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Wilby, Kevin
Modal Characterization of a Thin Flat Plate in the Free-Free Condition
with Non-Contact Particle Velocity Measurements
by
Kevin Wilby
A Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING in MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December 2012
i
CONTENTS
1. Introduction.............................................................................................................. - 1 2. Background .............................................................................................................. - 2 3. Theory/Methodology ............................................................................................... - 5 3.1
Theoretical Prediction of Natural Frequencies .............................................. - 6 3.1.1
Analytical Prediction of Natural Frequencies .................................... - 6 -
3.1.2
Finite Element Prediction of Natural Frequencies ............................. - 8 -
4. Results and Discussion .......................................................................................... - 10 4.1
Test Setup ..................................................................................................... - 10 -
4.2
Test Results .................................................................................................. - 13 -
5. Conclusions............................................................................................................ - 16 6. References.............................................................................................................. - 17 Appendix A – Analytical Prediction Spreadsheet ....................................................... - 18 Appendix B – Microflown Scan and Paint Inputs ....................................................... - 19 -
ii
LIST OF TABLES
Table 1 Input Parameters to Theoretical Predictions ..................................................... - 6 Table 2 Analytically Derived Natural Frequencies ....................................................... - 7 Table 3 Calculated Natural Frequencies from FEA ...................................................... - 8 Table 4 Measured Natural Frequencies from White Noise Excitation ........................ - 14 Table 5 Comparison of Calculated and Measured Natural Frequencies ..................... - 16 -
iii
LIST OF FIGURES
Figure 1. Picture of the Two Platinum Wires in the Microflown from [3] ................... - 3 Figure 2. Directionality of the Microflown PU Probe from [7] .................................... - 3 Figure 3. Comparison of the Point response from an Accelerometer, Laser Vibrometer
and a Microflown Sensor [3] ......................................................................................... - 5 Figure 4. Predicted Mode Shapes from [10] .................................................................. - 7 Figure 5. FEA Mode Shapes (1st to 6th Modes From Upper Left to Lower Right) ....... - 9 Figure 6. Picture Showing Test Setup ......................................................................... - 10 Figure 7. Shaker Test Line Diagram............................................................................ - 11 Figure 8. Typical Spherical Piston Calibration Setup ([13]) ....................................... - 12 Figure 9. Measured Velocity Response from a White Noise Excitation ..................... - 13 Figure 10. PU Mode Shapes (1st to 6th Modes from Upper Left to Lower Right) ....... - 15 -
iv
LIST OF SYMBOLS
i
horizontal half waves
j
vertical half waves
natural frequency (i horizontal and j vertical half waves) [Hz]
dimensionless frequency parameter (i horizontal and j vertical half waves)
length of the plate (meters)
modulus of elasticity (pascals)
ℎ
thickness of the plate (meters)
mass per unit area (kg/meter^2)
poisson’s ratio
displacement (meters)
scale factor (meters)
sqrt(-1)
radian frequency (radians)
time (seconds)
velocity (meters/second)
acceleration (meters/second^2)
π
pi
v
LIST OF ACRONYMS
PU
Pressure/Velocity
FEA
Finite Element Analysis
G
Acceleration due to Gravity (meters/second^2)
vi
ABSTRACT
Structural acoustics is the study of the vibrations, and their propagation through solid objects. There are many industries that use vibratory analysis to provide insight to the
key strengths and weaknesses of structures. This project describes a novel type of
measurement (the Microflown sensor) used to perform a structural characterization of a
flat plate and applies both analytical and finite element tools to validate the test results.
