the university of department of...the new t.heory and method of analysis ís applied Èo a number of...
TRANSCRIPT
THE UNIVERSITY OF ADELAIDE
Department of Mechanical Engineering
TIME-AVERAGED HOLOGRAPITY FOR THE STUDY OF
TIIREE-DIMENSIONAI VIBRATIONS
by
Renzo Tonín, B.Sc. (Hons.)
Thesis for the Degree of DocËor of PhÍlosophy
March 1978
' t ,. "
SI.]MMARY
The theory of time-averaged holography is extended to take
ÍnÈo account three-dimensional phasor vibratíons of a síngle frequency.
The vibration ís considered to be characterísed by a spatially varying
I'ourier expansíon of superposed spaÈial modes and temporally dependent
coupled modes in three dimensions leadi-ng to the derivation of the
general characterístic equation. The method of GeneraLízed Least
Squares is ínEroduced to solve the non-linear general characteristic
equation including phase wíÈhout the need to modulate the laser beam.
The new t.heory and method of analysis ís applied Èo a number
of vibrating objects including a clamped circular plate, wine glass,
st.ainless-steel beaker and cylinders of various materi-als wíth shear-
diaphragm ends. tr'or the first time the radial, tangential and
longitudinal vibration componenËs are determined experimentally and
compared with theoretical predictions.
The theory of vibraÈion of cylinders of varying wal1 thicknesses
is solved using the Rayleigh-Ritz method and the mode shapes experí-
mentally determined, for the case of a cylinder rnrith non-concentric
inner bore and outer surface and a cylinder with a thin longitudinal
strip, using the new holographic Ëheory.
A number of coupled modes in a near perfect cylinder are
analysed and the phase component determíned for the first, time wíthout
modulating the laser beam. Results are compared wÍth predictions for
one-dimensional phasor víbrations.
New experimental techniques such as the translating hologram
table, resonance una't and data system are described whích aíd in
the generation and analysis of time-averaged holograms of conplex rribratÍons.
Fina1ly, the sound radíation efficiency of various cylinders is
measured in a reverberation room and compared wíth theory.
TABLE OF CONTENTS
StatemenÈ of Originality
Acknowledgements
List of Figures
List of Tables
List of Symbols
ii
l-aa
vii
vt_al-
1
1
11
13
I4
t4
L4
18
2T
Chapter 1
1.
1.
1.
Chapter 2
2.
Chapter 3
3.1
Literature Survey
Holographic InÈerf erometry
Víbrations of Cylinders
Sound RadiaËion from Vibratíng Cylinders
Tirne-Averaged Holography
Theory of Time-Averaged Holography
2.I.L. The Argument of the CharacteristicFunction in Spherical Co-ordinates
2.L.2 Coupled and Superposed Modes ofVíbration
2.L.3 Theory of Tirne-Averaged Holography forOne-Dimens ional VÍbrati_on
Theory of Tíme-Averaged Holography forThree-Dímensional Víbratíon and
ApplicatíonThe Theory of GeneraLized Least SquaresSolutíon of the Characteristic Equation
by MeËhod of Least Squares
HolographyModal Driving SystemGeometry for the Analysis of CylíndersData Analysis
Pure and Superposed Spatial Modes
Claurped Cj-rcular Plate
Holographic and Geometric TheoryClamped Plate Experiment and Results
I.Iine Glass and Beaker
Holographic and Geometríc Theoryhline Glass and Beaker Experiments and
Results
1
2
3
1
2.L.4
2.2 Experimental Equipment and Data Analysis
2.L.52.L.6
2429
2.2.L2.2.22.2.32.2.4
31
33
33364L47
52
52
5256
57
57
3.1.13.r.2
3.2.L3.2.2
3.2
68
Chapter 4
4
4
Chapter 5
5
APPENDICES
Appendix I
II
III
VII
References
3.3 Cylinders
3.3.3
1
4.2
3
1
5.2
Page
7B
783.3.13.3.2
Holographic and Least Squares TheoryTheory of thin circular cylindrical
Shel1sCylinder Experiments and Results
848B
Cylinder wiÈh Non-Concentric Bore
Cylinder with a Thin Longitudínal Strip
Coupled Temporal Modes
Holographic Theory for Two Coupled Modes
Experiment.al Procedure and Results
Circular Cylinders with VaryÍng tr{all Thickness 111
Theory of Vibration of Cylinders wiÈh VaryingtrIal1 Thickness 111
L2I
]-34
L42
L42
].44
Flow Diagram of Microprogram for Data SysÈem
Matrix for the Normal EquatÍons
The Strain Energy Integral
Coefficients of the GeneraLized Stiffness andMass Matrices
Solution of the II Functions
Sound Radiation frorn Cylinders
46.1 Theory of Sound Radiation from VíbratingCircular Cylinders
A6.2 Experimental Analysis and Results
Published Work
156
]-57
158
163
L67
L7L
L7L
L75
186
195
IV
VI
V
l_ l_
ACKNOI^ILEDGEMENTS
To my supervisor Dr. David Bies I express my greatest thanks.
Hís ideas have been my major source of inspiration and ít has been a
pleasure workÍng under him as he is a fine gentleman and a good friend.
His dedicaÈion to his work and unselfish givíng of his time will have
a lasting impression.
Thanks goes to Professor Sam Luxton, Chaírman of the Department,
for arranging my candidature parÈicularly since my under-graduaÈe
work was involved ín another discípline. His belief in ny abílíty ís
híghly Lreasured.
Many thanks to Colin Hansen for hÍs t.utoring ín hologram
production and to Andrew Davis and Phil i{alker for help in setting up
the data sysÈem without which at leasÈ another year would have been
needed to eomplet,e this thesis.
I am also índebÈed to Dudley Morríson and Ron Parfitt for their
engineering excellence whieh made possible the manufacture of the
close-tolerance cylinders .
To all those fine people in the department whose names I did
not mention buÈ whose help I could not have done without, a special
word of thanks.
Finally I acknowledge the generous financíal assistance of the
Australían Research Grants Coumit.Ëee, wíthouË whích much of the exper-
imenÈal work would noË have been possible.
Figure Number
2-La
2-Lb
2-2
2-3
2-4
2-5
2-6
2-7
2-B
2-9
2-L0
2-Lt
2-L2
2-L3
3-1
3-2
3-3
FIGURES
Title
Co-ordinate system for the vibration componentsof a point on the surface of the object
Co-ordinate system of the optics for a pointon the surface of the objeet
Co-ordínate sysÈem for the object surface
Schematíc of holographic process
(a) Síngle coupled mode. (b) Síngle coupledmode with ouÈ-of-phase components. Símplest.kind of elliptic moÈion. (c) Two coupled modes.
OpÈical system
OptÍcs for the generation of holograms
The translatíon table (T), hologram stage (S),typical hologram (H) and mask (M)
Modal driving system
The electroníc equípment used to dríve andmonít.or víbration modes
Geometry for interpretation of hologramphotographs - Cylinder Vertical
Geometry for inÈerpretation of hologramphotographs - Cylinder HorizontaL
The data system
Schematic of the data sysËem
Circular plate with varying direction ofl-llumination
Circular plate with varying directíon ofobservation
Page
16
L7
L9
22
27
34
37
38
40
42
43
46
48
49
53
55
iii.
58
Reconstructions of holograms ofcÍrcular plate víbrating in Èheand íllumination angle c=.|50.5s. = -52.8" (bottom) .
a clamped(0,1) rnode" (top) and
Holographically determined amplitudes of clampedcircular plate vibrating in the (0r1) mode forvarious illumj-nation angles
3-4
59
Fisure Number
3-5
3-6
3-7
3-8
3-9
3-10
3-11
3-12
3-13
3-L4
3-15
3-16
3-I7a
3-17b
3-18
3-19
3-20a
3-20b
Tít1e
Reconstructions of holograms of a clampedcircular plate vibrating in the (0,1) rnodeand dífferent orientatíon angles.
Holographically determined amplitudes ofclamped circular plate vibrating in the(0,1) mode for various orientation angles
Schematic of two-view experiment using aplane mirror
Geometry for the imaged view
Itline glass and nodal driving system
Tíme-averaged hologram reconstruction ofthe n= 2 mode for the wine glass
Time-averaged hologram of wine glassvibrating in the n = 2 mode and the mirrorimage
Radíal components of a vibrating wine glass
Hologram reconsÈructions of two views of a
beaker vibrating in a Love mode of order n = 3
Theoretical and experimental deterrnÍnation ofradial and tangenÈial vibration components ofstainless sÈeel beaker
(a) Cylinder wiÈh a seam (b) Cylinder witha lumped mass
Co-ordinate system for a thin cylindricalclosed circular shell
lv.
Page
60
6I
64
6s
70
7L
72
73
75
76
79
B9
92
93
95
96
Least squares procedure applied to data ofstainless steel beaker, N1 = N2 = 3
Least squares procedure apllíed to data ofstainless sÉeel beaker, Nl = 2, N2 = 3
Least squares procedure applied to daÈa ofbrass "y1ittd.t-with
a seam' Nl = 2, N2 = 5
Details of end mount and suPPorÈ
86
91
Circumferential order n=2 (Surall SteelCyl inder)
Circumferential order n= 3 (Smal1 SteelCylínder)
Circumferential order n= 4 (Sma11 SteelCyltnder)
3-20c97
v
Figure Number
3-2L
3-22a
3-22b
3-22c
3-23
3-24a
3-24b
4-L
4-2
4-3
4-4
4-5a
4-5b
4-5c
4-5d
4-5e
Title
Time-averaged hologram reconstructions ofa small steel cylinder vibrating ín thefl=4, n=3 mode.
Longit-udinal order m= I (Alurnini-um Cylinder)
Longitudinal order m= 2 (Aluminium Cylinder)
Longitudinal order m= 3 (Copper cylinder)
Tíme-averaged hologram reconstructíons of asma1l steel cylinder vibrating in Èhe û=2,n = 2 mocle
Circumferentíal order n= 4 síng1e mode analysis(Copper Cylinder)
Multi modal analysis technique N1 = 2 andN2=4 (Copper CylÍnder)
Circular cylinder wíth varying angular wallthickness
Geometry for calculating the fouriercoeffici-ent.s of the dístortion
Theoretical frequency curves for dístortedcylinder. Symmetric soluÈions.
Theoretical frequency curves for distortedcylínder. Asymmetríc solutions.
Radial and tangential vibration componentsfor the princípal symmeËri-c mode (I12) ofa distorted cylinder
Radial and tangential vibration componentsfor the principal s¡rmmetric mode (1,3) ofa distorted cylinder
Radíal and tangential vibraËion componentsfor the princípal symmetric mode (2,2) of adistorted cylinder
Radial and tangential víbration componentsfor the príncípal symmetric rnode (2r3) ofa dístorted cylinder
Radial and tangential vibration componentsfor the principal symmetric mode (3,2) of. adistorted cylinder
Radial'and tangentíal vibratíon componentsfor the principal symmetríc mode (3,3) ofa distorted cylinder
Page
9B
99
100
101
LO2
L04
105
113
L22
l,25
126
L28
L29
130
131
r32
4-5f.
133
vr
Figure Number
4-6
4-7
5-1
5-2
5-3
s-4
A6-1
A6-2
A6-3
A6-4
A6-5a
A6-5b
A6-5c
A6-5d
A6-5e
Tirle
Geometry for calculatíng Ëhe fouríercoefficienrs of the distortion
Two views of a cylinder with atÈachedstrÍp vibrating ín (2,2) mode
Two views of a cylínder vibraËing íntwo coupled modes
Experimentally determined components oftwo coupled modes - perfect cylínder
ReposiÈioning the drivers almost eliminatesthe other coupled mode shown in Fig. 5-2
Components of Ëwo coupled modes - perfectcy1índer
Co-ordínate system f.or a vibrating cylínderof finite length wíth cylindrÍcal baffle
Schematic cross-section of cylinder andbaffles
Cylínder and supporÈing structure
The reverberatíon room, rotatíng vane andmicrophone traverse
Radj-ation efficíency of steel cylínders.Axlalorderm=1
Radiation efficiency of steel cylínders.Axial order m= 2
Radiation efficiency of steel cylinders.Axialorderm=3
Radiatíon effíciency of steel cylínders.Axíalorderm=4
Radiation efficÍency of sËeel cylinders.Axíalorderm=5
Page
135
136
L46
L47
r48
150
L72
L76
L78
L79
181
TB2
183
184
185
Table Number
3-1
3-2a
3-2b
3-2c
3-2c
4-L
4-2
4-3
4-4
5-1
5-2a
5-2b
5-2c
TABLES
Títle
Physical Properties of Test Cylinders
Theoretical and Experiment.al Results -Small Steel Cylinder
Theoretíeal and Experimental Results -Large SËeel Cylínder
Theoretical and Experimental Results -Aluminium Cylinder
TheoreËical and Experimental Results -Copper Cylinder
Resonant Frequencies for Some Modes ofDistorted Cylinder (Non-Concentric Bore)
Theoret.ical Superposed Syrnmetric Modes for aCylinder with Non-ConcenÈric Inner Bore
TheoreÈícal and ExperimenÈal Symmetric ModeAmplitudes of Distorted Cylinder (NorrnalisedMagnitudes)
Resonant Frequencies for Some Modes ofDistorted Cylinder (Longitudinal Stríp)
Comparison of Generalízed and Normal LeastSquares Procedures in the Det,ermínation ofComponents of Principal Modes (212) and(2,3) for a Dístorted Cylínder
vit.
Page
106
L07
108
109
110
138
139
r40
141
L52
153
154
155
Effect of IniÈia1 Conditions on SolutionPrincipal Mode (2,2) of Undistorted CylinderNl=2rN2=3
Effect of Initial Conditions on SolutionPrincipal Mode (2,2) of Distorted Cylindertll = 2, N2 = 3
Effect of Initía1 ConditÍons on SolutionPrincipal Mode (2,3) of DisÈorted CylinderNI=3rN2=4
viii.
A
AO
A
A
\A(H, r)
A(S)
AA
B
BB
c
cc
D
DP
D
E
F
G
H
Hn
I
SYMBOLS
Longitutiinal vibration amplitude
Unperturbed complex analytícal sígna1
Average complex analyticat signal
Matrix of the normal equations
Area of reverberaÈion room
complex analytical sígnal in the plane of the hol0gram
Complex analyti_cal sígna1 in the plane of the object
Distance from aperture Eo centre of obj ect
Tangential vibratíon amplitude
Distance defined ín Fig. 3-7
Radial víbration arnplítude
Distance defined in Fig. 3-7
Aperture diameter
Distance defíned Ín Fig. 3-1
Generalized stiffness matrix
Youngrs Modulus
Matrix defíned by equarion (2.52)
MatrÍx of differentials defíned by equatíon (2.43)
General hologram co-ordinate
Hankel functÍon of the first kínd of order n
Number of coupled temporal modes
Bessel functíon of the fírst kind of order n
Illumination vector
Observation vector
Geometrical factor defined by equatíon (2.25)
Non-dírnensional frequency parameter
Principal length of obj ect
Principal length of objecr in photograph
Power leve1 re 10-12 wattsfm2
Jn
5
\2
Kq
t
aK
L
Lv
LT'
l_x.
LP
Sound pressure level re 20 ¡rPa
General characterisEic function
Number of supe.rposed spatial modes
Origin - direct, view
Orl-gln - ímaged view
Distanee from centre of object to poínt illumínation
Function defined in equation (46.6)
Geometrical factor defined in equaÈion (2.25)
Generalized mass matrix
General space co-ordinate
Vector of amplitude parameters
Optimum vector of amplitude parameters
Radial harmonic of the pressure field
General surface co-ordinate
Geometrical factor defined in equatíor. (2.25)
Angle Èhrough ¡^rhich cylinder is rotated
Straj-n energy
Radiation area
Kinetic energy
Reverberation tíme
Time-varying 1-ongitudinal vj-bration amplitude
Time-varying tangential vibration amplÍtude
Reverberation room volume
Tíme-varyíng radial vibration amplítude
Cartesian co-ordinate
Cartesían co-ordinaÈe
Neumannrs functíon of order n
Length defíned Ín Fig. 2-LL
CartesÍan co-ordinate
M
N
oo
ot
S
S
so
St
P
Pnn
q.
R
R
R
Rnn
q
S
T
T
U
V
V
A
60
R
n
w
x
Y
Y
Y
z
D
x
a
Rav
Lav
al
^2
av
ao
Longitudinal vibration component
Right radíus of inaged cylinder in photograph
Left radius of imaged cylinder in photograph
Radíus of cylinder in photograph
Mean radius of eylínder
Inner radius of cylinder
OuÈer radius of cylínder
Tangential vj-bration component
Radial vibration component
PropagaËion speed
Total vibratíon phasor
Diameter of imaged cylinder in photograph
EccenËricity
Principal co-ordinates
Unit vectors
Resídual function
Frequency
Number defined in Appendix IV
Cylinder wall thickness
Príncípa1 hologram co-ordinates
Outer thickness measured from cylinder mídd1e surface
Inner thickness measured from cylinder middle surface
I,Iave number or propagation constant. k = ul/c_
Axíal wave number. k, = mn/L
Length co-ordínate along the surface contour
Longitudinal mode order
Círcumferential mode order
Order of distortion component
I,Ieight of amplitude parameter
o
b
c
c
e
êr t
êt t
f q
f
ê2'
êrt
_{g
dv
3
3
e
e
oÞ
h
h
h
h
k
km
9"
m
n
P
,I h2
+
D^ r^r
pr
q
r
sr'
s
xl-.
Sound pressure
Point on the surface contour
Two dímensional space co-ordinate
Non-dimensional axial co-ordinate. s = X/ao
Pri-ncípal surface co-ordinates
TÍme
Asymnetric longítudínal amplitude componenË
Symmetric longitudinal amplitude component
AsymneÈric tangential amplitude component
AsymrnetrÍc radíal amplitude component
Symmetric radial amplitude component
Synmetric Ëangential amplitude component
Phasor phase difference
Angle defíned in Fig. 3-7
Non-dimensional eigen frequency
Matrix defined by equaËion (2.5I)
Matrix differential operator
Acoustic pornrer
Integral functíon defined ín Appendix V
Total number of data poínts
Distance from aperture to screen
Argument of Ëhe characterístic function
Vector of frínge parameters
Average over space and time
Angle subtended by centre líne AA and illunination líne p.
Longitudinal component phase
Tangential component phase
Angle defined by equari_on (2.59)
t2
t
u
v
It
x
v
z
^V
^f
T
fl
'liT
X
s¿
CI
c[
Íd
ßr
I
st
IY
ð
e
K
À
u
v
E
1T
p
0
ot
xt_ l_
Radial component phase
Correction vector
Angle defíning orienÈation of mode around cylínder
Non-dimensional thickness parameËer. K = h2/L2a2o
Non-dimensional axial wavelength. ), = mnao/L
Angle defined in Fig, 4-6
Poi-ssonr s ratio
Angular co-ordinate along a surface contour
pi
Density
Angular co-ordinate
Angular co-ordinate in imaged view
Angle subtended by surface normal and illumination vector
Angle subtended by surface normal and observation vector
Radíation efficiency
Sum of squares of residuals
Least sum of squares of resíduals
Angular co-ordinate
Angle between di-rect and imaged víews of cylinder
Angular frequency
0t
02
o
Ç
e
0
rl
u)
N
CHAPTER 1
LITERATURE SIIRVEY
1.1 HOLOGRAPHIC INTEP.FEROMETRY
The díscovery of holographic inÈerferometry, notably by Powell
and Stetson tl] ín 1965, has Ied to a remarkable developmenÈ in the
field of vibration analysis. Here is a developmental tool with the
same impact on dynanr-ics as quantum theory had on mechanics - a contact-
less probe into the nlniscule víbratory motions of structures. Stetson
and PowelL l2l in L966 demonstrated the equívalence concept, whích is
basic to the theory of holographíc interferometry, that exposing t.he
holographic plate wíth the reference beam and object beam Ín sequence
gÍves the same result as simultaneous recording. Thís leads to the
concept of real-time holographíc ínterferomet.ry. Representíng the
electromagnetlc field of the laser radiation as an angular specËrum
of plane \¡raves [3], Brown et al derÍve semí-rigorous equations for
tÍme-averaged and real-time holographic ínterferometry of one dimen-
síonal símple harmonic vibration [4]. Butusov [5] presents the theory
for tíme-averaged holography in a maËhemaÈically more rigorous form
and Híldebrand t6l generalízes the theory of holography.
The basic requirements for rnaking a good hologram are coherent,
monochromatic light and freedom from extraneous vibrat,ions. Luríe 17l
shows in theory, íf Èhe reference beam is a plane vrave, the reconstr-
uction is sharp evon though the light is only parËially coherent. The
variation ín coherence over the object surface is a greater factor in
the loss of claríty. If the object ís moved linearly duríng the recording
the irnage inÈensíty is modulated by a sínc-function which ís responsible
2
for the perceived blurring tB]. For sinusoidal di-sturbance, the degree
of blurring is proportional to the víbration arnplítude t9].
The theory of time-averaged holography developed by powell
and SÈetson hras verified for one dimensional sÍmple harmonic víbration
by LurÍe and Zambuto t10]. The authors recognize thaÈ the ínÉegral
expIikc(t) (cos0, t cos0r) ]dr,
is valid for any motion c(t) no matÈer how complex it may be. In the
equation 0, and A, axe the angles subtended by the surface normal and
the illurnínation and observation vectors respectively and k is the
radiaÈíon wavenumber. The errors [11] involved in the det.ermin¿¡1s¡
of a dísplacement are príncipally
f. inaccuracy in determiníng 0, and 0, (1o error results in
about 17" error in displacement amplitude) and
2. inaccuracy ín determining the fringe order due to frínge
wídÈh whích is inversely proportional to signal to noise ratío (0.S
fringe error results in roughly 5% error in displacement arnplitude).
Nevertheless the analysis of sl-mple harmonic motion by time-averaged
holography can be trivial with the formula derived by Borza l,fzl.
One of the liníËations of holographic ínterferometry of
vibratíons is that only relatively sma1l amplitudes (1ess than about
2urn) can be anaLyzed since frÍnge intensity is inversely proportional
to fringe order which is deÈermíned by the argument of Èhe characteristic
fringe function. For tíme-averaged holography the intensity of the
fringes varies as Jf where the characteristic frínge function Jo is the
zeto order Bessel function. In thís case the tenth bright frínge is
less than 27" the intensity of the zeroth order bright fringe. For real-
time holography the contrast ís poorer since the íntensity varíes as
(1+Jo) t4l. one method of irnproving the dynamic range is to artificíally
3
increase the v¡avelength of the radiating light by a Moiré techníque
used in the study of stress deformations ín transparent objects t13].Rowe [14] describes a method of projectíng ínterference fringes onto an
object surface and recording a time-averaged hologram of a fixed rotationof the object in the usual way. The fringes ínterfere to form a Moiré
pattern analogous to the normal Bessel frínges except that quiÈe large
dísplacements are analysed. Hung et al [15] have extended. this method
to analyse vibrations of large anplítudes and Joyeux [16] descríbes an
on-line inst.rumenÈ which measures dísplacements directly usíng thisMoír6 technique.
The object under study ís usually required to scaÈter lightdlffusely. However, phase objects (usual ín gas dynamics problems) have
been analysed usíng the prínciples of holographic inËerferometry.
Ovechkin et al [17] descríbe a double exposure method wíth the object
in the path of rhe objecr beam. Another techni_que [18] ís ro project
the phase-varying object beam onto a flat diffuse surface which then
functÍons as a normal diffuse object.
Holographic ínterferometry requires only one reference beam.
However, Dähdliker et al [19] and Tsuruta er al [20] describe a method
of two reference beams as a means of adding flexibí1ity to convenËíonal
double exposure interferomeÈry. This procedure enables information of
objecÈs to be taken separately or with mutual interference. Tsuruta
et el use this principle, in lieu of the double exposure technique,
which has alignmenË problems, to ínvestígate an object whÍch has been
nodifÍed and replaced ín the object beam.
The theory of holographíc ínterfer,ometry of síurple vibrationsin one dimensíon has found widespread applications ín other dísciplínes.Bies Í2Ll shows how the normal component of vibration, as determined
from holograms, can be used to predict radiation efficiencies of vibratingsurfaces - an iurportant problem in acoustícs. Frankort [22] predÍcts
Ll
the radiation behaviour of loudspeaker cones in this way. Vasi-lryev
et al [23] arralyse víbration components of blades and discs of comp-
ressors ín aircraft engínes. Zakharov et al 1,241 describe a potentially
portable laser microprobe to analyse vibrations in the field using Ehe
principles of holographic interferometry. In the biomedícal scíences
Greguss [25] and Hogmoen and Gundersen [26] describe the use of inter-
ferometry ln the analysís of the vibration of the human tyrnpanic
menrbrane, stresses in teeth (holodontornetry), deformation of the femur
(orthopedics) and respíration contours of the human body. The poss-
ibilíty of colour interferograms has also been considered by Chernov
and Gorbatenko 1271. Fryer [28] and Rogent and Brown [29] have written
reviews on holographic vibration analysís covering almost every aspect
of holographic j-nterferometry.
