RESPONSE OF A CANTILEVER
BEAM UNDER STATIC AND
DYNAMIC LOADINGS
Tuesday March 4th, 2014
Xie Zheng – First Author, Introduction and Experimental Procedure
Kanchan Bhattacharyya – Discussion and Conclusion
Matthew Stevens – Results, Error Analysis
Ting Zhang – Abstract, Theory, List of Equipment
Abstract
This experiment studies responses of a cantilever beam under static and dynamic loadings. In
first part of this experiment, students use the strain gage and LVDTs to measure deflections of the a
cantilever beam under static loading, and then compare the data collected from strain gage with the
data calculated in theory to test the validity of beam theory. Second part of this experiment requires
students to measure dynamic strain of a vibrating beam at its natural frequency by using data
acquisition technology, and then compare the experiment data with data calculated theoretically in
order to test the validity of theory. Second part is designed to let students be familiar with the data
acquisition technology including how to use LABView data acquisition hardware and how to program
a Labview project. In addition, students will enhance the numerical analysis ability when doing
calculations of uncertainties of the experimental results. Results of the experiments depend on
measurement accuracy and correct experimental steps. From the results which are obtained through
computer aided program for both experimental parts, validity of theories are tested and verified
because experimental data matches very well with calculated theory data with acceptable accuracy.
Introduction
Material’s response to static loading is one of the material’s properties which are considered in
the design. The study of the deflection of a beam or structure which is made of a certain material could
significantly help the designers to take this property of the material into consideration. As for the
natural frequency, it is an important property of the certain size of the material in the natural and
industry since avoiding the natural frequency is always a caution for engineers during the process of
design. Although there are definitely plenty of ways to utilize the natural frequency, the first thing for
the engineers to consider for design would be doing the measurement and calculation for the natural
frequencies of the certain size of material which they are going to use. By measuring the weight, length,
active length, width and height which we measured in the experiment as well, engineers will be aware
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of the natural frequencies for their material. Then they can modify the amount for the specific
measurement to avoid the frequencies of the wind or some other natural effects in that area to ensure
the building or the object they are building could stand for a long period without most of the influences
from nature. One typical example would be the Tacoma Narrows Bridge Collapse. Due the windy
weather in local area, once the wind reached the natural frequency of the bridge, the resonance which
led to the final disaster happened. The bridge became famous as the most dramatic failure in bridge
engineering history. Now, it's also one of the world's largest man-made reefs.
In this experiment, we are given a physical view of the dynamic strain distribution along a
vibrating cantilever at its natural frequencies and calculate the deflection of a cantilever beam with
static loading. We calculate the deflection of the beam with a LABView program. As for dynamic loading,
we are using a DAQ to retrieve a large amount of dynamic data from a pre-loaded LABView program.
By observing the change of amplitude shows on oscilloscope screen, we can find out the frequencies
and look at the vibration mode shapes of the cantilever beam whose material is aluminum. To calculate
the theoretical natural frequencies, we are using the derived equation to do the calculation. Based on
the end condition which the end of the beam is fixed, the equation could be rewrite in a clearer form
which leads to three equations for the first three modes: ω1 = 3.516*(EI/ρA)1/2/l2, ω2 = 22.03*(EI/ρA)1/2/l2
and ω3 = 61.70*(EI/ρA)1/2/l2. Since the theory is based on some conditions such as the vibration should
be free undamped in flexure of beams which is assumed that vibration occurs in one of the principle
planes of the beam and then they can be neglected. Also, the end conditions required for the uniform
beam are simply supported, clamped or free. These are the restriction made upon the beam. After doing
the calculation, we are able to locate the approximate range of the natural frequencies. When the
natural frequencies are reached, the piezoelestric premounted on the beam will indicate the strain at
the certain location on the beam. By dealing with the relation between the location of the piezoelestric
and the corresponding strain, the responding under dynamic loading will be shown as curves.
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Theory
Part I (Deflection and strain measurements of a tip loaded cantilever beam)
In general, the equation for calculating a deflection curve of a beam can be derived from Bending
Moments Diagram Area Method
the original equation( Equation of the Elastic Curve):
d3 y
EI dx2 = −M(x)
(1)
where E is Young’s modulus, I is moment of Inertia and the product EI is known as the flexural rigidity ,in
addition, the moment M is seen as a function of x.
