Intern
ati
o
n
a
l
Jo
urn
a
l
o
f
Ad
va
nces
in Applied Sciences (IJ
A
AS)
V
o
l.
2, N
o
. 1
,
Mar
c
h
20
13
,
pp
. 41
~50
I
S
SN
: 225
2-8
8
1
4
41
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJAAS
Dispersi
on of Th
erm
o El
asti
c W
a
ves in a Rot
a
ting
Cylindrical Panel
R .
S
elv
a
m
a
ni
1
, P.
Ponnus
am
y
2
1
Departm
e
nt
of
m
a
them
atics K
a
r
u
n
y
a
Universi
t
y
,
Coim
batore
,
Ta
m
il Nadu,
India
2
Department of
mathematics, Go
vt Arts College (
A
utonomous),
Coimbatore, Tamil Nadu, India
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Oct 9, 2012
Rev
i
sed
Jan 13, 201
3
Accepte
d
Ja
n 27, 2013
The t
h
ree
di
m
e
nsi
onal
di
s
p
ersi
on
o
f
t
h
erm
o
el
ast
i
c
wave
s i
n
a
hom
oge
neo
u
s
i
s
ot
ro
pi
c r
o
t
a
t
i
ng cy
l
i
n
d
r
i
cal
panel
i
s
i
nve
st
i
g
at
ed i
n
the context
of t
h
e linear theory
o
f
th
erm
o
elasticity. Th
ree
displacem
ent pote
n
tial func
tions a
r
e introduce
d
to
uncouple the
equat
i
o
ns
o
f
m
o
ti
on
.
The
fre
que
ncy
e
q
u
a
t
i
ons are o
b
t
a
i
n
ed f
o
r
t
r
act
i
on f
r
ee b
o
u
n
d
ary
co
ndi
t
i
ons usi
ng B
e
ssel
funct
i
o
n s
o
l
u
t
i
o
ns.
I
n
o
r
d
e
r to
illu
strate th
eo
retical d
e
v
e
lop
m
en
t, n
u
m
erical so
l
u
tio
ns are
obt
ai
ne
d
an
d p
r
esent
e
d gra
p
h
i
cal
l
y
fo
r a
zin
c
m
a
terial. In
this stu
d
y
we foun
d
th
at th
e wav
e
ch
aracteristic
s are
more stable and realistic in
the prese
n
ce
of therm
a
l and t
h
e rotation
pa
ra
meters.
Keyword:
Bessel fun
c
tion
so
l
u
tio
n
D
i
s
p
er
s
i
on
an
alys
is
R
ot
at
i
ng
cy
l
i
n
dri
cal
panel
Therm
o
elasticity
Copyright ©
201
3 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
R
.Sel
vam
a
ni
,
Depa
rtem
ent of Mathem
atics,
Karun
y
a Un
iv
ersity,
Co
im
b
a
to
r
e
, Ta
m
il N
a
d
u
,
India, 64
114
.
Em
a
il: selv
a
m
1
729
@g
m
a
il.c
o
m
1.
INTRODUCTION
The
dispe
r
sion of dis
p
lacem
e
n
t, tem
p
erature cha
nge
in a
rotating cylindrical pa
nel is
plays a vital
ro
le in
sm
art
material ap
p
licatio
n
s
and
ro
tatin
g
g
y
ro
scop
e
.
Thi
s
t
y
pe o
f
m
odel
anal
y
s
i
s
i
s
very
im
port
a
nt
i
n
bi
o se
nsi
n
g a
p
pl
i
cat
i
ons i
n
n
u
cl
ear m
a
gnet
i
c
reso
na
nce (
N
M
R
), m
a
gnet
i
c
reso
na
nce i
m
agi
n
g (M
R
I
) a
n
d ech
o
pl
ana
r
im
agi
n
g
(EPI
). T
h
e an
al
y
s
i
s
of t
h
er
m
a
l
l
y
i
nduce
d
vi
brat
i
o
n o
f
r
o
t
a
t
i
ng cy
l
i
n
d
r
i
cal
panel
i
s
com
m
on
place in the design
of struct
ures
, atom
ic r
eactors, st
eam
turbi
n
es, s
upersonic aircra
ft, and
othe
r devices
ope
rating at el
evated tem
p
erature. At t
h
e
present tim
e ap
p
lied
m
a
th
e
m
a
tician
s
are exhib
itin
g
con
s
id
erab
le
i
n
t
e
rest
i
n
dy
nam
i
cal
m
e
t
hods
of el
ast
i
c
i
t
y
, si
nce t
h
e
usu
a
l
quasi
st
at
i
c
appr
oach
i
gno
res cert
a
i
n
ver
y
im
port
a
nt
feat
ures
of t
h
e p
r
o
b
l
e
m
s
un
der c
onsi
d
erat
i
o
n.
That
ap
pr
oach
i
s
based on t
h
e assum
p
t
i
on t
h
at
t
h
e
in
ertia ter
m
s
may b
e
o
m
i
tte
d
fro
m
th
e eq
u
a
tio
ns o
f
m
o
tio
n
.
