MODAL ACOUSTIC RADIATION CHARACTERISTICS
OF A THICK ANNULAR DISK
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Hyeongill Lee, M.S., B.S.
*****
The Ohio State University
2003
Dissertation Committee:
Approved by
Dr. Rajendra Singh, Adviser
Dr. Donald R. Houser
_____________________________
Dr. Ahmet Selamet
Dr. Robert G. Parker
Adviser
Department of Mechanical Engineering
ABSTRACT
New analytical and semi-analytical solution procedures for modal sound radiation
from a thick annular disk are proposed. Classically, thin annular or circular plate theory
has been used to describe sound radiation from normal surfaces while ignoring
contributions from the radial surfaces. But, in many practical cases, the disk thickness is
often beyond the thin plate theory limit and consequently a thick plate structural and
acoustic formulation must be employed, as illustrated in this study. Also, radiation from
in-plane vibration must be considered along with that from out-of-plane vibration to
properly estimate the total sound radiation. First, we consider purely modal radiations
from a disk with free-free and fixed-free boundaries. A new analytical formulation, based
on the thick plate theory, is proposed for radiation from out-of-plane flexural modes.
Further, the far-field sound pressures from in-plane radial vibration modes are obtained
by using two alternate analytical methods based on the Rayleigh integral technique and a
cylindrical radiator model. Analytical predictions are confirmed with measured data (with
free-free boundaries only) as well as computational results (with both sets of boundaries)
from finite element and boundary element codes in terms of structural eigensolutions,
accelerance, acoustic response function spectra, modal sound pressures in far-field and
modal directivity patterns. Selected parametric studies investigate the effects of disk
ii
geometry and vibrating frequencies on the radiation properties. Second, vibro-acoustic
response for a multi-modal case, given a multi-directional harmonic force, is formulated
based on the modal expansion technique. The analytical method employs the structural
eigensolutions (from an analytical or numerical method), measured damping ratios and
new analytical modal radiation solutions. This method is confirmed by comparing
predictions of acoustic frequency response function, sound power and radiation
efficiency spectra with those obtained using purely computational methods. The effects
of coupling between structural modes (gap between their natural frequencies) and
circumferential separation between two force excitation locations are investigated.
Finally, as an example, modal and multi-modal sound radiations from a simplified brake
rotor are expressed in terms of the characteristics of a generic thick annular disk having
identical geometric dimensions. Coupling between in-plane and out-of-plane vibration
modes that is introduced by the hat structure and boundaries of rotor is also investigated.
Accuracy of our semi-analytical method is confirmed by purely numerical analyses based
on finite element and boundary element models.
iii
DEDICATION
Dedicated to my parents
for their love and support throughout my life
iv
ACKNOWLEDGMENTS
I would first like to thank my advisor, Dr. Rajendra Singh for his continuous
support, patience, and guidance throughout this work. I would also like to thank Dr.
Donald Houser, Dr. Ahmet Selamet and Dr. Robert Parker for their careful examination
on this dissertation and kind suggestions and comments throughout this study.
I thank my fellow graduate students in Acoustics and Dynamics Laboratory for all
their assistance during this research. I also thank all Korean students in Mechanical
Engineering department at the Ohio State University for their advice and help.
Finally, I would like to thank my family especially my wife, Ji-Young, for her
continuous love and support. Without her support, this study would not have been
possible.
v
VITA
October 27, 1964 ............................... Born – Korea
1986 ................................................... B.S. Mechanical Engineering,
Seoul National University, Seoul, Korea
1993 ................................................... M.S. Mechanical Engineering,
The University of Toledo
1986 - 1998........................................ Hyundai Motor Company,
1999 - present .................................... Graduate Research Associate,
The Ohio State University
PUBLICATIONS
Journal Publications
1. H. Lee and R. Singh, “Acoustic radiation from radial modes of a thick annular disk”,
submitted to the Journal of Sound and Vibration, 2002.
2. H. Lee and R. Singh, “Acoustic radiation from out-of-plane modes of an annular disk
based on thick plate theory”, submitted to the Journal of Sound and Vibration, 2002.
Conference Presentations
1. H. Lee and R. Singh, “Vibro-acoustic analysis of a thick annular disk,” The 6th U.S.
National Congress on Computational Mechanics, 1 - 3 August 2001 Dearborn,
Michigan.
vi
2. H. Lee and R. Singh, “New calculation strategy for acoustic radiation from a thick
annular disk” The 143rd Meeting of the Acoustical Society of America, 3 - 7 June
2002 Pittsburgh, PA. Abstract published in J. Acoust. Soc. Am.,111(5-2), p. 2474,
May 2002.
3. H. Lee and R. Singh, “Vibro-Acoustics of a Brake Rotor with Focus on Squeal
Noise” The 2002 International Congress and Exposition on Noise Control
Engineering, paper # N613, Aug. 19 – 21, 2002 Dearborn, Michigan.
4. H. Lee and R. Singh, “Sound Radiation from a Disk Brake Rotor Using a SemiAnalytical Method” The 2003 SAE N&V Conference, 03NVC-229, May 5-8, 2003
Traverse City, Michigan.
FIELDS OF STUDY
Major Fields: Mechanical Engineering
Structural Dynamics and Vibration.
Vibro-Acoustics and Noise Control.
vii
TABLE OF CONTENTS
ABSTRACT ........................................................................................................................ii
DEDICATION ...................................................................................................................iv
ACKNOWLEDGMENTS................................................................................................... v
VITA ................................................................................................................................vi
LIST OF TABLES .............................................................................................................xi
LIST OF FIGURES..........................................................................................................xiii
LIST OF SYMBOLS ........................................................................................................xx
Chapter ..................................................................................................................................
1
Introduction ............................................................................................................. 1
1.1 Motivation ......................................................................................................... 1
1.2 Literature Review .............................................................................................. 1
1.3 Problem Formulation......................................................................................... 5
1.4 Organization ...................................................................................................... 9
REFERENCES FOR CHAPTER 1....................................................................... 12
2
Acoustic radiation from out-of-plane modes using thin and thick plate
theories .................................................................................................................. 16
2.1 Introduction ..................................................................................................... 16
2.2 Structural Analysis Based on Thick and Thin Plate Theories......................... 21
2.3 Acoustic Radiation Calculation....................................................................... 39
2.4. Computational and experimental Investigations of Sound Radiation............ 46
2.5 Effect of key Parameters on the Modal Sound Radiation ............................... 52
viii
2.6 Conclusion....................................................................................................... 61
REFERENCES FOR CHAPTER 2....................................................................... 62
3
Sound radiation in-plane vibration........................................................................ 64
3.1 Introduction ..................................................................................................... 64
3.2 Problem Formulation....................................................................................... 66
3.3 Structural Analysis .......................................................................................... 67
3.4 Acoustic Radiation Model............................................................................... 76
3.5 Method I: The Rayleigh Integral Approach .................................................... 78
3.6 Method II: Cylindrical Radiator ...................................................................... 81
3.7 Modal Radiation Results ................................................................................. 87
3.8 Parametric Studies........................................................................................... 96
3.9 Conclusion..................................................................................................... 103
REFERENCES TO CHAPTER 3 ....................................................................... 105
4
Multi-modal vibro-acoustic response.................................................................. 107
4.1 Introduction ................................................................................................... 107
4.2 Assumptions and Objectives ......................................................................... 109
4.3 Vibro-Acoustic Responses to a Harmonic Excitation................................... 109
4.4 Effects of Structural and Acoustic Modal Coupling on the Acoustic
Radiation ....................................................................................................... 127
4.5 Conclusion..................................................................................................... 133
REFERENCES FOR CHAPTER 4..................................................................... 134
5
Application to a brake rotor ................................................................................ 135
5.1 Introduction ................................................................................................... 135
5.2 Objectives and Assumptions ......................................................................... 137
5.3 Structural Modal Analysis............................................................................. 137
5.4 Sound Radiation from Structural Modes of Brake Rotor.............................. 140
ix
5.5 Vibro-Acoustic Response to a Multi-Directional Harmonic Force .............. 146
5.6 Conclusion..................................................................................................... 151
REFERENCES FOR CHAPTER 5..................................................................... 152
6
Conclusion........................................................................................................... 154
6.1 Summary ....................................................................................................... 154
6.2 Contributions................................................................................................. 157
6.3 Future Research............................................................................................. 158
BIBLIOGRAPHY ........................................................................................................... 160
x
LIST OF TABLES
Table
Page
1.1
Geometric dimensions and material properties of the example disks..................... 7
2.1
Disk examples with free-free or fixed (r = b) - free boundaries ........................... 18
2.2
List of models developed and methods employed for Disk I................................ 20
2.3
Eigensolutions for Disk I, II and III with free-free boundaries............................. 30
2.4
Measured modal damping ratios for Disk I........................................................... 31
2.5
Eigensolutions of Disk I with fixed-free boundary conditions ............................. 34
2.6
Comparison of directivity patterns for selected modes of Disk I.......................... 50
2.7
Comparison of modal acoustic power and radiation efficiency levels for
selected out-of-plane modes of Disk I................................................................... 51
2.8
Modal acoustic powers and radiation efficiencies for first four out-ofplane modes with fixed-free or free-free boundaries. ........................................... 59
3.1
Disk dimensions and material properties .............................................................. 69
3.2
Comparison of disk eigen-solutions for radial modes........................................... 74
3.3
Modal displacement amplitudes used for acoustic analysis.................................. 74
3.4
Comparison of directivity patterns for selected modes ......................................... 92
xi
3.5
Comparison of modal acoustic power and radiation efficiency levels.................. 94
4.1
Self and mutual radiation terms of sound power between elastic modes of
the sample disk .................................................................................................... 128
5.1
Geometric dimensions and material properties of the brake rotor ...................... 137
5.2
Selected structural modes of the brake rotor ....................................................... 139
5.3
Sound powers and radiation efficiencies for selected modes.............................. 145
xii
LIST OF FIGURES
Figure
1.1
Page
Thick annular disk example cases a) with free – free boundaries b) with
fixed – free boundaries c) disk with hat structure ................................................... 6
2.1
A thick annular disk with an excitation in the normal direction ........................... 17
2.2
Comparison of Disk I mode shapes given free – free boundaries. (a) (1, 0)
mode; (b) (0, 2) mode. Key:
plate theory (Model C);
2.3
WKLFNSODWHWKHRU\0RGHO$
WKLQ
ILQLWHHOHPHQWPHWKRG0RGHO' ...................... 28
/f(ω), acoustic
Vibro-acoustic experiment used to measure structural w
frequency response functions P/f(ω) and out-of-plane modal sound
radiations. .............................................................................................................. 29
2.4
/f(ω) at r = 151.5 mm and ϕ = 180° for Disk I
Structural accelerances w
with free-free boundaries. Key: - - -, computed using FEM (Model D); –––
–, measured (E). .................................................................................................... 32
2.5
Effect of geometry on the (0, 2) and (0, 3) modes of Disk I with free-free
boundaries. (a) Effect of radii ratio (β); (b) Effect of thickness ratio ( h ).
Key:
WKLFN SODWH WKHRU\ 0RGHO $
WKLFN SODWH WKHRU\ ZLWKRXW
rotary inertia effect (Model B); - - -, thin plate theory (Model C); measured (E). ........................................................................................................ 35
xiii
2.6
Effect of geometry on the (1, 0) and (1, 1) modes of Disk I with free-free
boundaries. (a) Effect of radii ratio (β); (b) Effect of thickness ratio ( h ).
Key:
WKLFN SODWH WKHRU\ 0RGHO $
WKLFN SODWH WKHRU\ ZLWKRXW
rotary inertia effect (Model B); - - -, thin plate theory (Model C); measured (E). ........................................................................................................ 36
2.7
Effect of geometry on the (0, 2) and (0, 3) modes of Disk I with fixed-free
boundaries. (a) Effect of radii ratio (β); (b) Effect of thickness ratio ( h ).
Key:
WKLFN SODWH WKHRU\ 0RGHO )
WKLFN SODWH WKHRU\ ZLWKRXW
rotary inertia effect (Model G); - - -, thin plate theory (Model H)........................ 37
2.8
Effect of geometry on the (1, 0) and (1, 1) modes of Disk I with fixed-free
boundaries. (a) radii ratio (β); (b) thickness ratio ( h ). Key:
mode with thick plate theory (Model F); ∆∆∆, (0, 0) mode with thick plate
theory without rotary inertia effect (Model G);
plate theory (Model H);
PRGHZLWKWKLQ
PRGHZLWKWKLFNSODWHWKHRU\0RGHO
F); ◊◊◊, (0, 1) mode with thick plate theory without rotary inertia effect
(Model G); - - -, (0, 1) mode with thin plate theory (Model H). ........................... 38
2.9
Sound Radiation from the out-of-plane vibration modes in the sherical
coordinate system. (a) a thin disk with an infinite baffle; (b) a thick disk
without baffle. ....................................................................................................... 41
2.10
Acoustic frequency response function P/f(ω) given unit impulsive force
excitation f(t) in the z direction at r =151.5 mm. (a) θ = π/2 and φ = 0; (b)
θ = 0 and φ = 0. Key: –––, analytical calculation (Model L); - - -,
PHDVXUHG+HUHT LVWKHUDGLDO
computed using BEM (Model M);
mode.
2.11
48
Directivity pattern Dmn(θ) given φ = 0 and R = 303 mm. (a) m = 0, n = 2
DQDO\WLFDOPHWKRGEDVHGRQWKLFN
mode; (b) m = 0, n = 3 mode. Key:
plate theory (Model J); ∆∆∆, analytical method based on thin plate theory
xiv
(Model K); - - -, semi-analytical method (Model L); , computed using
BEM (Model M);
2.12
PHDVXUHG ........................................................................ 49
Effect of radii ratio on the modal sound radiation based on alternate plate
2
theories. (a) Spatially averaged mean-square velocity < w > t , s ; (b)
Radiation efficiency σ. Key:
(Model J);
PRGH ZLWK WKLFN SODWH WKHRU\
PRGHZLWKWKLQSODWHWKHRU\0RGHO.
mode with thick plate theory (Model J); - - -, (0, 3) mode with thin plate
theory (Model K)................................................................................................... 53
2.13
Effect of radii ratio on the modal sound radiation based on alternate plate
2
theories. (a) Spatially averaged mean-square velocity < w > t , s ; (b)
Radiation efficiency σ. Key:
(Model J);
PRGH ZLWK WKLFN SODWH WKHRU\
PRGHZLWKWKLQSODWHWKHRU\0RGHO.
mode with thick plate theory (Model J); - - -, (1, 1) mode with thin plate
theory (Model K)................................................................................................... 54
2.14
Effect of thickness ratio on the modal sound radiation based on alternate
2
plate theories. (a) Spatially averaged mean-square velocity < w > t , s ; (b)
Radiation efficiency σ. Key:
(Model J);
PRGH WKLFN ZLWK SODWH WKHory
PRGHZLWKWKLQSODWHWKHRU\0RGHO.
mode with thick plate theory (Model J); - - -, (0, 3) mode with thin plate
theory (Model K)................................................................................................... 56
2.15
Effect of thickness ratio on the modal sound radiation based on alternate
2
plate theories. (a) Spatially averaged mean-square velocity < w > t , s ; (b)
Radiation efficiency σ. Key:
(Model J);
PRGH ZLWK WKLFN SODWH WKHRU\
PRGHZLWKWKLQSODWe theory (Model K);
mode with thick plate theory (Model J); - - -, (1, 1) mode with thin plate
theory (Model K)................................................................................................... 57
xv
2.16
Modal directivity patterns of Disk I with alternate boundary conditions. (a)
n = 2 modes; (b) n = 0 modes. Key: –––, fixed-free; - - -, free-free
boundary condition................................................................................................ 60
3.1
A thick annular disk with a radial force ................................................................ 68
3.2
L(λ) for q = 2 and q = 3 modes. Key: –––, q = 2 mode; - - -, q = 3 mode. ........... 73
3.3
Comparison of radial mode shapes Key: solid line, analytical solution
given by equation (3.13); discrete point, finite element analysis.......................... 73
3.4
Structural frequency response functions ü/f(ω) at j = 0. Key: ––––,
measured; - - - -, computed using BEM. ............................................................... 75
3.5
Spherical sound radiation from a vibrating disk. .................................................. 77
3.6
Sound Radiation from the radial vibration of a thick annular disk in
spherical coordinate system. ................................................................................. 79
3.7
Cylindrical radiator of length h using cylindrical coordinate system. .................. 82
3.8
Vibro-acoustic experiment used to measure structural u f (ω) and
acoustic P/f(ω) frequency response functions and in-plane modal sound
radiation. 88
3.9
Acoustic frequency response functions P/f(ω). (a) θ = π/2 and φ = 0; (b) θ
= 0 and φ = 0. Key: –––, measured; - - -, computed using BEM. ......................... 90
3.10
Acoustic frequency response functions P u (ω) . (a) θ = π/2 and φ = 0; (b)
θ = 0 and φ = 0. Key: –––, measured; - - -, computed using BEM. ...................... 91
3.11
Directivity pattern Dq(θ) given φ = 0 and R = 303 mm. (a) q = 2 mode; (b)
q = 3 mode. Key: - - -, Rayleigh integral method; , cylindrical
radiator method; – - –, computed using BEM; •, measured.................................. 93
xvi
3.12
Directivity pattern Dq(φ) given θ = π/2 and R = 303 mm. (a) q = 2 mode;
(b) q = 3 mode. Key: –––, Analytical and numerical methods; • • •,
measured................................................................................................................ 95
3.13
Effect of radii ratio on the amplitude ratio and phase difference between
PqI and PqO. (a) q = 2 mode; (b) q = 3 mode. Key: ο ο ο, Phase difference;
∆ ∆ ∆, Amplitude Ratio......................................................................................... 98
3.14
Effect of radii ratio on Πq and σq. (a) q = 2 mode; (b) q = 3 mode. Key:
, Acoustic power; ο ο ο, Radiation efficiency. ............................................. 99
3.15
Effect of ω on Πq and σq. (a) q = 2 mode; (b) q = 3 mode. Key: ,
Acoustic power; ο ο ο, Radiation efficiency. ..................................................... 101
3.16
Effect of thickness ratio on Πq and σq. (a) q = 2 mode; (b) q = 3 mode.
Key: , Acoustic power; ο ο ο, Radiation efficiency.................................... 102
4.1
Sound radiation from a vibrating thick annular disk........................................... 110
4.2
Vibro-acoustic experimental setup used to measure structural frequency
/f(ω) or u /f(ω) and acoustic P/f(ω) frequency
response functions w
response functions. .............................................................................................. 113
4.3
Structural frequency response functions with free-free boundaries
/f(ω) at r = 0.1515 and ϕ = 180°; (b) ü/f(ω) at ϕ = 0. Key: ,
(a) w
measured; - - -, computed using FEM; –––, analytical calculation..................... 115
4.4
Acoustic frequency response functions P/f(ω) due to radial excitation. (a)
θ = π/2 and φ = 0; (b) θ = 0 and φ = 0 Key: , measured; - - -, computed
using BEM; –––, analytical calculation. ............................................................. 117
4.5
Acoustic frequency response function P/f(ω) given impulsive force
excitation f(t) at r = a in the z direction. (a) θ = π/2 and φ = 0; (b) θ = 0
xvii
and φ = 0. Key: , measured; - - -, computed using BEM; –––,
analytical calculation........................................................................................... 118
4.6
Far-field sound pressure spectra P(ω) due to multi-modal excitation. (a) θ
= π/4 and φ = 0; (b) θ = 0 and φ = 0. Key: ––––, analytical calculation; - - , computed using BEM........................................................................................ 120
4.7
Acoustic radiation functions due to combined harmonic excitation. (a)
acoustic power spectra Π(ω); (b) radiation efficiency spectra σ(ω). Key: –
––, analytical calculation; - - -, computed using BEM........................................ 121
4.8
Example cases for calculation of vibro-acoustic responses due to multipoint excitations .................................................................................................. 124
4.9
Far-filed sound pressure spectra due to two identical harmonic forces with
specific circumferential distances. (a) P(ω) at R = 303 mm, φ = 0, θ = π/4;
(b) P(ω) at R = 303 mm, φ = 0, θ = π/2. Key: - - -, ∆ϕ = π/12; , ∆ϕ =
π/6;.
4.10
, ∆ϕ = π/4; –––, ∆ϕ = π/3. ................................................................... 125
Acoustic radiation functions due to two identical harmonic forces with
specific circumferential distances. (a) acoustic power spectra Π(ω); (b)
radiation efficiency spectra σ(ω). Key: - - -, ∆ϕ = π/12; , ∆ϕ = π/6; , ∆ϕ = π/4; –––, ∆ϕ = π/3................................................................................... 126
4.11
Effect of natural frequency separation on P/f(ω). (a) θ = π/2 and φ = 0; (b)
θ = 0 and φ = 0. Key: , modified; - - -, original case. .................................. 131
4.12
Effects of natural frequency separation on acoustic radiation functions. (a)
acoustic power frequency response functions Π/f(ω); (b) radiation
efficiency function σ(ω). Key: , modified; - - -, original case...................... 132
5.1
A thick annular disk with a hat structure simulates the brake rotor. Disk is
clamped at the inner bolts and free at outer edge. ............................................... 136
xviii
5.2
Finite element model of the brake rotor with 2010 solid elements..................... 138
5.3
Directivity patterns for selected modes ............................................................... 143
5.4
Finite element model used for forced vibration analysis given a multidirectional harmonic force................................................................................... 148
5.5
Far-field sound pressure spectra p/f(ω) for selected receiver positions rp1
and rp2. Key: Analytical
5.6
&RPSXWHGXVLQJ%(0----.................................. 149
Acoustic power spectra Π(w) and radiation efficiency spectra σ(w) of the
brake rotor. Key: Analytical
&RPSXWHGXVLQJ%(0----. .......................... 150
xix
LIST OF SYMBOLS
LIST OF SYMBOLS FOR CHAPTER 2
a
outer radius of annular disk (mm)
b
inner radius of annular disk (mm)
B
Hankel transform
c0
speed of sound in the acoustic medium (m/s)
Db
flexural rigidity of disk (Nm)
E
Young’s modulus of disk (N/m2)
f(t)
dynamic force on disk (N)
F
amplitude of applied force (N)
i
−1
k
acoustic wave number (rad/m)
h
disk thickness (mm)
h
disk thickness ratio (h/a)
m
number of nodal circles in the disk
Mr, Mϕ, Mrϕ
ending moment in the disk (Nm)
n
number of nodal diameters in the disk
p
far field sound pressure (Pa)
P
spatially-dependent far field sound pressure amplitude(Pa)
Pmn
far field sound pressure amplitude due to the (m, n)th out-of-plane mode
(Pa)
R
radius of sphere at the far-field location (m)
Qr, Qϕ
shear forces in the disk (N)
r, φ, z
cylindrical coordinates
R, θ, φ
spherical coordinates
xx
rf
position vector of the excitation force f(t) on the disk
rp
position vector of a receiver position sound pressure
rs
position vector of a sound source position
Ss
surface of the sound source
Sv
boundary surface of the acoustic control volume
[TTHICK]
characteristic matrix for the thick plate theory
[TTHIN]
characteristic matrix for the thin plate theory
V
acoustic control volume
w
transverse velocity in the disk (m/s)
w
transverse acceleration in the disk (m/s2)
W
spatial dependent transverse displacement in the disk (m)
W
spatial dependent transverse velocity in the disk (m/s)
β
radii ratio of the annular disk
γ
angle between the surface normal vector and the vector from source
position to receiver position (rad)
∆θ
increment in the cone angle θ (rad)
λmn
dimensionless structural eigenvalue for the (m, n)th flexural mode
ν
Poisson’s ratio of disk
Π
acoustic power from the disk vibration (W)
Πmn
acoustic power from the modal vibrations of the disk (W)
ρ0
mass density of the acoustic medium (kg/m3)
ρd
mass density of the disk (kg/m3)
σmn
sound radiation efficiency of normal modes of the disk
ϕ
azimuthal angle of the disk (rad)
∆φ
increment in azimuthal angle φ (rad)
ℜ
radiation resistance of disk
Φmn
flexural mode shape of the disk
Ψr, Ψϕ
bending rotations of the disk (rad)
ψr, ψϕ
spatially-dependent bending rotations of the disk (rad)
ω
angular frequency (rad/s)
ωmn
natural frequency of the (m, n)th out-of-plane mode (kHz)
xxi
ζmn
modal damping ratio of the (m, n)th out-of-plane mode (%)
Subscripts
d
disk
m, n
out-of-plane mode indices
0
acoustic medium
p
observation point in a far-field location
r
radial direction of the disk
s
source (radiator)
ϕ
circumferential direction of the disk
Abbreviations
BEM
boundary element method
FEM
finite element method
LIST OF SYMBOLS FOR CHAPTER 3
a
outer radius of annular disk (mm)
b
inner radius of annular disk (mm)
c0
speed of sound in the acoustic medium (m/s)
D
extensional rigidity of disk (N/m)
Db
flexural rigidity of disk (Nm)
E
Young’s modulus of disk (N/m2)
f(t)
dynamic force on disk (N)
F
amplitude of applied force (N)
g
free space Green’s function (m-1)
i
−1
k
acoustic wave number (rad/m)
kz
structural wave number in z direction (rad/m)
xxii
h
disk thickness (mm)
h
disk thickness ratio (h/a)
Hn
Hankel function of order n
m
number of nodal circles in the disk
n
number of nodal diameters in the disk
Nr
normal force in the radial direction (N)
Nϕ
normal force in the tangential direction (N)
Nrϕ, Nrϕ
shear forces in the radial direction (N)
N r , N rϕ
normalized forces in the radial and shear direction
p
far field sound pressure (Pa)
P
spatially-dependent far field sound pressure (Pa)
pq
far field sound pressure amplitude due to qth radial mode (Pa)
q
radial mode index
R
radius of sphere at far-field location (mm)
r, φ, z
cylindrical coordinates
R, θ, φ
spherical coordinates
rf
position vector of force f(t) on the disk
rp
position vector of sound pressure
rs
position vector of source
Ss
surface of sound source
Sv
boundary surface of acoustic control volume
[T(ξ)]
transfer matrix
u
radial displacement (mm)
uq
radial displacement for the qth radial mode (mm)
u
radial velocity of disk (m/s)
u
radial acceleration of disk (m/s2)
U
spatially-dependent radial displacement of disk (m/s)
U
spatially-dependent radial velocity of disk (m/s)
U
spatially-dependent radial acceleration of disk (m/s2)
[U(ξ)]
utility matrix
u
amplitude of radial velocity (m/s)
xxiii
u
amplitude of radial acceleration (m/s2)
v
circumferential displacement (mm)
V
acoustic control volume
Z(z)
variation of surface acceleration in z direction
β
radii ratio of annular disk
γ
angle between surface normal vector and vector from source point to
receiver position
∆θ
increment in cone angle θ (rad)
λq
dimensionless structural eigenvalue for the qth radial mode
λ2mn
dimensionless structural eigenvalue for the (m,n)th flexural mode
ν
Poisson’s ratio of disk
ξ
dimensionless radial coordinate for disk
Πq
acoustic power from the qth radial mode (W)
ρ0
mass density of acoustic medium (kg/m3)
ρd
mass density of disk (kg/m3)
σq
sound radiation efficiency of the qth radial mode
σ rr
normal stress of disk in radial direction (Pa)
σ ϕϕ
normal stress of disk in tangential direction (Pa)
σ rϕ
shear stress of disk in tangential direction (Pa)
ϕ
azimuthal angle of disk (rad)
∆φ
increment in azimuthal angle φ (rad)
ℜ
radiation resistance of disk
Φq
mode shape of the qth radial mode
ω
angular frequency (rad/s)
ωq
angular frequency (rad/s)
ω
natural frequency ratio of annular disk
ζq
modal damping ratio
Subscripts
d
disk
xxiv
I
inner radial edge
o
acoustic medium
O
outer radial edge
p
observation point in far field
q
radial mode index
r
radial direction of disk
s
source (radiator)
ϕ
circumferential direction of disk
Superscripts
1
first kind
2
second kind
~
Fourier transform
−
complex valued
Abbreviations
BEM
boundary element method
FEM
finite element method
LIST OF SYMBOLS FOR CHAPTER 4
a
outer radius of annular disk (mm)
b
inner radius of annular disk (mm)
c0
speed of sound in the acoustic medium (m/s)
fn(t), fr(t)
dynamic force on disk in normal and radial directions (N)
F
amplitude of applied force (N)
i
−1
k
acoustic wave number (rad/m)
h
disk thickness (mm)
xxv
m
number of nodal circles in the disk
n
number of nodal diameters in the disk
p
far field sound pressure (Pa)
P
spatial dependent far field sound pressure (Pa)
pq
far field sound pressure amplitude due to qth radial mode (Pa)
q
radial mode index
R
radius of sphere at far-field location (mm)
R, θ, φ
spherical coordinates
rf
position vector of force f(t) on the disk
rp
position vector of sound pressure
So
surface of sound source
Sv
boundary surface of acoustic control volume
u r
radial velocity of disk (m/s)
u r
radial acceleration of disk (m/s2)
U
r
spatially-dependent radial velocity of disk (m/s)
U
r
spatially-dependent radial acceleration of disk (m/s2)