Integral this equation, we can get
x
dy
EI dx = ∫0
M(x)dx + C1
(2)
dy
where C1 is the constant of the integration. dx on the left hand can be seen as a function of θ with
respect to x. As shown in the picture below,
dy
we can figure out that dx = tanθ, especially when θ is extremely small, tanθ=θ. As a result, the
previous equation yields to:
x
EIθ(x) = ∫0
M(x)dx + C1
(3)
Integral equation 3 again, we get
x
EIy = ∫0
x
(∫0
M(x)dx + C1 )dx + C2 (4)
3
In general, we can use euqation (2),(3) and (4) with their boundary conditions to get the constant C1 and
C2. After obtain the constants C1 and C2, we can get the general equation of deflection for a certain
beam.
For the model of the cantilever beam in this experiment, as shown below,
The general equation for this beam is,
y=
1 lx2
(
EI 2
−
x3
)
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(5)
Equation 5 is used to calulate the beam deflections theorically, in practice, strain gage and LVDT are
used to measure the strain of the beam under deflections.
Part two (Dynamic strain measurements of a vibrating cantilever at its natural frequencies)
Natural frequencies of Uniform Beams
In the process of deriving the eqauation governing free undamped vibration in flexture of beams. We
neglected shear deformation and rotatory inertia.
S and M are the shear force and bending moments at the dx
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From equations Sds + M − (M +
∂M
dx)
∂x
M = −EI
= 0, S =
∂M
∂S
and ∂x
∂x
∂2
= pA ∂t2 ,we can get
∂2 V
∂X2
where E is Young’s modulus
I is moment of inertia
the product of EI is known as flexural rigidity
∂2 V
Combing equation M = −EI ∂X2 and equation S =
∂2
∂2 V
Combing equation ∂x2 (−EI ∂t2 ) = pA
∂2 v
and
∂t
d4 v
Finally, we can get the general equation dx4 −
∂M
,
∂x
we can get
equation S =
pAw2
V
EI
∂2
∂2 V
(−EI ∂t2
∂x2
∂M
,
∂x
) = pA
∂2 v
∂t
∂4 v
we can getEI ∂x4 + pA =
∂2 V
∂t2
=0
we get the general solution :
V1 = B1 sinλx + B2 cosλx + B3 Sinhλx + B4 coshλx
To get the constants B1, B2, B3 and B4 , we have to use four end conditions
a. simply supported or pinned
displacement is zero and bending moment is zero which means:
Δy = 0 M=0
b. fixed or clamped
Δy = 0, θ=0
displacement and slope is zero which means
c.free
bending moment and shear force is zero which means
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M=0, S=0
From bondary conditions above, we can get
(sinλl + sinhλl − sinλl) − (cosλl + coshλl)2 = 0
Finally, we can get the shape of rth mode, in terms of a single constant
Vr (x) = Br [coshλr x − cosλr x − nr (sinhλr x − sinλr x)]
where nr =
cosλlr +coshλr l
,r=1,2,3
sinλr l+sinhλr l
List of Equipment
1. Strain indicator and multichannel strain conditioner
2. Data Acquisition Work Station
6
3. A beam with 8 premounted strain gages and a Piezoelectric
4. A shaker
6. A cantilever beam with a paramounted strain gage
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7. A oscilloscope
8. A digital function indicator and a power amplifier
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EXPERIMENTAL PROCEDURE
Part A. Deflection and strain measurements of a tip loaded cantilever beam.
1. Since the experiment is already setup in the lab, we check the connections of the equipment and
make sure we have all the instruments (four equal thickness metal blocks and a digital caliper to
measure the size of blocks) we need.
2. Turn on the power of the LVDT power supply and the strain indicator.
3. Adjust the strain indicator until the reading on the display screen is 0.
4. Take five zero readings of all the four LVDTs and also the strain gage by running the pre-loaded
LABView program in the computer as U0 for LVDTs and Ui for strain gage.
5. Repeat step 4 for three times and take the average as the initial reading for U0 and Ui.
6. Do the calibration of the LVDTs by inserting the metal blocks between the beam and the tips of the
LVDTs.
7. Adjust the strain reading which is displayed on the strain indicator screen to 2000 micro strain.
8. Run the pre-loaded program three times to get the calibration reading of the LVDTs and also the
strain gage: UC and Uc.