Th
is assum
p
t
i
o
n
ho
ld
s
g
ood
on
ly when
th
e
v
a
r
i
ation
s
in
str
e
sses and
d
i
splace
m
e
n
t
s, b
u
t
th
er
e ar
ise num
b
e
r
o
f
p
r
ob
lem
s
in
en
g
i
n
e
erin
g
and
technolo
g
y
,
wh
en
t
h
is assum
p
t
i
o
n
m
a
y n
o
t
ho
ld goo
d an
d th
e i
n
ertia t
e
rm
s in
th
e equ
a
tio
ns
o
f
m
o
tio
n
m
a
y h
a
v
e
lead
to
cases of c
o
nsi
d
era
b
le m
a
the
m
atical com
p
lications.
In
t
h
e
field of nonde
structive e
v
aluation, laser-ge
n
erate
d
wav
e
s h
a
v
e
at
tracted
great atten
tio
n
o
w
i
n
g to
th
eir
p
o
t
ential ap
p
licatio
n to
no
ncon
tact an
d
non
d
e
st
ru
ctiv
e
ev
alu
a
tion
o
f
sh
eet m
a
terials.
Th
e h
i
g
h
v
e
l
o
cities o
f
m
o
d
e
rn
aircraft g
i
v
e
rise
to
aero
d
y
na
m
i
c h
eatin
g
,
wh
ich
produces i
n
tense therm
a
l stresses, re
du
cing the st
rength of t
h
e aircraft st
ructure. In the
nuclear field,
the
extrem
ely high tem
p
eratures
and tem
p
erat
ure gr
adi
e
nt
s ori
g
i
n
at
i
n
g
i
n
s
i
de
nuclear
re
actors i
n
flue
nc
e their
d
e
sign
an
d
op
eratio
n
s
. Moreov
er, it is well reco
gn
ized th
at
th
e inv
e
stig
atio
n
of th
e t
h
ermal effects on
ro
t
o
tin
g
el
ast
i
c
wave
p
r
opa
gat
i
o
n
has
beari
n
g
o
n
m
a
ny
st
r
u
ct
u
r
al
a
ppl
i
cat
i
o
ns.
Th
e static an
alysis can
no
t
p
r
ed
ict th
e
b
e
h
a
v
i
or
o
f
t
h
e m
a
terial due
to t
h
e therm
a
l stresses cha
n
ges
very
rapi
dl
y
.
There
f
ore i
n
c
a
se o
f
s
u
d
d
e
n
l
y
appl
i
e
d l
o
a
d
, t
h
erm
a
l
def
o
rm
ati
on an
d t
h
e r
o
l
e
of i
n
er
t
i
a
are
get
t
i
ng m
o
re i
m
port
a
nt
. Thi
s
t
h
erm
o
el
ast
i
c st
ress res
p
o
n
se bei
ng si
gn
i
f
i
cant
l
eads t
o
t
h
e p
r
opa
gat
i
on
o
f
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
252
-88
14
IJA
A
S
Vol
.
2
,
N
o
.
1,
M
a
rc
h
20
1
3
:
4
1
–
50
42
therm
o
elastic
stress wa
ves i
n
solids.
The t
h
e
o
ry
of th
erm
o
elasticit
y is well estab
lish
e
d
by Nowack
i
[1
]. Lo
rd
an
d
Shu
l
m
a
n
[2
] an
d
Green
an
d
Li
n
d
say [3
]
m
o
d
i
fied
th
e
Fou
r
ier law and
con
s
titu
tiv
e relatio
n
s
, so
as to
g
e
t
hy
pe
rb
ol
i
c
eq
u
a
t
i
on f
o
r
heat
con
d
u
ct
i
on
by
t
a
ki
ng i
n
t
o
acc
ou
nt
t
h
e t
i
m
e
neede
d
f
o
r acc
el
erat
i
on
of
he
at
fl
ow
and rela
xation of stresses.
A special feature of the
G
r
een–
L
i
n
dsay
m
odel
i
s
t
h
at
i
t
does not
vi
ol
at
e t
h
e
classical Fo
u
r
i
e
r's h
eat co
nd
uctio
n
law.
Vibratio
n
of
fu
n
c
ti
o
n
a
lly g
r
ad
ed
m
u
l
tilayered
o
r
tho
t
rop
i
c cylin
drical
panel
under therm
o
m
echani
cal load
was an
alyzed
b
y
X.Wang et.al [4
]. Hallam
an
d
Ollerto
n
[5
]
in
v
e
stig
ated
th
e th
erm
a
l s
t
resses and
d
e
flectio
n
s
th
at o
ccurred
in
a co
m
p
o
s
ite c
y
lin
d
e
r du
e to
a u
n
i
form
rise in
t
e
m
p
erat
ure
,
expe
ri
m
e
nt
al
ly
and t
h
e
o
retically and com
p
ared the obtaine
d
res
u
lts by a
special applica
tion of
t
h
e f
r
oze
n
st
re
ss t
ech
ni
q
u
e
of
p
hot
o el
ast
i
c
i
t
y
. N
oda
[
6
] ha
s studied t
h
e t
h
erm
a
l-induce
d inte
rfacial cracking
of
m
a
gnet
o
el
e
c
t
r
o el
ast
i
c
m
a
t
e
ri
al
s u
nde
r
u
n
i
f
orm
heat
fl
ow.
