DYNAMIC BEHAVIOR OF A THREE DIMENSIONAL ALUMINUM TRUSS
IN FREE SPACE
by
Marcus R. A. Heath
B.Eng. Mechanical Engineering
Royal Military College of Canada (1989)
Submitted to the Department of Ocean Engineering
in Partial Fulfillment of the Requirements
for the Degrees of
MASTER OF SCIENCE
IN NAVAL ARCHITECTURE AND MARINE ENGINEERING
and
MASTER OF SCIENCE
IN MECHANICAL ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May, 1994
© Massachusetts Institute of Technology 1994. All rights reserved.
'7-72
/
/7
/
Signature of Author
Department of Ocean Engineering
May, 1994
Certified by
Professor Ira Dyer
7 Department of Ocean Engineering, Thesis Supervisor
Certified by_
Proressu, Zaichun Feng
Department of Mechanical Engineering, Thesis Reader
Accepted by
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Department GrajNm
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Professor A. Douglas Carmichael, Chairman
nitte, Department of Ocean Engineering
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DYNAMIC BEHAVIOR OF A THREE DIMENSIONAL ALUMINUM TRUSS
IN FREE SPACE
by
Marcus R. A. Heath
Submitted to the Department of Ocean Engineering on May 6, 1994
in partial fulfillment of the requirements for the degrees of
Master of Science in Naval Architecture and Marine Engineering
and Master in Science in Mechanical Engineering
Abstract
The United States Navy sponsored Truss Damping Program is a research study in
the area of submarine noise reduction. The research explores the feasibility of effectively
attenuating noise associated with vibration of rotating machinery by means of a truss
support structure. Such a structure occupies the entire machinery space by supporting all
internal equipment. This thesis explores the dynamics of a three dimensional aluminum
truss to determine the inherent type and degree of damping.
By comparing experiments to analytic predictions, I learn that attenuation is due to
strut radiation for all frequencies above a critical value. Close to the source of excitation
the measured attenuation rate is higher because two mechanisms occur simultaneously:
struts excited in flexure radiate effectively, and energy in the form of predominantly
compressional or torsional waves scatters to energy comprising a more equal balance of
compressional, torsional and flexural wave types. Other mechanisms of attenuation,
including radiation from joints, losses in the interfaces between components and losses to
ground through the supports, are negligible in comparison to the attenuation of scattering
and strut radiation.
From experimental results and supporting theory, I develop basic dynamic design
guidelines. In full scale, the architect must maximize the quantity of unobstructed struts
so that efficient radiation occurs. Maximizing the joints (minimizing cell size) near
vibration sources and designing strut connections at or close to right angles promotes
desirable scattering of wave types. Designs must consider global truss dynamic effects
associated with relatively low frequency excitation, especially where the truss is effectively
decoupled from the submarine's outer shell. These guidelines, when combined with
fundamental requirements of submarine design, are useful in developing a specialized truss
which provides an effective means of passively damping machinery-borne noise.
Thesis Supervisor: Dr. Ira Dyer
Title: Professor
Acknowledgments
I wish to extend my appreciation to Professor Ira Dyer for his guidance in
experimentation and interpretation of results. I would like to thank Dr. Yueping Guo for
his encouragement and contributions in numerical modeling. I am also grateful to
Professor Patrick Leehey for his thought provoking discussions and the assistance he
provided by relating my work to his research in similar areas of study. Successful
experimentation would not have been possible without the assistance of my colleagues in
the Acoustics and Vibrations Laboratory. Their interest in my experiments and helpful
advice inspired my research. Special thanks are due to Djamil Boulahbal, Dan McCarthy
and Rama Rao.
Table of Contents
Ab stract ................... ..................................
Acknowledgments............................................................................................
Tab le o f C o ntents .............................................. ....... .....................
.................
List of Figures.....................................
List of Tables ..................................
2
3
4
6
99.......................
Chapter 1
Introduction............................ ............................ 10
1.1
Objective ..........................................
............ 11
1.2
Approach............................
................................. 11
Chapter 2
Apparatus....................
.. ...
.......... 13
2.1
Design and Preparation of Truss ..................................... 13
2.2
Equipment Selection and Preparation..........................
. 17
Chapter 3
Steady
From
3.1
3.2
Chapter 4
Determination of Wave Speeds Using Temporal Analyses ............... 28
4.1
Procedure.............................
.......... 28
4.2
Results .... ............ . ...............................
................. 31
Chapter 5
Stop Band Analysis.............................................. 34
5.1
Procedure...........................
.......... ................. 34
5.2
Results .......................... .............
........ 34
Chapter 6
Physical M odels and Predictions.................... ...... ...................
6.1
Comparison of Measured and Predicted Group Speeds........
6.2
Prediction of Frequency of Transition Between Global
and Local Truss Dynamics ................... .......................
6.3
Calculation of Radiation Loss Factor of a Strut....................
6.4
Calculation of Radiation Loss Factor of a Joint ....................
6.5
Other Mechanisms of Attenuation .................... ...
6.6
Application of Loss Factors................... ......
6.7
Determination of Experimental Attenuation Slopes ..............
6.8
Comparison of Theoretical and Experimental Attenuation....
40
40
Conclusions......
70
Chapter 7
State Attenuation as a Function of Axial Distance
Source of Excitation ................................................. 20
Procedure ............
................
.......................
.... 20
Results ..........................................
23
..............................................................
R efe ren ce s ........................................................................................................
Appendix A
Truss Design Details................................
47
48
53
57
58
64
67
72
............... 73
Appendix B
Added Mass Effect of a Sensor on a Strut .................................... 79
Appendix C
Measurement of Strut Vibration.......................................... 83
Appendix D
Pulse Analysis Data......................
Appendix E
Group Delay Calculations Using Phase Information......................
98
Appendix F
MATLAB C odes....................................................
103
.. ................. 88
List of Figures
1.1
2.1
2.2
2.3
2.4
2.5
2.6
2.7
3.1
3.2
3.3
3.4
4.1
4.2
4.3
4.4
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
6.1
6.2
Schematics of typical submarine cross-section showing a conventional
10
and truss mount supporting the main engine......................
Side view of truss in the laboratory........................ ............................ 13
Assembly of the truss from single square-based pyramidal cells................ 14
Upper joint and connected struts showing epoxy at the interfaces................. 16
Joint labeling schem atics ................................
.................................... 16
17
Attachment of Bruel and Kjaer low frequency vibration generator ............
18
Attachment of Wilcoxon Research vibration generator............................
Equipment; from the top: PCB 483B07 signal conditioner, HP 3562A
dynamic signal analyzer, Wilcoxon Research model PA7D power
amplifier, and Precision low pass filter; on right: terminal for Concurrent
multi-channel data acquisition system; on left: HP 9872C plotter ............ 19
Plots of accelerometer power due to the vibration generator signal
and from background noise, at joint b13 in the 8 kHz octave................... 22
Attenuation of acceleration as a function of axial distance from force
excitation, for octaves: 125, 250, 500 and 1000 Hz................................... 24
Attenuation of acceleration as a function of axial distance from force
excitation, for octaves: 2, 4, 8, 16 and 32 kHz................................
. 25
Accelerance as a function of frequency across the first joint bl....................
26
View of hammer pulse oriented axially on joint bl................................... 29
Location of hammer and sensor for local first cell experiment ..................... 29
Response at e2 due to pulse at bl, displayed in the time domain.
Each division in time represents 0.1 msec. The frequency range
is 10 - 2 0 kHz ...................................................................
... 30
Figure 4.4: Apparent base group speeds determined at b joints for four
octaves: 4, 8, 16 and 32 kHz and for predicted compressional waves,
where the lengt,-, between sections equals length of base scrt ................. 32
Accelerance measured between joints b3 and bl for frequencies 0 - 1 kHz. 35
Accelerance measured between joints b3 and bi for frequencies 1 - 2 kHz. 35
Accelerance measured between joints b3 and bl for frequencies 2 - 3 kHz. 35
Accelerance measured between joints b3 and bl for frequencies 3 - 4 kHz. 36
Accelerance measured between joints b3 and bl for frequencies 4 - 5 kHz. 36
Accelerance measured between joints b3 and bl for frequencies 5 - 6 kHz. 36
Accelerance measured between joints b3 and bl for frequencies 6 - 7 kHz. 37
Accelerance measured between joints b3 and bl for frequencies 7 - 8 kHz. 37
Accelerance measured between joints b3 and bl for frequencies 8 - 9 kHz. 37
Accelerance measured between joints b3 and bl for frequencies 9 -10 kHz 38
Experimental and predicted group speeds for compressional, torsional
43
and flexural waves in octave with center frequency equal to 4 kHz........
Experimental and predicted group speeds for compressional, torsional
and flexural waves in octave with center frequency equal to 8 kHz........
44
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12
6.13
6.14
6.15
6.16
A. 1
A.2
A.3
A.4
B. I1
B.2
B.3
Experimental and predicted group speeds for compressional, torsional
and flexural waves in octave with center frequency equal to 16 kHz.......... 45
Experimental and predicted group speeds for compressional, torsional
and flexural waves in octave with center frequency equal to 32 kHz.......... 46
Bessel functions plotted as a function of non-dimensional frequency ka...... 50
Strut radiation loss factor vs. ka plotted for each frequency octave
band. The circles are points where rs is calculated and crosses
are points at frequencies below fc .............................
52
Spherical Bessel functions plotted as a function of non-dimensional
frequency ka ......
...........
......................... 54
Joint radiation loss factor rij vs. ka plotted for each frequency octave band. 56
Acceleration of truss at bungee chord connection divided by acceleration
of ground at support base...................................
57
Attenuation slopes as a function of non dimensional frequency ka
due to predicted: strut radiation, and structural damping in struts
due to flexural waves for r7d = 10- 6 10 - 5 and 10- 4 ...................................... 61
Attenuation slopes as a function of non dimensional frequency ka
due to predicted: strut radiation, and structural damping in struts
due to compressional waves for rd = 10- 4, 10- 3 and 10 - 2 ..... . . ... . . . 62
Attenuation slopes as a function of non dimensional frequency ka
due to predicted: strut radiation, and structural damping in struts
due to torsional waves for r7d = 10-4, 1 0- 3 and 10 - 2 ............................ 63
Best curve fits of the experimental data of Chapter 3, for octaves: 1, 2
and 4 kHz......................... .. .....
... ..
...................... 65
Best curve fits of the experimental data of Chapter 3, for octaves: 8, 16
and 32 kHz......................
...............................
66
Attenuation slopes plotted as a function of nondimensional frequency ka
for: predicted loss due to strut radiation plus flexural structural damping
(77d=10-5),and experimental total attenuation far from the source ......... 68
Attenuation slopes plotted as a function of nondimensional frequency ka
for: predicted loss due to strut radiation plus compressional structural
damping (7d = 10-3), predicted loss due to strut radiation plus torsional
structural damping (7d=10-3),and experimental total attenuation near
the source .................................... ............................. ...
69
Assembly of the truss from single square-based pyramidal cells................ 73
Dimensions of a truss cell; shown are dimensions between vertices (cm)..... 74
Joint Drawings..................
...............
.......... 76
Pattern of holes in and quantity of the six node types...............................77
Model of added mass on strut for compressional vibration added
. ................ 79
mass analysis........................................ .......
Model of added mass on strut for beam bending added mass analysis.......... 81
ADINA outputs for clamped beam bending for modes 1, 3 and 5;
left: unloaded beam; right: beam loaded with a 1 gram point mass
...................
................... 82
at m id length ................. ..............
C. 1
Accelerance versus position along strut a4b3 with total length 48 cm,
for octaves 125, 250 and 500 Hz............................. .............................. 84
C.2
Accelerance versus position along strut a4b3 with total length 48 cm,
for octaves 1 and 2 kHz ..................... ................... .. ............. 85
C.3
Accelerance versus position along strut a4b3 with total length 48 cm,
for octaves 4 and 8 kHz......................................
86
C.4
Interface of strut and joint with added epoxy............................................ 87
D. 1 Hammer pulse (upper) at joint bl and arrival of energy at b3 in the
32 kHz octave displayed in the time domain; the expected arrival of
compressional energy is marked by cl ............................. . .................. 89
D.2
Hammer pulse (upper) at joint bl and arrival of energy at b5 in the
32 kHz octave displayed in the time domain; the expected arrival of
compressional energy is marked by cl .....................
.....................90
D.3
Hammer pulse (upper) at joint bl and arrival of energy at b7 in the
32 kHz octave displayed in the time domain; the expected arrival of
compressional energy is marked by c l .....................................................
91
D.4
Hammer pulse (upper) at joint bl and arrival of energy at b9 in the
32 kHz octave displayed in the time domain; the expected arrival of
compressional energy is marked by cl ..............................
................... 92
D.5
Hammer pulse (upper) at joint bl and arrival of energy at bNl in the
32 kHz octave displayed in the time domain; the expected arrival of
compressional energy is marked by c ...................... .................. 93
D.6
Hammer pulse (upper) at joint bl and arrival of energy at b13 in the
32 kHz octave displayed in the time domain; the expected arrival of
compressional energy is marked by cl....................................................... 94
D.7
Hammer spectrum in 4 kHz octave..................
...
....... 95
D.8
Hammer spectrum in 8 kHz octave.......................................... 95
D.9
Hammer spectrum in 16 kHz octave.........................................96
D. 10 Hammer spectrum in 32 kHz octave...............................
96
D. 11 Hammer spectrum for frequency range: 100 Hz to 10 kHz....................... 97
D.12 Spectrum of background noise.......................
............... 97
E. I1 Group delay plotted as a function of shortest travel path.......................... 99
E.2
Plot of phase for accelerance measured between bl and b3.....................
100
E.3
Plot of phase for accelerance measured between bl and b5 ...................... 100
E.4
Plot of phase for accelerance measured between bl and b7................... 101
E.5
Plot of phase for accelerance measured between bl and b9................... 101
E.6
Plot of phase for accelerance measured between bh and b.......................
102
E.7
Plot of phase for accelerance measured between bl and b13.................... 102
List of Tables
3.1
Altered frequency octave band limits required for HP 3562A
signal analyzer ................................................................... .................... 21
Energy arrival times in ms at each b joint for octave bands: 4, 8, 16
4.1
and 32 kH z ................................
... .. ............. ...............
... 3 1
5.1
Comparison of expected and experimental stop band frequencies .......
. 39
6.1
Length Ratio: length corresponding to arrival of predicted wave speeds
over length of shortest path, presented for each wave type at b joints........ 42
6.2
Theoretical attenuation slopes for strut and joint radiation and
the respective values of non dimensional frequency ka for each
octave center frequency ........................................................ 59
A. 1 Lengths and quantities of strut types.......................... ... 75
B. I1 Difference between unloaded and loaded theoretical natural frequencies
of the strut under compressional vibration............ ............
