handbook of modern sensors

116 3 Physical Principles of Sensing
(A)
(B)
Fig. 3.49. Mechanical model of an accelerometer (A) and a free-body diagram of mass (B).
of newton meters (N m), the inertia has units of kilogram per meter squared (kg/m 2 ),
and the angular acceleration has units of radians per second squared (rad/s 2 ).
Let us consider a monoaxial accelerometer, which consists of an inertia element
whose movement may be transformed into an electric signal. The mechanism of conversion
may be, for instance, piezoelectric. Figure 3.49A shows a general mechanical
structure of such an accelerometer. The mass M is supported by a spring having stiffness
k and the mass movement is damped by a damping element with a coefficient
b. Mass may be displaced with respect to the accelerometer housing only in the horizontal
direction. During operation, the accelerometer case is subjected to acceleration
d 2 y/dt 2 , and the output signal is proportional to the deflection x 0 of the mass M.
Because the accelerometer mass M is constrained to linear motion, the system
has one degree of freedom. Giving the mass M a displacement x from its equilibrium
position produces the free-body diagram shown in Fig. 3.49B. Note that x 0 is equal
to x plus some fixed displacement. Applying Newton’s second law of motion gives
Mf =−kx − b dx
dt , (3.154)
where f is the acceleration of the mass relative to the Earth and is given by
f = d2 x
dt 2 − d2 y
dt 2 . (3.155)
Substituting for f gives the required equation of motion as
M d2 x
dt 2 + b dx
dt + kx = M d2 y
dt 2 . (3.156)
Note that each term in Eq. (3.156) has units of newtons (N). The differential equation
(3.156) is of a second order, which means that the accelerometer output signal may
have an oscillating shape. By selecting an appropriate damping coefficient b, the
output signal may be brought to a critically damped state, which, in most cases, is a
desirable response.