Issue 29

V. Zega et alii, Frattura ed Integrità Strutturale, 29 (2014) 334-342; DOI: 10.3221/IGF-ESIS.29.29 338 Consequently, the resonators, are subjected to an axial action: in particular, two of them, for instance resonators I and IV, are subjected to a compressive force, whereas the other two, for instance II and III, are subjected to a tensile force of the same intensity, N , 2 c II III I IV F N N N N        (7) where  is a force-amplification factor, depending on the geometry of the resonator and in particular on H . Because of the axial stress, the frequency of oscillation of the resonators subjected to compression decreases, whereas the frequency of oscillation of the resonators subjected to tension increases, according to the relation (4). Figure 3 : Schematic plan view of the structure: (a) detection of the yaw angular velocity Ω z , (b) detection of the linear acceleration a y Combining the readings of frequency of the four bending resonators from equation (4), linearized around the nominal frequency f 0 , one can compute the external yaw angular velocity Ω z 0 0 1 4.88 1 4.88 2 1 1 2 ( ) 2 ( ) 4.88 2 ( ) III II I IV z z m e m e z m e f f f f f m x m x h k k h k k f m x h k k                            (8) On the other hand, when an external linear acceleration a y is applied, as shown in Fig. 3(b), the inertial masses are subjected to inertia forces F a directed in the y direction, with the same sign and with modulus a y F ma  (9) As a result of the inertia forces F a , the bending resonators are again subjected to axial stresses, but in this case, these axial stresses are of the same sign in the resonators I and III, and they are of the same sign in the resonators II and IV. 2 a II III I IV F N N N N        (10) Combining the readings of the four resonators starting once again from equation (4) linearized around the nominal flexural oscillation frequency f 0 , one obtains a measure of the external acceleration a y : 0 0 1 4.88 1 4.88 4.88 2 1 1 4 ( ) 4 ( ) ( ) I III II IV y y y m e m e m e f f f f f m a m a f m a h k k h k k h k k                     (11) Therefore, via a different combination of the quantities supplied by the same resonator elements it is possible to detect simultaneously the yaw angular velocity Ω z and the linear in-plane acceleration a y ; both measures are of differential type. The measure of the roll angular velocity Ω y and of the linear out-of-plane acceleration a z is performed by means of the torsional elements labeled 1, 2, 3, 4 in Fig. 4, kept in resonance according to their torsional natural mode through the driving electrode located below them on the substrate. When an external roll angular velocity Ω y is applied, two Coriolis’ forces, labeled F c in Fig. 4(a), originate on the inertial masses which are oscillating in the x -direction. These forces are directed along the out-of-plane axis z , have opposite signs and the modulus is given by