MEC316 Lab#4
Natural Vibration Modes of a
Cantilever Beam
Group #9:
Kanchan Bhattacharyya – Results & Discussion; Conclusion; Synthesizing Paper
Matthew Stevens – Graphs & Error Analysis
Ting Zhang – Theory & Experimental Procedure
Xie Zheng – Abstract & Introduction
Abstract
The general goal of this testing experiment is first to be familiar with the operation of the
instruments since we are going to use them to generate and study vibrations. For this experiment,
we do the test of the natural frequencies of aluminum and steel beam respectively, and we have
a set of instruments which are piezoelectric sensor, oscilloscope, digital function generator, strobe
light and vibration exciter and power amplifier. In this experiment, these instruments are used to
set up a testing system to test the natural frequencies and then compare them with the theoretical
natural frequencies which are calculated by derived equation. For our experiment, the data we
record is slightly different from the theoretical ones, which means that the results are within the
range of error. Since we already have the derived equation for calculating the natural frequencies,
the parameters which we need to measure are the weight, length, active length, width and height
of the aluminum beam and steel beam respectively. First of all, we need these measurements to
define the specific beams those we are going to use for testing in the experiment since the results
for any experiments with different materials would be different. For the equation, we plug the
numbers into it and then we got the theoretical natural frequencies. Base on the theoretical part,
we can get the calculated natural frequencies which will be our standard conference when we
were doing the test.
The other goal in this experiment is understand the dynamical behavior of a vibrating
cantilever by observing the oscilloscope which is connected to the piezoelectric. For our
experiment, we test the first three natural frequencies by adjusting the gain of power amplifier and
also the frequency of the function generator. Since we test the first three natural frequencies,
there were three mode of the vibration of the two beams respectively. Considering one side of the
beam with active length, the beam could be vibrating as the part of a wave. For the first mode,
corresponding to the first natural frequencies, the side of the beam with active length is acting as
a quarter of a sine function wave with the original position stays as v=0 (the displacement at this
position is zero) throughout the whole time, and the end of the beam would reach the position
where v=vmax. Refer to the theoretical part with deriving the equation for the ω, the constant
coefficient for this mode is 3.516. Similarly, the beam would acting in another two ways which are
the three quarters of the sine wave and one and a quarter of the sine wave respectively with the
original point stays as v=0 in the second and the third mode of the vibration. The constant
coefficients for these two modes are 22.03 and 61.70. Since the beam is excited to vibrate in the
certain frequencies, we can tell that the reaction of the beam is quite different from each
frequencies we give to another. However, every time when we adjust the frequency to get close
to the natural frequencies, no matter the first, the second or the third one, the beam has the most
exciting reaction which is called resonance of the vibration. Also, the frequencies we give in these
situations are the testing natural frequencies we get.
Introduction
The natural frequency 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 of the natural frequencies for their material. Then
they can modify the amount for the specific measurements 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 natural of the dynamical behavior
of a vibrating cantilever by measuring the frequencies and observing the vibration mode shapes
of the cantilever beam whose material is aluminum and steel respectively. To calculate the natural
frequencies, we are using the derived equation to calculate the theoretical natural frequencies in
which the effects of rotatory inertial and of transverse shear deformation are neglected. The
gravity forces are also neglected since we can measure the displacement from the position of
static equilibrium of the beam. During the deriving process, we are using the moment equation
and the force equation first to the bending moment in related to the approximate curvature
displacement. Till we get a complex equation in the form V = B exp(λ 0x)for the motion equation.
Then, 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.
According to the equation of the natural frequencies, the natural frequencies are
determined by the dimensions we measure along with the Young’s modulus which is 10^6 and
30*10^6 for aluminum and steel respectively. As for the observation part, in the experiment, we
use the digital function generator to send the sinusoidal electric signal to the power amplifier, and
then the signal would be amplified by the power amplifier to carry out an enough power of to excite
the mini-shaker which is place in the middle of the beam. Also, we have a piezoelectric sensor
which is mounted near the fixed end of the beam to detect the vibration amplitude of the beam. It
will produce electricity when it is under dynamic loading since the beam is fixed to the mini-shaker
which is connect to the power amplifier and it will send out the signal to the display screen of the
oscilloscope. By observing the waves appear on the display screen, we can easily find the
approximate natural frequencies when the amplitude of the wave reaches the largest. 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.
Theory
Equation of Motion and its solution. When deriving the equation governing free undamped
vibration in flexure of beams it is assumed that vibration occurs in one of the principle planes of
beams. We can ignore the effects of rotatory inertia and of transverse shear deformation. From
Fig 4-1, we know that BC represents the centerline of the beam during vibration; the displacement
at any section x is shoed by v.