V
acoustic control volume
w
transverse displacement in the disk
w
transverse acceleration in the disk
W
spatial dependent transverse displacement in the disk
Z(z)
variation of radial surface acceleration in z direction
η
vector of structural modal participation factors
Π
acoustic power from the disk vibration (W)
Πm,n,q
acoustic power from the modal vibrations of the disk (W)
ρ0
mass density of acoustic medium (kg/m3)
σ
sound radiation efficiency of the disk
ϕ
azimuthal angle of the disk (rad)
∆ϕ
circumferential separation of two excitation on the disk (rad)
Φm,n,q
mode shape for mode (m, n, q)
Φ vm, n, q
velocity modal vector for mode (m, n, q)
xxvi
ω
angular frequency (rad/s)
ωm,n,q
natural frequencies of the disk (Hz)
ζm,n,q
modal damping ratios
Subscripts
m, n, q
combined mode indices
0
acoustic medium
p
observation point in far field
r
radial direction of disk
s
source (radiator)
ϕ
circumferential direction of disk
Superscripts
V
velocity
Abbreviations
BEM
boundary element method
FEM
finite element method
LIST OF SYMBOLS FOR CHAPTER 5
a
outer radius of annular disk
b
inner radius of annular disk
c0
speed of sound in the acoustic medium
E
Young’s modulus of the sample rotor
fn(t), fr(t)
dynamic force on rotor
F
amplitude of applied force
h
disk thickness
xxvii
H
i
Hat height
−1
I
acoustic intensity at a field point
Jn
Bessel’s function of order n
k
acoustic wave number
kmn
structural wave number of the (m, n)th out-of-plane mode
kq
structural wave number in z direction
l
tangential mode index
m
number of nodal circle
n
number of modal diameters
R, θ, φ
spherical coordinates for receiver positions
q
radial mode index
Sv
boundary surface of acoustic control volume
th
thickness of hat structure
uqI , uqO
acceleration on inner and outer radial surfaces of the qth radial mode
U
spatially-dependent radial velocity
U q
spatially-dependent radial velocity of the qth radial mode
V
acoustic control volume
w
transverse displacement of rotor
W
spatial dependent transverse displacement
Wmn
mode
spatial dependent transverse displacement of the (m, n)th out-of-plane
β
radii ratio of annular disk
η
structural modal participation factor vector
Γ
modal sound pressure of rotor
Π
acoustic power
Πmn, Πq
acoustic power from the single mode of the sample rotor
Πmnq,
acoustic power from the combined mode of the sample rotor
ρ0
mass density of acoustic medium
ρd
mass density of the rotor
σmn, σq
sound radiation efficiency of the single mode of the sample rotor
xxviii
σmn,q
sound radiation efficiency of the combined mode of the sample rotor
ϕ
azimuthal angle of rotor
Φ
velocity modal vector of rotor
ωj
natural frequencies of rotor
ζj
modal damping ratios of rotor
Subscripts
d
disk
j
mode number
I
inner radial edge
l
tangential mode index
m,n
out-of-plane mode indices
m,n,q
combined mode indices
0
acoustic medium
O
outer radial edge
p
observation point in far field
q
radial mode index
s
source (radiator)
ϕ
circumferential direction of rotor
Superscripts
1
first kind
2
second kind
Abbreviations
BEM
boundary element method
FEM
finite element method
xxix
CHAPTER 1
INTRODUCTION
1.1 Motivation
Annular and circular disk models have been widely used to describe many
practical sound radiators such as gears, brake rotors, clutches, flywheels, railway wheels,
circular saws, electrical machinery stators, and electro-acoustic transducers. Thickness of
such components are often beyond the thin plate theory limit and one must examine both
out-of-plane and in-plane vibrations to appropriately control sound radiation from these
thick bodies. Traditionally, thin annular disk radiations with infinite baffles have been
employed to describe the sound radiation from flexural modes. Consequently, the effect
of coupling between in-plane and out-of-plane modes has not been included in the
calculation of sound radiation. This research intends to study such unresolved issues and
it proposes new analytical and semi-analytical approaches for sound radiation from thick
annular disks.
1.2 Literature Review
1.2.1 Structural Dynamics of Annular Disks
There is a substantial body of literature on the out-of-pane (flexural) vibration of
thin and thick plates [1.1-1.12]. Leissa [1.1-1.4] has summarized natural frequencies and
1
modes of plates with various geometric configurations and boundary conditions. In
particular, Vogel and Skinner [1.5] investigated natural frequencies of an uniform annular
disk using the thin plate theory. Wang and Thevendran [1.6] used the Rayleigh-Ritz
method, based on thin plate theory, to analyze annular plates. Mindlin [1.7] and Mindlin
and Deresiewicz [1.8] proposed a sixth-order thick plate theory to describe flexural
vibration of thick circular disks. McGee et al. [1.9] used the same theory to solve free
vibrations of thick annular plates. Irie et al. [1.10] analyzed the vibration of annular
Mindlin plates with nine combinations of inner and outer radial edge conditions. For the
admissible functions of the displacement fields, they employed Bessel’s functions in the
radial direction and trigonometric function the circumferential direction. Out-of-plane
vibration of annular Mindlin plates of varying thickness has been studied using similar
approach [1.11]. Liew et al. [1.12] investigated the vibration of circular and annular
Mindlin plates of different boundary conditions with multiple internal ring supports.
Conversely, published literature on the in-plane vibrations of annular or circular
disks is relatively sparse [1.13-1.16]. For instance, Bhuta and Jones [1.13] studied
coupled symmetric and torsional vibrations of a thin, rotating circular disk. Burdess et al.
[1.14] generalized the analysis to consider asymmetric in-plane vibrations and the
properties of forward and backward traveling waves. Chen and Jhu [1.15] examined the
in-plane stability of a spinning annular disk and determined the effect of rotational speed
on natural frequencies. Irie et al. [1.16] calculated natural frequencies of annular disks
using the transfer matrix method. We will use this method for the structural analysis of
thick annular disks.
2
1.2.2 Acoustic Radiation from Annular Disks
Sound radiation from thin circular and annular disks has been examined by
several investigators [1.17-1.21]. For instance, Thompson [1.17] computed self and
mutual radiation impedances of a uniformly vibrating annular or circular piston by
integrating of the far-field directivity function. Lee and Singh [1.18] proposed a
polynomial approximation for modal acoustic power radiation from a thin annular disk
but this method was restricted to only out-of-plane modes. Levine and Leppington [1.19]
developed an analytical solution for active and reactive powers from a planar annular
membrane given axisymmetric motions. Rdzanek and Engel [1.20] suggested asymptotic
formulas for power from a thin annular disk with clamped edges. Finally, Wodtke and
Lamancusa [1.21] investigated a circular plate using finite element analysis and then
calculated the sound radiation via the Rayleigh integral formula.
Sound radiation from in-plane modes of thick annular disks has not been
investigated thus far. Conversely, many researches on the radiation from cylindrical
radiators have been executed using various approaches [1.22-1.27]. Williams et al. [1.22]
used semi-analytical method with finite series of eigenfunctions for boundary condition
to calculate the acoustic radiation from a finite cylinder. Sandman [1.23] investigated
sound radiation from finite cylindrical shells and found that cylindrical baffle has very
little influence on sound radiation and concluded that the baffled cylindrical geometry
may be assumed to be a reasonable approximation for this problem. Stepanishen [1.24]
combined Green’s function and Fourier integral technique to develop integral expressions
for the generalized radiation impedance and radiated power and applied it to an infinite
cylinder. Junger and Feit developed expressions for far-field sound pressures for finite
3
and infinite cylindrical radiators given arbitrary surface velocity distribution [1.25].
Williams solved the same problem using a 2-dimensional Fourier transform [1.26].
Finally, Wang and Lai calculated the modal-averaged radiation efficiency of a finite
length circular cylindrical shell [1.27].
Several studies on the multi-modal sound radiation have been executed so far. For
instance, Kelti and Peng [1.28] investigated the effects of modal coupling on the acoustic
power from simply supported or clamped unit width plate and concluded that
contributions due to the modal coupling may be important for off-resonant excitation
cases. Cunefare [1.29] developed a technique for deriving the optimal surface velocity of
a beam that minimizes the radiation efficiency of the beam. The author [1.30] also
developed analytical and computational tool to assess the contribution of individual
modes and interaction effects to the total sound power from planar structures. Cunefare
and Currey [1.31] investigated orthogonal acoustic modes of finite baffled beam using
singular value decomposition of the radiation operator. Here, acoustic modes are
particular velocity patterns on the radiator surface. The authors [1.32] obtained the
radiation modes of baffled finite plates using the same approach. Several researchers
applied the acoustic radiation mode concept to the active structural acoustic control or the
estimation of radiated acoustic power [1.33-1.34]. Multi-modal sound radiation and
modal coupling effects on the sound radiation from a thin annular disk have also been
investigated using the modal expansion technique [1.18].
1.2.3 Brake Rotor Dynamics and Squeal Noise
Several structural dynamic models have been used to explain the brake squeal
generation mechanism. Two most common approaches deal with either the self-excited
4
vibration generated by the “stick-slip” phenomenon that is related to the velocity
dependency of friction coefficient [1.35-1.36] or the modal coupling phenomena that
involves two system modes coupled together because of the friction interface [1.37-1.39].
Recently, non-linear transient analysis [1.40-1.41] and complex eigen-value problem
[1.42-1.46] have been implemented using the finite element method. Dihua and
Dongying [1.43] extracted component modal data from finite element models and
calculated squeal propensity using a modal synthesis method. They investigated the
contribution of each component mode and the sensitivity of each mode to the instability
of the system mode. Hamabe et al. [1.44] calculated the participation of an unstable
system mode for a drum brake assembly. Squeal was eliminated after they separated the
two highest participating drum modes. Kung et al. [1.45] applied the modal participation
factor concept to a disk brake and identified a source of a low frequency squeal. Dunlap
et al. [1.47] investigated brake squeal using various approaches including numerical and
experimental approaches and concluded that natural frequency separation between inplane and out-of-plane modes is critical to high frequency squeal. McDaniel et al. [1.48]
investigated coupling between in-plane and out-of-plane modes and concluded that the
coupling creates vibrational instability that is characterized by power flow through the
transverse motion of the rotor.
1.3 Problem Formulation
Almost all of the above mentioned studies have considered sound radiation only
from either flexural vibration modes or rigid body piston motions as evident from the
review of available literature as presented in Section 1.2.
5
(a)
(b)
Fixed boundary
r
r
ϕ
b
z
ϕ
b
a
z
a
h
h
(c)
disk
hat
r
z
b
a
th
h
H
Figure 1.1: Thick annular disk example cases a) with free – free boundaries b) with fixed
– free boundaries c) disk with hat structure
6
Outer radius (a)
151.5 mm
Inner radius (b)
87.5 mm
Radii ratio (β = b/a)
0.54
Thickness (h)
31.5 mm
Hat height (H)
24 mm
Density (ρd)
7905.9 Kg/m3
Young’s modulus (E)
218 GPa
Poisson’s ratio (ν)
0.305
Table 1.1: Geometric dimensions and material properties of the example disks.
In such studies, sound radiation from the in-plane modes of a disk has been assumed to
be negligible compared to that from the out-of-plane modes. But, if the thickness of a
disk is beyond the range of thin plate (shell) theory, in-plane vibration could generate
sufficient sound given proper excitation. Therefore, the chief goal of this research is to
develop analytical solutions for sound radiation from the normal modes of a thick annular
disk. Both analytical and semi-analytical procedures are proposed to calculate structural
velocities and radiated acoustic pressure due to uni-directional and arbitrary harmonic
excitation forces. Figure 1.1 illustrates the example cases for this study. Also, geometric
dimensions and material properties of these example disks are given in the Table 1.1.
7
The primary assumptions for this study are as follows:
1. Structural and acoustic systems are linear time-invariant systems. Non-linear effects
arising from the coupling of motion variables are ignored.
2. Structural velocities in the normal direction (z) of flexural modes and in the radial
direction (r) of radial modes vary sinusoidally in the ϕ direction.
3.
Far field sound pressure due to radial mode is generated by the structural motions
of the inner or outer radial surfaces exclusively.
4.
Coupling between in-plane and out-of-plane modes is assumed to be negligible in
the calculation of sound radiation from normal modes of the disk. But the same effect
is assumed to be critical in the sound radiated in the combined excitation problem.
Chief objectives of this study are as follows.
1. Develop analytical and semi-analytical solutions for sound radiation from modal
vibrations and validate analytical solutions using computational and/or experimental
vibro-acoustic methods.
2. Critically examine thick and thin plate theories and investigate the effect of rotary
inertia and shear deformations on the structural eigen-solutions and acoustic sound
radiation from flexural (out-of-plane) modes. Also, study the effects of the disk
geometry and boundary conditions on sound radiation using the proposed analytical
solutions.
8
3. Examine the effect of couplings between in-plane and out-of-plane modes as well as
couplings within the same type of modes on the total sound radiation considering
natural frequency separations. Develop a semi-analytical procedure for calculating
sound radiation from the disk (including a simplified brake rotor) given multi-modal
harmonic excitations.
1.4 Organization
Each chapter of this dissertation is self contained with its own objectives, problem
formulation, methodology, results and list of references. Given below is a brief
discussion of each chapter.
Chapter 2
Structural eigen-solutions for the out-of-plane modes of a thick annular
disk with free-free boundaries are calculated using both thick and thin
plate theories. New analytical formulation is then proposed for the sound
radiation problem. In addition, the same problem is solved by a semianalytical procedure in which the disk surface velocity is numerically
defined by a finite element model and sound radiation is then analytically
obtained using a modified circular radiator model. Also, the effects of
radii and thickness ratios on the structural dynamic and acoustic radiation
characteristics are investigated using the analytical procedure. Finally, the
effect of boundary conditions is briefly examined.
Chapter 3
In-plane modal vibration of a thick annular disk is analytically
investigated using the transfer matrix method and confirmed using the
results of a finite element model. Sound radiation from structural modes of
9
this type is obtained using two different analytical approaches: (i) the
Rayleigh integral formula, (ii) modified cylindrical radiators of finite
length along with Fourier series and Sinc function approaches. The
problem is also analyzed using a boundary element code. Relevant
computational and analytical results are successfully compared with vibroacoustic measurements. Acoustically efficient radial structural modes may
be determined based on modal radiation efficiency and acoustic power
calculations. Strategies for minimization of sound radiation are also
investigated by parametrically varying disk geometry and natural
frequencies.
Chapter 4
A new semi-analytical procedure for the calculation of sound radiation
from a thick annular disk, when it is excited by a multi-modal and multidirectional harmonic force, is introduced. Vibro-acoustic responses for a
specific force has been calculated by the modal expansion technique from
the structural modal dataset from analytical or numerical methods, and the
normal radiation solutions representing the far-field sound pressure due to
structural modal vibrations. This method is confirmed by comparing
predicted results for a single frequency excitation with numerical
calculations. Based on this procedure, acoustic power and radiation
efficiency spectra of the sample disk corresponding to a specific force
location and direction are obtained. Now, excitation can be applied at
multiple locations, with different frequencies. The effects of coupling
between structural modes and natural frequency separation on the acoustic
10
radiation are also investigated through the proposed procedure. This study
could lead to strategies that would minimize sound radiation as excited by
arbitrary harmonic forces.
Chapter 5
Modal and multi-modal sound radiation mechanisms from a brake rotor
are studied using analytical and semi-analytical procedure proposed earlier
in Chapters II – IV. In this study, structural eigen-solutions and modal
sound radiation of a brake rotor are expressed in terms of the vibroacoustic characteristics of a generic thick annular disk having similar
geometric dimensions. Accuracy of the proposed method is confirmed by
purely numerical analyses based on finite element and boundary element
models. Vibro-acoustic responses such as surface velocities and radiated
sound pressures due to multi-modal excitations are calculated from
synthesized structural modes and modal acoustic radiation of the rotor
using the modal expansion technique. In addition, acoustic power and
radiation efficiency spectra corresponding to a specific force excitation are
obtained from the sound pressure data. Predictions are also confirmed by
comparisons with analogous numerical analyses.
Chapter 6
Contributions of this study are highlighted and several areas of future
research are suggested.
11
REFERENCES FOR CHAPTER 1
1.1
A. W. LEISSA 1969 NASA SP-160 Vibration of Plates.
1.2
A. W. LEISSA 1987 The Shock and Vibration Digest 19(3), 10-24. Recent
Research and plate vibration, 1981-1985. Part 1: Classical theory.
1.3
A. W. LEISSA 1987 The Shock and Vibration Digest 19(3), 10-24. Recent
research and plate vibration, 1981-1985. Part 2: Complicating effects.
1.4
A. W. LEISSA 1993 Vibrations of Plates, New York: Acoustical Society of
America.
1.5
S. M. Vogel and D. W. Skinner 1965 Journal of Applied Mechanics December,
926-931 Natural frequencies of transversely vibrating uniform annular disk.
1.6
C. M. WANG and V. THEVENDRAN 1993 Journal of Sound and Vibration 163(1),
137-149 Vibration analysis of annular plates with concentric support using a
variant of Rayleigh-Ritz method.
1.7
R. D. MINDLIN 1951 ASME Journal of Applied Mechanics 18, 31-38 Influence of
rotatory inertia and shear on the flexural motion of isotropic, elastic plate.
1.8
R. D. MINDLIN and H. DERESIEWICZ 1954 Journal of Applied Physics 25(10),
1329-1332 Thickness-shear and flexural vibration of a circular disk.
1.9
O. G. MCGEE, C. S. HUANG and A. W. LEISSA 1995 International Journal of
Mechanical Science 37(5), 537-566 Comprehensive exact solutions for free
vibrations of thick annular sectorial plates with simply supported radial edges.
1.10
T. IRIE, G. YAMADA and K. TAKAGI 1982 Transactions of the American Society
of Mechanical Engineers, Journal of Applied Mechanics 49, 633-638 Natural
frequencies of thick annular plates.
1.11
IRIE, G. YAMADA and S. AOMURA 1979 Journal of Sound and Vibration 66 (1),
187-197 Free vibration of a Mindlin annular plate of varying thickness.
1.12
K. M. LIEW, Y. XIANG, C. M. WANG AND S. KITIPORNCHAI 1993 Computer
Methods in Applied Mechanics and Engineering 110, 301-315 Flexural vibration
of shear deformable circular and annular plates on ring support.
1.13
G. BHUTA and J. P. JONES 1971 Journal of the Acoustical Society of America 35
(7), 982-989. Symmetric planar vibrations of a rotating disk.
1.14
J. S. BURDESS, T. WREN and J. N. FAWCETT 1987 Proceeding of the Institution of
Mechanical Engineers 201, 37-44. Plane stress vibration in rotating discs.
12
1.15
S. CHEN and J. L. JHU 1996 Journal of Sound and Vibration 195 (4), 585-593. On
the in-plane vibration and stability of a spinning annular disk.
1.16
T. IRIE, G. YAMADA and Y. MURAMOTO 1984 Journal of Sound and Vibration 97
(1), 171-175 Natural frequencies of in-plane vibration of annular plates.
1.17
W. THOMPSON, JR. 1971 Journal of Sound and Vibration 17 (2), 221-233. The
computation of self- and mutual-radiation impedances for annular and elliptical
pistons using Bouwkamp integral.
1.18
M. R. LEE and R. SINGH 1994 Journal of the Acoustical Society of America 95 (6)
3311-3323. Analytical formulations for annular disk sound radiation using
structural modes.
1.19
H. LEVINE and F. G. LEPPINGTON 1988 Journal of Sound and Vibration 121 (5),
269-275. A note on the acoustic power output of a circular plate.
1.20
W. P. RDZANEK Jr. and Z. ENGEL 2000 Applied Acoustics 60 (5), 29-43.
Asymptotic formula for the acoustic power output of a clamped annular plate.
1.21
H. W. WODTKE and J. S. LAMANCUSA 1998 Journal of Sound and Vibration 215
(5), 1145-1163. Sound power minimization of circular plates through damping
layer placement.
1.22
W. WILLIAMS, N. G. PARKE, D. A. MORAN AND C. H. SHERMAN 1964 Journal of
the Acoustical Society of America 36 (12) 2316-2322. Acoustic Radiation from a
Finite Cylinder.
1.23
B. E. SANDMAN 1976 Journal of the Acoustical Society of America 60 (6), 12561264. Fluid loading influence coefficients for a finite cylindrical shell.
1.24
P. R. STEPANISHEN 1978 Journal of the Acoustical Society of America 63 (2) 328338. Radiation power and radiation loading of cylindrical surfaces with
nonuniform velocity distribution.
1.25
M. C. JUNGER and D. FEIT 1985 Sound, Structures, and Their Interactions. New
York: MIT Press.
1.26
E. G. WILLIAMS 1999 Fourier Acoustics. San Diego: Academic Press.
1.27
C. WANG and J. C. S LAI 2000 Journal of Sound and Vibration 232 (2), 431-447.
The sound radiation efficiency of finite length acoustically thick circular
cylindrical shell under mechanical excitation I: Theoretical analysis.
1.28
R. F. KELTIE and H. PENG 1987 ASME Trans. J. Vib. Acoust. Stress Reliabil. Des.
109, 48-53. The effect of modal coupling on the acoustic radiation from panels.
13
1.29
K. A. CUNEFARE 1991 Journal of the Acoustical Society of America 90(5), 25212529. The minimum multimodal radiation efficiency of baffled finite beams.
1.30
K. A. CUNEFARE 1992 AIAA J. 30, 2819-2828. Effect of modal interaction on
sound radiation from vibrating structure.
1.31
K. A. CUNEFARE and M. N. Currey 1994 Journal of the Acoustical Society of
America 96(4), 2302-2312. On the exterior acoustic radiation modes of structures.
1.32
M. N. CURREY and K. A. CUNEFARE 1995 Journal of the Acoustical Society of
America 98(3), 1570-1580. The radiation modes of baffled finite plates.
1.33
G. P. Gibbs, R. L. Clark, D. E. Cox and J. S. Vipperman 2000 Journal of the
Acoustical Society of America 107(1), 332-339. Radiation modal expansion:
Application to active structural acoustic control.
1.34
M. R. Bai and M. Tsao 2002 Journal of the Acoustical Society of America 112(3),
876-883 Estimation of sound power of baffled planar sources using radiation
matrices.
1.35
H. MURAKAMI, N. TSUNADA AND T. KITAMURA, “A Study Concerned with a
Mechanism of Disc-Brake Squeal,” SAE Paper # 841233.
1.36
H. MATSUI, H. MURAKAMI, H. NAKANISHI and Y. TSUDA, “Analysis of DiscBrake Squeal,” SAE Paper # 920553.
1.37
W. V. NACK AND A. M, JOSHI, “Friction Induced Vibration: Brake Moan,” SAE
Paper # 951095.
1.38
J. FLINT AND J. HULTÈN 2002 Journal of Sound and Vibration 254 (1), 1-21.
Lining-Deformation–Induced Modal Coupling as Squeal Generator in a
Distributed Parameter Disc Brake Model.
1.39
D. N. HERTING, MSC/NASTRAN Advanced Dynamic Analysis User’s Guide, pp.
157-173, 1997.
1.40
Y. K. HU, AND L. I. NAGY, “Brake Squeal Analysis by Using Nonlinear Transient
Finite Element Method,” SAE Paper # 971510.
1.41
O. N. HAMZEH, W. W. TWORZYDLO, H. J. CHANG AND S. T. FRYSKA, “Analysis
of Friction-Induced Instabilities in a Simplified Aircraft Brake, SAE Paper #
1999-01-3404.
1.42
G. D. LILES, “Analysis of Disc Brake Squeal Using Finite Element Methods,”
SAE Paper # 891150.
14
1.43
G. DIHUA AND J. DONGYING, “A Study on Disc Brake Squeal using Finite
Element Methods,” SAE Paper # 980597.
1.44
T. HAMABE, I. YAMAZAKI, K, YAMADA, H. MATSUI, S. NAKAGAWA AND M.
KAWAMURA, “Study of a Method for Reducing Drum Brake Squeal,” SAE Paper
# 1999-01-0144.
1.45
S. W. KUNG, K. B. DUNLAP AND R. S. BALLINGER, “Complex Eigenvalue
Analysis for Reducing Low Frequency Squeal,” SAE Paper # 2000-01-0444.
1.46
T. S. SHI, O. DESSOUKI, T. WARZECHA, W. K. CHANG, AND A. JAYASUNDERA,
“Advances in Complex Eigenvalue Analysis for Brake Noise” SAE Paper # 200101-1603.
1.47
K. B. DUNLAP, M. A. RIEHLE AND R. E. LONGHOUSE, “An Investigative Overview
of Automotive Disc Brake noise” SAE Paper # 1999-01-0142.
1.48
J. G. MCDANIEL AND X. LI, “Analysis of Instabilities and power flow in Brake
Systems with Coupled Modes” SAE Paper # 2001-01-1602.
15
CHAPTER 2
ACOUSTIC RADIATION FROM OUT-OF-PLANE MODES USING
THIN AND THICK PLATE THEORIES
2.1 Introduction
Acoustic radiation from thick plates or disks has not been adequately examined
though there is a substantial body of literature on the structural dynamics of thin and
thick plates [2.1-2.12]. Limited acoustic studies have considered either flexural vibration
modes or rigid body piston motions of thin disks [2.14-2.19]. For instance, Thompson
[2.14] computed self and mutual radiation impedances of a uniformly vibrating annular
or circular piston by integrating of the far-field directivity function. Lee and Singh [2.15]
proposed a polynomial approximation for modal acoustic power radiation from a thin
annular disk but this method was restricted to only out-of-plane modes. Levine and
Leppington [2.16] developed an analytical solution for active and reactive powers from a
planar annular membrane given axisymmetric motions. Rdzanek and Engel [2.17]
suggested asymptotic formulas for power from a thin annular disk with clamped edges.
Finally, Wodtke and Lamancusa [2.18] investigated a circular plate using finite element
analysis and then calculated the sound radiation via the Rayleigh integral formula. In this
16
chapter, new analytical and semi-analytical methods for sound radiation from a thick
annular disk will be proposed. In particular, we comparatively evaluate the merits of thin
vs. thick plate theories on the calculation of radiation from out-of-plane flexural modes.
Vibro-acoustic experiments and large scale finite and boundary elements codes are used
to validate the analytical formulation.
y
fn(t)
r ϕ
b
z
x
a
h
Figure 2.1: A thick annular disk with an excitation in the normal direction
17
Outer radius a (mm)
Inner radius b (mm)
Radii ratio β (= b/a)
Thickness h (mm)
Thickness Ratio h (=h/a)
Density ρd (Kg/m3)
Young’s modulus E (GPa)
Poisson’s ratio ν
Disk I
151.5
82.5
0.54
31.5
0.21
7905.9
218
0.305
Disk II
139.0
82.5
0.59
31.5
0.23
7905.9
218
0.305
Disk III
151.5
82.5
0.54
16.3
0.11
7905.9
218
0.305
Table 2.1: Disk examples with free-free or fixed (r = b) - free boundaries
Annular disk idealization can be used to analyze many real-life mechanical components
such as gears, brake rotors, clutches, flywheels, railway wheels, circular saws, and
electric motor. In many cases, thickness (h) is not negligible relative to other dimensions
of the component, and thus one must consider the thickness effects in structural dynamic
and acoustic radiation characteristics. Figure 2.1 illustrates the example case that is
assumed to be non-rotating and without the complicating effects of inner hub. Disk is
assumed to be of uniform h and made of an undamped, isotropic material. First, free-free
boundaries at the inner and outer edges are assumed. Then, the inner edge is assumed to
be ideally fixed but the outer edge is still free. Table 2.1 provides typical values of 3
disks. Disk I is used for all analytical, numerical, and experimental studies. Additionally,
Disk II and III are used for structural modal analysis to examine plate theories. For a
18
complete investigation of the vibro-acoustic characteristics of a thick annular disk, it is
necessary to simultaneously consider both in-plane and out-of-plane vibrations. But, the
current analysis focuses only on the out-of-plane modal vibration and the resulting sound.
Primary assumptions are as follows: (1) Structural and acoustic systems are linear timeinvariant systems and complicating effects such as fluid loading and acoustic scattering
from the disk edges are negligible. (2) Structural velocities in the normal direction (z)
vary sinusoidally in the ϕ direction. (3) Free and far field sound pressure at the
observation point (rp) is generated only by the structural motions of two normal surfaces
and the inner or outer radial surfaces at edges does not contribute to the far-field sound.
(4) Coupling between in-plane and out-of-plane modes is negligible.
Chief objectives of this chapter are as follows. (1) Critically examine thick and
thin plate theories and investigate the effect of rotary inertia and shear deformations on
the structural eigensolutions and acoustic sound radiation. (2) Develop analytical and
semi-analytical solutions for sound radiation from modal vibrations. (3) Validate
analytical solutions using computational and/or experimental vibro-acoustic methods. (4)
Study the effects of the disk geometry and boundary conditions on sound radiation using
the proposed analytical solutions. Only single mode excitations are considered here as the
multi-modal excitations and coupling issues will be considered in a future chapter. Table
2.2 summarizes various models or methods that will be employed in this study. For the
analytical
method,
procedure
includes
analytical
determination
of
structural
eigensolutions and resulting sound field. Conversely, the finite element method (FEM) is
used for structural analysis for the semi-analytical formulation though the sound field is
still computed using a modified circular disk radiator model.
19
Medium
Structural
Dynamics
Structural
Dynamics
+
Acoustic
Radiation
Model or
Method
Designation
Structural
Dynamics
Formulation
Acoustic
Radiation
Formulation
Disk
Boundaries
Method Type
A
Thick plate
-
Free-Free
Analytical
B
Thick plate without
rotary
inertia effect
-
Free-Free
Analytical
C
Thin plate
-
Free-Free
Analytical
D
Finite
elements
-
Free-Free
Computational
E
Experiment
-
Free-Free
Experimental
F
Thick plate
-
Fixed-Free
Analytical
G
Thick plate
without
rotary
inertia effect
-
Fixed-Free
Analytical
H
Thin plate
-
Fixed-Free
Analytical
I
Finite
elements
-
Fixed-Free
Computational
J
Thick plate
Thick plate
Free-Free
Analytical
K
Thin plate
Thin plate
Free-Free
Analytical
Thick plate
Free-Free
SemiAnalytical
Boundary
elements
Free-Free
Computational
L
M
Finite
elements
Finite
elements
N
Experiment
Experiment
Free-Free
Experimental
O
Finite
elements
Boundary
elements
Fixed-Free
Computational
Table 2.2: List of models developed and methods employed for Disk I
20
2.2 Structural Analysis Based on Thick and Thin Plate Theories
2.2.1 Thick Plate Theory
According to the procedure proposed by Mindlin and Deresiewicz [2.7] or Mcgee
et al. [2.8], the vibratory displacements of a thick annular disk are assumed as follows,
while recognizing the effects of shear and rotating inertia.
u = zΨr (r , ϕ, t )
v = zΨϕ (r , ϕ, t )
(2.1)
w = w(r , ϕ, t )
Here, u, v and w are components in the radial (r), circumferential (ϕ), and transverse
directions (z), Ψr and Ψϕ are the bending rotations of normal to the mid-plane in radial
and circumferential directions, respectively. Refer to the list of symbols given at the end
of this chapter for a complete list of symbols. The equations of motion in terms of the
stress resultants in polar coordinates (r, ϕ) are
ρ h 3 ∂ 2 Ψr
∂M r 1 ∂M ϕ 1
+ (M r − M ϕ )− Q r = d
+
r ∂ϕ
r
12 ∂t 2
∂r
∂M rϕ
2
1 ∂M ϕ 2
ρd h3 ∂ Ψϕ
+
+ M rϕ − Qϕ =
∂r
r ∂ϕ
r
12 ∂t 2
∂Q r 1 ∂Q ϕ 1
∂2w
+ Q r − Qϕ = ρ d h 2
+
r ∂ϕ
r
∂r
∂t
(2.2)
(2.3)
(2.4)
where ρd is the mass density of the annular disk. The stress resultants in terms of
moments Mr, Mϕ, and Mrϕ, along with shear forces Qr and Qϕ can be related to the
transverse displacements and bending rotations as:
21
∂Ψϕ
 ∂Ψ
ν
M r = Db  r +  Ψr +
∂ϕ
r
 ∂r
∂Ψϕ
1 
M ϕ = Db   Ψr +
∂ϕ
r 
M rϕ = M ϕr =