Chen et al
[7] a
n
al
yzed
t
h
e point tem
p
erature
so
lu
tion
for a p
e
nn
ay-sh
a
p
p
ed
crack
in
an
in
fi
n
ite
transv
ersely iso
t
rop
i
c th
erm
o
-p
i
ezo
-elastic m
e
d
i
u
m
subjecte
d
to a
concentrate
d
therm
a
l load applied ar
bitra
r
ily at the cra
c
k s
u
rface using
the gene
ra
lized
p
o
t
en
tial th
eory. Ab
ouh
am
ze
[8
]
d
i
scussed a m
u
lti o
b
j
ectiv
e op
timiza
tio
n
strateg
y
fo
r
op
tim
a
l
sta
c
k
i
ng
sequ
en
ce
o
f
lamin
a
ted
cylin
drical p
a
n
e
ls is
p
r
esen
ted
with resp
ect to
t
h
e first n
a
t
u
ral freq
u
e
n
y
and
critical
buc
kl
i
n
g l
o
a
d
usi
n
g t
h
e w
e
i
g
ht
ed s
u
m
m
a
t
i
on m
e
t
hod. He
use
d
t
h
e t
r
ai
ne
d ne
ural
net
w
or
k t
o
e
v
al
uat
e
t
h
e
fitn
ess fu
n
c
tion
in
th
e o
p
timizatio
n
p
r
o
cess an
d
in
th
is way increasing t
h
e proce
d
ure s
p
eed. Chadwic
k
[9]
stu
d
i
ed
th
e p
r
op
ag
ation
of
p
l
an
e h
a
r
m
o
n
i
c wav
e
s in
ho
m
o
g
e
n
o
u
s
an
iso
t
rop
i
c h
eat co
nductin
g
so
lid
s. Sh
ar
m
a
[1
0]
i
nvest
i
g
a
t
ed t
h
e t
h
ree
dim
e
nsi
onal
vi
b
r
at
i
on a
n
alysis of a transversely
isotropic therm
o
elastic
cy
l
i
ndri
cal
pa
nel
.
T
h
e ap
pl
i
cat
i
on o
f
po
w
e
rf
ul
n
u
m
e
rica
l to
o
l
s lik
e fi
n
ite el
e
m
ent or boundary el
e
m
ent
m
e
t
hods t
o
t
h
e
s
e pr
o
b
l
e
m
s
i
s
al
so bec
o
m
i
n
g
i
m
port
a
nt
. P
r
ev
ost
an
d Ta
o
[1
1]
carri
ed
o
u
t
an aut
h
ent
i
c
fi
ni
t
e
ele
m
ent analysis of
problem
s
includi
ng rela
xation e
ffects.
Esla
m
i
an
d
Vahed
i
[1
2
]
app
lied
th
e Galerk
in
fin
ite
ele
m
en
t to
th
e
co
up
led th
ermo
elasticity p
r
ob
lem
in
b
eam
s
.
Hu
ang
an
d Tau
c
h
e
rt
[1
3]
derive
d the
a
n
alytical
so
lu
tion
for cro
ss-p
l
y lam
i
n
a
ted
cylin
drical p
a
n
e
ls
with
fin
ite leng
th
sub
j
ected
t
o
m
e
c
h
an
ical and
therm
a
l
lo
ad
s
u
s
ing
the ex
tend
ed
p
o
wer series m
e
th
od
. Pon
n
u
s
amy an
d
Selv
aman
i [1
4
]
investig
ated
th
e
wav
e
propagation in a ge
neralized
therm
o
l elastic
plate em
bedd
ed o
n
el
ast
i
c
m
e
di
um
. Pon
n
u
sam
y
and Sel
v
am
ani
[1
5]
ha
ve st
u
d
i
ed t
h
e di
s
p
er
si
on a
n
al
y
s
i
s
of
gene
ral
i
zed
m
a
gnet
o
-t
her
m
o el
ast
i
c
waves i
n
a t
r
a
n
s
v
ersel
y
i
s
ot
ro
pi
c cy
l
i
n
dri
cal
pa
nel
u
s
i
ng t
h
e w
a
ve
pr
o
p
agat
i
o
n
app
r
oach
.Lat
er
,Sel
vam
a
ni
an
d P
o
n
n
u
sam
y
[1
6]
st
udi
e
d
t
h
e
da
m
p
i
ng o
f
ge
ne
ral
i
zed t
h
e
r
m
o
el
ast
i
c
waves
i
n
a
hom
oge
ne
ous
i
s
ot
r
o
pi
c p
l
at
e usi
n
g t
h
e
wave
pr
o
p
agat
i
o
n a
p
p
r
oach a
n
d o
b
tained t
h
e numerical result for Zinc
pl
ate. Since the s
p
e
e
d of the
disturbe
d
wave
s de
pe
nd
up
o
n
r
o
t
a
t
i
o
n
rat
e
, t
h
i
s
t
y
p
e
of st
udy
i
s
i
m
port
a
nt
i
n
t
h
e desi
g
n
of
hi
gh s
p
ee
d st
ea
m
,
gas
t
u
r
b
i
n
e an
d r
o
t
a
t
i
on rat
e
sen
s
ors
.
L
o
y
and L
a
m
[17]
di
scu
s
sed t
h
e vi
brat
i
on
of
rot
a
t
i
n
g t
h
i
n
cy
l
i
ndri
c
a
l
panel
usi
n
g L
o
ve fi
rs
t
app
r
o
x
i
m
at
i
o
n t
h
e
o
ry
.
