MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
MECH 364, Experiment 1
Engine Vibration Due to Unbalanced Excitations
OBJECTIVES
1. To observe the effect of excitation frequency on vibration response, notably the large
response at a resonant frequency.
2. To learn the measurement of the damping ratio of a lightly damped structure.
3. To observe the occurrence of multiple resonant/natural frequencies and vibration modes.
4. To observe how different unbalance forces can affect the vibration response.
5. To observe how the measured vibration response depends on the measurement location.
6. To gain experience in the use of typical vibration measuring instrumentation such as a
piezoelectric accelerometer, signal conditioner (charge amplifier), optical encoder
(motion/vibration sensor), data acquisition system, and LabVIEW.
BACKGROUND
Rotating parts are common components of many mechanical devices. Typical examples are
motors, pumps, shafts pulleys, gears, and wheels. With most of these devices, some care is
usually taken to ensure that the centre of mass is located at the axis of rotation. That way, one can
avoid the centripetal (centrifugal) forces generated by the mass eccentricity—the rotation
component is “balanced.”
In practice, it is not possible to balance a rotor perfectly. Furthermore, many rotating devices
have reciprocating devices connected to them (e.g., automotive engine) which are difficult to
balance fully. Some harmonically-varying excitation forces always exist. The main frequency of
variation of these forces equals the rotation speed of the device. The resulting effect depends
mainly on the magnitude, frequency, location, and direction of the forces, and the vibration
response of the device (which depends on its dynamic characteristics including the flexible
mounts). Of particular concern is the case when the excitation frequency is close to the resonant
frequency of the device. The resulting vibration can become quite large, and may cause
malfunctions, damage, or even catastrophic failure.
This experiment is designed to illustrate some features of the response of vibrating systems to
excitation forces created by the mass eccentricity of a rotating component. This situation is
common in an engine with rotating components. In addition to “dynamic balancing” proper
engine mounts (i.e., spring-damper devices) may be used to avoid the resulting adverse vibrations
(not considered in this experiment, but quite relevant in the study of mechanical vibration).
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
The experimental apparatus consists of a box-shaped structure mounted on four spring supports.
This structure is an approximate model of an automobile engine secured on engine mounts. Outof-balance forces are created by eccentric masses attached to two counter-rotating shafts in the
structure, in two separate planes. In the experiments you will learn about harmonically excited
(forced) vibrations, a damping measurement method, multiple resonances, vibration mode shapes
and vibration measurement instrumentation.
The main steps in the experiment are:
1. Measure vibration response (acceleration amplitudes) over a range of excitation
frequencies that include the resonant frequency.
2. Measure damping ratio by the logarithmic decrement method.
3. Measure vibration response for a different excitation type.
4. Observe different modes of vibration in the system.
Note: The underlying theory of this experiment is given in Appendix A, at the end of this
document.
EQUIPMENT OVERVIEW
The diagram in Figure 1 schematically shows the main components of the experimental set-up.
1. Engine model: The box-shaped structure supported on four springs is an approximate
model of an automobile engine secured on engine mounts. A dashpot attached to the
centre of the box provides damping, representing, for example, the damping present in an
automobile shock absorber . The box includes two counter rotating shafts with four discs
carrying eccentric masses (two separate planes of rotating disks; each plane has two
counter-rotating eccentric masses). These eccentric masses generate the excitation force
through their centripetal forces, as described in Appendix A.
2. Drive motor: All four eccentric masses (on the four disks) are driven by a single motor,
using proper gearing. The motor is the stepper category which is moved in response to a
sequence of pulses generated by the drive hardware (indexer) inside the speed control
box. As the speed control knob is turned clockwise, the pulse rate increases thereby
increasing the motor speed, and vice versa.
3. Motor speed sensor: The motor shaft carries an optical encoder. It generates an optical
pulse per rotation. The pulse count gives the angle of rotation (angular position) since
each pulse corresponds to a rotation of 2π radians. The pulse rate gives the speed. Both
readings are displayed, as determined by the electronics inside the control box. Note:
This sensor generates a pulse by optical means when the projection of the rotating shaft
reaches the sensor head. Since pulses are counted and converted (encoded) into
measurements of rotation angle and angular speed, it is an encoder. It is not a
conventional optical encoder, however, where many pulses are generated per revolution.