 ,


∂Ψ 
 + ν r ,
∂r 

(1 − ν) Db  ∂Ψϕ 1  ∂Ψr

+ 
− Ψϕ  

r  ∂ϕ
2

 ∂r
(2.5)
(2.6)
(2.7)
∂w 

Q r = κ 2 Gh Ψr +

∂r 

(2.8)

1 ∂w 
Q ϕ = κ 2 Gh Ψϕ +

r ∂ϕ 

(2.9)
where Db = Eh3 / 12(1 − ν2 ) is the flexural rigidity, E is the modulus of elasticity, ν is the
Poisson’s ratio, κ2=π2/12 is the shear correction factor, and G is the shear modulus of the
disk. Assume a harmonic variation with time,
Ψr (r , ϕ, t ) = ψ r (r , ϕ) cos ωt
Ψϕ (r , ϕ, t ) = ψ ϕ (r , ϕ) cos ωt
(2.10)
w(r , ϕ, t ) = W (r , ϕ) cos ωt
to reduce equations (2.2– 2.4) to
ω2 ρ d h 3
∂M r 1 ∂M ϕ 1
+ (M r + M ϕ )− Q r +
ψr = 0
+
r ∂ϕ
r
12
∂r
∂M rϕ
∂r
+
ω2 ρ d h 3
1 ∂M ϕ 2
+ M rϕ − Q ϕ +
ψϕ = 0
r ∂ϕ
r
12
∂Q r 1 ∂Q ϕ 1
+ Q r + ω 2 ρ d hW = 0
+
r ∂ϕ
r
∂r
22
(2.11)
(2.12)
(2.13)
The transverse deflection amplitude (W) and associated angular rotations (ψr and ψϕ) are
defined in terms of three potential functions (φ1, φ2, and φ3) as:
ψ r = (σ1 − 1)
ψϕ =
∂φ1
∂φ
1 ∂φ 3
+ (σ 2 − 1) 2 +
∂r
∂r
r ∂ϕ
(σ1 − 1) ∂φ1 (σ 2 − 1) ∂φ 2
+
∂ϕ
r
r
∂ϕ
+
∂φ 3
∂r
W = φ1 + φ 2
(2.14)
(2.15)
(2.16)
while introducing the following parameters:
σ 1 ,σ 2 = (δ , δ
2
2
2
1
{
)
 4 1
 Rλ − 
S

[
−1
λ4
2
δ ,δ =
R + S ± (R − S ) + 4λ4
2
2
2
2
1
R=
(2.17a-b)
]
−1 / 2
ρ ω2 h
D
h2
, S = 2 b , λ4 = d
12
Db
κ Gh
}
(2.18a-b)
(2.19a-c)
Substitution of equations (2.3–2.9) and (2.14–2.16) in (2.11–2.13) along with a series of
subsequent manipulations yields
(∇
(∇
(∇
2
2
2
)
+ δ )φ
+ δ )φ
+ δ12 φ1 = 0
2
2
2
=0
2
3
3
=0
(2.20)
where ∇2 is the harmonic differential operator, and another parameter is introduced as
follows:
23
1

δ 23 = 2 Rλ4 −  (1 − ν )
S

(2.21)
The solutions to equations (2.11–2.13) require the determination of the potential
functions φ1, φ2, and φ3 that must satisfy equation (2.20).
φ1 (r , ϕ) = Rn1 sin(nϕ)
φ2 (r , ϕ) = Rn 2 sin(nϕ)
(2.22 a-c)
φ3 (r , ϕ) = Rn3 cos(nϕ)
Introducing equation (2.22) into (2.20) yields
r2
(
)
d 2 Rni
dR
+ r ni + δ i2 r 2 − n 2 = 0; i = 1,2,3
2
dr
dr
(2.23)
where n is typically a positive integer. The general solutions to equations (2.23) involve
ordinary and modified Bessel functions of the first and second kinds and the six constants
of integration that are determined from the boundary conditions.
2.2.2 Thin Plate Theory
The thin plate theory essentially neglects the effects of rotary inertia and
additional deflections caused by shear forces [2.5]. Consequently, the governing
differential equation for transverse displacement w(r, ϕ, t) in the mid-plane of the plate is
2
 ∂2 1 ∂
1 ∂2 
∂2w

−
ρ
=0
Db  2 +
+ 2
w
h
d
r ∂r r ∂ϕ 2 
∂t 2
 ∂r
(2.24)
Solution to this equation is assumed as follows:
w(r , ϕ, t ) = W (r ) cos(nϕ)e− iωt
24
(2.25)
Using equations (2.24-2.25), the following Bessel’s equation is obtained.
 d 2 1 d n2
 2 +
+
r dr r 2
 dr
2

 W − λ4W = 0

(2.26)
General solution to this equation can be written as
W ( r ) = C1 J n (λ mn r ) + C 2Y n (λ mn r ) + C 3 I n (λ mn r ) + C 4 K n (λ mn r )
(2.27)
where Jn and Yn are the Bessel functions of first and second kinds and In and Kn are
modified Bessel functions of first and second kinds. Here n is the order of the Bessel
function representing the number of nodal diameters and m is the order of eigenvalues
representing the number of nodal circles [2.5].
2.2.3 Eigensolutions for Free-Free Boundaries
For the thick plate theory, free-free boundary conditions at the inner and outer radial
edges can be expressed as follows:
M r (a, ϕ) = M rϕ (a, ϕ) = Qr (a, ϕ) = 0,
M r (b, ϕ) = M rϕ (b, ϕ) = Qr (b, ϕ) = 0
(2.28a-b)
From the expressions of Mr, Mrϕ, and Qr as defined in (2.13-2.23) along with boundary
conditions defined by (2.28), one could formulate the following equation in matrix form.
[TTHICK ]{C}= {0}
(2.29)
Here, [TTHICK] is a 6 ×6 characteristic matrix with elements of various Bessel functions,
{C} is an arbitrary coefficients vector and {0} is a null vector. For the thin plate theory,
boundary conditions of (2.28) are simplified as follows:
25
M r (a, ϕ) = M rϕ (a, ϕ) = 0,
M r (b, ϕ) = M rϕ (b, ϕ) = 0
(2.30a-b)
Given the relations between w and bending moments (Mr, Mrϕ), the following equations
can be derived for the boundaries satisfying (2.30).
 1 ∂w 1 ∂ 2 w 
∂ 2w
+ ν
+ 2 2  = 0,
∂r 2
r
∂
r
r ∂ϕ 