We can ignore gravity force through measuring the displacement from the position of static
equilibrium of the beam. The elements of length dx of the forces and the moments are shown
also in Fig.4-1; S and M are the shear force and bending moments at section x; the inertia;
force on the element is ρA 2v/ t2, where ρ is the density of the material of the beam and A
is its cross sectional area. Taking moments about the center line of the element, resolving for
forces in the Y direction.
M
𝑆𝑑𝑥 + 𝑀 − (𝑀 +
)=0
(4.2)
X
Or
𝑀
𝑆=
𝑋
And
𝑆
𝑥
=ρA
2𝑉
(the force equation) (4.3)
𝑡2
The relationship between bending moment and curvature and the approximate curvature
displacement relation, used in finding static deflections of beams, we have
M=-EI
2𝑉
(4.4)
𝑋2
Where E is Young’s modulus and I is the relevant second of moment of area of the crosssection(𝑏ℎ3 /12). Combing both equations (4.2) to (4.4) we can get
2
2𝑉
(-EI
𝑥
) = ρA
𝑋2
2x
(4.5)
t2
Equation (4.3) can be used for uniform and non-uniform beams; for the latter the flexural rigidity
EI and the mass per unit length ρA are functions of the coordinate x, For a beam of uniform
cross-section, Equation (4.5) becomes:
2
𝐸𝐼
2
𝑉
+ ρA
𝑋2
𝑉
=0
(4.6)
𝑡2
the beam’s motion equation. For free vibration, v(x, t) must be a harmonic function of time, i.e.,
v(x,t)=V(x)sin (wt+α)
(4.7)
Substituting equation (4.7) in (4.6), we have
d4 V
ρAw2
−
V
dx4
EI
=0
(4.8)
A solution of Equation (4.8) of the form V = Bexp(λ0 x) is satisfactory if
λ40 = ρAw 2 /EI . From the equation, we can find four roots λ0 = ±λ, λ0 = ±iλ, where λ0 =
(ρAw 2 /EI)0.25
So, we can get solution for general is
V = B1 sinλx + B2 cosλx + B3 sinhλx + B4 coshλx
(4.9)
End conditions. The four constants are determined from the conditions; the standard end
conditions are (a) simply supported or pinned, where the displacement is zero and the bending
moment is zero; (c) free, for which the bending moment and shear force are zero. In terms of
function V(x) these conditions for a uniform beam are:
(a) simply supported
d2 V
V=0 and dx2 = 0
(b) Clamped
dV
V = 0 and dx = 0
(c) Free
d2
dx2
= 0 and
d3 V
dx3
=0
(4.10)
(4.11)
(4.12)
The Natural Frequency of a Cantilever: With the origin at the fixed end, as in Fig. 4-2, and using
Equation (4.11) and (4.12), the end conditions are:
At x=0, V=0, i.e. 0=B2 + B4 .
At x=0, dV/dx=0, i.e. 0=λB1 + λB3.
d2 V
At x=1, dx2 = 0, i.e.,
2
0=-λ B1 sinλl − λ2 B2 cosλl + λ2 B3 coshλl+λ2 B4 coshλl
d3 V
At x=1, dx3 = 0, i.e.,
3
0=-λ B1 sinλl − λ3 B2 cosλl + λ3 B3 coshλl+λ4 B4 coshλl
Hence,
B1 (−sinλl − sinhλl) + B2 (−cosλl − coshλl) = 0
and
B1 (−cosλl − coshλl) + B2 (−sinλl − sinhλl) = 0
Eliminating B1 /B2 , the frequency equation becomes,
(sinℎ2 𝜆𝑙 − 𝑠𝑖𝑛2 𝜆𝑙)-(cosh𝜆𝑙 + 𝑐𝑜𝑠𝜆𝑙)2 =0
i.e. cos𝜆𝑙 𝑐𝑜𝑠ℎ𝜆𝑙 + 1 = 0
(4.13)
The first three modes of vibration, and the associated natural frequencies, are given in Fig 4.2
Theory. Determine the natural frequency of a uniform cantilever beam
The experiment is designed to make students have knowledge of the nature of the
dynamical behavior of a vibrating cantilever by measuring the natural frequencies and observing
the vibration mode shapes of the cantilever beam. A piezoelectric sensor mounted near the fixed
end of a uniform cantilever beam, an oscilloscope, a digital function generator, a power amplifier ,
a mini-shaker, and a strobe light are to be used in this experiment. A piezoelectric sensor is used
to detect the vibration amplitude of the uniform cantilever beam. A piezoelectric sensor is a kind
of ceramic that exhibits the piezoelectric effect. When the ceramic is under dynamic loading, it
will produce electricity, or if excited by an AC signal it will produce dynamic force.
