BE309 FINAL PROJECT
VIBRATION ANALYSIS OF A SINUSOIDALLY DRIVEN SYSTEM
Group T1
Soe Y Ahn
Bethany Gallagher
Robert Pierson
Zack Shinar
Alexander Taich
Abstract
This experiment gave a thorough analysis of a system under a sinusoidal force. Using
light plastic bars of various materials (Acrylic, Polystyrene, HMWPE) and sound waves,
resonance phenomena was demonstrated. Two different transducers, piezoelectric accelerometer
and electret condenser microphone, were used to record the vibration of materials. Initially, the
impulse hammer was used to find the experimental fundamental frequencies of the materials, but
due to low Young’s moduli of the bars, they vibrated in low frequencies in addition to their
natural frequencies. This created excessive noise in the signal that interfered the determination of
the fundamental frequencies. Expected fundamental natural frequency for each bar was
calculated. For accelerometer the expected values for Acrylic, Polystyrene, HMWPE1, and
HMWPE2 were 12.52, 3.79, 5.91, and 6.36 (HZ) respectively. With the microphone these values
changed to 16.73, 6.52, 8.16, and 8.32 (Hz) as a result of lighter weight of the transducer.
Resonant frequencies were determined through plotting power vs. input frequency. When the
input frequency approached the resonant frequencies, the power of the vibration increased
dramatically. The accelerometer graphs displayed sharp peaks at resonant frequencies, but with
the microphone, continuous increase in power was observed. This is due to the fact that the
omnidirectional microphone recorded the sound waves from the speaker as well as the vibration
of the bars. Experimental resonant frequencies were found to be much greater than the expected
values. Therefore, they were attributed to harmonics of the system. Due to low power in low
frequency region, the transducers were not able to detect the fundamental frequencies. The
experimental values correlated with harmonics of the expected fundamental frequencies.
Deviations from the theoretical harmonics were greater for the accelerometer due to its heavy
mass. The average percent deviation for the accelerometer ranged from 14.99 to 34.43% and for
the microphone it ranged from 5.98 to 20.19 %. The HMWPE bars with different widths were
expected to have increase in natural frequency with increase in width. However, the trend was
unclear because of difficulty in determining which harmonics were recorded. Finally, a glass
slide cover with an expected fundamental frequency of 1.697KHz was subjected to input
frequency with high amplitude that was varied around the expected value in attempt to shatter the
glass slide cover. However, not enough energy was transferred to break it.
Background
Every object constantly vibrates at a certain frequency. That frequency is called the
natural frequency and depends on object’s composition and shape. The lowest such frequency is
called the fundamental frequency. Integer multiples of the fundamental frequency are called
harmonics. If a force is applied to an object with the same frequency as the object’s fundamental
frequency, then constructive interference is observed and the amplitude of the object’s vibration
increases. This increase in amplitude of vibration at natural frequency is called resonance. The
natural frequency can be theoretically calculated for any object by knowing it’s physical
properties. One way of generating a sinusoidal force is through a speaker. The force on the bar is
a product of the pressure generated by the waves sent by the speaker and the contact area on the
bar. The speaker creates regions of high and low pressure around the bar when generating the
sound by either moving a speaker membrane that is attached to an electromagnet (Figure 1).
When the membrane is pushed outwards pressure increases. The opposite is true when the
membrane is pulled back. This force, striking the bar at a certain frequency, causes sinusoidal
vibrations of the testing sample. An equation for such a wave is x(t)=sin(t) where  is the
circular frequency of the wave.
Figure 1: Schematic representation of sound waves created by a speaker
Motion of an oscillating system is governed by the following combination of forces:
F(total) = F(spring)+F(damping)+F(driving) and can be described by a following second order
linear differential equation:
a2y’’+a1y’+a0y = b0*x(t),
where a2 is the mass of the object, a1 is the expression for damping, a0 is the function of geometry
and elastic properties and b0 is amplitude of the sinusoidal force. No physical object can vibrate
at its theoretical natural frequency, but, instead, vibrates at a damped natural frequency which is
related to the natural frequency by the relationship:
d = n ( 1 – 2)1/2
where  is the damping factor. Undamped natural frequency can be determined from the
equation:
0=sqrt (a0/a2),
where, assuming that the object is a bar, a0 = (6EI)/(3Lz2 – z3) and a2 = LWT. I=WT3/12 is
the moment of inertia, is density, E is Young’s modulus of bar material and z is the distance
from fixed end of the bar to vibrating point of interest, L length of the bar, and T and W are
thickness and width, respectively.
Based on the 2nd order differential equation of motion, the frequency response of a system
with damping can be represented with the following equation that relates the vibration amplitude
with the frequency of the input force.
Amplitude 
F
( o  2 ) 2  4 2 2
2
F = amplitude of the input sinusoidal force
2 = damping constant/ mass = a1/a2
o = natural frequency
 = frequency of the input signal
(note that  = 2f where f is frequency in Hz)
By analyzing the above equation the frequency at which the maximum oscillation occurs can be
   o 2 2 2
found to be
If damping constant is 0,  will equal to o. At this frequency, the amplitude of the vibration
will maximum and the value can be represented with the following equation.
