Experiment
2
Winter 20
12
ME 411
last update:
11/16/2013
1
EXPERIMENT 2
:
ST
RA
IN GAGE
DYNAMIC
TESTING
Objective:
In this experiment you will use the strain gage installation from the prior lab assignment and test
the cantilever beam under
dynamic loading situations.
Theory:
S
train
will
be measured
using
a resistance strain gage
as one resistance arm in
a Wheatstone
bridge
. The gage will be mounted on a cantilever beam and subjected to
uniaxial tension or
compression. The formula for volt
age output is given in Eq. (
1) below
. For example for
=
1000
(microstrain)
,
an input voltage V of
10
volts
,
gage factor F of
2.005 and
a
n amplifier
gain
A
of
1
00 the
output voltage reading
would indicate V
o
=
5
00 mV
.
4
F
A
)
10
*
F
+
2
2
(
4
F
A
=
V
V
ex
O
*
*
6
(
1)
Strain
V
o
Output voltage
reading (
V)
V
ex
Excitation
(input)
voltage
(V)
F
Gage factor
–
provided by gage manufacturer (typ. ~ 2.0)
A
St
rain gage
circuit
amplifier
gain
(external amplifier unit, typ. ~100)
Solving Eq. (1) for strain,
we obtain:
F
A
V
V
4
=
ex
O
*
*
(
2)
Equipment:
Cantilever Beam
(Aluminum)
Strain gage
Strain gage application chemicals and supplies
Voltage
amplifier
Beam clamp
system
(steel
sandwich and C

clamp
)
Digital voltmeter
Precision 120

ohm resisters for bridge circuit
ProtoBoard
Data acquisition system and computer running LabView
Small weights
Experiment
2
Winter 20
12
ME 411
last update:
11/16/2013
2
Figure
1
.
S
upplies available for installing a strain gage.
These include an installation guide, the
gages, cleansing chemicals, abrasive sand paper, adhesive, a soldering iron,
and
a ProtoBoard
that is used to facilitate the creation of a bridge circuit.
Precision resistors and the amplifier needed
for the bridge
circuit are not shown.
Procedure:
Install
and Test
Strain Gage
Retrieve the instrumented aluminum beam from your prior lab assignment
.
Reconnect
the
gage
electrical connections
and check
by first verifying that the resistance across the gage is
approximately 120 Ohms. The beam should then be clamped in the steel cla
mping device as
shown in Fig.
2
.
The
center of the gage should be 0.75” from the edge of the clamping
fixture
. Be careful not
to clamp down on the gage itself!
Be sure that the cantilever edge is
perpendicular to the long edge of the beam
.
M
easure all beam dimensions
(thickness, width,
length)
including distances from cantilever edge to gage center and to
loading divot.
If you
haven’t done so already, measure the mass of the beam so that you can estimate the mass per
unit length value used in the analysis of beam vibrations.
The lead wires should be attached to a bridge circuit constructed with 3 other high

precision 120
Ohm res
istors mounted on the ProtoBoard shown in
Fig.
1. The circuit should be
wired as
illustrated in Fig.
3
, with the output voltage
wired into the input end of the
OMNI AMP
amplifier
and the amplifier output
initially measured by a digital voltmeter.
Be sure
the amplifier is
switched on a
nd set to a gain of
25
.
Experiment
2
Winter 20
12
ME 411
last update:
11/16/2013
3
Figure
2
.
The completed strain gage installation. The beam is shown mounted to the cantilever
structure (steel blocks and clamp).
Actual beams in this experiment may be different (e.g., hook).
Figure
3
. Wiring diagram for bridge circuit.
R
g
is the gage and R
1
, R
2
, and R
3
are all the same
resistance as the nominal gage resistance (120
).
Graphic from Omega.com.
Determine beam natural frequency
Write a virtual instrument
to
measure vibration
of
the beam. Note that the sensor output (a
resistance change) is converted into a voltage signal in the transducer stage (bridge circuit) of
this experiment. So, the virtual instrument is measuring voltage NOT strain
.
Set
the
sampling
rate to
5
0
0 Hz and
set
the sample period so that you collect
1
0
seco
nds worth of data for each
test (5,000 samples).
The virtual instrument should store the raw voltage values, but it should
also be able to determine and display a frequency
analysis
for the response
(ie the fre
quencies
within which most of the signal power is contained)
. This can be accomplished in a variety of
ways, but will likely include the use of an existing subVI that can be found in the Functions
Panel
either in the
SignalProcessing tool palettes. You wil
l need to explore LabView Help to
identify which resource is best suited to your particular need in this experiment.
Be sure to
record the results of the frequency analysis for every run that you conduct.
To start with you should see if there are any unde
sired vibrations being transmitted into the
beam from the table
—
lab floor
—
building. If there appears to be any such noise try to minimize it
through isolating the test apparatus (perhaps adjusting the rigidity of the table and cushioning
either the table f
eet or the clamp mechanism). Prior to testing for the natural frequency of the
Experiment
2
Winter 20
12
ME 411
last update:
11/16/2013
4
beam record at least one sample that will allow you to quantify the noise in the system.
That is,
run the vi without touching the beam. Name the resulting file “control.txt”.
To conduct the vibration experiment, r
un the VI, i
ni
tiate sampling and wait between 1 and 2
second
s
before plucking the beam. Pluck the free end of the beam with a
gentle
motion of the
index finger
. Use your fingernail to initiate the beam vibration
.
The
data gathered by the system
will provide a measure of strain, and hence deflection vs. time
.
These data can be analyzed
later to determine both the natural frequency of response and the damping ratio of the system.
Name the resulting file “vibe
500_
1.txt”.
Repeat
this step
2
additional times
(total of 3 replicates)
,
naming subsequent files “vibe
500_
2.txt” etc.
By conducting the same measurement
multiple
times you will be able to investigate measurement repeatability in your analysis.
To verify that
the sampl
ing rate is sufficient run the same expt. at twice the sampling rate (for same total
sample period), naming these files “vibe1000_1.txt”, etc.
Analysis and
Calculations:
T
he strain
can be calculated
as a function of
beam
deflection as predicted by linear b
eam
theory.
3
3
b
L
EI
P
(
3)
where P is the load, E is the modulus of elasticity, and I is the moment of intertia (be sure to
calculate this about the correct axis),
is the deflection, and L
b
is the length of the beam (from
cantilever poin
t to load).
Knowing the load and beam properties/dimensions you can calculate
strain
at the strain