(2.31a-b)
∂  ∂ w 1 ∂w 1 ∂ w  1 − ν ∂  ∂w w 
+

+ =0
+
+

∂r  ∂r 2 r ∂r r 2 ∂ϕ2  r 2 ∂ϕ2  ∂r r 
2
2
2
The characteristic matrix equation corresponding to thin plate theory, similar to equation
(2.29), is as follows.
[TTHIN ]{C}= {0}
(2.32)
The characteristic or frequency equations are obtained from equations (2.29) or (2.32).
2.2.4 Validation Studies
Analytical solutions for the free-free boundaries, as obtained by both thin and
thick plate theories, are compared in Table 2.3. Results of finite element analyses and
structural modal experiments are also provided for Disks I, II, and III. Only the first four
modes are listed in Table 2.3 since the relevant upper frequency for acoustic radiation
study for Disk I is 8 kHz. In the finite element method (FEM), 11 out-of-plane modes
have been obtained in the frequency range from 0 to 16 kHz with a model that includes
4,400 solid brick elements and 6,600 nodes [2.13]. In addition, mode shapes of Disk I
from alternate analytical approaches are compared with numerical analysis in Figure 2.2.
As shown in this figure, the mode shapes from alternate plate theories are very similar in
26
spite of differences in natural frequencies. In modal experiments, the excitation force f(t)
is applied in the z direction by an impulse hammer (PCB GK291C) at ϕ = 0° at the outer
edge of the disk. The set up for structural modal experiment is explained in Figure 2.3.
The frequency range and resolution (∆f) of this experiment are set as 16 kHz and 8 Hz
respectively. Natural frequencies (ωmn) and modal damping ratios (ςmn) are extracted
/f(ω) where w
is the acceleration and f is the applied force.
from accelerance spectra w
As shown in Table 2.3, the finite element predictions match well with measurements.
Analytical solutions based on the thick plate theory produce more accurate answers than
the ones that based on the thin plate theory. Yet, even the thick plate theory predictions
show significant errors over the higher frequency range. Also, it is obvious that
differences between eigenvalues based on two alternate plate theories are proportional to
the thickness ratio ( h = h/a). Modal damping ratios are estimated from measured
accelerance spectra using the half-power bandwidth method for every resonant peak and
( r , ϕ) is
these results are summarized in Table 2.4. In addition, accelerance spectrum w
calculated based on the numerical modal dataset using the forced vibration analysis in
FEM. These results are subsequently used as excitation to numerical or analytical
methods for the calculation of far-field sound radiation. Figure 2.4 compares computed
and measured accelerance spectra and a good agreement over the given frequency range
is observed. Dominant peaks in this figure correspond to the out-of-plane modes whose
frequencies are listed in Table 2.3.
27
0.4
Displacement (µm)
Displacement (µm)
0.1
0
-0.1
0.5
0.75
0.3
0.2
0.1
0.5
1
r/a
0.75
1
r/a
Figure 2.2: Comparison of Disk I mode shapes given free – free boundaries. (a) (1, 0)
mode; (b) (0, 2) mode. Key:
WKLFNSODWHWKHRU\0RGHO$
WKLQSODWHWKHRU\
(Model C);
, finite element method (Model D).
28
Microphone p
i φ = φi,
rp = 303 mm)
Anechoic
Chamber
φ
Accelerometer w
° °,
ru= 151.5 mm)
Impulse Hammer f
° °
rf = 151.5 mm)
Rotor
Signal
Conditioning
Unit
Signal
Conditioning
Unit
FFT
Analyzer
FFT
Analyzer
/f(ω), acoustic
Figure 2.3: Vibro-acoustic experiment used to measure structural w
frequency response functions P/f(ω) and out-of-plane modal sound radiations.
29
Disk
I
β=0.54
h =0.21
II
β=0.59
h =0.23
III
β=0.54
h =0.11
ωmn
(kHz)
Non-dimensional Eigenvalues
λmn2 = ωmn a2(ρd h/Db)1/2
Mode
Indices
m
n
Thick Plate
Theory
(Model A)
Thin Plate
Theory
(Model C)
Finite
Element
(Model D)
0
2
3.82
4.02
3.86
3.92
1.331
1
0
8.85
9.80
8.69
9.02
3.063
0
3
10.59
11.11
10.04
10.25
3.481
1
1
15.42
17.55
13.57
14.00
4.756
0
2
3.62
3.90
3.72
3.72
1.500
1
0
9.14
10.53
8.74
9.33
3.756
0
3
10.04
10.73
9.75
9.78
3.938
1
1
15.36
18.32
13.22
13.81
5.563
0
2
4.06
4.12
4.03
4.03
0.706
1
0
9.55
9.82
8.99
9.53
1.669
0
3
11.11
11.13
10.71
10.78
1.888
1
1
16.86
17.59
15.27
15.99
2.800
Experiment Experiment
(E)
(E)
Table 2.3: Eigensolutions for Disk I, II and III with free-free boundaries.
30
Mode Indices
m
n
Damping Ratio
(%)
0
2
0.62
1
0
0.34
0
3
0.26
1
1
0.26
Table 2.4. Measured modal damping ratios for Disk I.
31
150
(0, 4)
(0, 3)
(0, 2)
(1, 1)
2
dB re 20 µm/s -N
(1, 0)
100
50
0
2
4
6
8
Frequency (kHz)
/f(ω) at r = 151.5 mm and ϕ = 180° for Disk I with
Figure 2.4: Structural accelerances w
free-free boundaries. Key: - - -, computed using FEM (Model D); ––––, measured (E).
32
2.2.5 Effect of Fixed-Free Boundaries
Eigensolutions for a disk with fixed-free boundaries can be easily calculated from
the analytical solutions of Chapter 2.2.1. For the thick plate case, boundary conditions at r
= b (fixed) and r = a (free) edges are expressed as follows.
M r (a, ϕ) = M rϕ (a, ϕ) = Qr (a, ϕ) = 0,
W (b, ϕ) = ψ r (b, ϕ) = ψ ϕ (b, ϕ) = 0
(2.33a-b)
Likewise, for the thin plate case, these conditions are specified as
M r (a, ϕ) = M rϕ (a, ϕ) = 0,
Wr (b, ϕ) =
∂Wr
(b, ϕ) = 0
∂r
(2.34a-b)
From these boundary conditions, matrix equations similar to equations (2.29) and (2.32)
can be obtained and eigenvalues can be determined using the same procedure. In addition,
natural frequencies are also calculated using FEM and compared with alternate plate
theories in Table 2.5. As in the free-free boundary case, eigensolutions based on the thick
plate theory are much more accurate than those given by the thin plate theory. Note that
modal experiments are not attempted since it is difficult to experimentally simulate the
perfect fixed boundary condition at r = b.
2.2.6 Effect of Disk Geometry on Eigensolutions
As one can see from equations (2.20-2.23), structural eigensolution of a thick
annular disk are affected by the radii ratio (β = b/a), thickness ratio ( h = h/a), as well as
by material properties. In this Chapter, effects of β and h on the non-dimensional
33
eigenvalues are examined for selected modes of Disk I with free-free or fixed-free
boundaries. In our investigation, such non-dimensional parameters are controlled by
adjusting h and b for a fixed a. Eigenvalues are calculated using three alternate analytical
methods based on models A, B, and C as described in Table 2.2. Results are shown in
Figures 2.5 and 2.6 where differences in the alternate formulations are evident. For the
free-free disks, experimental and FEM results are also included in these figures for the
sake of comparison.
Non-dimensional Eigenvalue
λmn2 = ωmn a2(ρd h/Db)1/2
Mode
Indices
m
n
Thick Plate
Theory
Thin Plate
Theory
0
0
0
0
0
1
2
3
11.96
13.43
15.28
18.75
15.72
16.05
17.52
21.17
ωmn
(kHz)
FEM
FEM
13.61
13.63
14.28
16.81
4.623
4.628
4.849
5.709
Table 2.5: Eigensolutions of Disk I with fixed-free boundary conditions.
34
(a)
(b)
15
15
(0, 3)
(0, 3)
10
λ2
λ2
10
5
0
0.3
(0, 2)
0.5
5
0
0.7
β
(0, 2)
0
0.2
h
Figure 2.5: Effect of geometry on the (0, 2) and (0, 3) modes of Disk I with free-free
boundaries. (a) Effect of radii ratio (β); (b) Effect of thickness ratio ( h ). Key:
WKLFN
plate theory (Model A);
WKLFNSODWHWKHRU\ZLWKRXWURWDU\LQHUWLDHIIHFW0RGHO%- -, thin plate theory (Model C); PHDVXUHG(
35
30
(a)
(b)
20
(1, 1)
λ2
(1, 1)
λ2
20
10
(1, 0)
10
(1, 0)
0
0.3
0.5
0
0.7
β
0
0.2
h
Figure 2.6: Effect of geometry on the (1, 0) and (1, 1) modes of Disk I with free-free
WKick
boundaries. (a) Effect of radii ratio (β); (b) Effect of thickness ratio ( h ). Key:
plate theory (Model A);
WKLFNSODWHWKHRU\ZLWKRXWURWDU\LQHUWLDHIIHFW0RGHO%- -, thin plate theory (Model C); PHDVXUHG(
36
(a)
(b)
40
25
(0, 3)
20
30
20
(0, 2)
λ2
λ2
15
(0, 3)
10
10
0
0.2
(0, 2)
5
0.5
β
0
0.8
0
0.1
0.2
h
0.3
Figure 2.7: Effect of geometry on the (0, 2) and (0, 3) modes of Disk I with fixed-free
boundaries. (a) Effect of radii ratio (β); (b) Effect of thickness ratio ( h ). Key:
WKLFN
plate theory (Model F);
WKLFNSODWHWKHRU\ZLWKRXWURWDU\LQHUWLDHIIHFW0RGHO*- -, thin plate theory (Model H).
37
40
20
λ2
λ2
30
20
10
10
0
0.2
0.5
β
0
0
0.8
0.2
Figure 2.8: Effect of geometry on the (1, 0) and (1, 1) modes of Disk I with fixed-free
boundaries. (a) radii ratio (β); (b) thickness ratio ( h ). Key:
PRGHZLWKWKLFN
plate theory (Model F); ∆∆∆, (0, 0) mode with thick plate theory without rotary inertia
effect (Model G);
PRGHZLWKWKLQSODWHWKHRU\0RGHO+
PRGH
with thick plate theory (Model F); ◊◊◊, (0, 1) mode with thick plate theory without rotary
inertia effect (Model G); - - -, (0, 1) mode with thin plate theory (Model H).
38
Differences between the λ2mn values, based on alternate approaches, are proportional to
h for all modes. But as shown in Figures 2.5a and 2.6a, the effects of β are mode
dependent. For example, differences are proportional to β for (1, 0) and (1, 1) modes.
Conversely, for (0, 2) and (0, 3) modes, such differences are inversely proportional to β.
The comparison of natural frequencies based on models A, B, and C suggests that
differences between eigensolutions are mainly caused by the rotary inertia effect in (1, 0)
and (1, 1) modes and by the shear deformations in (0, 2) and (0, 3) modes. Similar
investigation has been executed for the fixed-free disk and results are given in Figures 2.7
and 2.8. Differences between eigenvalues based on thick and thin plate theories appear to
be caused by the shear deformations regardless of the mode type. Also, in this case, these
differences are proportional to thickness and radii ratios for all modes.
2.3 Acoustic Radiation Calculation
2.3.1 Formulation
Sound radiation from the flexural vibrations of circular plates or annular disks has
been examined by several investigators [2.14-2.18]. For instance, Thompson [2.14]
computed self and mutual radiation impedances of a uniformly vibrating annular or
circular piston by integrating of the far-field directivity function. Lee and Singh [2.15]
proposed a polynomial approximation for modal acoustic power radiation from a thin
annular disk using the far-field and radiation impedance approaches. Levine and
Leppington [2.16] developed an analytical solution for active and reactive powers from a
planar annular membrane given axisymmetric motions. Rdzanek and Engel [2.17]
39
suggested asymptotic formulas for power from a thin annular disk with clamped edges.
Finally, Wodtke and Lamancusa [2.18] investigated a circular plate using finite element
analysis and then calculated the radiation via the Rayleigh integral formula. However,
none of these studies have examined radiation from a thick annular disk.
If acoustic scattering from the edges of a vibrating structure is neglected, the far
and free field sound radiation from a planar radiator in an infinite baffle can be typically
calculated by the Rayleigh integral method as described below. With reference to Figure
2.9a, the sound pressure amplitude (P) is as follows where ρ0 is the mass density of air, c0
is the speed of sound, k is the acoustic wave number, rp and rs are the position vectors of
receiver and source positions, and W is the amplitude of vibratory velocity in the z
direction at rs.
ρ c k
P(r p ) = 0 0
2π
∫
e
ik r p − rs
W (rs )
r p − rs
Ss
dS (rs )
(2.35)
For an axially symmetric radiator, (m, n)th modal sound pressure can be expressed as
follows by simplifying equation (2.35) using the Hankel transform [2.19 - 2.20]. Here, Jn
is the Bessel function of order n, R = |rp| is the radius of sphere on which the observation
positions are defined, and θ and φ are the cone and azimuthal angles of the observation
positions.
ρ0c0 keik mn Rd
cos nφ(−i ) n +1 Β n [w (r )];
Pmn ( R, θ, φ) =
Rd
Β n [w (r )] =
∫
∞
0
w (r ) J n (kr r )rdr ; kr = k sin θ; Rd = rp − rs .
40
(2.36a-d)
z
(a)
Pmn(R, θ, φ)
Sv
rprs
W mn
rp
R
rs
Ss
(b)
Pmn(R, θ, φ)
z
Sv
rprs
W mn
γ
rp
rs
Ss
R
Figure 2.9: Sound Radiation from the out-of-plane vibration modes in the sherical
coordinate system. (a) a thin disk with an infinite baffle; (b) a thick disk without baffle.
41
For a non-planar source, the far and free field sound pressure can be expressed as
equation (2.37) based on the plane-wave approximation within the short-wavelength
limits [2..20] along with reference to Figure 2.9b:
ik r − r
ρ0c0 k e p s W (rs )
P ( rp ) =
(1 + cosγ ) dS
4π ∫S s
rp − rs
(2.37)
In our study, the observation positions are defined by a group of points having equal
angular increments (∆ϕ, ∆θ) on a sphere (SV) that is centered at the disk center and sound
pressures at all of the observation positions are calculated using equations (2.35), (2.36),
or (2.37). The modal directivity function Dmn(θ, φ) at frequency ωmn can be defined from
the modal pressure Pmn(rp) expression as follows.
Dmn (θ, φ) = RPmn ( R, θ, φ)eik mn R
(2.38)
From the far-field approximation, modal sound power Πmn of the (m, n)th mode is
calculated using the following equation.
2
Π mn = I mn SV
s
1 2π π P
= ∫ ∫ mn R 2 sin θ dθ dφ
2 0 0 ρ0c0
(2.39)
Here, Imn is the acoustic intensity, and Sv is area of the control surface. The modal
radiation efficiency σmn of an annular disk is determined from Πmn as follows where
< w mn
2
> t , s is the spatially averaged mean-square velocity on the two normal surfaces of
the annular disk.
42
σ mn =
Π mn
2
< w mn >t , s
2
< w mn >t , s =
;
(2.40a-b)
1
a
2π(a 2 − b 2 ) ∫b
∫
2π
0
W dϕ dr
2
mn
2.3.2 Thin Plate Approach
The classical method based on the thin plate theory does not consider the effect of
h in sound radiation calculation. In this case, h is assumed to be negligible relative to
other disk dimensions and sound radiation from the (m, n)th mode of a thick annular disk
is generally calculated by assuming the baffled condition. In addition, Rd in equations
(2.36) is approximated with R and equation (2.36) is simplified to yield the following
equation.
ρ 0 c 0 k mn e ik mn R
Pmn ( R, θ, φ) =
cos nφ( −i) n +1 Β n [w ( r )]
R
(2.41)
As an alternative to equation (2.36), the results of structural analysis of Chapter 2.2 may
be used. Define approximate mode shape Ψmn and modal surface velocity w mn as follows
where polynomial functions are used to describe the flexural vibrations.
N
ψ mn (r , ϕ) = cos( nϕ)∑ Cmn , s r s
s =0
(2.42a-b)
w mn (r , ϕ, t ) = W ( r , ϕ)eiω mn t = −ωmnCmn , s r s cos( nϕ)eiω mn t
Here, Cmn,s is an arbitrary constant. Substituting equation (2.37) into equation (2.36), the
far-field modal sound pressure Pmn is:
43
Pmn ( R, θ, φ) =
N
a
ρ 0 ck mn e ik mn Rd
cos nφ(−i ) n +1 ∑ C mn, s ∫ r s +1 J (k mn r sin θ) dr
b
Rd
s =0
(2.43)
By expressing the Bessel function of the first kind with corresponding power series,
modal sound pressure can be expressed as:
Pmn ( R, θ, φ) =
ρ0ckmneik mn Rd
cos nφ(−i ) n +1 ×
Rd
N
∞
∑∑
s =0 l =0
Cmn , s (−1)l (kmn sin θ) 2 m +1 b n + s + 2l + 2
l!(n + l )!2
2l + n
(n + s + 2l + 2)
(2.44)
(1 − β
n + s + 2l + 2
)
The analytical method based on the thin plate theory (Model K) uses equation (2.44) for
sound pressure calculations.
2.3.3 Thick Plate Approach
For the thick plate theory, we do not ignore the thickness (h) effect. Therefore,
sound radiations from two normal surfaces should be simultaneously considered. With
reference to equation (2.37) along with the far-field assumption, angle γ is approximated
by θ for the normal surface facing the field point and by -θ on the opposite side.
Consequently, sound pressure at rp is given by the sum as shown below:
ik r p − rs'


ik r − r

e
W (rs' )
ρ0ck  e p s W (rs )
P ( rp ) =
(1 + cosθ) + ∫
(1 − cosθ)  dS
 ∫S s
'
Ss
4π 
rp − rs
rp − rs



(2.45)
Here, rs’ is the position vector of a source point on the normal surface that is away from
the field point. In addition, |rp - rs| and |rp - rs’| are expressed as follows:
44
Rd = rp − rs =
h
hz
h
x 2 + y 2 + ( z − ) 2 ≈ R(1 − 2 ) ≈ R − cos θ;
2
2
R
Rd' = rp − rs' =
h
hz
h
x 2 + y 2 + ( z + ) 2 ≈ R(1 + 2 ) ≈ R + cos θ;
2
R
2
(2.46a-f)
R = rp ; x = R sin θ cos φ; y = R sin θ sin φ; z = Rcosθ;
According to the far-field condition, Rd and Rd’ in the denominator of equation (2.46)
can be approximated by R, but the numerator terms cannot be replaced with R especially
over the higher frequency range. Substituting (2.46) into (2.45) and employing the
Hankel transform used in equation (2.36), we find the following equations with a
simplified R expression in the denominator.
ρ0ckmn eik mn R − ik mn 2 cos θ
P ( R, θ, φ) =
e
cos nφ( −i ) n +1 Β n [w ( r )]
2R
h
s
mn
o
Pmn
( R, θ, φ) =
ρ0ckmn e
2R
ik mn R
e
ik mn
h
cos θ
2
(2.47a-b)
cos n(φ + π)(−i) n +1 Β n [w (r )]
As one can see from these equations, the disk thickness introduces a phase difference that
is equal to –kmn(h/2)cosθ for the surface facing the field point and is kmn(h/2)cosθ for the
surface away from the field point. The total far-field modal sound pressure is expressed
by a sum of sound radiations from two normal surfaces as follows:
o
Pmn ( R,θ , φ ) = (1 + cos θ ) Pmns ( R,θ , φ ) + (1 − cos θ ) Pmn
( R, θ , φ )
(2.48)
The semi-analytical (Model L) and analytical (Model J) methods based on the thick plate
theory consider the effect of h and use equation (2.48) for the calculation of sound
pressure.
45
2.4. Computational and experimental Investigations of Sound Radiation
Modal acoustic radiation properties such as acoustic frequency response functions
P/f(ω), modal acoustic power (Πmn), and modal radiation efficiency (σmn) of Disk I are
obtained using the analytical methods of Chapter 2.3. Furthermore, the same radiation
properties are calculated with uncoupled, direct, exterior, and unbaffled boundary
element analyses [2.21]. In the computational study (Model M), 6,146 acoustic field
points and 6,144 elements are defined on the sphere (Sv) surrounding the disk that is
represented by the finite element model for structural dynamics. The center of this sphere
coincides with the disk center. Excitations to this boundary element analysis (BEM) are
the normal velocity distribution W ( r , ϕ) on both normal surfaces that are obtained from
the forced vibration analysis using the finite element code (Model D). Analytical
predictions and numerical analyses are verified by comparing results with measured data
obtained from vibro-acoustic experiments conducted in an anechoic chamber as shown in
Figure 2.3. Far-field sound pressures are measured with a 6 mm microphone (MTS
L130C10 combined with pre-amplifier MTS 130P10) at predetermined field points on a
circle of R = 303 mm radius from the disk center in the plane of ϕ = 0° and θ = 90°.
Considering the symmetries of the pressure distributions, P is measured in the range of 0°
≤ θ ≤ 90° in the ϕ = 0° plane with an increment of ∆θ = 2.5° and over 0° ≤ ϕ ≤ 90° in the
θ = 90° plane with an increment of ∆φ = 5°. The same radius (303 mm) is used in
computational and analytical studies. Force and pressure signals are conditioned and
analyzed via a 2-channel dynamic signal analyzer (HP 35670A) to obtain p/f(ω) spectra
such as the one shown in Figures 2.10. The experimental directivity pattern D(θ, φ) on
46
the sphere SV is synthesized from measured P(θ, φ) data. Our analytical methods
accurately predict the far-field sound pressure distributions. This is illustrated in Figure
2.11 where P(θ, φ) results from alternate analytical procedures are compared with
measured and computed values for (0, 2) and (0, 3) modes.
Further, Table 2.6 compares directivity patterns in a pictorial form. Analytical
predictions of Πmn and σmn for two modes are compared with BEM code (Model C) and
measured results in Table 2.7.In the experimental case, results at the discrete points over
SV have been synthesized using the measured Pmn(θ, φ) data to yield Πmn along with σmn.
In this process, the measured Pmn(θ, φ) profile is assumed to have a perfect sinusoidal
variation in the φ direction. As shown in Tables 2.6 and 2.7, acoustic radiation properties
obtained using analytical solutions (Models J, K, and L) match well with computational
predictions (Model M) and measurements. The discrepancies between measured data and
analytical or numerical results can be explained by uncertainties in the acoustic
experiment. There are two important uncertainties in the acoustic experiment. The first
uncertainty is generated by frequency resolution in the experiment. Since this study
covers very wide frequency range (0 – 16kHz), relatively coarse frequency resolution (8
– 16 Hz) is used in the experiment. Consequently, measured sound pressure for a given
receiver position could lower than the actual sound pressure up to 6 dB. On the other
hand, analytical and numerical sound pressures are calculated at exact natural
frequencies. The other uncertainty is due to spatial resolution. Angular resolutions of 2.5
and 5.0° are used in the measurement that induces maximum 3.0 dB error in the sound
pressure. 2.5° angular resolution is used in the calculations and maximum error is 1.5 dB.
47
P/f (dB re 20µPa/N)
100
80
60
40
20
0
1
2
3
1
2
3
4
5
6
7
8
4
5
6
7
8
P/f (dB re 20µPa/N)
100
80
60
40
20
0
Frequency (kHz)
Figure 2.10: Acoustic frequency response function P/f(ω) given unit impulsive force
excitation f(t) in the z direction at r =151.5 mm. (a) θ = π/2 and φ = 0; (b) θ = 0 and φ = 0.
Key: –––, analytical calculation (Model L); - - -, computed using BEM (Model M);
measured. Here q = 0 is the radial mode.
48
Normalized Pressure
1
0.5
0
Normalized Pressure
0
30
60
90
30
60
90
1
0.5
0
0
deg
Figure 2.11: Directivity pattern Dmn(θ) given φ = 0 and R = 303 mm. (a) m = 0, n = 2
mode; (b) m = 0, n = 3 mode. Key:
DQDO\WLFDOPHWKRGEDVHGRQWKLFNSODWHWKHRU\
(Model J); ∆∆∆, analytical method based on thin plate theory (Model K); - - -, semianalytical method (Model L); , computed using BEM (Model M);
PHDVXUHG
49
Structural
Mode
(0, 2) Mode
+
-
-
(0, 3) Mode
+
+
+
-
+
Analytical
Method
Measured
Computed
using
BEM
Table 2.6: Comparison of directivity patterns for selected modes of Disk I.
50
Πmn, dB re 1 pW
Analytical
Thick Plate
Thin Plate
(Model J)
(Model K)
Measured
BEM
(Model O)
SemiAnalytical
(Model L)
(1, 1)
70.1
70.7
71.6
70.1
70.9
(0, 3)
75.3
76.3
76.2
75.1
74.8
Mode
(m, n)
Measured
BEM
(Model O)
SemiAnalytical
(Model L)
(1, 1)
0.77
0.75
1.08
1.23
1.06
(0, 3)
0.81
1.01
1.01
1.28
1.14
Mode
(m, n)
σmn
Analytical
Thick Plate
Thin Plate
(Model J)
(Model K)
Table 2.7: Comparison of modal acoustic power and radiation efficiency levels for
selected out-of-plane modes of Disk I.
51
2.5 Effect of key Parameters on the Modal Sound Radiation
As shown in Chapter 2.3, natural frequencies and modes of a thick annular disk
are affected by its geometry and boundaries. Furthermore, equations (2.44-2.45) illustrate
that Pmn depends on disk geometry, vibrating mode and frequency. In this Chapter,
effects of the radii ratio (β = b/a), thickness ratio ( h = h / a ), and boundary conditions on
modal sound radiation are studied through variations in < w mn
2
> t , s , Πmn and σmn as
introduced by changes in key parameters. As a first step, natural frequencies and mode
shapes corresponding to a specific geometric configuration are calculated using thin or
thick plate theory and then the modal surface velocities are defined from the
corresponding structural eigensolutions. Then, the modal far-field sound pressures are
calculated using equation (2.43) or (2.45). Finally, Πmn and σmn are obtained from the
sound pressure data using equations (2.39-2.40). In this particular study, the amplitudes
of modal vibrations are intentionally adjusted to get the same modal velocity amplitudes
regardless of variations in the natural frequencies for a given geometric configuration.
2.5.1 Effects of Radii Ratio
First, the effect of β is investigated using Disk I. The results of this investigation
are summarized in Figures 2.12 and 2.13 where < w mn
2
> t , s and σmn are significantly
affected by β. For a limiting case of β → 0 when the annular disk turns into a circular
disk, sound radiation can be solved using the same solution. And, for the other limiting
case when β → 1, the annular disk can be considered as a thin cylinder that cannot
generate sound with its out-of-plane vibration.
52
(b)
2
50
1
0
0.3
σ
2
2
(pm /s )
(a)
100
0.5
β
0.7
0
0.3
0.9
0.5
β
0.7
0.9
Figure 2.12: Effect of radii ratio on the modal sound radiation based on alternate plate
2
theories. (a) Spatially averaged mean-square velocity < w > t , s ; (b) Radiation efficiency
σ. Key:
PRGHZLWKWKLFNSODWHWKHRU\0RGHO- PRGHZLWKWKLQ
plate theory (Model K);
PRGHZLWKWKLFNSODWHWKHRU\0RGHO-- - -, (0, 3)
mode with thin plate theory (Model K).
53
(a)
(b)
3
2
2
2
(pm /s )
100
σ
50
1
0
0.3
0.5
β
0.7
0
0.3
0.9
0.5
0.7
0.9
β
Figure 2.13: Effect of radii ratio on the modal sound radiation based on alternate plate
2
theories. (a) Spatially averaged mean-square velocity < w > t , s ; (b) Radiation efficiency
σ. Key:
PRGHZith thick plate theory (Model J);
PRGHZLWKWKLQ
plate theory (Model K);
PRGHZLWKWKLFNSODWHWKHRU\0RGHO-- - -, (1, 1)
mode with thin plate theory (Model K).
54
As shown in Figure 2.12, < w mn
2
> t , s and σmn for (0, 2) and (0, 3) modes converge to 0
as β → 1 irrespective of plate theories. But, as shown in Figure 2.13, σmn values for (1, 0)
and (1, 1) modes based on thin plate theory significantly fluctuate with β even in the case
of β → 1 though the corresponding < w mn
2
> t , s values monotonically decrease.
Conversely, σmn values for (1, 0) and (1, 1) modes based on the thick plate theory do not
show much fluctuation with β. It is conceivable that Πmn values based on the thin plate
theory fluctuate with β but the same values based on the thick plate theory are stable.
2.5.2 Effects of Thickness
Next, h is selected as an independent variable and it is varied from 0.025 to 0.35
with a nominal value of h0 = 0.21. Results are summarized in Figures 2.14 and 2.15
where one can observe considerable variations in σmn. The < w mn
2
> t , s values based on
the thin plate theory are constant irrespective of the mode type. Conversely, the same data
based on the thick plate theory are mode dependent. For instance, < w mn
2
> t , s is
proportional to h for (0, 2), (0, 3) or (1, 0) modes, but is inversely proportional to h for
(1, 1) mode. As shown in Chapter 2.2, natural frequencies of the out-of-plane modes are
proportional to h. If the disk thickness is small enough such that the natural frequency for
a specific mode is below the critical frequency, the modal sound radiation is very low
[2.20]. For this reason, σmn values are very low in the region of small h regardless of the
mode type as shown in Figures 2.14 and 2.15.
55
(a)
(b)
1.5
1
2
2
(pm /s )
150
σ
125
0.5
100
0
0
0
0.2
Á
0.2
Á
Figure 2.14: Effect of thickness ratio on the modal sound radiation based on alternate
2
plate theories. (a) Spatially averaged mean-square velocity < w > t , s ; (b) Radiation
efficiency σ. Key:
PRGHWKLFNZLWKSODWHWKHRU\0RGHO- PRGH
with thin plate theory (Model K);
PRGHZLWKWKLFNSODWHWKHRU\0RGHO-- -, (0, 3) mode with thin plate theory (Model K).
56
(a)
(b)
1.5
1
2
2
(pm /s )
50
σ
25
0.5
0
0
0
0
0.2
h
0.2
h
Figure 2.15: Effect of thickness ratio on the modal sound radiation based on alternate
2
plate theories. (a) Spatially averaged mean-square velocity < w > t , s ; (b) Radiation
PRGHZLWKWKLFNSODWHWKHRU\0RGHO- PRGH
efficiency σ. Key:
with thin plate theory (Model K);
PRGHZLWKWKLFNSODWHWKHRU\0RGHO-- -, (1, 1) mode with thin plate theory (Model K).
57
Furthermore, for the thick plate theory, phase difference between sound pressures
radiated from two normal surfaces is proportional to h and the effect of this phase
difference should be considered in addition to the effect of natural frequency change. For
the thin plate theory that considers sound pressure from only one normal surface,
radiation is affected only by the natural frequency variation.
2.5.3 Effects of Boundary Conditions
The effect of fixed–free boundary conditions on sound radiation is finally studied.
Typical Πmn and σmn of two out-of-plane modes for Disk I with either free-free or fixedfree boundary conditions are listed in Table 2.8. The modal acoustic powers and radiation
efficiencies for modes with the same number of nodal diameters (n) significantly change
when the inner edge is clamped. For example, Π02 and σ02 increase with fixed-free
boundaries due to the increases in the corresponding natural frequencies. For instance,
ω02 goes up from 1.31 kHz (below the critical frequency that is around 2.0 kHz) to 4.85
kHz (above the critical frequency). Also, Πmn and σmn for n = 0 and n = 1 modes,
significantly increase due to the elimination of a nodal circle in the corresponding mode
shapes. In addition, directivity patterns of two sample modes with fixed-free boundaries
are numerically calculated and compared in Figure 2.16 with those with free-free
boundaries. The application of fixed-free boundary conditions increases the number of
ripples in the θ direction for both modes.
58
Boundaries
Free - Free
Fixed - Free
Mode
Indices
ωmn (kHz)
Πmn
(dB re 1 pW)
σmn
(0, 2)
1.31
70.9
0.263
(1, 0)
2.95
67.2
0.379
(0, 3)
3.41
76.3
1.030
(1, 1)
4.61
70.7
0.808
(0, 0)
4.62
69.7
0.900
(0, 1)
4.63
73.4
0.884
(0, 2)
4.85
73.2
0.878
(0, 3)
5.71
73.7
0.883
Table 2.8: Modal acoustic powers and radiation efficiencies for first four out-of-plane
modes with fixed-free or free-free boundaries.
59
Normalized Pressure
1
0.5
0
Normalized Pressure
0
30
60
90
30
60
90
1
0.5
0
0
θ (deg)
Figure 2.16: Modal directivity patterns of Disk I with alternate boundary conditions. (a) n
= 2 modes; (b) n = 0 modes. Key: –––, fixed-free; - - -, free-free boundary condition.
60
2.6 Conclusion
This chapter has proposed a new analytical solution that explicitly considers the
disk thickness effect on sound radiation from out-of-plane modes. In addition, our semianalytical procedure combines the computationally obtained disk surface velocities with
analytical solutions for sound radiation. A comparative evaluation of thin and thick plate
theories shows that the thick plate theory is more accurate when predictions are compared
with computational codes (such as FEM and BEM) and vibro-acoustic experiments. Our
procedure can be efficiently used to conduct parametric studies such as the ones reported
in this chapter by varying the radii or thickness ratio. In particular, one can easily analyze
the limiting cases of a circular plate and a thin cylinder considering only the out-of-plane
flexural modes. In a future chapter, we will simultaneously consider both out-of-plane
and in-plane components of the disk vibration. Modal interaction effects and sound
radiation from coupled modes will also be studied.
61
REFERENCES FOR CHAPTER 2
2.1 A. W. LEISSA 1969 NASA SP-160 Vibration of Plates.
2.2 A. W. LEISSA 1987 The Shock and Vibration Digest 19(3), 10-24. Recent Research
and plate vibration, 1981-1985. Part 1: Classical theory.
2.3 A. W. LEISSA 1987 The Shock and Vibration Digest 19(3), 10-24. Recent research
and plate vibration, 1981-1985. Part 2: Complicating effects.
2.4 A. W. LEISSA
America.
1993 Vibrations of Plates, New York: Acoustical Society of
2.5
S. M. Vogel and D. W. Skinner 1965 Journal of Applied Mechanics December,
926-931 Natural frequencies of transversely vibrating uniform annular disk.
2.6
R. D. MINDLIN 1951 ASME Journal of Applied Mechanics 18, 31-38 Influence of
rotatory inertia and shear on the flexural motion of isotropic, elastic plate.
2.7
R. D. MINDLIN and H. DERESIEWICZ 1954 Journal of Applied Physics 25(10), 13291332 Thickness-shear and flexural vibration of a circular disk.
2.8 O. G. MCGEE, C. S. HUANG and A. W. LEISSA 1995 Journal of Sound and Vibration
163(1), 137-149 Comprehensive exact solutions for free vibrations of thick annular
sectorial plates with simply supported radial edges.
2.9 T. IRIE, G. YAMADA and K. TAKAGI 1982 Transactions of the American Society of
Mechanical Engineers, Journal of Applied Mechanics 49, 633-638 Natural
frequencies of thick annular plates
2.10 IRIE, G. YAMADA and S. AOMURA 1979 Journal of Sound and Vibration 66 (1),
187-197 Free vibration of a Mindlin annular plate of varying thickness.
2.11 C. M. WANG and V. THEVENDRAN 1993 International Journal of Mechanical
Science 37(5), 537-566 Vibration analysis of annular plates with concentric support
using a variant of Rayleigh-Ritz method.
2.12 K. M. LIEW, Y. XIANG, C. M. WANG AND S. KITIPORNCHAI 1993 Computer
Methods in Applied Mechanics and Engineering 110, 301-315 Flexural vibration of
shear deformable circular and annular plates on ring support.
2.13 I-DEAS User’s manual version 8.2. 2000 SDRC, USA.
2.14 W. THOMPSON, JR. 1971 Journal of Sound and Vibration 17(2), 221-233. The
computation of self- and mutual-radiation impedances for annular and elliptical
pistons using Bouwkamp integral.
62
2.15 M. R. LEE and R. SINGH 1994 Journal of the Acoustical Society of America 95(6),
3311-3323. Analytical formulations for annular disk sound radiation using
structural modes.
2.16 H. LEVINE and F. G. LEPPINGTON 1988 Journal of Sound and Vibration 121(5),
269-275. A note on the acoustic power output of a circular plate.
2.17 W. P. RDZANEK Jr. and Z. ENGEL 2000 Applied Acoustics 60(5), 29-43. Asymptotic
formula for the acoustic power output of a clamped annular plate.
2.18 H. W. WODTKE and J. S. LAMANCUSA 1998 Journal of Sound and Vibration 215(5),
1145-1163. Sound power minimization of circular plates through damping layer
placement.
2.19 H. LEE and R. SINGH 2002 submitted to Journal of Sound and Vibration Acoustic
radiation from out-of-plane modes of an annular disk using thin and thick plate
theories, submitted to Journal of Sound and Vibration.
2.20 M. C. JUNGER and D. FEIT 1985 Sound, Structures, and Their Interactions. New
York: MIT Press.
2.21 SYSNOISE User’s manual Version 5.4. 1999 NIT, Belgium.
63
CHAPTER 3
SOUND RADIATION FROM IN-PLANE VIBRATION
3.1 Introduction
Many mechanical or structural components such as gears, brake rotors, clutches,
flywheels, railway wheels, circular saws, electrical machinery stators, and electroacoustic transducers can be idealized as annular disks. Often thick bodies are encountered
and one must examine radiation from both out-of-plane vibrations and in-plane vibrations
to appropriately control sound radiation from such components. Even though many
researchers have studied structural dynamics of annular disks, published literature on the
in-plane oscillations of an annular disk is relatively sparse [3.1 – 3.4]. Bhuta and Jones
[3.1] studied coupled symmetric and torsional vibrations of a thin, rotating circular disk.
Burdess et al. [3.2] generalized the analysis to consider asymmetric in-plane vibrations
and the properties of forward and backward traveling waves. Chen and Jhu [3.3]
examined in-plane vibration and stability of a spinning annular disk and determined the
effect of rotational speed on the natural frequencies. Irie et al. [3.4] calculated natural
frequencies of annular disks using the transfer matrix method. We will use this method
for the structural analysis of thick annular disks.
64
Sound radiation from thin circular and annular disks has been examined by several
investigators [3.5-3.9]. For instance, Thompson [3.5] computed self and mutual radiation
impedances of a uniformly vibrating annular or circular piston by integrating of the farfield directivity function. Lee and Singh [3.6] proposed a polynomial approximation for
modal acoustic power radiation from a thin annular disk but this method was restricted to
only out-of-plane modes. Levine and Leppington [3.7] developed an analytical solution
for active and reactive powers from a planar annular membrane given axisymmetric
motions. Rdzanek and Engel [3.8] suggested asymptotic formulas for power from a thin
annular disk with clamped edges. Finally, Wodtke and Lamancusa [3.9] investigated a
circular plate using finite element analysis and then calculated the sound radiation via the
Rayleigh integral formula. However, almost all of the above mentioned studies have
considered sound radiation only from either flexural vibration modes or rigid body piston
motions. In such studies, sound radiation from the in-plane modes of a disk has been
assumed to be negligible compared to that from the out-of-plane modes. But, if the
thickness of a disk is beyond the range of thin plate (shell) theory, in-plane vibration
could generate sufficient sound given proper excitation.
This chapter proposes analytical solutions for sound radiation from radial
vibration modes of a thick annular disk. Structural eigensolutions are calculated using the
transfer matrix method and then the surface velocities at outer and inner radial edges are
determined using the modal expansion. The far-field sound pressure distribution is
obtained using two alternate methods. In the first method, pressure is calculated using the
Rayleigh integral technique. The second method treats sound radiating radial surfaces as
cylindrical radiators of finite length. The Fourier series and Sinc function approaches are
65
employed for calculations. Acoustic powers and radiation efficiencies of radial modes are
also determined from the far-field sound pressure calculations. Analytical predictions
match well with measured data as well as computational results from a finite element
code in terms of structural eigensolutions and from a boundary element code in terms of
sound pressure, directivity etc. Selected parametric studies include the effects of disk
geometry and natural frequencies on the radiation properties. Limiting cases are also
examined.
3.2 Problem Formulation
Chief objective of this chapter is to develop analytical solutions for acoustic
radiation from the radial structural modes of a thick annular disk. Figure 1 illustrates the
example case that is assumed to be stationary and free at the inner and outer edges. Table
1 provides sample values used for analytical and experimental studies. Primary
assumptions are as follows: 1. Structural and acoustic systems are linear time-invariant
systems and the scattering effect is negligible. 2. Free field sound pressure at the
observation point is generated only by the radial velocities on inner (r = b) and outer (r =
a) edges and other surfaces do not contribute to the far-field sound pressure. 3. Vibration
amplitude of the radial surfaces is uniform in the z direction.
Structural eigensolutions are analytically obtained using the transfer matrix
method. Then modal expansion is used to determine radial velocities for a given
harmonic force fr(t). In the first acoustic radiation analysis method, the far-field sound
pressure is calculated from the Rayleigh integral formula. In the second method,
pressures are obtained by treating sound radiating surfaces as two cylindrical radiators of
66
finite lengths. Then Fourier series and Sinc function approaches are applied. The problem
is also analyzed using numerical (finite and boundary element) codes. Relevant
computational and analytical results are compared with vibro-acoustic measurements.
Acoustically efficient radial structural modes may be determined based on modal
radiation efficiency and acoustic power calculations. Strategies for minimization of sound
radiation are also investigated by parametrically varying disk geometry and natural
frequencies.
3.3 Structural Analysis
For in-plane vibration analysis of a thick annular disk, the cylindrical coordinate
system (r, ϕ, z) is employed where z is normal to the disk as shown in Figure 3.1. Normal
and shear stresses in the z direction (σzz = σrz = σϕz = 0) are ignored. The equations of
motion corresponding to the plane stress condition can be obtained from the following:
∂σ rr 1 ∂σrϕ σrr − σϕϕ
+
+
= ρ d u
r ∂ϕ
r
∂r
(3.1)
1 ∂σϕϕ ∂σ rϕ 2
+
+ σ rϕ = ρd v
r ∂ϕ
r
∂r
(3.2)
 u 1 ∂v 
E  ∂u
 +