The Microflown is a sensor that directly measures the particle velocity of the air moving
past it. The Microflown measurement system has been shown in this project to accurately capture the dynamic response of a structure.
vii
1. Introduction
Structural acoustics is the study of the vibrations, and their propagation through solid
objects. Understanding the structural dynamics of a structure can provide valuable
information on its strengths and weaknesses for any given application. Vibratory analysis is constantly being performed in industry from assessing structural integrity in the
aerospace industry [1] to minimizing the acoustic response of a structure in the automotive industry [2]. [1] discusses a computational technique for assessing the integrity of
structures. This is a very powerful diagnostic tool that can provide indications of machine and structure wear by monitoring the vibrations. Signals related to rotational
harmonics for example can indicate bearing failure before more damage is done to a
machine. [2] discusses the design of a car muffler that uses microperforations to increase
damping and in turn better absorb the sound waves. Understanding the structural response of the many moving parts in a car is very important as the perceived sound of a
car can be a major part of its value.
Generally speaking all structural acoustic analyses can be broken down into three
types: simplistic analytical, detailed finite element and full or scale model testing.
Typically analytical models are used wherever possible because of their ease of use,
however, most real-life problems are much more complex then these models can replicate. Finite element models give the ability to analyze a structure well before it’s
actually built. However, these models take longer to build and analyze than the analytical models and need a lot of validation for the results to be used for complex structures.
Scale model testing can be used as insight into the dynamics of the structure as well as
input into the finite element models, however can prove to be costly and may not fully
replicate the full scale structure. Lastly, full scale testing is typically the best to analyze a
structure, however, does not have the ability to be analyzed until the system is built, and
is usually the most costly solution. This project focuses on the testing aspect of structural
acoustics and uses both analytical and finite element tools to validate the test results.
-1-
2. Background
Modal testing is a way to quantify the dynamic response of a structure. Typically, this is
done by exciting the structure (possibly with a calibrated force hammer or a shaker) and
then measuring its response (usually with accelerometers). The procedure for these types
of tests can vary significantly based on the application and in some cases can involve
long setup times and/or long test duration. Accelerometers can be mounted in many
ways (e.g. threading them into a hole, glue, wax) which depending on the test can be
time consuming and potentially affect the structural response of the system (with the
added mass of the accelerometer itself). Non-contact vibration measurements have the
advantage that there is no influence on the structural response. This type of measurement
can be performed with a sensor called Microflown [3]. The Microflown system performs
non-contact measurements that are relatively easy to generate for both quick look and
more detailed analyses.
The Microflown is a sensor that directly measures the particle velocity of the air
moving past it by the use of two heated platinum wires (Figure 1). As outlined in [3], the
Microflown simply measures the difference in voltage caused by the heat of each
platinum wire that is lost due to convection. By using two wires instead of one, and
looking at the difference in voltage between these two wires, a highly directive measurement of velocity can be performed. The PU probe is a sensor package that combines
this Microflown technology with a small 1/8” pressure sensor. This allows for a measurement of intensity and acoustic impedance (beyond just the velocity and pressure
response). Additionally, the highly directional nature of the velocity probe (Figure 2)
provides the ability for sound source localization. This project focuses on the velocity
measurements, however, intensity and impedance measurements are discussed in more
detail in [4] and [5], respectively and sound source localization is discussed in [6] as
well as [3].
-2-
Figure 1. Picture of the Two Platinum Wires in the Microflown from [3]
Figure 2. Directionality of the Microflown PU Probe from [7]
The Microflown PU probe will be used in this project to measure the surface
velocity of a thin flat plate in the free-free condition. The Microflown Scan and Paint
system (which uses a camera in conjunction with the PU probe) was chosen for this
-3-
project because it provides a visualization of the velocity profile of the plate. These
measurements will be compared to both analytical and finite element methods to validate
the measurement technique. The free-free boundary condition was chosen because the
analytical and finite element methods should be very accurate for such a well researched
case. Specific applications to this project are any industries that perform modal analyses
(such as the aerospace and automotive industries).
-4-
3. Theory/Methodology
The Microflown is a sensor that directly measures the particle velocity of the air moving
past it and will be used in this project and compared to both analytical and finite element
prediction methods. This comparison involves the assumption that the particle velocity,
when measured in the very near field (where the air can be approximated as incompressible), coincides with the surface velocity of a structure. This assumption is explained in
detail in [8]. Additionally, this technique has been validated in [3] by comparing the
point response of an accelerometer, laser vibrometer and a Microflown probe. Figure 3
shows the test setup from [3] as well as the comparison of the three sensors.