9. Average them for the final value for later use.
10. Create the test program for deflection and strain measurements by adding a formula loop onto the
base program. And the formula we use for deflection of the LVDTs and strain are:
Eq.A
Eq.B
where 0.0671’’ is the thickness of the metal calibration block.
11. Add the load to the end of the beam with a 2kg weight.
12. Run the program for three times and record the testing readings UR and Ur. the testing strain reading
should be close to the reading displayed on strain indicator.
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13. Add another 2kg weight (4kg in total) and repeat step 12. Get 6 trials in total.
14. For every trial, the program calculate the deflection and strain automatically (record them as well).
The created program is shown as below:
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Part B. Dynamic strain measurements of a vibrating cantilever at its natural frequencies.
1. Since the experiment is already setup in the lab, we check the connections of the equipment.
2. Measure the size of the beam, the distance from center to the first piezoelestric and also the
distance between each piezoelestric.
3. The power amplifier and the digital function generator are on and the waveform is already set to
sine wave. Output and offset are also pre-set. Adjust the knob on digital function generator to set
the frequency at around 10Hz and the increment is 0.1Hz.
4. Check the piezoelestric if it is working by slightly hit the cantilever, the vibration is shown on the
oscilloscope screen.
5. Multichannel strain conditioner is on and it is also pre-set. Use the “Bal” adjustments and rotate
back and forth until two LED is not lighted which means the channel is balanced.
6. As for the calibration, the multichannel strain conditioner is pre-calibrated and the output level of
each channel is made as 2.5 volt.
7. Make sure the “Cal” switches to the mid position and all the output back to the channel board of the
data acquisition.
8. The program is already created and opened in the computer, set the “sample” as 100 and “rate” as
5000 for the first natural frequency.
9. Slightly turn up the gain of the power amplifier.
10. Adjust the frequency of the function generator starts from 10Hz and search for the first natural
frequency. Observe the oscilloscope to see if the vibration amplitude gets to the maximum value,
then that is the first natural frequency.
11. Record the frequency of function generator, run the program to get the save the data into an Excel
file and also plot the curves.
12. Turn down the gain of the power amplifier and use the “Bal” adjustments and rotate back and forth
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until two LED is not lighted which means the channel is balanced.
13. Set the starting frequency as 100Hz and increment as 1Hz, change the “rate” to 10000 and repeat
steps 10, 11 and 12 to search for the 2nd natural frequency and save the dynamic data.
14. Set the starting frequency as 100Hz and increment as 1Hz, change the “rate” to 15000 and repeat
steps 10, 11 and 12 to search for the 3rd natural frequency and save the dynamic data.
15. Save the data file to a CD for record.
Results
Fig.1 The measured values for the length l, width b, and height h of the beam cross section, the
computed moment of inertia Iz, the Modulus of Elasticity E, and the maximum theoretical deflections
δ2.0kg and δ4.0kg calculated using equation: δ = Pl3/(3EIZ)
Fig.2 The mean deflections computed by the LabVIEW program using Eq.A, the Theoretical Deflection
each position as per Equation (5), corresponding true and relative errors, and relative accuracy of the
experimentally found values with respect to the theoretically predicted values for the 2 kg mass trials.
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Mean Deflection vs. Position
0.05
Deflection (in)
0.04
yE = 0.0033x - 0.0053
0.03
0.02
0.01
0
-0.01
0
2
4
6
8
10
12
14
Position (inches)
Mean Deflection vs. Position
Linear (Mean Deflection vs. Position)
Fig.3 Graph plotting the experimentally determined deflections against the position from the fixed end
of the beam for the 2 kg mass trials. Our results rendered a linear form of yE = 0.0033x - 0.0053 in.
Deflection (in)
Mean Deflection vs. Position
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
-0.005 0
-0.01
yE = 0.0033x - 0.0053
yt = 0.0031x - 0.0047
2
4
6
8
10
12
14
Position (inches)
Mean Experimental Deflection
Theoretical Deflection
Linear (Mean Experimental Deflection)
Linear (Theoretical Deflection)
Fig.4 Graph plotting the experimentally determined deflections against the position from the fixed end
of the beam as well as the theoretically determined strains at each respective position for the 2 kg mass
trials. Comparing the two sets of results we graphically illustrate the marginal error in the experimental
data in comparison to the theoretically predicted values.