... ................. 80
B.2
Difference between unloaded and loaded theoretical natural frequencies
of the strut under flexural vibration .................................. ................... 81
C. 1 Comparison of experimental mode counts with hinged and clamped
................... 87
predictions in a base strut of length 48 cm..................
Chapter 1
Introduction
Since World War II the design of machinery mounts for submarines has progressed
from solid deck mounts, to raft isolation mounts, to box beam girders. This progression
of design was driven by the requirement to minimize the underwater noise signature
detectable by enemy forces and to provide isolation from shock. Isolation of vibration
sources, caused by rotating imbalances in machinery, has advanced from spring systems to
more elaborate passively and actively damped platforms.
Research in the United States Navy sponsored Truss Damping Program takes this
machinery isolation concept one step further. Instead of attempting to isolate a rotating
machine from the deck on which it is mounted, the machinery is effectively slung from a
truss. The truss provides a torturous path through which vibratory energy must travel
before it reaches the submarine's outer shell. The propagated energy is damped in the
components of the truss and through radiation into air inside the submarine. Added
damping can then be applied to the truss components and acoustic damping can be
mounted on the inner surface of the submarine shell. A simple schematic showing a
typical arrangement of the truss concept versus the conventional mount is shown in Figure
1.1.
Conventional Mount
Truss Mount
Figure 1.1: Schematics of typical submarine cross-section showing a conventional and
truss mount supporting main engine.
The design process of such a submarine is reversed. Conceptually, instead of a
conventional architecture where machinery and equipment are assigned to existing deck
space, the truss is designed around the various sources of vibration. The truss is
effectively tuned to passively attenuate energy for each different source. Unlike the
internal structure of a surface ship, the submarine internals need not contribute to the
global strength of the outer shell. The shell is a pressure vessel that, constructed with ring
stiffeners, is stable against collapse and buckling failure modes. Therefore, the truss is
isolated from the outer shell by isolation mounts at the limited number of attachment
points.
1.1
Objective
Before an effective machinery support truss is designed, the dynamics of such a
complicated three dimensional structure must be understood. The objective of this thesis
is to study the dynamics of a laboratory-scale three dimensional truss through
experimentation and application of analytic theory. I determine the magnitude of overall
spatial damping experimentally and compare results to predictions of analytic models. By
discovering the type of elastic waves present and understanding the scattering that occurs
between wave types, I am better able to understand the expected spatial damping
behavior. Of primary importance is the determination of applicable length scales in the
model and to quantify critical characteristics that affect its behavior. I determine
applicable physical and dynamic length scales in order to distinguish local from global
effects and efficient from inefficient attenuation domains.
1.2
Approach
An aluminum truss comprising simple, stable cells connected in series is
constructed and slung elastically to model free space boundary conditions. The transverse
dimensions are selected based upon a 1/15 scale model of a full size submarine (Trident
Class displacing 17,000 tons with cross-section 42 x 38 ft) where the cross-section of the
model is considered to fill the entire internal area of the submarine. The truss is not
damped other than that required for construction and is loaded only by its own weight.
I identify important characteristic lengths in the truss. Experimentally, the truss
experiences global dynamic behavior below 1 kHz. Confirmation with analytical
relationships shows that global dynamics of a full scale truss can not be overlooked. Low
frequency excitation, from the main shaft for example, can excite a truss in a low global
mode of vibration, especially as the truss is decoupled form the outer shell as much as
possible.
Acceleration is measured throughout the truss, when excited at one end, and
spatial attenuation as a function of axial distance is determined. The frequency range
covered is 100 Hz to 32 kHz which, given the 1/15 scale, corresponds to about 7 Hz to 2
kHz in full scale. This range is deemed to cover potentially useful applications of a truss
to submarine noise reduction. Comparisons of experiments and theoretical predictions of
strut radiation attenuation show that strut radiation accounts for the measured attenuation
for all frequencies above a critical value of 950 Hz. Close to the source, however, the
measured attenuation rate is high because two mechanisms occur simultaneously: struts
excited in flexure radiate effectively, and energy in compressional and/or torsional waves
scatters to energy comprising a more equal balance of compressional, torsional and
flexural wave types.
Other mechanisms of attenuation, including radiation from joints, losses in the
interfaces between components and losses to ground through the supports, are negligible
in comparison to the attenuation of scattering and strut radiation.
In full scale a design could maximize the quantity of unobstructed struts so that
radiation can occur. Furthermore, a stop band analysis reveals periodicity is important
when designing a structure to effectively control excitation at particular frequencies.
However, such a design is only useful if the quantity of same length repeated,
unobstructed strut is large. This is impractical in a submarine where the truss is nonperiodically loaded with massive equipment.
Because the losses due to scattering at joints are significant, a design should
maximize the number of joints. This is achieved by minimizing cell size within the
constraints of submarine design. Careful selection of angles between connecting struts is
important to maximize the scattering of wave types. Connections at or close to right
angles produce excellent scattering
I measure the group speed of energy traveling axially along the truss as a function
of frequency. I calculate expected group speeds for all wave types, as a function of
material, form and (for flexural waves) frequency. Comparison of measured and expected
speeds support my theory that wave scattering occurs. Predominant compressional waves
scatter into a combination of all wave types as energy travels along the truss.
Chapter 2
Apparatus
The apparatus discussed in this section include the truss model and the
experimental equipment.
2.1
Design and Preparation of Truss
The experimental model is a truss consisting of eleven square-based pyramids
joined in series. The truss is fabricated from 6061 T6 aluminum bar and tube and weighs
16 kg (35 1/2 lb.). Design details are included in Appendix A and a photograph of the
truss, as situated in the laboratory, is shown in Figure 2.1. Figure 2.2 shows pyramidbased single cells, the assembly of the cells in series and the complete construction with all
members assembled.
Figure 2.1: Side view of truss in the laboratory.
a. Square-coased \/
I
\\
-
-I
b, Pyramid a
c. Complete
Figure 2.2: Assembly of the truss from single square-based pyramidal cells.
The truss has overall dimensions 4.7 x 0.84 x 0.80 m (12 1/2 x 2 3/4 x 2 5/8 ft).
The transverse dimensions are selected based upon a 1/15 scale model of a full size
submarine where the cross-section of the model is considered to fill the entire internal area
of the submarine. The selection of strut diameter and wall thickness are also scaled
according to required strength.
Although scaling provides some degree of applicability to the model, it must be
noted that the truss is not architectured as a feasible internal structure. Instead, I designed
the model to best understand the dynamics of a three dimensional truss. Experimental and
theoretical factors drive all other selected sizes and forms. The length of the truss is
selected to maximize the axial dimension so that attenuation is readily measurable, but is
limited by practicality.
The square-based pyramidal cell shape is selected to maximize simplicity while
meeting several constraints. Cell simplicity facilitates construction, eases the
understanding of experimental results and ameliorates the feasibility of numerical
modeling. The pyramidal cell is repeatable forming a structure extended in one dimension,
whereas a simpler tetrahedral cell, for example, would not posses the same additive
simplicity. The cell can also be repeated in the transverse direction thus filling a volume.
The cell is stable, meaning that the ability to maintain shape does not depend upon the
bending strength of the joints.
Finally, cell shape is governed by the theory of designing a truss to maximize the
torturous path through which vibration energy must travel. For example, the truss is
designed to have no direct axial paths and it need not have direct transverse paths should
the cell be repeated in the transverse direction. Because the submarine internal structure
can be effectively decoupled from the outer shell, it is feasible to fix only selected joints to
the shell, thereby maximizing the number ofjoint-strut interfaces between a given source
and destination. The model is well designed in this respect.
The truss components include 109 struts of three different lengths and 35 joints of
six different hole configurations. The joints are machined from 2 1/2 inch aluminum bar.
Holes and slots are fabricated to provide a tight fit around the connecting struts. The
number of struts per joint vary from three to eight. The connection between components
is strengthened with the application of two part epoxy. Figure 2.3 shows the epoxy at the
interfaces between struts and joints. It is appreciated that the epoxy may provide damping
and that the exact amount of epoxy is different in each joint. However, the method of
attachment provides realism to the model and is practical for construction and destruction.
To facilitate experimentation, the truss components are labeled. The axial
direction of the truss is considered the X direction, Y extends in the horizontal transverse
direction, and Z represents the vertical transverse direction. The truss is segregated into
sections in the X direction. Each section, labeled 1-13, comprises three joints that form a
triangle in cross-section (excluding end sections 1 and 13 that have only one joint). In
accordance with Figure 2.4, the joints are labeled by a letter a-f indicating the transverse
location and a number 1-13 showing the section. For example the fifth central lower joint
is labeled b5. The struts are labeled by stating the endpoints such as b5a6 for example.
The truss is slung from the laboratory ceiling using two bungee chords. The
natural frequency of rigid body oscillation (zeroth mode vibration) is less than one Hertz
so that the truss is considered to be suspended in free space at frequencies of the order
100 Hz or more. The two supports are attached at intermediate positions along the truss
(joints e4 and e10O) in order to minimize hogging or sagging effects.
Figure 2.3: Upper joint and connected struts showing epoxy at the interfaces.
II
VeCr
Plan Vie
Prrg-:I
View
13
12
11
10
9
Figure 2.4: Joint labeling schematics.
8
7
6
Sections
5
4
3
2
1
2.2
Equipment Selection and Preparation
To simulate the vibration caused by a rotating machinery, a vibration generator is
fixed to an end joint (joint bl) by means of a mounting stud. Located at one extreme end
the shaker maximizes the available length for measurement and best achieves the unloaded
condition.
Two different vibration generators are used to excite the truss in frequency bands
over the desired frequency range of 100 Hz to 32 kHz. At frequencies below 10 kHz the
Bruel and Kjaer (B&K) type 4810 vibration generator is mounted as shown in Figure 2.5.
This shaker applies sufficient force (average 7 N rms) to maintain a signal to noise ratio of
at least 24 dB throughout the truss. At higher frequencies excitation is provided by the
Wilcoxon Research (WR) model F3/F9 vibration generator. It provides a signal to noise
ratio of at least 30 dB in the higher frequency bands. The WR shaker is mounted as
shown in Figure 2.6. The WR vibration generator is designed with a built in impedance
head (force and acceleration transducer) whereas the B&K shaker requires the attachment
of an external force transducer as shown in Figure 2.5.
Figure 2.5: Attachment of the Bruel and Kjaer low frequency vibration generator.
Twelve PCB model 309A internally amplified (voltage mode) accelerometers are
attached to the truss components with bees wax. Bee's wax provides a rigid connection to
the truss over the frequency range of interest, while facilitating setup. The PCB sensor,
weighing 1 gram, is selected to minimize the added mass effect when mounted on a strut,
while providing sufficient frequency range and sensitivity. The addition of one sensor to a
strut provides an added mass error not greater than 10% as is shown in the analysis in
Appendix B.
Figure 2.6: Attachment of the Wilcoxon Research vibration generator.
The twelve channel PCB 483B07 ICP signal conditioner, provides the voltage
supply and amplification for the twelve sensors. This equipment is the upper unit shown in
Figure 2.7. The twelve amplified channels become inputs to either the Concurrent multichannel data acquisition and processing computer (terminal shown in Figure 2.7) or to the
HP 3562A two channel dynamic signal analyzer (unit located below the PCB signal
conditioner). Although the multi-channel analyzer provides simultaneous processing, the
HP dynamic signal analyzer is preferred due to its speed, built in functions and high degree
of processing power. The unit located below the HP signal analyzer is the amplifier for
the vibration generator, and the lower unit is a Precision filter set that provides low-pass
filtering (anti-aliasing) for use with the Concurrent multi-channel system.
Figure 2.7: Equipment; from the top: PCB 483B07 signal conditioner, HP 3562A
dynamic signal analyzer, Wilcoxon Research model PA7D power amplifier, and Precision
low pass filter set; on right: terminal for Concurrent multi-channel data acquisition system;
on left: HP 9872C plotter.
Chapter 3
Steady State Attenuation of Acceleration as a Function of Axial Distance
from Excitation
The attenuation of acceleration is measured as a function of distance in the X
direction. The truss is excited at one end with the vibration generator mounted with force
vector in the Z direction, as discussed in Chapter 2. The vibrating truss reaches steady
state conditions before data are taken, meaning that sufficient time has passed for power in
the truss to build to equilibrium. Steady state is reached quickly because the transient
energy build-up (proportional to 1-exp[-rlot]) takes approximately t=1/(irc9), where t is
time, co is radian frequency and 77 is the loss factor. I do not know q; this is indeed what
the experiment is expected to yield, but it is no smaller than about 10- 2. Furthermore, 0o is
no smaller than about 2;rx102 rad. Thus, for t21/6 sec (approximately) I can expect
steady state conditions to prevail, and all data acquisition times meet this criterion quite
amply. Beyond such an initial time, the energy introduced into the system balances that
which is lost through the mechanisms of attenuation inherent in the untreated truss.
3.1
Procedure
A random force signal in proportional frequency bands ranging from the 125 Hz to
32 kHz octaves excites the truss. The excitation of only one band at a time permits data
acquisition within the band while maximizing the power available to the vibration
generator. The HP 3562A dynamic signal analyzer generates the band passed signals and
acquires and processes data. The quality of filtering provided by the HP analyzer is
precise to the extent that no roll-off effects are detectable when comparing filtered and
unfiltered input signals. The lower and upper limits of each octave band are slightly
modified from the standard in order to suit the capabilities of the analyzer. In all cases the
geometric mean remains unaltered. Table 3.1 summarizes the required alterations.
Octave Center Frequency
(Hz)
Standard Frequency Range
(Hz)
Altered Frequency Range
(Hz)
125
88 - 176
85 - 185
250
500
176-353
353 - 707
171 -366
342- 732
1000
707 -1414
2000
4000
1414 -2825
2825 - 5650
8000
16000
5650- 11300
11300 -22500
683 - 1464
1366 -2928
2732 - 5857
5464- 11714
11800 -21800
32000
22500 -45000
21000 -46000
Table 3.1: Altered frequency octave band limits required for HP 3562A signal analyzer.
I calibrate all sensors and obtain the signal to noise ratio at locations throughout
the truss. I identify the location showing the worst signal to noise ratio as joint b13 that is
the farthest joint from the source. The frequency at which the worst ratio is observed is
the 8 kHz octave using the B&K shaker (the WR shaker is completely unacceptable at this
frequency). When comparing power measured due to the intentional signal with that
measured due solely to noise in these worst case conditions, the signal to noise ratio is
24.2 dB. Figure 3.1 shows this comparison for the 8 kHz octave band. The values used
in the ratio are the linear average of all processed power data: -53.43 dB for signal and 77.62 dB for noise as shown at the top of each graph. Note that this is the worst case
scenario: the average signal to noise ratio over the entire truss and all frequencies is
approximately 40 dB.
