ME 4600:483 – Lab Manual
Revised August 2015
Vibration Measurement
WARNING: PART OF THIS LAB USES STROBE LIGHTS!
Table of Contents
Vibration Measurement ................................................................. Error! Bookmark not defined.
I. Objective ................................................................................................................................... 1
II. Apparatus .................................................................................................................................. 1
III. Principles and Background ....................................................................................................... 2
System Dynamics ........................................................................................................................ 2
Zero-Order Systems................................................................................................................. 2
First-Order Systems ................................................................................................................. 2
Second-Order Systems ............................................................................................................ 3
Free Vibration of a Single Degree of Freedom System............................................................... 4
Forced Vibration of a Distributed-Mass Cantilever Beam System ............................................. 5
IV. Procedure.................................................................................................................................. 7
A. Transient Decay of a Vibrating Cantilever Beam.............................................................. 7
B. Steady State Forced Vibration of Cantilever Beam ........................................................... 9
V. Required Data Analysis .......................................................................................................... 12
VI. References .............................................................................................................................. 13
VII. Figures ................................................................................................................................... 13
I. Objective
The purpose of this lab is to study and observe free and forced vibrations of simple cantilever
beams.
1. In the first part of the lab, the frequency and amplitude of oscillation will be recorded of a
free vibrating system. Using the acquired data and known information about the
cantilever beam, the characteristics of the vibrating system will be computed and verified.
2. In the second part of the lab experiment, the frequency and amplitude of vibration of a
continuous mass system will be measured. The first three natural frequencies will be
theoretically calculated, experimentally determined using several methods, and observed.
The three mode shapes will be subsequently graphed and comparisons between the
theoretical and the experimentally observed shapes will be discussed.
II.
Apparatus
Part 1:
1. Cantilever beam mounted in a clamp support,
2. C-clamp to hold the base to the table,
3. Two strain gages mounted to the beam in an axial direction,
4. Four pound set of weights with pan and hanger,
5. Ruler
6. 24-bit A/D converter and data acquisition software,
Part 2:
8. Cantilever beam with a mounting plate and an adjustable transducer mount,
9. B&K 4818 Mini-Shaker,
10. Bogen 60 watt Audio Amplifier,
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11. BK Precision Signal Generator,
12. Magnetic Reluctance Transducer MM0002,
13. Strobe light
III. Principles and Background
System Dynamics
All physical systems have a characteristic response to any forcing function. If you put a
thermometer in the sun, it heats up. If you hit a piece of metal, it makes a noise. These
characteristics can be modeled by differential equations. We can use these system models to
predict how a system will behave even if we have not specifically tested the system response.
This is true of systems we perform measurements on as well as systems we use to take the
measurements.
A general model for an output variable y can be defined as a system of nth order:
n
n-1
dy
d y
d y
+
an n an-1 n-1 + ..... a1 + a0 y = F(t)
dt
dt
dt
Where F(t) is a forcing function and n is the highest derivative. We will consider cases where
n=0, 1, and 2.
Zero-Order Systems
A zero-order system has the following equation and solution:
a0 y = F(t) ; y = KF(t)
where, 1/a0=K is the system gain. From the solution it is clear that the system response is simply
the input multiplied by K. All zero-order systems mimic their input with no time delays,
oscillations or other dynamics. The BAM is an example of a zero-order system (unless the input
frequency becomes high). A lever arm is another one (as long as the motion is slow enough to not
excite the lever arm bending mode).
First-Order Systems
A first-order system has a derivative in the model. The model equation and the solution for the
response to a step change in F(t) of amplitude A is:
a1
dy
+ a0 y = F(t) ;
dt
t
y(t) = KA + ( y0 - KA) e- 
where a1/a0=, the system characteristic time constant. A first-order system will respond to an
input with a multiplication by K and an exponential time delay. A thermometer is a first-order
system. Exposed to a temperature, its display exponentially approaches that temperature. One
may think of a first-order system as having the capability to store one form of energy.