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Fig.5 The mean deflections computed by the LabVIEW program using Eq.A, the Theoretical Deflection
each position as per Equation (5), corresponding true and relative errors, and relative accuracy of the
experimentally found values with respect to the theoretically predicted values for the 4 kg mass trials.
Mean Deflection vs. Position
0.09
0.08
0.07
yE = 0.0064x - 0.0096
Deflection (in)
0.06
0.05
0.04
0.03
0.02
0.01
0
-0.01
0
2
4
Mean Deflection vs. Position
6
8
Position (inches)
10
12
14
Linear (Mean Deflection vs. Position)
Fig.6 Graph plotting the experimentally determined deflections against the position from the fixed end
of the beam for the 4 kg mass trials. Our results rendered a linear form of yE = 0.0064x - 0.0096 in.
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0.09
0.08
0.07
yE = 0.0064x - 0.0096
Deflection (in)
0.06
yt = 0.0031x - 0.0047
0.05
0.04
0.03
0.02
0.01
0
-0.01
0
2
4
6
8
10
12
14
Position (inches)
Mean Experimental Deflection
Theoretical Deflection
Linear (Mean Experimental Deflection)
Linear (Theoretical Deflection)
Fig.7 Graph plotting the experimentally determined deflections against the position from the fixed end
of the beam as well as the theoretically determined strains at each respective position for the 4 kg mass
trials. Comparing the two sets of results we graphically illustrate the marginal error in the experimental
data in comparison to the theoretically predicted values, a margin smaller than that determined from
the 2 kg trials.
Fig.8 the maximum and minimum recorded strains (in micro-strain) and experimentally determined
natural frequency of the aluminum beam for (a) the first natural frequency (b) the second natural
frequency (c) the third natural frequency.
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Fig.9 The strain values (in micro-strain) associated with the maximum and minimum strain in the
aluminum beam at various positions from the fixed end of the beam for (a) the first natural frequency
(b) the second natural frequency (c) the third natural frequency.
Strain vs. Position
300
Strain (in/in)-6
200
100
0
1
2.5
4
5.5
7
9.5
11
12.5
-100
-200
-300
Position (inches)
Minimum Strain
Maximum Strain
Fig.10 Graph plotting the Strain (in micro-strain) against position from the fixed end of the aluminum
beam (in inches) for the first natural frequency.
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Strain vs. Position
200
150
Strain (in/in)-6
100
50
0
1
-50
2.5
4
5.5
7
9.5
11
12.5
-100
-150
-200
Position (inches)
Maximum Strain
Minimum Strain
Fig.11 Graph plotting the Strain (in micro-strain) against position from the fixed end of the aluminum
beam (in inches) for the second natural frequency.
Strain vs. Position
60
Strain (in/in)-6
40
20
0
1
2.5
4
5.5
7
9.5
11
12.5
-20
-40
-60
Position (inches)
Maximum Strain
Minimum Strain
Fig.12 Graph plotting the Strain (in micro-strain) against position from the fixed end of the aluminum
beam (in inches) for the third natural frequency.
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Fig.13 (a) Table of measurements taken for active beam length, width, and height, the calculated
moment of Inertia Iz and cross sectional area, as well as the known elastic modulus and density for
aluminum used in theoretical evaluation of the natural frequency of the beam (b) the theoretical natural
frequencies calculated from (ω1 = 3.516*(EI/ρA)1/2/l2, ω2 = 22.03*(EI/ρA)1/2/l2 and ω3 = 61.70*(EI/ρA)1/2/l2)
the experimentally determined natural frequencies, and the corresponding true and relative errors as
well as the relative accuracies of the experimental results with respect to the theoretically determined
frequencies.
DISCUSSION
PART I – STATIC LOADING
In Part I of this experiment, four LVDTs are laid at evenly spaced distances along a cantilever
beam alongside a strain gage. The initial zero reading for each LVDT was taken and then each was
calibrated using accurately machined metal blocks. Together, these LVDTs were used to measure the
deflection along the beam in the x-direction in response to loading of a 2 kg and 3 kg mass.
