E1C12 09/14/2010
13:54:10 Page 518
value of displacement recorded at time intervals dt. The velocity at any time ndt can be
approximated as
v t
ð Þ ¼ y
0
t
ð Þ ¼
y nþ1 À y n
dt
ð12:10Þ
where
y n+1 ¼ the (n+1) measurement of displacement at time n þ 1
ð
Þdt
y n ¼ the nth measurement of displacement at time ndt
v t
ð Þ ¼ velocity at time t
If the signal is sampled at 10 Hz, dt is 0.1 s. For the present, we assume that the uncertainty in time is
negligible, so that the uncertainty in v; u v can be expressed as
u v ¼
qv
qy nþ1
u y nþ1
! 2
þ
qv
qy n
u y n
! 2
(
) 1=2
ð12:11Þ
The uncertainties in the measured displacements, y n and y nþ1 , will be equal, and are listed in Table
12.2. Substituting these values for uncertainty in Equation 12.11 yields an uncertainty of Æ 3 cm/s in
the velocity measurement. Notice that this uncertainty magnitude is not a function of time or the
measured velocity. This corresponds to a minimum uncertainty of 30% in the velocity measurement.
To determine velocity from acceleration, the measured values of acceleration must be
integrated. Since we have a digital signal, the integration can be accomplished numerically as
v t
ð Þ ¼ y
0
t
ð Þ ¼
X
i
y
00
i dt
ð12:12Þ
Assuming the uncertainty in time is negligible, the uncertainty in velocity at any time t is simply
u v ¼ u y 00 t
ð12:13Þ
Clearly, the integration process tends to accumulate error as the calculation of velocity proceeds in
time, as illustrated in Figure 12.11, but for very short time periods can produce acceptable results.
Table 12.2 Specification and Uncertainty Analyses for Displacement and Acceleration
Measured Variable
Functional Form
Full-Scale Output Range
Displacement (cm)
y(t) ¼ 20 sin 2t
0 to 10 V
Acceleration (cm/s
2
)
y
00
¼ À
20
4
sin 2t
À5 to 5 V
Uncertainty Values for Displacement and Acceleration
Measured Variable
Uncertainty
Displacement
Accuracy: 1% full scale ¼ 0.2 cm
A/D 8 bit (0.04 V) ¼ 0.08 cm
Total uncertainty ¼ Æ0.22 cm
Acceleration
Accuracy: 1% full scale ¼ 0.05 cm/s
2
A/D 8 bit (0.04 V) ¼ 0.04 cm/s
2
Total uncertainty ¼ Æ0.064 cm/s
2
518 Chapter 12 Mechatronics: Sensors, Actuators, and Controls
13:54:10 Page 518
value of displacement recorded at time intervals dt. The velocity at any time ndt can be
approximated as
v t
ð Þ ¼ y
0
t
ð Þ ¼
y nþ1 À y n
dt
ð12:10Þ
where
y n+1 ¼ the (n+1) measurement of displacement at time n þ 1
ð
Þdt
y n ¼ the nth measurement of displacement at time ndt
v t
ð Þ ¼ velocity at time t
If the signal is sampled at 10 Hz, dt is 0.1 s. For the present, we assume that the uncertainty in time is
negligible, so that the uncertainty in v; u v can be expressed as
u v ¼
qv
qy nþ1
u y nþ1
! 2
þ
qv
qy n
u y n
! 2
(
) 1=2
ð12:11Þ
The uncertainties in the measured displacements, y n and y nþ1 , will be equal, and are listed in Table
12.2. Substituting these values for uncertainty in Equation 12.11 yields an uncertainty of Æ 3 cm/s in
the velocity measurement. Notice that this uncertainty magnitude is not a function of time or the
measured velocity. This corresponds to a minimum uncertainty of 30% in the velocity measurement.
To determine velocity from acceleration, the measured values of acceleration must be
integrated. Since we have a digital signal, the integration can be accomplished numerically as
v t
ð Þ ¼ y
0
t
ð Þ ¼
X
i
y
00
i dt
ð12:12Þ
Assuming the uncertainty in time is negligible, the uncertainty in velocity at any time t is simply
u v ¼ u y 00 t
ð12:13Þ
Clearly, the integration process tends to accumulate error as the calculation of velocity proceeds in
time, as illustrated in Figure 12.11, but for very short time periods can produce acceptable results.
Table 12.2 Specification and Uncertainty Analyses for Displacement and Acceleration
Measured Variable
Functional Form
Full-Scale Output Range
Displacement (cm)
y(t) ¼ 20 sin 2t
0 to 10 V
Acceleration (cm/s
2
)
y
00
¼ À
20
4
sin 2t
À5 to 5 V
Uncertainty Values for Displacement and Acceleration
Measured Variable
Uncertainty
Displacement
Accuracy: 1% full scale ¼ 0.2 cm
A/D 8 bit (0.04 V) ¼ 0.08 cm
Total uncertainty ¼ Æ0.22 cm
Acceleration
Accuracy: 1% full scale ¼ 0.05 cm/s
2
A/D 8 bit (0.04 V) ¼ 0.04 cm/s
2
Total uncertainty ¼ Æ0.064 cm/s
2
518 Chapter 12 Mechatronics: Sensors, Actuators, and Controls