Y--53.43 dBGQms
Y--77.B24 dBGrms
POWER
0.0
SPEC2
B4Avg
OXOv1p
Hann
10.0
/Div
dB
rms
G2
-80.0
Fxd
Y
L1~· · ~..,
L,,,
5.484k
iI
Hz
"
.
IO-N""
-4
'4g*W
-4 "" _ "I "
8 KHZ OCTAVE NOISE
.
Q UO...
i1.714k
Figure 3.1: Plots of accelerometer power due to the vibration generator signal and from
background noise, at joint b3 in the 8 kHz octave..
I measure the accelerance, or the acceleration sensor output divided by the shaker
force input, at points throughout the truss. I obtain data for sensors located on the joints
instead of struts. Strut responses are higher in magnitude but data collection at the joints
is simpler and more reliable. The response of the struts is not considered for this
particular experiment, but the process used to measure strut response and pertinent
observations are included in Appendix C.
I determine the optimum sensor placement and orientation on a joint. Although
preliminary experimentation indicates that in most cases axially positioned sensors provide
stronger signals than transverse orientations (up to 10% higher), I select a transverse
orientation. The selected position provides a signal of sufficient magnitude and is effective
in measuring global truss motions at lower frequencies. Accelerometers are positioned on
the rim of the joint, in the vertical direction, on the edge closest to the noise source.
Figure 2.3 shows this location.
I measure the steady state accelerance of every joint at all frequency octaves and
record in units: dB re 1 g/N. The data are presented in the frequency domain and many
sets of Fourier transformed data are averaged (frequency averaged) to improve signal
clarity. At high frequency octaves, 64 averages are conducted whereas at lower
frequencies the decreased range in frequency requires longer process times, so that shaker
overheating limits the number of averages to 32. Furthermore, practically obtaining a
single value at each octave requires taking a linear average of all processed accelerance
data within the octave band.
To obtain the overall effect of axial attenuation, I perform some simple processing
operations. First, I combine the responses of each three joint grouping per section by
linear averaging. This effectively smoothes variations over one section. Furthermore, I
normalize the accelerance data by the responses measured across the first joint bl. By
dividing each section output by that at the first section, the results indicate attenuation in
acceleration with reference to the first section, measured in dB.
3.2
Results
The processed results are shown in Figures 3.2 and 3.3. The attenuation is plotted
as a function of axial distance measured in meters where the distance between each section
is 38.4 cm. In Figure 3.4 the accelerance measured across the first joint is shown, where
the sensor is located on the joint surface opposite that of the shaker.
A fr
4U
I
I
II
II
Average of all Joints at Each Section
35
-
125 Hz octave
- - 250 Hz octave
.- .-500 Hz octave
30
.....
1 kHz octave
,25
o
,- 20
CO
0)
10
.........."._..............
~~-L.
-
..-.-
*-
~
.......
%
~
N
-
N
-
N
K
N
N
-N-
-
II
I
0.5
1
I
I
1.5
2
I
2.5
I
3
I
3.5
Axial Distance Along Truss (m)
I
4
Figure 3.2: Attenuation of acceleration as a function of axial distance from force
excitation, for octaves: 125, 250, 500 and 1000 Hz.
4.5
I
I
I
I
Average of all Joints at Each Section
2 kHz octave
- 4 kHz octave
- -8 kHz octave
16 kHz octave
3C
M
-
32 kHz octave
0r.
2C
15
1c
S.
'X
E
I
0.5
I
1
I
I
I
I
1.5
2
2.5
3
3.5
Axial Distance Along Truss (m)
I
4
Figure 3.3: Attenuation of acceleration as a function of axial distance from force
excitation, for octaves: 2, 4, 8, 16 and 32 kHz.
4.5
I
CI
10
I
I
I
rllrl
1
.
I
.
I
I
I
.
~
E
Z
0T)
U)
L.
f13
0
U)
"O
0
CU
-
I.-
C-)
C.
-5
-10
-
-
..
Ir
10
-1
.
. . I
I
I
0
I
..
1
10
10
Frequency (kHz)
Figure 3.4: Accelerance as a function of frequency across the first joint bl.
.
.
2
10
Figures 3.2 and 3.3 indicate that, for the higher octaves, the attenuation generally
increases as a function of axial distance from the vibration source. The magnitude of the
attenuation is greater for higher frequencies and the slopes of the attenuation curves are
higher accordingly. At lower frequencies the attenuation curves does not increase
continually with distance. The curve representing the 125 Hz data falls below zero at the
far joint, meaning that the accelerance with the sensor located on joint b13 is greater than
that with the sensor on bl. The curves representing the 125, 250 and 500 Hz octaves are
nearly symmetric about the midpoint of the truss. Consequently, I observe that the truss
vibrates in a global mode when excited by energy in each of these first three octaves, most
likely that associated with compressional waves. I observe no global modes at higher
frequencies. In addition, a similar experiment with all sensors oriented in the axial
direction produces the same trend. I explain the reason for this behavior and determine
the expected transition frequency in Section 6.2.
The next general observation of interest is the increase in slope of attenuation
curves with increasing frequency for the higher octaves. A single slope fitted to each
attenuation curve reveals that the slope is noticeably greater than zero only at octaves
equal to or greater than I kHz. In Section 6.3 I further explore this slope relationship by
predicting the effect of strut radiation, and use two slopes for each octave as a better
representation of the data.
The data representing 32 kHz octave excitation deviate from the visible trend with
increasing frequency. I believe this is due to operation above the usable frequency limit of
the WR vibration generator where insufficient energy is introduced into the truss.
Although the average signal to noise ratio is acceptable at this frequency range, the shaker
response fluctuates greatly with frequency. During further analyses I consider the 32 kHz
octave data questionable.
Chapter 4
Determination of Wave Speeds Using Temporal Analyses
I conduct experiments to determine the speed of energy propagation along the
truss. I compare the wave speeds experienced in the truss to expected speeds of the
dispersive and non-dispersive wave types that can exist in such a structure. The aim of
this study is to distinguish wave types in order to understand how energy is transmitted
through the truss. Knowledge of wave types is essential before damping systems can be
effectively designed.
The truss is a complicated structure with many paths through which energy travels.
The reverberation that exists in the truss when excited continuously makes wave speed
experiments difficult with the equipment available to me. Therefore, the experiments are
conducted in a small time period so that only the non-reverberant energy is detected. For
further discussion of the effect of reverberation, an analysis of the group delay is
conducted and is included inAppendix E.
4.1
Procedure
A single pulse of energy provided by the strike of a hammer excites the truss. The
energy is introduced at joint bl as is shown in Figure 4.1. Sensors located at points along
the truss detect the first arrival of energy. The force gauge fitted on the hammer head is
used to trigger the HP dynamic signal analyzer that starts the data collection process. The
time histories of the force gauge and the applicable sensor are collected simultaneously. In
less than 4 milliseconds the energy travels the full length of the truss along the shortest
path and energy occupies the entire truss.
Only the time of the first arrival of energy along the shortest path can be observed
in my experiment because all other energy arrives through a series of longer paths and
back paths. When excited at joint bl the shortest axial path is made up of lower base
struts which form two parallel paths of a zigzag pattern. Accordingly, sensors are located
on all b joints and on lower base struts.
I expect the first arrival of energy to be a compressional wave, while at later times
the energy associated with rotational and dispersive flexural waves arrives. Of course,
such behavior presumes sufficient energy in each of the wave types to be observed. To
exemplify the effect of arrival of more than one energy type, I conduct a local experiment
along one straight path in the first cell only. I position a sensor on joint e2 and introduce
the pulse at bl as shown in Figures 4.1 and 4.2. The frequency range for this particular
experiment is 10 - 20 kHz.
Fign•m, 4.1: Hammer tap oriented axially on joint h,.
Figure 4.2: Location of hammer and sensor for local experiment
The output data, in Figure 4.3, shows the time histories of the force at bl and the
acceleration at e2 in the upper and lower graphs respectively. This experiment is one of
the simplest cases where the arrivals of different wave types are readily detectable. Note
that the small time scale gives the impression that the pulse signal lacks sharpness.
Processing the observations reveals that the first arrival (denoted 1) corresponds to the
compressional wave speed and the second (denoted 2) is due to either a flexural wave
along the shortest path (at 10.5 kHz) or a flexural-compressional wave combination along
the next shortest path.
AVG
0.0
dB
N
-32.0
Fxd X
I
AVG
30.0
dB
G
-±0
Fxcd
X
- - _~- .`.-
~
~ ~
Figure 4.3: Response at e2 due to pulse at bl, displayed in the time domain. Each
division in time represents 0.1 msec. The frequency range is 10 - 20 kHz.
I conduct experiments on the entire truss in a number of frequency bands. The HP
signal analyzer is capable of maximizing resolution only when a large frequency span is
used. Octaves lower than 4 kHz provide unacceptable resolution so the selected
frequencies are: 4, 8, 16 and 32 kHz octaves.
4.2
Results
Initial experimentation proves that altering position and orientation of the sensor
on each joint makes little difference. I select orientation in the Z direction for all
experiments. For each octave and joint I measure the time of arrival from time history
curves. These data are presented in Table 4.1. Using the shortest path I calculate the
corresponding average group speeds and plot them in Figure 4.4. One sample of data for
the 32 kHz frequency range is included in Appendix D. In Figure 4.4, I also plot the
expected compressional wave speed of 5050 m/s as determined in Equation 6.1. I
compare the experiments to expected speeds for each wave type and draw conclusions in
Section 6.1.
Joint
Length
(m)
4 kHz
Octave
8 kHz
Octave
16 kHz
Octave
32 kHz
Octave
b3
b5
1.06
2.12
0.24
0.71
0.23
0.60
0.19
0.48
0.22
0.47
b7
3.18
1.00
0.90
0.76
0.71
b9
bl1
b13
4.24
1.40
1.19
5.30
2.24
1.43
6.36
3.60
2.00
Table 4.1: Energy arrival times in ms at each b joint for octave
kHz.
1.18
1.47
1.86
bands: 4, 8, 16
1.07
1.31
1.88
and 32
rrh~h
DVUU
N
E
500d
N
4000
CO
E
C3000
C)
0
................
..................
.
. .. .
-,.
..
L-
2000
32 kHz Experimental
-
*-.
1000
16 kHz Experimental
-
8
kHz
Experimental
34 kHz Experimental
*
A
3
Predicted Compressional
I
4
I
I
5
I
6
7
I
8
9
Station Number
I
I
10
I
11
12
Figure 4.4: Apparent base group speeds determined at b joints for four octaves: 4, 8, 16
and 32 kHz and for predicted compressional waves, where the length between sections
equals length of base strut.
13
Far from the source the data indicate that compressional energy is less significant.
The sample curves in Appendix D are marked to indicate the expected arrival times
corresponding to the compressional wave energy. These markings show that the
compressional energy is noticeable close to the source but its arrival far from the source is
buried in noise. This is true for all octaves and even when the accelerometers are
positioned axially on base struts in a conscious effort to distinguish the compressional
energy. I expect that compressional energy would be detectable if the truss were more
slender such that base struts joined at obtuse angles.
The accumulation of the above observations leads to an important result.
Scattering of wave types occurs in a truss. Compressional, torsional and flexural energy
multiply scatter until a balance of wave types is reached. This conclusion is applicable in a
continuously excited truss.
Chapter 5
Stop Band Analysis
Periodic structures are known to possess stop band characteristics. In a stop band,
frequencies corresponding to wavelengths equal to twice the length of the repeated section
appear as anti-resonances in transfer function data[ ]. The periodicity of the truss
possesses this behavior and provides valuable information about the presence of wave
types.
5.1
Procedure
With apparatus prepared as in the attenuation experiments of Chapter 3, I measure
the accelerance at particular locations in ten frequency bands of 1 kHz width. I examine
the detailed accelerance data over the entire frequency range and determine frequencies
where the results deviate from the norm. I compare the results of arbitrary sensor
positions and orientations on all truss components.
5.2
Results
I observe that accelerance data are relatively constant in each frequency band with
the exception of several significant deviations. These deviations represent frequency
bands as wide as 200 Hz where the accelerance values are lower than the norm. Similar
trends are present at positions throughout the truss, whether the sensor is located on joints
or struts.
The results at joint b3 are displayed in Figures 5.1 - 5.10. From these data four
prominent regions of low accelerance are identified. These four possible stop bands span
at least 100 Hz and indicate a decrease of at least 25 dB in relation to the norm. Their
center frequencies are: 3040, 5430, 8570 and 9850 Hz.
FREG
37.5
dB
N
-82.5
Figure 5.1: Accelerance measured between joints b3 and bl for frequencies 0 - 1 kHz.
FREG
40.0
dB
N
-24.0
Figure 5.2: Accelerance measured between oints b3 and bl for frequencies 1 - 2 kHz.
FREQ
32.0
dB
G
N
-32.0
Figure 5.3: Accelerance measured between joints b3 and bl for frequencies 2 - 3 kHz.
FREG
30.0
dB
G
N
-50.0
Figure 5.4: Accelerance measured between joints b3 and bl for frequencies 3 - 4 kHz.
FREG
25.0
dB
G
N
-- 15.0
Figure 5.5: Accelerance measured between joints b3 and bl for freqe-"cies 4 - 5 kHz.
FREG RESP
40.0
20Avg
OOvIp
Hann
f"
dB
N
-24.0
SI
1
I
Hz
I
I
I
I
I
E6k
Figure 5.6: Accelerance measured between joints b3 and bl for frequencies 5 - 6 kHz.
FREQ
48.0
RESP
20Avg
0%Ovlp
Hann
[
dB
G
N
-_18.0
I
_·
it
I
I
I
I
I
I
I
i
7k
Hz
Figure 5.7: Accelerance measured between joints b3 and bl for frequencies 6 - 7 kHz.
FREG RESP
50.0
20Avg
f"
OOvlp
Henn
dB
N
-30.0
7K
-
I
I
_ I_
I
--
I
Hz
l
I
I
I
I
Sk
Figure 5.8: Accelerance measured between joints b3 and bl for frequencies 7 - 8 kHz.
FREG
20.0
dB
G
N
-20.0
Figure 5.9: Accelerance measured between joints b3 and bl for frequencies 8 - 9 kHz.
FREG RESP
48.0
OOv1p
20Avg
Hann
dB
N
-1B. 0
..... L....1....
9k
I
I
I
1Ok
Hz
Figure 5.10: Accelerance measured between joints b3 and bl for frequencies 9 - 10 kHz.
To research the applicability of stop band theory, I identify possible length scales
of periodicity in the truss. The simplest repeated lengths are those of the struts.