Figure 3. Comparison of the Point response from an Accelerometer, Laser Vibrometer and a
Microflown Sensor [3]
The structure in this report will be a thin plate in the free-free condition. As defined
in [9], a thin plate is one with a thickness that is less than 1/10 the minimum lateral plate
dimension. The plate used in this project is made of steel and has the dimensions of 8in
by 12in by ½ in. The free-free condition of a thin flat plate was used to provide a condition that has been well investigated and the accuracy of the prediction methods is known.
The test setup is a plate hung with rope, excited by two shakers, and measured with the
Microflown Scan and Paint system. This method is validated using [9] and [10], using
theory of vibrations of plates and by creating the finite element model and solving for
the natural frequencies.
-5-
3.1 Theoretical Prediction of Natural Frequencies
In this section, the natural frequencies of the thin flat plate are calculated by a textbook
analytical method and by a finite element method. The parameters (geometry and
material properties) used in each prediction method are outlined in Table 1.
Table 1 Input Parameters to Theoretical Predictions
Parameter
length
width
thickness
Young's Modulus
Poisson's Ratio
Density
3.1.1
Value
0.3048
0.2032
0.0127
2.0684E+11
0.3
7854.17
Unit
m
m
m
Pa
kg/m^3
Analytical Prediction of Natural Frequencies
The vibration of a thin plate in the free-free condition has been well investigated. Some
examples include [11] which calculates the mode shapes of a free-free plate that is
antisymmetric with respect to the x and y axis but symmetric about the diagonals and
[12] which experimentally measures the frequencies and mode shapes for square brass
plates. [9] provides a method for determining the natural frequencies of the first six
modes of rectangular plates for all 21 possible combinations of the three elementary
boundary conditions on the four edges of the plates (free, simply supported and
clamped). The general assumptions of this method are:
•
The plates are flat and have constant thickness
•
The plates are composed of a homogeneous, linear elastic, isotropic material
•
The plates are thin (defined as the thickness of the plate is less than 1/10 the
minimal lateral plate dimension)
•
The plates deform through flexural deformation
•
The in-plane load on the plates is zero
Equation (1) provides an equation for calculating the natural frequency of the mode
sequence for a plate:
\
=
! "
; i=1,2,3… ; j=1,2,3…
-6-
(1)
Where fij is the natural frequency (in hertz) of the mode sequence of i horizontal half
waves and j vertical half waves, λij is a dimensionless frequency parameter, a is the
length of the plate, E is the modulus of elasticity, h is the thickness of the plate, γ is the
mass per unit area, and ν is Poisson’s ratio. The values for the square of λij were taken
from [9] for a length of width ratio of 1.5. The Table 1 geometry and material properties
were used in this analysis. Table 2 and Figure 4 show each calculated frequency and
mode shape, respectively. Note the different naming conventions used in [9] and [10] (i
and j vs. m and n) and that modes 1-6 using the [10] naming convention would be would
be [1,1], [2,0], [2,1], [0,3], [3,0], and [1,2].
Table 2 Analytically Derived Natural Frequencies
Mode
[i,j]
1 [2,2]
2 [3,1]
3 [3,2]
4 [1,3]
5 [4,1]
6 [2,3]
0
0
1
2
3
m
 n
1
Frequency
(Hz)
680.1
729.8
1576.2
1699.2
1966.4
2280.3
2
3
Figure 4. Predicted Mode Shapes from [10]
-7-
4
3.1.2
Finite Element Prediction of Natural Frequencies
A finite element analysis (FEA) was performed to predict the undamped natural frequencies and mode shapes of the thin flat plate in the free-free conditions. The model
was built in HYPERMESH v11.0 and the Table 1 geometry and material properties were
used. Once the model was built, the natural frequencies were calculated using the mass
and stiffness of each element by importing the model into VIBES-SABER. Table 3
shows the calculated natural frequencies for each mode and Figure 5 shows a snapshot
of the mode shape animation. Notice how each of the mode shapes correspond with the
analytical mode shapes identified in Figure 4.