+
ν

1 − ν 2  ∂r
 r r ∂ϕ 
(3.3)
σ ϕϕ =
E  ∂u u 1 ∂v 
+ +
ν

1 − ν 2  ∂r r r ∂ϕ 
(3.4)
σ rϕ =
E  1 ∂u ∂v v 
+
− 

2(1 + ν )  r ∂ϕ ∂r r 
(3.5)
σ rr =
67
fr(t)
u
r ϕ
b
z
a
v
h
Figure 3.1: A thick annular disk with a radial force
68
Outer radius (a)
151.5 mm
Inner radius (b)
87.5 mm
Radii ratio (β = b/a)
0.54
Thickness (h)
31.5 mm
Density (ρd)
7905.9 Kg/m3
Young’s Modulus (E)
218 GPa
Poisson’s Ratio (ν)
0.305
Table 3.1: Disk dimensions and material properties
where u and v are radial and circumferential displacements, σrr and σϕϕ are the normal
stresses and σrϕ is the shear stress. The components of in-plane force can be defined from
u and v as
 ∂u ν 
∂v 
,
N r = D  +  u +
∂ϕ 
 ∂r r 
1 
∂u 
∂v  ∂u
 + ν ,
N ϕ = D   u +
∂r 
∂ϕ  ∂r
r 
N rϕ = N ϕr =
(3.6a-c)

(1 − ν) D  ∂v 1  ∂u
− v ,
 + 
2

 ∂r r  ∂ϕ
where Nr and Nϕ are the normal forces in radial and tangential direction and Nrϕ is the
shear force. Also refer to the list of symbols given at the end of this chapter for the
identification of symbols. In case of the qth radial mode at natural frequency ωq (rad/s),
69
equations (3.1) and (3.2) may be rearranged to yield following equations in dimensionless
{
form, where {z(ξ)} = u , v, Nrϕ , Nr
}
T
and [U(ξ)] are the state vector and the coefficient
matrix respectively.
d
{z (ξ)}= [U (ξ)]{z (ξ)}
dξ
ν

−

ξ

q


ξ
[U (ξ)]=  q(1 − ν 2 )

ξ2

 (1 − ν 2 )
2
− λq

2
 ξ
qν
ξ
1
ξ
2
q (1 − ν 2 )
2
− λq
2
ξ
q (1 − ν 2 )
ξ2
−
(3.7)
0
2
1− ν
2
−
ξ
q
−
ξ



0 

qν 

ξ 
1− ν
−

ξ 
1
(3.8)
u = au cos qϕ, v = av sin qϕ, Nr = DNr cos qϕ, Nϕ = DNϕ cos qϕ,
Nrϕ = DNrϕ sin qϕ, Nϕr = DNϕr sin qϕ,
(3.9a-i)
ξ = r / a, λq = (2ρd La2ω2q / D)1 / 2 , D = Eh /(1 − ν2 )
In equation (3.8), the non-dimensional frequency parameter λq is introduced. One may
define the transfer matrix [T(ξ)] in the following form, where ξ and β = b / a represent
normalized radial coordinates at an arbitrary radial location and at the inner edge
respectively.
{z (ξ)}= [T (ξ)]{z(β)}
(3.10)
Using equation (3.7) and (3.10), transfer matrix at any given frequency can be obtained
from the following relation.
70
ξ
ξ
ξ
[T (ξ)]= exp ∫β [U (ξ′)]dξ′  = [I ]+ 1  ∫β [U (ξ′)]dξ′ + 1  ∫β [U (ξ′)]dξ′


1! 

2! 

2
+ ⋅⋅
(11)
By applying the boundary conditions at ξ = β and ξ = 1, [T(ξ)] of equation (3.10) is
reduced to a 2×2 sub-matrix. In our study, we employ the procedure of reference [3.4] to
find the eigensolutions. Given the free boundary conditions at both radial edges,
N r = N rϕ = 0 , equation (3.9) can be simplified as follows:
T31


T41
T32  u 
0
  =  
 
 


T42  v ξ =β 0
(3.12)
where Tij in the sub-matrix is the (i, j)th element of the original [T(ξ)] matrix. The
characteristic equation L(λ) is obtained from equation (3.12) and eigenvalues λq for inplane modes may be determined from the relation L(λ) = 0. Typical L(λ) curves for q = 2
and q = 3 modes are plotted in Figure 3.2. The analytical solutions obtained by the
transfer matrix method are compared in Table 3.2 with the results of numerical (finite
element) analysis and structural modal experiment. In the finite element analysis, 6 inplane modes (including 2 circumferential modes) are obtained with a model that includes
4,400 solid elements and 6,600 nodes [3.10]. In modal experiments, the radial excitation
force f(t) is applied by an impulse hammer at ϕ = 0° in the mid-plane of the disk. Natural
frequencies (ωq) and modal damping ratios (ςq) are extracted from structural frequency
response functions such as u /f(ω) where u is the acceleration and f is the applied force.
The upper frequency limit of finite element analysis and modal experiment is 16 kHz.
Excellent agreement is evident between analytical, numerical, and experimental
71
eigensolutions. According to the assumptions given earlier, radial displacement of two
edges for the qth mode can be expressed as follows where |uq| is the displacement
amplitude and ωq is the corresponding angular frequency.
uq (ϕ, t ) = U q (ϕ)e
iω q t
= uq cos(qϕ)e
iω q t
(3.13)
The above expression is consistent with the assumption of radial motion only and further
analyses of sound radiation are executed based on this equation. Radial component of the
computed mode shapes, from the finite element analysis, are compared in Figure 3.3 with
the assumed analytical solution given above. Considering the symmetry of modes, only
half of the vibration patterns are plotted in this figure from φ = 0° to φ = 180°. To
calculate the far-field sound pressure from the disk vibration, Uq for a specific excitation
should be defined. The harmonic response of the disk for a point force f r (t ) = Feiωt at rf =
(rf, ϕf) can be expressed as u (r , ϕ; t ) = U (r , ϕ; ω)e −iωt . In frequency domain, U (r , ϕ; ω) can
be theoretically calculated via the modal expansion as follows:
F ∞ Φ q (rf , ϕ f )Φ q (r , ϕ)
U (r , ϕ; ω) = −
∑
ρ d h q =1 (ω2 − ω2q ) + i 2ς q (ω / ωq )
T
(3.14)
where Φq is the eigenfunction and ςq is the modal damping ratio. In our study, the
displacement amplitudes are obtained from a numerical synthesis of forced vibration
response based on finite element results. Computed natural frequencies and mode shapes
are combined with experimentally obtained ςq values for modal expansion. To find the
frequency response function, unit harmonic concentrated force f r (t ) = e
iω q t
is applied to the mid-plane in the radial direction at ϕ = 0° (see Figure 3.1).
72
at a given ωq
Figure 3.2: L(λ) for q = 2 and q = 3 modes. Key: –––, q = 2 mode; - - -, q = 3 mode.
Figure 3.3: Comparison of radial mode shapes Key: solid line, analytical solution given
by equation (3.13); discrete point, finite element analysis.
73
Non-dimensional Frequency λq = ωq(ρdha2 /D)1/2
Mode
q
Transfer Matrix
Finite Element
Experiment
2
0.493
0.498
0.489
3
1.193
1.207
1.185
0
1.296
1.271
1.254
Table 3.2: Comparison of disk eigen-solutions for radial modes
Mode
q
Frequency (kHz)
|uq| (µm)
2
2.86
0.063
3
7.00
0.028
0
7.26
0.015
Table 3.3: Modal displacement amplitudes used for acoustic analysis
74
The result of forced vibration analysis is given in Table 3.3. The structural frequency
response functions u /f(ω) for specific locations are also obtained via numerical analysis.
Figure 3.4 compares computed and measured u /f(ω) spectra and a good agreement over
the given frequency range is observed. Each dominant peak in this figure corresponds to
radial mode whose frequencies are given in Table 3.2.
(dB re 20 µm/s2 N)
150
125
100
75
1.6
3.2
4.8
Frequency (kHz)
6.4
8
Figure 3.4: Structural frequency response functions ü/f(ω) at j = 0. Key: ––––, measured;
- - - -, computed using BEM.
75
3.4 Acoustic Radiation Model
The sound pressure at the observation location p(rp;t) can be expressed as P(rp)
e-iωt where P(rp) is the spatially-dependent pressure at ω. With reference to Figure 3.5,
P(rp) in the far and free fields due to a vibrating structure with normal surface
acceleration U (rs ) at source point rs can be expressed by the Helmholtz integral equation
[3.11].
 ∂g

P(rp ) = − ∫  P
+ ρ0U (rs ) g dS (rs )
ss
 ∂γ

(3.15)
In this equation, g is the free space Green’s function, ρ0 is the medium density and Ss is
source surface (see Figure 3.5). The first and second terms in equation (3.15) represent
the sound pressure generated at rp by the surface pressure at rs and surface acceleration ü
at rs respectively. If the field point is sufficiently far from the source (k|rp| >> 1), one can
express the amplitude of acoustic particle velocity of as P/ρ0c0 where k = ω/c0 is the
acoustic wave number and c0 is the sound speed of the medium. Furthermore, since
particle velocity is in-phase with the sound pressure in the far-field, the sound intensity at
the same location can be uniquely defined as I = P2/2ρ0c0. The sound power Π(ω) from a
structure vibrating with ω can be found by integrating the far-field sound intensity over
the surface Sv that surrounds the source. Acoustic radiation resistance ℜ(ω) can then be
obtained from Π(ω) and spatially-averaged mean square radial velocity < u 2 > t , s as
follows [3.12]:
ℜ(ω) =
Π (ω)
= σ(ω)ρ 0 c0 As
< u 2 > t , s
76
(3.16)
Sv
ü
γ
V
rp rs
rp
rs
z
Ss
Figure 3.5: Spherical sound radiation from a vibrating disk.
77
where σ(ω) is the acoustic radiation efficiency, As is the area of the radiator, and < >t,s
implies temporal and spatial averages.
3.5 Method I: The Rayleigh Integral Approach
Without restrictions on the source configuration and frequency range, the surface
pressure distribution must be obtained through a numerical calculation. If the frequency
range is restricted to short-wavelength limit, the solution of original Helmholtz integral
equation can be circumvented. With this assumption, equation (3.15) can be simplified as
the following equation that can be solved without using a numerical method [3.11].
ik r − r
ρ0ck e p s Ur (rs )
P ( rp ) =
(1 + cosγ )dS (rs )
4π ∫S s
rp − rs
(3.17)
In our study, sound radiation from the qth radial mode of a thick annular disk is calculated
by assuming the unbaffled condition. Using the mode shape of equation (3.13), normal
acceleration on the two radial surfaces is expressed as follows:
− iω t
− iω t
− iω t
uq (ϕ, t ) = U (ϕ)e q = uq cos( qϕ)e q = −ω q2 u q cos(qϕ) exp e q
(3.18)
If the sound generating surfaces are discretized into small elements dS of constant üq,
P(rp) from the structure can be easily calculated using equation (3.17). For the annular
disk case, dS(rs) in equation (3.17) can be expressed by dS(rs) = 2πa dϕ dz and dS(rs) =
2πb dϕ dz for the outer (r = a) and inner (r = b) radial surfaces respectively. The total
sound pressure P at rp can be calculated by integrating sound pressure generated by each
element over the entire source surface. In this study, numerical integration is used to
solve for the sound pressure distribution.
78
p(R, θ, φ)
z
rp
θ
R
y
h
ϕ
x
Figure 3.6: Sound Radiation from the radial vibration of a thick annular disk in spherical
coordinate system.
79
The size of dS element should be selected according to the frequency of vibration. If
characteristic dimension of the element is larger than 0.5λ, where λ = 2π/k is the acoustic
wavelength, the far-field sound pressure will have some errors and consequently acoustic
radiation properties including the directivity patterns will be distorted. In our study,
observation positions are defined by a group of points having equal angular increments
(∆ϕ, ∆θ) on a sphere that is centered at the disk center. Sound pressures at all of the
observation positions are calculated using equation (3.17). With computed far-field sound
pressure data, acoustic directivity patterns for all structural modes are obtained. For
spherical radiation, the modal directivity function Dq(θ, φ) at ωq can be defined from
modal pressure Pq(rp) as follows where R = |rp| is the radius of sphere on which
observation positions are defined.
Pq ( R, θ, φ) =
e
ik q R
R
D q ( θ, φ)
(3.19)
From the far-field approximation, modal power Πq for the qth radial mode is also
calculated from modal pressures on a sphere surrounding the disk by using the following
equation where θ and φ are the cone and azimuthal angles of the observation positions.
2
Π q = I sq S
s
=
1 2 π π Pq
R 2 sin θ dθ dφ
∫
∫
0
0
2
ρ0c0
(3.20)
Modal acoustic radiation resistance ℜq is calculated using equation (3.16) where
< u q
2
> t ,s =
h / 2 2π( a+b)
1
2
U q dl dz
∫
∫
h
/
2
0
−
4πh(a + b)
(3.21)
Based on equation (3.16), the modal radiation efficiency σq of an annular disk is
determined as follows where 2πh(a + b) is the total area of radiating surfaces.
80
σq =
ℜq
2ρ 0 c0 πh ( a + b)
(3.22)
3.6 Method II: Cylindrical Radiator
3.6.1 Formulation
Outer and inner radial surfaces of the annular disk are treated as two separate
cylindrical radiators of finite lengths that have uniform surface velocity amplitudes in the
thickness direction (z in Figure 3.7). Sound radiation from a cylindrical radiator has been
analytically studied by several researchers [3.11, 3.13-3.14]. Junger and Feit developed
expressions for far-field sound pressures for finite and infinite cylindrical radiators given
arbitrary surface velocity distribution [3.11]. Williams et al. [3.13] used semi-analytical
method with finite series of eigenfunctions for boundary condition to calculate the
acoustic radiation from a finite cylinder. Sandman [3.14] investigated sound radiation
from finite cylindrical shells and found that cylindrical baffle has very little influence on
sound radiation and concluded that the baffled cylindrical geometry may be assumed to
be a reasonable approximation for this problem. Stepanishen [3.15] combined Green’s
function and Fourier integral technique to develop integral expressions for the
generalized radiation impedance and radiated power and applied it to an infinite cylinder.
Williams solved the same problem using a 2-dimensional Fourier transform [3.16].
Finally, Wang and Lai calculated the modal-averaged radiation efficiency of a finite
length circular cylindrical shell [3.17]. In our approach, the far-field sound pressure has
been calculated based on the procedure proposed by Junger and Feit [3.11] along with the
approximation suggested by Sandman [3.14].
81
p(rp, φp, zp)
zp
rp
φp
z
h
Figure 3.7: Cylindrical radiator of length h using cylindrical coordinate system.
82
Therefore, we briefly summarize this procedure in order to clarify the proposed method.
Junger and Feit [3.11] analyzed a cylindrical radiator of length h that has arbitrary
acceleration distribution Z(z) in the z direction and a sinusoidal distribution (cosqϕ) in
the circumferential direction ϕ. Modal surface accelerations can be expressed in the
cylindrical coordinate system excluding time dependency as
Ur ( z, ϕ) = ur Z ( z ) cos nϕ
(3.23)
If the Fourier transform [ℑ] in z direction is applied to the Helmholtz equation that is
expressed in cylindrical coordinates, the partial differential equation governing the
~
Fourier transformed pressure P ( r, φ; k z ) = ℑ[P( r, φ, z )] can be obtained as follows:
 ∂2 1 ∂
1 ∂2  ~
2
2
 2 +
+ k − k z + 2 2  P (r , φ; k z ) = 0
r
r
r
r ∂φ 
∂
∂