B
h
i
m
arad
di
[
1
8]
de
v
e
l
ope
d a
hi
g
h
e
r
o
r
der t
h
eo
ry
f
o
r t
h
e
free
vi
b
r
at
i
o
n
analysis of circ
ular cylindrical she
ll. Zh
ang
[1
9
]
inv
e
stig
ated
th
e
p
a
ram
e
tr
i
c
an
alysis o
f
freq
u
e
n
c
y of
ro
t
a
tin
g
la
m
i
n
a
ted
co
mp
o
s
ite cylin
dri
cal sh
ell u
s
ing wav
e
prop
a
g
a
t
i
on
ap
p
r
oac
h
. B
ody
wave pr
opa
gat
i
o
n
i
n
r
o
t
a
t
i
n
g
therm
o
elastic
m
e
dia was investigated
by
Sha
r
m
a
and
G
r
o
v
er
[
20]
.
Th
e effect
of
r
o
t
a
t
i
on, m
a
gnet
o
fi
el
d
,
t
h
erm
a
l
rel
a
xat
i
o
n
t
i
m
e
and
p
r
essu
re
o
n
t
h
e
wave
pr
opa
gat
i
on
i
n
a
gene
r
a
l
i
zed vi
sc
o el
ast
i
c
m
e
di
um
un
de
r
t
h
e i
n
fl
ue
nce o
f
t
i
m
e
har
m
oni
c sou
r
ce i
s
di
scusse
d by
A
b
d
-
Al
l
a
an
d B
a
y
one
s [2
1]
.T
he pr
o
p
agat
i
o
n o
f
wav
e
s
in conducting
piezoelectric s
o
lid is
studied for the case
whe
n
the entir
e
m
e
d
i
u
m
ro
tates with
a u
n
ifo
r
m
ang
u
l
a
r
vel
o
ci
t
y
by
W
a
uer
[2
2]
. R
o
y
c
ho
u
d
h
u
r
i
an
d M
u
kh
o
p
ad
hy
ay
st
udi
e
d
t
h
e
ef
fe
ct
of
r
o
t
a
t
i
o
n
an
d
rel
a
xat
i
o
n t
i
m
es o
n
pl
a
n
e
wa
ves i
n
ge
neral
i
zed t
h
e
r
m
o
vi
sco el
ast
i
c
i
t
y
[23]
. Gam
e
r [2
4
]
has di
sc
usse
d t
h
e
elastic-p
lastic d
e
fo
rm
atio
n
of th
e ro
tating
solid
d
i
sk.
Lam
[25] has studie
d the fre
que
nc
y characteristics of a
t
h
i
n
rot
a
t
i
n
g
cy
l
i
ndri
cal
s
h
el
l
usi
n
g ge
neral
d
i
ffere
nt
i
a
l
q
u
ad
rat
u
re
m
e
t
hod.
In
t
h
i
s
pa
per,
t
h
e t
h
ree
di
m
e
nsi
onal
di
sp
ersi
on
o
f
t
h
erm
o
e
l
ast
i
c
wave
s i
n
a
h
o
m
ogene
ous
i
s
ot
ro
pi
c
ro
tating
cylin
drical p
a
nel is d
i
scu
s
sed
u
s
i
n
g
th
e lin
ear
t
h
ree-d
i
m
e
n
s
io
nal th
eo
ry
o
f
therm
o
elasticity. The
fre
que
ncy
e
q
u
a
t
i
ons a
r
e
obt
ai
ned
usi
n
g
t
h
e t
r
act
i
o
n
f
r
ee
b
o
u
n
d
ary
c
o
n
d
i
t
i
ons
. T
h
e B
e
ssel
f
unct
i
o
n
wi
t
h
com
p
lex argument is directly used to fi
nd t
h
e sol
u
tions
a
n
d are studied
num
e
rically
for the
m
a
terial Zinc. T
h
e
co
m
p
u
t
ed
n
on-d
i
m
e
n
s
io
n
a
l phase v
e
l
o
cities are
p
l
o
tted
i
n
the fo
rm
o
f
d
i
sp
ersion
cu
rv
es.
2.
FORMULAT
ION OF
T
H
E PROBLEM
C
onsi
d
er a cy
l
i
n
d
r
i
cal
panel
as sh
ow
n i
n
Fi
g.
1 o
f
l
e
n
g
t
h
L havi
n
g
i
n
n
e
r
and
o
u
t
e
r ra
di
us a an
d
b
wi
t
h
t
h
i
c
k
n
ess
h and
uni
fo
r
m
angul
ar
vel
o
ci
t
y
. The an
g
l
e subt
en
de
d b
y
t
h
e cy
li
ndri
c
al
panel
,
w
h
i
c
h i
s
kn
o
w
n as ce
nt
er an
gl
e, i
s
de
not
e
d
by
. The
cy
l
i
ndri
cal
pa
n
e
l
i
s
assum
e
d to be
hom
oge
n
e
ou
s, i
s
ot
r
o
pi
c and
l
i
n
earl
y
el
ast
i
c
wi
t
h
Yo
u
n
g
’
s
m
odul
us E
,
P
o
i
sso
n rat
i
o
a
n
d
den
s
i
t
y
i
n
a
n
un
di
st
u
r
be
d
s
t
at
e.
In cylindrical coordi
nate the three dim
e
nsiona
l stress e
quation
of motion, strain
displacem
ent
relatio
n
and
h
e
at co
ndu
ction
i
n
th
e absen
c
e of
b
o
d
y
force for a lin
early elastic ro
tatin
g m
e
d
i
u
m
.