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Second-Order Systems
A second-order system is modeled by the equation:
2
dy
d y
+ a1
+ a0 y = F(t)
a2
2
dt
dt
which can be rearranged as:
1 d 2 y 2  dy
+
+ y = KF(t)
 2n dt 2  n dt
where n is the natural frequency of the system and  is the system damping ratio. The solution
to this equation has three parts because complex numbers occur in the roots.
1) If  >1, then the homogeneous solution (solution for F(t)=0) is:
y(t) = C1 e1 t + C 2 e 2 t
2) If  =1, then
y(t) = C 1 e1 t + C 2 t e 2 t
3) If  <1, we get
y(t) = C e-   n t sin (  d t )
where
2
2
 1 ,2 = -   n +  n  - 1 ;  d =  n 1 - 
The first case gives exponential time delays similar to the first-order system and is called
overdamped. The second case is critically damped, and gives the smallest pure time-delayed
response possible for the system. The last case is underdamped. Here the solution is oscillatory
at the damped natural frequency  d. A second-order system can thus oscillate. Mechanical
oscillation is called vibration.
Vibrations are an important aspect of mechanical engineering. Vibration is present at some level
in all engineering machines and structures. Vibration can be either helpful, as in the case of
vibrating sifters and mixers, or harmful, such as in the case of the Tacoma Narrows Bridge, which
collapsed due to unstable torsional vibrations. One of the first steps required in the analysis of any
vibration problem is determining the natural frequencies of vibration of a part, or a system
containing the part, in question.
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Part A) Free Vibration of a Single Degree of Freedom System
Free vibration is a condition that exists when a system, starting from some initial condition, is
allowed to vibrate without the effect of an external forcing function. In the first part of this lab,
we will study the free vibration of a cantilever beam with a mass on the end, shown in Figure 1.
We will model the freely vibrating mass as a single degree-of-freedom system, as shown in
Figure 2. This is an approximation because the cantilever beam, although stiff and light
compared to the mass, may bend and introduce additional degrees of freedom. The degree of
freedom is equal to the number of independent coordinates that must be used to fully describe the
motion of the system as it vibrates. The spring-mass-damper model in Figure 2 is the simplest
model which can be used to represent all single degree-of-freedom (or point-mass) systems and is
the basis of the modeling equations given earlier.
Consider the spring-mass-damper in an initial state of equilibrium. When the system is perturbed
by external forces, it absorbs a certain amount of energy. The energy storage may be in the form
of potential energy of the spring, or kinetic energy of the mass. As the perturbing force is
removed, the system tends to return to its equilibrium position and dissipate energy in the
process. Depending on the damping in the system, this process may be characterized by a simple
asymptotic return to its original position (overdamped or critically damped) or by exponentially
decaying vibrations at a single characteristic natural frequency about the equilibrium position
(underdamped).
The underdamped system is characterized by the amplitude of vibration, the damping ratio and
the natural frequency. The amplitude depends on the initial condition and will decrease to zero as
equilibrium is approached, but the frequency and the damping ratio should remain constant. They
are not affected by the nature of the initial disturbance (this is true only for purely linear systems,
non-linear systems may change frequency or damping with amplitude but our experiment behaves
quite linear).
The deflection of the vibrating beam, or the spring-mass-damper system, in our experiment will
be characterized by a damped sinusoidal motion, shown in Figure 3, of the form:
-
x (t) = A e
1 - 2
d t
sin (  d t +  )
Where :
 d = damped natural frequency =  n
1 -  2 ; n=
k eq
M eq
: damping ratio
: offset phase angle
keq: equivalent spring rate ( or stiffness) = Ebh3/4l3)
Meq: equivalent mass = mass of weight + 33/140 * mass of the overhung portion of the
beam ( this means that 33/140=23.5% of the beam mass participate in the motion)
Note that as  approaches zero (i.e. the system is undamped)  d approaches  n, the undamped
natural frequency.