The extension of one-dímensíonal hologr:aphic interferometry
into three dimensions \¡ras a natural progression. Haines and Hildebrand
(1966) [30], to whom the discovery of hol-ographic interferometry is
also attributed, extended the theory to include sËatic rotation and
Èranslatíän of the object ín three dimensíons. The result:'-ng formulae
are dífficult to use in practice hence Sollid [31] developed two schemes -
the single hologram method utilízing parallax and fringe countíng and
a multiple hologram method using interference orcler assignment. The
flrst scheme ínvolves eounting the number of frínges that shíft past
an object point as the point of observation is moved from one position
on the holograrn to anoÈher. Dhír and Síkora l32l improve this technique
by expressíng the componenÈs of displacement as a set of lj-near equations
whích are solved by the leasÈ squares method and they note that only
the sígn of one component r.ras necessar:I a. priori to determine the others.
The second sctreme utili zed a set of holograms with several direcËions
of observation or illumíhation and was improved by Sciarnmarella and
Gilbert [33] r^¡ho determi.ned the displacement components by the method
5.
of least squ¿Lres. Although the first scheme has been automatecl for data
processing 134] its maín dj-sadvantage is that for small displacements
the fíeld of the hologram may not be large enough to enable at least
a ferv fringe counts. For this reason in this thesís ttre author prefers
the second scheme which is also easier to adapt Ëo daËa proeessing.
Stetson t:S1 and with Pryputnier,rícz [36] use Èhe second
technique to separate rígid-body motion and homogeneous deformation with
the application of the least squares method to an overdeÈermíned set of
línear equations. Hu eË al [37] describe a Moirá Èechnique which comp-
ensates for rigid-body motion usíng two holographic interferoElrams, one
on each side of the object, and reconsÈructed together to produce Moiré
Ísopaehic fringes of the stress displacernent only.
Ennos [38] makes use of the technique of multiple holograms
.to measure the strain in one direction only in the plane of a surface
under tension. Scíammarella and Gílbert [39] extend this method with
the aíd of a Moíré technique to optically separate tr/o components of
di-splacenent of a surface under compressíon.
Using light scattered by Ëhe interíor of a three dímensional
transparent object from a sheet of coherent light passíng through the
body Barker and Fourney t4O1 describe how displacement Ínformation may
be recorded in sectÍons to describe fully the deformatíon of the object.
Thís broad concept is used in this Èhesis to construct a fu11y three-
dimensional vibration map of any objecÈ surface in two-dimensional
sheets or slices, with the result that the mathenatics ís less complex
and hence more applicable to real analysís.
A rígorous theory of frínge formation and localízation has
been derived by Stetson as an ímprovement of more complex and approx-
irnegs descríptions; for example that of Tsuruta eÈ a1 I4f]. In a seríes
of excellently planned papers SteËson 142-48] derives a rigorous form-
6
ulation of the generalized frlnge function, discusses ín depth the
factors affectíng the argument of the fringe function, predícts frínge
locÍ- and localizatÍon for a combination of whole body rotations and
t.ranslatÍons and supports the theory by a large number of experiments.
The results most applicable to this thesis are
1. that fringe spacíng and IocaLizatíon are unaffected
by the curvature of an object surface and
2. the observer-projectíon theorem (fringes observed
localÍzed in any plane may be projected on to the objecÈ plane)
justÍfíes the use of a camera and photographic enlarger to produce
pícÈures of the holographic interferogram.
I^lalles [49] extends the concept of homologous rays to compute
and vísualize fringe localizatíon for arbítrary movements, Íncludi-ng
strain and shear.
Abramson [50-55] íntroduces the holo-diagram which is a
pictorial represenÈation of the formatíon of frínges Ín holographic
lnterferonetry. The distance bet¡¿een the point of ílluminatíon and
the hologram determínes the sensitivity. Hence the process makes
optimum use of the coherence length of the laser radiation and objecÈs
up to 2m in length may be studied. Abramson also Íntroduces the
sandwích hologram [56] to eliminate whole body movement ín holographic
interferograms of deformed objects.
Matsumoto et al [57] analyse the measuring errors of three-
dímensional displacements by holographic interferometry. In partícular,
for the case of the multiple hologram technÍque, the contrÍbutions of
optical errors and fringe-readÍng errors are least for orthogonal
systems of illu¡nination or observation directíons.
Holographic ínterferometry has been applíed to the measure-
ment of steady velocíty [58-59] where it 1s shor^m the characterístic
functlon 1s the sinc functÍon. Gupta and Singh [60-63] analyse the
7
characteristíc functions of time-averaged hoJ-ography for non-linear
vlbraÈíons of the form of Jacobian elliptics. such motíon is typical
of vibrations in a shaft connected to a crank and piston or in prop-
ellers. However, no experimental application of the theory has yet
been reported. Janta and Míler [64] calculaÈe the characteristíc
function of time-averaged holography for sinusoídal damped oscillations
and from frínge data devise a scheme to determine the damping coeffic-
lent. Zambuto and Lurie [65] derive the characÈeristj.c function for
various complex motions - consÈant velocity (rarrp) motíon, superpositíon
of ramp motíon and sÍnusoídal vibration and sEep motion - by considering
the effecË of motion on coherence. on the other hand, vikram [66-69]
and vikram and sirohi l7o-72] analyse the same motion using the phase
variation equaËíon [1] and suggest various schemes incruding Moírá
techníques, amplítude modulatíon and spatíal variation of the object
beam to separate the comporients of vibration.
In additíon to the study of hologram interferometry of
complex motíons, various workers have considered one dimensíonal víb-
rations of a number of modes of more than one frequency. I^Illson and
SÈrope [73] show that the characteristíc function of two ratÍonally
related modes with non-zero phase difference is a linear combination
of the producÈ of two Bessel functions of the first kind and of varÍous
orders. lJilson 174l extends the concept to include irratíonally
related modes and generaËes computer inages of Èhe fringe patterns of
a circular clamped plate víbrating ín such modes. A special case of
thÍs is discussed by Reddy 1,751. Stetson [76] íntroduces the rnethod
of stationary phase to prediet fringe patËerns of vibrations of modes
of dífferent frequencies and phases. The meËhod assumes that the main
contributíon of the tíme-varying propagation argument, f,¿, to points
ín the hologram which reconstruct the brightestroccurs at ðQ/àt = 0.
B.
The theory ís shor'rn to agree well with experimental observations of
two modes with rationally relaËed frequencies and of various phase
and is physically more meaníngful than that proposed by hrílson and
Strope.
vikram [77] suggests a stroboscopic technique to separate
two modes of different frequencies. The method consists of pulsíng
the laser lighË at. tímes when the amplÍtude of one mode is invaríant
and vice-versa for Èhe other mode whereby the separation of the two
modes is accomplished by analysing the two holograms.
The more general case of an unlimíted number of modes of
different frequencíes and varying phase has been considered by Dallas
and Lohmann [78]. The decipheríng concept uses a moving grating in
the Fourier plane to deconvolve Èhe coefficients of vibration from
the optical Fourier coefficients in the hologram. By "synchronizíngtl
the grating velocity with the frequency of vibration, the mode of
vibration for that frequency is separated from the rest. lÍilson [79]
derives the general characterístic functíon for any number of modes
with differenË frequencies and phase in terms of a linear combinaÈion
of a product of Bessel functions of the first kind and varíous orders.
Stetson [80-81] introduces the concepÈ of analysis by density functions.
The general characteristic functíon is expressable as a finite sum of
Fourier transforms of the vibration modes. Finally, vikram and Bose
[82] analyse damped oscillatíons ¡trith two frequencies by íllunínatÍng
the object from Èwo dífferent directíons resulting in a Moiré pattern
by whì.ch Èhe anplitudes of víbration and the damping coefficients are
determined.
The materral contained in thís thesis applíes to víbrations
of one frequency. The analysis of vibratíons involves ín part deter-
míning the geometry and phases of modes both of which contribute to
the resulting frínge pattern of a holographic ínterferogram. Firstly,
9
the geometry of Èhe vibration may be separated by considering the motíon
to consist of a number of línearly índependent superposed modes, a
method which dates back to Rayleigh (1894). The analysis then involves
finding the ampliÈudes of the contributíng modes. Thís concept vras
inÈroduced Èo holographíc ínËerferometry by Stetson and Taylor tB3].
The pure mode shapes of a clamped rectangular plate were analysed by
hologram ínterferometry and these ïrere used to predlcÈ the statíc
deflection resulting from the applicatíon of poínt forces to the plate.
In another paper, SteLson and Taylor [84] predict vibratíon patterns
that result from mode combinations in an asyrnmetrically loaded dísk
by applying the holographically determined pure mode data of an unloaded
disk. Evensen [85] determínes the amplitudes of the normal modes of
a flutteríng panel by a strobing technique.
The phases of the conÈributing modes are usually det.erur-ined
by beam modulation in contrast to the method presented in this thesís
which det.ermines both amplítude and phase informatíon from normal time-
averaged holograms. The combination of modes of different phase was
shown by Molín and Stetson [86] to result in the addítion of theír
corresponding fringe functÍons as if Ëhey weré phasors. Shajenko and
Johnson [87] introduced stroboscopíc holographíc interferometry to
"fteeze" the phasor moÈion at any portion of the cycle. Miler tB8]
ímproves the theory by assuming the object moves línearly during the
exposure period rather than remainíng statíonary. Takai eË a1 t89]
show thaÈ by sinusoidally modulaËing the amplitude of the reference
beam the characteristic fringe function is JrcosA where A is the phase
difference between the phasor vibration and sinusoidal rnodulation and
J, is the fírst-order Bessel function. Hence phase information ís
observable as a brightness varíatíon of the frínges.
Aleksoff [90] introduces phase modulatíon of the reference
10.
beam ín lieu of amplitude modulation and the theory and techníque are
analysed in detail by Neumann et al [91] and Aleksoff 1,921. The
characteristíc fringe function is essentíally Jo (kA) where A includes
the phase difference between the phasor vibratíon and the bean mod-
ulatíon and k ís Ëhe propagation coristant for Ëhe laser líght. Mottier
t9:1 shows that by phase modulating the reference beam wíth a tríangular
function rather than a sínusoídal functÍ-on, fri-nges near points vibrating
with Èhe sarne phase reconstruct brighter than others.
Gupta and Aggarwal [94] report on a scheme for deterrnining
Èhe direction of motion of statíc deflections by a triple exposure
technique wíth a change of phase of n radians for one exposure. Stetson
t95] sËudies the effects of beam modulation on fringe loci and local-
izatÍon in tíme-average hologram interferometry of phasor vibrations.
Lokberg and llogmoen t96] extend the theory of vibration phase rnappíng
to electronic speckle pattern inÈerferometry. Belogorodskii et al 1,971,
Butusov [98] and Yoneyama et a1 [99] use a small mirror on a point of
the vibrating surface to phase modulate Èhe reference beam thereby
separating rigid-body motion from oÈher vibration. Fínally, LevítË
and Stetson [100] describe a phase-mapping procedure to generate
vibration-phase conËour maps.
The theory of time-averaged holographíc interferometry hras
first applíed to three-dimensional vibrations by Líem et al [101].
They reported a sÈrange shift in the calculated geometrical vibration
over the surface of a cylinder when illuminated from various posítions.
Tonln and Bies [102] explaíned the anonaly by pointlng out that Ëhe
cylinder vibrated with components in three orthogonal dírections and
hence the application of the normal one dímensional theory of Powell
and Stetson r,{as erroneous. They subsequently extend the theory of
statíc holographíc interferometry of surface strains in two dÍmensions
11
[103] to simple harmonic motion in three dimensions and successfully
apply it to a vi-brating beaker and wine glass r¿ith no strange geom-
etrícal shíft. Tuschak and Allaire [r04] determíne rhe radial and
longitudinal components of an ultrasonic resonator using a sí.mp1e
extension of three dimensional statie holographic theory. Tonín and
Bies [105] improve Èhe method of analysis by assuming the víbration
Èo consist of a seríes of orÈhogonal superposed modes and determine
the amplitudes by the urethod of least squares.
Tonin and Bies [106] extend this to include phasor víbrations
1n three dimensions and show hov¡ the amplitudes and phases of super-
posed and coupled modes (phasor vibrations) are determined rrrithout
modulating the laser beam. The theory also obviates the need Ëo
arbitrarily assign signs to frÍnges, a meÈhod reported by vlasov and
Shtanrko [107]. Finally, Archbold and Ennos t1O8l consíder two-dimen-
sional víbrations of two frequencÍes.
The theory of holographíc interferometry applíed to general
three-dimensional surfaces requíres that the angles subËended by the
ill-unination and observaËion vecËors to the surface normal be knor,¡n.
The author believes the best way to achÍeve this ís by application of
contour holography. rf the surface conÈour and the geometry of the
optícs are known then the required angles may be determíned. Hence
a complete vibrat.ion analysis would requíre a set of hologram pairs -a contour hologram and a vj-bratj-on hologram - which uray be taken sequen-
tíally. The list of references includes methods of hologram contouring
using single frequency lasers [109-]-201, dual frequency lasers ]L2L-L231
and Íncoherenr light IL24l.
r.2 VIBRATIONS OF CYLINDERS
Most of the three dimensíonal holographic theory in this thesis
wlll be applied to víbrations of cylinders as they provide símple curved
12.
surfaces and three dimensional motíon which i.s analytically predictable.
The flexural vibrations of perfectly cylindri-cal shells is rqell docu-
mented by Leissa If25] who sunmarízes ten theories which generally
give results too síurilar to be experímentally distinguished. The
Arnold l{arburton theory ín Èhe authorrs mind is perhaps the best
reported in the literature [126-130].
The theory of vibrations of distorted cylínders is reported
using exact methods, semi-empírícal methods (e.g. Rayleigh-Ritz, Least
squares) or finíte eleme-nt methods. The methods consistent r¿ith the
holographic theory developed in thís thesis are those which assume
soluÈions which are eígen functíon expansions. Such methods applíed
to one-dlmensional beam problems are lísted in the references [131-135].
In particular, the Rayleigh-Rítz method is excellently described by
Hurry and Rubinstein [136, chapt. 4].
Firth t137] predicts the generation of extra-ordínary modes
(or superposed modes) by descríbing irregularities ín a shell. as a
Fourier s,eries in the radius and hence solvíng the Helmholtz equation
by assumÍ.ng solutÍons to be a combínation of the mode.s for an undisËorted
cylínder. In a símilar way, Rosen and Singer [138'139] solve the
Kármán-Donnell non-linear shel1 equatíons to study the effect on Ëhe
resonant frequency of the superposed modes. Yousri and Fahy [140] gen-
exa!íze Èhe distortion to include anisotropies ín the radius of the
cylinder, wall thíckness and Youngrs Modulus. Courb:'-níng all three
dlstortions into one term and solvíng the Reissner-Naghdi-Berry equaËÍons
Ëhey obtal-n an expression for Èhe radial displacement. The Rayleígh-
Ritz meÈhod ís used by Toda and Komatsu [141] and the finite difference
method by Br:ogan et al Í1421 to determine the resonance frequencíes
and mode shapes of a cylinder with cut-outs. Tonin and Bies [143]
determine the resonance frequencíes and mode shapes for a cylinder of
13.
variable hrall thíckness usíng the Rayleigh-Rltz method. and verífy the
theory usíng holographíc interferometry.
1.3 SOI]ND RADIATION FROM VIBRATING CYLINDERS
Holographic ínterferometry has been applied to the study of
sound radiation from plates by Hansen and Bíes lL44l and pipes by Kuhn
and Morfey [145]. Measurement of acoustíc power by holographÍc
methods Ís possible íf Èhe radial component of vibratíng modes is known
[146 chapt. 6, L47]. The theory of sound radiation from a cylinder
of finÍte length ís presented by Junger and Feit tl48l but no experi-
mental evídence is available ín the literature LL4g-r52] to support
thís theory. However, experiments have been conducted by the author
and results are presented in Appendix VI.
L4"
CHAPTER 2
TIME-AVERAGED HOLOGRAPIIY
The theory of time-averaged holography for staric displac.e-
nents ís extended Èo include three-dimensional vibrat.ions of the most
general kínd at a single frequency. Additíonal1y the Theory of
Generalízed Least Squares ís introduced to solve the characteristic
equatíon. The determination of component amplitudes and phase is shom
to be possible using thís method. Also the experimental equipment
used in thís research and method of analysis are described.
2.L THEORY OF TII'ÍE-AVERAGED HOLOGRAPHY
2,L.I The ArcumenL of the Characteristic Function ín Spherical
Co-ordi-nates
A spherical co-ordinate system ls used through the discussion
to follof as iÈ is r,rell suited to descríbe the optical arrangement and
is amenable for use ín descríbing the curved cylindrical surfaces of
principal concern in this work. Additionally the spherical co-ordinate
system, being curvilinear, is most convenient for describíng the small
scale víbratory motíon of concern here as ít reduces locally to a
Cartesian system. Finally the optícal process to be described requires
that a number of holograms of different orientations be taken and ínter-
preÈed. This enables the Least Squares procedure to converge with
confÍdence and hence use is made of a double turntable arrangement -
one turnt.able rotar-es the object about a horizontal axís and simult-
aneously rotates on anoLher turntable with vertical axis, not unsímilar
to a turret. This arrangement is besÈ descríbed by a spherical co-
ordinate system.
15.
consider any point on a surface at rest as the origin [see
Fig 2-1 (a) l. The co-ordínate system for the vibraríon Ís defined
such that the principal co-ordinate e, Ís the surface normal, a choice
díctated by the fact Èhat in acoustics (an area in v¡hich the author isconcerned), Ëhe vibraÈion component normal to the surface is responsible
for the generatíon of sound. rf a Èíme-varying force of constant
frequency ís applied to the object, the point under consideration
executes Ëhree-dimensíonal harmonic motion wíth a locus most generally
descríbed as an e11ipse. rn addiÈion, the vibration is assumed to be
statistically statíonary. The locus of víbration is then a spatiallyoriented plane ellípse. The time-varyíng vector which descríbes the
motion of the point is
g (r) c t e (2.L))( + a(t)t(b+ ) "2tJI
where the orthogonal components of vi-bration c(t), b(t) and a(t) are
time dependent. The choice of letters e,b,a and theír order isborrowed from shell theory for cylinders where they correspond to the
radial, tangential and longitudínal components of vibration respecËively.
Fig. 2-]- (b) shor,¡s the co-ordinate system for the opticalarrangement.
lr i" the illumínat.ion vector, K, the observatíon vector
and as ís usually the case ín time-averaged holographic theory 0, and
02, the angles subtended by !, ."a y t" the plane ere2 are measured
from the surface normal e, and positive in the directíons indicated.
The opÈical path difference ís t33l
o. p. d. (2.2)k
where k Ls the wavenumber (2r/X) of the laser radiatíon and fl is Ehe
argument of the characterisÈic function M(o) [33,86]. From the geometry
Kr)(K2 ICI
k
ê3
r6
Elliptic trajectory
b
d
d
e2
c \\
Surface normal
jr
FIG. 2-I (") CO-ORDINATE SYSTEM FOR THE VIBRATION
COHPONENTS OF A POINT ON THE SURFACE OF THE OBJECT
]-7.
9g
g0 2 0r
\
0t\
lz /e,2
! /¡
I
lr-
//
\l\l /_v
/
Surface normal
FIG, 2-I (t) CO.ORDINATE SYSTEI1 OF THE OPTICS FOR A POINT
ON THE SURFACE OF THE OBJECT
Arrows show posltíve increasing direcËion.
3
1B
of Fig. 2-I (b),
\ = -t cos0, "t.0r1, - ksínOrsinS, uz - k cos0, e, (2.3)
K2 = kcos0rsín{r..'t -ksinO, sínQre, * kcosôre, (2 .4)
Substituting equarions (2.T), (2.3) and (2.4) inro equation (2.2) gives
Ëhe general result
fl = .(t) (cos 0, sin 0, + cos 0, sínþr)
+ b (È) (sin 0, sín 0, - sín 0, sin g, )
+ a(t)(cosQ, * cosSr) (2.s)
2.I.2 Coupled and Superposed Modes of Víbration
rn general,vibration is a function of spatial varíables as
well as tíme dependent. rn order to make the analysis praetical, the
vibration is analysed on a seríes of surface contours whích are defíned
as the lntersection of the object surface and the hyperplane 0 = 0o
(see Fig. 2-2). rn the case of a cylínder, for example, the surface
contour Ís a principal circumference. Hence the location of any pointon the surface contour is vlrtually defíned by a síngular angular
co-ordinate f and the toËal vibration determi-ned "slice-wlse" over the
entire surface of the object. The vector d describing the vi-braËion
of points on a surface contour is
9(q,t) = c(E,t).J + b(6,t)e2+ a(t,r)es (2.6)
and the components of víbratíon may be expressed as eigen functions
[84,85] of the form
19.
z
I
ø\
I Surlace'contour
0
FIG, 2-2 CO-ORDINATE SYSTEH FOR THE OBJECT SURFACE
The surface contour is the intersection of the objecÈ surface and
the surface ô = 0o.
\Y
x
c (8, t)I
= I .t (6) cos (urt + yí¡i=1
20.
(2.7)
(2.8)
(2.L0)
(2.LL)
I.b(E,t) = I ¡i(E)cos(or+ßi)
í=1
(2.e)i=1
Taking the c component as an example the i-th term in equation
(2.7) is a mode and the I Èerms are referred to as COIIpLED TEMPORAL MODES
since they are generated by the clrivÍng force of constant frequency and
couple together with a time-invariant phase relationshíp, yí. The
anplitudes of rhe coupled teuporal modes - "í(E), tí(E) and ai(E) - vary
over the surface contour and may be expanded into a trigonometric series
[102] of order (N+ 1) rhus
Ia(q,t) = I "í(l)cos(t¡r+aí)
.N
"t(E) = I t{sir,1níg) + yico"(r,íE) ln=o
.Ntl(E) = I twlsin(r,Í6) + zicos(.,1g)l
n=o
ri(E) = I t";"rr,(r,ig) + icos (r,ig) ln=o
(2.L2)
The form of this expansíon is particularly relevant sínce the vibration
of any structure may be consídered as a combination of normal modes
[83,84]. The (N+1) terms ín each expansíon are ca1led supERposED
SPATTAL MODES since they do nor exist independently in the physical
sense but describe mathernatically a complex spatíal function of period
2r.
2I.
2,L.3 Theorv of Tíme-Aver aged Holosraphv for One-Dimensíonal Víbratíon
The following rígorous derívatj_on of the theory of time-
averaged holography for one-dimensíonal símple harmonic moÈion of a
constant frequency is due to Butusov t5]. The complex analyÈical
signal in the plane of the hologram H is t3l
dS
where A(S) is the anplitude of the laser light in the plane of rhe
object surface S, Z ís the separation between object and hologram and
(2.L4)
(refer to Fig, 2-3). If the vibration d(S) ís normal ro the surface
and executes sírnþle harmonic motion of frequency ul then
R Ro (II, s) + g (H, s) sÍnr¡r (2. ls)
Hence equaÈion Q.ß) becomes
-.i IA(H,t) = * )
A(H,t) = å lJ ocr "
S
iZ sin 5
(2.13)
(2.L6)
(2.L7)
J octl eikR
(H's ' t)
S
ft = lz2 + (trr-sr)2 ,- "")'1"(h+
ikRo (H,S) Íkd(H,S)sint¡te ds
where t4l
with 0, and 0,
SubsÈituting
d(H,S) = d(S) [cos0l (H,S) t cosOr(H,S) ]
defined in section 2.L.L.
e
equation (2.16> becomes
íj5 (2. 18)
22.
S,IL
//
ll llluminal ion/ Lz
/s
É(s)
S(s,.sr) H(hl.h2)
/ h
ßs
H
z
FIG. 2-3 SCHEIIATIC OF HOLOGRAPHIC PROCESS
-{llikR^(H,S)@A(H,t) =u;J.|oCtle - (,_l ..rs J=-'
jij ot(kd) e )¿s
23.
(2.Le)
(2.20)
(2.23)
(2.24)
I J (kd) e
A(H,t) = Jo(kd)Ao(lt) * i",r, (ra¡"ij'tAo(H)
#o
æij trlt -i
kz A(s) eikRo (H, S)
dSj =-- j
where the Èerm in the square brackets ís sírnply the origínal complex
wave amplj-tude denoted eo(n). Hence,
S
jj
where J-_ ís the n-th order Bessel function of the first kínd. Forn
time-averaged holography, Èhis fíeld is averaged over time t thus
1.
A(H) = i A(tt,t) dt (2.2L)
Assuming t>>2nfu then the second term of equation (2.20) is zero and
the average fíeld A(H) becomes
Ã1u¡ = Jo(kd)Ao(H) (2.22)
That is, the surface of the reconstructed object ís modulated by the
Bessel function of zero order which is also called the characteristicfunctíon for this víbration. In general the characËerístic function ísdefined as ta01
M(o) = + e dr
I
f iCI
whereupon equation (2.22) is
Ã1u¡ = M(ç¿) Ao(H)
The argument of the characterístic function denoted fl is equal to kd,
24.
r^ríLh d given by equatíon (2.L7), for the vibrat-ion consjdered here.