11
,
,,
,
,
((
)
2
)
t
rr
r
r
rz
z
r
r
t
t
rr
u
u
u
ur
ur
r
u
u
rr
11
,,
,
,
,
2
rr
r
z
z
z
z
r
t
t
rr
v
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
AA
S I
S
SN
:
225
2-8
8
1
4
Di
spersi
o
n
of
Ther
mo
El
a
s
t
i
c
Waves
i
n
a
Ro
t
a
t
i
ng C
y
l
i
n
dri
c
al
P
anel
(
R
.
S
e
l
v
am
ani
)
43
11
,
,,
,
,
((
)
2
)
t
rz
r
z
z
z
z
r
t
t
rr
u
u
w
ur
ur
r
u
u
rr
(
1
)
12
1
,,
,
,
,
0
,
,
,
,
rr
r
z
z
t
rt
t
t
tz
K
Tr
T
r
T
T
c
T
T
u
r
u
v
w
whe
r
e
is th
e
mass d
e
n
s
ity,
v
c
is
the specific he
at capacity,
/
Kc
is
t
h
e d
i
ffusiv
ity,
K
is th
e th
erm
a
l
co
ndu
ctiv
ity,
0
T
i
s
t
h
e uni
f
o
rm
reference
t
e
m
p
erat
ure ,
t
h
e di
spl
ace
m
e
nt
equat
i
o
n
of m
o
t
i
on has t
h
e
additional term
s
with a t
i
m
e
de
pe
nde
nt centripetal acceleration
()
u
and
,
2
t
u
whe
r
e,
(,
0
,
)
uu
w
is the dis
p
lacement vector and
(0
,
,
0
)
is a co
nstan
t
, th
e co
mma n
o
t
atio
n
u
s
ed
in th
e su
b
s
crip
t
d
e
no
tes
th
e
p
a
rtial d
i
fferen
tiatio
n wit
h
resp
ect t
o
t
h
e v
a
riab
les. The stress st
rain
relatio
n
s
are
g
i
ven
as fo
llo
ws
()
2
(
)
rr
rr
z
z
rr
ee
e
e
T
()
2
(
)
rr
z
z
ee
e
e
T
(
2
)
()
2
(
)
zz
rr
zz
zz
ee
e
e
T
Whe
r
e
ij
e
are the strain com
ponents,
is th
e th
erm
a
l
stress co
efficien
ts, T is th
e te
m
p
eratu
r
e, t is
th
e ti
m
e
,
and
are
Lam
e
’ c
onst
a
nt
s. T
h
e st
rai
n
ij
e
are
related to t
h
e
displacem
ents are give
n
by
rr
rz
r
z
zz
rr
u
e
r
1
u
rr
v
e
(
3
)
zz
w
e
z
1
r
v
rr
vu
r
rz
wu
rz
1
z
r
vw
z
(4)
Whe
r
e
,,
uvw
are
displacem
ents along
radial, circum
fe
rential and
axial di
rec
tions
respectively,
,,
rr
z
z
are t
h
e n
o
rm
al
st
ress com
ponent
s a
n
d
,,
rz
z
r
are t
h
e shea
r st
ress
com
pone
nt
s ,
,,
rr
zz
ee
e
are
norm
al strain com
p
onents
and
,,
rz
z
r
ee
e
are s
h
ea
r st
rai
n
c
o
m
pone
n
t
s.
Substituting the Eq. (3) a
n
d Eq. (2) in
Eq.
(1), gi
ves the followi
ng t
h
re
e
displacem
ent
equations
of moti
on
12
2
1
2
,,
,
,
,
,
,
2
,,
,
23
2
rr
r
z
z
r
rz
rt
t
t
ur
u
r
u
r
u
u
r
v
r
v
w
Tu
w
u
12
2
2
1
1
,,
,
,
,
,
,
,,
23
rr
r
z
z
r
z
tt
vr
v
r
v
r
v
v
r
u
r
u
r
w
Tv
,
12
1
1
,,
,
,
,
,
2
,,
,
2
2
z
z
zr
r
r
r
z
z
zt
t
t
ww
r
w
r
w
u
r
v
r
u
Tw
u
w
,
12
1
1
,,
,
,
,
,
2
,,
,
2
2
z
z
zr
r
r
r
z
z
zt
t
t
ww
r
w
r
w
u
r
v
r
u
Tw
u
w
12
1
,,
,
,
,
0
,
,
,
,
()
vr
r
r
z
z
t
t
r
t
t
t
z
cT
r
T
r
T
T
c
T
T
u
r
u
v
w
(5)
Th
e ab
ov
e co
up
led
p
a
rtial d
i
fferen
tial eq
u
a
ti
o
n
s
is also
subj
ected
t
o
th
e
fo
llo
wi
n
g
no
n-d
i
m
e
n
s
io
n
a
l
bounda
ry c
o
nditions at the
s
u
rfaces
,
ra
b
(i)
Th
e traction
free
no
n d
i
men
s
ion
a
l m
ech
an
ical bo
und
ary co
nd
itio
ns for a st
ress
free ed
g
e
are g
i
v
e
n by
0,
rr
r
r
z
(
6
a)
(ii). Th
e
no
n d
i
men
s
io
n
a
l i
n
sulated
or iso
t
h
e
rmal th
erm
a
l b
o
u
n
d
a
ry cond
itio
n is
g
i
v
e
n
b
y
,
0
r
Th
T
(6b
)
Whe
r
e
h is t
h
e surface
heat
trans
f
er coe
f
ficient .He
r
e
0
h
co
rresp
ond
s
to
th
erm
a
ll
y
in
sulated
surface a
n
d
h
re
fers
t
o
i
s
ot
he
r
m
al
one.