Table 3 Calculated Natural Frequencies from FEA
Mode
[i,j]
1 [2,2]
2 [3,1]
3 [3,2]
4 [1,3]
5 [4,1]
6 [2,3]
Frequency
(Hz)
678.1
722.3
1563.0
1680.8
1943.7
2258.7
-8-
Figure 5. FEA Mode Shapes (1st to 6th Modes From Upper Left to Lower Right)
-9-
4. Results and Discussion
Section 3 provided a prediction of the natural frequencies of the plate by two different
methods (analytical and finite element). The focus of this project is to use the
Microflown particle velocity sensor to measure the natural frequencies of this plate. The
non-contact measurements that are possible with the Microflown system provide an easy
to use tool that can quickly provide a detailed analysis of the dynamics of a structure.
4.1 Test Setup
A thin flat plate made of steel and with the geometry of 8in by 12in by1/2in was used in
this project. The plate had two holes drilled in it to provide a means for mounting it. The
plate was hung by a rope to approximate a free-free condition. Two shakers were attached to the structure and excited in parallel to reduce the risk of the shakers placement
on an anti-node for any given mode shape. The locations of the shakers can be seen in
Figure 6 and are generally indicated on each of the figures in Section 4.2 that show test
results. Figure 7 shows a line diagram of the test setup, where the entire test was run
using the Microflown Scan and Paint hardware and software.
Shakers
Figure 6. Picture Showing Test Setup
- 10 -
PU Pressure
PU Velocity
LEMO
LEMO
Signal
Conditioner
BNC
BNC
Camera
USB
Microflown
Digitizer
USB
CPU
BNC
Microdot
Shaker 1
Microdot
Shaker 2
Figure 7. Shaker Test Line Diagram
The Microflown Scan and Paint hardware includes a PU probe, a signal conditioner
(MSFC-4) and a digitizer (Scout 422). The PU probe simultaneously measures pressure
and velocity. The signal conditioner receives both the pressure and velocity measurements, and provides a calibrated correction to the signal. The digitizer has two functions:
it receives the pressure and velocity signals and sends them to the Scan and Paint software and provides the shakers with the excitation voltage. A camera is also used to
record a video of the PU probe measurement.
The Microflown Scan and Paint software takes in the pressure and velocity measurements and the video feed from the camera. The camera is set up perpendicular to the
surface of interest. The probe is then scanned by the user over the entire surface of
interest while maintaining a consistent distance (roughly 1-2 inches from the surface)
and aiming the probe perpendicular to the surface. Both the audio (pressure and velocity)
and video signals are recorded while the probe is scanned. Although there is no actual
painting occurring, the Microflown Scan and Paint software will follow the probe from
the video signal (typically by following a brightly colored object attached to the probe)
- 11 -
and synchronize it with the audio signal. Once the signals are synchronized, the measurements (such as those in Section 4.2) can be processed and analyzed.
The Microflown PU probe requires a special spherical piston calibrator to perform
an official calibration of the sensor (this is performed once a year). The procedure for
this test is generally to excite the calibrator with a known signal, and use a reference
microphone to relate this signal to both pressure and particle velocity [13]. Figure 8
shows the spherical piston calibrator, the Microflown probe and the reference microphone in a typical setup.
Spherical Piston
Reference Microphone
PU Probe
Figure 8. Typical Spherical Piston Calibration Setup ([13])
It is also good practice to do a quick bench top calibration to confirm that the probe
is measuring correctly and that the appropriate sensitivity is being applied. This is done
by taking a standard accelerometer calibrator that provides a G of acceleration (120dB
re: µG) on a flat vibrating surface at 100Hz. By using Equation (2), one can easily relate
acceleration response and velocity measurements.
