, a first resonator element approach the substrate and its gap reduces to 0 g R   while the other resonator element move away from the same substrate and its gap increases to 0 g R   , R being the distance between the a-a axis and the torsional resonator’s axis of rotation. The torsional resonators change their natural frequencies according to (5) as a consequence of their electrostatic stiffness variation   2 3 3 0 3 1 3 2 3( ) 1 2 m p mass L K V b c g R J            ,   2 3 3 0 3 2 4 2 3( ) 1 2 m p mass L K V b c g R J            (14) By combining the readings of the four torsional resonators and linearizing one obtains the frequency shift which provides a differential measure of the roll angular velocity Ω y : 2 1 4 3 0 0 3 0 0 6 18 G e e y m e m e m x R K K R R l K K g K K g G s t                 (15) In the presence of linear out-of-plane acceleration a z , (see Fig. 4(b)), two forces of inertia F a originate on the two inertial masses directed along the axis z , which have in this case the same sign. Because of these forces, one inertial mass rotates counterclockwise, while the other rotates clockwise by an angle: 3 3 2 z G m a R l G s t   (16) In this case, therefore, frequency variations of the same sign occur in the torsional resonator elements 1 and 4, and in the torsional resonator elements 2 and 3. Similarly to what done for the measure of the roll velocity, combining the readings of the four torsional resonators, the following expression is obtained, allowing for the measure of the external acceleration a z :