(3.24)
where kz is structural wave number in the z direction. The solution to this equation can be
assumed as
~
P (r , φ ; k z ) = AH n1 [(k 2 − k z2 ) 1 / 2 r ] cos nφ
(3.25)
where H n1 is the Hankel function of order n. The coefficient A in this equation can be
obtained from boundary conditions at r = a and r = b.
~
∂P (r , φ; k z )
~
= −ρ 0 u Z (k z ) cos nφ, at r = a and r = b,
∂r
h/2
~
where
Z(k z ) = ∫ Z ( z )e −ik z z dz
(3.26a-b)
−h / 2
Consequently, A can be expressed as follows where r̂ is the radius of given boundary
surface.
83
~
− ρ0 u Z (k z )
A=
′
2
2
(k 2 − k z )1 / 2 H n1 [(k 2 − k z )1 / 2 rˆ]
(3.26c)
(
[ ])
~
Next, P(r, φ, z) can be calculated by taking the inverse Fourier transform P = ℑ −1 P of
equation (3.25) and the final equation is given as
~
ik z
1
2
2 1/ 2
∞ e z H [(k − k )
ρ0
r ]Z (k z )dk z
n
z
P(r , φ, z ) = −
u cos nφ∫
−∞
′
2
2
2π
(k 2 − k )1 / 2 H 1 [(k 2 − k )1 / 2 rˆ]
z
n
(3.27)
z
After transforming this equation into spherical coordinates by setting r = Rsinθ and z =
Rcosθ, the following expression is obtained using the stationary phase approximation.
~
ρ 0 e ikR
Z (k z )(−i) n +1
cos nφ
P ( R, θ, φ) =
u
πkR sin θ H 1′ (krˆ sin θ)
n
(3.28)
As a special case, if Z(z) is a simple sinusoidal function as equation (3.29), Fourier
~
transform Z (k z ) and P ( R, θ, φ) can be obtained as equations (3.30) and (3.31) where km
= (1+2m)π / h for m =0, 1, 2,⋅ ⋅ ⋅.
Z ( z ) = cos km z,
z <h/2
= 0,
z > h/2
2k ( −1) m cos(k z h / 2)
~
Z (k z ) = m
2
2
km − kz
P ( R, θ, φ) =
n +1
2ρ0eikR km ( −1) m cos(kh cos θ / 2) u (−i ) cos nφ
′
πkR sin θ(k m2 − k 2 cos 2 θ)
H n1 ( krˆ sin θ)
84
(3.29)
(3.30)
(3.31)
3.6.2 Sinc Function Approach
Next consider a thick annular disk case in which the modal surface acceleration of
two radial surfaces is given as Uq (ϕ) = uq cos(qϕ) . In this case, variation in the z
direction can be expressed via a square pulse (rectangular) function. Accordingly, Z(z)
can be expressed as equation (3.32) and the Fourier transform of this equation is obtained
as equation (3.33).
Z ( z ) = 1,
z < h/2
= 0,
z > h/2
sin(k z h / 2)
sin(k z h / 2)
~
Z ( k z ) = ℑ[ Z ( z )] = 2
=h
= h Sinc( k z h / 2)
kz
kzh / 2
(3.32)
(3.33)
where the Sinc function is defined as Sinc(x) = sin(x)/x. Sound pressure PqO from the
outer radial surface and PqI from the inner radial surface are generated by diverging and
converging waves respectively. These are expressed by the Hankel functions of the first
and the second kind respectively [3.16]. Therefore, using equations (3.28) and (3.33), we
get
ik R
Sinc ( k q sin θ h / 2)( −i ) q +1
ρ0e q
PqO ( R, θ, φ) =
uq h
cos qφ
1′
πk q R sin θ
H (k a sin θ)
(3.34a)
ik R
Sinc ( k q sin θ h / 2)( −i ) q +1
ρ0e q
PqI ( R, θ, φ) =
uq h
cos qφ
2′
πk q R sin θ
H q ( k q b sin θ)
(3.34b)
q
q
And, the total modal sound pressure is the sum of sound pressures from two radial edges
as follows.
85
Pq ( R, θ, φ) = PqI ( R, θ, φ) + PqO ( R, θ, φ)
(3.35)
Other modal radiation properties such as Πq, ℜq and σq are calculated based on equations
(3.16), (3.20) and (3.22).
3.6.3 Fourier Series Approach
The far-field sound pressure can also be calculated via yet another procedure. In
the Fourier series approach the square pulse function Z(z) is approximated by a Fourier
series of 15 components.
4 14
1
Z ( z ) = ( ) ∑ (−1) m (
) cos km z ,
π m=0
2m + 1
= 0,
z < h/2
(3.36)
z > h/2
Using equation (3.31), the sound pressures generated by the mth component of equation
(3.36) are calculated as
PqOm ( R, θ, φ) =
Pq Im ( R, θ, φ) =
q +1
k m ( −1) m cos( k q h cos θ / 2) uq ( −i ) cos qφ
2
′
πk q R sin θ( k m2 − k q cos 2 θ)
H q1 ( k q a sin θ)
(3.37a)
q +1
k m (−1) m cos( k q h / 2 cos θ) uq ( −i) cos qφ
2
′
πk q R sin θ( k m2 − k q cos 2 θ)
H q2 ( k q b sin θ)
(3.37b)
2ρ 0 e
2ρ 0 e
ik q R
ik q R
where PqOm and PqIm are sound pressure from the outer and inner edges due to the mth
component. The total PqO (and likewise PqI) at a given rp is obtained from a summation of
PqOms and PqIms as
86
PqO ( R, θ, φ) =
14
∑P
qOm
( R, θ, φ),
m =0
PqI ( R, θ, φ) =
(3.38a-b)
14
∑P
q Im
( R, θ, φ)
m =0
As in the Sinc function approach, Pq, Πq, ℜq, and σq are calculated using equations (3.35),
(3.16), (3.20), and (3.22).
3.7 Modal Radiation Results
Modal radiation properties such as acoustic frequency response functions P/f(ω)
and P / u(ω) , directivity function Dq(θ,φ), Πq, and σq of the sample annular disk (Table 1)
are obtained by using analytical methods of Chapters 3.5 and 3.6. Further, the same
radiation properties are calculated with an uncoupled, direct, exterior, and unbaffled
boundary element analysis [3.18]. In the computational model, 4,400 nodes and 6,600
elements are used to describe the source. With this model, the number of elements per
acoustic wavelength exceeds 6, below 8 kHz. And 6,146 acoustic field points and 6,144
elements are defined on the sphere surrounding the disk. The center of this sphere
coincides with the disk center. Excitation to this boundary element analysis model is the
normal velocity on two radial surfaces that is calculated from the forced vibration
analysis as explained in Chapter 3.3.
Results of analytical solutions and numerical model are verified by measurement
data obtained from a vibro-acoustic experiment that is conducted in an anechoic chamber
as shown in Figure 3.8. In this experiment, the disk is excited by an impact hammer (PCB
GK291C) in the radial direction at mid-plane of the disk.
87
Microphone p
i, i,
r p = 303 mm)
Anechoic
Chamber
Impulse Hammer f
° °,
r f = 151.5 mm)
Accelerometer ür
° °,
r u= 151.5 mm)
Rotor
Signal
Conditioning
Unit
Signal
Conditioning
Unit
FFT
Analyzer
FFT
Analyzer
Figure 3.8: Vibro-acoustic experiment used to measure structural u f (ω) and acoustic
P/f(ω) frequency response functions and in-plane modal sound radiation.
88
Far-field sound pressures are measured with a 6 mm microphone that is used for out-ofplane modes at the predetermined field points on a circle of 303 mm radius that is
centered at the center of the disk in the plane of ϕ = 0° and θ = 90°. Field point mesh for
out-of-plane modes is used in this experiment since directivity patterns of in-plane modes
have similar symmetries to those of out-of-plane. Force and pressure signals are
conditioned and analyzed via a 2-channel dynamic signal analyzer (HP 35670A) to obtain
the P/f(ω) spectrum as shown in Figure 3.9. Sample FRFs at two rp locations (R = 303
mm, φ = 0, θ = 0 and π/2) are also calculated using BEM and then compared with
measured data from Figure 3.8. Excellent agreement between experimental and numerical
results is seen especially in the vicinity of the three peaks at 2.86, 7.00 and 7.26 kHz.
These three peaks correspond to the first three modes (q =2, 3, and 0) of Table 3.3.
Spectral contents of P/f(ω) of Figure 3.9 depend on the receiver positions rp suggesting
that the sound source is highly directive. The relation between surface vibration and Pq is
obtained in the form of P u (ω) FRF that is based on the u f (ω) and P/f(ω) data of
Figures 3.4 and 3.9. These FRFs for two rp locations are given in the Figure 10. Note that
P u (ω) spectra for two locations show considerable differences since P/f(ω) is affected
by a highly directive sound field but u f (ω) is controlled only by disk modes.
The modal directivity pattern Dq(θ, φ) on the sphere Sv is synthesized from
measured Pq(θ, φ) data. All of the analytical methods accurately predict far-field sound
pressure distributions. This is illustrated in Table 3.4 along with Figures 3.11 and 3.12
where Pq(θ, φ) results are compared with measured and computed values for q = 2 and q
= 3 modes. As shown in Figure 3.12, Pq shows a sinusoidal variation in the φ direction.
89
P/f(dB re 20 µPa/N)
100
75
50
25
P/f(dB re 20 µPa/N)
0
1.6
3.2
4.8
6.4
8
4.8
6.4
8
100
75
50
25
0
1.6
3.2
Frequency (kHz)
Figure 3.9: Acoustic frequency response functions P/f(ω). (a) θ = π/2 and φ = 0; (b) θ = 0
and φ = 0. Key: –––, measured; - - -, computed using BEM.
90
(dB re 20 µPas2/m)
100
75
50
25
(dB re 20 µPas2/m)
0
1.6
3.2
4.8
6.4
8
4.8
6.4
8
100
75
50
25
0
1.6
3.2
Frequency (kHz)
Figure 3.10: Acoustic frequency response functions P u (ω) . (a) θ = π/2 and φ = 0; (b) θ
= 0 and φ = 0. Key: –––, measured; - - -, computed using BEM.
91
q = 2 Mode
q = 3 Mode
Structural
Mode
Analytical
Method
Measured
Computed
using
BEM
Table 3.4: Comparison of directivity patterns for selected modes.
92
Normalized Pressure
Normalized Pressure
1
0.5
0
0
10
20
30
40
50
60
70
80
90
0
10
20
30
40
50
60
70
80
90
1
0.5
0
θ (Deg)
Figure 3.11: Directivity pattern Dq(θ) given φ = 0 and R = 303 mm. (a) q = 2 mode; (b)
q = 3 mode. Key: - - -, Rayleigh integral method; , cylindrical radiator method; – - –,
computed using BEM; •, measured.
93
Πq, dB re 1 pW
Analytical Methods
Mode
q
Measured
Computed
using BEM
Rayleigh
Integral
Cylindrical Radiator Model
Sinc
Fourier
2
66.5
66.5
66.8
66.0
66.0
3
68.0
67.5
67.2
67.5
67.5
Measured
Computed
using BEM
σq, dB re 1
Analytical Methods
Mode
q
Rayleigh
Integral
Cylindrical Radiator Model
Sinc
Fourier
2
-3.5
-4.0
-2.3
-4.0
-4.0
3
-1.5
-1.0
-2.2
-2.0
-2.0
Table 3.5: Comparison of modal acoustic power and radiation efficiency levels
94
Figure 3.12: Directivity pattern Dq(φ) given θ = π/2 and R = 303 mm. (a) q = 2 mode; (b)
q = 3 mode. Key: –––, Analytical and numerical methods; • • •, measured.
95
Also, one can observe from Figure 3.11 that the cylindrical radiator model is more
accurate than the Rayleigh integral method. The Πq and σq predictions for the first two
radial modes are compared with BEM code and measured results in Table 3.5. In the
experimental case, Pq over Sv has been synthesized using the Pq(θ, φ) data to yield Πq and
σq. In this process, the Pq(θ, φ) profile is assumed to have a perfect sinusoidal variation in
the φ direction based on the results of Figure 3.12. As shown in Table 3.5, acoustic
radiation properties obtained using analytical solutions exhibit good agreements with
computed results and measurements. Since predictions are within 6 dB for Πq and 13 dB
for σq, one may conclude that analytical solutions are sufficiently accurate in calculating
Πq and σq. These discrepancies can be explained by the uncertainties in the measurement
introduced by frequency and spatial resolutions as explained in Chapter 2.4.
3.8 Parametric Studies
Acoustic radiation properties from a vibrating disk depend on its geometry,
vibrating mode and frequency, and the ratio of structural to acoustic wave numbers. In
our study, effects of the vibration frequency (ωq), radii ratio (β = b/a), and thickness ratio
( h = h / a ) on modal sound radiation are investigated by calculating Πq and σq. In
addition, the relative contribution of sound radiated from the inner and outer radial edges
is investigated using the same procedure. Note that, in general, natural frequencies and
geometric dimensions of any structure are closely related and cannot be independently
controlled. Nonetheless, in our study, ωq and h are independently varied to explicitly
96
determine the effect of each on Πq and σq. For all parametric studies, the cylindrical
radiator method (Sinc function approach) is used for calculation.
First, we investigate the effect of β on the relative contribution of sound from
inner and outer radial edges to the total sound pressure. Therefore, we define the
amplitude ratio ( p qI
p qO ) and phase difference (∠PqI − ∠PqO) as a function of β, using
the formulation of Chapter 3.7. In this study, the maximum modal sound pressure
locations are chosen as the sample receiver positions (rp). Realistically, β cannot be
altered without changing the in-plane natural frequencies. However, in this investigation,
the vibration frequency and its amplitude are fixed and then β is altered. The radii ratio is
controlled by changing the inner radius (b) while the outer radius (a) is fixed, and the
range is 0.01 ≤ β ≤ 1.0 where the nominal value of β is 0.54. Results for two radial modes
are summarized in Figure 3.13. In the limiting case of β → 0, sound pressure is not
generated from the inner radial edge (PqI = 0). Consequently, p qI
p qO = 0 and ∠PqI −
∠PqO = −∠PqO. In the other limiting case where β → 1, one can see from equation (3.34)
that p qI = p qO , and ∠PqI = −∠PqO.
Next, the effect of β on Πq and σq is investigated. The results of this investigation
are summarized in Figure 14 where Πq and σq significantly fluctuate with β. For each
mode, Πq and σq show similar patterns with β. But the pattern of q = 3 mode fluctuates
more rapidly than that of q = 3 mode with an increase in β. The acoustic wavelength of q
= 3 mode is smaller than that of q = 2 mode and consequently interference between sound
waves from two surfaces is more sensitive to a change in b.
97
(a)
1.00
p qO
0.75
90
0.50
0
0.25
-90
0.00
0.01
-180
0.25
0.50
∠pqI − ∠pqO
p qI
180
0.75
Radii Ratio (β)
p qI
p qO
180
0.75
90
0.50
0
0.25
-90
0.00
0.01
-180
0.30
0.54
∠pqI − ∠pqO
(b)
1.00
0.80
Radii Ratio (β)
Figure 3.13: Effect of radii ratio on the amplitude ratio and phase difference between PqI
and PqO. (a) q = 2 mode; (b) q = 3 mode. Key: ο ο ο, Phase difference; ∆ ∆ ∆, Amplitude
Ratio.
98
(a)
5.0
0.8
4.0
0.6
3.0
0.4
2.0
0.2
1.0
0.0
0.01
Acoustic Power(µW)
Radiation Efficiency
1.0
0.0
0.25
0.50
0.75
1.00
Radii Ratio(β)
(b)
10.0
1.5
7.5
1.0
5.0
0.5
2.5
0.0
0.01
0.0
0.25
0.50
0.75
Acoustic Power (µW)
Radiation Efficiency
2.0
1.00
Radii Ratio (β)
Figure 3.14: Effect of radii ratio on Πq and σq. (a) q = 2 mode; (b) q = 3 mode. Key: , Acoustic power; ο ο ο, Radiation efficiency.
99
As a limiting case in which β → 0, sound pressure is generated only by the outer edge
and the far-field sound pressure can be determined by the PqO term of equation (32a). For
the other limiting case where β → 1, the annular disk can be considered as a thin cylinder
that can be solved using the method of reference [3.11]. The effect of ωq on radiation is
also investigated for q = 2 and 3 modes. In this study, the forced vibration amplitude and
geometric dimensions are fixed while ωq is varied by ±1,000 Hz about its nominal value
ωq0. Typical variations in Πq and σq are expressed as a function of the frequency ratio ω
= ωq/ωq0, as illustrated in Figure 3.15. Though Πq and σq show significant variations with
ω , their shapes are mode dependent. Since geometric dimensions (h, a and b) of the disk
are fixed in this investigation, the amplitude ratio PqI
PqO is constant at any frequency
but the relative phase ∠PqI − ∠PqO varies linearly with ω at a given receiver position.
Consequently, Πq and σq show sinusoidal variations with the frequency ratio.
Finally, the effect of disk thickness (h) is investigated. The thickness ratio
( h = h / a ) is selected as an independent variable and it is varied from 0.02 to 0.6 with a
nominal value of h0 = 0.21. Recall that ωq of the disk is independent of h. Results are
summarized in Figure 3.16 where one can observe an increase in Πq because of the larger
radiation area as h is increased. As one can see from equation (3.34), in the limiting case
of h → 0, both PqI and PqO vanish for all modes. Consequently, Πq = 0 and σq cannot be
defined for any q.
100
(a)
12.0
0.8
9.0
0.5
6.0
0.3
3.0
0.0
0.65
0.83
1.00
1.17
Acoustic Power(µW)
Radiation Efficiency
1.0
0.0
1.35
Frequency Ratio
12.0
1.5
9.0
1.0
6.0
0.5
3.0
0.0
0.86
0.93
1.00
1.07
Acoustic Power (µW)
Radiation Efficiency
(b)
2.0
0.0
1.14
Frequency Ratio ( ω)
Figure 3.15: Effect of ω on Πq and σq. (a) q = 2 mode; (b) q = 3 mode. Key: ,
Acoustic power; ο ο ο, Radiation efficiency.
101
(a)
20.0
0.9
15.0
0.6
10.0
0.3
5.0
0.0
0.0
0.02
0.10
0.18
0.26
0.34
0.42
0.50
Acoustic power (µW)
Radiation Efficiency
1.2
0.58
Thickness Ratio (h )
8.0
0.8
6.0
0.5
4.0
0.3
2.0
0.0
0.0
0.02
0.10
0.18
0.26
0.34
0.42
0.50
Acoustic Power (µW)
Radiation Efficiency
(b)
1.0
0.58
Thickness Ratio (h )
Figure 3.16: Effect of thickness ratio on Πq and σq. (a) q = 2 mode; (b) q = 3 mode. Key:
, Acoustic power; ο ο ο, Radiation efficiency.
102
In the other limiting case where h → ∞, the annular disk can be treated as an infinite
cylinder where PqI does not contribute to the total sound pressure Pq that has uniform and
sinusoidal (cosqφ) distributions in the z and circumferential directions respectively. In
~
this case, Z (k z ) in equation (3.27) can be expressed as 2πδ(kz) where δ is the Dirac delta
function. The integral can be performed without the stationary phase approximation to
yield:
Pq (r , φ , z ) = − ρ 0 urq cos qφ
H q1 (kr )
′
kH q1 (ka)
(39)
This equation is identical to the far-field sound pressure expression from an infinite
cylinder with standing wave configuration that is reported by Junger et al. [3.11].
3.9 Conclusion
This Chapter has introduced new analytical solutions for sound radiation from the
radial modes of a thick annular disk. Neglecting the effect of scattering, two simplified
calculation procedures have been proposed. Based on comparisons with computed (using
FEM and BEM) and measured vibro-acoustic data, both analytical solutions are deemed
to be sufficiently accurate though the cylindrical radiator method (using either the Sinc
function or the Fourier series approach) appears to be better in predicting the modal
radiation. Limiting cases of disk radiator have also been evaluated, including the infinite
cylinder expression that is identical to one given in the literature. Therefore, the radiation
problem, given an arbitrary radial force excitation, can be analytically solved using the
103
structural mode synthesis procedure and the Rayleigh integral or cylindrical radiator
method. When the proposed theory for radial modes is combined with the analytical
radiation solutions for in-plane circumferential and out of plane flexural modes as well as
their interactions, the total sound radiation from a thick annular disk having a multidimensional force excitation can be formulated in an efficient manner. This work is in
progress. Also, future research should incorporate other boundary conditions (including
the fixed inner edge) as well as structural imperfections within the disk body. Finally,
real-life noise control problems such as the brake squeal may be addressed using
analytically rigorous procedures.
104
REFERENCES TO CHAPTER 3
3.1 G. BHUTA and J. P. JONES 1971 Journal of the Acoustical Society of America 35 (7),
982-989. Symmetric planar vibrations of a rotating disk.
3.2
J. S. BURDESS, T. WREN and J. N. FAWCETT 1987 Proceeding of the Institution of
Mechanical Engineers 201, 37-44. Plane stress vibration in rotating discs.
3.3
S. CHEN and J. L. JHU 1996 Journal of Sound and Vibration 195 (4), 585-593. On
the in-plane vibration and stability of a spinning annular disk.
3.4
IRIE, G. YAMADA and Y. MURAMOTO 1984 Journal of Sound and Vibration 97 (1),
171-175 Natural frequencies of in-plane vibration of annular plates.
3.5
W. THOMPSON, JR. 1971 Journal of Sound and Vibration 17 (2), 221-233. The
computation of self- and mutual-radiation impedances for annular and elliptical
pistons using Bouwkamp integral.
3.6
M. R. LEE and R. SINGH 1994 Journal of the Acoustical Society of America 95 (6)
3311-3323. Analytical formulations for annular disk sound radiation using
structural modes.
3.7 H. LEVINE and F. G. LEPPINGTON 1988 Journal of Sound and Vibration 121 (5),
269-275. A note on the acoustic power output of a circular plate.
3.8
W. P. RDZANEK Jr. and Z. ENGEL 2000 Applied Acoustics 60 (5), 29-43.
Asymptotic formula for the acoustic power output of a clamped annular plate.
3.9 H. W. WODTKE and J. S. LAMANCUSA 1998 Journal of Sound and Vibration 215 (5),
1145-1163. Sound power minimization of circular plates through damping layer
placement.
3.10 I-DEAS User’s manual version 8.2. 2000 SDRC, USA.
3.11 M. C. JUNGER and D. FEIT 1985 Sound, Structures, and Their Interactions. New
York: MIT Press.
3.12 C. E. WALLACE 1970 Journal of the Acoustical Society of America 51 (3), 946-952.
Radiation resistance of a rectangular panel.
3.13 W. WILLIAMS, N. G. PARKE, D. A. MORAN AND C. H. SHERMAN 1964 Journal of
the Acoustical Society of America 36 (12) 2316-2322. Acoustic Radiation from a
Finite Cylinder.
3.14 B. E. SANDMAN 1976 Journal of the Acoustical Society of America 60 (6), 12561264. Fluid loading influence coefficients for a finite cylindrical shell.
105
3.15 P. R. STEPANISHEN 1978 Journal of the Acoustical Society of America 63 (2) 328338. Radiation power and radiation loading of cylindrical surfaces with nonuniform
velocity distribution.
3.16 E. G. WILLIAMS 1999 Fourier Acoustics. San Diego: Academic Press.
3.17 C. WANG and J. C. S LAI 2000 Journal of Sound and Vibration 232 (2), 431-447.
The sound radiation efficiency of finite length acoustically thick circular cylindrical
shell under mechanical excitation I: Theoretical analysis.
3.18 SYSNOISE User’s manual Version 5.4. 1999 NIT, Belgium
106
CHAPTER 4
MULTI-MODAL VIBRO-ACOUSTIC RESPONSE
4.1 Introduction
In many practical conditions, mechanical excitation is not limited to single
direction and frequency does not always coincide with the natural frequency of the
component. In these cases, several structural modes are simultaneously excited and sound
pressures from individual modes contribute to total sound pressure. Furthermore, these
modal sound radiation solutions interact with each other and final acoustic field is
affected by these interactions. In addition, total acoustic power should include power
from coupling effects between the acoustic field generated by different modes as well as
sound powers from individual modes. Consequently, a procedure for the calculation of
structural and acoustic responses for multi-modal excitations should be introduced to
properly control sound radiation from these practical components.
Several studies on this subject have been executed so far. For instance, sound
radiation from the multi-modal vibration and the coupling effect of the structural modes
has been investigated with analytical modal sound pressure of simple structures such as
beams or rectangular plates [4.1-4.3]. Also, exterior acoustic radiation modes of simple
beams and baffled finite plates are defined from expressions for far-field acoustic power
107
[4.4-4.5]. Several researchers applied the acoustic radiation mode concept to the active
structural acoustic control or the estimation of radiated acoustic power [4.6-4.7]. Multimodal sound radiation and modal coupling effects on the sound radiation from a thin
annular disk have also been investigated using the modal expansion technique [4.8].
Contrary to the thin plate cases where total sound radiation can be represented by
the radiation from out-of-plane vibrations without a considerable error, sound radiation
from in-plane and out-of-plane vibrations should be considered simultaneously to
appropriately control the sound radiation from a thick annular disk. In addition, effect of
coupling between in-plane and out-of-plane modes should be included along with the
effect of coupling within out-of-plane modes. But, neither the multi-modal sound
radiation nor modal coupling effect of complex structures such as a thick annular disk has
been clearly defined thus far.
In this study, structural and acoustic responses of a thick annular disk to an
arbitrary multi-modal or multi directional excitation are investigated using modal
expansion technique based on structural modal participation factors along with modal
sound radiation data defined by analytical solutions that have been introduced in Chapter
2 and 3. In addition, vibro-acoustic response of the sample disk to a multi-points
excitation is studied using the matrix of modal participation factors corresponding overall
excitations. The effect of relative circumferential distance between two identical
harmonic forces are investigated in this study. Finally, the inter-modal coupling effect on
the total sound radiation from the disk is investigated along with the effect of coupling
within the same type modes using the same procedure. The disk for modal sound
radiations described in Figure 2.1 and Table 2.1 is used as an example case for this study.
108
4.2 Assumptions and Objectives
Sound radiation from a thick annular disk is conceptually shown in the Figure 4.1
in the context of source and receiver positions. In this study, in-plane and out-of-plane
vibrations of a disk are considered simultaneously to completely investigate the vibroacoustic properties of the disk along with their interactions. The scope of this study is
strictly limited to the frequency domain analysis of a linear time-invariant (LTI) system
with free-free boundaries. Complicating effects such as fluid loading and scattering at the
disk edges are not considered.