2.1.1+ Theorv of Time-Averased Holoqraphv for Three-Dimensicnal
Vibratíon and Application
In section 2.L.1 the argumenË of the fringe functíon Q was
derived for three-dimensional vibration. 0n substituting equations
(2.7), (2.8) and (2.9) into equations (2.5) and (2.23) and denoting
the geometrícal faetors ín equatíon (2.5) as
KO = cos0rsinó, * cos0rsin0,
SO = sinOrsin{, - sinOrsinO,
Qn=cos0r*cosQ,
flrsín rrrt ]
(2.2s)
(2.26)
(2.27)
then,
where,
1 ft ík[Qrcos or -"
_ _ jo"
(Bi)sq + aisín(ot)QolI
n, = ,1' ["t"ir,(yt)Kq + basl-=I
dt
OS+ bicKr¿qI
I= I l.tcos(yt)i=1
( ß1) sq + alcos(oi)Qol
t_n
whích is of the same form as deríved by Molin and Stetson [86] for
two one-dimensional vibrations in phase quadrature. The solutíon of
equation (2.26) as deríved in the latter paper is
M(CI) = Jo{k(a1 + a3)%}
Hence for every point q on the surface contour,
(2.28)
25
Õ2
__9.
kz
+[
I
i--1
1
rT (.í"os (yi) r (ßa) sq
)sq
OS
q
K = cosO * cosOq I
S = sÍnO - sin0q I
=0
+ "i"o=(oi) Qo2+ bic
q
f,.t"rr'(yt)r + bisín(ßi , i .íf a sl_n(c )Qql2l=
which is the general solution for time-averaged holography of three-
dlmensi-onal vibrations of a single frequency where the spatiallyvarying amplitude".i,bi
"rd ri are given by equations (2.10) to (2.r2),
The varíables whích are deÈermined from tíme-averaged holograilìs¡ are
nO' *q, Sq, Qn and 60. The other variables are the unknowns.
EquaËion (2.29) will now be applied to some cases of part-
icular interest. For simpliciÈy, the surface contour is assumed to
IÍe on the XY plane (see Figs. 2-L and 2-2) and hence óo = 0 t = þ2 = n/2.
Equations (2. 25) become
(2.2e)
(2. 30)
2
2
Qn
SPECIAL CASE 1
The sirnplest ki-nd of simple harmonic motion in one dimension
is sÍngle mode recÈilínear vibration. An example of this is the vibration
of a cantilever or simply-supported beam. I^Iíth r= 1 and. bl = al = 0,
equation (2.29) reduces to
c (cos0 * cos02k )I
which is the classícal resulr t4] (the subscrlpt q is assumed).
(2.3r)
26.
SPECIAL CASE 2
Consíder a number of one-dimensional modes with non-zero phase
dífferences. For this c.ase, bí = ri = 0. Equation (2.2g) becomes
ç2
yz [(clcosyl + czcosyz + ...)2
f (clsín11 + c2siny2 + .. )\(coso l. cosO 2 (2.32,>2I )
Èhe term in the square brackets ís also the square of the phasor sum of
Ëhe índividual vibration components. Ilere is proof of the iurplication
of the statement made by Stetson and Taylor [84] that "the magniÈude of
the phasor sum of the argument functions corresponding to e.ach of the
component motions (is) the argument function for the combíned moÈíontt.
SPECIAL CASE 3
The simplest kind of sírnple harmonic motion ín three dimensíons
is single mode recÈilinear víbration. An example of this Ís the vibraÈion
of a perfect cylj-nder. I^Iíth I= 1 and yl= ß1, equation (2.2g) becomes
* = .t(coso, * cosor) + bl{sirre, ) (2.33)- sinO2
which is exactly the expression derived by Tonín and Bies [102] and applied
to a beaker vibrating in a Love mode of order t= 2. Fig. Z-4 (a) is a
pictoríal represent.ation of the way 1n which the varíables of equatíon
(2.33) contribute Èo CI/k whích determínes Ëhe order of Ëhe fringes on
the hologram.
SPECIAL CASE 4
The motíon havíng the next highest degree of complexity is
simple elliptíc motion which may also be descrfbed as a single coupled
mode (I=1) with the c and b components out of phase (yI# gl). Hence
equation (2.29) becomes
27.
cl Kq
Lqk
ds
(a)
(b)
q c K
P(a)
A B
(b)
.'Kg+b2S q
9qk
FIG. 2-4
(c)
Single coupled mode.
Single coupled mode with out-of-phase componefits.
Simplest kind of elliptic motion.
Two coupled modes.(c)
28.
with a=yl-ßI 12. :S)
2-4 (b) shows how cl, bl and A in partícular ínfluence the frínge
determined bV OO/k. It is assumed that the phase difference A
ín equatíon (2.35) remains invariant over the vibratíng surface. This
is represenÈed by an angle subtended from a chord AB in a circle of
dÍameter 2bl (or equivalently 2cl) which is the maximum value of
br(sine, - sín0r). The order of the frínges on Èhe object surface is
represented by the position of poínt P on the clrcumference of the círcle.
This sirnple phasor-líke picture clearly shows the role played by A. If
A ís zero or a multiple of n then Fig. 2-4 (b) reduces to Fíg. 2-4 (a)
and the variation of An/k wíth A is small and hence errors in the deter-
mination of A from CIn/k will be largest for these cases.
It is interesting to note that the characterisÈic equation
(2.34) for a single coupled mode r¿ith Èhe c and b components out of phase
is sirnilar to that for two one-dimensional vibraÈions of the kínd
considered in Special Case 2. Equatíon (2.32) becomes
n2__9. (crro) 2 + {c2xr¡2 + 2eosa (clKo) {c2ro) (2.36)
(2 .34)
Fig.
order
k
with L=yr-12 (2.37)
The vlbraÈions corresponding to equations (2.34) ar.ð, (2.36) are dist-
lnguished by varying 0, and 0rr the íllumínation and observation angles
and hence Èhe deÈermination of the motion is unambíguous.
SPECIAI CASE 5
ç2__q-
k2
The next order of
is two coupled modes (I = 2)
(c ltco) 2 + (u t to, ' * 2cosa {c lro) (b l sq)
complexity of three-dimensional vibration
for which the phases of conponents wíthin
2
29.
a mode are identical but there exists a phase dÍfference between the
modes l-.e. yl= ßL, "(2 = ß2, \r +"(2. The path trajectoïy is elliptic
as in special case 4 but the total motion is the phasor additíon of
four components. An example of such motíon ís Èhe simultaneous
excitatíon of two modes in a cylinder. Equatíon (2.29) becomes
Q2_9.y2
("rr +bls )2 + ("2xqqq +u2s )2q
* 2cosa("tKq + blsq) ("'Kq + b2sq) (2. 3B)
with a=yr -.(2 (2.3e)
Again, the method of formation of the fringe orders is represented by
the phasor-líke diagrarn in Fíg. 2-4 (c) which is a combination of Fig.
2-4 (a) and 2-4 (b). As in Special Case 4, a one-dimensíonal phasor
vlbration considered ín Special Case 2 exists which ís described in
a simílar way Èo equations (2.38) and (2.39) but the two vibrations
are dÍstínguished by the geornetrical factors of equation (2.30).
For three-dÍmensj-onal phasor vibration the statement of Stetson
and Taylor quoted previously nay be generalízed to:
The magnitude of the phasor sum of the argument functions corresponding
to each of the component motíons of three-dimensional víbratíon ís the
argument functÍon for the combined motion.
Hence providíng a number of holographic interferograms are taken of
the vibrating surface there is no ambiguiÈy in resolving the motion.
2.L.5 The Theorv of Genera]-ized Least Squares
The theory of Generalízed Least Squares presented here ís due
to Spendley [153] and Powe11 [154]. Given a function O(R,Ç) invotving
parameter values R = R(x, , "rr) and independent variables
E = E(81, , Ek) the deviatíon of an observed value yo from a
predicted value 0(R,E ) is of the formq
R) ly - 0(R,6 )l(f
30.
(2.40)
(2.4l-)
(2.42)
(2.43)
q q
where f is called the Residue Funetíon. rt is required to find the
set of parameters R that wíl1 minimize
q
T
ç = I trq=1
âf(R) --s.
aR.J
, i = 1,
(R) t'q
where t ís the number of data poínts.
TNow
Thus
â6 =2LfâRj q=l q
afq
,n
rt is assumed that E is a quadratic function ín the R. and hence fo
must be linear in Rr. EssenÈíally the derÍvation assumes that the
functíon þ has been línearízed about approximate values of the R.J
However as the derivation shows this ís not, essential. rt is worth
notíng that monotonic convergence Ís by no means assured.
(R)
consË - ^(q)JãR.
J
sâY, and if 6 is a correctíon vect.or whích is the dífference between
vector R and Ê. tor which Ç is minimum
1.e. ô = R R (2. 44)
then f. (R)-f (R)qq *r!ro' t(n)
and substituting ín equarion (2.42),
(2.45)
31
T n+ I ô.c!q)1t_l-=Iq=zLtf (R)
R=R q=l
since ç(R=Â)
Rearranging, we have
= 0 (for all j)
Iô
)( âeaR.
J
F(j)
^ (q)J
R) . cj(q)
(2.46)
(2.47)
(2.48)
(2 .4e)
(2. s0)
(2. sl)
(2.52)
' trto'
=iísaminímum.
n T
IT
-Irqlr q9=1
Io .(s). .(a)r-J cn> . cr(e)i=1 l_
or ín matrix notation
F
Iô f F
where f(i,i)
and the jth element of the colunn vector F is
= T .Ío).q=r
T
-olrto'
Thus, given F and f, the correction vector ô is determíned and the
optimum solution ff. att"itt"d in a seríes of iterations.
2.L.6 Solution of the CharacËerístic Equation by MeÈhod of Least
Squares
As equation (2.29) is non-linear in the parameters, the con-
ventíona1 theory of least squares ís not dlrectly applícable. Linear-
ization of the characteristíc equation results in an exÈremely large
number of unknor^m coefficients which require an even greater number of
data points for a solution. By using the method of GenetaLLzed Least
32
Squares, however, li-nearízation ís not required and the available data
ls used to its fullest advantage.
The resídr.re function, from equarions (2.10) , (2.1.l-), (2.L2),
(2.29) and (2"40) is
INtI{Ii=l n=o
+[
âfrc*(9) = 4= zl. IJ ,*o i=
n
Next, the y componenÈs,
for j=1, r(N+ 1)
f ,jsirr(rritn) + yicos (oign) Ìcos (yi¡rq+. . . +. . .12
+ yícos (rriEn) Ìsin(yí)rq+. . .+. . . l2
Q2q
y2
íi-c cos(y )1
*o*. . .+. . . lsin(nlEo)cos (1ø¡rn
"i"l-rr(yi)Kq+. . .+. . . lsin(n¿Eo) sin(vg)r,
q
INI t f xasín(na"nl-=I n=o
E)q
Note there is no need to allocate a sign to fJ as ín references"qand [107] sínce it appears as a square in equatíon (2.53). The
of rnatrix G are the derivatives of equation (2.53) with respect
unknown and the ro\Áts correspond to each datum, q. Fírstly, for
componerrts,
(2.s3)
[102]
columrs
to each
the x
(2.54)I
+21,Li=1
afc(q). _ q = 2["j+r(N+1) ,rl ,lr.t.o"(ví)Kq+.
. .+. . . lcos(nrEo)"o"{y¿¡ro
"i"irr(yí)Kq*. . .+. . . lcos {n¿Eo)sin(y¿)xoI
+2tlí=1
33.
(2.ss)
for j=1, , r(N+1)
And, for y the phase,
(q)j+2r (N+1)
âfG
âY
o= --i,í"o" (yi)rq+. . .+. . . l.[rn"i.r(y¿)
I__21)cí=1
(2. s6)
for j=1, ,Ï
Síurílar equaËions for wrzr$ ar.d urvrq complete matrix G.
If the number of data points in t, the number of superposed
modes (N+ 1) and the number of coupled modes I then, G ís of order t
by 3I(2N+ 3), f is a 3I(2N+ 3) square matrix and F a colurnn vecÈor of
order 3I(2N+3). The vector R describíng the parameters is
R = ("1,"å,...,y| ,yr2,...,yÌ ,f2,...,r| ,rtr,...,or) (2.57)
2.2 EXPERTMENTAL EQUIP},IENT AND DATA ANALYSIS
2 2.I HOLOGRAPHY
A schematic of the optical system used Èo generate holograms
is shown ín Fíg. 2-5, this system being common to almost all experiments
reported in thís thesis. Depending upon availability at the tíme of
the experiment any of the following Helíum-Neon (63281) lasers were
I+zt. l
i=1císin(yí)Kq*. . .+. . . 1"lro"o" {"¡,[)
Mirror
5pat ialf ilter
FIG. 2-5
Tu r n table
\
Camera
Spatial
Laser
34.
Beamspl itter
Mirror
f ilt er
Holog raphicplate
OPTICAL SYSTEM
E lect r omag net
C yl i nder
35.
used - Lasertron LE105 (15mw) , LE2O5 (5mhr) or Merrologic MV910A (lml^t).
The optical powers were monitored with a Tektroníx J16 dígital photo-
meter. Each spatial filter is fitted wíth a 1.0¡r pin hole which was
measured by ray-tracing to be 35 tlmrn behind the lens. Holograms are
recorded on Ilford He-Ne/1 (5Oerg/cm2), Agf^ 10E75 (50erg/cm2) or
Kodak L3L-02 (Serg/cm2) plates nomÍ-nally lmm thick and 120mm by gOmrn
in slze. Exposed plates are developed for 7 minutes in Ilford Microphen,
Agfa G5c or Kodak Microdol-x developer. Exposure times varíed from
20 sec to 120 sec depending on laser por¡rer, plate and developer. The
best combínaËion is Kodak plate with Agfa G5c developer. After devel-
oping, the plates are immersed in a L% soluËion of Acetic Acíd stop
bath for 5 seconds and fixed i¡ a l/L solution of llfofix Acid Hardening
Fixer for 10 minutes. The plaËes are r,rashed in running water for 15
minutes, ímmersed in an B0Z solution of industrial alcohol for 5 ninutes
to remove the antí-halo coating and then dríed.
Duri-ng exposure, the beam splítter ís set for a 4lL ratio of
reference to object beam intensity. On reconstruction, all the laser
light is direcÈed to the reference beam and the ímage photographed with
a Minolta sRT303 35um camera loaded wirh rlford Hp4 (400ASA) or Fp4
(125ASA) film. The aperture setting on the camera is important - too
low a setting wíl-l result in broad fringes due to varíation of the
observation angle and conversely a high seÈting resulÈs in very long
exPosure Èímes. The f5.6 sÈop results in less than .02rad observation
variation and maximum exposure tíme of 60 sec and was used throughout
the experiments.
The object under invesÈígation is lightly spray-painted wíth
natt white primer which ensures an optícally diftuse surface. In some
experíments retro-reflectíve paint was used but with a different optj-cal
system. The object Ís placed on a lovr-profíle turnÈable graduated
in degree increments. A translation table and hologram mask were
36.
constructed to enable six holograms to be recorded on one plate which
avoids the problem of cutting the plates. The Ëranslation table and
t'windows" of the mask are so arranged that each hologram is taken from
the same position ín space which enables the same geometry to be used
in Èhe analysis. Fig. 2-6 is a photograph of Lhe optics and Fig. 2-7
of the translatíon table and mask with a typical hologram Ín the fore-
ground. Unless othenrrise sÈated, seven holograms are taken of each
mode of the vibratíng object at 15" angular increments of the turntable,
a procedure which uses türo plates.
2.2.2 Modal Drivíng System
To prevent distortíon of the modes electromagnets r¿ere used to
excite the object withouË contact.. These are bar magnets wound with a
coil of wlre and positioned close to the surface of the object hence
provídíng a steady force which is then modulaÈed. For Ëhe ferromagneti-c
objects no problem hras encountered, however, for the non-ferromagnetic
objects the steady biasing force could only be índuced by fastening a
sroall piece of ferromagnetíc materíal to the object aÈ the point to be
driven. Thus although a metallíc surface may be driven with an elect-
romagnet making use of the fíeld of Índuced eddy currents [155] the
resulting forcíng function wíll alr"rays be unidírectÍonal, one of
attractÍon or repulsion dependíng upon whether the surface is para-
mâgnetic or diamagnetic and in consequence the r¿ave form of the forcing
functíon will always be rÍch ín harmonícs whích is not desired.
The spatíal dístríbuÈíon of the excitation force is often an
lmportant facÈor in determining the mode shape. Shírakawa and Mizoguchí
t1-56] determine the mode shapes of a cylinder excited by a periodíc
point force. For a slightly imperfect cylinder for example the node will
usually orient iÈself wlth Èhe asymmetries no maÈter where the force ís
applied. It is not necessary to excite the modes at the antl-nodes. By
37.
FIG, 2-6
Optics for the generatíon of holograms'
rl=tIIII
FIG. 2-7
The translation table (T), hologram stage (S), typical hologram (H)
and rnask (M).
(,oo
39.
contrast, for a neai-perfect cylínder a large degree of modal coupling
is evident for antinodal excitatíon and is only reduced by posítioning
Lhe drivers close to the shear-diaphragm ends thereby decreasing the
magnitude of Èhe cross-correlaLion between the node to be driven and
the spatial fourier components of the forcing function.
As the holographic exposure time could be as long as 60 sec
the mode under study must be stable for that perÍ_od of tíme. For
modes of high Q use of a dríving oscillaËor is ímpossible due to problems
of drift. Ilence use is made of a technique employ,íng a nethod of
posítive feedback to create an electro-mechanical oscillator which is
self-exciÈed and very stable [157]. A schematic of the system is shoum
Ín FÍg. 2-8. The heart of the feedback system is the resonance unit
designed by the author. Thís is essentÍ-al1y a voltage controlled
arnplifier whose gaín is determined by the 1evel of the input signal.
If thís increases then the gain of the unít decreases and vice-versa.
Hence the resonance unit provides the required level of output signal
whích will rnaíntain a constant ínput level. Additionally a 0-360 degree
phase shifË circuit ís incorporated into the uníÈ to enable correct
phase matching.
A Bruel & Kjaer (B & K) miniature accelerometer type 8307
monitors the vibration 1evel wíth a minimurn of loadíng (the weíght of
the accelerometer is only 0.4 grams). The signal is anplified by a
B & K spectrometer type 2L12, fed to the resonance unit and then to a
power anplif ier and the elecËromagnet. The gain of the por¡rer arrplif ier
is set such that the system gain ís greater than unity hence ensuring
sustained oscillation. However, Íf Ëhe system gain is too hígh the
vibration wil-l- hunt. To select any mode, the spectrometer thírd-octave
band filter is set to include the frequency of that mode. The phase and
gaín of the resonance unit and povrel: amplífier are then adjusted for
optimum oscj-llatory stability. Finally, the leve1 of víbration ís set
r
40
electromagnet
cylinder
accelerometer
t-
gate
feed backamplifier
resonance unit
freq.counter
30 wamplifier
DC shifto ut putlevelVCA
rms-DC
phaseshift
inputattenuator
B&Kspect romete r
2tt2
FIG. 2-8 I'1ODAL DRIVING SYSTETI
4L
by adjustíng the DC conditíons at the gate of t,he voltage controlled
amp lifier ín the resonance uníÈ. The system ís so successful that a
mode may be kept stable for hours wiÈh no frequency or amplítude
varíation (as limíted by the accuracy of the instrumenÈs). A photo-
graph of the equípment is shovm ín Tíg , 2-9.
2.2.3 Geometrv f or the Analvsis of Cvlinders
As the analysis of the reconstructed holograms is cornrnon to
all experiments excepting those in sections 3.1 and 3.2 ít wÍll be
descríbed here. The basic geomeÈry is due Èo Líem et al [101]. A
cross-sectíon through the optical plane for a vertically standing
cylinder of outer radius a, ís shown in tr'ig. 2-10 where the angles
0, and 0, are to be calculated for dark frínges on a photograph of the
hoJ-ogram reconsÈructíon. The perspectíve of the object is taken into
account by nakíng use of the pin-hole camera concept. The process of
phoÈographing the reconstruction and printing the photograph with an
enlarger is considered equívalenË to phoËographing the reconstruction
with a pÍn-hole camera consisting of an aperture of diameter D and a
screen. If D is made smal1- compared to the distance AÀ by suit.able
choice of the camera f-sËop and the size of the image formed on the
Ímaginary screen ís rnade exactly equal to the size of Èhe image in
the photograph by suitable choice of the distance X then fírstly
considering a plane photograph of the cylinder
& =^u =^'XAA(2. s8)tany
where a__ ís Èhe radius of the cylínder ín the photograph and thev
asÈerísk indicates the maxímum value of the angle y. Now, for any
other poínt on the surface of the cylinder
!ãC
t:)uål Powêr AñFl¡fiaFæ¡, tr tsa
3
"
f*t¡
€r rB
¡L
['l
(,(' e- C- CtoÕe
FI G. 2-9F.l.J
The electronic equipment used to drive and monitor vibration modes.
Þ
43.
Poi ntillumination
0¡
02\AA
D
Apert ure
x
tt--* H t+lr
5c reen
FIG. 2.IO (JEOI'IITRY FOR INTERPRTTATION OF HOLOGRAIV]
PHOTOGRAPllS
Cylinder Vertical
e2
II
/
44
H H'^,tanY ='v
^ AA.av(2.se)
by equati-on (2'58). The angle 0 is calculated by applicarj.on of the sinerule for triangles hence,
AA^2 (2.60)
ofr
IfPis
and o is
rule,
orr
sÍn[n - (tr/2 - 0) - V] síny
o = cos 1(aesiny/ar) *1 (2.6r)
the distance from the cylinder axis to the poínt of illuninaËionthe angle subtended by thÍs 1íne and AA then again by the sine
^2sin0 I sÍn (0 Iu- (r/2 - o)l) (2.62)
P
I
or = tan t{r"o"(a+0) /laz - psín(o.+e)l} (2.63)
and 0, ís gíven by
0,=n/2-0+y (2.64)
Hence using equarions (2.59), (2.6r), (2.63) and (2.64) rhe angles g
lwhich is equivalent ro I in equation (2 .2g)], 0, and 0, are determined
for each fri-nge along the surface contour which is the íntersection of thecylinder and the optical plane. rf the cylinder ís rotated an amount
so radian" (so ís positive antíclockwíse in Fig. 2-Lo) then the values of0, and 0, determíned from equations (2.63) and (2,64) pertain to thepoínt (0+ SD) radians from the orígÍ_n.
For this arrangement the geometric tern QO is zero for all pointsalong Èhe surface contour and reference to equatío,- (2.29) shows Èhe
45.
component" "i ""rrrrot be deÈerrníned. This is overcome by positioning the
cylinder on its side wíÈh its axis in the optlcal plane. The surface
contour ís now a straíght line paralle1- to the generator. Reference to
Fig. 2-11 shows
L/2 L /2.*EanY
v (2.6s)AA- a
2 X
where L is the length of the cy1-inder in the photograph without rot-v
ation (Sl = 0) and the asterisk again indicates the maximum value of
the angle y. Now, for any other poínt on the surface contour,
H H.LfanY=-=-' X L__. (AA - ^z)v
(2.66)
(2.67)
(2.70)
(2.7L)
which is true even íf the geometric centre of the cylinder does not
intersect the axis of the turntable. From Ëhe sine rule,
Y AAD_=
siny
A1so,
and 0r=y-lSo
srnl(nlz * So - o - 0) + 0rl2 - Y - SD)l
and considering the ríght angled tríange of which YO ís the hypotenuse
Yo= ar/cos(So - 0 - 0) (2. 68)
Combining equatíons (2.67) and (2.68) and solving for þ gives
-tÖ=tan'arsín(s+y) - AAsíny cos(SO-c)
(2.6e)AAsinysin (So-cr) - a2cos(a+ y)
g, = arl ltan(rl2* So- d + 0) l
where 1, ís a length co-ordinate along the surface contour relaÈed to the
I0
46.
tltl
a*tøe
Çe
\oòe(
S¡
Y Cytindersu rface
Poi ntillumination
X
ì- t>c f een
Hl+
GEOIVlETRY FOR INTERPRETATION OF HOLOGRAIT
PHOT0GRAPHS
AA
I
->l
FIG. 2-II
Cylinder llorízontal
angular co-ordinate E of equatíon (2.29) by the transformation
E = rt'/t
which ís solved for 01
to gíve
e = tan-t a2tan(a+ ó - tO) - Psin(o-sO)
47.
(2.72)
and the circular mode number n is replaced by the longítudinal mode
number m ín equatlons (2.10) Ëo (2,L2>. Also from the sine rule,
sin[n - Gr/2+ar+r/2*So-cx- 0+0)] sin(x/2n So- o- O+ n/2+o)
PYD
(2.7 3)
(2.7 4)I"z-Pcos(a-So)
Hence using equations (2.66), (2.69), (2.7O), (2.7I), (2,72) and (2.74)
the angles 0,, 0, and co-ordínaËe 6 are deÈermined for each frínge
along the surface contour.
For the cylinder horizontal, a vertícal poínting rod at the
centre of the turntable índicaËed the posítion of the origin j-n the
hologram reconstructions. In this case, P ís the distance from the
orígín to the point of illuminaÈion and o is the angle subtended by
this líne and AA.