Amplitude(max) 
F
2  o  2
2
When damping constant approaches 0,  will also approach 0 and in turn, the amplitude will
approach infinity.
Figure 2. Relationship of amplitude vs. frequency with varying 
In this experiment, the samples are various types of polymers. Polymers today are often
incorporated in several bioapplications and implants. For example, PVC, acrylic, and
polyethylene are often utilized for ear parts, facial prosthesis, lung, kidney, and liver parts, and
bones and joints. Often, plastics are used in situations where corrosion is an extreme problem.
The physical characteristics of the polymers are of great importance when deciding the material
for the implant. Polymers are composed of long-chained molecules that are capable of assuming
many conformations through rotation of valence bonds. Plastics are found in an either
amorphous or semicrystalline state. They are never completely crystalline due to the lattice
defects that form disordered, amorphous regions. Primarily, plastics have lower elastic moduli as
compared to materials such as metals and ceramics. This low elastic modulus effects such
properties as the fracture strength of the material, the tensile strength, and the resonance
frequency of the material. The lower elastic modulus is directly related to the natural frequency
of the material and therefore plastics have significantly lower natural frequencies. Within the
lattice, the freedom of motion of the polymer chains is retained at the local level while the chain
interaction prevents large scale movement or flow.
The testing of the resonance frequency of materials is of vital importance in the field of
implants and bioengineering. For example, in the ear implant, the material is vulnerable to all the
frequencies of sound in the surrounding environment. If the sounds approached the natural
frequency of the implant material, deformation and damage could
occur due to the dramatic increase in the amplitude of the vibrations of the molecules. In
addition, these vibrations could cause injury to the delicate bones of the inner ear.
Picture 1: Middle ear implant
Also, implant materials would be subject to possible vibrations caused by such innocent
actions as riding in a car or airplane, possibly even the high impact of running. The vibrations
sent through the implants such as joint or bone could cause such violent increase in amplitude that
extensive deformation and possible cracking could occur. Therefore, the resonance frequency of
the material must be great enough to ensure that the activities the implant will be subject to will
not generate vibrations significant enough to equal it.
Accelerometer and Microphone
Vibrations of the bars are measured by a transducer which converts the amplitude of
motion into a proportional voltage or electrical output. Two transducers were used for each bar.
The first transducer was a piezoelectric accelerometer. The mechanism contains a piezoelectric
crystal that gives off a charge when pressure is applied. The accelerometer is equipped with a
piezoelectric crystal, a mass (to apply pressure in relation to the relative movement of the
accelerometer) and an FET circuit. The FET circuit converts the charge amplitude of the crystal
into a relative voltage or electrical signal. The other transducer was an omnidirectional condenser
microphone.
A microphone is a voltage generator that has an optimum load impedance. The
microphone used in our experiment had an impedance of 1 k, which is not considered “low
impedance” but is significantly lower than a “high impedance” microphone. The lower the level
of impedance allows longer distance transmission of the signal. The condenser microphone acts
as a capacitor that changes its capacitance in respect to pressure variations. The condenser, or
electret, is an insulator that has a quasi-permanent static charge trapped on it. (Schetgen) The DC
power supply acts as an FET impedance converter. The schematics of the microphone and the
power supply are near identical. A voltage is sent to the microphone and to an output via a
capacitor. The voltage sent to the microphone is then sent to the ground. The change in
capacitance of the condenser is accompanied by a change in voltage from the power supply of the
output signal.
Materials
1. Impulse Hammer
2. Speaker
3. Piezoelectric Accelerometer
4. Panasonic Electret Condenser Microphone
5. Function Generator
6. 100 Watt amplifier
7. Power supply
8. Bar support with clamp and clamping plate
9. Pyrex Glass Slide Cover
10. Bars of plastic cut from sheets: HMWPE, Acrylic, Polystyrene
Table 1: Physical Properties of Plastic Bars
Bar
Material
E, Mpa
Length,
z, cm
cm
Width,
Thickness,
cm
cm
1
Acrylic
3450
53
32.8
5.9
0.35
2
Polystyrene
3200
52
32.8
4.8
0.15
3
HMWPE
690
52
32.8
5.9
0.35
4
HMWPE
690
52
32.8
7.8
0.35
Procedure:
1) Expected values: Before starting testing the bars expected fundamental frequencies
were calculated using the equations given in background section. This values will be
compared to the experimental values.
2) Impluse Hammer: This part of our lab was done using an experiment nearly identical
to the previous vibration lab. An impulse hammer was used to find the resonant
frequency of the bars. Plastic bars were clamped to the table and accelerometer and
microphone were used as transducers. Attachment of the accelerometer was done
similar to the vibration lab done earlier this semester. Attachment of the
accelerometer was also be done similar to the previous vibration lab and the
microphone was also be attached directly on the bar. The fundamental frequency of
each bar will be determined using power spectra.