Chief objectives of this study are: (1) Develop analytical and semi-analytical
procedures for calculating sound radiation from the disk given multi-modal harmonic
excitations. (2) Examine the effect of couplings between in-plane and out-of-plane modes
as well as couplings within the same type of modes on the total sound radiation. (3)
Investigate the effect of natural frequency separation on the sound radiation. (4) Study
sound radiation due to multi-point excitations and investigate effect of circumferential
distance between two identical harmonic forces on the sound radiation.
4.3 Vibro-Acoustic Responses to a Harmonic Excitation
4.3.1 Modal Formulation for Structural and Acoustic Responses
If a disk is excited by a harmonic excitation with an arbitrary frequency and
direction, several modes, including both in-plane and out-of-plane modes, are excited
109
P(R, θ, φ)
z
Sv
W
γ
Ssn
Ssr
R
U
ϕ
Figure 4.1: Sound radiation from a vibrating thick annular disk.
110
simultaneously. Based on the modal expansion technique, velocity distribution (v) on the
disk surfaces can be expressed in terms of elastic velocity mode shapes of the disk.
{v}= {η}T {ΦV }
{Φ }= {Φ
{η}= {η
V
V
0 , 2 , −1
0 , 2 , −1
, Φ1V, 0, −1 , ΦV0,3, −1 ,, ΦVm, n , −1 , ΦV−1, −1, 2 , ΦV−1, −1,0 ,, ΦV−1, −1, q
, η1, 0, −1 , η0,3, −1 ,, ηm, n , −1 , η−1, −1, 2 , η0, 2, 0 ,, η−1, −1, q }
}
(4.1)
where ΦVm, n, q is a velocity modal vector of the disk that can be expressed as mode shapes
multiplied by corresponding natural frequencies and ηm,n,q is corresponding modal
participation factor. In this expression, new modal index (m, n, q) has been introduced
which is the combination of the out-of-plane mode index (m, n) and radial mode index q.
The value –1 in the indices is used to represent null index that means the corresponding
velocity mode shape corresponding to that index does not contribute to the newly defined
modal velocity vector. For instance, ΦV0, 2, −1 is the velocity mode shape of the (0,2) out-ofplane mode and ΦV−1, −1, 2 is the velocity mode shape of the q = 2 radial mode. Structural
modal participation factors due to a harmonic excitation with frequency ω on a given
structure can be obtained from the modal data set (natural frequencies, mode shapes, and
damping ratios) as follows:
ηm, n, q = ∑
Φ m, n, q (rf , ϕ f )Φ m, n, q (r , ϕ)
T
(1 − ω2 / ωm2 , n , q ) + i 2ς m, n, q (ω / ωm, n , q )
(4.2)
Lee and Singh expressed far-field sound pressure from a thin annular disk due to a multimodal excitation using structural modal participation factors and modal sound pressures
[4.8]. Applying the same procedure to an acoustic problem, the far-field pressure on a
111
sphere (SV) surrounding the disk due to surface velocity of equation (4.1) can be
expressed as follows:
P = {η} {Γ}
T
{Γ}= {Γ0, 2, −1, Γ1,0,−1, Γ0,3, −1,, Γm, n, −1 , Γ−1,−1, 2 , Γ−1, −1,0 ,, Γ−1, −1, q }
(4.3)
where, Γm,n,q is the modal sound pressure obtained from equation (2.48) and (3.35).
Acoustic power (Π) from the disk and corresponding radiation efficiency (σ) due to an
arbitrary harmonic excitation f(t) is also calculated from far-field sound pressures on a
sphere surrounding the disk as follows:
Π = IsS
σ=
where, < v >t , s =
2
s
=
1 2π π P H P 2
R sin θ dθ dφ
2 ∫0 ∫0 ρ0c0
(4.4 a b)
Π
< v >t , s
2
1
4πh(a + b) + 2π(a 2 − b 2 )
{∫
h/2
∫
2π(a +b)
−h / 2 0
a 2π
U 2 dl dz + ∫ ∫ W 2 dϕ dr
b
0
}
4.3.2 Structural Response
Utilizing the procedure given in the Section 4.3.1, structural response to a specific
multi-modal harmonic excitation has been calculated based on the analytical modal
datasets given in Chapter 2 and 3 or numerical eigen-solutions along with experimental
modal damping ratios. Analytical responses of the sample disk with free-free boundaries
for the normal and radial excitation are validated with numerical analyses and structural
experiments.
112
Microphone p
i φ = φi,
rp = 303 mm)
Anechoic
Chamber
, u
Accelerometer w
° °,
ru= 151.5 mm)
φ
Impulse Hammer f
° °
rf = 151.5 mm)
Rotor
Signal
Conditioning
Unit
FFT
Analyzer
Signal
Conditioning
Unit
FFT
Analyzer
Figure 4.2: Vibro-acoustic experimental setup used to measure structural frequency
/f(ω) or u /f(ω) and acoustic P/f(ω) frequency response functions.
response functions w
113
In the structural experiments, excitation force f(t) is applied on a normal surface in the z
direction at the outer edge of the disk (out-of-plane) or on the mid-plane of the outer
radial edge in the radial direction (in-plane) by an impulse hammer (PCB GK291C) at ϕ
= 0°. The set up for the structural experiment is explained in Figure 4.2. The upper
frequency limit and resolution (∆f) of this experiment are set as 16 kHz and 8 Hz
respectively. Natural frequencies (ωmn and ωq) and modal damping ratios (ςmn and ζq) are
/fn(ω) or u /fr(ω) using
extracted from measured structural frequency response function w
and u are the accelerations and f is applied force
the half-power point method where w
in the corresponding directions. In the finite element method (FEM), structural frequency
/fn(ω) or u /fr(ω) have been obtained in the frequency range from 0
response function w
to 8 kHz with a model that was used in the Chapter 2 and 3 [4.9]. In addition, harmonic
( r , ϕ) and u( r , ϕ) given a unit harmonic force in single direction with
acceleration w
arbitrary frequency is calculated based on the numerical modal dataset using a forced
/fn(ω) spectra for a normal force
vibration analysis. Figure 4.3 compares uni-directional w
and u /fr(ω) spectra for a radial force via analytical methods with computed and measured
spectra for given excitation and accelerometer positions. Excellent agreements among the
results from three approaches can be observed for both cases over the given frequency
range. Dominant peaks in these figures correspond to natural frequencies for out-of-plane
and in-plane modes of the sample disk respectively.
114
(a)
150
(0, 4)
/f (dB re 20µm/s2N)
(0, 3)
(0, 2)
(1, 1)
(1, 0)
100
50
1
2
3
4
5
6
7
8
Frequency (kHz)
(b)
150
ü/f (dB re 20µm/s2N)
q=0
q=3
q=2
100
50
2
3
4
5
6
7
8
Frequency (kHz)
/f(ω)
Figure 4.3: Structural frequency response functions with free-free boundaries (a) w
at r = 0.1515 and ϕ = 180°; (b) ü/f(ω) at ϕ = 0. Key: , measured; - - -, computed using
FEM; –––, analytical calculation.
115
4.3.3 Acoustic Response
Elementary radiation properties such as acoustic frequency response function
P/f(ω), acoustic power Π(ω), and radiation efficiency σ(ω) spectra of the sample annular
disk (Figure 2.1 and Table 2.1) are obtained by the analytical method given in Section
4.3.1. The same properties are calculated with uncoupled, direct, exterior, and unbaffled
boundary element analyses [4.10]. The field point model for modal sound radiation
solutions is used for this calculation along with the finite element model that has been
used in the numerical structural analyses. Excitations to this boundary element model are
the normal velocity distribution on the external surfaces obtained from the forced
vibration analysis as explained in Section 4.3.2. Analytical and computational P/f(ω) at
two rp locations (R = 303 mm, φ = 0, θ = 0 and π/2) due to single directional force
(normal or radial) are verified by the vibro-acoustic experiments conducted in an
anechoic chamber as shown in Figures 4.4 and 4.5. The excitation for structural
experiment has been used in the acoustic experiment. Far-field sound pressures are
measured with a 6 mm microphone (MTS L130C10 with pre-amplifier MTS 130P10) at
the predetermined field points on the sphere Sv. P/f(ω) given unit harmonic force in
single direction with arbitrary frequency is calculated based on the numerical modal
dataset using numerical forced vibration analysis in FEM. Figures 4.4 compares unidirectional acoustic response function P/f(ω) due to a radial force from analytical
methods with purely numerical analysis and measured data for two rp locations. Figures
4.5 compares the same data due to a normal force for the same rp locations. Dominant
peaks in these figures correspond to natural frequencies of the corresponding modes of
the sample disk.
116
(a)
P/f (dB re 20µPa/N)
100
50
0
2
3
4
5
6
7
8
5
6
7
8
(b)
P/f (dB re 20µPa/N)
100
50
0
2
3
4
Frequency (kHz)
Figure 4.4: Acoustic frequency response functions P/f(ω) due to radial excitation. (a) θ =
π/2 and φ = 0; (b) θ = 0 and φ = 0 Key: , measured; - - -, computed using BEM; –––,
analytical calculation.
117
(a)
P/f (dB re 20µPa/N)
100
(0, 3)
(0, 2)
(1, 1)
(1, 0)
50
0
1
2
3
4
5
6
7
8
(b)
100
P/f (dB re 20µPa/N)
(0, 4)
(1, 0)
q=0
50
0
1
2
3
4
5
6
7
8
Frequency (kHz)
Figure 4.5: Acoustic frequency response function P/f(ω) given impulsive force excitation
f(t) at r = a in the z direction. (a) θ = π/2 and φ = 0; (b) θ = 0 and φ = 0. Key: ,
measured; - - -, computed using BEM; –––, analytical calculation
118
Excellent agreements among analytical, experimental and numerical results can be found
over the given frequency range for both receiver positions, especially in the vicinities
natural frequencies of the disk.
In addition to acoustic responses for the single directional forces, P(ω), Π(ω) and
σ(ω) for the multi-directional harmonic force of f n = f r = 1N at a single location are
calculated using the proposed analytical procedure and compared with the corresponding
numerical investigation results. The combined force is applied at ϕ = 0 location on the
mid-plane of the disk. Measured data have not been included in this study because of
practical difficulties. Analytical P(ω) data for the receiver locations that have been used
in the cases of uni-directional force are compared with numerical results in Figure 4.6.
These spectra have dominant peaks at the every frequency where uni-directional p/f(ω)
have peaks as shown in Figures 4.4 and 4.5. Also acoustic power and radiation efficiency
spectra from two approaches are given in Figure 4.7. Acoustic responses from analytical
calculation match relatively well experimental and numerical data for all cases.
119
(a)
P(ω) (dB re 20µPa)
100
50
0
1
2
3
4
6
7
8
5
6
7
8
(b)
100
P(ω) (dB re 20µPa)
5
50
0
1
2
3
4
Frequency (kHz)
Figure 4.6: Far-field sound pressure spectra P(ω) due to multi-modal excitation. (a) θ =
π/4 and φ = 0; (b) θ = 0 and φ = 0. Key: ––––, analytical calculation; - - -, computed
using BEM
120
(a)
Π (dB re 1pW)
100
50
0
1
2
3
4
5
6
7
8
5
6
7
8
(b)
1.5
σ
1
0.5
0
1
2
3
4
Frequency (kHz)
Figure 4.7: Acoustic radiation functions due to combined harmonic excitation. (a)
acoustic power spectra Π(ω); (b) radiation efficiency spectra σ(ω). Key: –––, analytical
calculation; - - -, computed using BEM
121
4.3.4 Responses for Multiple Excitations
Many practical mechanical components are simultaneously excited by several
forces on several different locations with different frequencies. Analytical approach
developed in the previous sections can be expanded to this kind of practical problems
using additional modal participation factors. If a number of harmonic excitations are
applied on the disk simultaneously, surface velocity and far-field sound pressure can be
expressed as combination of those for individual excitation as follows:
{v}= ∑ {ηi }T {ΦV }
i
{P}= ∑ {ηi }T {Γ}
(4.5)
i
{η }= {η
i
i
0 , 2 , −1
, η1i ,0, −1 , η0i ,3, −1 , , ηim , n, −1 , ηi−1, −1, 2 , ηi0, 2, 0 , , ηi−1, −1, q
}
where {ηi } is the vector of modal participation factors corresponding to the ith excitation.
As in the cases of single excitation, acoustic power radiation efficiency can be calculated
2
from far-field sound pressure using equation (4.4). In this case < v >t , s should be
calculated from {v} that is obtained using equation (4.5)
As an example case, sound radiation from the sample disk due to two identical
harmonic forces in normal direction applied at different locations having specific
circumferential separation (∆ϕ) on the disk (rad). Figure 4.8 explains sample cases used
in this study. As shown in the figure first force is applied at r = 0.1515 m, ϕ = 0 and
second force is applied at r = 0.1515 m, ϕ = π/12, π/6, π/4, or π/3 according to the
specific case. Far-field sound pressure spectra P(ω) at two receiver positions (R = 303
mm, φ = 0, θ = π/4 and π/2), due to identical harmonic forces at two circumferential
122
locations are calculated using the analytical procedure. This result is summarized in
Figure 4.9. In addition, acoustic power spectra Π(ω) for above four cases are calculated
from previously obtained far-field sound pressure distributions and compared each other
in Figure 4.10 (a). Finally, radiation efficiency spectra σ(ω) corresponding to above four
2
2
cases are calculated from Π(ω) and < v >t , s using equation (4.5). Here, < v >t , s is
calculated using equation (4.6) from surface velocity obtained by modal dataset and
modal participation factors corresponding to two excitations.
As shown in the Figure 4.9 and 4.10, far-field sound radiation is significantly
affected by the circumferential separation between two excitations. Also, as one can see
from the comparison of this result with sound radiation due to a single excitation that has
been introduced in Figures 4.4 and 4.5, sound radiation at a specific frequency can be
controlled using additional forces at proper location. For instance, sound radiation at 6.15
kHz or 3.4 kHz can be significantly reduced by adding an identical force with ∆ϕ = π/4
∆ϕ = π/3 respectively.
123
f2
∆ϕ
f1
Z
Disk
Figure 4.8: Example cases for calculation of vibro-acoustic responses due to multi-point
excitations
124
(a)
P(ω) (dB re 20µPa)
100
50
0
1
2
3
4
6
7
8
5
6
7
8
(b)
100
P(ω) (dB re 20µPa)
5
50
0
1
2
3
4
Frequency (kHz)
Figure 4.9: Far-filed sound pressure spectra due to two identical harmonic forces with
specific circumferential distances. (a) P(ω) at R = 303 mm, φ = 0, θ = π/4; (b) P(ω) at R =
303 mm, φ = 0, θ = π/2. Key: - - -, ∆ϕ = π/12; , ∆ϕ = π/6;.
, ∆ϕ = π/4; –––, ∆ϕ =
π/3.
125
(a)
90
Π (dB re 1pW)
80
70
60
50
40
1
2
3
4
5
6
7
8
5
6
7
8
(b)
1.5
σ
1
0.5
0
1
2
3
4
Frequency (kHz)
Figure 4.10: Acoustic radiation functions due to two identical harmonic forces with
specific circumferential distances. (a) acoustic power spectra Π(ω); (b) radiation
efficiency spectra σ(ω). Key: - - -, ∆ϕ = π/12; , ∆ϕ = π/6; , ∆ϕ = π/4; –––, ∆ϕ =
π/3.
126
4.4 Effects of Structural and Acoustic Modal Coupling on the Acoustic Radiation
4.4.1 Effects of Structural Modal Coupling on the Acoustic Radiation
If several structural modes are excited simultaneously, total sound power should
include power from coupling effects between the acoustic field generated by different
modes as well as sound powers from individual modes. For instance, Keltie and Peng
[4.1] found that the coupling between two structural modes is as important as the
individual mode when both natural frequencies are much lower than the excitation
frequency. Cunefare [4.2] developed a quadratic expression for the radiation efficiency of
a beam under multi-modal excitation using the far-field intensity integration technique.
Lee and Singh [4.7] investigated modal coupling effects on the sound radiation from a
thin annular disk and found that nonzero sound power due to the coupled modes exists
only when the coupled modes have the same number of nodal diameters.
As explained in Chapter 4.3, overall sound power from a thick annular disk
excited by a harmonic force can be expressed as equation (4.4). Consequently, total
sound power can be decomposed into two groups: 1) Sound power from self-radiations of
individual modes, 2) Sound power from mutual radiation between arbitrary couple of
structural modes. In a thick annular disk case, being evident from the equation (4.4),
modal coupling effects exist between out-of-plane modes and in-plane modes as well as
between any two out-of-plane modes and any two in-plane modes. Sound power
generated by the interaction between mode (mi, ni, qi) and mode (mj, nj, qj) can be
obtained with following equation.
Π
m j n j rj
mi ni ri
R2
=
2 0 c0
2π
π/2
0
0
∫ ∫
Pmi ni ri P * m j n j rj sin θdθdφ
127
(4.6)
Here, Π mijni rj i j is sound power from a self-radiation when mi = mj, ni = nj and qi = qj,
m n r
otherwise Π mijni rj i j is that from a mutual radiation between (mi, ni, qi) and (mj, nj, qj) modes.
m n r
Acoustic powers due to modal coupling between any two structural modes are calculated
using equation (4.6). By repeating this procedure, radiated powers associated with
individual modes of the disk can be obtained along with sound powers due to the
coupling effects between two structural modes of different types. The results are
summarized in Table 4.1.
0,2,-1
1,0,-1
0,3,-1
1,1,-1
0,4,-1
1,2,-1
-1,-1,2
-1,-1,3
-1,-1,0
0,2,-1
2.1E+7
0
0
0
0
1.4E+6
2.6E+7
0
0
1,0,-1
0
9.0E+7
0
0
0
0
0
0
1.0E+7
0,3,-1
0
0
2.0E+8
0
0
0
0
5.3E+7
0
1,1,-1
0
0
0
1.7E+8
0
0
0
0
0
0,4,-1
0
0
0
0
4.7E+8
0
0
0
0
1,2,-1
1.4E+6
0
0
0
0
3.7E+8
9.1E+6
0
0
-1,-1,2
2.6E+7
0
0
0
0
9.1E+6
7.8E+7
0
0
-1,-1,3
0
0
5.3E+7
0
0
0
0
6.0E+8
0
-1,-1,0
0
1.0E+7
0
0
0
0
0
0
1.2E+9
Table 4.1: Self and mutual radiation terms of sound powers between elastic modes of the
sample disk.
128
As shown in the table, acoustic power due to the coupling between out-of-plane
modes exists only when two modes have the same nodal diameters (ni = nj). Similarly,
sound powers from the coupling between two modes of in-plane type with different q
numbers are negligible. Finally, sound powers from coupling between in-plane and outof-plane modes exist only when n = q otherwise, coupling powers are negligible. For
example, (0, 2) out-of-plane mode has non-zero coupling effects with (1, 2) out-of-plane
mode and q = 2 in-plane mode. Also, total acoustic power due to multi-modal excitation
can be obtained as the linear combination of the sound powers from self and mutual
radiations as follows:
Π = ∑∑ ηmi ni qi ηm j n j q j Π mijni qj i j
m n q
(4.7)
4.4.2 Effects of Structural Natural Frequencies Separation on the Acoustic Radiation
In the thick annular disk case, any two normal modes of the same type (in-plane
or out-of-plane) have enough natural frequencies separation and generally two modes are
not excited simultaneously by a single frequency harmonic excitation. But the natural
frequencies separation between in-plane and out-of-plane modes can be very small
according to the geometric configuration and material properties of the disk. In this
section, the effect of natural frequency separation between any combination of an inplane mode and an out-of-plane mode is investigated using the proposed analytical
solutions and the results are compared with the results of boundary element analysis.
129
As shown in equation (4.2), modal participation factors for individual structural
mode due to a harmonic excitation are functions of frequency ratio (ω ⁄ ωr) between
excitation frequency ω and natural frequency of the given mode ωr. Consequently,
acoustic power from a thick annular disk, including powers from self-radiation and
mutual radiation, is affected by the structural natural frequencies separation. In the
analytical investigation, natural frequency of one of the neighboring modes is changed to
adjust natural frequency separation between the neighboring modes while other structural
natural frequencies, mode shapes, and modal damping ratios of all the disk modes remain
constant. Based on these data, structural and acoustic response to a harmonic excitation
with the frequency between the two natural frequencies of the neighboring modes. As a
sample case, the natural frequency of the (0, 2) out-of-plane mode is increased to
2.85kHz so that the natural frequency separation between (0, 2) out-of-plane mode and q
= 2 in-plane mode is 10 Hz. P/f(ω) at two rp locations (R = 303 mm, φ = 0, θ = 0 and
π/2), Π/f(ω), and σ(ω) for the modified case are calculated using the proposed analytical
procedure and compared with the results of original case. As shown in Figure 4.11 and
4.12 considerable increases in P/f(ω)s and Π(ω), at the vicinity of modified natural
frequency (2.85 kHz). But, radiation spectra σ(ω) at the same frequency range remains
almost constant in spite of the modification in natural frequency.
130
P(ω) (dB re 20µPa)
100
50
0
1
2
3
2
3
4
5
6
7
8
4
5
6
7
8
P(ω) (dB re 20µPa)
100
50
0
1
Frequency (kHz)
Figure 4.11: Effect of natural frequency separation on P/f(ω). (a) θ = π/2 and φ = 0; (b) θ
= 0 and φ = 0. Key: , modified; - - -, original case.
131
Figure 4.12: Effects of natural frequency separation on acoustic radiation functions. (a)
acoustic power frequency response functions Π/f(ω); (b) radiation efficiency function
σ(ω). Key: , modified; - - -, original case.
132
4.5 Conclusion
This study has introduced analytical and semi-analytical procedures for the sound
radiation from multi-modal vibration of a thick annular disk. Sound radiation modes from
out-of-plane and radial modes of the disk using analytical and numerical investigations.
Sound radiation from the disk excited by an arbitrary harmonic force is obtained from
modal sound radiation and structural modal participation factors using the modal
expansion technique. Based on the comparison with numerical analysis results, it is
evident that the proposed procedure has sufficient accuracy in predicting sound radiation
from a thick annular disk excited by arbitrary harmonic forces. Sound powers from the
modal coupling between two structural modes of the same type (in-plane or out-of-plane
mode) as well as between one in-plane mode and one out-of-plane mode have been
studied using proposed analytical solutions for sound radiation modes. According to the
results of analytical and numerical investigations, sound powers from modal coupling
between two out-of-plane modes exist only when two modes have same number of nodal
diameters n. In case of radial modes, sound power due to modal coupling doesn’t exist
between two modes with different radial mode index q. Furthermore non-zero sound
power due to modal coupling between in-plane mode and out-of-plane mode exists only
when n = q. If natural frequency separation between two neighboring modes is small
enough to excite two modes with significant modal participating factors, both modes
significantly contribute to the total sound power when the excitation frequency is close
enough to the natural frequencies of the modes.
133
REFERENCES FOR CHAPTER 4
4.1
R. F. KELTIE and H. PENG 1987 ASME Trans. J. Vib. Acoust. Stress Reliabil. Des.
109, 48-53. The effect of modal coupling on the acoustic radiation from panels.
4.2 K. A. CUNEFARE 1991 Journal of the Acoustical Society of America 90(5), 25212529. The minimum multimodal radiation efficiency of baffled finite beams.
4.3 K. A. CUNEFARE 1992 AIAA J. 30, 2819-2828. Effect of modal interaction on sound
radiation from vibrating structure.
4.4 K. A. CUNEFARE and M. N. Currey 1994 Journal of the Acoustical Society of
America 96(4), 2302-2312. On the exterior acoustic radiation modes of structures.
4.5 M. N. CURREY and K. A. CUNEFARE 1995 Journal of the Acoustical Society of
America 98(3), 1570-1580. The radiation modes of a baffled finite plates.
4.6 G. P. Gibbs, R. L. Clark, D. E. Cox and J. S. Vipperman 2000 Journal of the
Acoustical Society of America 107(1), 332-339. Radiation modal expansion:
Application to active structural acoustic control.
4.7
M. R. Bai and M. Tsao 2002 Journal of the Acoustical Society of America 112(3),
876-883 Estimation of sound power of baffled planar sources using radiation
matrices.
4.8
M. R. LEE and R. SINGH 1994 Journal of the Acoustical Society of America 95(6),
3311-3323. Analytical formulations for annular disk sound radiation using
structural modes.
4.9 I-DEAS User’s manual version 8.2. 2000 SDRC, USA.
4.10 SYSNOISE User’s manual Version 5.4, NIT, Belgium, 1999
134
CHAPTER 5
APPLICATION TO A BRAKE ROTOR
5.1
Introduction
Several structural dynamic models have been used to explain the brake squeal
generation mechanism based on the self-excited vibration [5.1-5.2] or modal coupling
phenomena [5.3-5.5]. Recently, non-linear transient analysis [5.6, 5.7] and complex
eigen-value method [5.8-5.12] have been implemented using commercial finite element
software. Also, Dunlap et al. [5.13] investigated brake squeal noises in various frequency
ranges using appropriate approaches and concluded that natural frequency separation
between coupled flexural and tangential modes is critical in generating high frequency
squeal noise. McDaniel and Li [5.14] investigated coupling between in-plane and out-ofplane modes and concluded that the coupling creates vibrational instability that is
characterized by power flow through the transverse motion of the rotor. But, most prior
studies have focused on the structural dynamics of brake rotors and related components.