The natural log of the amplitude decay of each successive period in Figure 3 is called the
logarithmic decrement, , and it is related to the damping ratio,  , by the equation
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 = ln [
Revised August 2015
x (t)
1
x (t)
2 
] = ln [
]=
x (t + T d )
n
x (t + n T d )
1 - 2
Where: x(t): displacement at time t
Td: period of the damped sinusoidal wave
n: any arbitrary integer
Thus, the damping ratio,  , can be calculated by measuring the decay in amplitude after any
number of oscillations, n , and plugging it into the above equation. Notice that as the damping
ratio approaches unity,  goes to infinity. This corresponds to the critical damping ratio.
Part B) Forced Vibration of a Distributed-Mass Cantilever Beam System
Forced vibrations occur when a system is subjected to some type of continuing external excitation
that adds energy to the system. In this lab, we will examine the effect of a steady-state sinusoidal
excitation of a distributed-mass cantilever beam system. With the hanging mass removed, a
cantilever beam will exhibit higher modes of vibration just like the guitar string in our first lab.
The higher modes of vibration had a negligible effect on the initial overhung mass system, and
thus were neglected in the single degree-of-freedom analysis. In the second part of the lab,
however, you will see that these higher modes have a significant effect on the dynamics of the
distributed-mass cantilever beam. Continuous systems like this possess an infinite number of
degrees-of-freedom. In this lab we will examine the characteristics of the first three natural
frequencies. Infinitely many higher modes exist, but their amplitude becomes too small to
measure.
An electrically driven mini-shaker will provide excitation to the system by vibrating the end of
the cantilever beam. The vibration is produced by applying a sinusoidal alternating current to an
inductive coil surrounding a central plunger. By adjusting the frequency of oscillation of this
current with a sine-wave signal generator, different forcing frequencies can be applied to the
beam.
The vibration of the beam used in this lab will be characterized by the first three mode shapes
shown in Table 1, plus an additional spurious mode ???, which may occur (or may not, depending
on boundary condition on your particular system) somewhere between modes 1 and 2. This extra
mode is not predicted by the first-order analysis of the cantilever beam. It is caused by the fact
that the shaker mount does not exhibit perfect translation, but allows for a certain amount of
rotation. The location of the mystery mode will be dependent on the stiffness of the bearing
system - a boundary condition. You may not observe this mode on your particular setup. You
should always think about the effects of system boundaries when designing an experiment as they
can modify what you think you're measuring. Note that the mystery node is not really a
“mystery”, it is simply not predicted by the theory, because the theory assumes a perfectly
clamped boundary condition.
The primary natural frequencies of the mode shapes (except spurious mode) are determined by
(use consistent units, that means force, mass, time and length unit can not be independently
chosen and are related via Newton’s Law)
2
 n = ( n l )
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EI
 A l4
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Where: n: nth - harmonic natural frequency [ rad / sec ]
: density of the beam
A: cross-sectional area of the beam
l: length of the beam
E: Modulus of Elasticity
I: Area Moment of Inertia
n: mode number ( n = 1 for the 1st fundamental mode )
n: mode constant for mode n given in terms of the product of (n l ) in Table 1
In general, the vibration will depend upon
1) the natural frequencies of the system,
2) the damping inherent in the system, and
3) the frequency and amplitude of excitation.
One phenomenon, which you will notice is that the magnitude of the oscillation becomes very
large when the frequency of the excitation matches one of the natural frequencies of vibration of
the beam calculated above. In addition, the cantilever beam will visibly assume each of the above
mode shapes at the respective natural frequencies. This phenomenon can be observed with the
use of the strobe light. Note that in the second and third mode shapes, there are points where
there is no displacement; these points are called nodes (denoted N). The points corresponding to
maximum displacement are likewise called antinodes (denoted A).
n
n l
1
1.8751
?????
??????
2
4.6941
3
7.8548
Mode Shapes (N: node, A: anti-node)
Table 1. A schematic of the mode shapes for a cantilever beam subjected to forced
vibration with the corresponding nodal and anti-nodal positions.
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IV. Procedure
In Part A, we will use the resistance strain gage
transducer to monitor the time-dependent deflection
of a cantilever beam with a mass hung from the free
end. The vibration is sensed with a strain gage. The
change in resistance of the strain gage is measured
with a Wheatstone bridge circuit similar to the strain
experiment. The strain output is recorded with the
data acquisition system so that the decay rate of the
vibration can be plotted and evaluated. After setting
up the data acquisition, the beam will be given an
initial disturbance and allowed to come to rest while
the strain versus time is recorded.