3) Sinusoidal force: the goal of this section was determine the natural frequency of the
bar under the sinusoidal force. A speaker was used to provided a sinusoidally varying
force. The speaker was mounted to a stool and fixated with clamps and tapes. And
the speaker was placed about 1~2cm away from the plastic bars. The input frequency
was generated with the virtual bench and ranged from 1Hz and included two resonant
frequencies. The geometry(lengths, width, and thickness) of the bars were varied
also. Compare the results to that of the impulse hammer and expected values.
4) Application: The final part of our experiment consisted of using a glass cover slide.
The resonant frequency of this slide was be determined by the equations mentioned
above. To subject the slide to maximum energy, the glass was fixed firmly directly in
front of the speaker. The method of fixation with a clamp reduced any possible
energy loss because vibrations were limited to the glass only. After determination
was complete, the amplitude of the input frequency was increased around the vicinity
of the expected natural frequency.
Figure 3. Setup for Sinusoidal Experimentation
RESULTS
A theoretical natural frequency of each bar with the accelerometer or microphone
attached was calculated. Then each bar was subjected to a sinusoidal input force of equal
amplitude. The input frequency ranged from 1Hz to the second harmonic frequency of each bar.
The FFT data for each test frequency were recorded. The power of the peak in the FFT plot
corresponding to each input frequency was then determined. The power vs. input frequency was
plotted to find the experimental natural frequency (see graph 1 and 2). Each peak in the graph
represents natural frequency of the bar. The lowest natural frequency is the fundamental
frequency. Table 3 shows the experimental natural frequencies that we found through plotting
graphs.
Table 2: Theoretical fundamental natural frequencies of the bars
Bar
Theoretical natural frequency
Theoretical natural frequency with
with accelerometer (Hz)
microphone (Hz)
1
12.52
16.73
2
3.79
6.52
3
5.91
8.16
4
6.36
8.32
Table 3: Experimental natural frequencies of the bars
Bar
Experimental natural frequencies Experimental natural frequencies
with accelerometer (Hz)
with microphone(Hz)
1
16.5, 51
34, 50
2
16.5, 65.5
64.5, 100.5
3
19.5, 41.5
65.5, 82.5
4
14.5, 40
80, 100
Graph1: Power vs. input frequency of bar 3 with the accelerometer attached
Graph 2: Power vs. input frequency of bar 3 with the microphone attached
Since the experimental resonant frequencies were much higher than the expected
fundamental natural frequencies, we calculated harmonics of the expected values and compared
them with our results for both the accelerometer and microphone. Table 4 presents the calculated
harmonics of natural frequencies. Only the harmonics that correlate to our experimental values
are included in the table. The deviations from the expected harmonics were determined and the
percent deviation for each bar is shown in Table 5.
Table 4: Theoretical harmonics of natural frequency
Bar
Expected harmonics with
Expected harmonics with
accelerometer (Hz)
microphone(Hz)
1
12.52, 50.09
33.46, 52.54
2
15.88, 67.49
63.92, 102.08
3
17.73, 41.37
65.28, 81.6
4
12.72, 38.16
83.2, 99.84
Table 5: Average percentage deviation from the theoretical natural frequencies
Bar
Avr % deviation from theoretical
harmonics with accelerometer
19.52
34.43
14.99
28.46
1
2
3
4
Avr deviation from theoretical harmonics with
microphone
5.98
7.67
7.48
20.19
Table 6. Natural Frequency variance with width of bar.
Bar
Width
Accelerometer (Hz)
Microphone
0.059
0.078
5.91
6.36
8.16
8.32
0.059
0.078
19.5,41.5
14.5, 40
65.5,82.5
80, 100
Calculated
HMWPE
HMWPE
Experimental
HMWPE
HMWPE
Discussion:
First we used the impulse hammer to determine the fundamental frequency of each bar.
However, peaks that correspond to natural frequencies could not be determined from power
spectra data. Low Young moduli of the plastic bars created unwanted multiple vibrations. These
vibrations made determination of the resonant frequency difficult by creating “noise” all along
the spectrum of frequencies. To circumvent this problem, it would be necessary to increase the
relativity resistivity to motion. This could be done through a) using a material of higher modulus
of elasticisty or b)increasing the thickness of the bar.
For the sinusoidally driven system, the theoretical fundamental natural frequencies were
first calculated as shown in Table 2. The theoretical fundamental natural frequencies are lower
for the bars with the accelerometer attached due to its heavier mass (90 grams). By increasing the
mass of the object, the resonant frequency will decrease. This can be shown through the
equation:
 = (a0/a2)
Since a2 is equal to the mass of the object, then as the mass increases from the addition of the
accelerometer the resonant frequency will decrease.
To determine the natural frequency for each bar, power vs. input frequency graphs were
plotted. With the microphone, we were able to obtain data that corresponded to expected
harmonics. Sharp peaks representing resonance phenomena were clearly visible. However, extra
“noise” made the entire graph appear less than ideal. Unlike with the accelerometer, we saw a
continuous increase in power with an increase in frequency with the microphone. We believe this
is due to the microphone’s detection of the speaker sound in addition to the vibration of the bar.