This sensor is a "low-resolution" encoder where only one pulse is generated per
revolution.
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
4. Accelerometers: These devices contain a piezoelectric element (a crystal) that produces
an electric charge when subjected to a load (strain). When accelerated, an inertia load
(proportional to the acceleration) is generated on the element, which measures the local
acceleration. It has a magnetic base to allow easy and secure attachment at various points
on the vibrating box.
5. Signal conditioning electronics: The amplifier is a charge amplifier, which essential for
a piezoelectric sensor (accelerometer in the present experiment). The data acquisition
(DAQ) board has analog-to-digital conversion hardware. The software band-pass filter
and Fast Fourier Transform (FFT) implemented in the LabVIEW program allow accurate
measurement of the vibration (accelerometer) signal.
Figure 1: Schematic diagram of the experimental setup.
Experimental Setup
Prepare the experimental equipment as follows:
Accelerometers: Attach the accelerometer to location A (for in-phase excitation) or B (for outof-phase excitation) of the upper surface of the vibrating box.
Important Note: The accelerometer cables are very fragile. When moving an accelerometer,
hold the body of the device without touching the cable. Avoid dropping an accelerometer or
knocking it against a hard surface. Don’t allow the magnetic mounts to pull the accelerometer to
the mounting points as this will result in a “knock.” Instead slide the magnetic mount into place.
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
Signal conditioning electronics: Turn on the motor and signal conditioning and computer power
supplies and start LabVIEW on the PC.
EXPERIMENTS
“In-Phase” Vibration Response Measurements
1. Make sure that all eccentric masses are securely fasted and are in phase; i.e. they all reach
their uppermost positions at the same time. Attach the accelerometer to point “A.”
2. Adjust the rotation speed to 19 Hz (1140RPM). Read the rotation speed from the front
panel (in rpm) and convert it to Hertz (Hz). The indicator on the front panel of the
apparatus reads two values: a count of the number of rotations (a C shows on the display)
and the rate of rotation (an R shows on the display)
3. Set the filter frequency on the computer screen equal to the rotation speed. The signal
from the accelerometer contains higher harmonics and noise. The filter extracts the
response component whose frequency is equal to the excitation frequency. Observe and
compare the pre-filtered and post-filtered accelerometer signals. Also observe the signal
from the optical sensor. You may want to down load a time trace including the prefiltered signal, post-filtered signal, together with the tachometer signal for your lab report.
4. Record the vibration amplitude and phase indicated on the computer screen. The phase
here represents the phase difference between the upward zero-crossing of the post-filtered
acceleration signal and the front edge of the tachometer signal in the response graph. The
TA will show you how to identify the relationship between the signal from the tachometer
and the position of the discs. Remember that Figure A3 shows the phase angle between
displacement and the force. In this experiment we are measuring acceleration.
5. Repeat steps 2-3-4 for rotation speeds at 1 Hz intervals in the rage 19 Hz down to 4 Hz
record your readings in a table. You may notice several resonances. Do some extra
measurements around the resonances to reveal the details of the vibration response near
these important frequencies, especially around 10 Hz for System A, and 11Hz for System
B.
6. On separate graphs, plot vibration amplitude and phase measured at point A vs. rotation
speed. Compare that with Figure A3.
Damping Ratio Estimation
The damping ratio z is a measure of how quickly energy is dissipated from a vibration system. In
reality, all physical systems possess some amount of damping. If one looks at a time domain trace
of a simple spring-mass-damper oscillator, as obtained by displacing the mass through an initial
distance yo and releasing, the “free vibration” decay would look similar to Figure 2. (Note: We
are now dealing with a homogeneous version of the general equation of motion (A1). The right
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
hand side of (A1) will be zero for this situation since there is no driving force). The decay is
governed by an equation of the form (see equation (A6)):
y (t ) = yo e -zwnt cos wd t
(1)
where the “damped natural frequency” wd is influenced by the damping and is given by:
wd = wn 1 - z 2
(2)
As the damping ratio increases the decay also increases. An example is the automobile shock
absorbers where the damping ratio ranges between 0.1 (old shock absorber, relatively ineffective
at damping out vehicle oscillations) and 0.4 (new and relatively effective at damping out
suspension oscillations). In many applications it is desirable to have a large damping ratio to
quickly eliminate undesirable motions. In other applications, high damping ratios translate into
high energy dissipation (losses) and hence low efficiencies. Also, this can create thermal
problems due to the heat generated by the damper. Therefore, measuring the damping ratio is
critical to understanding the behaviour of a dynamic mechanical system.