Beam Tip
Oscillation
Added
Mass
A. Transient Decay of a Vibrating Cantilever Beam
1. Connect axial half bridge strain gage wires to the data acquisition system (refer back to
strain lab)
2. Modify the looping version of the VI to read strain inputs
a. Open the DAQ Assist properties window.
b. Under “Channel Settings”, click the red “X” to remove the voltage input channel.
c. Click the blue “+” to add a strain channel (9219, ai0).
d. Set gage factor to 2.09, gage resistance to 120.5, and configuration to Half Bridge II.
e. Add about 3 lbs of weight to the beam.
f. Calibrate the system the same way as in the strain lab.
g. Resave the VI as FreeVibration.vi
3. Assemble transient vibration experiment apparatus
a. Verify that the beam is clamped in the base perpendicular to the edge with the
weight-pan eyebolt located 6 1/2 inches from the edge of the base
b. Measure the physical dimensions of the beam. You'll need these values to find the
natural vibration frequency for the beam.
c. Identify and record the material composition of the beam
4. Record vibration decay data for beam
a. Set the data acquisition to take 300 samples at 100 samples/sec.
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b. Run the VI.
c. Quickly push the beam down with a hard metal object (screwdriver) and release
sharply, however, click the Stop button right before pushing on the beam. This will
freeze the graphs on the damped vibration trace. Clicking stop afterwards will freeze
the graphs one data period too late.
Don't push it too hard or the beam will undergo plastic deformation and you'll
have to start all over again with a new apparatus!
d. Repeat this process until you acquire a good trace of the decaying oscillation of the
system. Adjust the DAQ parameters or rezero the bridge as necessary.
e. Export the data from the two graphs to Excel.
5. Evaluate potential errors in the measurements
a. Look at the equations on page 4. Identify and evaluate possible sources of error.
Remember the model is for a cantilever beam, but your experiment has more
complex boundary conditions. Can you change the frequency of oscillation by
adjusting any uncontrolled boundary condition in the experiment? Can you affect the
rate-of-decay of the oscillations?
6. Cleanup experiment A. Carefully dismantle and store all equipment. Close the VI.
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B. Steady State Forced Vibration of Cantilever Beam
WARNING: THIS PART OF THE LAB USES STROBE LIGHTS!
In Part B, we will use a strobe light
Magnetic Reluctance Velocity
and a magnetic-reluctance velocity
Transducer, measures
velocity along the beam axis
transducer to monitor the forced
vibration of a long, distributedmass, cantilever beam.
The
magnetic transducer is effectively a
small magnet with a surrounding
coil of wire. When a ferrous beam
breaks the magnetic forcelines of
Shaker,
the sensor, it induces a voltage in
drives
the coil, just like an electrical
beam
generator. The voltage generated
motion
is proportional to the rate at which
forcelines are broken, so the
transducer actually responds to the
velocity of the beam in the direction normal to the magnet surface. It cannot distinguish between
velocity away from or toward the sensor (sign of velocity). Because the beam is metal and moves
in the magnetic field of the transducer, the transducer disturbs the system. This is a common
concern with any measurement system; the sensor may actually interfere to some degree with
what you are trying to measure. Always try to keep this effect to a minimum.
The transducer signal will be recorded with the data acquisition system, and the fundamental
vibration frequency and any harmonics will be evaluated with the FFT function of LABVIEW as
in the first lab. Since the beam is in simple harmonic motion, dividing the magnitude of the
beam's maximum velocity at each point by the fundamental frequency will yield the maximum
displacement of the beam at that point. The conversion from measured voltage to velocity
depends on the gap between beam and transducer and is not linear, so the calibration curve given
in Figure 4 must be used. The transducer will be moved along the length of the beam in one inch
increments so that data can be taken along the length of the beam at each of the three lowest
natural frequencies of vibration of the beam. This data can then be used to plot the mode shapes
of vibration at these three natural frequencies. In addition, the strobe light will be used to view
the beam's vibration at these three resonant conditions in a darkened laboratory. The frequency
of the beam vibration can be found by "stopping" the motion with the stroboscopic illumination.