This coincides with the increased amplitude with increased frequency and could possibly show
much b better with some sort of filter to eliminate this affect. With added power from the speaker
sound, the peaks for resonant frequencies were more difficult to distinguish.
The microphone has an omni-directional pick up pattern, which means it detects sound
from all directions. This type of microphone is applicable to our experiment, because omnidirectional microphones are extremely effective at relatively close ranges. At close ranges only
pressure waves directly in front of the microphone are detected. The method of attaching the
microphone pick up face directly to the bar allows a thin space of air to couple the condenser to
the bar. When the microphone is air-coupled to the bar, the bar acts as a speaker when it begins
to vibrate. The vibrating bar creates pressure waves in the air-couple. Consequently, these
pressure waves are detected by the microphone.
The noisy data suggests the microphone responded to more than just the vibration of the
bar. The continued increase in output amplitude with respect to increase in frequency input
suggests the graphs contain the frequency response of the microphone. The microphone was
manufacture tested to contain a relatively constant response for the frequency range, 40Hz-4kHz,
with a minimum response to 20Hz. The increase of amplitude on the microphone graphs suggests
this increase in power. Energy increases for constant amplitude as frequency increases. This
phenomenon might have attributed to the continual increase in amplitude as frequency increased.
Also, the ability for the microphone to detect sound in multiple directions may have cause noise
from other sources (i.e., the speaker).
The nature of the air-couple is also questionable. The motion of the bar might have
possessed a horizontal motion in addition to the visually observable translational vibration. The
piezoelectric accelerometer only deciphers translational movement. This inherent filtering
mechanism may have been the reason why the data acquired from the accelerometer contained
distinct peaks and relatively no noise.
The data from the microphone did possess reproducible characteristics that are relevant to
the calculated data suggesting that the bars did influence the response of the microphone. Had we
had better foresight, a noise trial or an equivalent over-damped trial would have been performed.
This noise data could be used to create a filter for our experimental data.
The theoretical fundamental natural frequencies ranged from 5.91-16.73 Hz. However,
the experimental resonant frequencies determined from the graphs of power vs. input frequency
range from 14.5 – 80 Hz. An explanation for this deviation is that high frequency sound carries
higher energy than lower frequency sound of the same amplitude. Therefore, due the sensitivity
of the transducers, only the higher energy signals were detected. This coincides with the
published sensitivities of the accelerometer (lowest detectable frequency = 3 Hz) and
microphone(lowest detectable frequency = 20 Hz). Between the accelerometer and the
microphone, the setup using the microphone produced higher frequency results. This is due to the
microphone’s low sensitivity, which results in a higher frequency threshold for picking up
vibrations in the bar. The smaller magnitude of peaks in power spectrum of microphone data
compared to that of the accelerometer also shows that the microphone has lower sensitivity than
the accelerometer.
We calculated the theoretical harmonic frequencies for each bar and found the ones
closest to the experimental values (Table 4). Then we compared these harmonics to the
experimental results. The average percent deviation was determined (see Table 5) and the bars
with the microphone attached showed smaller deviation from the theoretical harmonics of the
natural frequency. The average percent deviation of the accelerometer ranged from 14.99 to
34.43% and that of the microphone ranged from 5.98 to 20.19 %. The smaller deviation for the
microphone is due to the microphone’s small mass and small physical dimensions. Attachment
of the microphone does not affect the natural frequency of the bar as much as the accelerometer
does. This would make it an advantageous instrument for performing that measurement; however
the decreased sensitivity of the microphone are a detriment. The microphone should be adequate
for measuring the natural frequencies of objects that posses higher natural frequency such as a
wine glass.
Although the accelerometer has better sensitivity compared to the microphone, the
addition of the accelerometer will affect the resonant frequency by changing the mass of the
object and the geometry of the object. The geometry of the object will also change from the
addition of the accelerometer. First of all, the moment of inertia will change. A second effect
will be the Young’s Modulus. The accelerometer has a significantly different Young’s Modulus
than the bar, and therefore the section of bar surrounded by the accelerometer will be affected.
All of these affect the resonant frequency of the combination bar/accelerometer.
The accelerometer is very accurate and precise in measurement due to its high sensitivity.
However, its relatively high mass (even greater than some of the plastic bars) alters the natural
frequency of the bars from the theoretical values. Therefore the accelerometer will be ideal for a
system with heavier mass and high Young’s modulus. The microphone is less accurate and less
precise because of its low sensitivity, but due to its small mass, the measured natural frequency
does not deviate much from the theoretical value. So the microphone is more ideal to measure
undamped natural frequency of a system with a light mass, low Young’s modulus and high
natural frequency. A transducer with a high sensitivity and a light mass will be an ideal apparatus
to measure the natural frequency of our system with high precision and accuracy.