'
Since: &'%(
'
Since: &'%(
= #
=
=
=
=−
$%
#
- 12 -
$%
#
$%
=
(2)
Where x is displacement, A is a scale factor, j is the sqrt(-1), ω is the radian frequency, t is time, ẋ is velocity, and ẍ is acceleration. Therefore, by placing the PU probe
directly above the calibrator, and converting the velocity reading by simply multiplying
by ω or 2*π*100Hz, the probe can be quickly calibrated to the signal.
4.2 Test Results
The two shakers that were mounted to the plate were first excited with white noise and
scanned using the Scan and Paint system. Figure 9 shows the measured velocity response of the plate (an averaged linear spectrum of each location). Each of the peaks
shown represents a natural frequency of the plate. Table 4 shows the measured natural
frequency for each of the first 6 modes (i.e. the first 6 peak frequencies).
Total Average Linear Spectrum of U
PVL (re 50e-9 m/s) [dB]
70
60
50
40
30
20
10
10
2
3
10
4
[Hz]
10
Figure 9. Measured Velocity Response from a White Noise Excitation
- 13 -
Table 4 Measured Natural Frequencies from White Noise Excitation
Mode
[i,j]
1 [2,2]
2 [3,1]
3 [3,2]
4 [1,3]
5 [4,1]
6 [2,3]
Frequency
(Hz)
684.8
726.3
1505
1682
1913
2207
Once the natural frequencies were identified with the white noise, the shakers were
then excited with a sine wave at each of the frequencies of the peaks shown in Figure 9.
Figure 10 shows the measured velocity profile for each of the first 6 modes. The colors
on the graph correspond to the areas where the velocity is highest (red) and lowest
(blue). Note that these measurements are taken as a steady state (i.e. the velocity profile
is the absolute value of the velocity) as opposed to Figure 5 which is the instantaneous
modal distribution of the plate. With this in mind, Figure 10 can be directly compared to
the modal patterns in Figure 5 (e.g. 1st mode in the upper left of each figure should show
the same pattern). The expected mode shapes (from both the FEA and the analytical
solutions outlined in Section 3) as well as the shaker locations are also shown on each
graph in Figure 10.
The first, second and third modes (upper left, upper right and center left of Figure
10) show very clean mode shapes that closely follow the expected patterns. The fourth,
fifth and sixth modes (center right, lower left and lower right of Figure 10), resemble the
expected patterns, however they are not as clean. This could potentially be due to many
factors. The shakers were randomly placed in an attempt to reduce the risk that they will
both be placed on a nodal pattern line. However, it can be seen in the 5th and 6th modes
(lower left and lower right of Figure 10) that the shaker locations appear to be located
very close to a nodal line. Using a larger plate or more shakers could potentially have
further reduced the risk of placing a shaker on a nodal line. Additionally, as can be seen
in the 4th mode (center right of Figure 10), while the center nodal line can be easily seen,
the edges do not show the detailed character of the expected pattern. This could potentially be improved upon by using a larger plate and/or using a finer grid in the Scan and
Paint software.
- 14 -
Shaker Location
Shaker Location
Nodal Lines
Nodal Lines
Shaker Location
Shaker Location
Nodal Lines
Nodal Lines
Shaker Location
Shaker Location
Nodal Lines
Nodal Lines
Figure 10. PU Mode Shapes (1st to 6th Modes from Upper Left to Lower Right)
- 15 -
5. Conclusions
The Microflown measurement system has shown the capability to accurately measure
the dynamic response of a structure. Table 5 shows the comparison of each of the three
methods outlined in this project. The test was shown to be within 5% of either prediction
method for each of the first 6 modes. The Microflown measurements were also shown to
generally follow the mode shapes that were predicted from both the analytical and finite
element methods. Improvements upon the test setup could potentially have shown a
cleaner measurement for the higher modes (as well as the modes above the 6th).