A digital function generator is a digital is a digital display frequency and amplitude
adjustable function generator. The cantilever beam is excited by a mini-shaker. A sinusoidal
electric signal produced by the digital function generator is sent to the power amplified. The signal,
after amplified by the power amplifier, will carry enough power to excite the mini-shaker. The minishaker follows the external signal to vibrate. The vibration tip of the mini-shaker is placed under
the end of the beam. The output of the piezoelectric is connected to a oscilloscope. The deflection
of the cantilever is always higher when the beam is vibrating at one of the natural frequencies. As
a result the output from the piezoelectric is also higher at these frequencies. Watch the amplitude
of the output signal from the piezoelectric on the display screen of the oscilloscope as the vibration
frequency is the natural frequency. Use a strobe light to watch the vibration mode. A strobe light
is frequency adjustable flashing light. When the cantilever is excited at one of its natural frequency
of the strobe is adjusted to be same, twice, or a simple friction of the natural frequency of the
cantilever, the vibration modes can be seen clearly.
LIST OF EQUIPMENT: Sensors and Instruments
1) Piezoelectric sensor
2) Oscilloscope (pa-135)
3) Digital function generator (2003 Synthesized Function Generator)
4) Strobe
5) Vibration exciter and power amplifier
Procedure
1. Check and make sure all equipment is powered off. Follow the experimental setup and
instructions shown in Fig. 4-3 to complete the connections of the equipment. There are two
specimen of cantilever beams, one is made of steel and the other is made of aluminum in this
experiment.
2. Turn on the power amplifier: (1) switch the “attenuator” to the “o” position; (2) set “Current
limit” to “1.8 A RMS”; (3) set the “Gain” to “0 POSITION.” (Switch the knob all the way to the
left). Here turning the knob all the way to the left is making sure that the gain could be increased
slowly enough to make sure the amplitude of the wave would not be too large when the power
amplifier is turned on.
3. Since we already got every set up for the first part of the experiment (aluminum beam and
also the function generator) when we arrive at the lab, we do not need to worry about placing
the beam and also the setting for function generator, power amplifier. In addition, professor
already calculated the first three natural frequencies for us, so we do not need to do the
measurement for the aluminum beam.
4. Turn on all the equipment.
5. For setting the frequencies of the function generator. Set frequency to “10Hz”. And then the
display shows “Select Function”. Press “Freq” and we found the display reads “10Hz” (The
theoretical natural frequency is 21.12). For changing the frequency to observe the wave
displayed on the oscilloscope display screen, we press “+/-” until the cursor lies under the first 0
after the decimal. Then we slightly rotate the knob clockwise to increase the frequency in 0.1 Hz
increments and to adjust the frequency.
6. Slowly adjust the “Gain” of the power amplifier by turning it to the right which is increasing the
gain of the power amplifier, we observe the dynamic output of the piezoelectric on the screen of
oscilloscope. When the reading on the function generator screen of the frequency comes closer
to 20 Hz, the amplitude of the wave which is displayed on the oscilloscope about reaches the
maximum. At this point, we try to find out the first natural frequency by adjusting the frequency
of the function generator back and forth until the amplitude of the wave on the oscilloscope is
the maximum by observing. Then we record the number as the first natural frequency.
7. Set the “Gain” of the power amplifier back to “0” again. Repeat steps 5 and 6, but set the
starting frequency of the function generator as 120 Hz since we already got the theoretical value
of the second natural frequency which is 132.30 Hz. So, here we just start with a closer value of
the frequency which would save our time by skipping turning the knob for a long time. After
repeating the step 6, we record the frequency on the function generator screen when the
amplitude of the wave on oscilloscope reaches the maximum as the second natural frequencies.
8. For the third natural frequencies, we repeat the steps 5 and 6 again, but set the starting
frequency of the function generator as 340 since we already got the theoretical value of the
second natural frequency which is 370.6 Hz. For testing the third natural frequency, we found
out that the testing value is way smaller than the theoretical one which would not be considered
as within the error range. We tried to tighten the screw of on the mini-shaker to make sure the
beam is tightly enough to the mini-shaker. By doing another testing for the third natural
frequency, it turns out that the testing value get closer to the theoretical one and we think that
value would be considered within the range of error.