2.2.4 Data Analysis
The measurenent of fringe posítíons on photographs is a
laborious process with only the aíd of a straight-edge. Hence a data
system was assembled by the author to digitise rringe co-ordínates,
store and edit the information on cassette tape and transmít the data
to a central computer.' .Fíg. 2-12 is a photograph of the data system,
Fig. 2-L3 a schematic of the system. In Appendix I a flow dÍagram of
'-:
FI G. 2_I2
II
þ
i'A.I
c
F-Oo
The Data System.
49.
¡-rúc-('-t-¡o
ovrJ-o>
-oflt
r!rú
,!r!o
rú.c:TiÁJ r-
=3-cr lJ
=x
flt
rúu
-oo
o¡-
cou
-c¡{
(lrúp
-ct@
lil llill IIITI
D ATATABLET
Aport Ñ BPortAport Cport BPort
(\¡
l--É.
UI)
l-É.
UI
P.P. l. {f2PP.T.+2
INTEL 8O8O
MICROPROCESSOR
cDc 6400COMPUTERV.D.T
FIG,2-I3SCHEIVIATIC0FTHEDATASYSTEM
50.
the microprogram is given.
Data is taken fr:om the photographs usíng a Summagraphícs
HI^I-L-l4-TT Data Tablet having two 11 bit bínary ouËputs corresponding
to the. X and Y c.o-ordínates of a cursor whích is moved over the tablet
surface, the resolution being .01 inch. The outputs are connected
to the second Programmable Perípheral fnterface (P.P.I ll2) of an Intel
8080 Microprocessor f itted with 4K of E.P.R.O.M. (Erasable Programrnable
Read Only }fernory) and lK of R.A.M. (Randon Access Memory) . Data ís
recorded with a Dígideck PI-70 cassette rrnit whích is Ínterfaced to
P.P.I. llL, A Video Display Terminal (V.D.T.) enables comnands to be
fed to Ëhe system via U.S.A.R.T. lfl (Universal Synchronous /Asynchronous
Receíver / Transmítter) at 2400 Baud (bits/second). Addítíonally
U.S.A.R.T. ll2 is connected to a CDC6400 courputer vía a 300 Baud Intercom
líne.
The sysËem has fíve modes of operation which are selected
sequentially with the ESCAPE key and the system responds by printing
Èhe mode title on the right hand side of the VDT screen. These are as
follorvs: '
1) ..INTBRCOM.. A two-way link with Èhe CDC6400 computer.
FÍles are created and programs run using keyboard c.ommands.
2) ..SEND DATA.. Informatíon recorded prevíously on cassette
ís Èransmitted to Ëhe computer at a rate of 300 Baud.
3) ..RECORD.. Keyboard characters or co-ordinates from the
data tablet are recorded on tape and dísplayed on Èhe screen aË the
rate of 50 characters per second with seven co-ordinate pairs (corres-
ponding to X and Y) per 1íne.
4) ..R-EPLAY.. Recorded information is revíewed on the screen
on1y.
5) ..EDIT.. Enables one character at a tíme to be reviewed
with the SPACE key or one line at a time with the CARRIAGE RETURN (C/R)
51.
key. To enÈer the EDrr mode from the REpLAy mode key E ís depressed.
To alter a character Ëhe ESCAPE key is depressed which transfers the
system to the RECORD mode.
!ü1th this data system, the digitising of holographic inform-
atlon is sirnplÍfied enormously. rn addition the data tablet has a
STREAM MODE selector by which the 'co-ordinates of the cursor are
transmítted at the system rate of 50 char./sec. Hence a fringe may
be digitiseci ín a matter of seconds sÍmply by tracing ít h7tÈh the
cursor on the data tablet.
52.
CI]APTER 3
PURE AND SUPERPOSED SPATIAL MODES
The theory of time-averaged holography for three-dinensional
vibrations outlined ín chapter 2 is applied to a vibrating clamped
circular plate, wine glass, stainless-steel beaker and four cylinders
with shear-diaphragm end condíÈlons. rn addition the analysis is
applied to a vÍbrating cylinder wíth a seam. The application of the
method of least squares will be shoum to make optimal use of the data.
This chapter also r¿í11 show step by step the ideology leading to the
formulation of the general analysis descríbed in Chapter 2.
3.1 CLAMPED CIRCULAR PLATE
3.1.1 Holosraphic and Geometric Theo ry
The study of vibration of clamped círcular plates by holography
is reported by Hansen and Bies lL44l. The nodes are charaeÈerized by
a sÍngle component of víbration normal to the plate with a complex
Bessel distribution over the surface. For ÈhÍs case equatíor. (2.29)
becomes
o
Ë = "(E). [cos0, * cosOr] (3.1)
The mode shape c([) as determined by time-averaged holography has been
shor,m tT44l to agree r,rell with Ëhe theory for 0, = 0, = 0 (retro-reflective
illumination). An experíment to be described will demonstrate the
validity of equatíon (3.f) for various illumínation and observation angles.
Consideríng fÍrst a change of 1llumínatíon directíon, Fig. 3-1
shows the geometry Ín the optical- plane. For fixed angles of il-luurin-
atíon and observatÍon, 0, and 0rr the resulÈíng surface contour is a
53.
+l<-d.'lPtate
a2
AArP
Normat
Poi ntittu minat ion
ll"-+t H t'<-
CIRCULAR PLATE hIITH VARYING DIRECTION OF
I LLUM I NAT I ON
I
l*lI
DP
1
x
FIG. 3-I
a straight líne and usíng again the concept of the pin-hole camera,
H9"tanY=t=AA
s4.
(3.2)
(¡. ¡)
(3. s)
(3.6)
(3.7)
(3. s)
If L is the plate diameter then
L/2+VtanY^ =
-X
Ll2AA
where the asterísk denotes the maximum value of y and L.,, is the plate
diameter 1n the phoËograph. Using símpIe geometry ánd combining equations
(3.2) and (3.3),
r. = H.LIL (3.4)v
2-t0
0
= tan (Ll AA)
(tana - h/Dp)and I
Consider novr a change of observation direct,ion accomplíshed
by rotatíng Èhe plate abouÈ a vertícal axis through lts centre.
Reference to Fig. 3=2 shows
-1= tan
HtanY = -
and from the sÍne rule for triangles,
Ll2 AA AA
siny* sÍ:r-(r/2- [St+yo])
Hence the angle of rotation of the plate as determíned from the photo-
graphs is-lsD = .o"-'(2aasiny*/L) -.y* (:.s¡
cos (So + yo)
where if SD = 0 and L.,, ís the plate díameter in the photograph then ï
\
55.
Poi ntittu m ination
eP
AA
tl+,1 H k-
CIRCULAR PLATE l^lITH VARYING DIRECTION OF
OBSERVATION
x
FIG. 3-2
is determined from
and hence
L/2
AA
¿ -t
L/v 2
56.
(3.10)
(3.11)
(s. rz)
(3.13)
(3.14)
(3.1s)
(3.16)
Y = tan
The other constants are determined as fo11or¿s using the sine rule,
L AA
siny sín(tr12 - [SO+y])
I = AA siny /cos (So + y)
and L P
sín( [o - to] - er ) sin(r/2 + e )I
hence P sin(a- SO) - L
0 = tan-tI P cos (a - So)
and ty
or
o2 = so
where y ís determined from equation (3.7)
3.L.2 Clamped Plate Experiment and Results
A steel plate of thíckness 1.96rnm and radius 175.5mrn was
bolted to a steel structure using a circular rÍ-ng clamp. Since the
edges of the plate r¡rere sometimes obscured by thís ring,measurement
of the plate radius in the photograph,which is requíred ín the analysís,
was replaced by measurement of a 200rnm long Letraset arrow horizontally
oríentated and mounted on the plate whíIe another vertical arrohT marked
the centre of the plate. The plate was driven at íts lowest resonant
57.
(0'1) mode of frequency 150 Hz by an electrouagnet and oscillator (the
resonance unit had not been developed at this stage) and the vibratíon
monitored r¿ith an acceleromeËer and B & K spectrometer 2LI2.
Six time-averagecl holograms were taken r'¡ith different illumin-
atíon posítions (three positions on the righÈ of the holographic plate
and three on Èhe left) ¡,rríth the angle a varyíng from *50.5o to -52.8".
ReconsÈructions of holograms for these extreme cases are shown in
Fíg. 3-3. The results of the analysis ouÈlíned ín section 3.1.1 are
shown in Fig. 3-4 where the consístency of the experíment.al poinËs
clearly demonstraËes the valfdíty of the holographic theory.
Seven time-averaged holograms rnrere taken of the plate of
varying orientation but vibrating Ín the same mode. one hologram was
recorded with the angle sD = 0, three wíth so positive and maximum
angle near hf4 and three with SO negatíve and maxímum angle near
-T/4. Two reconstructions of the holograms are shown in Fíg. 3-5.
The results of the analysis outlined in sectíon 3.1.1 are shown in
Fíg. 3-6 where again there is favourable consisteney except for one
photograph for which it ís suspecÈed the oscillator drifÈed s1íghtly
1n frequency resulting in a drop in víbration anplítude.
Hence the theory of one-dimensional vibratíon, being a specíal
case of the general theory, is supported by analysís of a clamped
clrcular p1ate. Equation (S.f) ís demonstraËed to be completely suff-
icienÈ to describe this vibration.
3.2 I^IINE GLASS AND BEAKER
3.2.1 Holoeraohic and Geome tric Theory
Time-averaged holography was fírst applied to vibrating curved
surfaces by Liem et al [101] who analysed the vibrations of a circular
cylinder using the formula of equation (3.f) and reported a shift in
the aurplitude plots when the lllurnination and observaÈion vectors are
58.
FIG. 3-3
Reconstructions of holograms of a clamped círcular plate vlbrating in
the (0,1) mode and illumination angle a =* 50.5" (top) and q = - 52-8"
(bottom) .
oco
.vE
(¡,'rJJ
o-E
0
0.9
0. I
0
0.8
0
0 6
0.5
0 L
7
0.3
0.2
- 120 -80 0
Radius (mm)_ 1.0 /.0 80 120
FIG. 3-4 HOLOGRAPHI CALLY DETERI1I NED AI1PLITUDES OF CLAÍTPED CI RCULAR
PLATE VIBRATING IN THE (0,]) ITODE FOR VARIOUS I LLUÍVIINATION ANGLES,
(-¡l\o
^{oAA
xxo o
ôç¡{ro
A
xa
ox o.ro
^ÂI
Âxo
qX
¡ t
oqx ¡Alo
Aoï I ¡44þ
60.
FIG. 3-5
Reconstructions of holograms of a clamped circular plate vibrating ín
the (0,1) mode and different orientation angles.
t.2
30
oco
.9E
0,EJ
o-E
0.9
0.6
0- t20 -80 0
Radius ( mm)-40 lr0 80 t?0
FIG. 3-6 HOLOGRAPHICALLY DTTERIVIINED A|\IPLITUDES OF CLAlvlPED CIRCULAR PLATE
VIBRATING IN THE (0,1) IVIODE FOR VARIOUS ORIENTATION ANGLES.o\ts
aI
xÂ
v Ârao xO¡
I6X + Â
oo 9 ð+9
x Âf A
++ oL
I9
xA
ôt eo ìl¡ ++ I
Ac AX
0ol
t 49.oð + ôi a(
62.
varied and attempted to explain this in terms of localizatíon of
fringes and parallax. IË r"¡as subsequentJ-y shown by Tonín and Bies
[102] Èhat the shift was in fact due to a misinterpretation of the
rnotion of Èhe cylinder which is three-dimensional rather than one-
dimensional .
The surface contour of a vertícally mounted cylinder in the
optical plane ís a circle and the co-ordinate f ís equivalent to the
angular co-ordinate 0. Considering only this surface contour in the
analysis Ëhen the geometrícal factors are given by equatíons (2.25).
Addítionally if the radíal and tangential components of vibration are
respecÈively of Ëhe form
c(Ert) = C cosn(0 + e)cosrrlt
(3.17)
b(E,t) = Bsinn(0+ e)cost¡t
Èhen Ëhe characteristic equaríon Iequation (2.29)l ís
f = "(0) (cos0, + cos0r) + b(O) (sínO, - sínOr) (3.18)
where c(0) = Ccosn(e+e)(3. le)
and b(O¡ = B sínn(e+e)
The experimenÈs of Liem et a1 were repeated for the case of a wine glass
and staj-nless-steel beaker, which are manufactured easÍly with high
accuracy.
rn the case of the wi-ne glass, freedom from distortíon ísreadily demonstrated by rubbing a moj-st finger around the líp which
excites the glass ín its lowest energy n= 2 flexural mode. Additionally
63.
the mode rotates around the círcumference if the fÍ-nger is removed
which clearly shov¡s the degeneracy of t.he mode. A scheme was devised
whereby the two holographic views could be recorded on the same hologram
using a plane mírror which was positioned such that the image of the
object was fully visible buÈ the Ílluminating beam díd not reflect onto
the object from the mirror.
Referríng Èo Fíg.3-7, whích shows the geometry in Èhe optical
plane, the centre of the imaged cylínder in the photograph (or on the
ímaginary screen defined ín Chapter 2, see f ígure) is at an angle ri.r
with respecË to the centre of the cylinder ín the direct view. The
corresponding poinÈ T, on the mirror is ín practice marked with a small
piece of masking tape such that ít coincides with the centre of the
ímaged cylinder when viewed Ëhrough the camera eyepíece. Also shor,¡n
fn the figure is the relative illumination direction K, and the relative
centre line È for Ëhe image and the orígins OO and O, which are defíned
for the dírect and inaged views respectíve1y. Fíg. 3-8 shows the geo-
metry for Èhe imaged view drarnrn inverted for comparison with Fig. z-LO.
In the case of the dírect view the geometrical analysis is Èhat
given in section 2.2.3, noËably equations (2.59), (2.6L), (2.63) and
(2.64). For the lmaged view it is clear that the photograph ís dístorted
due to perspective. From the sine rule for triangles (see Fj_gs. 3-7
and 3-8),
H X/cosV
- =
-
(3.20)sr-nY sinfn /2 + (v - v) ]
Solving for y,
-t H cos2VY=tan (3. zr¡
X - H cosVsinV
where V is compuÈed
mined. Denote "R ""v
from the lengths M,BB and CC and X is to be deter-
the radÍus from the image centre to the ríght edge
64.
I maged v tew
Direct view
AA
+Directcentre
SCHEI4ATIC OF
PLANE
\ poi ntitt um inat ion
0e
0
m
M i rror
ûI
irBg
\\['
\
X
\
IImagecentre
Imaginary screen(def ined in ch.2 )
Tl,^lO-VIET^l EXPERII'{ENT USING A
I\1I RROR
FI G. 3.7
65.
-0¡
BB+Qç L
1t"93\
centre
+
FIG. 3-8 GEOIIETRY FOR THE IIÏAGED VIEW
66.
of the cylinder in the photograph, r" "" the radius to the left edge
and d- as the diameEer of the cylinder in the photograph. Hence,v
L (3.22¡v
J
x srnY
cosV cos(v - y* )(3.23)
L¿
x s]-nYand (3.24)
cosV cos(V+ y*)
where * I/AA) (3.2s)Y = tan (a
2
and the asÈerisk denotes the maximum value of y. Combining equatíons
(3.22) ro (3.24) gives
a+advR
v
and from equation (3.20),
Rav
av
-1
X=d cosVv_
&sr-nY
I 1+
cos(v - y*) cos(v+ vo)(3.26)
The angle 0, ís determíned ín a siurilar way to equation (2.61) wíth
the resulÈ that
or = .o"-t ( tut + ccl síny /a") + y (3.2t¡
where, with respect to OO the origin of the dÍrect view,
0=0 -rþ (3.28)I
Hence using equatíons (3.21) and (3.25) to (3.28) the angle 0 is
computed for frínges in the ímaged view with respect to the origín of
the dlrect view. The observation angle 0, ís given by equation (2.64)
and the íllumination angle 0, by equatíon (2.63) bur with rhe angle o,
67.
replaced by (a - ú) thus
0r = Ëan-t{P.o" (a -ü + O)/[^z-psín(a- ú + O)]] (3.2e)
For n = 2 the radíal component of vibratíon is determined from
equation (3.18) as
oR-1"(e) = * [ cosor* coso, -f.."[n(e+ e)](sÍno, - sinor)J (3.30)
for the direct víew; for the ímaged view (e - rl) is substiÈuted for e.
The coefficients B/C and e, the latter whích determínes the orientation
of the mode around the circumference, must be estimated in order to use
equatÍ-on (3.30). This technique is obvíously not practical in general
and hence a refinement is added in the following analysis of a beaker.
For the case
or 0 i r/2 (3. 31)2
equation (3.30) reduces to
c(O¡ = ç¿/ (zkcos 0r) (3.22¡
Thus, íf the beaker is set on a turntable with its central axj-s corres-
ponding to the turntable axis which is also normal to the opÈical plane
then a number of poínts on the cylinder may be analysed for whÍch
equation (3.31) ís true and the radial component determined separately
using equation (3.30).
Reference to Fig. 2-L0 and application of the sine rule to the
two trÍangles wíth sídes AA and P gives
AA a2
sin(n - 0r) sin[0 r- (r/Z - e)](3.33)
68.
P^2and (3. 34)
sin(n - 0r) sín[0t - (o - lr12 - 0]) l
whereupon elirnínating 0 by combining equations (3.33) and (3.34) rhe
result is1 I
-f"ir,(o+2e)=0 (3.35)¡ cos0 +fr cos(a+ 0) a2
which ís easily solved for 0 by Newtonrs method [158, page 26]. The
corresponding point on the photograph is, from equation (2.59) and
(2.60),
AA.a .cosOv (3.:o¡(AA - a'sín0)
and the íllumínation angle
-1AA cos 0
0 = tan (3.37)I A.A,sin0- a2
Hence a number of experimenËal points are obtaíned for the radial
componenÈ through whích a sinusoidal curve of best fít ís drar,¡n and
hence the coefficients C and e determined. The tangential points are
thus calculated from
b(e) = {n/t - c cos n(e + e) [coso * cosO I ]/ (síno - sin02
(3.38)2
ll=
I )
whích follows from equatíons (3.fa¡ and (3.19).
3 .2.2 l^líne Glass and Beaker Experiments and Results
A wíne glass of outer radíus 27.5nm at the rip sprayed matt
white, $ras fixed to a steel base and positíoned wíth the rím in the
optical plane. A cluster of four bar electromagnets arranged in quad-
rature I¡las suspended from above and lowered a short r4ray into the glass
69.
\ùithout touching the sides. The electrornagnets r{ere connected such
that opposÍtes \4rere in phase but adjacenEs in antiphase which corresponds
to the circumferential mode n= 2. Four tiny píeces of iron vrere then
placed around the circumference of the glass and held in place by the
attraction of Èhe electronragnets. In Èhis way the electromagnets and
pieces of iron (v¡hich we r¿i11 call the drivers) ".. Oo"ìaioned for
opt.irnum coupling with the mode being driven.
Fig. 3-9 is a picture of the wine glass and electromagnets. The
mode was excited at a frequency of 1.34kli.z using an audio' oscíllator
and 30!I amplifier. The level of víbration was monitored wíth a Bruel
and Kjaer (¡ t f) spectrometer type 2LL2 and a B & K half inch micro-
phone type 4133 placed near the glass at a radial antínode. A photo-
graph of a time-averaged hologram is shor,rn in Fig. 3-10.
For the purpose of the analysís a plane back-silvered mirror
was placed in the fíeld of view of the camera ü/íth a fair degree of
overlap of points on the circumference of the glass in the direct and
imaged views. A typical photograph of a hologram reconstruction is
shor"m ín Fíg. 3-11. Wíth AA=502.5nrn, P=432.5mrn, q=0.474rad,
rl = 0.876xad, V = 0.224rad and BB+ CC = 558.5uun the radial component
is fírstly determined from equation (3.1) and is shor¿n in Fig. 3-I2
(a) as circles for the direct víew and triangles for the imaged vÍew
where Ëhe amplitude shift first reported by Liem et al [101] Í-s quite
apparent. Next the analysis of orthogonal vibrations is used to cal-
culate the radial component using equatíon (3.30). The ratio B/C is
estirnated as \/n or 0.5 which is borrowed from Èhe theory for cylínders
wíth shear-diaphragm ends ÍL25, page 311 for which ít is a good approx-
i.mation and the angle determining the orientatíon of the mode around
the circumference, e, is estimated as 0.36rad. The result ís shown in
Fig. 3-I2 (b) which, a""iit. the crude approximations, ís extremely
70.
FIG. 3-9
l,Iine glass and modal driving system.
7r.
FIG. 3-TO
Time-averaged hologram reconstruction of the n = 2
mode for the wine glass.
FIG;3-II
Time-averaged hologran-of wineglass
vibrating in the n=2 mode and the
mirror image.
!N)
(a)
o
o
A
o
A
o
0.6
r/tco¡-
':E
0.2A)!)=o_
E
-0.2 -0.¿ 0 0.t 0.8 1.2 -0.1 0 0-L 0.8 1.2
Theta ( rad ians )
FIG, 3-I2 RADIAL CO|IPONENTS 0F A VIBRATING WINE GLASS, (a) calculareà ,'o* rheory
used by Liem eÈ al; (b) Theory extended to include orthogonal componenÈs of vibraËion; o, experimental
points, dírect view of Èhe cylínder; ^,
experimenÈal points, imaged vÍew of cylinder; -¡
sinusoidal
curve fitted Èo data poinÈs in (b).
0.1
0
!(,
(b)
A
A
74. .
favourable.
Next a staínless-steel beaker of radíus ar.= 44.2mm, wall thick-
ness h=l.23mm and length L=101.3mm \¡/as sprayed matt whi-te and fastene<l
Èo a base plate by means of a bolt through its bottom. The modal driving
system was similar Èo that used in the wine glass experiment except that
three electromagnets were used arranged symmetrícally and wíred ín-
phase and the drivers were slightly heavier tiny steel bolts. The
driving frequency üras measured as 1011H2 for the n = 3 mode.
The beaker and driving mechanism were placed on a. turntable
with the beaker centreline coíncident \,üith the Ëurntable axís. In all,
seven holograms were taken of the vibrating beaker at 15o rotation
intervals Èwo views of which are shovm in Fig. 3-13. I^Iith p =486.2mm,
AA= 749.2rnm, ar= 44.2rnm and o, = 0.5492rad the solution of equation (3.35)
ís 0 = L.29L5'rad at which poínt the sensitivity vector is normal to
the surface of the beaker. However, if the beaker ís sprayed only
líghtly with white paint, specular reflection of the i1lumínating
beam occurs at thís value for 0 and this shows on the photographs of
Fíg. 3-13 as a thin white line along the length of the beaker.
Solving for the radÍal component using equation (3.32) requires
that a dark fringe coíncide wíth the poínt where the sensitivíty vector
ís normal to the surface. Thís is not usually the case, however, and
raÈher than interpolate a value for O, a method employed by Tushak and
Allaire t1O4], the two adjacent fringes are analysed. This procedure
1s justified in view of Ëhe excellent agreement of the data points so
obtained onto a sine curve of best fit as shown in Fig. 3-L4. From the
figure the radial vibration component is
c(0) .74 eos[3(0 - 1.s3)] (3.3e)
From equations (3.38) and (3.39) the tangential component of vibration
75.
FI G . 3-T3
Hologram reconstructíons of two views of a beaker vibrating in a
love mode of order n= 3.
I6
L
0.2
0
20
0
0
0
-0. t,
otro(J
E
oEa
=o_
E
-0.6
- 0.8
0 0.1 0.8 1-2 1.6 2.0 2.1 2.8 3.2Theta ( radians )
FIG. 3-14 THEORETICAL AND EXPERIIVIENTAL DETERI1INATION OF RADIAL AND TANGENTIAL VIBRATION
COIV|PONENTS OF STAINLESS STEEL BEAKER
- Theoretical curves; Cl , expe-rimentally determined radíal components; *rxrorArorVrA, experlmentally
determined tangential components from different views of beaker.
!o\
ôA
t
¡
ItA
/
o
tr
A
+
x
/
+
oô
XA
x
Ib*Xx
+
l
o
tr¡
vO
77.
b(0) ' is calculated for all dark fringes and the results are shor¡n in
!'ig. 3-14 as well. The sign of the fringe order, o, ís chosen as is
usually the case in time-averaged holography such Èhat adjacent groups
have alternatíng signs and correspond to the sign of c(0) wtrich is the
larger component in terms of absoluÈe magnitude.
The theoretical curve for the tangential motion is determined
on the assumption that the cylinder vibrates in a Love mode (see
reference [125], page 125) whích arises from ínextensional theory. Thís
ís confirmed by the equal spacíng between fringes (see Fig. 3-13) along
the length of the reconstructed beaker indicating that the sides remain
straight. The bottom of the beaker acÈs effectivery as a shear dia-
phragm satisfyíng the boundary conditions at the consÈrained end of the
beaker. The frequency of vÍbration (Í1251, page 125) of rhe mode
considered is
¡2$2
E
p(3.40)
(3.41)
L2(L- r,)"å
In this case, assuming E/p = 2.445 x 107 ^2"-2, Poissonts ratio v=0.3
and n = 3 the resulÈ is f = LL83Hz which is to be compared wíth the
experimental value of l2L6Hz.
