E1C11 09/14/2010
13:14:3 Page 480
Evaluating the appropriate partial derivatives from Equation 11.17 yields
dE 0 ¼ E i
R 2 dR 1 À R 1 dR 2
R 1 þ R 2
ð
Þ
2
þ
R 3 dR 4 À R 4 dR 3
R 3 þ R 4
ð
Þ
2
"
#
ð11:19Þ
Then from Equations 11.2 and 11.11, dR i ¼ R i e i GF i , and the value of dE 0 can be determined.
Assuming that dR i ( R i , the resulting change in the output voltage, dE 0 , may now be expressed as
dE 0 ¼ E i
R 1 R 2
R 1 þ R 2
ð
Þ
2
e 1 GF 1 À e 2 GF 2
ð
Þ þ
R 3 R 4
R 3 þ R 4
ð
Þ
2
e 4 GF 4 À e 3 GF 3
ð
Þ
"
#
ð11:20Þ
If R 1 ¼ R 2 ¼ R 3 ¼ R 4 , then
dE 0
E i
¼
1
4
e 1 GF 1 À e 2 GF 2 þ e 4 GF 4 À e 3 GF 3
ð
Þ
ð 11:21Þ
It is possible and desirable to purchase matched sets of strain gauges for a particular application, so
that GF 1 ¼ GF 2 ¼ GF 3 ¼ GF 4 , and
dE 0
E i
¼
GF
4
e 1 À e 2 þ e 4 À e 3
ð
Þ
ð 11:22Þ
Equation 11.22 is important and forms the basic working equation for a strain gauge bridge circuit
using multiple gauges (compare this equation with Eq. 11.15).
Equation 11.22 shows that for a bridge containing one or more strain gauges, equal strains on
opposite bridge arms sum, whereas equal strains on adjacent arms of the bridge cancel. These characteristics can be used to increase the output of the bridge, to provide temperature compensation, or to
cancel unwanted components of strain. Practical means of achieving these desirable characteristics
will be explored further, after the concept of the bridge constant is developed.
Bridge Constant
Commonly used strain gauge bridge arrangements may be characterized by a bridge constant, k,
defined as the ratio of the actual bridge output to the output that would result from a single gauge
sensing the maximum strain, e max . The output for a single gauge experiencing the maximum strain
may be expressed as
dR
R
¼ e max GF
ð11:23Þ
So that, again for a single gauge,
dE 0
E i
ffi
e max GF
4
ð11:24Þ
The bridge constant, k, is found from the ratio of the actual bridge output given by Equation
11.22 to the output for a single gauge given by Equation 11.24. When more than one gauge is used in
the bridge circuit, Equation 11.15 becomes
dE 0
E i
¼
kdR=R
4 þ 2dR=R
¼
kGFe
4 þ 2GFe
%
kGFe
4
ð11:25Þ
480 Chapter 11 Strain Measurement
13:14:3 Page 480
Evaluating the appropriate partial derivatives from Equation 11.17 yields
dE 0 ¼ E i
R 2 dR 1 À R 1 dR 2
R 1 þ R 2
ð
Þ
2
þ
R 3 dR 4 À R 4 dR 3
R 3 þ R 4
ð
Þ
2
"
#
ð11:19Þ
Then from Equations 11.2 and 11.11, dR i ¼ R i e i GF i , and the value of dE 0 can be determined.
Assuming that dR i ( R i , the resulting change in the output voltage, dE 0 , may now be expressed as
dE 0 ¼ E i
R 1 R 2
R 1 þ R 2
ð
Þ
2
e 1 GF 1 À e 2 GF 2
ð
Þ þ
R 3 R 4
R 3 þ R 4
ð
Þ
2
e 4 GF 4 À e 3 GF 3
ð
Þ
"
#
ð11:20Þ
If R 1 ¼ R 2 ¼ R 3 ¼ R 4 , then
dE 0
E i
¼
1
4
e 1 GF 1 À e 2 GF 2 þ e 4 GF 4 À e 3 GF 3
ð
Þ
ð 11:21Þ
It is possible and desirable to purchase matched sets of strain gauges for a particular application, so
that GF 1 ¼ GF 2 ¼ GF 3 ¼ GF 4 , and
dE 0
E i
¼
GF
4
e 1 À e 2 þ e 4 À e 3
ð
Þ
ð 11:22Þ
Equation 11.22 is important and forms the basic working equation for a strain gauge bridge circuit
using multiple gauges (compare this equation with Eq. 11.15).
Equation 11.22 shows that for a bridge containing one or more strain gauges, equal strains on
opposite bridge arms sum, whereas equal strains on adjacent arms of the bridge cancel. These characteristics can be used to increase the output of the bridge, to provide temperature compensation, or to
cancel unwanted components of strain. Practical means of achieving these desirable characteristics
will be explored further, after the concept of the bridge constant is developed.
Bridge Constant
Commonly used strain gauge bridge arrangements may be characterized by a bridge constant, k,
defined as the ratio of the actual bridge output to the output that would result from a single gauge
sensing the maximum strain, e max . The output for a single gauge experiencing the maximum strain
may be expressed as
dR
R
¼ e max GF
ð11:23Þ
So that, again for a single gauge,
dE 0
E i
ffi
e max GF
4
ð11:24Þ
The bridge constant, k, is found from the ratio of the actual bridge output given by Equation
11.22 to the output for a single gauge given by Equation 11.24. When more than one gauge is used in
the bridge circuit, Equation 11.15 becomes
dE 0
E i
¼
kdR=R
4 þ 2dR=R
¼
kGFe
4 þ 2GFe
%
kGFe
4
ð11:25Þ
480 Chapter 11 Strain Measurement