t1
t1 +
2p r
wd
Figure 2: Free vibration with damping (from Reference [1]).
Logarithmic Decrement Method
In this experiment we use the logarithmic decrement method to estimate the damping ratio.
7. Switch the LabVIEW program to “Damping Factor” and adjust the rotation speed to read
the maximum amplitude near the 10 Hz resonance for System A, and 11 Hz resonance for
System B.
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
8. Switch off the motor power supply. The motor will stop rapidly, and the vibration
amplitude will diminish exponentially. Your trace should look approximately like Figure
2. Save the trace. The first part of the trace is the response when the motor is still running
although slowing. Focus on the later part of the trace which should correspond to the
acceleration history when the motor has completely stopped.
Identify a reasonably clean integer number (r) of cycles of decay from the time response trace
2p r
(Figure 2) and measure the start and end amplitudes (Ai and Ai+r) in this time interval
.
wd
Hence, from equation (1) we have:
é
ù
æ wn
ö
Ai + r
z
ê
= expç -z
2pr ÷ = exp 2pr ú
Ai
è wd
ø
êë 1 - z 2
úû
(3)
By definition, the logarithmic decrement d is given by (per unit cycle):
d=
1 æ Ai ö
2pz
lnç
÷=
r è Ai + r ø
1-z 2
(4)
Then using equation (4) we can estimate the damping ratio from the logarithmic decrement as:
z =
1
1 + ( 2p / d )
2
»
d
2p
(5)
“Out-of-Phase” Vibration Response Measurements
9. Unscrew the two eccentric masses on the front side of the vibrating box. Insert and
securely fasten them so that they are out of phase relative to the masses on the other side;
i.e., the two masses are at their lowest positions when the other two are at their highest
positions. Move the accelerometer to the front, location B.
10. Repeat steps 2-3-4-5-6.
11. Repeat steps 7-8, but adjust the motor speed to achieve the maximum response near 6 Hz
resonance for System A, and 7 Hz resonance for System B. Remember the accelerometer
should be at location B. Save the time trace.
Things to bring to the lab
You should prepare your own
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
(1) Ruler or tape measure: To measure the distance between the front and the back disks,
distance between front and back springs, and the eccentric radius e.
(2) Protractor: The technician has adjusted the rotating shaft such that the phase is measured
between the upward zero crossing of the acceleration signal and the front edge of the
signal from the tachometer. The TA will show you how to relate the tachometer signal to
the angular position of the discs.
(3) Flash disk: To store a data file of your experimental result.
(4) Digital camera or camera-enabled phone: To take pictures of the experimental set-up from
different views. You should include some photographs to help you describe the
experimental set-up in your report.
(5) Calculator: Always bring a calculator to laboratories in case you need to do some simple
calculations.
Observe the experimental set-up carefully
The experimental set-up and the instrumentation are designed and assembled by others. When
you work in your station you should observe the experimental set-up closely. Take pictures.
Where are the motor, gear train, motion encoder, accelerometer, dashpot, charge amplifier, motor
drive hardware, and data acquisition hardware? Find out what the model number of the
accelerometer, motor, spring, and the dashpot are. Measure the important lengths between front
and back disks, and between springs. Record the total mass and each of the eccentric mass values
(they will be provided for you). Find out the calibration constant which allows you to calculate
the acceleration signal from the voltage signal. Check the phase angle set-up between the
eccentric mass and the shaft. What is the highest speed the motor can go? Where are the cables
connected to? Prepare to describe these details in your report with pictures you take.