Locate the equipment for experiment B, place it on top of the counter.
1. Measure beam Geometry
a. Measure the beam length, width and thickness as accurately as possible. The first
two theoretical resonant frequencies will be calculated in the data analysis from the
given equation and Table 1. During lab, assume the two frequencies are 20Hz and
150Hz.
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2. Assemble equipment for force vibration equipment.
a. Refer to figure 5 for a schematic of the experiment. Connect the function generator to
the Bogen Audio Amplifier (AUX1). Connect its output to the mini-shaker. The
magnetic pickup connects back to the data acquisition board.
b. Remove the cardboard shim on the shaker.
c. Position the magnetic transducer at the end of the beam and use the gap gauge to set
the transducer-to-beam spacing. Measure and record the gap gauge thickness for use
in Data Analysis Question 3.
d. On the amplifier, turn bass to cut and turn treble to zero. Maximize the signal from
signal generator so you don’t have to increase the amplification too much.
3. Modify the looping version of the VI to display the velocity amplitude
a. Find the maximum and minimum of the voltage signal.
Array → Max and min
Wire into data out of DAQ Assist before the Vrms block
b. Add a formula to calculate velocity amplitude.
Mathematics → Script and Formula → Formula
Change the X1 label to max, and the X2 label to min
Subtract max from min and divide by 2 to get the amplitude of the signal
c. Wire the maximum and minimum values of the array into the formula.
d. On the front panel, add a numeric indicator and rename it “Velocity Amplitude”
e. Wire the result of the formula to the indicator. The result is in V.
f. Rename the VI ForcedVibration.vi
New
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4. Observe vibration of beam
a. Turn on the equipment (sine wave on function generator at 10 Hz) and begin
continuous data acquisition. Start with a frame size of 2000 at 2000 Hz but you may
want to change this later.
b. Set the signal generator to 10 Hz. Adjust the Aux 1 setting on the amplifier until you
see a corresponding frequency spike on the FFT display.
c. Slowly increase the frequency of the Signal Generator to approach the Mode 1
resonant frequency assumed above. Stop when you see a significant increase/decrease
in the amplitude as you sweep the frequency. You can judge the amplitude visually
but to fine-tune the amplitude, use the data acquisition display and look for the
maximum amplitude on the signal (see next step). Go beyond the calculated resonant
frequency if necessary.
d. Determine the actual resonant frequency by fine-tuning the frequency of the signal
generator to obtain the maximum amplitude on the display. Adjust AUX 1 if
necessary to prevent the beam from hitting the transducer as the oscillation increases.
e. Record the frequency and amplitude of the fundamental vibration indicated by the data
acquisition display. Also record the indicated frequency on the signal generator.
f. Consider the boundary conditions of the experiment. Does the indicated resonance
frequency (the frequency of maximum vibration amplitude) change, if the transducer
is moved nearer or farther from the beam or along the beam length? (find the
resonance for the following modifications: change the gap between transducer and
beam and in a second step move the transducer to about the middle of the beam).
Record any effects and the estimated range of uncertainty for use in the error analysis.
g. Use the strobe light to observe the beam motion at the resonance. First, move the
transducer away from the beam in case it alters the resonant frequency. You can adjust
the strobe frequency to stop the image at even integer multiples of the actual
frequency, with a doubled image of the beam visible, or at even integer fractions of
the actual frequency, with a single image visible. To find the true frequency, you must
start high, with a doubled image, and then decrease the flash rate until the first single
image is observed. Turn off the lights in the lab to adequately observe the
phenomenon. Think about how strobing the beam is really the same as “visually
sampling” it. By flashing it twice as fast you see two images, top and bottom. By
flashing it with the vibration frequency you make it stop in time, that is, you
“visually” alias the vibration into zero frequency (remember the folding diagram).
h. Sketch the observed beam shape in your notebook.