The four positions tracked by the LVDTs were at x = 0.625, 4.625, 8.625, and 12.625. In Fig. 2,
the average experimental deflection and theoretical deflection for each position are tabulated along
with absolute error and relative error/accuracy calculations. The experimental deflections match very
well with theory for the most part, with 2-3 significant figures matching perfectly for x = 4.625 and x =
8.625, and x = 12.625 having a theoretical deflection within 9% of the actual value. The only main outlier
is at x = 0.625 where the actual deflection is nearly 3.6x what is predicted by the theory. It’s important
to consider the fact that the x = 0.625 measurement comes far closer to the fixed end of the beam than
the other positions which are at least 4 inches away. Being so close to the fixed end, experimentally or
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theoretically, means the deflection will be exceeding small in this area. Both the experimental value and
theoretical values are in the ten thousandths region.
Variations in deflection on that magnitude can be more of a limit in experimentation than actual
mismatch with theory. For example, voltage signal the LVDTs produce when the pin is displaced should
be quantized since there is a limit in terms of resolution of the DAQ. However, acquired voltage values
had several significant figures in the decimal region, often the first 2-3 held steady over several trials but
the latter few digits changed significantly from trial to trial. Hence, these are uncertain digits but are still
very important in calculating deflection when it as small as the ten thousandths range. Thus, to truly
accurately measure deflection at positions very close to the fixed end, it may be important to find a
more precise way of calculating it – keeping only significant figures that are certain lead to round-off
error while using the full number of digits given by LabView may also lead to some error.
QUESTION: What is the basic concept of computer aided testing?
Computer aided testing in this experiment uses a PC, DAQ, strain indicator, strain gage, 4 LVDTs
for loading, and a power supply. LVDT deflection and strain from the piezoelectric strain gage register as
voltage signals which are sent to the DAQ (Data-Acquisition-System) and passed to the computer which
analyzes and displays the voltage readings using LabVIEW, a VI program designed to interact with the
DAQ, supplying commands and reading data. By measuring deflections indirectly via voltage signals and
using conversion factors or calibration to correlate voltage with physical deflection, it is possible to
register very small deflections not seen by the human eye. Using Eq.A, it’s possible to compare the
voltage for any given loading with the voltage output of an LVDT during calibration with the metal block
as a multiplier, and then this multiplier can be multiplied with the thickness of the block for calibration,
to get the deflection for a given load.
QUESTION: Describe the need to make calibrations during the test.
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The results of this experiment rely on the LVDTs being calibrated so their voltage output (in
volts) when the beam is subject to a given load can be correlated with a deflection that makes sense, in
terms of some unit of length. Calibration has two aspects to it, (1) recording the initial voltages of each
LVDT when no load is applied to the beam – this voltage must be subtracted from all subsequent voltage
readings because this is misleading voltage, since ideally with no load, no deflection, they should be all 0
V. (2) Each LVDT can have variation in sensitivity which can affect results. To avoid this, each LVDT is
displaced by the same amount using a highly accurately machined block of known thickness, with the
voltage readout of each LVDT in response being recorded. Any subsequent voltage recording for a load
can be compared by how many times greater or smaller it is to this initial voltage reading. Once that
multiplier is found, i.e. for a given load, the voltage = 2V, where V is the voltage the LVDT outputted for
being displaced by the block  therefore, the deflection at that point is 2x(thickness of block).
Essentially Eq.A is just a scaling equation.
Going back to why calibration is necessary, let’s think about what would happen if each
calibration was not done. If the “zero” voltage readings are not subtracted from later voltage readings,
shown in the equation as U0, then all recorded voltages will be higher than they should be, predicting a
deflection that is higher than it actually is. Secondly, if each LVDT is not calibrated to a certain fixed
displacement, then the sensitivity of each LVDT is being ignored. Also with no initial measurement to
compare voltage output with a known deflection, there is no way of telling what a voltage of a certain
magnitude means in terms of the real world deflection.
PART II – DYNAMIC LOADING
In Part II, there is a switch to an oscilloscope, shaker, and multi-channel strain conditioner setup,
which are used to test the dynamic loading of a cantilever beam via strain gage readings to map out the
modes of vibration and therefore find the natural frequencies of the beam. The strain is mapped out
over several x-values along the beam and how that strain changes for that given x over several instances
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in time is tabulated, with the rows with the highest and lowest values at the farthest strain gage taken
to represent the curves of natural frequency.