Report Requirements
You need to submit your report in a “ready-to-publish” professional format. Each group will
submit one report. The report should include the following:
(1) Abstract: Summarize the methods used and your experimental results, the comparison
between the measured and the calculated natural frequencies. How good or how bad is the
comparison? Keep this to around 300 words or less.
(2) Introduction: Feel free to copy and edit the BACKGROUND section of this hand out,
preferably adding your own personal experience on the vibration due to rotating
unbalanced machinery. Include a brief description of what you were asked to do in this
lab.
(3) Methods: Describe in detail what you see in the experimental set-up, the parameters you
measured, or the information that you obtained with the help of the TA. Identify the
important components in the pictures you take. Provide a schematic diagram of your own,
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
similar to the Fig. 1 of this hand-out but with additional detail. Describe the experimental
procedure for the in-phase, out-of-phase, and damping ratio experiments.
(4) Results for In-Phase vibration response measurements: This section and the next section
are the essential parts of the report. Describe this in as much detail as possible. They
should include (a) the time trace used to estimate the damping ratio, (b) a typical prefiltered signal, post-filtered signal, and a tachometer signal, (c) the displacement
amplitude versus rotation speed graph and phase versus rotation speed graph. (d)
Compare these experimental results with the theoretical results based on equations (A7)
and (A8). Try to reproduce one curve of Figure A3 (for your damping value) for both
experimental and theoretical approaches. For instance, how does the measured resonant
speed compare to the theoretical one? Note that in Eq. (A7) we have used the
displacement amplitude instead of acceleration. If the displacement amplitude of a
harmonic motion is yo, then the acceleration amplitude should be w 2 y0 .
Note: y (t ) = yo sin wt
y& (t ) = w yo cos wt
&&
y (t ) = -w 2 yo sin wt .
Therefore, if acceleration is used as the vibration measurement (not displacement), then to
compare with the experimental results you should multiply both sides of Eq. (A7) by w 2 .
Remember, we do not expect perfect agreement between the experiment and theory. Also,
LabVIEW may double-integrate the acceleration signal, thereby giving the displacement
signal (in that case, the above adjustment should not be made). Check this.
(5) Out-of-phase vibration response measurement: Do the same as in section (4) above ,
except that we are unable to compare the experimental result with the theoretical one
because we do not know the moment of inertia of the apparatus. However, can you use
the resonant frequency of this experiment to estimate the moment of inertia of the block?
Better yet, can you derive the theoretical prediction on the amplitude versus rotation
speed relation as we have for in-phase excitation in Appendix A? If you are willing to go
the extra mile and try to explore this, you will receive bonus points.
(6) Discussion and conclusion: Summarize the measurements that you made. Describe the
physical meaning of the resonant frequencies. Explain why the damping ratio for the 7 Hz
resonance is much smaller than that for the 11 Hz resonance. Try to identify the source of
error causing the discrepancy between the experiment and the theory. Comment on any
anomalous results that you observe. Suggest any changes or follow-up measurements that
could shed light on additional aspects of the vibration measurement.
How we will grade your reports
You will have two weeks following your experiment to prepare your report. We will grade your
report based on the following guidelines. (1) Format and tidiness (10%). (2) Experimental set-up
description (10%). (3) Experimental procedure and results (in-phase 20%, out-of-phase 20%). (4)
Comparison with theoretical results (20%). (5) Discussion and conclusion (20%).
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
Appendix A. RELEVANT THEORY
The experimental apparatus is represented schematically in Figure A1. Each of the counterrotating discs incorporates an unbalance represented by an eccentric masses, mo with eccentricity
e.
Referring to Figure A2:
(a) Unbalance in a rotating disk due to mass eccentricity
(b) Rotating vector (phasor) of centrifugal force generated by unbalance.
(reproduced from [1]) we note that each rotating eccentric mass (me) will generate a (radial)
centripetal acceleration of meeω2.
Here:
ω = constant spinning velocity of the eccentric mass
m
me
O
me
e
e
ω
ω
O
y
b
Figure A1: Schematic diagram of the experimental apparatus. In the configuration shown,
a harmonic vertical force is generated which causes vertical vibrations in the system. Each
of the counter-rotating discs is turning at a constant angular speed ω. Note: The magnitude
of each eccentric mass is me and the mass of the entire system is m.
