Page 393 - The Mechatronics Handbook

If a single-degree-of-freedom system behaves linearly in a time invariant manner, the basic second-
order differential equation describing the motion of the forced mass-spring system can be written as

2
d x dx
f t() = m -------- 2 + c ------ + kx (19.4)
dt dt
where f(t) is the force, m is the mass, c is the velocity constant, and k is the spring constant.
Nevertheless, the base of the accelerometer is in motion too. When the base is in motion, the force is
transmitted through the spring to the suspended mass, depending on the transmissibility of the force to
the mass. Equation (19.4) may be generalized by taking the effect motion of the base into account as

2 2
d z dz d x 1
m ------- 2 + c ----- + kz = mg cos q m ---------- (19.5)
–
2
dt dt dt
where z = x 2 – x 1 is the relative motion between the mass and the base, x 1 is the displacement of the base,
x 2 is the displacement of mass, and θ is the angle between the sense axis and gravity.
The complete solution to Eq. (19.5) can be obtained by applying the superposition principle. The sup-
erposition principle states that if there are simultaneously superimposed actions on a body, the total
effect can be obtained by summing the effects of each individual action. Using superposition and using
Laplace transforms gives
X s() 1
---------- = -------- 2 + cs + k (19.6)
Fs() ms

or

X s() K
---------- = ---------------------------------------------- (19.7)
2
2
Fs() s /w n + 2zs/w n + 1
where s is the Laplace operator, K = 1/k is the static sensitivity, ω n = k/m is the undamped critical
frequency (rad/s), and ζ = (c/2) km is the damping ratio.
As can be seen in the performance of accelerometers, the important parameters are the static sensitivity,
the natural frequency, and the damping ratio, which are all functions of mass, velocity, and spring
constants. Accelerometers are designed and manufactured to have different characteristics by suitable
selection of these parameters. A short list of major manufacturers is given in Table 19.1.

Vibrations
This section is concerned with applications of accelerometers to measure physical properties such as
acceleration, vibration and shock, and the motion in general. Although there may be fundamental
differences in the types of motions, a sound understanding of the basic principles of the vibration will
lead to the applications of accelerometers in different situations by making appropriate corrective
measures.
Vibration is an oscillatory motion resulting from application of varying forces to a structure. The
vibrations can be periodic, stationary random, nonstationary random, or transient.

Periodic Vibrations
In periodic vibrations, the motion of an object repeats itself in an oscillatory manner. This can be
represented by a sinusoidal waveform

xt() = X p sin ( w t) (19.8)

©2002 CRC Press LLC