Frequencies corresponding to the applicable wavelengths in struts are calculated for
flexural waves using Equation 6.5 and compressional waves using:
f
AC
c__
2L
(5.1)
where cl is 5050 m/s for the tube, L is the strut length in meters andf is in Hz. Similarly,
for torsional waves, where the torsional wave speed in a tube is 3115 m/s (Equation 6.2),
the stop band frequency is
f = C
2L
(5.2)
The frequencies of expected stop band behavior are compared with the
experimental regions of low accelerance as displayed in Table 5.1. The expected stop
bands of the flexural waves in all struts are not detected experimentally, suggesting the
lack of energetic waves of this type in the truss. On the other hand, the first two
experimental frequencies, 3040 and 5430 Hz, correspond to expected stop bands of
compressional waves in the diagonals and base struts respectively. In addition, 3040 Hz
nearly corresponds to torsional waves in the base struts. The mid-length support struts
produce no stop band perhaps because they are few in number (less than ten percent of
total struts).
Strut
Wave Type
Expectedf (Hz)
Corresponding
Experimental Centerf (Hz)
Base
Flexural
137
None
Support
Flexural
64
None
Diagonal
Flexural
47
None
Base
Torsional
3200
3040
Support
Torsional
2200
None
Diagonal
Torsional
1900
None
Base
Compressional
5430
5300
Support
Compressional
3600
None
Diagonal
Compressional
3100
3040
Table 5.1: Comparison of expected and experimental stop band frequencies.
The two higher stop bands experimentally identified, 8570 and 9850 Hz, are a
result of some small length scale such as that of the joints. However, this is difficult to
verify with certainty.
Larger scales of periodicity, such as cell length, also could display significant stop
bands in the truss. For example, the cell periodicity scale for compressional waves is
21/ 2L where L is the length of the base strut. This suggests
f= 2J2L
(5.3)
where cm is a phase speed corresponding to the group speeds observed in Figure 4.4.
However, larger length scales produce lower stop bands that are not detected in the
experiment.
The results from my experiments confirm that stop band behavior can be applicable
to a truss. The effect of periodicity is important when designing a structure to effectively
control excitation at particular frequencies. In practice, however, no truss will be periodic
because it will be non-periodically loaded with massive equipment.
Chapter 6
Physical Models and Predictions
Having conducted experiments to determine the actual attenuation and behavior of
wave speeds in the truss, I study the expected truss behavior by determining applicable
length scales and critical frequencies in the structure. I calculate the expected loss factors
for truss components and apply them to the truss using analytical methods. I compare
results with experimental data in order to quantify all mechanisms of attenuation.
6.1
Comparison of Measured and Predicted Group Speeds
In this section I examine the group speeds, measured in Chapter 4, from a different
perspective. I no longer assume the lengths are those of the shortest path (through base
struts only) as is done to achieve Figure 4.4. Instead I calculate the path lengths required
for the first arrival of each wave type where the arrival times are the experimental data
presented in Table 4.1. This process allows me to determine what a path length should be
for each wave type and to predict the wave types dominating along the truss.
First I calculate the group speeds for each wave type. The group speed (equal to
phase speed) of compressional waves cl in a strut is[ 2]
c, =
-=
5050 m/s
(6.1)
where E is Young's modulus and p is the density of aluminum. Similarly, the group speed
of torsional waves ct in a strut is
c =
= 3115 m/s
(6.2)
where G is the shear modulus of aluminum. The flexural wave speed is dispersive. In a
strut, phase speed is related to frequency using[ 3 ]:
1 rouer
Cb =2=
2
rner
in+ .
(6.3)
where the radius of gyration i is equal to 0.004 m for a strut. The group speed of flexural
waves is equal to twice the phase speed.
I present again the four experimental curves of group speed, from Figure 4.4, with
the predicted group speeds for each wave type in Figures 6.1 - 6.4. These figures show
the measured group speeds are bounded by the fastest and slowest predicted wave speeds.
In all four frequencies shown, the experimental group speeds suggest waves are
predominately compressional waves near the source and some combination of wave types
as distance from the source increases. (The reader should not misinterpret the
predominance of compressional waves near the source. Experimentally they arrive first
and are therefore readily detected before other wave types can overwhelm them. Thus in
a steady state experiment, other wave types could be much more energetic near the source
than compressional waves.)
I predict that the first arrival of energy, found experimentally, is a compressional
wave. However, in my experiments the compressional wave traveling along the base
struts is not detected because its magnitude is small compared to either compressional
waves arriving through other paths or due to other wave types. If the former applies the
compressional wave must travel some path longer than the shortest path between base
joints. I examine the possibility of compressional wave arrival along other paths such as: a
path of connecting diagonals along the centerline or paths comprising direct and back path
combinations. To accomplish this I calculate the required accumulated length of strut
corresponding to a purely compressional wave. I present this length as a non-dimensional
ratio LRcomp of the assumed base strut length by simply computing
LRcomp =
1
(6.4)
where cm represents all measured group speeds. I conduct this ratio for all data and
average the four frequencies at each joint. The results are factors by which the base strut
lengths, at each joint, must be multiplied if the first arrival of energy is purely a
compressional wave. For comparative purposes I conduct the same calculations for
torsional and flexural group speeds. The results are presented in Table 6.1.
Wave
b3
b5
b7
b9
bll
b13
1.9
1.5
1.5
1.3
1.3
1.1
Compressional
1.2
1.0
0.9
0.8
0.8
0.7
Torsional
0.9
0.7
0.7
0.7
0.6
0.5
Flexural
Table 6.1: Length Ratio: length corresponding to arrival of predicted wave speeds over
length of shortest path, presented for each wave type at b joints.
From Table 6.1 I observe the first arrival of a compression wave is through a path
of length 1.3 - 1.9 times the base strut length. A path comprising connecting diagonals
along the centerline has a length ratio of 1.7 and thus provides another feasible path
option. However, I believe some back path or combination of wave types better explains
the increased travel length.
At increasing distances from the source, Figures 6.1 - 6.4 and Table 6.1 indicate
that experimental group speeds approach, and in some frequencies coincide with, the
predicted torsional wave speed. Experiments approach but never meet predicted flexural
wave speeds. This behavior supports my belief that wave scattering occurs in the truss.
Compressional, torsional and flexural energy multiply scatter until a balance of wave types
is reached.
6000
-
·
1
1
1
I
I
r
I
5000
-
4 kHz Experimental
*
4000
--
-.-
E
a 3000 - ------------ ---
Predicted Compressional
Predicted Torsional
Predicted Flexural
------------------
o0n
2000
1000
5
6
7
8
9
Station Number
10
11
12
Figure 6.1: Experimental and predicted group speeds for compressional, torsional and
flexural waves in octave with center frequency equal to 4 kHz.
6000
500o0
£
4000
3000 F-
8 kHz Experimental
m Predicted Compressional
1000
O
- - Predicted Torsional
- - Predicted Flexural
'
U3
3
I
4
•
II
5
.
I
6
•
I
I,
•
I
,
9
7
8
Station Number
.I
10
.W
11
I.
12
Figure 6.2: Experimental and predicted group speeds for compressional, torsional and
flexural waves in octave with center frequency equal to 8 kHz.
6000
5000
*
*
-X
4000
E
- - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - --
L3000
U)
0
2000
*
1000
E
--
16 kHz Experimental
Predicted Compressional
Predicted Torsional
- - Predicted Flexural
·
--
I
7
8
9
Station Number
I
I
I
10
11
12
Figure 6.3: Experimental and predicted group speeds for compressional, torsional and
flexural waves in octave with center frequency equal to 16 kHz.
600C ·I ___
X
W
500o
4000
3000
E
2000
E
--
1000
32 kHz Experimental
I
Prc-•c-.ed Compressional
--
Predicted Torsional
E
- - Predicted Flexural
I
I
4
5
6
7
8
9
Station Number
I
I
10
11
--
12
Figure 6.4: Experimental and predicted group speeds for compressional, torsional and
flexural waves in octave with center frequency equal to 32 kHz.
13
6.2
Prediction of Frequency of Transition Between Global and Local Truss
Dynamics
I consider the truss to experience global dynamics when it vibrates as a beam in a
low order mode. In this case I treat the truss as a single component with amorphous
internals. When the truss loses this low mode beam behavior, I attribute the more
complex motions to local vibration effects. The transition between the two effects
depends on the wavelength of a particular wave type in comparison with the length of the
struts. In this section I predict this transition for each wave type.
First consider global flexural motion. Combining Equations 6.3 with the simple
relation cb =f b , frequency as a function of flexural wavelength is:
=(11.21
(6.5)
wheref is in Hz and ,b is in meters.
Classical beam bending theory is applicable if the global flexural wavelength is
larger than about 8 times the beam thickness, which I take for the present case to mean
that the global wavelength should be about 8 times the diagonal strut length. Because the
truss has a length to height ratio equal to 5 1/2, I expect the above criterion to correspond
to the first mode of truss flexure. The transition frequencyft below which global behavior
might be anticipated is then:
f
t,
(6.6)
where Ld is the diagonal length. From this I obtainft=3 Hz, which shows that nowhere in
the frequency range of interest is flexural global motion to be expected. Indeed when
Xb=2Ld (about 50 Hz) the diagonals are in their first flexural mode, further emphasizing
the lack of global flexural motion.
I next consider global dynamics of the truss associated with compressional and
torsional waves. In analogy with Equation 6.6, the transition frequencies for each,
respectively, are:
f = c
8Ld
(6.7)
f = ct
8Ld
(6.8)
which work out to beft= 780 Hz andft=480Hz respectively. Thus, global dynamics can
be expected for octaves up to 500 Hz center frequency. In all likelihood, the attenuation
curves of Figure 3.2, which show high relative accelerance at each end of the truss, are
caused by the low order free-free modes of the compressional and torsional waves. The
first modes of each are calculated to be 537 and 331 Hz respectively, with the use of
Young's and shear modulii, and density of aluminum without regard to their substantial
reduction associated with the rather open truss cross-section. Perhaps some comfort for
this disregard of reality is that at least the compressional wave speed observed near the
source is not far from that of solid aluminum.
6.3
Calculation of Radiation Loss Factor of a Strut
Flexural waves traveling in the strut radiate energy into the surrounding fluid. The
radiation loss factor rs is a measure of this coupling and is independent of strut length. It
is affected by three related characteristics: the radius and mass of the strut, and the
frequency of vibration.
For the strut to radiate efficiently the flexural wave speed cb must not be smaller
than the speed of sound in air c. The flexural wave speed depends on the cross section of
the strut and frequency. Equating the two speeds gives the critical frequency below which
no radiation occurs[ 4 ]:
fc
=
(6.9)
The critical frequencyfc is 950 Hz. Although there is an asymptotic effect near the critical
frequency, it suffices to assume an efficiency Y of zero below and unity abovefc.
Furthermore, strut radiation varies with the non-dimensional frequency ka where k
is the wave number of sound in air and a is the outer radius of the strut. This product is
simply a ratio of circumference 2=a to the wavelength of sound.
I model the strut as a vibrating wire with radial radiation as a function of angle 0
from the plane of radiation. The strut has velocity component Uocos re-i2nft
cylindrical coordinates the wave equation is a Bessel equation. From the Bessel equation,
the pressure p of the radiated wave is[ 5]
P = a[JI (lkr) +iY,(kr)] cosOe-'
2
(6.10)
where Jl(kr) and Yl(kr) are the first order Bessel functions of the first and second kinds
respectively, a is a constant and r is a radius not less than a. The velocity of air at radius r
is
1 p
ur
(6.11)
i2;afppc
where p is the density of air. Differentiation of Bessel functions yields zero and second
order functions. The form of these functions is shown in Figure 6.5. Solving for velocity
reveals
u, = a [(Yo(-)- Y,(kr))-i(Jo(kr) - J,(kr))]cos e - '2
2cp
(6.12)
where c is the speed on sound in air. The constant a is solved by equating ur to the
surface velocity of the strut. The sinusoidal terms cancel and by letting B equal the entire
complex Bessel term calculated at r=a, a becomes
a=
B
Uo.
(6.13)
The next step involves calculating the radiation intensity y as a function of the far
field pressure and velocity. When r is large the Bessel functions present in pressure and
velocity equations are replaced by sinusoidal approximations. The simplified equations
are[5]
pco=a
-cosqfe
4
.
a
c
PC
-ft
osek(r-ct)-
(6.14)
(6.14)
C
0
0.5
1
1.5
2
2.5
3
ka
Figure 6.5: Bessel functions plotted as a function of non-dimensional frequency ka.
3.5
4
Radiation intensity is the time averaged product of pressure and velocity:
1
Y= (pu ), = IRe(p"u.)
(6.15)
Pressure, velocity and a are substituted and exponential terms cancel leaving the following
relation for far field intensity per unit strut length:
2c&pU 2 cos2 q
fr BB*
r7'
0
27
c=pUo
(6.16)
Next the radiated power per unit strut length is calculated. Power is the time
averaged product of force and velocity. Given the relationship for intensity, power per
unit strut length is
rf
-I=
Hr =
ydo=
2c 2p U 2
o.
(6.17)
The resistive impedance per unit strut length is then calculated using the relation:
2HI
4c
p
4C2p(6.18)
R=2
u 02
7f BB
Finally, dividing R by com where m is mass per unit strut length gives the nondimensional strut radiation loss factor 77s. The term B is multiplied by its conjugate B*
and the product is replaced by the appropriate Bessel functions. Including the radiation
efficiency T the strut radiation loss factor becomes
8 pn 2
77 = 8p
1
1
r m (ka) (Jo-JY)
I.
(6.19)
The equation indicates the dependence on the mass ratio, non-dimensional
frequency, and critical frequency. The frequency term in the denominator is misleading
because the behavior of the Bessel functions make /sincrease with frequency. I calculate
the strut radiation loss factor for each frequency octave and plot results in Figure 6.6.
-3
10
-4
00
10
+P
L
0
t.t•
+I
O
+d
0
__j
-6
10
-71
't"'
iU
10
-2
10
-1
10
0
ka (k = k of air, a = radius of strut)
Figure 6.6: Strut radiation loss factor vs. ka plotted for each frequency octave band. The
circles are points where ris is calculated and crosses are points at frequencies belowfc.
6.4
Calculation of Radiation Loss Factor of a Joint
The joint radiation loss factor is determined using a process similar to that of the
strut. I model the joint as a sphere where radius a equals the actual joint radius. The joint
radiation also is affected by three related characteristics: the radius and mass of the joint,
and the frequency of vibration, but the radiation efficiency T' is always unity.
The joint surface has radial velocity Uocosp ri2nft. The equation of pressure at
radius r is determined by solving the wave equation using spherical coordinates:[ 5]
P = a(P,cos()[j, (kr) + iy,(kr)]e - i2x
(6.20)
where jl(kr) and yl(kr) are the first order spherical Bessel functions of the first and second
kind respectively, and Plcosp is a Legendre function. The spherical Bessel function is
related to the standard Bessel function by
j, (kr) = /•
,_(kr).