In Particular, the radial, tangential and longitudinal vibraÈíon
components are respectively of the forms
c(0rXrt)
b (e,x, t)
a(0 rX, t)
= nXC cos n0 cos ¿ot
= XC sín n0 cos r¡t
= (a/n)C cos n0 cos ot
where X ís the lengËh co-ordinate from the shear diaphragm, 0 is Èhe
78.
angular co-ordinate and C is an amplit.ude constant. Note that the
radial and tangential components vary linearly wiÈh X. Comparing
equatíons (3.17) and (3.41) gives
B/c=L/n=l/J (3.42)
The theoreti-cal curve for b(0) 1s thus determined and shornm in Fig.3-L4.
The scatter of points requires coûunent. The denominator in
equation (3.38) becomes very smal1 when the sensitivíty vector approaches
Ëhe surface normal. Hence slight errors ín the numerator are magnífied
enormously. Thus extraordinary deviations from the theoretical curve
occur at points where Èhe sensitivity vector is near normal to the surface
(and hence normal to the tangenÈial component of vibration). Reference
to Fí9. 3-14 shows that for the points rnarked o, for example, agreemenÈ
is poor at about 0=1.8 radian. In the lower photograph of Fíg.3-13
this corresponds to the region jusE right of the centre of the photo-
graph and this ís where the sensítivity vector is normal to the surface.
Hence a conclusion ís that íf an accurate determination of the
tangential component is required, the range of 0 and the positioning
of the illumination and observation vectors should be such that the
sensitivíty vecÈor ís never normal to the surface. I,Ihen it is near
normal the curve of best fit would probably be the best one coul-d do.
3.3 CYLINDERS
3.3.1 Holographic and Least Squares Theory
IE would be useful to apply the nethod of the last section to
calculate the vibration componerits of the superposerl modes of a distorted
surface. Two examples would be the vibration of a cylinder rvíth a seam
and a cylinder with an attached lump nass for which typical hologram
reconstructions are shov¿n in Fig. 3--15. In the figure the cylínder in
(a) is rolled from a thin sheet of brass, soldered at Ehe seam ancl
79
FIG. 3-I5
(a) Cylinder with a seam
(b) Cylinder with a lumped mass
80.
supported in a steel structure which j-s visible in the fígure and (b)
ís a steel cylinder with a brass mass g1ued on to the surface.
The characterísÈic equation is equation (3.18) with the radía1
and tangential vibratj-on components given by equations (2.10) and (2.1j.).
using these equations the following system of equations for .r points
results
"tK + blstI I
a
n
/t<
/t<"2K + b2s2 2
(3.43)
.tK + brs =Çl/kT T T
where the geometrical coefficients KO and Sn are defíned by equatíon (2.30)
for a surface contour in Èhe optical plane. It must be emphasised that
the bounds of the sumration in equations (2.L0) and (2.11) are quite
arbitrary. In fact these are generalized by replacing the bounds O and
N by Nl and N2 which are ïespectively Ëhe least and greatest orders
considered. For example if the fourth order of vibratíon is being invest-
ígated but because of as)rmmet.ry in the forcing field the lower second.
and third orders r¡rere present then Nl would be set aE 2 anð, N2 woutd be
set at 4. On the other hand if only one term is necessary for an adequate
description, NI and N2 would be identical. The seríes would. conÈain
only the single Èerms as for the analysis in the last section.
Hence, substituËing equations (2.r0) and (2.11) into equatíon
(3.43) gives for each point q corresponding to a dark fringe
2
B1
N2f x sÍn(n 6
,,!N I tN2
*r,l*r. Yn co"(t Eq) KqK)q q
N2*r,lr'tt"tn(n E
N2+ I z cos(nEI ,rl¡1 n
) )sqqS alkq
(3.44)
(3. 4s)
(3.46)
q
where the E , K ., S_ and fl_ are deÈermined from the tirne-averaged holo-'q,' q' q q
grams. These form a set of t linear equations in 4(N2-NI+1) parameters
or unknovms X', Ír,, r' and zn which are determined by the normal method
of least squares as follows. Assume that the left hand side of equat,ion
(3.44) is exacrly equal to CIå/k but slíghtly dífferenr ro Aq/k, Èhe
value determined by Ëhe data. The square of the residual ís
N2 N2 N2
lol"r*r"tn(n En) *o *rrl*, Yrrcos(nEo)ro ."1",
N2+T
n=Nrf tCIå/k - aq/kl2
r.¡n sin(n EO) tn
z cos(nã )S - An -q-q q /t<
t...ry r'Inl r...r\^7 ,Z ,...r2 )N2 Nr N2 Nr N2
DenoÈíng
f(x ,...rxNI N2
tYN I
T
--lq=1
(o;/k - ç¿q/k)2
then by the usual method ft = 0 ís required where the prime indícates
dífferentíation with respect to each variable in turn.
The NORMAL EQUATIONS are no\¡r derived using the matrj.x method
of Buckingham [f59]. Firstly set up the matríx A of order T by
4(¡z - Ul + t) (see Appendix II for partial expansion)
A(q,4 (n - nr + 1) - 3) = K'sÍtt(r tn)
A(q,4(n - ul + 1) - 2) = Kncos(n tn)
A(q,4(n - Nr + 1) - 1) = Snsin(n EO)
A(q,4 (n - NI + 1) ) = Sncos (o EO)
forq=ltoT andn=NltoN2
and let R and fl be the colu¡nn vectors
82
(3.47)
xN I
v.Nr
!TN
zNl
xN2
I
o t
fl,
R o
aT
vN̂¿
s7
N2
z^Nz
Then equation (3.44) is simply expressed as
(3.48)
83.
The NORì4AL EQUATIONS are 11591
(3 .4e)
where AT A í" a 4(N2 -tll+t) square matríx. The solution of R in
equation (3.49) is found by the Gauss el-imínation method [158, page 398].
As is usually the case in time-averaged holography the sígns of the QO
arguments are arbítrarily chosen for each group of fringes and for the
modes of vibration considered here, adjacent groups take alternatíng
sígns. In this way, the magnitudes and slgns of the parameters x, y, \^7,
z are automatically computed to gíve the best fit.
The sÈandard deviaÈion of each parameter of vector R is
){n'/k-n/k)z,2
aTan=Jatnt<-
defíne ¿,Tn(f) = 1 and eTn(Z) E ...
for parameter x , ,t ;sual way.¡1
P!r - 4(¡z-Nl+1)l
= Arn[4(N2 - Nl+ 1) ] 0 and solve
s.dT
where p, í" Èhe weíght of the coeffícient and is calculated in the
following way by the method of Bartlett [160]. In the normal equations
gíving the weight po,
The weight of this coefficient is
The
p-- = l/x .. Simílarly for the weíght of x , define ATQ(Z) = 0 and'w Nl " Nl+l¿rn{r) = dn(r) = ¿T&t4(¡z-ul+t) - 0 and solve tor **r*r_
= L/x , and so on for the other parameters.Nl+1
Èotal standard deviation s.d." for the vibration component c(f) is then
N2
ls.d.c(E) l' = T.(sin(nE)s.d. )2 + (cos(nE)s.d.xn Y¡
)' (3. s0)
n=N1
where s.d. and s.d. are the standard deviations of parameters x- andxnYnnv with símilar equations for the other components.,n
84.
3.3.2 Theorv of Thin Círcular Cvlíndrical Shells
Consider a thin circular cylindrical shell of length L, mean
radius a^ and wall thickness h (see Fig. 3-16).o
The cyllnder is supported by thin círcular end caps or shear-
diaphragrns at whích the boundary condítions are
b=c=M__=N__=O at X=O,L (3.51)xx
r¿here M-- ís the bending moment and N the longítudinal membrane forceX-X
ín the she11. If d = (arbrc) is the displacement vector for the víbration
with longitudínal, t,angentíal and radial components respectively of the
forma(X,0) = A cos(Às) cos(n0) cos(urt)
b(X,e) = Bsin(Às) sin(n0) cos(urt) (3. s2)
c (x, O) = C sin ( trs) cos (n0) cos (or)
mfiawhere ¡ = --o and s = x then the equaÈÍ.ons of motion may be r^rrítten
L
in matrix form as
-$vd=o (3. s3)
where $" is the matrix dífferentíal operaÈor deríved from the Donnell-
Mushtarí Èheory whích takes the form 1L25, Chapt. 2l
ao
85.
'¿2 'ò2, (1-v)-l.
2
1* v ð2 d
òs2 à02 2 âsã0uâ"
(L - .,2\ a2 'ò2o
-p E at2
(rt v) à22 ðsà0
aua"
â
ð0
fiu - (3. s4)
E
â
a0
àt2
1r r.v4 + p (r - v2) "3 ,'E atz
In the equation the non-dímensional thickness parameter K =h2/L2a2
andvq - ( 4 * ð'= )'
âsz àg¿
The theoríes of Love - Tj-nioshenko, Goldenveízer - Novozhilov
(also Arnold - Warburton) , Houghton - Johns, Flügge - Byrne - Lur I ye
(also Bíezeno - Grammel), Reissner -Naghdi - Berry, Sander:s, Vlasov,
Epstein - Kennard and a sirnplífied theory due to Kennard are all
modelled on the Donnel - Mushtari theory and in fact may be represented
by the addítion of a modifyíng matríx åUOO "" follows
KIMoD (3. ss)
all of r'¡hich are l-isted by Leíssa 1L25, page 33]. llence the eighth
order system of equations represented by equatíon (3.53) becornes
l=lor*
Td=0 (3. s6)
86
CO-ORDINATE SYSTEM FOR A THIN CYLINDRICAL
CLOSED CIRCULAR SHELL
L
z
h
FIG. 3-16
v-l\0
87.
which has a non-Ërívial soluÈion if and only if
detT:0 (3.s7)
This results in a third order polynomial of the form
¡6 (r<r+r Arr) nq + (KI+rcArr) À2 - (*o*rcArco ) ='o (3.s8)
where the rco, rc, and Kz are Donnell Mushtarí constarits and Ârco, A"l
and Ar, are the modifying constants for the other theories and are
tabulated by Leíssa 1J'25, page 45). The roots of equation (3.58) are
the eigen-values Â2 where
h2 = p(l- v2) ^2
,2/E (3. se)
v¡hich determine the resonant frequency o.
Equatl"on (3.56) cannoË be solved for the component amplítudes
A, B and C by Crarnerrs method [158, page 382] due to the constraÍnt
imposed by equatÍon (3.57). However, inspecÈion of maÈrix T reveals
it has rank ordex 2 ar.d hence one ror^r is superfluous [158, 5 7.6].
Equation (3.56) becomes
T11
T t3=Q (3. 60)
T2I 22T
23
orr díviding by C,
T T^/c
-T t311(3.61)
B/c
1r,
T
A
B
C
t 2
T T2I 22 23-l
whích ís solved for component ratios A/C and B/C by Cramerrs method.
For example, the form of equaEíon (3.61) for the Donnell-Mushtari
theory ís
_x2^ (1- v)2
n?- + L2
).n
(1+v)2 -vÀ
BB.
(3 .62)
Àn A/c
¡2-n2¡¡2 Blc(1- v
2n
For each mode there corresponds t.hree eígen-values Â2 which
are solutions of equation (3.58) of which Ehe two largest eigen-values
result in ratios A/C or B/C greater than unity and frequencies ín the
untrasonic range for ordínary cylinders. In Èhe case of the smallest
eígen-values the ratios A/c and B/c are less than unity (the radial
comPonenÈ is greatest in magnitude hence the modes are termed flexural)
and frequencies fall in the audio range for at least the lower order
modes which are of prímary interest.
Differences between theoríes amount to less than 2"Á ín resonant
frequencies and ratios B/C ar.d A,/C, determined for a number of cylinders
of typícal dímensíons used in the experiments to be described. As Èhe
accuracy of the experiments is at best of this order, it ís not possíble
to distinguish between Èheories aË present. However, the ratíos B/C
and A/C of flexural modes of a vÍbraËing cylínder are experimenÈally
neasured for the first tlme and reported by Tonin and Bies [105].
3.3,3 Cylinder Experiments and Results
The least squares analysis described in section 3.3.1 ís applied
to the data of the staínless-steel beaker of sectíon 3.2.2. Fig. 3-I7
(a) shows the radial and tangential components o-l vÍbratíon using a
síng1e mode in the expansíon i.e. Nl = N2 = 3. The solid línes are ì
the radial and tangential curves of best fl-t. For each point corres-
pondl-ng to a dark fringe the círcles o are values of b computed from
0.8
0.6oco
.:E
0.t
0.2
c,!J.o-E
-0.2
0
FIG,3-I7 (")
0
0
-0. I
l.
6
0 0.4 0.8 1.2 1.6 2.0Theta ( rad ians)
2.1 2.8 ' 3.2
LEAST SAUARES PP.OCTDURE APPLIED TO DATA OF STAINLESS STEEL BEAKER,
Nl=N2=3co\o
- Least squares curves; - - - standard deviation; À experimentally determined radial component.;
o experirnentally determlned tangenÈial eomponent.
oo%
o
ooo
oO6o
ooo
90
equation (3.38) using the least squares fit curve for c ancl the
experimental values of 0, 0, and 0r. Similarly the triangles A
are values of c computed from Èhe least squares fit curve for b using
equatíon (3.38). These component points are termed "experÍ.mentally
determined" since they cannot be measured singularly wiÈhout further
experimental effort and are in fact a best estimate of what one would
measure by ensuring Èhe sensitivíty vector to be in turn raclial and
tangential aË Èhose points.
Comparíng Fig. 3-L7 (a) to Fig. 3-14 shows Èhe irnprovement
of the least squares procedure over the previous method. The improved
consistency of the data points is due to a more accurate determination
of the radíal component. In Fig. 3-I7 (b) Ít ís assumed Ëhat a small
contríbution of the n = 2 mode is present and the consístency of the
experímenÈally determíned points is further ímproved. However, the
sÈandard devíation for both components, shown by the broken curve, ís
greater since the number of unknovrn parameters has doubled. Fig. 3-18
shows the results of the least squares proeedure applied to the brass
cylínder wíth a seam shown in Fíg. 3-15 (a) for r.rhich the analysis
íncludes four modes (Nt = 2, N2 = 5).
HavJ-ng thus proven the superíority of the least squares procedure,
it is applied to four cylinders of various diameters, wa11 thicknesses
and materi-als. The ends of each cylinder were machined flat rnríth a
líp on the inside edge to hold an end piece of thickness similar to
Ëhat of the cylinder. The end pieces \^rere cut so that only one edge
touched the cylinder lip around Ëhe circumference. An electromagnet
was used to excite the cylínder and both electromagnet and cylinder
wet:e supportecl by a steel structure whích held the latter by two probes
each neatly fitting into a small hole at the centre of each end plate
as shown in Fig. 3-19
o
o
oo
ocot-.:E
o!J:o-E
0.8
0.6
0.1
0.2
-0. I
0
- 0.2
-o.L
60
0 0.1 0.8 1.2 1.6 2.0T h eta ( rad ians )
2.8 3-22./.
FIG, 3-I7 (u) LEAST SOUARES PROCEDURE APPLIID TO DATA OF STAINLESS STEEL BEAKER, Nl = 2, N2 =3
- Least squares curves; --- standard deviation; A experimentally determined radial component;
o experimentally determined tangential component.
\oH
oo
I^
R ad ial
/
t
o
o
o
oo
ïangential
Sea m
oo
o
0.8
0.6
0.1oco
.:E
(.,E==o_
E
0.2
0
0
0
0
2
L
6
-0.8
2.t 2.9 3.2
FIG. 3-I8 LEAST SOUARES PROCEDURE APPLIED TO DATA OF BRASS CYLINDER I^IITH A SEAIÏ, Nl=2 ,I]2=5
- LeasÈ squares curves; A experimenÈally determined radial component; o experimentally deËerrnined
tangential component.
0 0.t 0.9 1.2
Theta1.6 2.0( radians )
\oN)
93.
End cap P rong
Cyl i n derwall
St eelst ructure
FIG. 3.I9 DETAILS OF END tvlOUNT AND SUPPORT
94.
Table 3-1 lísts the physical properlies of the cylínders and
Tables 3-2 (a) to 3-2 (d) list the theoretical resonant fr:equencies
and the ratios B/C ancl A/C basecl on these propertíes and calculated
by the method described in sectíon 3.3.2 using the Reissner --Naghdi -
Berry theory. The experimental equi-pment and experimental method ís
descríbed in sectíon 2.2 along with Ëhe method of analysis.
Fígs. 3-2O (a), (b) and (c) show the tangentÍal and radial
componenËs of three mocles in graphical form for the case where the
cylinder ís verËícal. Fígure 3-21 shows two views of time-averaged
hologram reconstructions of a sma11 sÈeel cylinder vibrating in the
n=4, n=3 mode. Figs.3-22 (a), (b) and (c) show the longitudinal
and radial components of three modes in graphícal form for the case
r¿here the cylinder is horízontal. Figure 3-23 shows trnro viervs of
time-averaged hologran reconstructions of a sma1l steel cylinder
vibratíng ín the n=2, n= 2 mode. In Fig. 3-22 note that the abcissa
labelled "Normalized LengÈh" refers to the distance from the centre
of the turntable to any point along the centre axis of Èhe cylínder
normalized with respect to the length of the cylinder. There is no
need to align the midpoint of the cyli.nder with the axís of the turn-
Èable. The least squares procedure auÈomatícally fits the data no
matter where the mídpoint of the cylinder ís.
The solution vector, R, is calculated for a number of modes
of each pipe wíth Nl = N2 (í.e. only a single order leasÈ squares
approximation) and hence the ratíos B/C and A./C are determíned. Results
are shown ín parenthesis in Tables 3-2 (a) to 3-2 (d) which are Ëo
be compared wíth the theoretical predíctions. The more important
source-s of error are probably anísoEropies in the cylinder, variations
in thickness ín the cylinder walls, variations in Youngrs modulus and
inaccuracíes in the end Or"a." which all conËribute to dístort the
oooo
o
o
oo
oo
Â
Ao
oo
A
oo
0
-0.¿
r/)co
.9E
q,!J-o_
E
0.8
0.6
a.L
0.2
-0.2
-0.8
60
-0.L 0 0.t 0.9 1.2 1.6 2.0 Z-t ?.9 3.2 3.6Theta (radians)
FIG, 3-2C (') CIRCUI1FERENTIAL 0RDER ru = 2 (SIVIALL STEEL CYLINDER),
Least Squares Procedure Determination of Radial and Tangential Components of a Circular Cylinder.
^ Experimentally Determined Radial Component. o ExperimenÊally Determíned Tangential
Component Least Squares Curves.
t\o!n
ooo
oo
o
ooo
o
o
ooo
o
o
oo
oþo
0.6
0.8
0.t
0.2
0
0.2
0.4
0.6
ulCouE
oE==o_
E
- 0.8
-0.t 0 0.t 0.9 1.2 1.6 2.0 2.t, 2.9 . 3.2 3.6T heta ( rad ians)
FIG, 3-20 (¡) CIRCUIVIFERENTIAL 0RDER N = 3 (SI{ALL STEEL CYLINDER) ,
Least Squares Procedure Determination of Radial and Tangential Components of a Circular Cylinder.
A Experímentally Determined Radial Component. o Experimentally DetermÍned Tangential
Component Least Squares Curves.
\oo\
0.8
0.6
0.L
0.2
0
- 0.2
mco
.:E
oE)=o_
E0.1
0.6
-0.8
-0.1 0 0.1 0.8 1.2 1.6 ?.0 2-L 2.8 3.2 3.6Theta ( radians)
FIG, 3-20 (") CIRCUIiIFERENTIAL 0RDER N = 4 (SIVIALL STEEL CYLINDER) ,
Least Squares Procedure Determínation of Radial and Tangential Components of a Circular Cylinder.
À Experímentally Determined Radial Component. o Experimentally Deterrnined Tangential Component.
LeasÈ Squares Curves
\o!
oo
oo
o
o
o
o
oo
oo
o
o
o
oo
A
o
o
@o
o
98.
FIG, 3-2ITime-averaged hologram reconstructions of a small steel cylinder vibrating
in the m= 4, n = 3 mode.
The cylinder was rotated 15" between holograms. The white line shows the
surface contour along which data points are taken (line of varying E).
Â
Vtu\
(r!coL'
tr
(¡,
!J
=o-E
0
-0.2
-0.¿
0.8
0.6
0.t
0.2
0.6
-0.8
-0.5 -0.1 -0.3 -0.? -0.1 0 0.1 0.2 0.3 0.¿Normalized length
FIG, 3-22 (') LONGITUDINAL 0RDER M = 1 (ALUITINIUIvI CYLINDER),
Least Squares Procedure Determination of Radial and LongiÈudinal Components of Circular Cylinder.
A Experimentally Determined Radial Component.
o Experímenta1ly Determined Longitudinal Component.
Least Squares Curves.
0.5
\o\o
o8oo
o
A
o
ooooo
o
ooo
o
o
Â
oo oo oOO
0.8
0.6
0
0
0
-0
0
ocolÉuE
oEf,
=o_
E
l.
2
0
2
L
6
I-0
-0.5 -0.¿ -0.3 -0.2 0.2 0.3 0 -L 0. s
FIG. 3-22 (¡) LONGITUDINAL ORDER 1,1 _ 2 GLUI'IINIUI{ CYLINDER),
Least Squares Procedure Determination of Radíal and Longitudinal ComponenÈs of a Circular Cylinder.
 Experimentally Determined Radial Component.
o Experinentally Determined LongiÈudinal Component.
- Least Squares Curves.
-0.1 0
Normal ized0.1
length
ts.Oo
ø 0.Lcoü o.zEv0c,
i -o'to -0.¿E
-0.6
0.8
0.6
-0.8
-0.5 -0.¿ -0.3 -0.2 -0.1 0 0.1
Normalized length
FIG, 3-22 (") LONGITUDINAL ORDER M = 3 (C0PPER CYLINDER),
Least Squares Procedure Determination of Radial and LongÍÈudínal Components of a Circular Cylinder.
A Experimentally Determined Radial Component.
e Experímentally DeÈermined Longitudinal Component.
- Least Squares Curves.
0.2 0.3 0.t 0.5
HO!-
oOoo
Â
o
A
ooo
o
o
o
o
o
o
o
o
Ao
Á
o
oo
A
A
oooo
o
102.
FIG, 3-23
Time-averaged hologram reconstructions of a small steel cylinder
vibrating in the fr= 2, n = 2 mode.
The cylinder was rotated 30o between holograms. The whíte line shows
the surface contour along which data points are taken (line of varying 6) '
10 3.
mode shape. Other factors include the point. source excitation which
generates coupled spatial modes and spurious effects from the cylinder
support. Al1 these factors cont::ibute to modal coupling aÈ a single
resonant frequency. Reference to Fig. 3-24 shows how these are
accounted for by expanding the least squares procedure into ot.her orders.
Fíg. 3-24 (a) shows Èhe initial analysis with Nl = N2 = 4 and Fig. 3-24
(b) shows the improved result with Nl = 2 and N2 = 4 Í.e. orders 2, 3
and 4 are present in the second analysis.
AnoÈher source of error could be due to internal damping whích
it is thought causes elliptic moËion and needs to be analysed by the
more complex procedure described in sectíon 2.L.
Lastly some errors are aÈÈributed t,o the fact that not all
surface contours are on the optical plane, especially for the cases
where the cylinders are upríght. For longitudínal mode numbers m=2 and
4, for example, the surface contours are analysed slíghtly above or
belor¿ the optical plane t.o coincíde wíth the radial anÈínodes. Hence
Ëhe geometrical facËor Q of equation (2.30) is non-zero.
Reference to Tables 3-2 (a) to 3-2 (d) show that frequencies
have been predicted wíth an accuracy better than 10%. Values of
component ratios are not as precise, however, probably due to the
reasons ouÈlined above. Nevertheless, the least squares procedure is
an invaluable tool for time-averaged holographic analysis of coupled
spatial modes and pure modes.
ocot-uE
QJ
T)f
=o-E
0
0
0
0
0
0
0
-0
I6
L
2
0
2
L
6
I
-0.¿ o o.L 0.8 1.2 1.6 2.0 2.t, 2.8 3.2 3-6
T het a ( rad ians )
FIG, 3-24 (") CIRCUI{FERENTIAL 0RDER N = 4 SINGLE l'100E ANALYSIS, (C0PPER CYLINDER) ,
Least Squares Procedure Determination of Radial and Tangential Components of a Circular Cylinder
Víbrating in Coupled Modes.
A Experimentally Determined Radial Component. o Experimentally Determined Tangential Component.
Least Squares Curves.
HO5.
oo
ooo
o
o
oo
o
oo
o
ooo ooc
o oo
gO
o
A
o
A
A
o
A
O9ooOg
Âr
o
o
o
o
o
oo
o
A
A
o
o
o
o
oo
o
o
A
oo
%
ooo
AAó
so o
o
o
A
â
ooooo
oo
ô
AÉ
o
oo
o
oo
o
o
b
6A
Â
A
t/)^cuot-
.:05
l.