Part of our hypothesis was the positive relationship between width of the bar and its
natural frequency. We tested two HMWPE bars with identical length and thickness and different
widths. The results of that test are presented in Table 6. Since, as discussed earlier, we could not
measure the fundamental natural frequency but only the harmonics it is impossible at this point to
correlate width and frequency data. It is apparent that there is a possibility from the table that
different harmonics were measured. Since we do not know exactly which harmonics it was that
we were able to detect, no width-based comparison can be performed.
Finally, a glass slide cover of expected natural frequency of 1.697 kHz was subjected to
input frequencies with large amplitude. This range of the input signal varied around the
fundamental frequency of the glass in an attempt to generate resonance. This resonance would
increase the molecular vibrations to the point of shattering the glass in the ideal case.
Unfortunately, significant energy was unable to be generated by the speaker to create resonance
in the glass.
Appendix
Here we wish to present a proposal for a modified version of the vibration analysis
laboratory that builds upon the data and skills acquired through our project. In addition to the
proposed lab, there are several additions that could make the experiment better.
The first addition that would definitely benefit the results of the lab is a light-weight
transducer with a sensitivity on the order of magnitude of the current accelerometer. The
proposed cost of this would be around $350 and is susceptible to damage. These are the two
biggest disadvantages to using this for the lab.
A second addition would be a high mass, high resonant frequency material. This, of
course, is counter-intuitive, because mass and resonant frequency are commonly inversely
proportional. However, such a material would greatly increase the results of the lab using the
following experiment. The high natural frequency would allow the microphone and
accelerometer to measure that frequency more accurately. The high mass would counter-act the
affects of heavy accelerometer.
A third addition that may improve the results is to increase the width of the bar.
Increasing the width will allow for the pressure generated by the speaker to affect the bar along a
greater surface area. This would increase the mass and may prove to be detrimental through the
decreased resonant frequency and greater force needed to move that mass.
A fourth addition is decreasing the microphone detection of the speaker. This could be
done through several deviations of the method we employed. First, the affix of the microphone
needs to seal the microphone to the bar. This will decrease transient air from entering the
microphone. Second, the surrounded noise (from speaker and lab) needs to be decreased. This
can be done by damping the microphone with light weight foam. Affix the microphone properly
to the bar will also help. A final improvement would to use a unidirectional microphone. This
would decrease unwanted noise and increase detection of the air pressure caused by the bar.
There are benefits and drawbacks of the new vibration lab. The trade-off is steering the
students to gain a maximum amount of knowledge in an allowable amount of time. These two
ideas are in direct conflict. Therefore, judgements must be made of which concept is most
beneficial to one’s education. In the new lab, the concepts of damping factor and geometry are
de-emphasized in favor of learning the use of an accelerometer versus a microphone, the
utilization of sinusoidal driven waves, and bars of materials more suitable to bioengineering
purposes.
Modified Experiment for BE309 Fall 1999
Vibration Analysis
OBJECTIVES
A. Experimental Goals
1. To measure the vibration response of metal bars to an impulsive load and determine
the sensitivity, damping factor, damped and undamped natural frequency of the bars;
2. To measure the response of a sinusoidal force on plastic bars and plot a frequency
versus relative power.
3. To measure the amplitude variance using a microphone versus an accelerometer as
the transducer.
B. Educational Goals
1. To understand the relationship between harmonics and fundamental frequency.
2. To understand the impulse response of structures in both the time and frequency
domains.
3. To gain a tangible representation of FFT graphs and utility in engineering purposes.
BACKGROUND
A. Biomedical Relevance and Importance
When subjected to appropriate mechanical forces, objects can vibrate at specific resonant
frequencies. Under appropriate conditions, sufficiently high amplitudes of the vibrating
wave can damage an object, even to the point of destruction. We are all familiar with the
example of a wineglass shattered by a soprano’s high frequency note. Resonant
vibrations can also adversely affect the mechanical integrity of the human body. A very
significant example today is the damage done to the inner ear by today’s widespread fad
to listen to music at extreme sound volumes. Since the cochlea of the inner ear is
sensitive to a very wide frequency range from about 20 Hz to at least 15 kHz, high
amplitude sound vibrations anywhere in that range can irreversibly damage the delicate
hair cells, particularly those having resonances at high frequencies. Vibration
considerations are also very important in ergonomics, the design of systems in which
humans interact with their environment, in applications ranging from office furniture
design to vehicular suspensions to wild amusement park rides.