(6.21)
The velocity is determined by differentiation as is done for the strut. The term Plcosqo
equals cos~p because the order is one. Solving for velocity reveals
u, = --a[(yo(kr)-2y2 (r))-i(j 0 (kr)-2j2(kr))]cospe-,z2.
jcp
(6.22)
The form of the spherical Bessel functions is shown in Figure 6.7. Arbitrarily, the
simplification term B replaces the entire complex spherical Bessel term and a is solved by
equating ur to the surface velocity of the joint:
3cp U
B
where B is evaluated at r=a.
(6.23)
t-0
O
Co
C.)
t-
ci
:3
LL
0)
7a-
a,
c.
t,o
-r
-1)
0
5
10
ka
Figure 6.7: Spherical Bessel functions plotted as a function of non-dimensional frequency
ka.
15
The equations for pressure and velocity using far field approximations are
c
cos(pe
PcO= -a2fr
uC= -a
1
ik(r-ct)
(6.24)
-k(r-ct)
47rpfr
cosPe
The joint radiation intensity of one entire joint is
9c3 pU0 2 cos2 p
,,Cs(6.25)
y161ýf2-rf
BB*
Y=
The power is calculated by integrating intensity around the sphere's surface:
J= P
yr2 sin(pdp= 4
B Uo.
(6.26)
The resistive impedance for the joint is
R=
21I
Uo
3cjp
3
2 nf 2 BB*
(6.27)
Finally, the non-dimensional joint radiation loss factor rr is determined by dividing
the resistance by caM where M is the total mass of one joint:
93 3 1
91
2
M
1
13(6.28)
(ka)3 [(jo-2j2)2+(yo-2Y2)']
2
I calculate the joint radiation loss factor for each frequency octave and plot the results in
Figure 6.5. The non-dimensional frequency is higher than that of the strut (because a is
larger); the presence of oscillation in the curve is due to the oscillating spherical Bessel
functions shown in Figure 6.8. There is no cutoff frequency for the joint.
I
1
77
1rl1
1
1
It717
1
I
I1
17r1
I
1
-r7/
-/a
/c
/~
0I`
-5
10
/I
0d
10
10
0I
-6
-8
A
-2
10
10
0
-1
10
ka (k = k of air,
Figure 6.8: Joint radiation loss factor
a =
10
radius of Joint)
vs. ka plotted for each frequency octave band.
vsj
2
10
Other Mechanisms of Attenuation
6.5
Other mechanisms of attenuation not yet determined analytically are losses:
through structural damping, to ground through truss supports and in the interfaces
between truss components.
The loss factors due to internal structural damping in a strut and joint are not
calculated. Instead, in Section 6.8 I determine a feasible structural damping rid (order of
magnitude) in a strut due to comparison with predicted and measured attenuation.
I predict that the attenuation due to energy propagation through the truss support
springs is negligible. The natural frequency of rigid translation of the slung truss is less
than one Hertz so that the bungee chords provide excellent isolation at frequencies above
100 Hz. Furthermore, Figure 6.9 shows the difference in acceleration measured at either
end of one support. At all frequencies the acceleration of the pipe from which the truss is
slung is at least 35 dB below acceleration at the support connection, when the truss is
excited by the vibration generator.
C
c)
0
a,
61
a)
<
0I
0
5
10
15
20
Frequency (kHz)
25
30
Figure 6.9. Acceleration of truss at bungee chord connection divided by acceleration of
ground at support base.
r%
35
The loss to heat in the interfaces is considered. As shown in Figure C.4, the tight
aluminum to aluminum contact between a strut and joint is limited to a very small area
causing possible friction losses. Otherwise, metal to metal contact in the interface is too
loose to provide excellent coupling. In addition, the epoxy provides a path through which
energy can pass. Although the loss factor due to the damping of epoxy is as high as 0.5 at
low frequencies[ 6], the actual applicable loss factor associated with the irregular form and
interface is difficult to determine analytically.
6.6
Application of Loss Factors
The loss factors calculated in the preceding sections are applied to the truss so that
the expected structural attenuation of mean square acceleration is determined. My aim is
to predict attenuation A along the truss and the expected slope of attenuation (A /x) as a
function of non dimensional frequency ka. These predictions can then be compared to
experimental data. Because the loss factors are non dimensional quantities, I introduce
appropriate quantity scales to account for the additive effect of all components.
To determine the attenuation due to radiation from the joints, I multiply the loss
factor by the number ofjoints at each section and convert to a function of axial distance.
The slope of joint attenuation is
A
x
-
3J(6.29)
0.384
where the axial distance between sections is 0.384 m. The slope of attenuation for each
octave is presented in Table 6.2.
To determine the attenuation due to radiation from the struts, I assume an
exponential relationship with strut length. The root mean square acceleration as as a
function of cumulative strut length is[7]
a, = a0 e -
kbL
(6.30)
where kb is the flexural wave number and L is the path length of strut. In calculating
attenuation As , the root mean square acceleration at a given location is normalized by that
at the first joint. This ratio is
As = as ejkb(
(6.31)
or, presented in dB and absolute value, becomes
A, (dB) = 8.686r7,kbL.
(6.32)
By combining the above with Equation 6.5 and converting from total strut length to axial
distance, the slope of attenuation is determined. The axial distance between sections is
0.384 m and the total strut length between sections is 6.686 m. After converting to axial
distance the slope of attenuation as a function of frequency is
As = 84.85 r J.•
(6.33)
The slopes are presented in Table 6.2 and plotted in Figure 6.10. Table 6.2 indicates that
attenuation due to radiation from joints is negligible when compared to the effects of strut
radiation. This is because the total strut surface area is much greater than that of the
joints.
Octave Center
Frequency (Hz)
Strut ka
A /ix for Strut
Radiation (dB/m)
Joint ka
A /ix for Joint
Radiation (dB/m)
125
250
0.0145
0.0000
0.0723
0.0000
0.0289
0.0000
500
1000
0.0578
0.1156
0.0000
0.0578
0.1446
0.2891
0.5782
0.0000
0.0000
0.0003
2000
4000
8000
0.2313
0.4626
0.9252
0.3369
1.8568
1.1565
2.3129
4.6259
0.0014
0.0020
0.0011
.
5.5680
9.2518
0.0006
5.6903
1.8504
18.5035
0.0026
3.7007
4.1240
32000
Table 6.2: Theoretical attenuation slopes for strut and joint radiation and the respective
values of non dimensional frequency ka for each octave center frequency.
16000
I also calculate the attenuation of flexural waves in the struts due to three possible
structural damping loss factors: 17d = 10- 6, 10-5 and 10- 4 . The results in Table 6.2 allow
me to neglect joint structural damping. Again, I assume the form of exponential decay as
a function of strut length and examine structural damping due to flexural waves. For
flexural waves Equation 6.33 is modified by inserting the structural loss factor:
Adflx
= 84.8517d
.
(6.34)
The slopes of attenuation due to strut radiation and structural damping are plotted in
Figure 6.10. The radiation curve is low at low frequencies as influenced by the loss factor
r/s. The curve drops asymptotically to zero at ka=O0.11 corresponding to the critical
frequency determined in Equation 6.9. The slopes of flexural wave structural damping
(order 10- 6, 10-5 and 10-4 ) increase as a function offl / 2 . In Section 6.8, comparison of
experimental and predicted attenuation allows selection of the most suitable r7d.
Similar calculations lead to predicted slopes for structural damping of
compressional and torsional waves, respectively:
Ad,comp = 0.19ddf
Ado,,
= 0. 3 1d7df .
(6.35)
(6.36)
These are plotted in Figures 6.11 and 6.12, again for rid parameterized as: 10-4 , 10-3 and
10-2. Inasmuch as the wavelengths of these waves are much larger than Ab, larger values
of r7d are used here to fit within the figures, i.e. to give slopes comparable to the measured
ones.
102
-Strut
+
101
Radiation
*
Structural: Loss Factor = 10"-4
Structural: Loss Factor = 10"-5
x
Structural: Loss Factor = 10^-6
100
+X
+
co
*
+
X 101-
+
*
10-2
x
x
x
x
x
-3
*
X
*
10
x
' " '
I
I
10
I
*
I
I
*i
1
)
(
10
ka
Figure 6.10: Attenuation slopes as a function of non dimensional frequency ka due to
predicted: strut radiation, and structural damping in struts due to flexural waves for rd =
10 - 6, 10 - 5 and 10 - 4 .
101
102
101
L.
CD)
*10
E
0
CL
co)
i 0n
0`
102
10-1
100
Figure 6.11: Attenuation slopes as a function of non dimensional frequency ka due to
predicted: strut radiation, and structural damping in struts due to compressional waves for
-2
rid = 10- 4 10 - 3 and 10 .
101
102
101
0
a)
• 10
E
L..
co
x 10-1
10-2
1-
3
102
100
101
I
Figure 6.12: Attenuation slopes as a function of non dimensional frequency ka due to
predicted: strut radiation, and structural damping in struts due to torsional waves for r7d
10-4, 10-3 and 10-2 .
101
6.7
Determination of Experimental Attenuation Slopes
To determine experimental attenuation slopes, I fit linear curves to the data in
Figures 3.2 and 3.3 using a least squares approach. Visual inspection of the data in each
frequency band reveals that perhaps two different slopes exist, one near and another far
from the source. The apparent transition from compressional to torsional and/or flexural
waves found in the wave speed experiments of Chapter 4 supports this approach.
Therefore, using an optimization code, included in Appendix F, I search for the best fit of
either one or two curves. Where two curves prove to be better than one based upon a
minimization of least squares error, the optimization also finds the best position where the
change in slope occurs. I call this point the transitionpoint. The data at end sections 1
and 13 are ignored because the structure differs and end effects may distort the data.
These best fit results are displayed in Figures 6.13 and 6.14, for those bands in
which non-global response is dominant. For all frequencies two curves produce the best
fit. Except for the 2 kHz band, the transition point generally occurs around the 1.5 meter
axial distance (at b5) for all frequencies, as is shown in Figures 6.13 and 6.14. This
observation suggests that wave type scattering is frequency independent at this distance. I
accept this observation knowing that, by the time energy reaches the fifth section,
sufficient scattering has occurred as is shown by the group speed curves in Figure 4.4.
I kHz
0
cD
5
2 kHz
0
0
(]
nr
j
4 kHz
M
0
C
-O-
)
~-^""^"""'
Axial Distance •luriy
I iuo,
II
4
data of Chapter 3, for octaves: 1, 2 and
experimental
the
of
fits
curve
Best
6.13:
Figure
kHz.
8 kHz
-40
v
0
20
I
c o•L
(
0
0.5
1
I
I
1.5
2
~~-:I~
2.5
I
3
3.5
4
4.5
3
3.5
4
4.5
1.5
2
2.5
3
3.5
Axial Distance Along Truss (m)
4
4.5
16 kHz
- 40t-40
S20
0
0.5
1
2
2.5
32 kHz
C
:
1.5
40
_----
tO
0
0.5
1
Figure 6.14: Best curve fits of the experimental data of Chapter 3, for octaves: 8, 16 and
32 kHz.
6.8
Comparison of Theoretical and Experimental Attenuation
I present the slopes of experimental data alongside the theoretical predictions. The
experimental slope represents the total loss per axial length of the truss and comprises the
following presumed significant losses:
a.
b.
radiation from flexural waves in struts, and
structural damping of flexural, compressional and/or torsional waves in
struts.
I plot the experimental far and near attenuation slopes, determined in Section 6.7,
as a function of ka in Figures 6.15 and 6.16 respectively. The magnitude of experimental
slopes near the source is higher than that farther away. To understand this behavior I
hypothesize, based upon wave speed experimental results in Figure 4.4, that attenuation is
greater near the source because of the high degree of scattering from predominantly
compressional and/or torsional to flexural waves. To substantiate this I compare the
magnitudes of predicted attenuation with experimental results.
In Figure 6.15, I plot the slopes of experimental attenuation far from the source
and compare with predicted losses of radiation plus flexural structural damping. From
Figure 6.10, I add the curve of predicted strut radiation to one of the flexural structural
damping curves; I select the structural damping curve representing ?rd=10 -5 because the
sum produces the best fit to the experimental results. Similarly in Figure 6.16, I compare
experimental attenuation slopes near the source to two predicted slopes: strut radiation
plus the appropriate compressional structural damping and strut radiation plus torsional
structural damping. The best fit is achieved with id=J10-3 for both the compressional and
torsional cases.
The structural damping near the source due to compressional and torsional waves
is much greater (two orders of magnitude) than the flexural structural damping far from
the source. This physical result supports my prediction of scattering near the source as the
beginning of the attenuation process. The scattering, which is conservative within the
truss, can be considered a true loss via radiation of flexural waves. Thus, loss near the
source is caused by both scattering and structural damping of value rld=10-5 for all wave
types.
2
E
CL
m
x
1
10
-2
10
-1
10
0
I
Figure 6.15: Attenuation slopes plotted as a function of nondimensional frequency ka for:
predicted loss due to strut radiation plus flexural structural damping (i7d=10-5), and
experimental total attenuation far from the source.
10
1
2
10
S
l
opI
IStu
R
P
C om
Is
i
S t
ca
- Slopes of Predicted Strut Radiation Plus Compressional Structural Loss
-- Slopes of Predicted Strut Radiation Plus Torsional Structural Loss
.. Experimental Slopes Near Source
1
10
E
L_
X10I
-o
[
A^-3
f-I LIIU
-2
10
I I
-1
I
10
10
0
ka
Figure 6.16: Attenuation slopes plotted as a function of nondimensional frequency ka for:
predicted loss due to strut radiation plus compressional structural damping (7ld=10-3),
predicted loss due to strut radiation plus torsional structural damping (7d=10- 3 ),and
experimental total attenuation near the source.
Chapter 7
Conclusions
I identify important characteristic lengths in the truss. These are:
a.
b.
c.
lengths corresponding to global vs. local dynamics,
wavelengths corresponding to the transition from inefficient to efficient
strut radiation, and
periodic strut lengths resulting in stop band behavior.
Experimentally, the truss experiences global dynamic behavior below 1 kHz.
Analytically, I determine that this transition between global and local effects is due to the
requirement that compressional and torsional wavelengths be much larger than the cross
dimension of the truss. The transition for flexural waves from global to local behavior
begins at about 3 Hz and is complete at about 5 Hz.
By comparing the flexural wavelength in a strut to the wavelength in air I
determine that efficient strut radiation occurs above 950 Hz. Such acoustic radiation is an
important element of truss damping, at least for one without added treatment. The data
are constant with such an analysis.