QUESTION: Describe the strain calibration method in this experiment.
In this experiment, the strain calibration method’s purpose was to correlate microstrain
readings to voltages. If it was not calibrated, for whatever voltage produced by the strain gage that is
taken as the reading, it would be uncertain what that means in terms of the microstrain the strain gage
is experiencing as the beam is vibrating. The multi-channel strain conditioner takes in strain readings
and correlates microstrain with a readable voltage given to the DAQ and then to LabView. It defines the
resolution by dividing up the -5 to 5 V range of the DAQ into -5000 mV to 5000 mV range for a total of
10,000 mV. The DAQ is a 12-bit board with 2^12 storage capabilities, which results in a resolution of 2.5
mV = 1 microstrain. This means that microstrain values in between cannot be registered. There is a limit
in accuracy from the number of strain gages as well and this effect creates segmented graphs.
In order to assign the resolution of 2.5 mV as 1 microstrain, the channel outputs of the
multichannel strain conditioner were disconnected from the DAQ, connected to the digital display unit
and then adjusted using the gain control for each channel until 2500 mV was displayed. Then the output
channels were reconnected to the DAQ. In this way, this correlation is set. Also, while searching for
natural frequencies, between trials we had to zero and balance each channel (no red light in each
channel) before raising the frequency to search for the next natural frequency, because the channel
reading will go off balance once the gain is changed.
Comparing Natural Frequencies:
In Table 13, the 1st, 2nd, and 3rd modes of vibration are listed along with the theoretical natural
frequency and experimental natural frequency for each. For n = 1, the difference between the two is
very small, on the order of tenths with theor = 25.838 Hz and the exp = 26.7 Hz. However for n = 2
(theor = 161.890 Hz, exp = 164.0 Hz) and n = 3 (453.410 Hz, 449.0 Hz), the difference is in the single
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digits. Also the difference between the two is growing larger it seems as the natural frequency goes up.
To be more specific, the theoretical natural frequency which starts out as actually less than the
experimental for n = 1 and n = 2, (likely due to damping coefficient being more significant at lower
frequencies) suddenly ends up being a bit higher for n = 3. It would be interesting to see the results from
a few more natural frequencies if that were possible but going off of what we’ve obtained from this
experiment it is possible to predict that for higher natural frequencies, the theoretical natural frequency
predicted will be consistently higher than the experimental natural frequency.
This can be easily explained by the fact that the theoretical natural frequency equations were
developed simply by increasing the number of nodes that define each natural frequency. It’s possible to
write an equation for an infinite number of higher natural frequencies. However, in reality, it is not
possible to have more than 3 or 4 at most natural frequencies. To explain this, consider that higher
natural frequencies means for a given fixed active length, the beam must cross the axes more times to
produce those extra nodes. Therefore there are regions of severe curvature that need to be sustained as
the order of natural frequency goes up. Eventually, in a real life situation where geometry and the
material properties are limiting factors, the beam would break before higher natural frequencies could
be reached.
Error Analysis
The sources for error in the values obtained include improper calibration, inaccurate
measurements, and misconception of observations. The results obtained during the static loading trials
and their accuracies depend on the accuracy of the calibration method, namely in this case the accuracy
of the LVDT and strain gage output voltages that were used to calculate the deflection and true strain. In
considering the significance of accuracy in measurements taken by hand for the features pertinent in
calculating the cross-sectional area and second moment of inertia, we find that errors made in these
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features propagate in determination of the theoretical values which are used to validate the accuracy of
experimental methods. We can also consider the accuracy of our own observations, specifically with
regards to dynamic loading and the experimental determination of natural frequencies. The accuracy of
the experimental method with respect to the theoretically determined natural frequencies was
dependent on our own interpretation of the occurrence of each natural frequency, as precise
identification of such an occurrence can at times be subjective to the observer.
We can initially consider the mean static deflections obtained via calibration of the LVDT and
their dependence on position, which took a linear form 𝑦 = (3.337 𝑥 10−03 )𝑥 − (5.304 𝑥 10−03 ) with
uncertainties of ua = 4.64 x 10-3 in the independent variable and ub = 5.809 x 10-4 in the dependent
variable for 2 kg. and 𝑦 = (6.379 𝑥 10−03 )𝑥 − (9.563 𝑥 10−03 ) with uncertainties of ua = 2.849 x 10-2 in
the independent variable and ub = 3.565 x 10-3 in the dependent variable for 4 kg. The relatively small
uncertainties considering the small sample size are a very good indication that the values obtained were
following the anticipated linear deflection trend and calibration was performed relatively accurately.