2
0
-0.2o¡of,
=o_
E
0.8
0.6
-0.6
l.0
-0.8
-0.¿ 0 0.1 0.8 1.2 1.6 2.0 2.t 2.9 ' 3-2 3.6Theta ( radians )
FIG, 3-24 (t) lvlULTI ÍTODAL ANALYSIS TECHNIQUE Nl = 2 AND N2 = 4, (COppER CyLINDER),Least Squares Procedure Determination of Radial and Tangential Components of a Circular Cylinder
Vibrating in Coupled Modes.
A ExperimenÈally Determined Radial Component. o Experimentally Determined. Tangential Component.
Least Squares Curves.
HOt¡
o
o
o
o
oo
o
oo
o
o
ooo
OO
oo
o
o
ooo
oo
106.
TABLE 3-1
PHYSICAL PROPERTIES OF TEST CYLINDERS
NOTE: Values of v, E and p are due to Ïaires [161].
Property Sma1lSteel
LargeSteel Aluminíum Copper
Length (nm)
Externaldiameter(rnm)
Ìüa11thfckness(nrn)
PoissonConst. v
E x 10-13p(*t21"e"2)
400
75.L
L.2
0.27
2.633
398. s
115. 1
4.4
o,27
2.633
403. s
91. 3
1.8
0. 33
2.702
402.5
101. 6
L.7
0. 33
L.222
LO7.
TABLE 3-2 (a)
THEORETICAL AND EXPERI}ÍENTAL RESULTS
(EXPERIMENTAL RESULTS IN PARBNTHESIS)
SI.IA]-L STEEL CYLINDER
m = axial order, [ = circumferential order,
A = longitudínal eomponent, B = tangentíal component,
C = radial component.
m=1 m=2 m=3
LlC
rt=2 B /C
FREQ (HZ)
^/cn=3 B/C
FREQ (HZ)
^/ctt=4 B/C
FREQ (Hz)
.0693 (.12s0)
.5028 (.485r)
7L7.4 (7L4.4)
.0317 (.0919)
.3346
1660. O (L617.3)
.0181 (.0s39)
.zsLO (.L707)
3L49.7 (3rs7 .9)
.L2r3 (.1931)
.5076 (.4549)
L64e.4 (1s87.3)
.0s96 (.09s3)
.336s (.26e5)
1856.s (1Bs7.s)
.0349 (.ls21)
.2sL9 (.1861)
3228.0 (3234.7)
.1489
.5086
3L94.4
.0809
.338s (.2877)
2406.L (2378.4)
.0494 (.ls8r)
.2s3L (.1933)
342r.6 (342s.9)
10 B.
rABLE 3-2 (b)
THEORETICAL AND EXPERIMENTAI RESULTS
(BxpenTMENTAL RESIILTS IN PARENTHESIS)
LARGE STEEL CYLINDER
m = axíal order, n = circumferential order,
A = longítudinal component, B = tangenÈíal
component, C = radial- component.
m=1
tt=2
a./c
Blc
FREQ (Hz)
.0990
.5065
1159.5
(.5161)
(.47 65)
(1073. s)
109.
TABLE 3-2 (c)
TIIEORETICAL AND EXPBRIMENTAL RESULTS
(EXPERI}4ENTAL RESULTS IN PARENTHESIS)
ALUMINIUM CYLINDER
n = axíal order, n = circumferentíal order,
A = longitudinal component B = tangential component
C = radial component.
m=1 m= 2
Alc
n=2 B/C
FREQ(nz)
^lcn=3 BIC
FREQ (Hz)
Alc
n= 4 B/C
FREQ (Hz)
.081s (.l1s8)
.s047 (.4717)
79s.4 (7s8.9)
.0378 (.0364)
.33s4 (.2s85)
L764.4 (1682.3)
.0217 (.028s)
.2sL6 (.ls73)
3336.2 (3L73.r)
.L346 (.r46e)
. s118 (.4:sS)
L902.r (1811.2)
.0690 (.09s2>
.3387 (.2777)
2029 .0 (1932.0)
.o4L2 (.1406)
.2532 (.L799)
344e. B (3286.7)
TABLE 3-2 (d)
THEORETICAL AND EXPERIMENTAL RESULTS (UXPSNT}ÆNTAL RESI]LTS IN PARENTHESIS) COPPER CYLINDER
m=5
.L032
.4455
5013.1
.09s8
.3313
3386.3
.07 49
.2s54 (.183s)
2949 . s (3013.3)
.05.70
.2056
3462.3
m= 4
(.2715)
(2s63.7)
( .13s3)
(2467 .8)
.1331
.4835
382r.8
.1001
.3393
25A8.7
.0708
.2564
2413.2
.05il
.2050
3t47 .9
m= 3
.1513
.5069
2553.9
.0942 (.7260)
.34rs (.26s3)
L72L.6 (r772 .9)
.0610 (.093s)
.25s4 (.24s7)
2020.3 (207s.4)
.a420
.2038
2932.8
m= 2
.L4L7
.sLzs (.3173)
L349.4 (136s.0)
.0748 (.0696)
.3392 (.3116)
LLs2.8 (1190.2)
.04s1 (.03e0)
.2s34 (.2s77)
L7es.r Q842.2)
.0300 (.0916)
.2024 (. 1793)
2806.0 (2864.8)
m=1
.0896 (.L422)
.s0ss (.488e)
486.s (s08.3)
.0419 (.0s12)
.33ss (.3191)
gLL.7 (933. e)
.024r (.1343)
.25rs (.2226)
170s.7 (L749.8>
.0ls6 (.2890)
.2013 (.322s)
274s.0 (2803.4)
^/cn=/. B/c
FREQ (Hz)
Ã/c
n=3 B/C
FREQ (Hz)
Alc
n= 4 E/C
FREQ (Hz)
Ã/c
n=5 B/C
FREQ (Hz)HFP
m=axialorder¡ D= circumferenËialorder, A= longitudinalcomp.onenÈ, B= tangentíal component, C= ¡¿día1 component.
rll.
CHAPTER 4
CIRCULAR CYLINDERS I^IITH VARYING I^IALL THICKNESS
The least squares procedure has been shor"m to be a powerful
Èool for the analysis of vibratíons and, i-n particular, the amplítudes
of the superposed spatíal modes of any vibrating surface may now be
determined. The result thaË vibration may be expressed as a combination
of normal modes has exciting possibilities for the study of vÍbratíng
objects wíth perturbatíons in some quantity, for example, in the
geometry, force or elast.icíty properties. The Fouríer components of
the perturbation may be thought of loosely as exciÈing the corresponding
vibration components in the structure although Èhere is no dírect
relatíonship between their relaËive magnitudes. In Lhis chapter the
problem posed is the Èheory for the flexural vÍbrations of a fínite
length circular cylinder wíth shear-diaphragm ends, s)nnmetric thíckness
variations about the cenËral plane and const.ant axial thíckness. The
solutíon is arrived aÈ using the Rayleigh - Rítz method and the mode
shapes so obtaíned are compared with experiment using the leasÈ squares
procedure.
4.T THEORY OF VIBRATION OF CYLINDERS I^IITH VARYING I^TALL THICKNESS
The theory to be described considers only thickness variations
in the wall of a círcular cylj-nder as ín practíce these would be the
principal sources of anisoÈropy. The procedure is a three-dimensional
extension of the analysis outlíned by Hurty and îubinstein ( [136] ,
sectíon 4.5) and uses the Rayleígh-Rítz method to solve the elasticíty
equatíons of Arnold and l^Iarburton [126].
Consider a cylíndrical shel1 of length L supported at the
II2.
ends by shear-diaphragms (see ltig. 4-1). Assume there exists an
unstraíned circular middle surface of radius ao and thal- the- thíckness
of the shell is symmetrical about the angular co-ordinate 0:0 and
constant along the generator X but rvhose outer and inner surfaces are
independently variable. The outer surface ls descrlbed by Èhe function
ar(O) and the inner surface by at (0) with the unstrained middle surface
as the origin and hence the rroutert' thíckness and ttinner" thickness are
respectively
(0)"h*
a, (0) - aoo
(4. 1)
aoh-(0) = a, (0) -a o
These conditions ímposed upon Èhe Èhickness variable serve only to
síroplify Èhe mathematics as ín principle one could solve the elasticity
equatíons for the most general form of Ëhickness variable.
Expressing the thíckness variation in Èerms of a one-dimen-
sional Fourier series gives
ah'(e) h+ cos(p0)
P
(4.2)
cos (p 0)
Clear1y, if the unstrained middle surface is to remain circular then
¿h-(0) = h-(0) í.e. the distortion is syrrnetríc about the rníddle surface.
Hence for non-symmetric disËortíons this analysis will only be valíd Íf
the dístorfion is small whereas for s¡rmmetric distortions the only
lírnitatíons on the Ëheory are those of the Arnold-tlarburton theory for
the perfect cylinder (i.e. thin walls).
Ip
h-(e) = I h;p
113.
L
zX
aoh-(0) Unstrained midd le surf ace
aoh+( 0)
FIG. 4.1: CIRCULAR CYLINDER hlITH VARYING ANGULAR
IIt
T\
I
\
tI
I
WALL THICKNESS.
114.
The symmetric solutions of the longitudinal, tangential and
radíal components of vibration are assumecl to be the sum of t.he normal
modes of a perfecÈ cylinder with no pert.urbations and are respectively
a(x, e, t) I ÀXcos -- cos n0 cos uJta
oa
AnnIo m n
b (x, 0, r)
c(xro,t)
o mrn
m n
1a ln .Àxsi-n- sin n0 cos t¡tmna
o
(4.:)
(4.5)
(4.6)
l:'l: l-.tc#,'*ril,'*
aijId
1 . ).xs1n- cos n0 cos (l)ta
od
cmn
o
where, (rn,n) is
the mode of víbratíon. The asymmetric solutions are obtained by sub-
stíÈuting sínn0 for cos n0 and více-versa ín equaÈions (4.S). The
total kínetic energy ís IL26l
À = mnao/L ís the non-dimensional axial wavelength and
oa5T= o
'2 ,*1,'] do dxdz (4.4)
where p is the densíty of the cylinder material. It will be useful to
note Ëhe following
t-(o)
(0)dZ=
.J
i (h; - ho)cos p0p
and to Íntroduce the following notation
r2r
'ii::: = Jo
cos iX cos jX . . . sin kX sín .Q,X dX,
4..r_J
=1 ua
o
1c
ooand U ij COS trtt, V íj cos ot, I^líj ij cos oË (4.7)
115.
Substitutíng equatíons (4.S) inËo equation (4.4), makíng use of equations
(4.S), (4.6) and (4.7) then considering the first kinetic energy term
of equation (4.4)
rripa
[,
(0)
o)
5 IT
oo
o âa)2 d0 dx dz( (4"8)
(4. e)
(4. 10)
(4.11)
(4.L2)
2 ar
where Imn
ÀXcos _- cos nua
oInn
and the superscrípt s índicates s)rmmetríc solutions. Hence,
pa (0)2
d0 dx dzmfl
Ua
L'f"o
ro
5o
2ri
(0)
* 1""o
tIú "o"4 cos no ]omn
and Èhe general Ëerm in ff is
oa5'o .2r1..L
.J" Ui¡Ukl.o" n. 0 cos nu 0 cos pe (h' - hp) de
tix rtxCOS.: COS--å . dXâo ao
2
L
Noting thaÈ
0 i#kT.X À. Xr- l(
COS-......- COS..- dX =aaoot, /2ao í= k
due to the orthogonality of the cosine funcËions, then the general term
in Ts becomesI
l"
Lao
IU. . U." (fr' - h1-'l Lx, p
L
2ão
o"5'o2
cos n. 0 cos nUO cos p0 d0 (4.13)
Using the notation of equation (4,6)
116.
(4.L4)
(4.ls)
(4. ro)
T U ij h+p
j-cp-h-)rp'
where the following notation has been used.
The second kínetlc energy term of equation (4.4) Ís
urn (
where
IIIIi j e, P
I=ij sp
L
,: = S l'" f' fn*tu' r*þr, do dxdz2 2 Jo io ,h- (o)
b=lInn
V.Àxsan
- si_n n0
a (4.L7)
sÍn no l2 ¿e dxdz (4.18)
(4.le)
0 i+k
mno
Lh+ (0)âo
ü sínmn
Àxd
o
The general term in t! is
Hence
pa 5o
x
and usÍng the Ídentity
t Ih (0) mn
2 t5vk.r,"ít n.0 sín nuo cos p e (h; - n;) de
L. Àix Àtx
sin- sin.- . dXâo to
L
I % ÀiX À.xI sl-n--- - sin--5- ¿¡ =Jo "o "o
Ll2ao
f=k
(4.20)
due to the orthogonaliÈy of the sine fulrctions, then
pa4L
T; o )r p
4 jL
The equatíon in W is solved in a símilar way. Hence the total
expression for the kinetic energy for the symmetric solutions is
,.5fni':ü'u(nl - n;
ililll * ùr¡ùroniuo) (nn - n;)
(ùr:ùrunTu * ür¡ùrunjrn * irjùrunTul cn* - n;)
LL7.
(4.2L)
(4.22)
(4.23)
+
oa4L-soI=- + ü..ü
r_JI rù..û. ^njøPijip r-J L)L4
and for Ëhe asymmetric solutions,
pa4L
TAo
4
lL26l
=- Iij lp
The total sÈraÍn energy with the cylinder in vibration is
L
Ea3St= o
2(t- v2)
6+
t"' riå r ,, #.#. ,,(#->'2
h-
. (*F,' - r"* + c2 - 2z(# - ")(#i * # ) + z2cffi>, *
+ 222 p*+ z2cj$)' * 2u[,* - r#) (åå -c - r# -'ji,]
. s;ù [,* )'* z# jå * ,#,' - 4z(# + #) (#,ï * Sr *
+
+ 422((#fu )2 + z## # * ,#,',] de dx dz (4.24)
r^rhere E ís Youngrs I'fodulus and v is Poissonts ral-io. Agaín naking
use of equatíons (4.:), (4.5), (4.6), (4.7) and (4.fS¡ Lhe inregral
ís evaluated and equation (4.24) reduces to
and o"4L'o=
2
118.
(4.2s)
(4.2ø¡
(4.27)
(4.28)
(4.2e)
(4.30)
(4.:r)
Ea2Lc* -
OJL - (r-v2) j
31
L St j-1
The thirty one terms in the summaEion are given in Appendíx III for the
symmetric SÈs and asymmetric Sta modes of vibratíon.
Applicatíon of Lagranges equation to thå independent
varíables U, V and lal gives
AT AT asrddr
( )AU AUnn
nn
h-¡ ¡nj Ip
mïl
and two símilar equations tr U*r, and hI Using equatíon (4.22)
s o"4L'oâT
å ù,",tnl
âUmn
2
+I ü .(h'- h-)rrnJPi; rnJ p p-
From equation (4.7)
hence
ü = -u2uMJ MJ
I u .(t+ -iimr p
-+) v .(h' - h )i; mJ p p-
h --njpIt)
P
-p"4Lur2lP.nJ
Similarly
2
and
119.
(4.s2)
(4.3s)
9= t â,T">oE aI^Inrt
I,I .(t+ - t-) ntjPmJpp-p a4Lrr.'2
o
2 jpI
Correspondíng equations for Èhe asymmetric solutions are easily written.
NoÈing thataT=âT.âT
au av = rri-= o (4'33)mn mn nn
completes the left hand side of equaËÍon (4.26). The righÈ hand side
is evaluated in a simílar way where for the first strain energy term
shown in Appendix III
ast!
ôur'(4. 34)
s sâSÈ AS tAV âI^I
mn nn
and for the second
ASS
t
= ,al x2v.(h+ - rr;) n"jeJ;MMJ P
hp¡3 u . (t+rr+MJ Pq
0
2
àuo-,4
jIpq
w ' (tr+ tt+ - h- h- ) nniPqmJ pq pq
h ) nniPq
¡3m
âss2
âvrnn
ASs2
âI^lmn
-kljpq
0
m q
and so on for the oËher terms.
Defíning the frequency parameter /\ by
L2 = p ^ïrr(L- .v2) lE (4.36)
then the three Lagrange equatíons of the form of equation (4,26)
become
nR=rt2gn
L20.
(4.2t)
where D and Q are the generali-zed stiffness and mass mati:j-ces respect-
ively and R ís the vector of coefficients of order 3M(N + J ) , where M
arrd N are the límiting orders Ëo which the analysis is taken, and is
defined as
R[3(k- 1) (N+ 1) + i + 1] = Akí
R[3(k-1)(N+1) + i + N + 2] = 8..ka
(4.38)
R[3(k-1)(N+1) + i + 2N * 3l = cu.
where k=1r..., M and i=0r..., N.
The uratrices D and Q are 3M(N+ 1) square matríces with non-
zero coeffícients about the leading diagonal and zero coeÍficíents
elsewhere. Lísted i-n Appendix IV are Ehe coefficients of these matrices
for both the symmetric and as)rmmeËric solutions and Appendíx V lists
the solutions of Èhe II functions of the form shov¡n in equBtion (4.6).
Equati-on (4.37) is simply an eígen-value problen and may
be sol-ved by expressing D ín upper Hessenberg form and Q in upper
triangular form lL62l. To each of the 3M(N + 1) eigen-values there
corresponds an eígen-vector which defínes the component amplítudes of
the contríbuting modes. The corresponding resonant frequency is cal-
culated from the eigen-value using equation (4.36).
As in the classical solutíon for the vibration of a non-
dÍstorted cylínder there corresponds three solutions or eigen-vectors
for each mode. In two cases the longitudinal components and tangential
components contríbute mostly to the vibratíon and the frequencies are
I2T.
usually very hígh (well out of the audio range). In the third case
the radial component is Èhe largest and the frequencies for the
lowest modes a::e well withín the audÍo range. Although the present
analysis solves for al1 the modes, the only ones of interest to this
study are the M(N+ 1) modes ¡.rith the lowest frequency parameters. Also,
sínce a number of modes contribute to the vibration, reference to a
single mode hereafÈer is taken to mean the principal mode unless oÈher-
wise qualífíed.
However equation (4.37) cannot be solved in iÈs existing
form. For the symmetric case N = 0 the cylinder ís vibrating ín a
breaËhing mode and hence the tangential component b is zero. I{ence
there are M columns in matrices D and Q which are zero corresponding
to the zero coefficients Vr.', Vr'r..., V"O which must be removed else
the solution to equatíon (4.37) is trivial. The corresponding rows
created by differentiation with respect to Èhese coefficíents
(a/â Vl o, ð/ â vr', . .. , à/ â V"o) must also be removed. Hence matrices
D and Q and vect.or R are reduced to order M(3N + 2). For the asymmeÈric
case N = 0 the cylinder vibrates in a purely twistíng mode and Èhe
longitudinal and radj-al components a and c are zero. The 2M columns
in matrices D and Q correspondíng to Ur', U2O,...,UMO and W10, W20,
..., WMO and the corresponding rows which are created by differentiation
with respect to Èhese coefficienËs are removed resulting in the maÈrices
and vector R having order M(3N+1).
4.2 CYLINDER WITH NON-CONCENTRIC BORE
A circular cylínder with a non-concentric bore is one of
Èhe simplest syÍìneÈric clistortions and can be readily manufactured.
The Fourier coefficients of the distortion are calculated as follows.
Reference to Fig. 4-2 shows that if the radius of the ínner bore is
4!, the radlus of the outer surface arr the radius of the unstrained
1e
e
I
â¡
L22.
Unstrainedmiddte surface
aoh
.lIQoh *I
?,2
FIG. 4-22 GEOIVIETRY FOR CALCULATIN(ì THE FOURIER
COEFFICIENTS OF THE DISTORTION.
ee
/
\
II
I
\
míddle surfac.e ao and the displacenent of the centres e then
,1= "" + a2(t+h-)2 + zeao(1*h-) coso
^î. = ", + al(r+n+)2 - 2 eao(1+h+) cose
Hence,
aoh-(O) = -êcoso * (eZcos2e - e2 + ^!¡', - ^,t' o
"ot+10) = ecoso + (e2cos20-e2+a|)'-.
I23.
(4.3e)
(4.40)
-a o
The Fouríer coefficients are thus
h I-17
12no
n:=hf'_1I
h (o) de
(4.+r¡f
h'(o) de
forp=g
and
h-(e)cos pg d0
(4. +z)
h+ (0) cos p0 d0
forp>0
which are evaluated numerically as'Èhe lntegrals are difficult to solve
explicitly.
A steel cylinder was fabrícated wíth dimensions ao = 39.29rnm,
ar= 37.83mm, ar= 40.75mm, e = 0.5mm and L = 398.Bunn. The ends of the
r.
r24.
cylinder were machíned flat with a lip on the inside edge r¿hich coin-
cides with the unstrained middle surface. Two 1.6mm sheet steel end
caps' with the círcumference machined to a thin edge, fit into the 1ip
one on each end of the cylinder (see Fig. 3-19) rvhích satisfies the
requirements of shear-díaphragm end conditions. Two electromagnets
were positioned at 0 = n or the thinnest part of the cylínder and near
the ends of the cylinder and the techniques of sectlon 2.2 applied to
excite and analyse the modes of vibration.
The frequencíes of the symmetric modes \^/ere experimentally
measured using a frequency counter and these are shown in Table 4-1
together with the theoretically calculaÈed frequencies for the syrmnetric
modes. Also shornm for comparíson are the theoretical resonant freq-
uencies for the same modes buË r¿ith no distortion. This data is shornm
in graphícal form in Fig. 4-3 from which it is clear that j-ncreasing
distortion lowers the frequencíes of the modes but only for this type
of distortíon. rn general the dístortion may lower or raise the
resonant frequencies of the modes [139].
The modes were idenÈified with the aid of a flexible plastie
tube functioning as a stethoseope. The antinodes and nodes could
be easíly detected from the loudness of the tone as the tube is moved
over the surface of t.he cy1índer. Table 4-1 shov¡s the experímental
error in the frequencies to be less than 2% due príncipally to
inaccuracÍes in machining, other modes could noÈ be excited due to
lj-mitatíons of the electromagnetic drivers.
The frequencies of the as¡rmmetric soluÈions are shovm in
graphical form in FÍg. 4-4. These are little different to, except
for large distortions, the frequencies of the symmetric modes and t.he
difference is greatest for the lowest axial and círcumferentíal orders.
Table 4-2 shows the theoretical superposed symmeÈríc modes
L25.
7
6
NI=
(,cq,f(tt¡,
LL
5
l.
3
2
0 0.2
Axial waveo.l.
length0.6 0.8 t.0
factor À (= Eo)L
FIG. 4-3 THEOREIICAL FREOUENCY CURVES FOR DISTORTED
CYLINDER, Symmetric solutions ê = O, e= 0.5, --- e= 1.0,
o experímental points for case e= 0.5.
r--OO--o-n=t1
o-n =3 /- /
---ê- //
,
n=2
5
L
N-=
(,cc,)(t(¡,t-l!
6
126.
1.0
3
2
I0 0.2
Axial wave0.¿
length0.6
f ac tor0.8
À (= IrfulL
FIG, 4-4
CYLI NDER.
THEORETICAL FREOUENCY CURVES FOR DISTORTTD
Asymmetric solutions. -
e = 0, e= O.S,
7
n=4--
- -
-"-n=3 /
¿2
oOê---/
//
/
î=2
e= 1.0.
t.27 .
of víbratiorr for four modes of the distorted cylinder. The asyünetric
solutions are nearly identícal in magnitude and so are not tabulatecl.
Clearly the greater the number of c:l-rcumferentíal modes included in
the analysis (i.e. the larger the value of N) the more exact r¿ill be
the theoretical- solutíon. It is found that N= 6 is sufficient to ensure
excellent agreemenË with experiment and favourable ín computing time.
The solution of the eigen equations is well behaved. If the distortíon
is seË to zeÍo, for example, then the modes decouple completely leaving
only the principal mode.
To determine the tangential and radj-al symmetric components
the cylinder was set upright on Èhe turntable and for the longitudínal
components it was posit.íoned on its side and holograms taken using the
techniques described in section 2.2. Table 4-3 shows the theoretically
calculated and experimenËally determined vibratíon compone-nts for Èhe
distorted cylínder. Due to the mechanícal arrangement it is not
possíb1e to analyse the circumferenËíally varying longitudinal modes.
The double Ëurntable scheme descríbed ín section 2.L is required.
Hence only the ratío of longitudinal to radial vibration is considered
which is
Am r (l ^âì lqlcfu) r
nn(4.43)
and varies sinusoidally along the length of the cylínder as given by
equaÈion (4.3). The theoreti.cal and experimental amplitudes for Èhe
B and C components are scaled by the factor which makes Èhe C component
of the príncipal mode unity. rn additíon sínce the modes are not
excj-ted exactly symmetrically due Èo influence of the magnets then
some asynmetric components are present in the experimenËal analysis.
Hence only the rnagnitudes of the components are shovm ín Table 4-3.
Figs. 4-5(a) to 4-5(f) show the mode shapes calculated from
7,1
/c^ =
cocoo-EotJ
fltãtr0J
orc(út-
0
cAtcoo-Eo(,
rrt
Erú
OL
60
0.3
'-2.0 -1.6 -1.2
r28.
