Issue 29

V. Zega et alii, Frattura ed Integrità Strutturale, 29 (2014) 334-342; DOI: 10.3221/IGF-ESIS.29.29 337 When, due to an external acceleration or yaw angular velocity, the in-plane inertia force or the Coriolis’ force is applied to the proof masses, an axial force N is exerted on the resonators and their frequency changes according due to an additional geometric stiffness term G k   2 0 4.88 h G k N dy N h      (3) In the case of a tensile axial load N the beam oscillation frequency increases while in the case of a compressive load it decreases [14-16] according to the relation: 0 1 4.88 1 2 ( ) m G e m e k k k f f N m h k k        (4) Since the resonating beam is very slender and the gap d 0 is very small, nonlinear effects can appear due to both geometrical and electrostatic effects. However, the presence of the little horizontal part of length H , see Fig. 2(a), partially releases the axial constraint, thus lowering the higher order mechanical stiffness terms and widening the regime of linear oscillation. The electrostatic stiffness nonlinearities can be reduced by lowering the actuation voltage v a ( t ); the sensor proposed in this work will be actuated in the linear regime. A detailed discussion on the nonlinear behavior of the resonator can be found in [13,17]. The other resonators included in the device are torsional resonators constituted by a small mass of in-plane dimensions 2 b L  and out-of-plane thickness s connected to the proof mass by two folded torsional springs, which allow for its out- of-plane rotation ( , ) x t  (see Fig. 2(b)). When in the rest position, the mass is at distance 0 g from both the electrodes (see Fig. 2(b)) and the torsional resonator has the nominal frequency 0  0 1 2 m e mass K K J      (5) with   2 3 3 0 3 0 2 2 , 3 m e p L GJ K K V b c l g     where J is the torsional momentum of inertia, mass J  is the centroidal mass moment of inertia of the rigid mass, l is the total length of one of the folded torsional springs and G is the shear elastic modulus. The sensing electrode is usually connected to a virtual ground at zero voltage while the mass is kept at the polarization fixed voltage V p (see Fig. 2(b)). By applying a small variable voltage v a ( t ) to the driving electrode the mass is electrostatically actuated and can dynamically vibrate. When, due to external actions, the proof masses tilt, there is a change of gap 0 g between the resonators and the electrodes and the frequency changes according to equation (5). As for the bending resonators nonlinear effects can appear for high values of v a ( t ) due to higher order terms in the electrostatic stiffness. The nonlinear behavior and the range of actuation available to ensure operation in the linear regime are studied in [18]. Device operation principle During the gyroscope-accelerometer functioning the inertial proof masses are kept in resonance according to their in- plane translational mode in x direction by means of electrostatic driving implemented by the respective driving electrodes (see Fig. 1). Also the bending resonator elements (labeled I, II, III, IV in Fig. 3) are kept in resonance according to the first flexural mode in the plane along the first axis x, by means of electrostatic interaction with the driving electrodes. In absence of an external angular velocity or linear acceleration, all resonators have the same nominal frequency 0 f , see eq. (1). When an external yaw angular velocity Ω z is applied, two Coriolis’ forces F c originate on the two inertial masses; they are directed along the axis y , have opposite signs (see Fig. 3(a)) and have modulus: 2 c z F m x    (6) where x  is the linear velocity of the inertial masses due to resonant driving and m is their mass.