To s
o
lve
E
q
.
(
5
)
,
w
e
take
[
1
0]
1
,,
r
u
r
1
,,
v
r
,
z
w
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
252
-88
14
IJA
A
S
Vol
.
2
,
N
o
.
1,
M
a
rc
h
20
1
3
:
4
1
–
50
44
U
s
ing
Eq
. (5
) in
Eq
. (1
),
w
e
f
i
n
d
th
at
T
,
,
satisfies th
e eq
u
a
tion
s
.
22
2
2
22
1
22
2
((
2
)
)
(
(
)
2
)
(
)
T
zt
zt
z
(
7
a)
22
2
22
2
1
22
1
((
2
)
)
(
(
)
2
)
(
)
T
rt
zt
(7b)
22
2
2
1
22
()
0
zt
(7c)
22
22
0
11
22
()
1
()
0
V
Ti
TT
T
kt
C
K
zz
(7d)
Eq
.
(7
c) in
term
s
o
f
gi
ves a
pu
rel
y
t
r
ans
v
e
r
se wa
ve,
w
h
i
c
h
is not affected by tem
p
erature. T
h
is
wav
e
is po
lari
zed
in
p
l
an
es
p
e
rp
en
d
i
cu
lar to
th
e z-ax
is.
We assu
m
e
th
at th
e d
i
stu
r
b
a
n
ce is ti
me h
a
rm
o
n
i
c
th
ro
ugh
th
e facto
r
e
i
t
.
3.
SOLUTION
TO THE PROBLEM
The E
q
s.
(
7
)
a
r
e co
u
p
l
e
d
pa
r
t
i
a
l
di
ffere
nt
i
a
l
eq
uat
i
ons
o
f
t
h
e t
h
ree
di
spl
a
cem
e
nt
com
ponent
s
.
T
o
unc
o
upl
e Eqs
.
(7
), we
ca
n w
r
i
t
e
t
h
ree di
spl
a
cem
e
nt
fu
nct
i
o
ns whi
c
h
sat
i
s
f
i
es
t
h
e
si
m
p
l
y
sup
p
o
rt
e
d
bo
u
nda
ry
co
nd
itio
ns fo
ll
o
w
ed
b
y
Sh
arma [10
]
(,
,
,
)
(
)
s
i
n
(
)
c
o
s
(
/
)
it
rz
t
r
m
z
n
e
(,
,
,
)
(
)
s
i
n
(
)
s
i
n
(
/
)
it
rz
t
r
m
z
n
e
(
8
)
(,
,
,
)
(
)
s
i
n
(
)
s
i
n
(
/
)
it
rz
t
r
m
z
n
e
F
(,
,
,
)
(
,
,
,
)
s
i
n
(
)
s
i
n
(
/
)
it
Tr
z
t
Tr
z
t
m
z
n
e
Wh
ere m
is th
e circu
m
feren
t
i
a
l
m
o
d
e
and
n
is th
e ax
ial
m
o
d
e
,
ω
i
s
t
h
e an
gul
a
r
f
r
eq
uency
of
t
h
e
cylin
d
r
ical p
a
nel
m
o
tio
n
.
By
in
tro
d
u
c
ing
t
h
e
d
i
m
e
n
s
io
n
l
ess qu
an
tities
'
r
r
R
'
z
z
L
0
T
T
T
n
L
mR
L
t
4
1
2
2
1
2
C
22
2
2
1
R
C
22
2
R
(9)
After su
b
s
titu
tin
g Eq
.
(9) and
Eq
.8
i
n
Eq
.
(7
), we ob
tain
t
h
e
fo
llowing
system
o
f
equ
a
tio
n
s
2
2
21
()
0
k
(
10
a)
2
21
2
4
()
0
gg
g
T
(
1
0b)
22
23
2
()
(
1
)
(
2
)
0
4
gg
T
(10c)
22
2
2
2
22
3
1
2
1
)0
(
LL
ti
T
i
t
i
(
1
0d)
w
h
er
e
2
2
2
22
2
1
rr
r
r
,
2
0
1
2
1
V
TR
CC
K
2
1
2
V
C
CK
1
3
CR
K
2
1
2
(2
)
(
)
L
gt
24
(1
)
L
L
g
ti
t
2
2
34
()
L
gt
2
0
4
2
TR
g
51
g
1
C
wav
e
v
e
l
o
city o
f
th
e cylin
d
r
ical p
a
n
e
l. A no
n-triv
ial
so
lu
tion
of th
e alg
e
b
r
aic syst
e
m
s (1
0)
ex
ist on
ly wh
en
th
e d
e
term
in
an
t of
E
q
s
.
(10) are
equal t
o
z
e
ro.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
AA
S I
S
SN
:
225
2-8
8
1
4
Di
spersi
o
n
of
Ther
mo
El
a
s
t
i
c
Waves
i
n
a
Ro
t
a
t
i
ng C
y
l
i
n
dri
c
al
P
anel
(
R
.
