88
3 Forced Vibration of Single Degree of Freedom System
3.10.2 Accelerometers
Basically, vibrometers and accelerometers are similar in appearance. The main difference lies in the stiffness of the spring of both instruments. Accelerometers record the
acceleration of the body.
If η << 1, 1/
( 1 − η 2 ) 2 + ( 2ηζ ) 2 or μ, the magnification factor tends to
unity (See Fig. 3.3) and φ tends to zero (Fig. 3.4). Therefore, for η << 1,
Equation (3.40) becomes
z =
m
k
ω
2 y 0 sin ω t
(3.68)
Now,
y = y 0 sin ω t
(3.69)
Therefore,
¨
y = − ω
2 y 0 sin ω t
(3.70)
Equation (3.68) can be written as
z = −
1
p 2 ¨
y
(3.71)
Equation (3.71) reveals that the relative displacement to some scale represents the
acceleration of the vibrating body.
For η << 1, the system will have a very high natural frequency, which means that
the spring will be very stiff. As a result of which, the instrument is very rugged and
is very much suited for measuring high accelerations, such as in strong earthquakes.
But p being high, Eq. (3.71) indicates that the output will be of a low order, and as
such, it is to be sufficiently amplified. Figure 3.3 indicates that the useful frequency
range for an undamped accelerometer is very small. From Fig. 3.18, it is seen that
at ζ = 0.7, μ = 1 for 0 ≤ η ≤ 0.2; the maximum error being less than 0.01%.
For an accelerometer with natural frequency 200 Hz, the useful range of frequency
is 0–40 Hz.
Introduction of damping in accelerometer is due to another reason as well. The
motion to be recorded is impure, in the sense that it contains harmonics higher than
the fundamental frequency. As the fundamental frequency is on the lower side, it
may be probable that one of these harmonics may be close to the natural frequency
of the instrument. The only way to take care of it is to introduce damping in the
instrument.
3 Forced Vibration of Single Degree of Freedom System
3.10.2 Accelerometers
Basically, vibrometers and accelerometers are similar in appearance. The main difference lies in the stiffness of the spring of both instruments. Accelerometers record the
acceleration of the body.
If η << 1, 1/
( 1 − η 2 ) 2 + ( 2ηζ ) 2 or μ, the magnification factor tends to
unity (See Fig. 3.3) and φ tends to zero (Fig. 3.4). Therefore, for η << 1,
Equation (3.40) becomes
z =
m
k
ω
2 y 0 sin ω t
(3.68)
Now,
y = y 0 sin ω t
(3.69)
Therefore,
¨
y = − ω
2 y 0 sin ω t
(3.70)
Equation (3.68) can be written as
z = −
1
p 2 ¨
y
(3.71)
Equation (3.71) reveals that the relative displacement to some scale represents the
acceleration of the vibrating body.
For η << 1, the system will have a very high natural frequency, which means that
the spring will be very stiff. As a result of which, the instrument is very rugged and
is very much suited for measuring high accelerations, such as in strong earthquakes.
But p being high, Eq. (3.71) indicates that the output will be of a low order, and as
such, it is to be sufficiently amplified. Figure 3.3 indicates that the useful frequency
range for an undamped accelerometer is very small. From Fig. 3.18, it is seen that
at ζ = 0.7, μ = 1 for 0 ≤ η ≤ 0.2; the maximum error being less than 0.01%.
For an accelerometer with natural frequency 200 Hz, the useful range of frequency
is 0–40 Hz.
Introduction of damping in accelerometer is due to another reason as well. The
motion to be recorded is impure, in the sense that it contains harmonics higher than
the fundamental frequency. As the fundamental frequency is on the lower side, it
may be probable that one of these harmonics may be close to the natural frequency
of the instrument. The only way to take care of it is to introduce damping in the
instrument.