B. Scientific Background
1. Wave Motion
Given a sinusoidal wave x(t), the wavelength  of the wave is defined as the distance, and
the period T as the time, between two successive identical points on the wave. The
velocity of the wave is defined as v =  f, where v is velocity and f is frequency. A good
model for understanding wave motion is a vibrating string, such as a violin string. If a
string of given length is fixed at one end and started vibrating at the other, a continuous
wave will travel down the string to the fixed end and be reflected back. As the string
continues to vibrate, waves will travel in both directions; the waves traveling down the
string will tend to destructively interfere with the reflected waves producing waves of
relatively low amplitude. However, if the string can be vibrated in such a fashion as to
assure that forward and reflected waves interfere constructively (the peaks and troughs of
both forward and reflected waves occur at the same time) one will observe large steady
peaks and troughs. These waves are called standing waves because they appear to be
standing still relative to the string. Areas of destructive interference (areas of no string
motion.) are called nodes, while areas of constructive interference (the peaks and trough
areas) are called antinodes. A string can have different frequencies of vibration that will
set up standing waves; such frequencies at which standing waves are produced are the
resonant frequencies of the string. The lowest frequency that causes a standing wave on
the string is known as the fundamental. Higher resonant frequencies are called
harmonics. For example, when the A string above middle C on a piano is correctly tuned
to the proper length and tension, the fundamental frequency will be 440 Hz when the key
is struck. Some of the energy will go into higher harmonics or overtones, for instance, at
880 Hz, or A above high C (one octave higher). Each higher harmonic above the first (or
fundamental) is an integral multiple of the fundamental. Thus when the string vibrates its
energy is distributed among a specified set of resonant frequencies. Every material object
behaves this way when given the appropriate energy.1 The specific vibration or motion
observed depends on the properties and geometry of the system, as well as the means of
excitation.
2. Second Order Systems
The type of vibrating system we shall consider, which is of great importance in modeling
many systems of bioengineering interest, is the second order system, so-called because
it can be described in terms of a second order ordinary differential equation. To
appreciate the significance of this model, we consider the various forces of importance in
doing work on the system and distributing the system energy:
1
Such phenomena can be best understood by considering their behavior in both the time and frequency
domains. If you have not yet done experiment #1, read the section about frequency domain in the
background section of that write-up.
1. To a system initially at rest, an external applied force does work on the system,
accelerates it and gives it a displacement y(t) from the rest position In mechanical
terms, the system experiences an inertial force which depends on its mass and
acceleration which in turn depends on the second derivative of y.
2. The system can lose energy as a result of dissipation due to frictional forces, also
called resistive or damping forces. Such forces tend to lower the displacement or
amplitude of vibration. Damping forces are 90o of phase with the displacement, which
means they depend on the velocity or first derivative of y. If there were no damping
forces, an object would continue to vibrate forever at constant amplitude and at a
frequency called the undamped natural frequency; of course, all real systems must
have some damping. However, the concept of an undamped system as an idealized
model is very useful.
3. In order to oscillate, a system must have the ability to store energy, or exchange the
energy form between kinetic and potential. Mechanical systems can store energy by
virtue of their elasticity, characterized by a spring constant or elastic modulus such as
Young’s modulus. The force applied to do the work for storing energy is directly
proportional to the displacement y itself.
To set a system into motion, some type of external force must be applied. Engineers use
three idealized models to characterize the input force applied. The response of all real
motions can be represented through combinations of these models, which can be
generally expressed in terms of a forcing function b0x(t). The models are:
(a) Impulse Response: In an ideal impulse, the time function x(t) is given by a delta
function (t), defined as
(t) = 0, t<0;
(t) = 1, t=0 ;
(t) = 0, t>0
This means the system is excited with an instantaneous force b0, and then left alone.
Whether it continuous to vibrate depends only on the properties of the system itself.
(b) Step Response: The ideal step is given by
x(t) = 0, t<0 ; x(t) = 1, t0
At time zero, a force is applied and maintained. The step and impulse are related, since
the derivative of the step function is obviously the impulse function.
(c) Sinusoidal or frequency response: The time dependent function is given by
x(t) = sin(2ft)
That is, a force that varies sinusoidally with frequency f is applied and maintained. Such
inputs are useful for studying system resonances, since if the forcing frequency f matches
one of the system’s natural undamped frequencies, the system will resonate.
To understand the motion of 2cd order systems quantitatively, we must analyze the forces
involved. Applying Newton’s second law of motion to the system, we get
a2
d 2 y (t )
dy (t )
 a1
 a 0 y (t )  b0 x (t )
2
dt
dt
(1)
The various coefficients in equation 1 reflect the force relationships discussed above. The
first term is the inertial force, so that a2 is equal to the mass. The second term is the
damping force; a1 will depend on the type of damping- e.g., for viscous damping it will
be proportional to the viscosity of the fluid surrounding the system. The third term is the
energy storage or elastic term; a0 is a function of the geometry and elastic properties; the
term on the right is the forcing function that starts and may contribute to maintaining the
motion. We can rewrite this equation in the form:
 D 2 2D 
 1 y (t )  Kx(t )
 2 
n
 n

(2)
where D is a shorthand notation for the nth order derivative of y with respect to t. The
parameters of this relation are defined by:
K
n 

b0
a0
a0
a2
a1
2 a0a2
the sensitivity of the system
the undamped natural frequency
the damping factor.
The damping factor  is the primary parameter for understanding second order response.
There are four values or ranges of , each with different significance, which can be
understood mathematically by examining the roots of the term in brackets in equation 2.
Since in this experiment we will only measure the system response to an impulse
function, we will only discuss the solution to these equations for an impulse.2
The ranges of  are:
2
The behavior of a vibrating system to a sinusoidal input force is much different than to an impulse, since
oscillations may be sustained even in highly damped systems because of the continuous input of energy.