S
e
l
v
am
ani
)
45
2
12
4
2
22
22
3
4
22
2
2
2
52
5
2
2
3
()
(1
)
(
)
(
2
)
(
,
,
)
0
()
LL
gg
g
gg
T
ig
ig
t
t
i
(
1
1)
Eq
. (1
1), on
si
m
p
l
i
ficatio
n
red
u
c
es to th
e
fo
l
l
o
w
ing
d
i
fferen
tial eq
uatio
n
:
64
2
222
0
AB
C
(12
)
Whe
r
e,
2
2
5
12
3
4
2
3
(1
)
L
Ag
g
g
g
g
i
t
i
22
2
55
5
13
1
4
2
4
1
2
3
4
2
2
2
3
2
23
2
3
1
3
2
(2
)
(
)
(
1
)
(
1
)
()
()
L
l
L
g
g
gg
i
g
gg
i
t
g
g
g
g
g
i
t
g
g
i
gt
g
i
g
i
Bg
22
2
1
3
3
2
345
()
(
2
)
LL
gt
i
i
g
g
g
t
Cg
Th
e so
lu
tion
of Eq. (11
)
is
3
1
()
(
(
)
(
)
)
ii
i
i
i
rA
J
r
B
Y
r
3
1
()
(
(
)
(
)
)
ii
i
i
i
i
rd
A
J
r
B
Y
r
3
1
()
(
(
)
(
)
)
ii
i
i
i
i
Tr
e
A
J
r
B
Y
r
(
1
3)
Here
,
2
i
r
are the
non-zero
roots
of the
alge
braic
equation
64
2
0
ii
i
rA
r
B
r
C
The a
r
bi
t
r
a
r
y
c
onst
a
nt
i
d
and
i
e
is ob
tain
ed fro
m
1
22
2
23
1
(2
)
(2
)
ii
i
i
g
d
gg
44
1
3
4
2
1
3
2
0
22
2
41
3
2
43
4
2
(1
)
2
)
ii
i
i
i
gg
g
g
g
e
TR
g
g
gg
(1
4)
Eq
. (9
a) is a B
e
ssel equ
a
tion
with
its
p
o
s
sib
l
e so
l
u
tio
n
s
is
4
''
11
2
41
4
1
1
2
41
2
44
1
()
()
,
0
,0
()
(
)
,
0
A
J
kr
BY
kr
k
Ar
B
r
k
AI
k
r
B
K
k
r
k
(
1
5)
Whe
r
e
2
'2
11
kk
and
J
and
Y
are Besse
l functions of the first a
n
d
second
kinds
respectivel
y
wh
ile,
I
and
k
are
m
odi
fi
ed B
e
ssel
fu
nct
i
ons
o
f
fi
rst
an
d sec
o
nd
ki
n
d
s
res
p
e
c
t
i
v
el
y
.
,
1,
2
,
3
,
4
ii
AB
i
are
th
e arb
itrary co
n
s
tan
t
s.
Gen
e
rally
2
1
0
k
, so
t
h
at th
e situ
atio
n
2
1
0
k
will n
o
t
b
e
d
i
scussed
in
th
e
fo
ll
o
w
i
n
g
.
For c
o
nve
n
ience, we
consi
d
e
r
the
case
of
2
1
0
k
and the
de
rivation for the ca
se
of
2
1
0
k
is si
m
i
lar.
Th
e so
lu
tion
of Eq. (10
a
) is
41
4
1
()
(
)
(
)
rA
J
k
r
B
Y
k
r
(
1
6)
Whe
r
e
22
2
1
(2
)
L
kt
4.
SPECI
A
L CASES
4.
1
T
h
erm
o e
l
asti
ci
t
y
B
y
t
a
ki
ng
0
th
e
m
o
tio
n
co
rresp
ond
ing
to
t
h
e
ro
tation
a
l m
o
d
e
d
ecoup
le from th
e rest of
m
o
t
i
o
n
an
d th
e
v
a
riou
s resu
lts red
u
c
es to
t
h
e th
erm
o
elasticity
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
252
-88
14
IJA
A
S
Vol
.
2
,
N
o
.
1,
M
a
rc
h
20
1
3
:
4
1
–
50
46
3
1
()
(
(
)
(
)
)
s
i
n
s
i
n
it
ii
i
i
i
rA
J
r
B
Y
r
m
z
n
e
3
1
()
(
(
)
(
)
)
s
i
n
s
i
n
it
ii
i
i
i
i
rd
A
J
r
B
Y
r
m
z
n
e
(
1
7)
3
1
()
(
(
)
(
)
)
s
i
n
s
i
n
it
ii
i
i
i
i
Tr
e
A
J
r
B
Y
r
m
z
n
e
41
4
1
(
)
()
()
s
i
n
c
o
s
it
rA
J
k
r
B
Y
k
r
m
z
n
e
Wi
t
h
2
1
2
(2
)
(
)
L
gt
22
2
1
(2
)
L
kt
(18)
Eq
s. (1
7)& (18) co
n
s
titu
te the so
lu
ti
o
n
fo
r the ho
m
o
g
e
no
us iso
t
ro
p
i
c cylind
r
ical
p
a
n
e
l
wi
th
traction
free
b
oun
d
a
ry
co
nd
itio
ns. It i
s
n
o
ticed
th
at
Eq
.
(18
)
is si
milar to
th
e p
a
rticu
l
ar case
ob
tain
ed
and
d
i
scu
ssed
by Sharm
a
[10] in case
o
f
th
erm
o
elasticit
y.
