MAE 170

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MAE 170 Experimental Techniques Lecture
7 Strain measurement Nov.10, 2008
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Announcements

• There will be no lab sections on Tuesday (Nov.
  11) due to Veterans day
• - Makeup lab in week 9 on Tuesday (Nov. 25)
• Practice labs
• Monday (Nov. 24), 11-4
• Drop in basis
• Make up labs with prior approval
• - Wednesday (Nov. 26), 10-3
• In-lab final (lab practical final exam)
• Tues. - Fri., Dec 1-5, during your regularly
  scheduled lab section

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Format of written final exam (Dec. 11)

• 10 of grade
• Same format as mid-term
• Multiple choice, true/false
• Closed book, closed notes
• Between 40-50 questions
• Material covered will come from lecture notes,
  lab write-ups and your understanding of the
  experiments
• You will have 1 hour to complete the exam

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Format of in-lab (practical) final

• 10 of grade
• Perform individually (not with your lab partner)
• Time 90 min. total during regular lab hours
• Open book and open notes
• Electrical circuits (black box)
• e.g. op-amps, filters
• (7) Experimental
• Simplified version of one of the eight labs
• (3) LabView practical
• Next week questions will be given
• Turn in during in-lab practical
• Partial credit given if whole assignment isnt
  completed

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Objectives of experiment this week

• 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 for the beam
• elastic modulus
• Poissons ratio
• compare to published values
• Compare strain measurements with theoretical
  values
• Determine the gauge factor of one strain gauge

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Concepts

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

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Review - Hookes Law

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

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Poissons ratio (?)

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

transverse
deformed
F
longitudinal
original undeformed
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Fixed cantilever beam analysis
x

• Beam deflection
• F k y
• k spring constant
• The maximum deflection (ymax) at the free end of
  the beam (x 0)

y
F
h
b
L
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More on fixed cantilever beam analysis
The maximum value of stress ?(x,y) occurs at any
point x emax esurface? s/E
spring constant
The elastic modulus, E, can be calculated from
the spring constant
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Vibration of cantilever beam
The oscillation of a fixed cantilever beam is
F
h

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

b
L
For a simple harmonic oscillator
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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
Change of resistance
Change of R with A
Change of R with r
Change of R with L
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Strain gauges
Change of R with A
Change of R with r
Change of R with L
For a cylinder, A ?r2 and dA/A 2(dr/r)
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Strain gauges
0
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Gauge Factor (G.F.)
measured from bridge voltage
given
calculated
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
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A strain gauge is the unknown resistor in a
Wheatstone bridge
Rx is the STRAIN GAUGE, generally VB ? 0
Unknown R (STRAIN GAUGE connected here)
VB
In lab, you are using 120W resistors and a 10V
power supply
Rx changes due to strain ?VB changes
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Relationship between VB and strain
VB is the bridge voltage
VB
But all strain gauge signals are amplified by
100x and you use an input voltage of 10 V
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A strain gauge
Made of Cu-Ni or Nichrome alloys
Too much load (large deflection) will cause
plastic deformation of wires

• Strain gauges are small stripes or wires whose
  resistance change with a
• change in their dimension

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Strain rosette
A strain rosette can be used to measure the
general state of strain at a point.
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A 90 strain rosette
Poisson's ratio
SG 4
Free end
SG 2
SG 1
SG 3
SG 5
Strain Rosette configuration
Underneath bar
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Laboratory experiment
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Experimental set-up
Amplifier
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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Beam dimensions
Using calipers, measure beam dimensions
x
h
z
y
fixed end
b
free end
L
? 2.77 gm/cm3 V bhL
Moment of inertia
m ?V
I bh3/12
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Determine k and E
D E F L E C T I O N
Add weights and measure deflection (y)
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Determine k and E
No strain gauges Measure deflection at end of
beam with meter stick
F kymax

F (N)

slope k



y (m)
Lab report
DO NOT EXCEED 1 KG!
Lab report appendix
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Measure x for each strain gauge
Strain Rosette configuration
SG 4
Free end
SG 2
SG 1
SG 5
SG 3
Underneath bar
x
xSG1
xSG2 xSG3
xSG4 xSG5
Using calipers, measure the distance of the
strain gauges from free end of beam
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(1) Measurements using strain gauges

• For EACH strain gauge (1-5)
• Use 5 different weights and combinations of
  weights (available 101.2, 147.3, 248.3 gms)
• Include 0 to null
• Use larger mass to minimize error DO NOT EXCEED
  1 KG
• Zero bridge without any weight
• Measure and record voltage for each weight