gage location
using beam theory:
I
PLc
E
(
4)
where L is the distance from the point load to the center of the strain gage.
From the
calibration data
and equation
2 the strain gage conditioning circuit gain (G) can be
determined. This same equation can then be used with output voltage data to determine the
transient strain signal during the vibration tests.
As the beam vibrates the out
put voltage from the strain gage system will oscillate and decay.
This oscillation occurs at the natural frequency of the beam. While the natural frequency is the
dominant term dictating the beam oscillatory frequency other higher order frequencies of the
system
may also be evident in the signal. A representative trace of the system re
sponse is
illustrated in Fig.
4.
Figure 4
. Sample trace of the response of the vibrating beam showing the natural frequency and
the signal decay associated with dampin
g.
Time (ms)
Output Voltage
t
Time (ms)
Output Voltage
t
Experiment
2
Winter 20
12
ME 411
last update:
11/16/2013
5
The
angular frequency
is given by
= 2
/t
.
To obtain a
n
estimate of natural frequency one can
measure
t over many (N=5) periods and determine the individual frequency by dividing by N.
Assuming there is no damping t
he natural frequency can be obtained
from:
4
2
'
)
875
.
1
(
b
n
L
m
EI
(
5)
where m’ is the mass per unit length of the beam
and L
b
is the beam length
from the cantilever
edge to the free edge
. See an appropriate textbook to investigate the theory behind this
equation and the determ
ination of th
e constant (1.875). Note that the natural
frequency resulting
from eqn (
5) is in the units of
radians per second
.
This equation for natural frequency assumes a uniform beam (without the hook at the end). An
alternative development found in some vibrations
texts allows one to consider the mass of the
hook at the end. Such an analysis is optional.
Note that in a real cantilever beam system the oscillations are damped. In such a case the
response of the beam is sinusoidal with frequency
d
and an amplitude tha
t decays as:
t
n
Ce
t
Amplitude
)
(
(
6)
To obtain the damping ratio (
) one can simply take the natural log of the above equation and
plot Ln(amplitude) vs. t. The resulting plot should have a slope of

n
.
The relationship between the natural
frequency and the damped frequency that you actually
measure is given by:
2
1
n
d
(
7)
ASSIGNMENT
:
You must write a f
ull report
(no more than
6
pages
)
. You should also include an
appendix
(no more than
4
pages)
of raw data &
calculations.
The following list of questions is intended as a
general guide
for students as they consider their
report write

up. Students should NOT set out to answer this specific set of questions, but rather use
them as a guide as they consider the anal
ysis of their experimental data.
How do the strain measurement results compare with beam theory predictions for the static
loading?
How
repeatable are the results of the calibration experiment? What factors contribute to
any variability?
How does the theor
etical natural frequency of this beam compare to the results of the
transient measurements? What factors contribute to any differences?
How does sampling frequency affect your results?
What is the damping ratio of the system?
Experiment
2
Winter 20
12
ME 411
last update:
11/16/2013
6
References:
Figliola, R.S. a
nd Beasley, D.E.,
Theory and Design for Mechanical Measurements
.
(chapter 11)
4
th
Edition. John Wiley and Sons.
2006
.
Robert L. Norton ,
Machine Design

An Integrated Approach
,
(chapter 4),
Prentice Hall, 1997
.
Shigley, J.E.,
Mechanical Engineering Design,
(3
rd
edit.), McGraw

Hill Book Co. 1977
.
Meirovitch, L.,
Elements of Vibration Analysis,
McGraw

Hill Book Co.,
1975
.
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