Table 5 Comparison of Calculated and Measured Natural Frequencies
Mode
[i,j]
1 [2,2]
2 [3,1]
3 [3,2]
4 [1,3]
5 [4,1]
6 [2,3]
Analytical
Frequency
(Hz)
680.1
729.8
1576.2
1699.2
1966.4
2280.3
FEA
Frequency
(Hz)
678.1
722.3
1563.0
1680.8
1943.7
2258.7
Test
Frequency
(Hz)
684.8
726.3
1505.0
1682.0
1913.0
2207.0
Three different types of modal analyses were used in this project: analytical, finite
element and test methods. Depending on the application, anywhere from one to all three
of these methods may need to be used to fully understand the structure. Each of the
analyses has their own strengths and weaknesses. The Microflown particle velocity
sensor has advanced measurement technology. The results of this project have implications in any of the many industries that currently perform vibratory analysis. Non contact
velocity measurements can be used to define the dynamics of a structure, provide insight
to proper accelerometer placement and/or provide important input into analytical or
finite element models.
- 16 -
6. References
[1]
Shepherd, M; Conlon, S. “Structural Intensity Modeling and Simulations for
Damage Detection”. ASME Journal of Vibration and Acoustics. October 2012.
Volume 134 Issue 5
[2]
Allam, S; “A New Type of Muffler Based on Microperforated Tubes”. ASME
Journal of Vibration and Acoustics. June 2011. Volume 133 Issue 3
[3]
de
Bree,
H-E.
“The
Microflown,
E-Book”.
<http:/www.microflown.com/library/books>. Microflown Technologies. Arnhem, The Netherlands. 2012.
[4]
Tijs, E; Nejade, A; de Bree, H-E. “Verification of P-U Intensity Calculation”.
Microflown Technologies
[5]
Tijs, E; de Bree, H-E. “Recent Developments Free Field PU Impedance Technique”. Microflown Technologies.
[6]
Wind, J; de Bree, H-E. “Sound Source Localization and Sound Mapping Using a
PU Sensor Array”. Microflown Technologies.
[7]
“Microflown Technologies – PU Regular Probe Manual”. V1.0 2009-03
[8]
de Bree, H-E. “The Very Near Field Theory, Simulations and Measurements of
Sound Pressure and Particle Velocity in the Very Near Field”. University of
Twente. International Conference on Sound and Vibration. June 2004
[9]
Blevins, R. “Formulas for Natural Frequency and Mode Shape”. Krieger Publishing Company. Copyright 1979
[10]
Waller, M. “Vibration of Free Rectangular Plates”. Proc. Phys. Soc. (London),
ser. B, vol. 62, no. 353, 1949, pp. 277-285
[11]
Bazley, N. W.; Fox, D. W.; Stadter, J. T. “Upper and Lower Bounds for the
Frequencies of Rectangular Free Plates. Tech. Memo TG-707, Appl. Physics
Lab., The Johns Hopkins University., August 1965
[12]
Waller, M. “Vibrations of Free Square Plates. Proc. Phys. Soc. (London), Vol.
51, January 1939, pp. 831-844
[13]
de Bree, H-E; Basten, T. “A Full Bandwidth Calibrator for a Sound Pressure and
Particle Velocity Sensor”. German Annual Conference on Acoustics. March
2008
- 17 -
Appendix A – Analytical Prediction Spreadsheet
Table A1 Values of Lambda Squared Used in Analytical Analysis ([9])
lambda^2
1
20.13
2
21.6
3
46.65
- 18 -
4
50.29
5
58.2
6
67.49
Appendix B – Microflown Scan and Paint Inputs
Figure B1 Parameters Used in Microflown Analysis
Parameter
Value
Sample Rate (Samples/Second)
50000
Number of FFT Points
8192
Overlap (%)
50
Number of Averages
5
Window Type
Hanning
Number of Samples per Probe Position 24576
- 19 -
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