Strain Gages Electrical resistance in material changes when the material is deformed R –...
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Transcript of Strain Gages Electrical resistance in material changes when the material is deformed R –...
Strain Gages
• Electrical resistance in material changes when the material is deformed
RA
R – Resistance
ρ – Resistivityl – LengthA – Cross-sectional area
log log logR A
dR
R
d d A
A
Taking the differential
Change in resistance is from change in shape as
well as change in resistivity
For linear deformations
R
RSs
ε – strainSs – sensitivity or gage factor(2-6 for metals and 40 – 200 for
semiconductor)
• The change in resistance is measured using an electrical circuit
• Many variables can be measured – displacement, acceleration, pressure, temperature, liquid level, stress, force and torque
• Some variables (stress, force, torque) can be determined by measuring the strain directly
• Other variables can be measured by converting the measurand into stress using a front-end device
Outputvo
Direction ofSensitivity
(Acceleration)
Strain Gage
Housing
SeismicMass
m
Base MountingThreads
Strain Member Cantilever
Strain gage accelerometer
Direction ofSensitivity
FoilGrid
BackingFilm
Solder Tabs(For Leads)
Single ElementTwo-Element Rosette
Three-Element Rosettes
Nickle-PlatedCopper Ribbons
WeldedGold Leads
Doped SiliconCrystal
(P or N Type)
PhenolicGlass
BackingPlate
Strain gages are manufactured as metallic foil (copper-nickel alloy – constantan)
Semiconductor (silicon with impurity)
Potentiometer or Ballast Circuit
vo
Output vref
(Supply)
Strain Gage
+
-
Rc
R
• Ambient temperature changes will introduce error
• Variations in supply voltage will affect the output
• Electrical loading effect will be significant
• Change in voltage due to strain is a very small percentage of the output
refvRR
Rv
co
Question: Show that errors due to ambient temperature changes will cancel if the temperature coefficients of R and Rc are the same
Wheatstone Bridge Circuit
vref
(Constant Voltage)
-
+ R1
A
R2
R3
R4
B
RL
vo
- +
Load (High)
Small i
1 3 1 4 2 3
1 2 3 4 1 2 3 4
( )
( ) ( ) ( )( )ref ref
o ref
R v R v R R R Rv v
R R R R R R R R
R
R
R
R1
2
3
4
When the bridge is balanced
True for any RL
Null Balance Method
• When the stain gage in the bridge deforms, the balance is upset.
• Balance is restored by changing a variable resistor
• The amount of change corresponds to the change in stain
• Time consuming – servo balancing can be used
Direct Measurement of Output Voltage
• Measure the output voltage resulting from the imbalance
• Determine the calibration constant
• Bridge sensitivity
v
v
R R R R
R R
R R R R
R R
o
ref
2 1 1 2
1 22
4 3 3 4
3 42
To compensate for temperature changes, temperature coefficients of adjacent pairs should be the same
The Bridge Constant
• More than one resistor in the bridge can be active
• If all four resistors are active, best sensitivity can be obtained
• R1 and R4 in tension and R2 and R3 in compression gives the largest sensitivity
• The bridge sensitivity can be expressed as
4o
ref
v Rk
v R
k bridge output in the general case
bridge output if only one strain gage is activeBridge Constant
Example 4.4
A strain gage load cell (force sensor) consists of four identical strain gages, forming a Wheatstone bridge, that are mounted on a rod that has square cross-section. One opposite pair of strain gages is mounted axially and the other pair is mounted in the transverse direction, as shown below. To maximize the bridge sensitivity, the strain gages are connected to the bridge as shown. Determine the bridge constant k in terms of Poisson’s ratio v of the rod material.
vref
+−
+
−
vo
1 2
3 4
1
Axial Gage
2 Transverse Gage
Cross SectionOf SensingMember
3
4
Transverse strain = (-v) x longitudinal strain
Calibration Constant
v
vCo
ref
Ck
Ss4 4
o
ref
v Rk
v R
R
RSs
k – Bridge Constant
Ss – Sensitivity or gage factor
Example 4.5A schematic diagram of a strain gage accelerometer is shown below. A point mass of weight W is used as the acceleration sensing element, and a light cantilever with rectangular cross-section, mounted inside the accelerometer casing, converts the inertia force of the mass into a strain. The maximum bending strain at the root of the cantilever is measured using four identical active semiconductor strain gages. Two of the strain gages (A and B) are mounted axially on the top surface of the cantilever, and the remaining two (C and D) are mounted on the bottom surface. In order to maximize the sensitivity of the accelerometer, indicate the manner in which the four strain gages A, B, C, and D should be connected to a Wheatstone bridge circuit. What is the bridge constant of the resulting circuit?
vref
+−
+
−
δvo
A
B
C
D
W
Strain Gages A, B
C, D
l
b
h
AB
CD
Obtain an expression relating applied acceleration a (in units of g) to bridge output (bridge balanced at zero acceleration) in terms of the following parameters:
W = Mg = weight of the seismic mass at the free end of the cantilever elementE = Young’s modulus of the cantileverl = length of the cantileverb = cross-section width of the cantileverh = cross-section height of the cantileverSs = gage factor (sensitivity) of each strain gagevref = supply voltage to the bridge.