However, we will not carry out such tests in this experiment. A further discussion of such phenomena is
given in the appended notes on “Basics of Vibration.”
(a) If =0, the roots are imaginary, and the system is undamped. The system response is
given by a sinusoidal function
A= Ao sin(nt)
The system will oscillate forever at constant amplitude A0 with a frequency equal to the
undamped natural frequency. Ideally the amplitude A0 is related to the impulse b0.
(b) If 0< > 1, the roots are complex, and the system is underdamped. This is the
situation of greatest interest in bioengineering. In response to an impulse, an
underdamped system will vibrate with amplitude given by:
A(t) = A0 e-nt sin (dt)
(3)
This very important relation states that the system will be observed to oscillate with
a frequency d, called the damped natural frequency, which is related to the
undamped natural frequency by the relation:
d = n ( 1 – 2)1/2
(4)
However, the amplitude of the oscillation decreases exponentially with time. If one
joined each oscillation peak (at which sin dt = 1) together to give a continuous curve, it
would be an exponential decay called an envelope. If one plotted the logarithm of the
envelope curve vs time, the slope would be negative and equal to n, and the intercept
would be equal to A0. In this case, A0 is the amplitude of the output at t=0. It is dependent
on the impulse strength b0, the damped natural frequency d, and the efficiency of
applying the impulse.
(c) If =1, the roots are real and both equal to –1. The system is said to be critically
damped. For this situation, the system does not oscillate. It responds to the impulse by
moving from the rest position, and then returns to the rest position asymptotically.
Critical damping can be useful in designing a system for moving people. Ergonomically,
one might wish to critically damp systems such as elevators and cars to avoid possible
injury from jerky motions. However, it would take some time to come to rest.
(d ) If >1, the roots are real and unequal. The system is overdamped. In this case there
are also no vibrations, and the system will come to rest more rapidly than when critically
damped. An overdamped car suspension system might produce a bouncy ride.
C. Equipment and Experimental Background
1. Accelerometers
An accelerometer is a transducer that detects the components of acceleration of an object
(or force if you know the mass). They can be placed on different parts of a moving object
and by suitable calibration and knowledge of their properties the vibration response of‘
the object is studied.. In this experiment we use a piezoelectric accelerometer to measure
the vibrations. In such transducers, a piezoelectric crystal is used to transducer motion
into an electrical potential.3 Because the accelerometer has mass, it will distort the natural
frequency of the bars being studied. Also the cables are delicate and particular attention
must be paid to avoid breaking them- we only have one unit on hand.
2. Microphone
A microphone is a voltage generator that has an optimum load impedance. The
microphone used in our experiment had an impedance of 1kohm, which is not considered
“low impedance”, but is significantly lower than a “high impedance” microphone. Lower
the level of impedance allows longer distance transmission of the signal. The condenser
microphone acts as a capacitor that changes its capacitance in respect to pressure
variations. The condenser, or electret, is an insulator that has a quasi-permanent static
charge trapped on it. (Schetgen) The DC power supply acts as an FET impedance
converter. The schematics of the microphone and the power supply are near identical. A
voltage is sent to the microphone and to an output via a capacitor. The voltage sent to the
microphone is then sent to ground. The change in capacitance of the condenser is
accompanied by a change in voltage from the power supply of the output signal.
3. Underdamped Impulse Response of Metal Bars
In this experiment, we will study the underdamped vibrations of metal bars that have
been given an impulse by striking with an impulse hammer, a fancy hammer equipped
with a transducer that can give an output proportional to the magnitude of the impulse.
For a bar of length L, width W, and thickness T vibrating either in air (air damped) or in
liquid (liquid damped), relationships can be derived between the various coefficients such
as:
I moment of inertia = WT3/12
mass density of bar material
E Young’s modulus of bar material
 viscosity of damping liquid
z distance from fixed end of the bar to vibrating point of interest; 0 z  L
a0 = (6EI)/(3Lz2 – z3)
a1 = C where C is a constant
3
A good introduction to such accelerometers, as well as related transducers which measure pressure and
force, is appended. Read this carefully, paying particular attention to the cautions given.
a2 = LWT or the mass of the bar (plus accelerometer!)
B0  measured amplitude of impulse actually applied
S  measured sensitivity = A0/B0
Using these relationships, one can calculate the undamped natural frequency n, and
parameters proportional to the damping factor . By observing the output of the impulse
hammer in the frequency domain, one gets a measured estimate of the impulse magnitude
B0. From the frequency domain output of the accelerometer, the damped natural
frequency can be taken as the value of frequency (in the neighborhood of the estimated
undamped natural frequency) with the highest energy. From the time domain output of
the accelerometer, a semilogarithmic regression of the amplitude peaks vs. time will give
an estimate of , n, and A0. By doing regression, you also get estimates of precision
from the confidence intervals. Use the experimental measure of sensitivity S, defined
above, rather than the theoretical value K given earlier.