9. Redo the steps from 5 to 8 for another two times to get another two sets of the testing value.
10. Turn off all the equipment and take the aluminum beam off the mini-shaker.
11. After observing the vibration of the aluminum beam, we are about to switch gear to the steel
one. We measure the steel beam first for the width, length, weight, height and also the active
length after placing the beam on the mini-shaker (we made sure that we make the screws tightly
enough).
12. After measuring the size of the steel, we put the measurements into the program and then
run the program which would calculate the first natural frequencies for us.
13. Start to test the natural frequencies of the steel beam, we repeat the steps from 4 to 9 to get
9 values of the natural frequencies for the steel beam and record the value every time when we
get the maximum amplitude on the oscilloscope screen.
14. Turn off all the equipment.
EXPERIMENTAL RESULTS, ERROR ANALYSIS, STATISTICS, DISCUSSION:
PART I: Experimental Data for 1st, 2nd, 3rd Natural Frequencies for Aluminum & Steel
Table 1: Input Beam Dimensions & Material Properties  Theoretical Nat. Freqs.
Input Data
Aluminum
Young's Modulus (psi)
1.000E+07 3.000E+07
Beam Active Length (in)
13.750
13.750
Beam Weight (grams)
168.000
473.000
Beam Thickness (in)
0.125
0.125
Beam Width (in)
1.000
1.000
Beam Overall Length (in) 30.000
Steel
30.000
Caption: This table lists all of the essential parameters used in the three equations listed in the
‘Theory’ section that give the theoretical natural frequency for each mode of vibration. Those
values are tabulated accordingly in subsequent tables alongside experimental read-outs.
Table 2: Raw Experimental 1st, 2nd, 3rd Natural Frequencies for Al & St. and Theoretical Nat.
Freqs.
First Natural Frequency
Material
Trial 1
Frequency (Hz)
Trial 2
Frequency (Hz)
Trial 3
Frequency (Hz)
Theoretical Natural
Frequency (Hz)
Aluminum
21.300
21.200
21.200
21.120
Steel
21.700
21.400
21.550
21.800
Second Natural Frequency
Material
Trial 1
Frequency (Hz)
Trial 2
Frequency (Hz)
Trial 3
Frequency (Hz)
Theoretical Natural
Frequency (Hz)
Aluminum
130.300
130.200
130.100
132.320
Steel
132.200
133.906
133.053
136.590
Third Natural Frequency
Material
Trial 1
Frequency (Hz)
Trial 2
Trial 3
Frequency (Hz) Frequency (Hz)
Theoretical Natural
Frequency (Hz)
Aluminum
359.700
359.600
359.300
370.6
Steel
359.800
367.506
363.653
382.55
Caption: This joined tables list the 1st, 2nd, 3rd experimental natural frequencies for both
aluminum and steel and for each mode of vibration, the measurement was repeated three
times, which provides room for in-depth analysis for how aligned the theoretical predictions and
the experimental results are, ultimately allowing us to make some confident statements about
how this sort of testing can be used.
Error Analysis (For Results Part II – Tabulated Statistics & Discussion, Which Follow)
Sources for Error include those attributed to instrumentation, experimental setup, and
inaccuracy in measurements and readings. The instruments used in this experiment were a
power supply and amplifier, an electronic shaker, a piezoelectric material, and an oscilloscope.
We can consider errors arising in each instrument; both the power supply and amplifier could
have had inaccurate outputs for voltage corresponding to our selected settings, the electronic
shaker could have been vibrating with inconsistent frequency, the piezoelectric could have had
an responded with an inaccurate output voltage in response to vibration, and the oscilloscope
could have displayed inaccurate frequencies.
When considering sources of error arising from experimental setup, we consider mainly
those arising from improper beam fastening as well as those attributed to faulty electrical
connections. The beam was fastened via four socket-cap screws which were tightened with
Allen wrenches, and because the natural frequency of the cantilever is calculated assuming one
completely rigid end any inadequacies in tightening will have a direct impact on the measured
natural frequency. We noticed these effects during our trials, as we encountered relatively
inaccurate results for the natural frequencies for the steel beam after the first trial. When the
variance of the measured Steel frequencies was greater than that of the Aluminum frequencies,
our first consideration was that the beam was not tightened properly. We went and tightened the
bolts, and proceeded to find that the measured natural frequencies were more accurate. As
expected, our initial lower values for the measured natural frequency increased. Similarly, we
can consider losses in the electrical wiring because the Power Supply, Amplifier and Shaker.
Any losses will affect the resulting frequency measurements, and subsequently would increase
the relative error of our measurements from the theoretical values. Similarly, we can consider
errors that would arise from now properly fastening the piezoelectric material. Errors could arise
from both improper fastening of the piezoelectric to beam as well as improper electrical
connections between the piezoelectric and the oscilloscope. Any deficiencies in either of these
connections could give frequencies that vary from the actual frequency of the electrical
response to the dynamic load.