Upon further comparison with the both the maximum theoretical deflection and deflections at various
cross-sections with theoretically predicted values, we find generally accurate results with error of
fractional order. The mean 2 kg. Deflections taken from the three LVDTs farthest from the fixed end of
the cantilever (0.04049 in., 0.01986 in., and 0.0064 in.) were extremely accurate in comparison to the
theoretically predicted values (0.0371 in., 0.0198 in., and 0.0064). These experimentally determined
values possessed relative accuracies near unity (1.098, 1.001, and 0.998), indicating a high degree of
relative accuracy. The 4 kg deflections (0.0769 in., 0.0398 in., 0.0133 in.) taken from the three LVDTs
farthest from the fixed end of cantilever possessed similar relative accuracies (1.0354, 1.0037, 1.0392),
also indicating a high degree of relative accuracy of the experimental method with respect to the
theoretical predictions. However, for both masses we found that the readings taken from the LVDT
closest to the fixed end of the cantilever were substantially larger than those predicted by theory. The
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variances in these values accumulated to relative errors of 2.7 for the 4 kg. Deflections and 3.6 for the 2
kg. Deflections. While the magnitude of the error associated with these values is a cause for concern,
the degree of accuracy maintained throughout the remaining cross-sections is enough to prove validity
in the experimental approach.
Considering the results obtained during dynamic loading, we found experimental natural
frequencies of 26.7 Hz, 164 Hz, and 449 Hz, respectively. The dependence on the actual natural
frequency on the order of the natural frequency was found to be linear, following the form
𝑦 = 211.2𝑥 − 209.1 𝐻𝑧 with uncertainties of ua = 92.1 in the natural frequency order and ub = 42.6 in
the frequency. These errors while considerably large, must be considered with attention paid to the
limited number of samples used in linear regression calculations. These uncertainties would be reduced
had we included additional measurements. In comparison to the theoretically predicted natural
frequencies (25.8 Hz, 161.9 Hz, 453.4 Hz), our experimentally determined values (26.7 Hz, 164.0 Hz,
449.0 Hz) were relatively accurate, with true relative errors of fractional order (-0.03, 0.01, 0.01) and
relative accuracies approaching unity (1.03, 1.01, 0.99). The uncertainty in the strain measurements
obtained from each natural frequency was also considered, as we used the experimentally determined
strain values to find the strain induced at various cross sections of the cantilever. Linear regression of
the maximum and minimum recorded strain per position of the first natural frequency rendered linear
𝑖𝑛 −6
𝑖𝑛 −6
results of 𝑦 = 21.375𝑥 − 34.924 (𝑖𝑛) for the maximum strain and 𝑦 = 37.080 − 21.292𝑥 (𝑖𝑛)
for
the minimum strain, with uncertainties of ua = 9.379 in, ub = 1.315 in and ua = 10.221 in, ub = 1.433 in
respectively. The results of the second natural frequency rendered approximations of the form 𝑦 =
𝑖𝑛 −6
𝑖𝑛 −6
11.068𝑥 − 107.259 (𝑖𝑛) for the maximum strain and 𝑦 = 112.928 − 11.324𝑥 (𝑖𝑛)
for the
minimum strain, with uncertainties of ua = 54.574 in, ub = 7.651 in and ua = 54.799 in, ub = 7.68 in. Finally,
the linear regression of the results from the third natural frequency rendered expressions of the form
24
𝑖𝑛 −6
𝑦 = 23.574 − 2.741𝑥 (𝑖𝑛)
𝑖𝑛 −6
for the maximum strain and 𝑦 = −21.813 − 2.607𝑥 (𝑖𝑛)
for the
minimum strain, with uncertainties of ua = 15.657 in, ub = 2.195 in and ua = 18.143 in, ub = 2.544 in
respectively. These errors are relatively acceptable, considering the limited sample size and the limited
precision used in measuring strain gage positions.