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
(a)
(b)
y
Eccentricity = e
v
(vector e
or complex
number in
the x-y plane)
fo sinwt
w
q = wt
v
e
Center (Axis)
of Rotation
Figure A2:
mew2e
x = real axis
y = imaginary axis
Center
of Mass
x
fo coswt
fo = mew2e
= centrifugal forcing
amplitude
(a) Unbalance in a rotating disk due to mass eccentricity
(b) Rotating vector (phasor) of centrifugal force generated by unbalance.
Referring to Figure A1 and Figure A2, and writing Newton’s second law in the vertical (y)
direction we have:
my&& = -ky - by& + 4mew 2e sin wt Þ
my&& + by& + ky = 4mew 2e sin wt Þ
&&
y + 2zwn y& + wn 2 y = uow 2 sin w t
(A1)
Equation (A1) is the general governing equation of motion. The number “4” in Eq. (1) accounts
for the number of eccentric masses in total.
Note:
4m
uo = e e = normalized (static) amplitude of the forcing excitation
(A2)
m
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
ω = constant spinning velocity of the eccentric mass = frequency of the forcing excitation
wn =
k
= undamped natural frequency
m
(A3)
k
b2
- 2 = 1 - z 2 wn = damped natural frequency
m 4m
b
b
z =
=
= damping ratio
2mwn 2 km
wd =
(A4)
(A5)
Equation (A1) is a non-homogeneous (i.e., there is a non-zero forcing function) linear second
order ordinary differential equation. The general solution to such an equation consists of two
parts:
1. Homogeneous Solution: Solution to the homogeneous equation: my&& + by& + ky = 0 , which
represents the “natural” unforced response of the system (depends on the initial
conditions)
2. Particular Solution: One solution of the non-homogeneous equation (A1), assuming zero
initial conditions.
The total solution then takes the form:
y (t ) = e -zwnt [ A cos wd t + B sin wd t ] + yo sin(wt - f )
1) Homogeneous Solution
(transient response)
2) Particular Solution
(steady-state solution)
(A6)
Here, A and B are constants to be determined from initial conditions, and f is the phase angle
between the forcing excitation and the response (see below).
Due to the “exponential decay” term (i.e., negative exponent in time) in the homogeneous
solution, this part of the total response will eventually decay to zero (after several motion cycles,
depending on the level of damping in the system). For this reason it is called the transient
response.
Eventually, only the particular solution part in (A6) will remain in the overall response. It
represents the “steady-state” vibration response of the system. In this experiment we are
interested in the steady state vibration.
By definition, the particular solution satisfies the non-homogeneous equation (A1). Hence, by
substituting only the particular solution part of (A6) into (A1) and equating the coefficients on
the two sides of the equation we can obtain the following results:
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
yo =
w2
(wn 2 - w 2 )2 + (2zwnw ) 2
uo =
r2
(1 - r 2 )2 + (2z r )2
uo
(A7)
and
tan f =
where:
r=
2zwn
2z r
=
2
2
wn - w
1 - r2
(A8)
w
= nondimensionalized excitation frequency
wn
These equations produce the familiar magnitude and phase curves with respect to frequency (also
known as Bode plots, when log scales are used for magnitude and frequency) as shown in Figure
A3.
Note the resonant frequency ωr which is the frequency where the magnitude of the frequency
response is a maximum. From Figure A3 one can observe the role of the damping ratio on
reducing the overall magnification response near the natural frequency as well as shifting the
frequency of peak responses away (to the right) from the undamped natural frequency w n (i.e., r
=1). The curves also indicate that resonance occurs at a phase angle of approximately 90°.
Note: Strictly speaking it is when the excitation frequency ω equals the undamped natural
frequency that the phase angle f is 90°. However, for low damping (i.e., for small ζ compared to
1) the damped natural frequency wd and the resonant frequency w r both become approximately
equal to the undamped natural frequency w n .
At this phase angle, for example, the eccentric masses me would be at the angular position wt =
180° (90° to from the vertical) as the displacement of the system, y reaches its maximum vertical
position (see Figure A2).