When considering the uncertainty of each of our measurements, we applied Statistical
Analysis to determine mean values, standard deviation, as well as mean deviation and
uncertainty for each natural frequency for each material. For Aluminum we found Population
Mean values for the first three natural frequencies to be 21.333 Hz, 130.200 Hz, and 359.333
Hz, respectively. For the standard deviation of each natural frequency measurement we found
values of 0.058 Hz, 0.100 Hz, and 0.208 Hz. These values gave us the basis for our same
uncertainty, which were calculated as 0.248 Hz, 0.4303 Hz, and 0.896 Hz. More importantly
these standard deviations gave us the basis for mean standard deviations and subsequently
mean uncertainties for our measurements, assuming a confidence interval of 95%. Our mean
uncertainties for aluminum were calculated as 21.333 Hz±0.143 Hz, 130.200 Hz ±0.248 Hz,
and 359.533 Hz ± 0.517 Hz.
Similar calculations were performed for the Steel, resulting in populations mean values
for the first three natural frequencies as 21.550 Hz, 133.05 Hz, and 363.653 Hz, respectively.
These correlated to Standard Deviations of 0.150 Hz, 0.853 Hz, and 3.853 Hz; which were
expectedly higher as a result of the different applied torques to the bolts fastening the beam.
The standard deviations rendered mean standard deviations and uncertainties of 0.087 Hz,
0.492 Hz, 2.225 Hz and 0.373 Hz, 2.119 Hz, and 9.572 Hz, respectively. These values gave us
the criteria for the Uncertainty of the Mean for the Steel for each natural frequency which
were calculated as 21.550 Hz±0.373 Hz, 133.053 Hz ±2.119 Hz, and 363.653 Hz ± 9.572 Hz.
These substantially larger uncertainties can be primarily attributed to the variance in torque
applied to the bolts fastening the beam, and emphasizes why the rigidity of the joint is critical
when accurately assessing natural frequencies.
Finally, we performed both Absolute and Relative error analyses for each measured
natural frequency versus the theoretical value for each material. For the first natural frequencies
we found maximum absolute errors of -0.180 for aluminum and 0.400 for steel , which resulted
in maximum relative errors of -0.009 for the aluminum and +0.018 for the steel. These relative
errors gave us the least accurate of the measurements, calculated as 1.009 for the aluminum
and 0.982 for the steel. For their successive natural frequencies we found greater Absolute
and Relative Errors which resulted in less accurate results, as indicated by minimum relative
accuracies of 0.983 for steel and 0.968 for steel at the second natural frequency and
0.970 for aluminum and 0.941 for steel at the third natural frequency. This drastic variance
in accuracy is what we have come to expect, at greater frequencies the margin for error
increases with the rate of a cycle. Similarly, we found our results were more accurate after we
re-tightened the bolts fastening the beam in place, and that degree of variance was also
dependent of the magnitude of the frequency in question.
RESULTS PART II - Tabulated Statistics
PART II (A): Arriving At Experimental Mean + Uncertainty Values For Al & St. at Each
Natural Frequency
Table 3: Population Mean, Sample Standard Deviation, Sample Uncertainty, Mean
Standard Deviation, Mean Uncertainty  Experimental Values w/ “Tolerance”
Population Mean
Material
First Natural Frequency
Population Mean
μ = (1/N)∑xi
Second Natural Frequency
Population Mean
μ = (1/N)∑xi
Third Natural Frequency
Population Mean
μ = (1/N)∑xi
(Hz)
(Hz)
(Hz)
Aluminum
21.233
130.200
359.533
Steel
21.550
133.053
363.653
Sample Standard Deviation
First Natural
Frequency Standard Deviation
Second Natural
Frequency Standard
Deviation
Sx = [(1/n-1)∑(xi-μ)2]1/2
Third Natural Frequency Standard
Deviation
Sx = [(1/n-1)∑(xi-μ)2]1/2
2 1/2
Sx = [(1/n-1)∑(xi-μ) ]
0.058
0.100
0.208
0.150
0.853
3.853
Sample Uncertainty
Let v = n-1 = 2; P = 95%; then t95 = 4.303
Material
First Natural
Frequency
μ = tv,P Sx
Second Natural
Frequency
μ = tv,P Sx
Third Natural
Frequency
μ = tv,P Sx
Aluminum
0.248
0.430
0.896
Steel
0.645
3.670
16.579
Mean Standard Deviation
First Natural
Frequency Mean Std.