All non-global motion curves of attenuation are best fit by two slopes, with
attenuation greater near the source. Predicted attenuation due to strut radiation accounts
for the measured attenuation for all frequencies above 950 Hz, far from the source of
excitation. Close to the source, the attenuation per unit axial distance occurs at a higher
rate because two mechanisms occur simultaneously: the struts excited in flexure radiate
efficiently, and energy in compressional and/or torsional waves scatters to energy in
flexural waves. Also, a more equal balance of compressional, torsional and flexural waves
is established at some distance (5 sections in my experiment) away from the source,
because flexural waves rescatter to the others.
I conclude other mechanisms of attenuation, including radiation from joints, losses
in the interfaces between components and losses to ground through the supports, are
negligible in comparison to the attenuation of scattering and strut radiation.
A temporal analysis provides the experimental axial group speed in the truss as a
function of frequency. I calculate expected group speeds for all wave types, as a function
of material, form and (for flexural waves) frequency.
The results of experimentation and supporting theory allow me to offer some initial
truss design ideas. In a full scale prototype of my 1/15 scale model, global truss dynamics
must be considered, especially where the truss is effectively decoupled from the
submarine's outer shell. The transition between global and local effects depends upon
compressional and torsional wavelengths which scale in proportional tof In full scale I
predict transition frequency is between 10 and 100 Hz (using Equations 6.7 and 6.8),
which is above the rotational frequency range of the main shaft.
Because the losses due to scattering at joints are significant, a design should
maximize the number ofjoints. This is achieved by minimizing cell size within the
constraints of submarine design. It is true that large free spaces must be present in a full
size truss for machinery and access, but the truss can be designed around these
requirements.
Selection of angles between connecting struts is important to maximize the
scattering of wave types. Connections at or close to right angles produce excellent
scattering.
Designs that maximize the quantity of unobstructed struts allow the largest degree
of attenuation due to strut radiation. This is important because strut radiation is the
largest mechanism by which flexural waves are attenuated in an untreated truss. Scaling
lowers the critical frequency, proportionally with strut radius, from 950 to 60 Hz.
Furthermore, if periodicity exists and quantity of same length unobstructed struts is large,
then the design can benefit by considering the damping provided by stop bands.
The design of such a truss would not be easy. By designing a truss around the
demands of equipment and personnel rather than fitting things into predetermined deck
space, an effective structure that maximizes the attenuation of machinery noise can be
developed.
References
[1]
Ingard, K.U. Fundamentals of Waves and Oscillations, Cambridge University
Press, Cambridge, 1988.
[2]
Norton, M.P. Fundamentals of Noise and Vibration Analysis for Engineers,
Cambridge University Press, Cambridge, 1989.
Mark's Standard Handbook for Mechanical Engineers, 1978.
[4]
Fahy, F. Sound and Structural Vibration - Radiation, Transmission and Response,
Academic Press, London, 1985.
[5]
Morse, P.M. Vibration and Sound, McGraw Hill Company, New York, 1948.
[6]
Snowdon, J.C. Vibration and Shock in Damped Mechanical Systems, John Wiley
& Sons, New York, 1968.
[7]
Rao, S.S. Mechanical Vibrations, Addison-Wesley Publishing Company, New
York, 1990.
[8]
Bruel and Kjaer, Mechanical Vibrations and Shock Measurements, K. Larsen and
Son, Denmark, 1984.
Appendix A
Truss Detailed Design
A truss consisting of a repeated square based pyramid cell is constructed using
6061 T6 aluminum. Eleven cells in total comprise the structure; six are upright and five
are inverted. Lengths of aluminum tube form the struts and sections of aluminum bar are
machined to form joints. The struts are fit into drilled holes in the joints and are affixed
with two part epoxy. Figure A.1 shows pyramid-based single cells, the assembly of the
cells in series and the complete construction with all members assembled.
o,. Square-ioase
pyrarncJ cel1s 1
/
/
-
N
b, yyrar id a
c. Complete
Figure A.1: Assembly of the truss from single square-based pyramidal cells.
Dimensions
The overall truss is designed to have a square cross-section and a length to width
ratio of greater than five. These guidelines are based upon the distances between vertices
(not extremities of the joints). A single cell forms a cube with sides equal to 0.775 m (30
1/2 inch), as shown in Figure A.2. The overall truss dimensions measured at the
extremities are:
length: 4.713
width: 0.838
height: 0.799
(x 15 1/2 ft),
(• 2 3/4 ft), and
( 2 5/8 ft).
77,5
-7.-7
-
-
-
-
77.5
/
77 F,5
j
7 /
Figure A.2: Dimensions of a truss cell; shown are dimensions between vertices (cm).
Components
The truss comprises a total of 109 struts and 35 joints. The three different lengths
of strut are identified and quantified in Table A.1. The shortest struts on the horizontal
plane are termed base struts, and the tube that bisects the square base is called the support
strut. The struts in the vertical plane are diagonals. The struts are cut from stock
aluminum tube that has the following characteristics:
material:
material density:
Young's modulus:
outside diameter:
wall thickness
cross-sectional area:
mass/length:
6061 T6 aluminum,
2700 kg/m 3,
6.89x10
11
N/m2 ,
1.27 cm (1/2 inch),
0.165 cm (0.065 inch),
0.573 cm 2 (0.09 inch 2), and
0. 154 kg/m.
Strut Type
Quantity
Cut Length
44
50.8 cm (20.0 inch)
Base
Support
11
73.7 cm (29.0 inch)
Diagonal
54
83.8 cm (33.0 inch)
Table A.1: Lengths and quantities of strut types.
Exposed Length
48.4 cm (19.1 inch)
70.0 cm (27.6 inch)
81.0 cm (31.9 inch)
The joints are machined from 2 1/2 inch aluminum bar in accordance with the
drawings of Figure A.3. Although all joints are the same size, there are six different
configurations of hole locations as indicated in Figure A.4. Characteristics of the joint
include:
material:
material density:
Young's modulus:
outside diameter:
average mass:
6061 T6 aluminum,
2700 kg/m 3,
6.89x101 1 N/m2 ,
6.35 cm (2 1/2 inch), and
0.12 kg.
1/2 oriL
-
N
26,
3/4
1/4i
-Joint Dimensions
II Units: inch
Marcus Heath
Sep 93 1Scate: 1:1 1Al 6061 T6
Figure A.3: Joint Drawings.
/Type
1
Quant ty
2
Type 4
Quantity: 18
/Type
2
Quan;tty' 2
Type 5
Quantty.
2
i
ype. 3
Quant:ty
9
Type 6
Quantity.
2
Figure A.4: Pattern of holes in and quantity of the six node types.
To obtain a tight interface between the joints and struts the diameter of holes
drilled equals the average actual diameter of tube stock, with a tolerance of ± 0.5 mm. In
addition, the inner surface of each hole is roughened with a punch so that the struts must
be forced into position during assembly.
Assembly and Testing
The tight fit between components makes assembly rather difficult. The ends of
each strut are lightly filed to remove any burrs and the components are laid out in
preparation for assembly. Two part epoxy is applied to the ends of each strut before
insertion into the joints. All diagonal struts are fitted first. They are inserted as far as
possible into the joints, until they make contact with the ends of adjacent diagonals.
When all diagonals are correctly inserted, the entire assembly should be left for
twelve hours, to allow the epoxy to set. When dry, the truss is oriented so that the base
and support struts are lowered into the prepared slots, allowing a 0.5 cm overhang.
Additional epoxy is applied at the interface to ensure a solid connection. The truss should
again be allowed to dry before turning it over to insert the remaining base and support
struts.
The final product, weighing 16 kg (35 1/2 lb.), is tested for strength. The entire
truss is to be supported by a single strut and no cracking noises are to be detected. This
test is to be repeated for several struts. Although the assembly should survive a two foot
drop test onto concrete, this test should not be conducted!
Appendix B
Added Mass Effect of a Sensor on a Strut
An initial analysis is conducted to determine the error in frequency obtained when
a sensor weighing 1 gram is mounted on a strut at mid-length. The first analysis explores
longitudinal effect while the second studies beam bending.
The simplicity of the longitudinal study permits an analytical solution to be applied.
The strut is assumed to be a beam fixed at one end as shown if Figure B.1. For the
unloaded case the wave equation for compressional vibrations is
(u
1 6u
&2
C12 072
(B.1)
where u is axial velocity and cl is the compressional wave speed in aluminum. The
solution to the wave equation is
u(x, t) = A cos
+ B sin( •-))(Dcos(cot) + E sin(cot)).
(B.2)
With boundary conditions fixed at one end and free on the other, the natural frequencies
are
con =
(2n + 1)=n
21
Figure B. 1: Model of added mass on strut for compressional vibration added mass
analysis.
(B.3)
For the mass loaded case of longitudinal vibration the free end boundary condition
is modified to
= -M
AE
(B.4)
where M is the added mass of the sensor. Applying the boundary conditions leads to the
following frequency dependent equation:
tan
)
fAEM --0.
(B.5)
The natural frequencies con corresponding to the first five compressional vibration modes
in both the unloaded and loaded conditions are calculated. The percentage difference
between the two is determined and the results are shown in Table B. 1. The errors, less
than 10% in all cases, are acceptable for this first order approximation. The error
decreases with increasing frequency. Note that for lower modes the error increases
approximately linearly with the magnitude of the added mass.
Difference (%) I
Loaded cn
7.1
46,080
6.7
77,120
6.2
108,500
115,700
5.7
140,200
148,700
5.3
172,200
181,800
Table B. 1: Difference between unloaded and loaded theoretical natural frequencies of the
strut under compressional vibration.
Mode
Mode
Unloaded o),
49,580
82,630
The effect of added mass on natural frequency is more applicable when considering
the strut bending case. The solution is most easily determined using a finite element
analysis. The model consists of a strut clamped at both ends with a 1 gram point mass
applied at mid length as shown in Figure B.2.
Figure B.2: Model of added mass on strut for beam bending added mass analysis.
The natural frequencies aon corresponding to the first four odd modes are
computed for the unloaded and loaded cases using the ADINA finite element software
package. The two cases are identical for even modes because the mass is added at an antiresonance. The natural frequencies and differences are displayed in Table B.2 and ADINA
outputs for modes 1,3 and 5 are given in Figure B.3. Again, the small error created by
adding a mass is acceptable.
Mode
1
Unloaded co
102.2
Loaded con
101.2
Difference (%)
1.0
3
551.8
547.7
0.7
5
1360
1350
0.7
2522
2504
0.7
7
Table B.2: Difference between unloaded and loaded theoretical natural frequencies of the
strut under flexural vibration.
ADINA
MODE SHAPE
MODE 1
F = 102.2
REFERENCE
L, ~
0.1387
ADINA
REFERENCE
MODESHAPE I - j
MODE 3
0.1676
F = 551.8
MODESHAPE r-l!XVMIN
3.304
MODESHAPE
-XVMAX
3.134
0.000 r ADINA
XVMAX 0.8400 MODE-SHAPE
YVMIN -0.1850 MODE 1
F = 101.2
YVMAX 1.000
REFERENCE
L0.1387
MODESHAPE
XVMIN 0.000
0.8400
YVMIN -0.2084
YVMAX 1.224
REFERENCE
- J
0.1669
MODESHAPE XVMIN 0.000
-XVMAX
0.8400
3.130
YVMIN -0.2030
YVMAX 1.223
ADINA
MODESHAPE
MODE 3
F = 547.7
..
3.275
XVMIN 0.000
XVMAX 0.8400
YVMIN -0.1850
YVMAX 1.000
..
.......
.
ADINA
MODE-SHAPE
MODE 5
F = 1360.
REFERENCE
MODESHAPE
XVMIN
0.1676
3.138
XVMAX
YVMIN
YVMAX
L-
0.000
0.8400
-0.2082
1.223
--
ADINA
MODE_SHAPE
MODE 5
F = 1350.
REFERENCE
L- j
C.1676
MODESHAPE
..
3.133
XVMIN 0.000
XVMAX 0.8400
YVMIN -0.2083
YVMAX 1.224
....
......
..
.....W
-
II
Figure B.3: ADINA outputs for clamped beam bending for modes 1, 3 and 5; left:
unloaded beam; right: beam loaded with a 1 gram point mass at mid length.
Y
Appendix C
Measurement of Strut Vibration
Measurement of acceleration on struts rather than joints provides the same
information but the signal to noise ratio is higher and that data acquisition is more tedious.
Because the mode of vibration changes with frequency and strut length, the location of
accelerometers along the struts must change if the peaks are to be identified. The modal
response of a typical strut at each octave is presented in Figures C.1, C.2 and C.3. In this
experiment a sensor is positioned vertically on base strut a4b3 (48 cm in length) and the
B&K shaker positioned at joint bl. The sensor is shifted along the strut by intervals of 2
cm and the accelerance is determined at each location. The graphs show the importance
of accelerometer position and illustrate that at higher frequencies averages of several
peaks must be taken.
These curves are useful in verifying predicted strut mode shapes with given
boundary conditions. Realizing that accelerometers produce only absolute values, the
sinusoidal wave forms are readily identified. The experiments confirm that the number of
modes increases with increasing frequency. As predicted in Chapter 6, the 125 Hz
response is relatively flat indicating that the flexural wavelength exceeds twice the strut
length. Only the zeroth mode is excited.
Furthermore, I use the experimental mode shapes to classify the type of end
conditions. The wavelength for each frequency is modified Lor diiTerent end conditions by
including a factor 0:
Ab =
11.2
.
(C.1)
where 0 equals unity for hinged and 2.27 for clamped end conditions[ 8]. Using this
relation I calculate the wavelengths for both hinged and clamped end conditions, and by
comparing the wavelengths with the strut length, I predict the number of modes expected
in each case. I compare predictions to the experimental results, found by counting the
number of evenly spaced lobes on each curve of Figures C.1, C.2 and C.3. The
comparisons for the first seven octaves are shown in Table C.1.
250
Hz
octave
0
-5
S-
50
//
Hz octave
"
-10
-1s
I
r%
0
--
5
10
15
35
30
20
25
Position on Strut a4b3 (cm)
40
45
Figure C.1: Accelerance versus position along strut a4b3 with total length 48 cm, for
octaves 125, 250 and 500 Hz.
84
50
f-%
14
12
4
2
0
0
5
10
15
20
25
30
35
Position on Strut a4b3 (cm)
40
45
Figure C.2: Accelerance versus position along strut a4b3 with total length 48 cm, for
octaves 1 and 2 kHz.
50
ID
I
I
I
I
I
I
kHz octave
-4
- - 8 kHz octave
14
12
10
4
-
I
I. /\
\
I~
/i
I'
I.
I
I\
I.'I
r1
*1
\*
/
I
I
I
5
10
15
I
I
\
'I
\
I
25
30
20
35
Position on Strut a4b3 (cm)
I
I
40
45
Figure C.3: Accelerance versus position along strut a4b3 with total length 48 cm, for
octaves 4 and 8 kHz.
