system and circuit design for a capacitive mems gyroscope - Aaltodoc

10 Micromechanical Gyroscopes
be written. From it, the formulae for the Coriolis signal resulting from angular velocity
will be derived. Finally, the dynamic operation will be considered.
All the analysis in this section will be performed for a linear resonator, or a system
of linear resonators. However, the formulae can be straightforwardly generalized to tor-
sional resonators as well, by replacing masses with moments of inertia, displacements
with angles, and forces with torques. Naturally, this also requires various parameters,
such as spring constants and damping coefficients, to assume proper units (for example,
Newton-meters per radian instead of Newtons per meter for a spring constant).
2.1.1 1-Degree-of-Freedom Mechanical Resonator
A schematic drawing of a 1-degree-of-freedom (DoF) mechanical resonator is shown
in Fig. 2.1. It is formed by a mass m, which is supported in such a way that it can move
only in the x-direction, a massless spring k, and a dashpot damper D.
k
D
Figure 2.1 A 1-DoF resonator, formed by a mass m, a spring k, and a damper D.
If the mass is displaced from its rest position by a distance x (the positive direction
of x is defined in the figure), the spring causes a restoring force
m
x
Fk = −kx, (2.1)
where k is the spring constant. Next, if it is assumed that the damping is purely viscous,
then if the mass moves with a velocity v = ˙x (the dots denote derivatives with respect
to time), the force exerted by the damper is
FD = −Dv = −D ˙x, (2.2)
where D is the damping coefficient. Newton’s second law of motion states that in an
inertial frame of reference, the sum of all the forces acting on a mass is the mass times