Deviation
Second Natural
Frequency Mean Std.
Deviation
Third Natural
Frequency Mean Std.
Deviation
Material
Sx = Sx / n1/2
Sx = Sx / n1/2
Sx = Sx / n1/2
Aluminum
0.033
0.058
0.120
Steel
0.087
0.492
2.225
Third Natural
Frequency tv,P Sx
Mean Uncertainty
Let v = n-1 = 2; P = 95%; then t95 = 4.303
First Natural Frequency
Material
tv,P Sx
Second Natural
Frequency tv,P Sx
Aluminum
0.143
0.248
0.517
Steel
0.373
2.119
9.572
Caption: This series of tables list all of the intermediate statistical parameters required to arrive at
the final calculation of an experimental mean and uncertainty range within a standard 95%
confidence interval. All used equations and relations between tables are listed in the formulas
beneath the column headers.
PART II (B): Relative Accuracies of Experimental Nat. Freq. wrt. Theoretical Predictions
Table 4: Relative Error, Absolute Error, and Relative Accuracy
First Natural Frequency
Material
Theoretical
Frequency
(Hz)
Measured
Frequency
(Hz)
Absolute
Error ε
Relative
Error εr
Relative
Accuracy
A
Aluminum
21.120
21.300
-0.180
-0.009
1.009
Aluminum
21.120
21.200
-0.080
-0.004
1.004
Aluminum
21.120
21.200
-0.080
-0.004
1.004
Steel
21.800
21.700
0.100
0.005
0.995
Steel
21.800
21.400
0.400
0.018
0.982
Steel
21.800
21.550
0.250
0.011
0.989
Second Natural Frequency
Material
Theoretical
Frequency
(Hz)
Measured
Frequency
(Hz)
Absolute
Error ε
(Hz)
Relative
Error εr
(Hz)
Relative
Accuracy
A (Hz)
Aluminum
132.320
130.300
2.020
0.015
0.985
Aluminum
132.320
130.200
2.120
0.016
0.984
Aluminum
132.320
130.100
2.220
0.017
0.983
Steel
136.590
132.200
4.390
0.032
0.968
Steel
136.590
133.906
2.684
0.020
0.980
Steel
136.590
133.053
3.537
0.026
0.974
Third Natural Frequency
Material
Theoretical
Frequency
(Hz)
Measured
Frequency
(Hz)
Absolute
Error ε
(Hz)
Relative
Error εr
(Hz)
Relative
Accuracy A
(Hz)
Aluminum
370.6
359.700
10.900
0.029
0.971
Aluminum
370.6
359.600
11.000
0.030
0.970
Aluminum
370.6
359.300
11.300
0.030
0.970
Steel
382.55
359.800
22.750
0.059
0.941
Steel
382.55
367.506
15.044
0.039
0.961
Steel
382.55
363.653
18.897
0.049
0.951
Caption: This series of tables lists some final values to ascertain the relative accuracies in
experimental natural frequencies with respect to the theoretical predictions. There are some
notable trends in experimental accuracy as we go up to higher natural frequencies which will be
discussed down below.
FINAL RESULTS OF STATISTICAL ANALYSIS OF EXPERIMENTAL RESULTS:
The Uncertainty of the Mean – Processed Experimental Natural Frequencies
Aluminum:
First Natural Frequency 21.233 Hz ± 0.143 Hz (95%)
Second Natural Frequency 130.200 Hz ± 0.248 Hz (95%)
Third Natural Frequency 359.533 Hz ± 0.517 Hz (95%)
Steel:
First Natural Frequency 21.550 Hz ± 0.373 Hz (95%)
Second Natural Frequency 133.053 Hz ± 2.119 Hz (95%)
Third Natural Frequency 363.653 ± 9.572 (95%)
Theoretical Natural Frequencies:
Aluminum:
First Natural Frequency 21.120
Second Natural Frequency 132.320
Third Natural Frequency 370.6
Steel:
First Natural Frequency 21.800
Second Natural Frequency 136.590
Third Natural Frequency 382.55
Outlier Relative Accuracies:
Aluminum:
First Natural Frequency 1.009
Second Natural Frequency 0.985
Third Natural Frequency 0.971
Steel:
First Natural Frequency 0.995
Second Natural Frequency 0.980
Third Natural Frequency 0.961
DISCUSSION:
At the end of Part II – Tabulated Statistics, (just above this) there is a listing of all
theoretical, experimental, and relative accuracies for each beam at each natural frequency.