50
Frequency
Hinged:
Clamped:
Experimental:
Hz
No. Modes
No. Modes
No. Modes
Best Model
125
0
0
0
Either
1
Hinged
0
1
250
Hinged
2
1
2
500
1000
3
2
2
Clamped
2000
4
3
4
Hinged
5
Hinged
4000
5
3
8000
7
5
6
Either
Table C. 1: Comparison of experimental mode counts with hinged and clamped
predictions in a base strut of length 48 cm.
Table C. 1 indicates that the experimental mode count does not match pure hinged
nor clamped end conditions. Instead, I conclude that the end conditions fall somewhere
between these two extremes. An illustration of the interface, shown in Figure C.4, shows
how the combined hinged and clamped model is applicable. To ensure a tight fit between
components, the inner surface each joint hole is roughed with a punch. This possibly
produces a hinge effect while the remaining aluminum contact and the epoxy couples the
strut to the joint (mass ofjoint >> mass of strut) and provides the clamped behavior.
eroxy
roughened
surface
strut
Figure C.4: Interface of strut and joint with added epoxy.
Appendix D
Pulse Analysis Data
Figures D. 1 - D.6 represent the raw data for each joint in the 32 kHz octave. The
curves are marked with an arrow to indicate where the data is taken and the time when
purely compressional wave energy is expected to arrive is identified with the symbol cl.
Figures D.7 - D. 11 show hammer spectra in the corresponding frequency ranges, and
Figure D. 12 shows the spectrum of background noise.
tz
O
(T)
m
I
1
r-I
0
110
U)
0)<
XID0
0H
0l L
011 >C
II a <Q
X--
C
Z
*
1Il3 a
TJ 11>O
O xn<
Xm (0
I IL>-
I
O0
)
0
ID x
1I LL
Figure D. 1: Hammer pulse (upper) at joint bl and arrival of energy at b3 in the 32 kHz
octave displayed in the time domain; the expected arrival of compressional energy is
marked by cl.
a
H
O
0
0
O
O
0
O
0)
U)
*
NU
w
aI
Nu
II
SI I
n
X> H
0-
w
r*% U
CUW
NUI
0*
4*10
NI
XUO
0
>0
El
0
O
0
XDM
LL>-
LO
ID
m
o
C
1I
1)
X
IL
Figure D.2: Hammer pulse (upper) at joint bi and arrival of energy at b5 in the 32 kHz
octave displayed in the time domain; the expected arrival of compressional energy is
marked by cl.
a
H
0
0,
0)
(U'-I
(D
U)
*(0
I (
)(-)
E
0( .1
*vim
(DO
XQ
in
IO
< ~1
Z
X1 O0
0
DU>O
XDl>0
I LL>-
Lq
I
Figure D.3: Hammer pulse (upper) at joint bl and arrival of energy at b7 in the 32 kHz
octave displayed in the time domain; the expected arrival of compressional energy is
marked by cl.
(
E
TI II
o II
ar>
III0
X0
E
OWJ
Orqw
U
ID
mC
Xi
>N
11( <
t II >)0O
X.L0
1.-
Xt·
Z
I
0
O0
O
0
0I
Figure D.4: Hammer pulse (upper) at joint bl and arrival of energy at b9 in the 32 kHz
octave displayed in the time domain; the expected arrival of compressional energy is
marked by cl.
r-f
(U
Uno
X>-
t<I<
I'o,,t
*'
,,
i
UJ
(T)JW0 .
O] >XU
U < %• M
X>-
V
Z
X
O
a• I
,, nl >0
0
M xn<Mm
I LL.-
13
0O
0
0 1'
0 x
1
Figure D.5: Hammer pulse (upper) at joint bi and arrival of energy at blN in the 32 kHz
octave displayed in the time domain; the expected arrival of compressional energy is
marked by cl.
Y
0.
Ht
0
0Q
0U)
IE
0
ENl
II
0t
)<I
X)-
mgl-
0 LOW
,oWI
Lf)lON 0
UrII > O
C"J
MW
>0I
4
0
0
I
Xr
O
II a>
LL>-
0
0
0\
N
Figure D.6: Hammer pulse (upper) at joint bi and arrival of energy at b13 in the 32 kHz
octave displayed in the time domain; the expected arrival of compressional energy is
marked by cl.
X=2.732kHz
Ye--79. 449
-- ''----POWE
-64.0
09kHz
AX=3.
AYa-1. 485
~
Y=-79.244 dBNrms
--
4.0
/Div
dB
rms
N
-96.0
Figure D.7: Hammer spectrum in 4 kHz octave.
X-5. 464kHz
AX-
.
242kHz
Ya--77.581
AYa-2.251
Ci
POWE
20Avg
-64.0
Y--78. 158 dBNrms
O%Ovlp
4.0
/Div
dB
rms
N
-96.0
5.
Figure D.8: Hammer spectrum in 8 kHz octave.
Hann
Ov
Y--76.257 dBNrms
-
POW
-
-64.
84 .
/Di
dB
rms
N
-12
Fxd
Figure D.9: Hammer spectrum in 16 kHz octave.
Y--77. 18
POWE
-48.0
8.0
/Div
dB
rms
N
-1 12
Fxd
X
Figure D.10: Hammer spectrum in 32 kHz octave.
dBNrms
AX=5. 062kHz
X=100 Hz
AYa=i-6.77
Ya=-42.553
SPEC i
20Avg
POWE
-40.0
O%Ovlp
Hann
4.0
/Div
dB
rms
N
-72
HAMMER SPECTRUM
Fxd
Figure D. 11: Hammer spectrum for frequency range: 100 Hz to 10 kHz.
Y--84.975
SPEC
Figure D.12: Spectrum of background noise.
dBNrms
Appendix E
Group Delay Calculations Using Phase Information
The group delay is provides an indication of the time it takes for energy to travel
along the truss. However, it should not provide the same information as the temporal
analysis using a hammer pulse. Instead, the group delay is an indication of the number of
resonances and anti-resonances in the system. Although it is not intended to conduct a
detailed analysis of the unwrapped phase information, the results are shown to verify that
reverberation complicates the expected structural behavior.
Using the same preparations as in the attenuation experiments of Chapter 3, the
phase information of the accelerance function is plotted for all b joints and is included in
this appendix. The slopes of these plots are used to calculate the group delay rg with the
following relationship:
Tg
(o
A=
360Af
(D.1)
The change in phase AO, measured in degrees, and the change in frequency Af is read from
the curves. The group delay as a function of shortest path length is plotted in Figure E. 1
and Figures E.2 - E.7 show the unwrapped phase data from which the group delay is
measured.
Z). !)
r_ C.
i
I
-
5
4.5
4
3.5
2.5
2
2
3
4
5
Shortest Travel Length (m)
Figure E. 1: Group delay plotted as a function of shortest travel path.
6
7
X=-kHz
Yb--734.43
FREG RESP
2.5
k
AX-15 .OkHz
AYb-12.09kDg
128Avg
.w.
2.5
k
/Div
O%Ov2p
Hann
-2,
Phase
Deg
-17.
iý
,
Fxd
I
J
0
Hz
I
I
I
I
-
I
PHASE AT B3
I
Figure E.2: Plot of phase for accelerance measured between bi and b3.
X--ikHz
Yb -- 9683 .71
FREG
4.0
k
AX-I15. 0kHz
AYb-21.09kDg
O%Ovlp
Hann
4.0
k
/Div
Phase
Deg
-28
Fxd
Hz
PHASE AT B5
Figure E.3: Plot of phase for accelerance measured between bi and b5.
100
20k
X=IkHz
Yb--752.88
FREG
5.0
k
/AX
15 .0kHz
kDg
.
AYb-24
5.0
k
/Div
Phase
Deg
-35
Fxd
0
Hz
PHASE AT B7
Figure E.4: Plot of phase for accelerance measured between bi and b7.
X-lkHz
Yb'=-566
FREG
16.0
k
8.0
AX-i15. OkHz
AYb-25.38kDg
k
/Div
Phase
Deg
-48
Fxd
Figure E.5: Plot of phase for accelerance measured between bi and b9.
101
20k
X=ikHz
Yb--.
AX-i15. 0kHz
1325k
AYb-27.7BkDg
FREG
16.0
k
O%Ovlp
Hann
8.0
k
/Div
Phase
Deg
-48.
Fxd
0
Hz
PHASE AT B11
Figure E.6: Plot of phase for accelerance measured between bi and bli.
AX-15 .0kHz
X-IkHz
Yb--i.0425k AYb-26.O9kDg
FREG
±6.0
k
8.0
k
/Div
Phase
Deg
-48
Fxd
PHASE AT 813
Figure E.7: Plot of phase for accelerance measured between bl and b13.
102
20k
Appendix F
MATLAB Codes
clear
clf
% Generates coordinates for x and y axis for 3D plotting
% f,ff
%N
% x,xx
%z
-frequency octave band
-section number
-axial distance along truss (global)
-strut length
f-[125 250 500 1000 2000 4000 8000 16000 32000]';
N=[1 23456789101112 13];
x=(N-1)*0.3843;
xx=zeros(1,13);
ff=zeros(1, 13);
for i=1:9,
xx(i, 1:13)=x(1: 13);
end
forj=1:13,
ff(l:9j)-f(l:9);
end
% Experimental Results of Total Attenuation at Each Joint
%a
-Raw Measured D
t, ansfered from al0secav in dB
i,
% am -attenuation of acceleration due to measured data
a=[-10.1100 -13.0100 -13.5900 -15.8933 -13.6733 -15.3200 -14.6033 -18.0433 -19.8167 -15.3467 12.5133 -10.5267 -8.7300
-5.9200 -7.3667 -10.1067 -11.4433 -13.2067 -11.6467 -12.1200 -12.1233 -12.9400 -13.1467 10.7300 -7.5700 -7.3700
2.7800 0.3233 -0.8867 -1.5667 -2.9433 -0.6100 -2.5367-1.7867 -1.7267 -2.7733 -3.4600 1.1367 1.6200
8.3200 5.0133 3.1433 2.8833 3.0867 3.1367 1.9000 1.9567 1.9833 1.1167 0.6033
1.7733 2.1900
6.7000 4.4700 4.3200 1.7933 1.0567 1.7233 0.3733 -1.1500 -0.7667 -0.9600 -1.4867 1.0100 -2.2100
9.3700 4.7867 0.8667 -0.9900 -4.9200 -4.1700 -6.6200-8.0367 -8.4033 -9.2600 -10.5100 11.1233 -13.1000
13.7000 7.4233 1.2733 -2.0833 -9.2700 -8.4900 -11.0800-13.6700 -15.1233 -14.7400 -17.8167
-16.1867 -25.1400
9.4100 1.9067 -0.8667 -3.9667 -8.6433 -10.3633 -15.1067 -15.8567 -19.5867 -19.6067 -24.0733
-25.5567 -26.4300
103
1.2400 -1.1900 -1.6100 -3.0567 -5.2967 -6.2867 -10.0667 -12.2467 -15.8300 -18.8500 -23.2700
-24.9467 -26.3700];
b=zeros(9,13);
forj=1:13,
%normalize by first section
b(1:9,j)=a(l:9,1);
end
am=-l*(a-b);
%plot3(ff,xx,am)
%title('Comparison of Total Experimental (upper) and Calculated Radiation (lower) Attenuation')
%xlabel('Frequency (Hz)')
%ylabel('Axial Distance Along Truss (m)')
%zlabel('Attenuation (dB) ')
%hold on
% Theoretical Calculation of Strut Radiation Loss
%etas -strut radiation loss factor
%as
-attenuation of acceleration due to strut theory
%z
-accumulated length of exposed strut
z=[0 409.8 1078.4 1747 2415.6 3084.2 3752.8 4421.4 5090 5758.6 6427.2 7095.8 7273.6];
z=z/100;
ka=2*pi*f/345*6.35e-3;
JO=besselj(0,ka);
J2=besselj(2,ka);
YO=bessely(0,ka);
Y2=bessely(2,ka);
c=345; ro=1.2232; m=0.1547;
etas=(2*c^2*ro) ./ (pi"2*f.2*m) ./ ((JO-J2).^2 + (YO-Y2).^2):
fcutoff=[0 0 0 1 1 1 1 1 1]';
doeta=0;
if doeta==1,
loglog(ka(l:3),etas(1:3),'+');
hold on
%/oetas plot (f < fcrit)
etass=etas .* fcutoff;
loglog(ka,etass,'o',ka,etass,':');
%/oetas plot (f> fcrit)
xlabel('ka (k = k of air, a = radius of strut)')
ylabel('Loss Factor ')
end
etas=etas .* fcutoff;
as=4.873*(etas.*sqrt(f)) * z;
% column X row = matrix
%surf(ff,xx,as)
% Theoretical Calculation of Joint Radiation Loss
%etaj -joint radiation loss factor
%aj
-acceleration attenuation of the joints
% nj
-accumulated number of joints
dojoint=l;
if dojoint==l,
kaj=2*pi*f/345*3.175e-2;
104
j0=sqrt(pi / 2 ./ kaj) .* besselj(0.5,kaj);
j2=sqrt(pi / 2 ./ kaj) .* besselj(2.5,kaj);
y0=sqrt(pi / 2 ./ kaj) .* bessely(0.5,kaj);
y2=sqrt(pi / 2 ./ kaj) .* bessely(2.5,kaj):
M=O. 12;
etaj=(3*cA3*ro) ./ (4*piA2*f.^3*M) ./ ((j0-2*j2).^2 + (y0-2*y2).A2),
doetaj=0;
if doetaj==1,
loglog(kaj,etaj,'o',kaj,etaj,':'); 0/oetaj plot (no fcrit for sphere)
%title('Calculated Radiation Loss Factors vs. ka')
xlabel('ka (k = k of air, a = radius of Joint)')
ylabel('Loss Factor ')
%gtext('strut loss factor')
%gtext('joint loss factor')
end
nj=[1 4 7 10 13 16 19 22 25 28 31 34 35);
% column X row = matrix
aj=etaj*nj;
%surf(ff, xx,aj)
end
% 2-d Plots Comparing Attenuations vs Section for each Frequecy Octave
as l25=as(1,1:13);
am125=am(1,1:13);
as250=as(2,1:13);
am250=am(2,1:13);
am500=am(3.1:13);
as500=as(3.1:13);
asl=as(4,1:13):
aml=am(4,1:13);
as2=as(5,1:13):
am2=am(5,1:13);
as4=as(6,1:13):
am4=am(6,1:13):
as8=as(7,1:13),
am8=am(7,1:13);
as16=as(8,1:13);
am16=am(8,1:13);
am32=am(9,1:13);
as32=as(9,1:13);
do2dplot=0;
if do2dplo = = 1,
plot(x,am8,':',x,as8,'-')
axis([1 13 0 401)
title('Comparison of Total Experimental (upper) and Calculated Radiation (lower) Attenuation')
xlabel('Axial Distance Along Truss (m)')
ylabel('Attenuation of Acceleration (dB)')
gtext('Frequency = 8 kHz octave band')
end
% Polyfit Each Frequency With One or Two Slopes
% omit sections 1 and 13 always
% a, b -first and second slopes fit respectively
-corner between two slopes
%q
-coefficients of the polifit (first order= 2 coeff.)