Conclusion
The use of LVDTs in static loading of the cantilever beam yielded excellent results especially when
testing distances of 4 inches and up from the fixed end, often matching in several significant figures
after trailing zeroes. However, the measurements appear to break down at x-values very close to the
fixed end as was the case for x = 0.625 where the experimental value appeared to be 3.6x the predicted
value. This can be partially attributed to the scale of the deflection now being in the ten thousandths
and the resolution of the voltage signal that can be picked up by the DAQ, which may not justify the
extra digits provided for the voltage readings in LabView. However, whether only certain significant
digits were used or whether the extra uncertain digits were kept, the fact remains that since the
deflection is so small, extra precision is required to know the voltage to a greater precision and thus
yield more accurate readings of the deflection. This precision can be provided using LVDTs with direct
digital readouts accurate to those extra digits or it may be possible to attach more strain gages at a
farther distance and then extrapolate what the deflection might be at smaller displacements, which
should not be of huge importance given that those deflections quickly diminish as we approach the fixed
end.
In the dynamic loading, the natural frequencies matched up exceedingly well and the deviation at
the higher natural frequencies such as at n = 3, can be attributed to limits of the geometry and material
in achieving that natural frequency. As was discussed, it was hypothesized that the reason why no more
25
than 3 or perhaps 4 natural frequencies are observable is that the number of times the beam would
have to cross the axis to produce the nodes required would result in severe curvature that the beam
could not sustain without fracture or respond with severe dampening of some kind. To improve the
dynamic loading with regards to the calculation of natural frequencies, extra strain gages can be
attached in order to smooth out the curves for each natural frequency and provide more accurate
readings which may reduce the error in as much as half, given that they tend to correlate very well even
as is.
REFERENCES
1. F. P. Chiang and T. Y. Hsu, Manual for experiments in solid Mechanics, 2014
26
APPENDIX A
Uncertainty Analysis
Part One – Response of a Cantilever beam under Static Loadings
Fig.A1 (a) The experimentally determined natural frequencies, calculated products for linear regression
analysis, linear approximation, and error individual error terms (b) the calculated summations used in
linear regression analysis (c) the calculated values for a and b forming the linear approximation
expression y = a + bx and their associated uncertainties
Part Two – Response of a Cantilever beam under Dynamics Loadings
Fig. A2 (a) the experimentally determined positions and induced strains, calculated products for linear
regression analysis, linear approximation, and error individual error terms calculated using the
maximum strain data set for the first natural frequency (b) the calculated summations used in linear
regression analysis (c) the calculated values for a and b forming the linear approximation expression y =
a + bx and their associated uncertainties.
27
Fig. A3 (a) the experimentally determined positions and induced strains, calculated products for linear
regression analysis, linear approximation, and error individual error terms calculated using the minimum
strain data set for the first natural frequency (b) the calculated summations used in linear regression
analysis (c) the calculated values for a and b forming the linear approximation expression y = a + bx and
their associated uncertainties.
Fig. A4 (a) the experimentally determined positions and induced strains, calculated products for linear
regression analysis, linear approximation, and error individual error terms calculated using the
maximum strain data set for the second natural frequency (b) the calculated summations used in linear
regression analysis (c) the calculated values for a and b forming the linear approximation expression y =
a + bx and their associated uncertainties.
28
Fig. A5 (a) the experimentally determined positions and induced strains, calculated products for linear
regression analysis, linear approximation, and error individual error terms calculated using the minimum
strain data set for the second natural frequency (b) the calculated summations used in linear regression
analysis (c) the calculated values for a and b forming the linear approximation expression y = a + bx and
their associated uncertainties.
Fig. A6 (a) the experimentally determined positions and induced strains, calculated products for linear
regression analysis, linear approximation, and error individual error terms calculated using the
maximum strain data set for the third natural frequency (b) the calculated summations used in linear
regression analysis (c) the calculated values for a and b forming the linear approximation expression y =
a + bx and their associated uncertainties.
29
Fig. A7 (a) the experimentally determined positions and induced strains, calculated products for linear
regression analysis, linear approximation, and error individual error terms calculated using the minimum
strain data set for the third natural frequency (b) the calculated summations used in linear regression
analysis (c) the calculated values for a and b forming the linear approximation expression y = a + bx and
their associated uncertainties.
30