These fruits of our statistical analysis can be used to evaluate our theoretical model of beam
bending and where it might begin to fail as we approach higher frequencies as the boundary
condition assumptions made about what’s negligible become less valid.
As succinctly stated in the error analysis section, which we will repeat here, it is visible in
our data that as we ascend to higher natural frequencies, the relative error between the
theoretical natural frequencies and the experimental natural frequencies increases. This is more
visibly shown in the ‘Outlier Relative Accuracies’ where we have chosen to tabulate the most
extreme relative accuracies for discussion, where we can see a steady drop in relative
accuracy, which means our theoretical values are growing father apart from our experimental
values at higher natural frequencies. In aluminum, we go from 1.009, to 0.985, to 0.971, while
in steel we go from 0.995, to 0.980, to 0.961. Clearly, for both beams, accuracy of the
theoretical model to reality drops, however, by how much the relative accuracy drops from
natural frequency to natural frequency depends on the material.
The Young’s Modulus for Aluminum is 1.000E+0.7 while the Young’s Modulus for Steel
is 3.000E+0.7. Since the modulus is a measurement of how stiff the material is, it’s safe to say
that this difference in respective drops of relative accuracies between aluminum and steel result
from their differences in stiffness, as steel is 3x stiffer than aluminum.
In the error analysis, we agreed that with regard to the boundary conditions, at higher
natural frequencies, where there are more nodes of vibration and as such the assumed fixed
end of the beam comes closer to a site of major deflection and oscillatory wobble, the fixed end
boundary condition assumed in solving for our equations of natural frequency can become
invalid.
However, we can understand some more nuanced issues with beam bending theory,
stemming from simple beam bending under load and what general assumptions are made in
that case – these can partially explain why reality deviates from theory at higher, more excited
frequencies.
General Assumptions of Beam Bending Theory:
 Material of beam is homogenous and isotropic  constant E in all
 direction
 Young’s modulus is constant in compression and tension  to simplify
 analysis
 Transverse section which are planar before bending, after bending remain
 planar after bending.  Eliminate effects of strains in other direction
 Beam is initially straight and all longitudinal filaments bend in circular arcs
 simplify calculations
 Radius of curvature is large compared with dimension of cross sections =>
 simplify calculations
 Each layer of the beam is free to expand or contract  Otherwise they will
 generate additional internal stresses.
The first of these assumptions is generally assumed to be true for metals, that is
although on a molecular level, the metal may have several crystal grains in varying directions,
which may suggest it being anisotropic, on the macro-scale level, these tend to equal out. Of
course, all such beams tend to have treatment of some kind in manufacturing to obtain desired
mechanical properties, such as cold work which strengthens by plastic deformation, annealing
to increase ductility to make the metal easier to work with, precipitation hardening to increase
yield strength, and so on. Particularly for the heat treatments, a lot of recrystallization and
recovery can happen where a lot of defects disappear and result in larger crystals which tend to
lower yield strength. From a materials science perspective, this means that the Young’s
Modulus for steel can be quite variable considering how it’s processed and that for segments of
the beam the isotropic assumption may not hold up perfectly.
The second assumption is fairly simple to see, consider a brittle material like concrete,
its strain response to a tensile load is far different than its strain response to a compressive
load. While metals are not so extreme in comparison, the assumption that the Young’s Modulus
E is constant for compression and tension may not hold up perfectly especially considering that
it depends on isotropicity which we discussed earlier.
The remaining considerations include assumptions made when developing beam
bending, the most important of which to us is the idea that each layer of the beam is free to
expand or contract, thereby generating no additional internal stresses, such as from
transverse shear stress. While the beams we used in this experiment where fairly thin in
height with respect to the effective length of the beam, however, that height thickness matters
as stiffness (E) increases and the material becomes more resistant to bending and thus
internal transverse shear increases.
It’s simple to think of how a beam of rubber will bend rather freely with little transverse
shear. In comparison, visualizing a very stiff beam which is bending, the transverse shear is
quite high. We have to consider that while the shaker and piezoelectric provides a constant
frequency of vibration, the material itself wishes to reduce oscillatory motion and so its vibration
should have a transient exponentially decaying term which reflects that dampening. It’s
something that’s visible if we induce a displacement and watch the beam vibration dampened to
zero, but it’s present even under continuous excitation and is more prominent if the beam is very
stiff.
Returning to Exp. Vs Theor. Values – Explain Why Exp. Nat. Freq. < Theor. Val.
The reason why we bring up the general assumptions of beam bending theory and
the issues of some anisotropic aspects in the macroscopic level as well as how beam stiffness
leads to larger transverse shear and dampening resistance is to explain why the experimental
natural frequencies are consistently lower than the theoretical natural frequencies.