0 0.1 0.8 1.2 1.6 2.0( rad ians )
0
-0.3
0.6
1.0
0.5
0 .5
-1.0
-2.0 -1.6 -1.2 -0.8 -0.¿ 0 0.1 0.8 1.2 1.6 2.0Theta ( rad ians)
-0.8 -0.tTheta
FIG. 4-5 (") RADIAL AND TANGENTIAL VIBRATION COIÏPONENTS
FOR T]lE PRINCIPAL SYIiI{ETRIC I.IODE (I,2) OF A DISTORTED
CYLINDER,
-
theory experiment.
\\
\
\/ \/ \
¿n -
\\
\ \.- _,
\
/\ \
-
I2.9.
0.6
0.3
0.3
-2.0 -1.6 -1.2 -0.9 -0.t 0 0.L 0.8 1.2 1.6 2.0Theta ( radians)
0
0.5
- 0.5
-1 .0
-2.0 -r .6 -1.2 -0.8 -0.t 0 0.t 0.9 1.2 1.6 2.0Theta ( radians )
FI G, 4-5 (¡) RADIAL AND TANGENTIAL VIBRATI0N C0I\iPONENTS
FOR THE PRINCI PAL SYÍ\1I4ETRI C IV]ODE (I,3) OF A DISTORTED
CYLINDER, -
theory. ---- experíment.
0
cocoo-Eou
rrl
;cA)olcrlt]-
0
c0,coo-EoC'
nt
õrú
É.
-0.6
I
\\\\
130.
0
cocoo-Eo(,
flt;c0,glcrúF
0.6
0.3
- 0.3
- 0.6
-2.0 -1.6 -1.2 -0-8 -0.t 0 0.L 0.8 1.2 1.6 2.0Theta ( rad ians )
r.0
0.5Ec,coo-Eo(, 0
ãu -O.Srlt
É-
-t .0
-2.0 -1.6 -1.2 -0.8 -0.¿ 0 0.¿ 0.8 1.2 1.6 2.0Theta ( radians)
FIG. 4-5 (") RADIAL AND TANGENTIAL VIBRATION COIIPONENTS
FOR THE PRINCIPAL SYI1II4ETRIC I4ODE (2,2) OF A DISTORTED
CYLI NDER , -
theory experí.ment.
//
cúcoo-Eot,
rl;coC'ìcrút-
0.6
0.3
-0. 6
0
30
-2.0 -1.6 -1.2 -0.8 -0.4T heta
131.
0 0.¿ 0.8 1-2 1.6 2.0(radians)
0
c0,cooEoL'
ñt
orlt
É.
1.0
0.5
0.5
-1 .0
-2.0 -l .6 -1.2 -0. B -0.1Theta
0 0-¿ 0.8 1.2 1.6 2-0( rad ians )
FIG, 4-5 (g) RADIAL AND TANGENTIAL VIBRATION COI4PONENTS
FOR THE PRINCIPAL SYI'II'IETRIC I/IODE (2,3) OF A DISTORTED
CYLINDER, -
theory experiment.
\
\
cc,cooEo(,
rú.;c(,(tlcfút-
0.6
0.3
-0.6
0
-0.3
-2.0 -1.6 -1-2 -0.8 -0.¿Theta
r32.
0 0.t 0.8 1.? 1.6 2 .0( rad ians )
1.0
0.5
rú
õ -o.srúÉ.
-1 .0
-2.0 -1.6 -1.2 -0 g -0.t o 0.¿ 0.8 1.2 1-6 2-0Theta ( radians )
FIG, 4-5 (.) RADIAL AND TANGENTIAL VIBRATION CONPONENTS
FOR THE PRINCIPAL SYI1I1ETRIC IVIODE 3,2) OF A DISTORTED
CYLINDER, -
theorY exPeríment'
0
cQ,cooEo(,
\
\\
\\
\
/
//
/,
13 3.
0.6
-0. 6
-2.0 -1.6 1 .2 -0.8 -0.1 0 0.t 0.8 1.2 1.6 2.0Theta ( radians)
t.0
0.5
-0.5
-1.0
-2.0 -1 .6 -1.2 -0.8 -0.1 0 0.1 0.8 1.2 1.6 2.0Theta ( radians )
FIG. 4-5 (T) RADIAL AND TANGENTIAL VIBRATION COIi1PONENTS
FOR THE PRINCIPAL SYI1|VIETRIC lVlODE (3,3) OF A DISTORTED
CYLI NDER, -
theory experiment.
0.3
0
30
c@cooEo(,
rlt
=c0)grCqt
0
c0)coo-EoIJ
6!rú
cÊ
a,
\\
\
/
/
\I
II
\\
L34.
the theory and those derj-ved experimentally. Evident from the exper-
imental curves ís the small contribution of the asymmetric modes
discussed above. Nevertheless the agreement is excellent for all
modes except the lowesË (1r2) node due to the absence of a significant
contribution of the n=1 mode (see Table 4-3). NoËe that the origin
of the holographic analysis, being different to that of the theory
of sectÍon 4.1, is Èransposed for dírect comparíson.
4.3 CYLINDER I^]ITH A THIN LONGITIJDINAL STRIP
A steel cylinder was manufactured with length L = 398. Brnm,
mean radius a = 39.29rnm and wall thickness h = 2.05mm. A nominalo
Vg" sq.t"re-section length of steel rod was attached t.o the cylinder
along a generaÈor with an extremely thin layer of LS-12 adhesive.
Assuming that the distorting strip is sufficíently small to neglect
errors of curvature, the Fouríer components of the distortion are
calculated with the aj-d of Fig. 4-6. Using equations (4.4I),
þ = _h/ 2ao o
(4.44)I
h' = h/2ao
* DU/naoo
and from equatíons (4.42),
h =0p(4.4s)
h+p
forp>0
where D = 3.21rnur and p = 0.03876 rad.
Fig. 4-7 shows reconstructions of tíme-averaged holograms
2D=-Sl-nDU
1IDA'o
135.
-lr-0
- '- ? T -f - \.t
htz
rI
I
,
I
\Unstrained
I
I
tm idd le surf ace
âg
FIG. 4-6 GEOI'IETRY FOR CALCULATING Tl-lE FOURIER COEFFICITNTS
rs1| FI
,
I
I
I
t
I
I
t
OF THE DISTORTION.
136.
FIG.4-7
Two views of a cylínder wíth attached strip
vibrating in (2,2) mode'
r37.
of the cylínder víbrating in the fr=2, n= 2 mode. Table 4-4 shorus the
resul-ts of measured and ËheoreÈícal frequencies with Èhe disËortíon
and without it. For the undistorted cylinder agïeement is excell.ent
but with the distortion the predictions are totally ínaccurate. The
strip \^ras re-attached usíng a number of fine bolts but wiÈh the same
results, índicating the adhesíve is not an j-nfluencíng facÈor. RaÈher
iÈ ís thought Èhat the sudden discontinuity of the aÈÈached strip
dictates the use of functions which satísfy the strict boundary
condit.íons at the stríp which Èhe compariËi-vely slowly-varying sínu-
soidal functions do not. Put another way, the Rayleigh-Ritz method
ttspreads" the díscontinuiÈy over the surface of the cylinder and does
not necessarily account for sharp díscontinuitíes.
138.
TABLE 4-1
RESONANT FREQUENCIES FOR SOME MODES OF
DISTORTED CYLINDER
(non-concentric bore)
PrÍncípalMode
(mrn)
Theoretical Freq. Exp.
Freq.
Distorted
ol
Error inExp. Freq.Undistorted Dístorted
1 t 2
1'3
L14
212
213
2 4
312
313
314
1340
3553
677 4
2r05
37 40
6905
3598
4204
7]-59
1316
3375
646s
2042
3562
6594
3524
4045
6847
1330
3442
6495
2063
3627
66L7
3463
4085
6861
1.1
2.O
0.5
1.0
1.8
0.3
r.7
1.0
o.2
139.
TABLE 4-2
THEORETICAL SI]PERPOSED SY}tr{ETRIC I"IODES }'OR A CYLINDER I^IITH NON_
CONCENTRIC INNER BORE
MODE AMPLITUDES II =
These coefficients are preset to zero before the eigen equations
are solved.
FREQ
Hz
PRINC -IPAL
MODE
(mrn)
COEFF.
0 1 2 3 4 5 6
1315. 6
337 s .4
204r.5
(r,2)
(1, 3)
(2,2)
In
In
C n
In
nI
CnI
A2n
B2î
A2n
c2n
A
B
A
B
crn
B2n
-.027
.002
.003
0*
O*
0
0
0*
0
0
0
0*
.]-54
.606
.s96
.001
.001
-. 015
-.o28
-.026
.001
.003
.003
0
-o ,72
-. 504
-1.000
-. 002
-.010
-.019
.r27
.509
1.000
.005
.o22
.042
.006
.064
.190
-.034
-.336
-1.000
-.013
-.o76
-.225
.064
. 339
1.000
-. 007
-.028
.006
.081
.32L
.001
.009
.038
-.or2
-.084
0
331
.001
.004
-.001
-. 014
_.o72
-.001
-. 006
.002
.015
.075
0
0
-.013
.oa2
.or2
.001
0
0
002
0
0
0
0
0
*
140.
TABLE 4-3
TI{I{O]IETICAL AND EXPEIII}IENTAL SY}ß{ETRIC MODE AMPLITUDES OF DISTORTED
CYLINDER (NOR}ÍALISED MAGNITUDES)
PR].NCIPAL
MODE (rn, n)ït
RADIAL COMPONENT
cnn
TANGENTTAL COMPONENT
Bmn
LONGITUDTNAI
RATIO
Alcmm
Theory Expt. Theory Expt. Theory Expt.
Lr2
1r3
2 , 2
213
3 t 2
a)JrJ
1
2
3
3
4
5
2
3
3
4
5
2
3
4
2
3
4
.596
1.000
.190
1.000
.32t
.072
1.000
.225
1.000
.331
.075
1.000
.4t5
.101
.238
1.000
.362
.108
1.000
.163
1.000
.268
.034
1.000
.191
1.000
.386
.083
1.000
.318
.090
,L39
1.000
.282
.606
.504
.064
.336
.081
.014
.509
.076
. 339
.084
.015
.509
.140
.o25
.L22
.34L
.092
.178
.496
.110
.228
.079
.054
.346
.044
.2L6
.100
.034
.2L7
.2r7
.r28
.064
.29L
.r47
.L46
.033
,L26
.062
.L44
.087
.091
c40
.186
.105
.186
.L69
L4I.
TABLE 4.4
RESONANT FREQUENCIES FOR SOME }ÍODES OF DISTORTED CYLINDER.
(Longitudinal SLrip)
PRINCIPAL
MODE
(m,n)
I]NDISTORTED CYLINDER DISTORTED CYLINDER
THEORY EXPERIMENT Z ERROR TIIEORY EXPERIMENT
1 t 2
1'3
2 t 2
213
312
3'3
413
990
2500
1871
2689
343L
3195
40sB
1007
2552
r852
2735
3328
3225
4056
L.7
2.L
I
1
3
1
0
0
7
0
0
0
TL2B
3647
1939
37l-7
3336
4r93
4636
999
25]-7
1859
2705
3343
32L3
4067
r42.
C]ìAPTER 5
COUPL]]D TEMPORAL MODES
Ii- was shor¡n in Chapter 4 that the effect of any dístortions
in the wall thickness (r,¡hich may be genetaLízed Èo include anisotropies
in radius or Youngr s modulus for example) is to couple modes but at
Èhe same frequency; such modes are called superposed spatial modes.
Alternatívely, for a non-distorËed cylinder, the applied time-varying
point force induces modal coupling with phase-shíft between the modes
which was demonstrated for the two-dimensional case by Stetson and
Taylor tB4]. Such modes are termed Coupled Temporal Modes. In this
light it ís Ëhought that the analysis of Shirakawa and Mizoguchi [156]
could be extended to predict coupled temporal mode response for
cy1-inders exciËed by poínt forces.
In thís chapter the theory of Generalízed LeasÈ Squares
is applied to solve the characterj-sËic equations for the case of
cylinders vibratíng in two coupled modes, one essentially pure mode
and two superposed modes at a single frequency. The temporal phase
difference is determined without the need for modulating the laser
beam.
5.1 HOLOGRAPHTC THEORY FOR TI,IO COI]PLED MODES
The characÈeristic equation for two coupled modes with
phase difference A was determined in sectíon 2.L.4 (special case 5).
The argument function for Èhe combined motÍon \^/as shown to be the
phasor sum of the argument functions of each component. Assuming that
each coupled mode is a singular superposed mode (i.e. pure mode) then
all terms in equations (2.10) to (2.I2) are zeto except for one - order
143.
Nl for one mode and N2 for the other. Hence equation (2.53) becomes
to = t(>d, sinlll Eo * ricosNlEo)*o + (risinNrqo * ,rTcosNlEq)sql2
+ t (4 sin N2Eo + yfr cos n2qo)Kq + (rft sin N2Eo + zfr cos u2tn) snJ 2
* 2coslt ({ sin *tEe vfr cos tllEo)Kq + (r* sinNrEo + zfr cos NIEo)soJ
. t({ sinN26n + yfr cosu26n)Kq + (rft sínN2Eo + zfr cosN2Eq)sql
-ç2 /xzq
(s.1)
The equations of the form of equatíons (2.54) to (2.56) are
aÍo) =
ffi = 2(c]ro + llso) sín(t'llEo)*n * 2cosa(.tKq+ b2sq)sin(tllEn)
.lo)- # = 2?2xr+ b2sq)sin(tl2Eo)Ko * 2cosa cIKn+tIso)sin(t'l2Eo)roN
a(o)-*+ = 2(clro+ blsq)cos(NlEn)*o * 2cosa(.rKq+ b2sq)cos(NtEo)*o" âyt
. (a) - a fg = z ( czxr+ b2sq) cos (tt2En)Ko * 2cos (" lKq + b lsq) cos (Nt 6n)*nt ,"1
aÍo)=# = -2(clro +blsq) ("2Kq+ b2sq) sin^
Kq
(s.2)
InIlìer e
L44.
(s .4)
{ sinNlE + vfr "o" t'tl6
", = * sin N2Ç + yrt cos N2E
(s.3)
b1 wfr sinNIt + zfr costllE
b2 wfr síntl2E + zfr cosll2E
Hence matríces f and F are calculated usíng equations (2.5L)
and (2.52) and the correctíon vecËor ô from equation (2.50). The
ínverse rnatrix I-l is calcul-ated by the Gauss-Jordan method with
elÍmination by partial pivoÈing [163]. The vector describing the
parameters
c
vi,{*R ( v ,2N
rrT, rfr, "i, 2rt, ^)
is arbitrarily set for the ínitial iteratíon but as shown in the next
section, local hígh-order mínima could terminate the procedure pre-
maturely. In practice a number of starting points are selected to
guard against this possíbility.
5.2 EXPERIMENTAL PROCEDURE AND RESULTS
To test the convergence of the method of Generalízed Least
Squares, five modes of two cyli-nders r¡rere considered. The first cylinder
of length 398.Brnn, mean radius 39.29mt and thickness 2.050 t .005nrn
was carefully rnachined to make it as near perfect as possible. The
second cylinder is the one considered in Chapter 4 wíth the centre of
the inner bore dísplaced from the centre of the outer circular surface
by lnm for which the Ëheory of víbration predicts a number of super-
posed modes.
L45.
To generate coupled modes in the perfect cylinder two elect-
romagnetíc drivers were positioned at. the radial antinod.es of vibration.
Each point force is a disturbance with spatial Fc¡urier components r,rhich
excite oÈher ntodes in the cylinder and these couple in some phase rela-
ti-onship resulting in phasor vibratíon. The clegree of couplíng of the
spatial Fourier components of the force to the cylinder modes depends
on the position of the force relative to the modes. Hence movíng the
point excítation towards the radial nodes (at the cylinder ends)
reduces the degree of coupling wíth the result that only a single mode
is induced in the cylinder. This behaviour was confírmed in the
experíment.
Fig. 5-1 shows Ëwo views of the perfect cylínder víbratíng
in two coupled modes (3,2) and (3,3). The predominant mode ís (3,2>.
The contríbution of the (3,3) mode ís seen as a coupling of the fringe
groups. This is to be conpared with Fig. 3-21 whích shows two views
of the cylinder vibrating j-n an essentially pure (4r3) mode. The
decoupling of the fringe groups is clear in this instance.
In the analysis of FÍg. 3-2I it is normal Ëo allocate a
sign to fringe groups, adjacent groups taking alternating signs. Thís
procedure j-s clearly irrelevant in t.he analysis of Fíg. 5-1 for since
the frínge order appears squared Í.n equaÈion (2.29) Èhen only the order
of the frÍnge is recorded.
Fig. 5-2 shows Èhe components of t¡^¡o coupled modes for the
perfect cylinder. stetson and Taylor [84] predict that if the reson-
ances are reasonably sharp then the response of a mode at its resonant
frequency is close to r/2 in phase to any other mode that may be
excited in combination. The temporal phase difference A was determined
as 1.96 racl a; the resonant frequency 7B52Hz for the coupled modes
(2,2) and (2,3) shov¡n in Fíg. 5-2. Fig. 5-3 is rhe result of reposír-
1-46
FIG.5-1Two views of a cylínder vibrating in two coupled modes '
0
cc,coo_
Eou
rú
;trA)qrcrúF
c3 0.soo-Eo0l.,
ã
6
0.3
0.3
-2.0 -1.6
1.0
0
-0.6
L47
1.2 1.6 2.0
D -0.5rú
E.
-1.0
-1.2 -0-8 -0.¿ 0 o.L 0.8Theta ( radians)
-1.2 -0.9 -0.t,Theta
0 0-t 0.9 1-2 1.6 2.0(radians)
-2.0 -1.6
FIG. 5-2 EXPERII1ENTALLY DETERIV|INED COI{PONENTS OF Tr^lO
COUPLED IVIODES PERFECT CYLINDER
Mode (2,2)
Mode (2,3)
\
\\\ /
\ \ /\ \
\\-/
//\
0
cG,coo-Eot,
frt
;trc,(tlErltF
0
c0,coo-Eo(,
¡ó
þrú
É.
-0
0.6
0.3
0.3
6
-2.0 -1.6 -1.2
r.0
0.5
- 0.5
r.0
-2 .0 -1.6 -1.2
r4B.
0 0.¿ 0. B 1.2 1.6 2.0( radians )
-0. B -0.¿Theta
-0.8 -0.¿Theta
0 0.¿ 0.8 1 .2 1.6 2 .O( rad ians )
FIG. 5-3 REPOSIIIONING THE DRIVERS ALI1OST ELIIVIINATES
THE OT|1ER C()UPLTD IÏODE SHOl/lN IN FIG. 5-2,
Mode (2,2)
Mode (2,3)
---
r49.
íoning the electromagnets near the ends of the cylinder, the frequency
remaining unchanged. The relative level of the coupled ruode (2,3) issubstantial-ly lorver in the case of the radj-al component but uncertain
ín the tangential component. The principal mode (212) however is well
behaved. The raÈio of the tangentíal to radial amplitude (B/c) is0.436 which is ro be compared wj.th 0.503 predicÈed by the Arnold-
I{arburton theory for cylinders [126) for a pure mode. To be noted
also is the radj-al component which is in phase quadraËure spatía1ly
with respect to the tangential component, also predicted from the
pure mode theory.
Fig. 5-4 shows the components of two ot,her coupled modes
for the perfect cylinder at a frequency of 3332H2. For this case,
A was determined as 1.53 rad compared with the predicted vaLue n/2
t84:1. Again the principal mode i.s well behaved r^rith B/c = 0.397
compared with the theoretical value of 0.509 and again the rad.ial and
tangentÍal components are in spatial phase quadrature. The amplitudes
of the components of the coupled mode seem to be uncertain due to
their low rnagnitudes, a sítuatíon whích again could be improved ifmore data poinËs were available.
The analysis was also applíed to the dÍstorted cylinder
described ín chapter 4 for the prÍ.ncipal modes (2,2) and (2,3). Table
5-1 shows the anrplitudes of the coefficienEs determined by the
Generalized Least squares Èheory to be nearly identical to those
determined by the normal leasÈ squares procedure of chapter 3. More
Ínteresting i-s the value determined for A which, theoretÍcally, should
be zero or a multiple of n. As explained in section 2.L.4 (special
case 4) however the determination of A for these cases is bound to be
somewhat unctrrt-ain.
Tables 5-2(a), (b) and (c) show rhe effecr of ínírial
cotrooEo(,
.!c0,C'lcrÎtt-
5
0
0cocoo-Eo(,
rtt
Erú
0Ê.
0.6
0.3
-0.3
0
-0.6
-2 .0 -1 .6 -1.2 -0.8 -0./.Theta
1.0
0.s
-t .0
-2.0 -1 6 -1 .2 -0.8 -0 .4Thet a
150.
0 0.4 0.8 1.2 1.6 2.0( radians)
0 0.¿ 0.8 1.2 1.6 2.0( rad ians )
FIG. 5_4 COIÏPONENTS OF Tl'^lO COUPLED I/IODES PERFECT
CYLI NDER
Mode (3,2)
Mode (3,3)
r'/
//
/
ft
/ \
/\_ / \
151.
conditions on the final solution for three cases. The starting
magnitudes ôf the amplitude païameters are equal but are varied with
respect to the phase parameter. Hence the notation [1r.1] in Tables
5-2 is Èaken to mean [1,1r1r1r1r1r1r1r.1] which is the starting vector.
All other starting condítions trled but not shovrn in Tables 5-2
resulted in exactly similar solutions. In the case of iable 5-2 (a)
the optimum solution, oscillated beËween that of the second and third
col-umns. For Tables 5-2(b) and (c) Ít is clear that the hígher order
solution (represented by Ëhe larger residual I (fq)2) is incorrect
in the phase Â. Hence the opÈimum solution is that \rith the least
residual.
L52
TA3LE 5-1
COMPARISON OF GENERALIZED AND NORMAL LEAST SQUARES PROCEDURES IN THE
DETERMINATTON OF COMPONENTS 0F pRrNCrpAL IÍODES (2,2) AND (2,3) FOR
A DISTORTED CYLINDER
PARAMETER
PRINCIPAL MODE (2,2)
NI=2 N2=3PRINCIPAL MODE (2,3)
Nl=3 N2=4
GENERALIZED NORMAL GENERALIZED NORMAL
I\
2\
v
v
IN
2N
"'i,fr
"nl
zK
^
[ {rr) 2
-.07 4
.236
.77L
.023
183
-.o22
.008
.o52
2.4LL
.o82
-.062
.1s0
.784
.002
.272
-. 031
.008
-. 015
?T
.467
.633
-.032
-. 131
-.202
-.o28
081
-.r77
.002
2.702
.101
.683
-. 083
-.L67
-.258
-.002
-.070
-.L52
-.010
'lt
064
153.
T^BLE 5-2 (a)
EFFECT OF rNrTrAL CONDITIONS ON SOLUTTON PRTNCTPAL MODE (2,2) OF
UNDISTORTED CYLINDER Nl = 2, N2 = 3
PARAMETER
TNTTIAL CONDITTONS IRr -RB, *ul
[ 1,1] [.1,1] [1,10-s]
I\
5i
vi
vfr
-"i
''l,i
"K
A
[ {ra) 2
-.163
.025
1.037
.004
460
.003
.07s
-. 086
T.26L
.300
-.163 -. 165
-.084 026
1.037 1. 038
.o34 .022
.463 .4s4
-.oo2 .016
.070 .o75
.L70 .t26
4.693 4.s66
.283 .294
L Solution _1OscillaÈed
r'54.
TABLE 5-2(b)
EFFECT 0F rN-rTrAL CONDTTTONS ON SOLUTTON PRTNCTPAL MODE (2,2) OF
DISTORTED CYLINDBR Nl = 2, N2 = 3
PARAMETER
TNTTIAL CONDTTIONS [Rr - R8, Rs]
[1,1][1,10-s][.5,1]
[.1,1] [1,.1]
-tr[ .5 ,10 ']
[.1, .1]
4xfr
vi
v
\t
2N
IN
''fr
,i
"fr
^
| {rc) 2
-.o29
-.o27
.622
.236
.2r8
.r25
.038
-. 113
L.434
.254
-.07 4
.236
.77r
.023
.183
-.022
.008
.052
2.4LI
.082
rs5.
TABLE 5-2(c)
EFFECT OF TNTTIAL CONDTTIONS 0N SOLUTTON PRTNCIPAL MODE (2,3) OF
DISTORTBD CYLINDER Nl = 3, N2 = 4
PARAMETER
INITTAL CONDTTIONS [Rr-R8, R9]
[.5,10-s] [ .5,1] [.5, .5]
[5,10-s] [1,r]
LL,2l [ 1,3 ]
[1,]-0 "l
4u2,II
IvN
viItN
''ft
zi
"2N
A
[ {ra) 2
.466
.L67
-.097
.038
-. 038
.032
-.203
-. 178
4.86L or I.424
.593
.633
-.032
-. 131
-.202
-.028
081
-.r77
.002
2.7O2 or 3.581
.101
156.