•If M = 5 gm, E = 5x1010 N/m2, l = 1 cm, b = 1 mm, h = 0.5 mm, Ss = 200, and vref = 20 V, determine the sensitivity of the accelerometer in mV/g.•If the yield strength of the cantilever element is 5xl07 N/m2, what is the maximum acceleration that could be measured using the accelerometer? •If the ADC which reads the strain signal into a process computer has the range 0 to 10 V, how much amplification (bridge amplifier gain) would be needed at the bridge output so that this maximum acceleration corresponds to the upper limit of the ADC (10 V)?•Is the cross-sensitivity (i.e., the sensitivity in the two directions orthogonal to the direction of sensitivity small with this arrangement? Explain.•Hint: For a cantilever subjected to force F at the free end, the maximum stress at the root is given by
6
2
F
bh
Mechanical Structure
Signal Conditioning
MEMS Accelerometer
Applications: Airbag Deployment
Data Acquisition
ACBridge
CalibrationConstant
Oscillator Power Supply
AmplifierDemodulator
And FilterDynamic
StrainStrain
Reading
• Supply frequency ~ 1kHz
• Output Voltage ~ few micro volts – 1 mV
• Advantages – Stability (less drift), low power consumption
• Foil gages - 50Ω – kΩ
• Power consumption decreases with resistance
• Resolutions on the order of 1 m/m
Semiconductor Strain Gages
Single Crystal ofSemiconductor
Gold Leads
ConductorRibbons
Phenolic GlassBacking Plate
• Gage factor – 40 – 200
• Resitivity is higher – reduced power consumption
• Resistance – 5kΩ
• Smaller and lighter
Material Composition Gage Factor(Sensitivity)
Temperature Coefficient of
Resistance (10-6/C)
Constantan 45% Ni, 55% Cu 2.0 15
Isoelastic 36% Ni, 52% Fe, 8% Cr, 4% (Mn, Si, Mo)
3.5 200
Karma 74% Ni, 20% Cr, 3% Fe, 3% Al
2.3 20
Monel 67% Ni, 33% Cu 1.9 2000
Silicon p-type 100 to 170 70 to 700
Silicon n-type -140 to –100 70 to 700
Properties of common strain gage material
Disadvantages of Semiconductor Strain Gages
• The strain-resistance relationship is nonlinear
• They are brittle and difficult to mount on curved surfaces.
• The maximum strain that can be measured is an order of magnitude smaller 0.003 m/m (typically, less than 0.01 m/m)
• They are more costly
• They have a much larger temperature sensitivity.
−3 −2 −1 1 2 3
−0.2
−0.1
0.1
0.2
0.3
0.4
−0.3
Strain
×103
ResistanceChange
= 1 Microstrain = Strain of 1×10-6
−3 −2 −1 1 2 3
−0.2
−0.1
0.1
0.2
0.3
0.4
−0.3
Strain×103
ResistanceChange
R
R
R
R
P-type
N-type
For semiconductor strain gages
R
RS S 1 2
2
• S1 – linear sensitivity
• Positive for p-type gages
• Negative for n-type gages
• Magnitude is larger for p-type
• S2 – nonlinearity
• Positive for both types
• Magnitude is smaller for p-type
Linear Approximation
Strain
Change inResistance
QuadraticCurve
max
−max
LinearApproximation
0
R
R
R
RS
Ls
Error eR
R
R
RS S S
Ls
1 22
21 2sS S S
J e d S S S ds 2
1 22 2
max
max
max
max
Quadratic Error
Minimize Error 0.s
J
S
max
max
2221)2(
dSSS s = 0
sSS 1
Maximum Error
e Smax max 22
Range – change in resistance
R
RS S S S
S
1 22
1 22
12
max max max max
max
Percentage nonlinearity error
22 max
1 max
max error100% 100%
range 2p
SN
S
2 max 150 %pN S S
Temperature Compensation
CompensationFeasible
CompensationNot Feasible
CompensationFeasible
(−β)
Concentration of Trace Material (Atoms/cc)
Tem
pera
ture
coe
ffic
ient
s (p
er °
F)
0
1
2
3α = Temperature Coefficient of Resistanceβ = Temperature Coefficient of Gage Factor
α
1 .oR R T
1 .s soS S T
Resistance change due to temperatureSensitivity
change due to temperature
R4
R1 R2
R3
δvo
+
−
CompensatingResistor
Rc
vref
+−
vi
RR−
vref
+
−
vi
RR
+
Rc
Self Compensation with a Resistor
vR
R Rvi
c
ref
v
v
R
R R
kSo
c
s
ref
4
1 .1 .
1 .oo
so soo c o c
R TRS S T
R R R T R
For self compensation the output after the temperature change must be the same
R R R R To c o c ( )
c oR R
Possible only for certain ranges