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(1) Measurements using strain gauges
Lab report appendix
Load and unload!
mass VB (volts) (gms) SG1 SG2 SG3 SG4 SG5
0 0 0 0 0 0 202.4 404.8 552,1 653.2 8
09.6
VERY IMPORTANT TABLE This is the basis for all
your analysis!
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For SG1 and SG2
Load and unload!
Lab report appendix
mass VB ?exp ?theo 202.4 V1 404.8 V2 552,1 V3
653.2 V4 809.6 V5
(gain 100 and input voltage 10 V)
Calculate ? from
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Determine Poisson's ratio, ?
You can only determine ?exp by looking at the
gauges 4 and 5
mass ?SG5 ?SG4 ?exp 202.4 ?SG5 ?SG4 -?SG5/?
SG4 404.8 552.1 653.2 809.6
Lab report appendix
lt 0
gt 0
gt 0
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Determine unknown G.F. (SG3)
-VB





F (newtons)
For gauge 3 x is known
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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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Dynamic spring constant

• Using the data collected on w
• Plot 1/w2 vs. load mass
• Determine k
• Calculate E

M mass of beam m applied mass
?-2
mgtgtM slope ?1/k
m
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Typical output
In appendix
period T 2???
Find angular frequency, ?, from the plot
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Compare k

• Which method (static or dynamic) is more accurate
  and why?
• Error analysis is necessary!

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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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Laboratory report Tables in appendix 6 Results
and Discussion figures in appendix
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Questions

• Question 1
• 20 pts On the same graph, plot force vs.
  vertical deflection and from this graph
  determine
• 10 pts The spring constant k (in N/m) of the
  beam. F (Newtons) k y(meters)
• 5 pts The Young's modulus E in GPa
• 5 pts How does E calculated above compare to
  the published value?

Figure 1


Multiply k (gm/cm) x 0.98 N/m
F (N)

Ealuminum 69 GPa 69 x 109 Pa 69 x 109 N/m2


slope k
deflection (m)
Error analysis! X Y GPa
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Questions
Question 2 5 pts Why should the load on the
beam not exceed 1000 gms? Question 3 15 pts On
a single graph, plot bridge voltage, VB vs. load
for all strain gauges 4a, 5 pts. What does
the shape and slope of the plot tell you about
the strain gauges 4b, 5 pts? How does the
plot for SG3 differ from that of SG2 and why
4c, 5 pts?
SG4
SG2

SG1
Figure 2
VB
Load (kgs)
SG5
-
SG3
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Questions
Question 4 15 pts For SG1 and SG2, plot
experimental and theoretical strain as a function
of load 5a, 5 pts. Which strain gauge would
best serve the purpose of an inclinometer 5b, 5
pts? For SG1 and SG2, compare experimental
and theoretical values and comment on
discrepancies 5c, 5 pts.
Figure 3
SG2
exp
theo
? (strain)
exp
SG1
Error analysis!
theo
F (N)
Multiply gm x 9.8 x 10-3 N
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Questions
Question 5 10 pts Determine the G.F. of SG3.
Hint you will have to use the best estimate of
the theoretical strain.
distance from free end
Error analysis!
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Questions
Question 6 10 pts Plot the experimental strain
of SG5 versus the experimental strain of SG4
for each load mass 7a, 5 pts. Determine
Poissons ratio for aluminum and compare to the
published value 7b, 5 pts.
Figure 5
Poisson's ratio for Al 0.33



? (SG4)


load (kg)



Error analysis!

? (SG5)

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Questions

• Question 7
• 10 pts On the same graph, plot 1/?2 vs. load
  mass for
• Measured data
• Theoretical values from Eq. (1), using the k
  calculated in Question 1(a)
• Theoretical values from Eq. (2), using the k
  calculated in Question 1(a)
• Find the range of load mass normalized to beam
  mass in which the error (in b and c) exceeds 10.
  Comment on the application of the two theories.

Error analysis!
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Questions
Take the proportional difference between static
and dynamic values. When is the difference 10?
Theoretical static and dynamic values are very
close
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Questions

• Question 8
• 10 pts
• Calculate the spring constant of the beam by
  fitting a line to the plot of 1/?2 vs. load mass
  5 pts
• State which method, 8(a) or 1(a), you think is
  more accurate and why 5 pts.

Figure 6


1/?2


Error analysis!
m (kg)
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Table I
What is the error in the mass of the loads? What
is the error in determining the deflection?
What is the error ot the strain gauge reading?
Table II
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SG1
SG2
mass VB ?exp ?theo ?exp ?theo 202.4 V1 404.8 V2
552,1 V3 653.2 V4 809.6 V5
Table III
Table IV