The acoustic radiation mechanism has not been adequately examined. To fill this void,
we investigate sound radiation from a simplified brake rotor. A semi-analytical procedure
is proposed that is based on structural eigen-solutions from finite element analysis and
analytical modal sound radiation solutions developed for a thick annular disk.
135
hat
disk
r
z
b
a
th
h
H
Figure 5.1: A thick annular disk with a hat structure simulates the brake rotor. Disk is
clamped at the inner bolts and free at outer edge.
Outer radius (a)
Inner radius (b)
Radii ratio (β = b/a)
Disk thickness (h)
Hat thickness (th)
Hat height (H)
Density (ρd)
Young’s modulus (E)
Poisson’s ratio (ν)
151.5 mm
87.5 mm
0.54
31.5 mm
6 mm
24 mm
7905.9 Kg/m3
218 GPa
0.305
Table 5.1: Geometric dimensions and material properties of the brake rotor.
136
Figure 5.1 describes the geometric configurations of the rotor example used in this study.
In addition, geometric dimensions and material properties are given in Table 5.1.
5.2 Objectives and Assumptions
Chief objectives of this chapter are as follows. (1) Develop semi-analytical solutions for
sound radiation from modal vibrations of a brake rotor. (2) Employ a modal synthesis
procedure to calculate the vibro-acoustic response to an arbitrary harmonic excitation. (3)
Validate proposed analytical procedures using computational vibro-acoustic methods. As
evident from Figure 5.1 and Table 5.1, disk thickness (h) is not negligible compared to
other dimensions of the disk. Consequently, for a complete investigation of the vibroacoustic characteristics of a brake rotor, it is necessary to simultaneously consider both
in-plane and out-of-plane vibrations. Primary assumptions are as follows: (1) Structural
and acoustic systems are linear time-invariant systems and complicating effects such as
fluid loading and acoustic scattering from the disk edges are negligible. (2) Sound is
radiated by only the thick annular disk area, and the hat structure of Figure 5.1 does not
contribute to the radiated sound pressure. (3) Boundary conditions for the mounting bolts
can be accurately simulated by the idea clamped boundaries at the same locations.
5.3 Structural Modal Analysis
The structural dynamics of the rotor of Figure 5.1 have been investigated using a
finite element model with 2010 solid elements and 3960 nodes [5.15].
137
Figure 5.2: Finite element model of the brake rotor with 2010 solid elements.
138
a. Out-of-Plane Modes (m, n)
Mode no.
Freq. (Hz)
1, 2
900
(0, 1)
Mode Shape
4, 5
1700
(0, 2)
9,10
3500
(0, 3)
6
1920
l=0
20
7240
q=0
13
4000
(1, 0)
-
b. In-Plane Modes (l, q)
Mode no.
Freq. (Hz)
Mode Shape
c. Combined Modes (m, n and q)
Mode no.
Freq. (Hz)
7, 8
2500
m = 0, n = 1
q=1
11, 12
3800
m = 1, n = 2
q=2
18, 19
7120
m = 1, n = 3
q=3
+
+
+
Mode Shapes
-
Table 5.2: Selected structural modes of the brake rotor
139
To simulate realistic boundary conditions, clamped nodal restraints have been applied at
the locations of mounting bolts. A schematic of this finite element model is given in
Figure 5.2. As many as 44 structural modes up to 16 kHz have been defined in our
analysis. Selected natural frequencies and mode shapes are listed in Table 5.2. In our
study, structural modes are described using four modal indices such as m (number of
nodal circles), n (number of nodal diameters), q (radial mode index) and l (tangential
mode index). Furthermore, structural modes of a brake rotor are categorized into three
types: out-of-plane modes described by (m, n), in-plane modes given by (l, q), and
combined modes given by (m, n and q). As shown in the table, the qth radial modes are
always coupled with out-of-plane modes having the same number of nodal diameters (n)
as q, due to the hat structure and clamped boundary conditions.
5.4 Sound Radiation from Structural Modes of Brake Rotor
As shown in Table 5.2, mode shapes of pure out-of-plane (flexural) modes of a
brake rotor can be expressed with the same modes of a generic annular disk with identical
geometric dimensions. Consequently, sound radiation from these modes can be expressed
in terms of the modal sound radiation for the corresponding annular disk. In this study,
far-field sound pressure due to the (m, n)th out-of-plane mode of sample rotor is
calculated using equation (5.1) that is introduced in Chapter 2. Refer to Chapter 2 for the
identification of symbols.
140
o
( R, θ, φ)
Pmn ( R, θ, φ) = (1 + cos θ) Pmns ( R, θ, φ) + (1 − cos θ) Pmn
ρ 0 ck mn e ik mn R −ik mn h2 cos θ
e
cos nφ( −i ) n +1 Β n [w (r )]
P ( R, θ, φ) =
2R
ρ 0 ck mn e ik mn R ik mn 2h cos θ
o
Pmn ( R, θ, φ) = −
e
cos nφ( −i ) n +1 Β n [w ( r )]
2R
s
mn
(5.1)
∞
where Β n [w ( r ) ] = ∫ w ( r ) J n (k r r ) rdr ; k r = k sin θ
0
Sound power (Π) for mode (m, n) is calculated from the far-field sound pressure Pmn
using the following equation.
2
Π mn = I mn SV
1 2π π P
= ∫ ∫ mn R 2 sin θ dθ dφ
2 0 0 ρ0c0
s
(5.2)
Here, Imn is the acoustic intensity on a sphere Sv where Sv is the control surface. The
modal radiation efficiency σmn of an annular disk is determined from Πmn as follows
2
where < w mn >t , s is the spatially averaged mean-square velocity on the two normal
surfaces of an annular disk.
σ mn =
Π mn
2
< w mn >t , s
2
< w mn >t , s =
;
(5.3)
1
a
2π(a 2 − b 2 ) ∫b
∫
2π
0
W dϕ dr
2
mn
Next, we consider combined modes. As one can see in the Table 5.2, the qth radial
modes are always coupled with (m = 0, n = q) or (m = 1, n = q) out-of-plane modes
except for the q = 0 radial mode. Since the thickness (h) of disk is beyond the thin plate
theory limit, sound radiation from in-plane (radial) modes should be included in the
calculation of total sound radiation from such modes. Modal velocity distributions on the
141
normal and radial surfaces have been defined from numerically estimated natural
frequencies and modes shapes. Consequently, the total far-field sound pressure at a given
receiver position is expressed as sum of sound pressure from normal surfaces due to the
out-of-plane vibration as given by equation (5.1) and that from radial surfaces due to the
radial vibration. Sound pressure from radial surfaces can be calculated by following
equation that is introduced in Chapter 3.
Pq ( R, θ, φ) = PqI ( R, θ, φ) + PqO ( R, θ, φ)
ik R
Sinc (k q sin θ h / 2)( −i) q +1
ρ0 e q
PqO ( R, θ, φ) =
uqO h
× cos qφ
1′
πkq R sin θ
H q (k q a sin θ)
Sinc ( k q sin θ h / 2)(−i )
ρ0e q
PqI ( R, θ, φ) =
uqI h
′
πkq R sin θ
H 2 (k b sin θ)
ik R
q
(5.4)
q +1
× cos qφ
q
Modal acoustic power for combined mode (m, n, q) can be obtained using an equation
similar to equation (5.2) from the total sound pressure that is sum of Pmn and Pq. In
addition, modal radiation efficiencies for the combined mode are obtained using
following equation.
σmnq =
Π mnq
(5.5)
2
< vmnq >t , s
2
In this case, < vmnq >t , s should be calculated and averaged over the entire radiating
surface. It can be obtained using the following equation.
vmnq
2
=
t,s
1
4πh(a + b) + 2π(a 2 − b 2 )
{∫
h/2
∫
2 π( a + b)
−h / 2 0
142
a 2π
2
U q2 dl dz + ∫ ∫ Wmn
dϕ dr
b
0
}
(5.6)
Mode
BEM
Analytical
m =0
n=2
m=1
n=2
q=2
q=0
Figure 5.3: Directivity patterns for selected modes
143
For pure in-plane modes, the far-field sound pressure can be calculated by using only
equation (5.4). Acoustic power and radiation efficiency can be obtained by the same
expressions as equations (5.2) and (5.5). Since normal surfaces do not contribute to the
2
modal sound pressure, < vq >t , s should be calculated over radial surfaces only using the
following equation.
vq
2
=
t,s
h / 2 2 π( a + b)
1
U q2 dl dz
∫
∫
h
/
2
0
−
4πh(a + b)
(5.7)
The accuracy of this procedure has been confirmed through a comparison with a purely
numerical analysis in terms of the directivity patterns, sound powers and radiation
efficiencies. In the numerical analysis, velocities on the rotor surface are calculated from
predicted natural frequencies and mode shapes. And, sound radiation data have been
calculated using uncoupled, direct, exterior, and unbaffled boundary element analyses
[5.16]. The analytical directivity patterns for selected modes are compared with
numerical prediction in Figure 5.3. As one can see, analytical and numerical directivity
patterns for 3 types of modes are consistent with each other. In addition to the directivity
patterns, modal sound powers and radiation efficiencies are listed in Table 5.3. Like the
directivity patterns, modal sound radiation data from analytical solutions match
numerical data quite well. Therefore, analytical modal radiation basis can be used to
calculate sound radiation from a brake rotor when it is excited by a multi-directional
harmonic force. This procedure is explained in the next section.
144
Mode
Efficiency (σ)
Power (dB)
Indices
Type
Out-ofPlane
Analytical
BEM
Analytical
BEM
-
74
72
0.15
0.05
-
-
83
86
0.37
0.44
3
-
-
92
92
0.77
0.56
-
-
-
0
0
0
-
-
-
-
0
-
99
99
0.52
0.69
0
1
1
-
87
88
0.40
0.31
1
2
2
-
91
91
0.43
0.34
1
3
3
-
96
95
0.71
0.44
m
n
q
l
0
1
-
0
2
0
In-Plane
Combined
Table 5.3: Sound powers and radiation efficiencies for selected modes.
145
5.5 Vibro-Acoustic Response to a Multi-Directional Harmonic Force
If a rotor is excited by a multi-directional harmonic force of arbitrary frequency,
several in-plane and out-of-plane modes are simultaneously excited. Based on the
procedure introduced in Chapter 4, velocity distribution (v) on the disk area of a rotor
surfaces can be expressed in terms of the structural modes of the disk.
{v} = {η}T {Φ V }
{Φ }= {Φ
V
V
1
, Φ V2 , Φ V3 , , Φ V44
{η} = {η1 , η 2 , η3 , , η 44 }
}
(5.8)
where Φj is the jth velocity modal vector of the disk and ηi is the corresponding modal
participation factor. Also, the far-field pressure (P) on the sphere (SV) surrounding the
rotor due to surface velocity of equation (5.8) is expressed as follows:
P = {η} {Γ}
T
{Γ}= {Γ1 , Γ2 , Γ2 , Γ3 ,, Γ44 }
(5.9)
where, Γj is the modal sound pressure for the jth mode obtained from equation (5.1) or
(5.3). Sound power (Π) from the rotor due to an arbitrary harmonic force (f) is also
calculated from far-field sound pressures, on a sphere surrounding the disk, as follows:
Π = ISV
s
=
1 2π π P 2 2
R sin θ dθ dφ
2 ∫0 ∫0 ρ0c
(5.10)
Corresponding radiation efficiency (σ) is calculated using Π from equation (5.10) and
2
< v > t , s that is obtained from velocity on the total radiating surfaces using an equation
similar to equation (5.7). As in the case of modal sound radiation, vibro-acoustic
146
responses to a multi-directional force are obtained using the proposed analytical
procedure and compared with prediction of a numerical analysis. In the purely
computational analysis, structural responses for the excitation are obtained from the
forced vibration analysis with the finite element model and acoustic responses are
obtained by boundary element analysis given structural velocities. In this example, unit
harmonic forces in the normal and tangential directions are applied in the mid-plane of
rotor at ϕ = π/2 as shown in Figure 5.4. Far-field sound pressure spectra p/f(ω) are
obtained using above procedure from 0.8 to 8 kHz. Results for two receiver positions rp1
(R = 1m, θ = 0, φ = 0) and rp2 (R = 1m, θ = π/4, φ = 0) are compared with purely
numerical results in Figure 5.5. In addition, sound power and radiation efficiency spectra
have been determined. A comparison of analytically obtained spectra and numerical
results is given in Figure 5.6. As shown in Figure 5.5, p/f(ω) from the analytical approach
match purely computational predictions quite well for both cases. Furthermore, these
results depend on the receiver positions indicating that the source is highly directive.
Also, analytical acoustic power and radiation efficiency spectra show relatively good
agreement especially below 5 kHz. As seen from Figures 5.5 and 5.6, the proposed
analytical procedure has sufficient accuracy in calculating the sound radiation due to a
harmonic force. Sound pressure at a given receiver position, sound power or radiation
efficiency for a given harmonic excitation can be easily calculated using this process.
Furthermore, this process may be easily extended to multi-location or multi-frequency
excitation cases by considering the modal participation factors. Even though numerical
structural modal data have been used, experimental data could be utilized instead of
numerical data.
147
fn
fr
Figure 5.4: Finite element model used for forced vibration analysis given a multidirectional harmonic force.
148
100
dB re 20µpa/N
80
60
40
20
0
1
2
3
1
2
3
4
5
6
7
8
6
7
8
100
dB re 20µpa/N
80
60
40
20
0
4
5
Frequency (kHz)
Figure 5.5: Far-field sound pressure spectra p/f(ω) for selected receiver positions rp1 and
rp2. Key: Analytical
&RPSXWHGXVLQJ%(0----.
149
Π (dB re 1pW)
100
80
60
40
20
1
2
3
1
2
3
4
5
6
7
8
6
7
8
1
0.8
σ
0.6
0.4
0.2
0
4
5
Frequency (kHz)
Figure 5.6: Acoustic power spectra Π(w) and radiation efficiency spectra σ(w) of the
brake rotor. Key: Analytical
&RPSXWHGXVLQJ%(0----.
150
5.6 Conclusion
Modal sound radiation from a brake rotor is successfully calculated by using a semianalytical procedure based on structural modal response yielded by a numerical code and
analytical solution for sound radiation from a generic, thick annular disk. Vibro-acoustic
response for a harmonic force is synthesized quite reliably from modal sound radiation
data and the structural modal participation factors using the modal expansion technique.
This procedure could utilize either computational or experimental modal data such as
natural frequencies, mode shapes and modal damping ratios. The modal expansion
technique used to calculate sound radiation due to a harmonic excitation could be easily
generalized to multi-locations or multi-frequencies excitations. One would, however,
need to define additional modal participation factor vectors. Effects of geometric
modifications on vibro-acoustic characteristics can be easily investigated using this
procedure. Also, this procedure could be combined with pre-developed methods using
numerical and experimental approaches. Finally, it is possible to develop vibro-acoustic
design guidelines using this procedure.
151
REFERENCES FOR CHAPTER 5
5.1 H. Murakami, N. Tsunada and T. Kitamura, “A Study Concerned with a
Mechanism of Disc-Brake Squeal,” SAE Paper # 841233.
5.2 H. Matsui, H. MURAKAMI, H. NAKANISHI and Y. TSUDA, “Analysis of Disc-Brake
Squeal,” SAE Paper # 920553.
5.3 W. V. Nack and A. M, Joshi, “Friction Induced Vibration: Brake Moan,” SAE
Paper # 951095.
5.4 D. N. Herting, MSC/NASTRAN Advanced Dynamic Analysis User’s Guide, pp.
157-173, 1997
5.5
J. FLINT AND J. HULTÈN 2002 Journal of Sound and Vibration 254 (1), 1-21.
Lining-Deformation–Induced Modal Coupling as Squeal Generator in a Distributed
Parameter Disc Brake Model.
5.6 Y. K. Hu, and L. I. Nagy, “Brake Squeal Analysis by Using Nonlinear Transient
Finite Element Method,” SAE Paper # 971510.
5.7 O. N. Hamzeh, W. W. Tworzydlo, H. J. Chang and S. T. Fryska, “Analysis of
Friction-Induced Instabilities in a Simplified Aircraft Brake, SAE Paper # 1999-013404.
5.8 G. D. Liles, “Analysis of Disc Brake Squeal Using Finite Element Methods,” SAE
Paper # 891150.
5.9 G. Dihua and J. Dongying, “A Study on Disc Brake Squeal using Finite Element
Methods,” SAE Paper # 980597.
5.10 T. Hamabe, I. Yamazaki, K, Yamada, H. Matsui, S. Nakagawa and M. Kawamura,
“Study of a Method for Reducing Drum Brake Squeal,” SAE Paper # 1999-010144.
5.11 S. W. Kung, K. B. Dunlap and R. S. Ballinger, “Complex Eigenvalue Analysis for
Reducing Low Frequency Squeal,” SAE Paper # 2000-01-0444.
5.12 T. S. Shi, O. Dessouki, T. Warzecha, W. K. Chang, and A. Jaya sundera,
“Advances in Complex Eigenvalue Analysis for Brake Noise” SAE Paper # 200101-1603.
5.13 K. B. Dunlap, M. A. Riehle and R. E. Longhouse, “An Investigative Overview of
Automotive Disc Brake noise” SAE Paper # 1999-01-0142.
5.14 J. G. McDaniel and X. Li, “Analysis of Instabilities and Power Flow in Brake
Systems with Coupled Modes” SAE Paper # 2001-01-1602.
152
5.15 I-DEAS User’s manual version 8.2, SDRC, USA, 2000.
5.16 SYSNOISE User’s manual Version 5.4, NIT, Belgium, 1999.
5.17 M. R. Lee and R. Singh, “Analytical formulations for annular disk sound radiation
using structural modes”, Journal of the Acoustical Society of America 95(6), pp.
3311-3323, 1994.
153
CHAPTER 6
CONCLUSION
6.1 Summary
This study has resulted in the development of new analytical and semi-analytical
procedures for the calculation of modal and multi-modal sound radiation from thick
annular disk. In-plane and out-of-plane modal sound radiations of the disk have been
obtained using analytical solutions based on cylindrical and modified annular plate
models respectively. Multi-modal sound radiation has been calculated from modal sound
radiation and structural modal participation factors using modal expansion technique.
Accuracy of these analytical solutions has been confirmed by numerical analyses and
vibro-acoustic experiments. As a practical example, whole procedure applied to a brake
rotor and successfully predicted sound radiation from a brake rotor.
In Chapter II, structural eigen-solutions for the out-of-plane modes of a thick
annular disk with free-free boundaries have been calculated using both thick and thin
plate theories. The differences between two approaches are clarified in terms of natural
frequency and mode shapes. New analytical formulation for the modal sound radiation
from a thick annular disk has been proposed with the consideration of the disk thickness.
154
Two approaches, analytical and semi-analytical procedures have been developed
combining this analytical formulation for modal sound radiation with structural modal
response yield by an analytical calculation or a numerical prediction. In addition, the
same problem has been solved by a purely numerical procedure in which the disk surface
velocity is numerically defined by a finite element model and sound radiation is then
obtained using a boundary element model. Also, the effects of radii and thickness ratios
on the structural and acoustic radiation characteristics have been investigated using the
analytical procedure. Finally, the effect of boundary conditions has been examined using
the semi-analytical method.
In Chapter III, in-plane vibration of a thick annular disk has been investigated
using Transfer Matrix Method. Structural modal characteristics from this method have
been confirmed by numerical and experimental results with excellent agreements. Three
analytical approaches based on modified cylindrical model – Rayleigh integral method,
Fourier series method, Sinc function method – have been employed to calculate sound
radiation from the vibration of this type. The results also have been confirmed numerical
analysis using finite and boundary element models and vibro-acoustic experiment in an
anechoic chamber. Excellent agreements among these results could be obtained. The
effects of vibrating frequency and geometric configuration on the modal sound radiation
have been investigated through parametric studies.
Chapter IV proposes a new semi-analytical procedure for the calculation of sound
radiation from a thick annular disk when it is excited by a multi-modal and multidirectional harmonic force. Structural response for a specific force has been calculated by
155
the modal expansion technique from the structural modal dataset for both in-plane (radial)
and out-of-plane (flexural) modes that have been obtained using analytical or numerical
methods. In addition, acoustic responses for various harmonic excitations have been
estimated by the same technique utilizing the structural modal participation factors and
the normal radiation solutions representing the far-field sound pressures due to the
structural modal vibrations. These methods are confirmed by comparing predicted results
for a single frequency excitation with numerical calculations. Based on this procedure,
acoustic power and radiation efficiency spectra of the sample disk corresponding to a
specific force location and direction are obtained. This study has been extended to multipoints and multi-frequencies excitation cases for greater generality of this procedure. The
effects of coupling between structural modes and natural frequency separation on the
acoustic radiation are also investigated through the proposed procedure. This study could
lead to strategies that would minimize sound radiation from a thick annular disk excited
by arbitrary harmonic forces.
The sound radiation from a brake rotor has been calculated using the whole
procedure proposed in previous chapters. Structural modes of the rotor have been
expressed by structural modes of a generic annular disk having similar geometric
configuration. Modal sound radiations of a brake rotor have been synthesized using those
of corresponding generic annular disk. Vibro-acoustic response of the rotor to a multidirectional harmonic excitation has been calculated using the procedure introduced in
Chapter IV. Modal and broadband vibro-acoustic characteristics of a brake rotor could be
calculated very efficiently using this procedure.
156
6.2 Contributions
This study has significantly advanced the literature (based on the thin plate model)
on the vibro-acoustic characteristics of a thick annular disk by including sound radiation
from radial edges and the effect of disk thickness. In particular, the following major
contributions emerge.
1. Analytical and semi-analytical solutions for sound radiation from modal vibrations of
a thick annular disk have been introduced and validated using computational and/or
vibro-acoustic experimental methods. In the radial mode case, total sound radiation is
expressed as a sum of radiation from two radial edges that have been obtained using
the modified cylindrical radiator method. Proposed solution has been successfully
compared with the classical approach for several limiting cases. Far-field sound
pressures from two normal surfaces are combined by considering the disk thickness
effects to obtain the total sound radiation from out-of-plane modes. Proposed theory
yields more accurate results than the classical solution based on a thin plate model.
Modal sound radiation solution for each mode over the given frequency range has
been calculated by using these two analytical solutions.
2. This study critically examined thick and thin plate theories and investigated the effect
of rotary inertia and shear deformations on the structural eigensolutions and acoustic
sound radiation from flexural (out-of-plane) modes. The effects of geometric
configuration on the modal sound radiations have been clarified by studying key
normalized parameters such as radii ratio and thickness ratio, based on the proposed
analytical solutions. Appropriate geometric modification to control sound radiation at
157
any specific frequency can be proposed based on the results of these parametric
studies.
3. Analytical and semi-analytical procedures for calculating sound radiation given
multi-modal harmonic excitations have been developed. Broadband vibro-acoustic
characteristics and responses to an arbitrary harmonic force excitation vector can be
easily calculated using analytical or numerical modal sound radiation solution and
structural modal participation factors. These procedures can be applied to a thick
annular disk to define sound radiation from couplings between in-plane and out-ofplane modes as well as couplings within the same type of modes. Also, acoustic
power from self and mutual radiation could be conveniently calculated using these
procedures with a reasonable accuracy.
6.3 Future Research
Since analytical and semi-analytical solutions for sound radiation from a thick
annular disk have been successfully introduced in this study, application of such solutions
to practical components such as gears, brake rotors, or clutches could be the main task of
future research work. Some specific issues have been identified and proposed for future
studies:
1.
Investigate the vibro-acoustic characteristics of annular disks with annular or radial
slots. Analytical and semi-analytical solutions and procedures proposed in this work
could be used in this investigation.
158
2.
Investigate the effect of a hat structure on the vibro-characteristics of a brake rotor
through a parametric study using the procedures proposed in our study. This study
could suggest design guidelines for a hat structure.
3.
Examine the role of material damping and sliding friction on the radiation
properties of rotors.
4.
Conduct a complete investigation on the brake squeal noise by combining analytical
solutions within a large scale computational code.
159
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