E1C11 09/14/2010
13:14:4 Page 491
Uncertainties in Multichannel Measurements
Uncertainties caused by an automated multichannel strain measurement system can be estimated in
a system calibration. Example 11.7 describes an initial test of such a system, often called a
shakedown test, and discusses the information that results from this test.
Example 11.7
The strain developed at M different locations about a large test specimen is to be measured at each
location using a setup similar to that shown in Figure 11.18. At each location similar strain gauges
are appropriately mounted and connected to a Wheatstone bridge that is powered by an external
supply. The bridge deflection voltage is to be amplified and measured. A similar setup is used at all
M locations. The output from each amplifier is input through an M-channel multiplexer to an
automated data acquisition system. If N (say 30) readings for each of the M setups are taken while
the test specimen is maintained in a condition of uniform zero strain, what information is obtained?
KNOWN Strain setup of Figure 11.18
M (j = 1, 2, . . . , M) setups
N (i = 1, 2, . . . , N) readings per setup
ASSUMPTIONS All setups are to be operated in a similar manner. Gauges are operated in their
linear regimes.
SOLUTION Consider the data that will become available. First, each setup is exposed to a
similar strain. Hence, each setup should indicate the same strain. We can calculate the pooled mean
value of the M Â N readings to obtain
h ei ¼
1
MN
X M
j¼1
X N
i¼1
e ij ¼
1
M
X M
j¼1
e j
The difference between the pooled mean strain and the applied strain (zero here) is an estimate
of the systematic uncertainty that can be expected from any channel during data acquisition.
Random errors manifest themselves through scatter in the data set. The pooled standard
deviation of the means
hs e i ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P M
j¼1
P N
i¼1
e ij À e j
À
Á 2
M N À 1
ð
Þ
v
u
u
u
t
0 ffiffiffiffiffiffiffiffi
MN
p
provides a representative estimate of the uncertainty due to random error to be expected in any
channel that is due to the data acquisition and reduction instrumentation.
COMMENT This test yields an estimate of the uncertainties in systematic and random error
due to propagation of the elemental errors amid M setups due to
excitation voltage errors (differences in settings, variations),
amplifier error (differences in gain, noise),
analog-to-digital (A/D) converter, multiplexer, and conversion errors,
computer errors (noise and roundoff),
11.6 Apparent Strain and Temperature Compensation 491
13:14:4 Page 491
Uncertainties in Multichannel Measurements
Uncertainties caused by an automated multichannel strain measurement system can be estimated in
a system calibration. Example 11.7 describes an initial test of such a system, often called a
shakedown test, and discusses the information that results from this test.
Example 11.7
The strain developed at M different locations about a large test specimen is to be measured at each
location using a setup similar to that shown in Figure 11.18. At each location similar strain gauges
are appropriately mounted and connected to a Wheatstone bridge that is powered by an external
supply. The bridge deflection voltage is to be amplified and measured. A similar setup is used at all
M locations. The output from each amplifier is input through an M-channel multiplexer to an
automated data acquisition system. If N (say 30) readings for each of the M setups are taken while
the test specimen is maintained in a condition of uniform zero strain, what information is obtained?
KNOWN Strain setup of Figure 11.18
M (j = 1, 2, . . . , M) setups
N (i = 1, 2, . . . , N) readings per setup
ASSUMPTIONS All setups are to be operated in a similar manner. Gauges are operated in their
linear regimes.
SOLUTION Consider the data that will become available. First, each setup is exposed to a
similar strain. Hence, each setup should indicate the same strain. We can calculate the pooled mean
value of the M Â N readings to obtain
h ei ¼
1
MN
X M
j¼1
X N
i¼1
e ij ¼
1
M
X M
j¼1
e j
The difference between the pooled mean strain and the applied strain (zero here) is an estimate
of the systematic uncertainty that can be expected from any channel during data acquisition.
Random errors manifest themselves through scatter in the data set. The pooled standard
deviation of the means
hs e i ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P M
j¼1
P N
i¼1
e ij À e j
À
Á 2
M N À 1
ð
Þ
v
u
u
u
t
0 ffiffiffiffiffiffiffiffi
MN
p
provides a representative estimate of the uncertainty due to random error to be expected in any
channel that is due to the data acquisition and reduction instrumentation.
COMMENT This test yields an estimate of the uncertainties in systematic and random error
due to propagation of the elemental errors amid M setups due to
excitation voltage errors (differences in settings, variations),
amplifier error (differences in gain, noise),
analog-to-digital (A/D) converter, multiplexer, and conversion errors,
computer errors (noise and roundoff),
11.6 Apparent Strain and Temperature Compensation 491