5. Recording mode shapes with magnetic transducer
a. Place the magnetic transducer centerline 1/2 inch from the free end of the beam.
b. Verify that the gap is set with the plastic gap gauge. Measure and record the gap
thickness.
c. Set the AUX 1 dial on the amplifier to a low value
d. Set the FREQUENCY of the signal generator to obtain the mode 1 resonance found in
step 3e and fine-tune the frequency to obtain the maximum amplitude as before.
e. Increase AUX 1 to obtain good signal amplitude, but be careful to maintain the setting
low enough to prevent the beam from striking the magnetic pick-up. Adjust the gain
of the DAQ system if necessary to get a good signal. If you have to move the
transducer closer or farther from the beam, make sure you measure the gap.
f. Record the velocity amplitude (in V).
g. Turn the amplifier OFF, but do not change the FREQUENCY or AUX 1 settings.
Move the magnetic transducer towards the shaker at one-inch interval and repeat the
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velocity amplitude measurement for a total of seven to eight positions along the beam.
At each position verify that the gap is maintained with the test gauge.
6. Repeat steps 3 and 4 for Mode 2
7. Clean up
a. Replace the cardboard shim on the shaker.
b. Carefully dismantle the equipment and return it to its original state for use by the next
laboratory group
V. Required Data Analysis
1
a. Plot the time-series strain gage output data stored in Step A4. Measure the amplitude
of the first two successive peaks of the decaying vibration of the approximate pointmass system. From the amplitude of the peaks find the logarithmic decrement  and
thus the damping ratio . Then calculate the undamped natural frequency using the
damped natural frequency from the FFT measurement and the damping ratio  just
found.
b. Compare the experimental natural frequency to the theoretical natural frequency of the
beam.
c. Repeat the above calculation for two other successive peaks near the end of oscillation.
d. Evaluate the uncertainty in the damped natural frequency  d, resulting from
uncertainties in the beam dimensions and  measurements. Don't forget boundary
conditions that affect  d and .
2
a. Calculate the first two theoretical mode resonance frequencies of the continuous mass
system using Table 1. Assume that  = 0.283 lbm/in3 and modulus is 29x106 psi. Use
consistent units in your calculation. Compare these values to the measured resonant
frequencies.
b. The actual resonant frequencies can be evaluated from the readout of the signal
generator, from the FFT analysis of the magnetic reluctance transducer measurement,
and by stopping the beam motion with the strobe light. Discuss the effectiveness of
these measurements by listing the potential errors in each. Consider how each device
is calibrated, the effect each device has on the experiment itself, and any boundary
conditions. Evaluate these errors from your data if possible. Evaluate which of the
readings is most trusted.
3.
a. Convert your readings of beam velocity amplitude in millivolts to absolute velocity in
cm/s using Figure 4. Divide each beam velocity by the resonant frequency  (in
radians) to get beam deflection at each point the transducer was placed. Plot the
displacement as a function of length along the beam for each of the three modes on the
same plot. Note that you will have to determine the sign (+/-) of each point by
comparison to Table 1 because the sign of the displacement is not given by the
measurement. Compare your results to the mode shapes you sketched in step 3h.
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VI. References
1. Theory and Design for Mechanical Measurements, R.S. Figliola and D.E. Beasley,
Wiley, (1991)
2. Theory of Vibration with Applications, W. T. Thomson, Prentice Hall, 2nd edition,
1981, Appendix D, pp. 464 - 474.
3. Vibration of Mechanical and Structural Systems, M. L. James, Harper & Row, New
York, (1989).
4. Vibration Problems in Engineering, S. Timoshenko, Wiley, New York, (1974).
VII. Figures
Figure 1. Schematic of a cantilever beam with overhanging weight.
Figure 2. Schematic of an equivalent SDOF spring-mass damper system.
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Figure 3. Free vibrating oscillation of a damped single degree of freedom (SDOF) system.
Figure 4. Conversion chart from voltage to velocity as a function of spacing for the
magnetic velocity transducer.
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NI 9219 Data Acquisition Module
Half-Bridge Configuration
Part 1
LabVIEW VI
Part 2
Figure 5. Schematic of the overall experimental apparatus.
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