COMMENT
Equation (A7) represent a frequency transfer function (frequency response function) which is
somewhat different from that of the traditional damped oscillator (mass-spring-damper system) in
that in the numerator it has a quadratic term in excitation frequency. The quadratic term in
frequency corresponds to a double integration. Consequently, the frequency at which the
magnitude peaks (i.e., the resonant frequency) has a slightly different analytical expression (as
obtained by differentiating the expression (A7), equating it to zero, and solving for the frequency
ω or frequency ratio r). Some pertinent results are given in Table A1 (adopted from Table 3.5 of
Reference [1]). The corresponding frequency response function curves are given in Figure 3.15 of
Reference [1]. For low damping (i.e., for small ζ compared to 1), all these cases give practically
the same results, as clear from Table A1.
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
yo
uo
Figure A3: Plots of equations (A7) and (A8) for forced vibration due to rotating unbalance.
(Adopted from [2])
Table A1: Some Practical Frequency Response Functions (FRFs) and Their Peaks
(Adopted from [1], Table 3.5).
Response/Excitation
Displacement/Force
Frequency Response
Function (Normalized)
Magnitude
1
Normalized Resonant
Frequency
(rp)
Peak Magnitude
(Normalized)
1 - 2z 2
1
1
2z 1 - z 2
1
2z
r2
1
1
(1 - r 2 )2 + (2z r )2
1 - 2z 2
2z 1 - z 2
(1 - r ) + (2z r )
2 2
2
r
Velocity/Force
(1 - r 2 ) 2 + (2z r ) 2
Acceleration/Force
Note: Any of the frequency response functions given in Table A1, whether determined
analytically or experimentally, may be conveniently converted into one of the other forms simply
by multiplying or dividing by the excitation frequency.
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
DAMPING MEASUREMENT
Once a frequency response function is determined experimentally, in the case of low damping
(i.e., when a prominent resonant peak is present) the corresponding damping ratio ζ can be
estimated by the Q-factor method (or half-power method), as described in reference [1].
Specifically, consider Figure A4, which gives the Displacement/Force frequency response
function of a mass-spring-damper (damped oscillator system)—Also see the first entry of Table
A1.
The Q-factor, which measures the sharpness of resonant peak, is defined by:
Q-factor =
Where:
wn
1
=
Dw 2z
(A9)
Dw = half-power bandwidth = frequency interval where the signal magnitude drops to
1/ 2 of the peak value (or, signal power drops to ½ of the peak value)
w n = undamped natural frequency ≈ resonant frequency w r
ζ = damping ratio
Amplification
1
2z
1
2 2z
1
0
w1 wn w2
Frequency w
Dw
Figure A4: The Q-factor (or Half-Power) Method of Damping Measurement
(Adopted from Reference [1]).
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MECH 364 Engine Vibration Due to Unbalanced Excitations (Experiment 1)
Note: This method is not used for damping measurement in the present experiment, particularly
in view of the difficulty of getting accurate frequency response measurements in the
neighbourhood of a resonant peak. Instead, the logarithmic decrement method, in the time
domain, is used.
Some parameter values for the experimental setups are given in Table A2.
Table A2: Parameter Values for the Engine Experiment.
System A (with perspex glass) System B (with metal guards)
Mass of Accelerometer
34.554 lbs or 15.673Kg
33.054 lbs or 14.993 Kg
Mass of Damper Piston
0.250 lbs or 113.4 g
0.222 lbs or 100 g
Spring Constant (Each)
82 lb/in = 14365.3 N/m
106.5 lb/in = 18277 N/m
Eccentric Mass
15.2 g
15.2 g
2
Calibration Constant for
98 mV/g (g = 9.81 m/s )
99 mV/g (g = 9.81 m/s2)
Accelerometer
REFERENCES
1. de Silva, C.W., Vibration—Fundamentals and Practice, 2nd Edition, CRC
Press/Taylor&Francis, Boca Raton, FL, 2007.
2. Thomson, W.T., Theory of Vibration with Applications, Prentice Hall, Englewood Cliffs, NJ,
1988.
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