Looking at aluminum’s 2nd nat. freq. the theoretical value is 132.320 and its
experimental value is 130.200. At its 3rd nat. freq, the theoretical value is 370.6 and its
experimental value is 359.533. That’s a difference of almost 10 Hz, while comparing first
natural frequencies, the differences were around 0.1 Hz. If we look at steel, these
differences are even more pronounced, where in the 3rd natural freq., the theoretical is
363.653 Hz while the experimental is 382.55 Hz, a difference of about 19 Hz which is
substantial compared to its differences in 1st natural freq. of 0.7 Hz.
Bringing back our discussion of how stiffness (E) increases transverse shear and
dampening at higher frequencies and thinking of the beam under a strobe light will have a
profile matching the frequency it’s being excited at, we can think about how a natural
frequency for a very stiff material will need to be much lower. At low frequencies, the
nodes are far apart and the material’s displacement in the y direction is not significant so
we don’t have to think much about stiffness and transverse shear in deviating from
theoretical predictions. But at much higher frequencies where the beam will have several
nodes and the distance between nodes is shorter, the beam has to be displaced a distance y
up and down from left to right several times and that leads to a lot of transverse shear and
desire to dampen. So if a natural frequency will occur, it will occur at a lower frequency
than predicted for stiff materials – lower frequency less internal stresses.
Is This Difference in Exp & Theoretical Acceptable?
We can see for the first natural frequency, the deviation from theory is almost negligible
but for the second and third it’s higher especially for stiffer materials with higher E, as we can
see for steel. Taking a look at our relative accuracies: in aluminum, we go from 1.009, to 0.985,
to 0.971, while in steel we go from 0.995, to 0.980, to 0.961. Although we have a drop in
accuracy with increasing frequency, it’s important to point out that at its worst this inaccuracy
between theory and reality is 4% for steel’s 3rd natural frequency. It’s unlikely that we will design
a structure whose beams are expected to easily reach 4th or 5th natural frequencies in response
to sudden loads. Some oscillatory motion is good but too much and the structure does not have
much integrity. So if we look at our 4% for steel and consider that in real world design we will be
designing far thicker beams with larger dimensions, we can perhaps lower or at least limit our
inaccuracy to a number around 4%. So as far as modeling how a structure’s beams will bend
under load, our theory provides good predictions and only deviates when we deal with very stiff
materials with high induced transverse shears. In other words, for most metals and alloys with a
balance between ductility and yield strength, our model works extremely well.
CONCLUSION:
In summary, this experiment was conducted firstly to become adjusted to the
commonplace tools in measuring vibrations in beams and secondly to understand the dynamic
behavior of the beam, see what natural frequencies it has and by knowing so we can watch out
for frequencies that could be induced by natural forces, which if they come close to the natural
frequency, resonance is induced which can result in massive wobble until the beam snaps, as
we see in some historical examples of the first bridges.
In general from experimental data we see that the real-world values for natural
frequencies at each mode of vibration matches up fairly well with the theoretical predictions,
resulting in differences of less than 5% at most in the third natural frequency, beyond which it is
unlikely the metal will ever be excited in real life.
However it is important to note how this inaccuracy is coming about. From the boundary
conditions we see that the assumption of a rigid fixed end for the beam to oscillate about
becomes less tolerable as the frequency increases, resulting in more nodes of vibration and
bringing the fixed end close to a node of vibration. It’s hard to assume that the piece of metal
fixed by a press or whatever will not have some wobble, especially if it is simply bolted down.
We also discussed the increase in transverse shear stress at higher frequencies as the
stiffness of the material increases which can lead to a desire for the metal to dampen or resist
oscillation. There is also the issue with the isotropic assumption leading to slight differences
when we considering manufacturing treatments of metals. But the major result of this is the fact
that for a particular beam, the real natural frequency is consistently lower than the theoretically
predicted one, since the latter does not consider dampening from stiffness. In order to reach a
stable natural frequency, you’d have to go lower than the predicted, in order to reach an
acceptable wavelength of oscillation which the beam can handle well.
In general, these sorts of experiments and assumptions hold up reasonably well at low
frequencies with high cross sectional area and not excessively stiff materials. For very flexible
metals like aluminum, it works exceptionally well, for metals stiffer than steel, it can be
suggested that our model may not work so well. That being said because we will probably never
expect or want our beams to go beyond a 3rd natural frequency or at least not several natural
frequencies higher, our theoretical predictions coincide with a very small percentage of error
with the real natural frequencies, which means they can be reliably used for modeling and
engineering design.