%c
% p -linear curve constructed from c(1) and c(2)
-error of approximation in the least squares sense
%e
105
% First the best fit and best corner location is calculated (or no corner)
or put one line
%loop starts at 1 to fill a matrix with no zeros
for i=1: 11,
q125=i+1;
%but want to curve fit only between 3 and 1
cl25a=polyfit(x(2:q125),am125(2:q125), 1); p125a=c125a(1)*x(2:q125)+c125a(2);
cl25b=polyfit(x(ql25:12),am125(q125:12), 1); pl25b=cl25b(1)*x(ql25:12)+c125b(2):
e125(i)=sum((am1l25(2:q125)-p125a).^2)/(q125-2) + sum((am125(q125:12)-p125b).A2)/(12-q125);
if finite(el25(i))==0,
e125(i)=sum((am125(2:q125)-p125a).^2)/(q 125-2):
end
if finite(el25(i))==0,
el25(i)=sum((am125(q 125:12)-p125b).^2)/(12-q125);
end
q250=i+1;
c250a=polyfit(x(2:q250),am250(2 :q250), 1); p250a=c250a(1)*x(2:q250)+c250a(2);
c250b=polyfit(x(q250:12),am250(q250:12), 1): p250b=c250b(1)*x(q250:12)+c250b(2);
e250(i)=sum((am250(2:q250)-p250a).^2)/(q250-2) + sum((am250(q250:12)-p250b).^2)/(12-q250);
if finite(e250(i))==0,
e250(i)=sum((am250(2:q250)-p250a).A2)/(q250-2);
end
if finite(e250(i))==0,
e250(i)=sum((am250(q250:12)-p250b).^2)/(12-q250);
end
q500=i+ 1;
c500a=polyfit(x(2:q500),am500(2:q500), 1); p500a=c500a(1)*x(2:q500)+c500a(2);
c500b-polyfit(x(q500:12),am500(q500:12), 1); p500b=c500b(1)*x(q500:12)+c500b(2):
e500(i)=sum((am500(2:q500)-p500a). 2)/(q500-2) + sum((am500(q500:12)-p500b).^A2)/(12-q500);
if finite(e500(i))==0,
e500(i)=sum((am500(2 :q500)-p500a). ^2)/(q500-2);
end
if finite(e500(i))==0,
e500(i)=sum((am500(q500:12)-p500b).^2)/(12-q500),
end
ql=i+l;
cla=polyfit(x(2-ql),aml(2:ql),1); pla=cla(1)*x(2:ql)+cla(2);
clb=polyfit(x(ql: 12),aml(ql:12), 1);
plb=clb(1)*x(ql: 12)+clb(2);
el(i)=sum((aml(2:q1)-pla).^2)/(ql-2) + sum((aml(ql:12)-plb).^2)/(12-ql);
if finite(el(i))==O,
el(i)=sum((aml(2:q1)-p1a). 2)/(q 1-2);
end
if finite(el(i))==O,
el(i)=sum((aml(q l:12)-p lb).^2)/(12-ql);
end
q2=i+1;
c2a=polyfit(x(2:q2),am2(2:q2), 1); p2a=c2a(l)*x(2:q2)+c2a(2);
p2b=c2b(1)*x(q2:12)+c2b(2);
c2b=polyfit(x(q2:12),am2(q2:12), 1);
e2(i)=sum((am2(2:q2)-p2a). ^2)/(q2-2) + sum((am2(q2:12)-p2b). 2)/(12-q2);
if finite(e2(i))==0,
e2(i)=sum((am2(2:q2)-p2a).A2)/(q2-2):
end
if finite(e2(i))==O,
e2(i)=sum((am2(q2:12)-p2b). 2)/( 12-q2):
end
q4=i+1;
106
c4a=polyfit(x(2:q4),am4(2:q4), 1); p4a=c4a( 1)*x(2:q4)+c4a(2);
p4b=c4b(1)*x(q4:12)+c4b(2):
c4b=polyfit(x(q4:12),am4(q4:12), 1);
e4(i)=sum((am4(2:q4)-p4a).^2)/(q4-2) + sum((am4(q4:12)-p4b).'2)/(12-q4);
if finite(e4(i))==O,
e4(i)=sum((am4(2:q4)-p4a). 2)/(q4-2);
end
if finite(e4(i))==O,
e4(i)=sum((am4(q4:12)-p4b).^2)/(12-q4);
end
q8=i+l;
c8a=polyfit(x(2:q8),am8(2:q8), 1); p8a=c8a(1)*x(2:q8)+c8a(2);
p8b=c8b(1)*x(q8:12)+c8b(2);
c8b-polyfit(x(q8: 12),am8(q8: 12), 1);
e8(i)=sum((am8(2:q8)-p8a).^2)/(q8-2) + sum((am8(q8:12)-p8b).^2)/(12-q8);
if finite(e8(i))==O,
e8(i)=sum((am8(2:q8)-p8a).A^2)/(q8-2);
end
if finite(e8(i))==0,
e8(i)=sum((am8(q8:12)-p8b). 2)/(12-q8);
end
q16=i+l;
pl6a=cl6a(1)*x(2:ql6)+cl6a(2);
cl6a=polyfit(x(2:ql6),aml6(2:q16), 1);
cl6b=polyfit(x(ql6:12),aml6(q16:12),1); pl6b=cl6b(1)*x(ql6:12)+cl6b(2);
el6(i)=sum((aml6(2:ql6)-pl6a).^2)/(ql6-2) + sum((aml6(ql6:12)-p16b).A2)/(12-q16);
if finite(el6(i))==0,
el6(i)=sum((aml6(2:q16)-pl6a).^2)/(q16-2);
end
if finite(el6(i))==O,
el6(i)=sum((aml6(q 16:12)-p16b).^2)/(1 2-q16);
end
q32=i+1;
c32a=polyfit(x(2:q32),am32(2:q32), 1);
p32a=c32a(1)*x(2:q32)+c32a(2);
c32b=polyfit(x(q32:12),am32(q32:12), 1): p32b=c32b(1)*x(q32:12)+c32b(2),
e32(i)=sum((am32(2:q32)-p32a).^2)/(q32-2) + sum((am32(q32:12)-p32b) A2)/(12-q32);
if finite(e32(i))==O,
e32(i)=sum((am32(2:q32)-p32a).^2)/(q32-2);
end
if finite(e32(i))==0,
e32(i)=sum((am32(q32:12)-p32b). 2)/( 12-q32);
end
end
ml25=min(e125);
m250=min(e250);
m500=min(e500);
ml=min(el);
m2=min(e2);
m4=min(e4);
m8=min(e8);
ml6=min(el6);
m32=min(e32);
/oe=[el25' e250' e500' el' e2' e4'e8' el6' e32']
for i=l:length(el25),
if m125==e125(i), q125=i+1; end
if m250==e250(i), q250=i+1; end
if m500==e500(i), q500=i+l; end
107
if ml==el(i). ql=i+l; end
if m2==e2(i), q2=i+1; end
if m4==e4(i), q4=i+1; end
if m8==e8(i), q8=i+l; end
if ml6=--el6(i), q16=i+l; end
if m32==e32(i), q32=i+1; end
end
cl25a=polyfit(x(2:ql25),am125(2:q125),1); p125a=c125a(1)*x(2:q125)+c125a(2);
cl25b=polyfit(x(ql25:12),am125(q125:12),1); p125b=c125b(1)*x(q125:12)+c125b(2);
c250a=polyfit(x(2:q250),am250(2:q250), 1); p250a=c250a(1)*x(2:q250)+c250a(2);
c250b=polyfit(x(q250:12),am250(q250:12), 1); p250b=c250b(1)*x(q250:12)+c250b(2);
c500a=polyfit(x(2:q500),am500(2:q500), 1); p500a=c500a(1)*x(2:q500)+c500a(2);
c500b=polyfit(x(q500:12),am500(q500:12), 1); p500b=c500b(l)*x(q500:12)+c500b(2);
cla=polyfit(x(2:ql),aml(2:ql).1); pla=cla(1)*x(2:ql)+cla(2);
clb=polyfit(x(ql:12),aml(q1:12),1);
plb=clb(1)*x(ql:12)+clb(2);
c2a=polyfit(x(2:q2),am2(2:q2),1); p2a=c2a(1)*x(2:q2)+c2a(2);
c2b=polyfit(x(q2:12),am2(q2:12), 1);
p2b=c2b(1)*x(q2:12)+c2b(2);
c4a=polyfit(x(2:q4),am4(2:q4), 1); p4a=c4a(1)*x(2:q4)+c4a(2);
c4b=polyfit(x(q4:12),am4(q4:12), 1);
p4b=c4b(1)*x(q4:12)+c4b(2);
c8a=polyfit(x(2:q8),am8(2:q8), 1); p8a=c8a(1)*x(2:q8)+cSa(2);
c8b=polyfit(x(q8:12),am8(q8:12), 1);
p8b=c8b(l)*x(q8:12)+c8b(2);
cl6a=polyfit(x(2:ql6),aml6(2:ql6), 1);
pl6a=cl6a(1)*x(2:ql6)+cl6a(2);
cl6b=polyfit(x(ql6:12),aml6(q16:12),1); pl6b=cl6b(l)*x(q16:12)+cl6b(2);
c32a=polyfit(x(2:q32),am32(2:q32), 1);
p32a=c32a(1)*x(2:q32)+c32a(2);
c32b=polyfit(x(q32:12),am32(q32:12), 1); p32b=c32b(1)*x(q32:12)+c32b(2):
fstplot=0;
if fstplot== 1,
subplot(3,1,1); plot(x,aml25,':',x(2:q125),p125a,x(q125:12),p125b)
axis([O 5 0 101)
title('125 Hz')
ylabel('Attenuation (dB)')
subplot(3,1,2); plot(x,am250,':',x(2:q250),p250a.x(q250:12),p250b)
axis([0 5 0 10])
title('250 Hz')
ylabel('Attenuation (dB)')
subplot(3,1,3); plot(x,am500,':',x(2:q500),p500a,x(q500:12),p500b)
axis([0 5 0 101)
title('500 Hz')
xlabel('Axial Distance Along Truss (m)')
ylabel('Attenuation (dB)')
end
sndplot=0;
if sndplot == 1,
subplot(3,1,1); plot(x,aml,':',x(2:ql),pla,x(ql:12),plb)
axis([0 5 0 101)
title('l kHz')
ylabel('Attenuation (dB)')
subplot(3,1,2); plot(x,am2,':',x(2:q2),p2a,x(q2:12),p2b)
axis([0 5 0 10])
title('2 kHz')
ylabel('Attenuation (dB)')
subplot(3,1,3); plot(x,am4,':',x(2:q4),p4a,x(q4:12),p4b)
axis([0 5 0 401)
title('4 kHz')
108
xlabel('Axial Distance Along Truss (m)')
ylabel('Attenuation (dB)')
end
trdplot=0;
if trdplot==1,
subplot(3,1,1); plot(x,am8,':',x(2:q8),p8a,x(q8:12),p8b)
axis([0 5 0 40])
title('8 kHz')
ylabel('Attenuation (dB)')
subplot(3,1,2); plot(x,aml6,':',x(2:q16),pl6a,x(q16:12),pl6b)
axis([0 5 0 40])
title('16 kHz')
ylabel('Attenuation (dB)')
subplot(3,1,3); plot(x,am32,':',x(2:q32),p32a,x(q32:12),p32b)
axis([0 5 0 40])
title('32 kHz')
xlabel('Axial Distance Along Truss (m)')
ylabel('Attenuation (dB)')
end
% 2-D Plot log(A/x) vs. log(ka)
% asx -calculated strut radiation slope (as / metre)
% amxa -experimental total first slope (am / metre)
% amxb -experimental total second slope (am / metre)
% adf -calculated structural damping slope for flexural waves
% adc -calculated structural damping slope for compressional waves
% adt -calculated structural damping slope for torsional waves
% etad -structural damping loss factor for aluminum
% aconv-compresional wave attenuation due to conversion to flexural waves
asx=7.48*(etas.*sqrt(f));
asx(l:3)=[le-100 le-100 le-100];
ajx=3*etaj/0.3843;
amxa=[cl25a(1) c250a(1) c500a(1) cla(1) c2a(1) c4a(1) c8a(1) cl6a(1) c32a(1)];
amxb=[cl25b(1) c250b(1) c500b(1) clb(1) c2b(l) c4b(1) c8b(l) cl6b(1) c32b(1)];
etad=0.001;
adf=-7.48*etad*sqrt(f);
adc=0.017*etad*f;
adt=0.027*etad*f;
aconv=[l1 9.5 8.5 7 6 5.1 4 3.1 2.8];
aconv=-10*log10(1- 10.A(-aconv/10))/.3843;
doaxka=l;
if doaxka==1,
loglog(ka,asx,ka,amxa,'--',ka,amxb,'-.',ka,aconv,':');
%loglog(ka,asx,ka,adf,'--',ka,adc,'-.',ka,adt,':');
axis([.01 10.001 100])
%title('Slope (A/X) vs. (ka) for Experimental and Calculated Radiation Attenuation')
xlabel('ka')
ylabel('A / X (dB per metre) ')
%legend('Radiation','Flexural','Compressional','Torsional')
%gtext('- Predicted Strut Radiation')
%gtext('.. Experimental Far Slope')
109
%gtext('-- Experimental Near Slope')
end
% Plot of Experimental Attenuation Curves
firstone=0;
if firstone==1,
plot(x,aml25,'-',x,am250,'--',x,am500,'-.',x,aml,':')
axis([0 5 0 40])
gtext('Average of all Joints at Each Section')
%legend('125 Hz octave','250 Hz octave','500 Hz octave','1 kHz octave')
%gtext('
125 Hz octave')
%gtext('-- 250 kHz octave')
%gtext('-. 500 Hz octave')
%gtext('.. 1 kHz octave')
end
secdone=0;
if secdone = = 1,
plot(x,am2,'-',x,am4,'--',x,am8,'-.',x,aml6,':',x,am32,'*')
axis([O 5 0 40])
gtext('Average of all Joints at Each Section')
%legend('2 kHz octave','4 kHz octave','8 kHz octave','16 kHz octave','32 kHz octave')
%gtext('_ 2 kHz octave')
%gtext('-- 4 kHz octave')
%gtext('-. 8 kHz octave')
%gtext('.. 16 kHz octave')
%gtext('* 32 kHz octave')
end
%xlabel('Axial Distance Along Truss (m)')
%ylabel('Attenuation (dB) ')
%title('Attenuation of Acceleration vs. Distance From Force Excitation')
110