Measurement of Strain and Force

1
Lecture 7 Measurement of Strain and
Force Strain Gauges May 14, 2006
2
Objectives of Experiment

• Determine the spring constant of a cantilever
  beam using 2 different methods
• Static
• Dynamic
• Evaluate the accuracy of approximating the beam
  as a simple harmonic oscillator with negligible
  mass
• Determine the elastic modulus and Poissons ratio
  and compare to published values
• Compare strain measurements with theoretical
  values
• Determine the gauge factor of one strain gauge
• Calibrate the beam to measure an unknown mass
• Explore signal amplification using multiple
  strain gages within a single bridge

EXTRA CREDIT
3
Concepts

• Hookes Law
• Relates stress and strain
• Mechanics of beam bending
• Gauge factor for a strain gauge (G.F.)

4
Review - Hookes Law

• Stress (s) is proportional to strain (e)
• s E e
• E elastic modulus (Youngs modulus)

5
Poissons ratio (n)

• For a 3-dimensional solid, we have to consider
  the strain in the transverse direction
• Most materials n 0.3-0.5

transverse
longitudinal
6
Pinned cantilever beam analysis

• Beam deflection
• ymax is maximum deflection
• at end of beam

x
y
F
h
b
L
Theoretical strain,
Moment of inertia,
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Pinned cantilever beam analysis

• Force and displacement
• We also know

x
y
F
h
b
L
Spring constant ,
Young Modulus,
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Vibration of cantilever beam
The oscillation of a pinned cantilever beam is

• Where
• w oscillation angular frequency
• lL 1.88 for first natural frequency
• M beam mass
• m load mass

F
h
b
L
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Principle of experiment
mass
aluminum cantilever beam
Strain due to cantilever deflection is sensed
electrically, through a strain gauge The strain
gauge is the unknown resistor in a Wheatstone
bridge configuration
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Principles of strain gauges
A length change of a wire causes a resistance
change, which is measured by a strain gauge
R resistance r resistivity (material
property) L length A area
Taking the logarithms of both side and
differentiating
(1)
Change in area
Change in resistivity
Change in length
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Strain gauges
Logarithmically differentiating
(2)
db/b is known as transverse strain ?t
(3)
? is a property of material known as Poissons
ratio
12
Strain gauges
Combining equation 1, 2 and 3
0
is called Gauge Factor
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Gauge Factor (G.F.)
The G.F. relates a change in resistance with
strain For most elements, G.F. ranges from
2.0-4.0 e.g., constantan 2.0, Nichrome
2.2
14
Review - Wheatstone bridge
Recall the value of an unknown resistance (R4)
can be found using a Wheatstone bridge
Adjust R3 so that VAB 0 (VA VB 0) VA - Vi
VB - Vi i1R1 i3R3 i2R2i3Rx
VAB Vg The voltage from the strain gauge
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An unbalanced Wheatstone bridge
Rx is the STRAIN GAUGE, generally VAB ? 0
Unknown R (STRAIN GAUGE connected here)
VAB
Rx changes due to strain ?VAB changes
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Unbalanced Wheatstone bridge

Rx is the unknown resistance of the strain gauge
In lab, you are using 120W resistors and a 10V
power supply Vgaugu/10 0.5 - Rx/(120 Rx)
Unknown R
Vstrain gauge
http//en.wikipedia.org/wiki/Wheatstone_bridge
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A strain gauge

• Stain gauges are etched from thin foil metal
  sheets that are bonded to
• plastic backing
• The dimensions of strain gauges could be as
  small as 0.2 mm

• Semiconductor gauges are
• are commonly used in
• pressure transducers and
• accelerometers

Made of Cu-Ni or Nichrome alloys
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Strain rosette
A strain rosette can be used to measure the
general state of strain at a point.
For your experiments, you will use a 45 strain
rosette
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A 45 strain rosette
Y
ec
eb
ea
X
SG 4
Free end
SG 1
SG 2
SG 3
SG 5
Strain Rosette configuration
Underneath bar
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Experimental set-up
Wheatstone bridge
TENSION
y deflection
COMPRESSION
SG 4
Free end
SG 2
SG 1
SG 5
SG 3
Underneath bar
Strain Rosette configuration
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1- Measurements using strain gauges

• For EACH strain gauge
• Use 5 different weights and combinations of
  weights (available 101.2, 147.3, 248.3 gms)
• Including 0 load
• Use larger mass to minimize error
• Zero bridge without any weight
• Measure and record deflection (y) and voltage for
  each weight
• Measure deflection and voltage for an unknown
  weight
• Record data

Please do not exceed weight above 1000 g
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2. Measure oscillation frequency
Press gently on free end and let go

• Using a single strain gauge
• Use 6 different loads
• 0 load 5 other weights
• Measure the oscillation frequency
• Measuring change in voltage gt change in
  resistance
• Convert to angular frequency, w 2pf
• Recall period, T 2?/?

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Static spring constantdata from (1)
D E F L E C T I O N
Add weights and measure deflection (x), from
graph of force vs. displacement determine k, F
kx from k 3EI/L3 find E I is moment of
inertia, I bh3/12
End View
h
b
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Dynamic spring constant data from (2)

• Using the data collected on w
• Plot 1/w2 vs. load mass
• Determine k
• Calculate E
• Which method is more accurate and why?
• Error analysis is necessary!

25
Relationship between dV and strain data from (1)
Plot dV (change in voltage across Wheatstone
Bridge) vs. mass for each strain gauge (except
3) and comment on shape (positive or
negative) Vo is bridge supply voltage
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Determination of GFdata from (1)
Determine the G.F. for strain gauge 3
Calculate strain, ? , from theory e(x)
xhF/(2EI) Plot dV vs. ? and find G.F. for 3
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To do before the experiment

• Understand the terms stress, strain, Poissons
  ratio, gauge factor
• Review principle of Wheatstone bridges
• Read pages 222-232 and 388-392 in your text,
  Introduction to Engineering Experimentation
• Review the experimental procedure

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Next week

• Last lab!
• CD diffraction experiment