APPENDIX I
FLOT,] DIAGIìAM OF MICROPROGRA}1 FOR DATA SYSTEM
NO
YES
s
STARIINITIALIZE PPI I,2
PRINTREPLAY
sr0P cÂssElf E
PR¡NI ..INIERCOM
vDl)
CHAR ¡NUSART I
CHECK FOR CHARlN usARl r (v0T)CHECK FOR CHAR
lN usaRl 2 lcDc)GEI I BYTE
FROM CÀSSEfIE
sEN0 fo vtJTECH0 l0 vDlPRINIEDII. PRINT
SENO OATASENO CHAR VIA
USART 2 TO CDCGEI CHÀR
FROM VOI
YES
SENO L/FIO VDI
6ET I BYIEFROM CASSETfEGET 1 BYIE
FROM CASSEIf E
EC80 t0 vDl ECH0 l0 v0T
WAII TILLL/F ECHOBY COC
GEI I EYTEFR0M CÀSSEtt
SENO CHAR VIAusaRl 2 l0 c0c
ts cHA C/R
ECHO tO VDI
SEND L/Ff0 vDl A C/R
IS CH
GEI CHFROM VOICAP E
GET CHARFROM VDI
SENO L/Ft0 vDt
SEND L/Fl0 vDt
PRINIRECORD
IS CHARA C/R
sEl cassEllE f0INC. REC. MOOE
E'I REGISTERB= 6SET REGISIER
s=7
PRINI CHARDECREMENI
REG B
NODATA AVÀILÀ8LEFROM IABLEÌ
CHECK FOR CHARIN USARI I (VDT)
CONVERI XTYTO OECIMAL
PRfNfIXXX,YYYY
SENO C/R I L/FI0 v0t
YES
NO
YES
A_
APPENDIX II
MATRIX FOR THE NORMAL EQUATIONS
Krsin(N1€r), Krcos(NItr), Srsin(N1Er), Srcos(NlEt) s- - - -,. Krsin(N2Er), Krcos(tt2tl), Srsín(N2Er), Srcos(tt2Er)
Krsin(Nl62), Krcos(Nlg2), srsin(NtEr), srcos(Nrt2) -,Krsin(Nzlr), Krcos(tt26z), srsin(Nz€r), srcos(N2E2)
Krsin(N18"), K.co"(u16t), srsin(NlEr), s.cos(NlEr)r - - - -, Krsio(li2Et), Krcos(N2Er), Srsín(N2Er), s"cos(N2Er)
I
Ht.¡l!
158.
APPEND]X ]TI
THE STRAIN ENERGY INTEGRAL
Evaluati-on of the strain energy integral, equation (4,24)
ín the main texÈ. There are 31 terms in the s¡rmmetric solut.ions
(denoted Stl, tt|, etc). The noËatÍon ís defined ín Èhe main text.
(denote these Stf, ta;, etc) and 31 terms in the asymmetric solutíons
n-., n j ln
p'h+p
x?u..u."(a aJ r_J¿ '
for srl, replace njrnor uy nflt
-l-
.n^ (h' -JJC P
',[ít"V
Iij op
for srl, replace nisP uv n!¿
srf = -'-u I l? u. _,w, , ( rr]r,] - n_ n- ¡ nj ølc, ijipq t r-J 1r¿' p q P q'
for sr!, replace njrnc ry nlfi
stl=t
+ [ wij.e,pqr
êstã = i:wilÀl(r'f n
k, IíiLp
+ hqr+ ) nj lPqr
pqr-h h h
êS.; V..V. ^nl-J Lv,
r.; =
hp ) njrP
for srf,, replace njsP ur nfs
-11 !r1j lp rJ
(h+p P
h ) nj¿P
for st!, replace nj uP uv -nfu
tt; = k.l"-"ri"ru(nl - h;) IIjøeaJ J¿p
for stf, reptace njsP ur n!¿
159.
vi¡wir,'3'fr (st"7
-L, f
ij rpq
st! = -'" Iíj ¿pq
h+p ¡ ¡j rn+
p
hh ) nirPqpq
hh ¡ ¡j snc
¡ ¡j rnl
hh j opqr)nqr
)r j lpqrp qr
h+q
hhp
for st|, replace nj'ona u" -n!T
I w . .w. ^n?n?(r,+rr+ n+ - n-ijipqrrJaJ¿JilPqr P
s.T,
(n+ r,+ r,+'p q r -h h hL
nvr.L
Ir.n*(tri -
faV. .V. ^n-n^ (h' h'
r_J aJ¿ J J¿' p q
wijwin'ît4ti -
If..V. ^n^ (h' h'LlLv,)Lpq
êstio
for srfr, replace njf'lc ty nll
s.; = -r4 Iij !.pq
=Lu I wÍj Î.pq
for stf, replace njl'lc o, npqje.
pp
hhPq
f or sr]o , reptace nj llc ry -n!sq
t\2
for stl' replace njllcr ur nfoot
I6 I r-J i9. ?n
JI^f
ij.{,pqr
a j lpqrfor St replace II bvn13t
v I U
ijLp ij í9.
for stl, , replace nj llcr tr -n
160.
ST
STst7
v4
sI8
SI I9
I5 2
,.¡ luoou':"'u
ui¡wiuÀi(n; - h-; ¡jrn
for stlu, replace nj llc by nll
-t- -t-u..v.^À.n-(h'h' - hr_J r-J¿ l_ J¿- p q
r?n^ (rr+ n+r-J¿ P q -hhv4
r? (n+ rr+r- pq
ttlo = å rrlno," r.i* ru^?"î,t; d nl - n
f.or star, replace nf u uv ni lP
s v tíj.0p
for stlu, reptace nj sP ur nfs
ttTu = T ¡,.r,? (rr+ n* - n- n- ¡ nj llca J¿'P q p q-
for stlr, replaee nj'cnc uy -n![
ST I iSuin
i1j-cpq
ij ¿pq
h ¡ nj snapq
) nj sPqI^l pq
for srf* replace nj lne ty -n![
-v4
I w..w. ^
ij ipq tJ LY'
j lpq- h h )rrqP
for stl' replace nj llc by n|¿q
h ) nj ¿Pqrpq hr
for stlo , replace nj'one r or nllt
t'l' = - å i:lno,"r:u, o^?nu,{tini - n; n;n; ¡ njrner
for stl' replace ¡jlncr ut -nflt
S
22(t - v)
B ,]unu':u'ullcr'f - r' )p
nfJ
St9,
161.
Sts23
uisÀi'¿(ni - r'-)n!ø
ui3ui.l,'3'ocnf - r'o > n?J
stäu =
ur:"ru^l"u (nj
Ionur:urur? cr'ir'f - r'- tro > n
replace nlf ur nj Î'nl
ui¡wi.tt5^r"u(4 d
IIu. .v. ^n.).. (h-h- - tr-t- )nf1r-J aJc J r. P q p q J)L
n
162.
pqpqrj1,r I)v..v. ^r? (t+h*h*
1Jr_Jtr_PqrS
sÉgt
for stlr, replac. n!;t by -njrPqr.
hhh_ (t - v) I6 ij l,pqr
t_63,
APPENDIX IV
COEFFICIENTS OF THE GENERALIZED STIF}'NESS AND MASS MATR]CES
Coefficients of matrix D. Denote Ds
and Da for aslnnmetric soluÈions.
k=lrM i=OrN j=grN
for symmetric solutions
P=OrP
The matríx is a 3M(N+ 1) square matríx. Put g = 3(k- 1) (n+ 1).
The non-zero coeffici-ents of matrix D are:
Ds[g+í+1, g+j+1]
= | {rflniir . a* i: ilPi: I rtf - nn IP
o"[g+í+1, g+ j +N+2]
(-v Àr. j n
t"[g+ í+ 1, g+ j + 2N+ 3]
+'l.1 (v Ànj nijpq + (1- v) Àuí nll I cnini - r;";, ]q
)ph=lt ijp (1- v)
2Àki nÏj ) (hl
hP
=l {"PI¡,. nliP(n+t(p )
->.I [*¿.vÀujz¡¡íirc - (1-v)Àuirrï,l](r;4 - r;r;, i
L64 "
o"[g+i+N+2, g+j+1]
=IP
)h-l-(h'p
r I [",trnijpq + (1-v) rf.nn]><nln+ - r'-no I
(-vI i niin l=U rn: npr, >It
k p
+%l (vruinijp( + (t-v)\irïl> <r,f ni - nnr;, Ì
SD [e+í+N+2, B*j +N+2]
t
o"[g+i+N+2,Eij+2N+3]
=lp
tt
(i iníjP . G;+ ç nl, > cr,l
*Iå(r5¡ijrcr + 2(L- v) À3 nllt l (r,+ n+ h* - h- h- h- ) IK r-J P q r P q r-J
)hp
I)
Ph=l
P t-t ntio <ni )
+[q
,, t(iiz +t+vrfli¡¡ijre + (r-v)rfr j nil r <nid- nnr,nl
- I + t(ij2 + vrflr¡nijnor + ze-v)rfr: n!rq'tcr'fr
t* h*qr htq-h h )I
p
165.
u"[g+i+2N+3, g+j+1]
)P
h=lP
v Àk rrÍjP (h+
Iq
(-{+ v Ào i2) niiPq -
o"[g+i+2N+3, g+ j +N+2]
- j nijp,nJ - nn ,
qhh-
p,2+
>,1çi2+i +vrfl j )niipa + (r-v)ifrtnTl r (4r'+ - r,nnol
)t-h h hq
(r - v) r,ni: nll} ,rtr, )
=Ip
+lp
- I + t(r2j +v rfri)nijpqr + 2(L- v)rfli nprf'rrr'ir'{ni
- h-h- )P 9.'
p
¡"[g+i+2N+3, g+j + 2N +3]
=lp
nriP(r,* - h )'p P'
+[q
*tåT
(,.GJz- i2 -v rfr ) nijPqcnjr'i
(ril + l2j2 + v rfr tr2 + jzl) ntjPq't
l ;,Ì-h h hqp
h*h*qr+ 2(L- v) rfr ij nl; I(h'
P
L66.
The asymnetric solutions Da are obtainecl by firstly changing the
i,j notatíon in the II funct.ions from subscript to superscript and
vÍce versa and changing the signs of the following groups of
coeffícíents
n"[g+i+1, c+j+N+2]
t"[g+i+N+2, g+j +1]
t"[g+Í+N+2, g+j+2N+3]
Dslg+i+2N+3, g+j+N+21
Sírnilarly, the coeffícients of matríces Qs and Qa which are also
3M(N+1) square matrices are as follows
Q"Ig+i+1, h+P
- n- ) nijPp
Q"[e+i+N+2, s+j+N+2] = I Cnl - rr^l n!.PppaJ
s+j+r1 = [(P
Q"[g+ i +2N+3, g+j +2N+ 3]
For the asyuunetríc coefficients, simply change the irj notation in
the II functions from subscrÍpt to superscript and vice-versa.
= I (t+ -h- ¡ niil;P p
APPENDIX V
Solution of the iI functions of the form shor^rn in equation
167.
(45.1)
(As. 2)
(4.0) of the text.
TíJ cos íX cos jX cos pX dXp t"o
=f +I +I +II
which are all zero unless
i+j*p=0
í-j-P=o
i+j-p=0
i-j*p=0
2n
32 4
fr= n12
T2= nf2
r, = t12
T, = r/2
,
,
t
,
il
whfch are all zero unless
Í+JtP=0
1-j-P=o
í+j-p=0
í-jfp=0
p cos pX sin iX sin jX dXijo
+ +I 3II=f +I
2 4
1, = -1/2
T, = *rl2
T, = -n/2
I, = *r/2
-ijpq -TI
which are all zero unless
cos iX cos jX cos pX cos qX dX
2 + +I I
=Q
=0
=0
=0
=Q
=0
-0
=Q
It /2
Tr /2
r/2
r/2
r/2
rl2
Tr /2
rl2
r+II t
1t
168.
(As. 3)
(As.4)
í+í+i+í+t--
1-
i-
l--
Í+i+f+l+i-
i-
1-
1-
j
j
j
J
j
j
J
j
+
+
+
+
P
p
p
p
p
p
p
p
+
+
+
+
q
q
q
q
q
q
q
q
,
,
t
t
,
t
Tz=
rt
rs
f=4
sin iX sín jX cos pX cos qXfi
I t
I5
ro
I
I7
I
'i;
+
which are all zero unless
+I2
=0
=0
=0
=Q
=0
=Q
=Q
=Q
dx
= -T/2
= -Tr 12
= -n/2
= -Tt /2
= +n/2
= *¡/2
-_ +r/2
= tr/2
+I I
j
j
j
j
j
j
j
j
+
+
+
+
P
P
P
p
p
P
p
p
+
+
+
+
q
q
q
q
q
q
q
q
I
2
3
4
I
I
I
ï
,
,
ru
6I
I
I7
I
2rilijPqt = i "o"
iX cos jX cos pX cos qX cos rX dXt
L69.
(A5. s)+
zero unless
j+p*q*
j-p-q-
j+p-q-
j-p*q*
j+p*q-
j-p-q*
j+p-q*
j-p-lq-
j-p+q*
j+p-q-
j-p-q-
J+p*q*j-p+q-
j+p-q*
j-p-q*
j+p*q-
I I I+2
+It6
which are all
1+
Í-
i+
i-
i+
i-
i+i-
i+1-
i+i-
i+i-
i+f-
r=0
r=0
r=0
r=0
r=0
r=0
r=0
r=0
r=0
r=0
r=0
r=0
r=0
r=0
r=0
r=0
I
I
I
I,4
T2
3
I
r/2
r/2
r¡ /2
r/2
r/2
t/2
r/2
Tr /2
r/2
rl2
rl2
Tt /2
rl2
rl2
¡/2
nl2
,
ru
r7
r8
r9
rro
ruT,,
f=13
I
I
I
t4
15
16
II Pqrij l'"sín iX sin jX cos pX cos qX cos rX dX
o
I1 2
+ I + ... + I
which are all zero unless
I6(As.6)
i+Í-
í+1-
l+
i-
Í+a-
i+
Í-
í+i-
í+i-
t+1-
170.
j
j
j
J
j
j
j
j
j
j
j
j
j
j
j
j
+
+
+
+
+
+
+
+
P
P
P
p
p
P
p
p
P
P
p
p
p
p
p
p
+
+
+
+
+
+
+
+
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
q
+
+
+
+
+
+
+
+
r
t
t
t
r
T
r
T
r
r
r
t
t
r
x
r
=0
-0
=0
=0
=0
=0
=0
=0
=0
=0
=0
=0
=0
=0
=0
=0
= -Tr /2
= *n/2
= -Tr/2
= *r/2
= -r/2
= *r/2
= -r/2
= *r/2
= -r/2
= *r/2
= -rl2
= *n/2
= -r/z
= *n/2
= -r/2
= *n/2
II
I
rs
2
4I
I
I
,
t
,
5
6
I I
9
rro
12
r7
I
rn
I
rrs
t4
t5
t6
I
I
I
t7r.
APPENDIX VI
SOI]ND MDIATION FROM CYLINDERS
The theory of tine-averaged holography has been developed ín
this thesis to such a sÈage that the víbraÈion components of an excite<l
arbitrary surface at a single frequency may be determined r¿ithout
difficulty. In particular the component normal to the surface ís of
ínËerest to the study of sound radiation from vibrating surfaces. In
this Appendix the radiatíon efficj-ency of three steel cylinders is
measured and compared with the theoretícal predictíons of Junger and
Feit [148] assuming normal mode shapes.
A6.1 THEORY OF SOIJND RADIATION FROM V]BRATING CIRCULAR CYLINDERS
The theory of sound radíation from círcular cylinders presented
here ís taken from Junger and Feit [148]. Consider a cylinder of length
L and mean radius ao wíth shear-diaphragm ends. The acceleratíon of
the vibration component normal to the surface may be expressed as a
combination of the normal modes
ii (x, e) I IÀI cos k X cos n0 (A6.1)mn mItrrI
M1lrrhere k* = ï , 0 is the círcumferential co-ordinate and X the co-ord-
inate along the generator measured from the body centre of the cylinder
as shown in Fig. A6-1.
The pressure field may be expressed in Èhe same form as Ehe
seríes
p I P R (r)cosk Xcosn0lnnmnmt(rrxr o)
ErD(tø.2¡
z
rR
-qz
X +Lt2
FIG, A5-1 CO-ORDINATE SYSTET! FOR A VIBRATING CYLINDER OF FINITI
Y
F!l.J
þ \ l¡
>(\\¿
--
LENGTH I^IITH CYLINDRICAL BAFFLE.
r73.
where P are coeffí-cienEs to be determined from the boundary condítionnn
(e0.:)f=âo
The radial harmonic n*r(r) ís found first by notíng that for steady
st.ate conditions Ít satísfies the three-dimensional HelrnholÈz equation
which reduces to
þ ,',x,0) = -p i; (x,o)
ð2 1a[_+àr2 r âr
n2
-+ (k
,2 '-uî>ìnr,(r)=o (A6.4)
(A6.6)
cosk X cosnO
This Ís Besselrs differential equaÈion r¿hose solutions are
l-inear combinations of Bessel functions of the fírst and second kind
and for outgoing vraves
n o(r) = Jrr[(k2 - ú^f.¿ + iyn t(k2 - uz^)'-"r]
= H- [ (k2 - x?)4t]n-' m'(A6. s)
r¿here H is Ëhe Hankel function of the fírst kind.n
Hence cornbining equations (46.5) and (A6.2) and substiÈuting
the general solution into the boundary condition, equation (A,6.3),
and solvíng for P,,, gir""
Hence Ëhe pressure field is
i,i--H-t(k2- u?i4'lnn n' m'Pr(r,x, o) = -p Im, n (k2 - k2)L'H' [ (k2 - U2^)%
^oJm
(ao. z¡
174.
Considering now a fínite number of stan<ling \raves cos knX confined to
the region lXl < L/2 and applyíng the stationary-phase approximarion
to the far field gives
pr (R, e,0)zp.ik\-(-1)'-tr î?l (kLcos þ /2) jm sl_n
=n k RsinQ (t<fr - t<2cos2q)
n-lI^I (-1) cosn0n
(A6. B)H'(k a sinQ)
n o
where cosine is used if rn is odd and sine ís used íf m is even.
The total sound po\¡rer II is obtained by integrating the radial
componenL of the sound intensity vector over a large sphere of radius
R and hence
xln
t{:::;(kL coso / z)lst-n-
sino I Hl(kaosinO) I
2 tr - k2cgs2$,
k2m
ocoso. w2 SI
.2n lr
t t P (R, e, o) 2sinq
do do (A6. e)
(A6.10)
and on substiÈuting equat.ion (46.8) ínto equation (46.9) gives
d0
I
f.or a single mode. Using the defínitíon for radiation efficíency lJ46l,
IIo= (ao. n¡
where . t¡2 t-- is the mean square velocity normal to the surface averagedST
over space and Èime and SO is the radiation area very nearly equal Èo
2raoL, and noting that
L7 5.
(A6. 12)
d0 (A6.13)
(%)
u2B.I^I2t
1ìS f
which resulÈs from averaging over two anguJ-ar co-ordínates and Lime,
then substituting equatíon (46.10) into equation (46.11) gives the
fína1 result also derived by Rennison t164]
16L
mn b,1T'm-a
, :î:: (K. L cosþ /2ao) la
K LcosQo a
aomïr
where K is the non-dimensional frequency parameter given by Ka0âo
d co
Ã6.2 E)GERIMENTAL ANALYSIS AND RESULTS
Three steel cylínders of length L = 39B.Bmm, mean radius
ao = 39.29nm and wal1 thÍcknesses 2.050rnm, 2.845mm and 4.597mm were
machined as in Fig. A6-2. Two stock cylínders of lengÈh 600mm and
thíckness 6umr acted as baffles and were machined aË one end to form
a thin ridge v¡hich fíts perfectly into the líp at the edge of each
cylinder thereby ensuring shear-díaphragm end conditíons. Four 6rnm
diameter supporting rods held the structure in place with a 1ittle
tensíon from springs mounted on one end as shown in Fíg. L6-2. Pieces
of mineral fibre were stuffed into the ends of the baffles to ensure
an anechoic termination and eliminate- the possibílity of standing r^raves
in the medíum insido the strucËure. AlÈhough the theory outlined in
the last section requires infjnitely long baffles (see Fig. A6-2) the
structure is at leasÈ five wavelengths in dimension at the lowest
frequency considered and hence the error is assumed neglígible. A
sinþ lH' (K"sinþ) I 2 tr - { \2
Cyl i nder Supporting rodSpring
Anechoictermination
Baf f le Sh ear - d iaphragmsupport
Clampedring
FIG. A6-2 SCHE|IATIC CROSS-SECTION OF CYLINDER AND BAFFLES,
F!o\
L77.
photograph of the apparatus is shown in Fig. A6-3.
The cylínder and supportíng structure were placed in a
reverberation room of volume VR = 181.5m3 and surface area \ = L95m2.
The cylinrler was excited usíng the modal driving system described
in Sectíon 2.2.2 and the acceleration monitored usÍ-ng a second B & K
spectromel-er and acceleroneËer.
The sound level in the reverberation room was measured
usíng a B & K %t' mícrophone type 4133 and, 2IO7 spectronìeter connected
to a daËa acquisitíon system [165]. The microphone scans the rever-
l¡eration room using a traverse system and the sound field ís furÈher
diffused by a rotaÈing vane assembly (see Fig. A6-4). The data
acquisiËion sysÈem samples and sÈores the ouÈpuÈ of the 2107 spectro-
meter at íntervals of 1 second and at, the end of one traverse of the
room calculates the mean and sÈandard devíation.
The reverberatíon time TuO of the room \,ùas measured ín the
normal way in Èhírd-ocËave bands and hence Èhe Power Level calculated
from [166]
t, = tO * 101ogVR - 101oBTuO f ^A- c
10 log (1+ J-9¡ - 13.s (A6.14)SfvR
where L_ in Éhe sound level averaged over space and tíme determinedp
usÍng the data acquisitíon system. Hence, usi-ng equation (46.11) the
experimental radiatíon efficiency Ís
f2 10(h- rzo)/ro
c = 7.545 (A6.ls)!t
where tr{ is the peak acceleration of the cylinder surface at a radial
2
antinode.
I
FIG. A6-3
Cylinder and supporting structure.
H!
,]
I
FIG. A6-4
The reverberation room, rotating vane
and mierophone traverse.
F{\o
180.
Figs. A6-5(a) - (e) show the experimental results together
with the ÈheoreÈical curves calculate.d from equation (46.13) using
numerical integratíon. In general the experímental data are 5-10d8
down from the theoretical curves which canrìot be explaíned. To eliminate
the possíbility that the baffles are too sma1l they were completely
removed and new data taken. As expected, sound radíation from Èhe
lower order modes was dramatícally reduced but for the higher order
modes Èhe results r^rere identical . The error is noÈ ín the frequency
parameter K" as measured frequencies are less than 57" in error from
theoretically determined frequencíes.
However, assuming that Ëhe problem ís resolved then the
radiatíon efficiency of the distorted pipe may be calculated using
the solution presented ín Chapter 4 and compared with measured values.
1Bl.
Í
10
10
10
10
l0
0
-l
-2
-3
-l-t0
-Rl0 "
1 0-2 10 -l 010
Ka
FIG. A6-5 (') P.ADIATION IFFICIENCY OF STEEL CYLINDERS.
Theoretical curves. o, E , A Experimental points fox n= 2, 3
and 4 respectivel;,
l0
n=2
o
o
n=3
m=1
f'¡ =1
DA
n= t,
A
Cr
t0
r0
10
I
0
-l
-2
-3
-4
-5
r82.
10
0I
10
10
1 0
10-2 0
0I 10K¿
FIG, A6-5 (t) RADIATI0N EFFICITNCY 0F STEEL CYLINÐERS,
- Theoretical curves. o, E , A Experimental points fot n= 2, 3
and 4 respectívely.
n=2 n=3
oo
o
n=4
A
A
m =2
n=1
18 3.
01
10
100
-2
-3
-1.
T
10
10
10
-E10 "
10 -2 10 -t 100 10Ka
FIG. A6-5 (") RADIATION EFFICIENCY OF STEEL CYLINDERS.
Theoretical curves. fJ, A Experimental points for n=3 and
I
n=¿
AA
n=2 n=3
4 respectively.
184.
cr
10
10
r0
10
0
-t
-2
10
l0
10
3
t,
-5
10 1 0-l 102 0
10Ka
FIG, A6-5 (¿) RADIATI0N EFFICIENCY 0F STEEL CYLINDTRS,
Theoretical curves. !rA Experimental point.s for n= 3 and 4
n=2 n=3 It= l+
A
^
respectively.
10
10
I
0
l8s.
t0
CT
1 0-l
1trz
I 0-3
-tl0 "
-E't0 "1 o-2 10-{ 010
Ka
FIG. A6-5 (.) RADIATION EFFICIENCY OF STEEL CYLINDERS.
TheoreÈical curves. n , A Experímental poínts for n = 3
and 4 respectively.
n=2 n=3 n= l+
A
Tonin, R., and Bies, D.A., (1977) Time-averaged holography for the study of three-
dimensional vibrations.
Journal of Sound and Vibration, v. 52 (3), pp. 315-323.
NOTE:
This publication is included on pages 186-194 in the print copy
of the thesis held in the University of Adelaide Library.
It is also available online to authorised users at:
http://dx.doi.org/10.1016/0022-460X(77)90562-4
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