PRELABORATORY WORK- must be completed prior to lab
A. Individual Work
1. Read the addendum on accelerometers;
2. If you have not yet done Exp. # 2, read the background about signal analysis and FFT
and the frequency domain analysis of signals;
B. Group Work
1. Become familiar with the Labview VI to be used and become familiar with those
features you will need to analyze your data.
2. Estimate the expected sensitivity and undamped natural frequency expected for the
various bars for the different setup conditions to be used.
3. Using the relations given in the scientific background for impulse response of an
underdamped second order system, plot the expected time domain response curve of
Bar#1 for damping factors of 0.05, 0.1, 0.2, and 0.5.
APPARATUS and EQUIPMENT
1. Accelerometer
2. Impulse Piston (Hammer)
3. 2 Accelerometer Power Supplies
4. Bar support (attached to lab bench) with clamp and clamping plate
5. Three test metal Bars with following dimensions and materials:
BAR
1
2
3
METAL
Al
Stainless
Al
L, mm
915
915
915
W, mm
38
38
51
T, mm
3
3
5
6. Cables
7. Large graduated cylinder
8. Damping solutions: water, 20 weight% sucrose, and carboxymethylcellulose
solution(same viscosity as sucrose solution and same density as water)
9. 40 Watt Speaker
10. Panasonic Electret Condenser Microphone
11. Labview Data Acquisition program BE shared\309\vib.VI
SPECIFIC PROCEDURES
A. Impulse Bar Tests
Mount each bar in turn in a vertical position in air using the bar mount clamping
apparatus. We suggest a 50 mm distance from the end of the bar to the floor as the default
position. Be consistent between bars to make comparison easier since all bars have the
accelerometer mount hole the same distance, 480 mm, from the bottom. Start with Bar#1
and designate it as the control bar, used for comparisons with the others.
To appreciate some of the unexpected problems with measuring vibrations, while
carrying out the trial below, we suggest that each group member try placing his/her hand
on the table top near the bar mount, and see if you can detect the table vibrating.
1. Remove the accelerometer from its storage box. Gently attach one end of the
recording/power cable to the accelerometer and the other end to the power
supply/amplifier. Connect the output of the power supply/amplifier to channel one of
Labview, and the impulse hammer to channel 2. Check that the power
supply/amplifier battery is good using the battery tester on the supply. Set the Lab
View acquisition program to sample 3000 samples a second for 2 seconds.
2. Securely fasten Bar#1 to the support bracket using the clamp and clamping plate
supplied. The effective vibrating length L of the bar is from the bottom of the
clamping plate to the lower end. Measure it accurately. The distance z is from the
same point to the accelerometer hole. Attach the accelerometer to the central hole of
the bar with the ¼ inch fine thread screws provided. Tap the bar lightly with your
hand to check that everything is working properly. Holding the impulse hammer at
right angles to the bar strike the bar lightly to set it into vibration. Practice hitting the
bar until you can get reasonable amplitude each time. It is not necessary to get the
same amplitude each time since amplitude can normalized to a selected value, and all
others tabulated as a percentage of that value. Do not hit too hard! The impulse
hammer transducer may be damaged (value $1000)! Repeat for 5 trials.
3. Repeat with the bar length reduced to 2/3 of its initial length.
4. Remove the accelerometer from the bar and mount it in the hole provided on top of
the clamping apparatus. Carry out a few trials measuring the vibrations picked up in
the table.
5. Repeat steps 2-3 with bars # 2 and 3 respectively.
6. Hang Bar#1 with string from the clamp instead of clamping. Try a few trials.
B. Sinusoidal Driven Force
1. Affix the plastic bars (acryllic, polysterene, and HMWPE) similar to the metal bar
section.
2. Adjust the speaker to a maximum of 2 inches from the bars. Record this distance for
use with all bars.
3. Record amplitude peaks using intervals of 10 Hz for frequencies. When you see the
amplitude begin to rise lower the interval to 1 Hz.
4. Repeat this for each of the three bars.
WORKUP OF RESULTS
1. For each trial determine the damped natural frequency, undamped natural frequency,
damping ratio, and sensitivity. Average the values for the multiple trials.
2. Tabulate the results for the various bars and air damping conditions. Quantitatively
explore the effect of bar geometry and material. Compare to expected values both in
absolute terms and as ratios to the control bar. What differences are significant?
3. Tabulate the results of using the sinusoidal force as the generator of motion. Pay
particular attention to how your data compares with the data calculated using the given
equations.
4. Report the values found for the table vibration and compare with the bar results.
5. Compare impulse response of clamped bar with hanging bar.
REFERENCE
Biriukov, S.V. Surface Acoustic Waves in Inhomogeneous Media.
Berlin, New York: Springer – Verlag, 1995.
Kleppner, Daniel, Robert J. Kolenkow. An Introduction to Mechanics.
New York: McGraw-Hill, Inc., 1973.
Fall 1998 Bioengineering 309 Lab manual
Febella, B., Fernandes, S. James, A., Musikabhunna, A.P. “ Study of Bivration Response
of a Sinusoidal Oscillating Driving Function” : BE309, 1997