4.
2
E
l
as
to
ki
n
e
ti
c
In
th
e present an
alysis if
we tak
e
th
e co
up
ling
p
a
rameter fo
r ro
tatio
n
a
l an
d
t
h
erm
a
l field
1
0
th
en th
e equ
a
tio
n
s
will redu
ces to
t
h
e classical case in
elast
o
k
i
n
e
tic.
2
23
2
22
22
1
,0
1
gg
g
(19)
42
23
2
3
0
AB
G
(
2
0)
13
2
12
3
cc
Ag
g
g
1
13
g
B
g
2
1
()
(
)
()
ii
i
i
i
A
Jr
B
Y
r
r
2
1
(
)
()
()
ii
i
i
i
i
rd
A
J
r
B
Y
r
(2
1)
2
2
3
1
i
i
i
r
d
rg
Eq
s. (1
9)& (20) co
n
s
titu
te the so
lu
ti
o
n
fo
r the ho
m
o
g
e
no
us iso
t
ro
p
i
c cylind
r
ical
p
a
n
e
l
wi
th
traction
free
bo
und
ary
co
nd
itio
ns.
It is no
ticed
t
h
at
Eq
. (1
9) and
Eq
.2
0 are simila
r to
on
e as ob
tain
ed
and
d
i
scussed
b
y
Chen et al [19] in
case
of elas
tokinetics.
5.
FREQ
UEN
C
Y
EQ
UATI
O
N
In
th
is section
we sh
all d
e
rive th
e secu
lar eq
u
a
tion
for th
e th
ree d
i
m
e
n
s
io
n
a
l
v
i
b
r
atio
n
s
cylin
d
r
ical
panel
subj
ected to traction
free boundary c
o
nd
itions at t
h
e
uppe
r a
n
d lower surfaces at
,
ra
b
'
'
'
'
)
)
)
sin(
)
s
in(
si
n(
)
c
os(
co
s
(
)
s
i
n
(
L
it
it
it
e
um
z
r
vm
z
e
r
wt
m
z
e
(,
,
,
)
(
,
,
,
)
s
i
n
(
)
s
i
n
(
/
)
it
Tr
z
t
Tr
z
t
m
z
n
e
'
22
''
'
22
22
2
2
11
22
(
)
sin(
)
c
os(
)
rr
iL
it
t
rr
r
r
rr
r
r
mz
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
AA
S I
S
SN
:
225
2-8
8
1
4
Di
spersi
o
n
of
Ther
mo
El
a
s
t
i
c
Waves
i
n
a
Ro
t
a
t
i
ng C
y
l
i
n
dri
c
al
P
anel
(
R
.
S
e
l
v
am
ani
)
47
'
22
'
2
22
2
12
2
2(
)
s
i
n
(
)
c
o
s
(
)
it
r
i
mz
e
rr
r
rr
r
''
2
c
os(
)
s
i
n(
)
it
rz
L
tm
z
e
r
(
2
2)
Wh
ere p
r
ime d
e
no
tes th
e
d
i
fferen
tiatio
n with
resp
ect to
r
,(
,
,
)
i
i
uu
R
i
r
z
ar
e thr
ee n
on–
dim
e
nsional di
splacem
ents and
,,
rr
z
rr
r
r
z
rr
are
three
non-dim
ensional
stresses
Usi
n
g t
h
e
res
u
l
t
obt
ai
ne
d i
n
t
h
e E
q
s.
(
1
)
-
(
3
) i
n
Eqs
.
(
6
) w
e
can
get
t
h
e
f
r
eq
ue
ncy
eq
ua
t
i
on
of
fre
e
v
i
br
atio
n as
f
o
l
l
o
w
s
0
,
1
,
2
,
.
..8
ij
E
ij
(23
)
The
values
of t
h
e
ij
E
are
defi
ne
d
i
n
Ap
pe
ndi
x.
6.
NU
MER
I
C
A
L
RES
U
LTS AN
D DIS
C
US
SION
The f
r
eq
ue
ncy
Eq. (
2
2) i
s
n
u
m
e
ri
cally solved
for Zinc
material
. For the purpose of num
erical
com
putation we consi
d
er the
closed circ
ul
ar cylindrical shell with the center angle
2
an
d th
e in
teg
e
r n
m
u
st be even
since the s
h
ell vibrates in ci
rcum
fe
rent
i
a
l
f
u
l
l
wave
. T
h
e fre
que
ncy
e
quat
i
o
n
fo
r a
cl
osed
cylin
d
r
ical sh
ell can
b
e
o
b
t
ai
n
e
d b
y
settin
g
1
,
2
,
3
...
..
ll
whe
r
e
l
is th
e ci
rcu
m
feren
tial
wav
e
nu
m
b
er i
n
Eq.
(
1
4
)
.
The
m
a
t
e
ri
al
pro
p
er
t
i
e
s of a
Zi
nc
i
s
t
a
ke
n f
r
o
m
[10]
f
o
r
i
s
ot
r
o
pi
c m
a
t
e
ri
al
33
7.14
10
k
g
m
11
2
0.
38
5
1
0
Nm
11
2
0.50
8
1
0
Nm
0.3
rps
62
1
5.75
1
0
d
e
g
Nm
0
29
6
TK
21
1
1.
2
4
10
de
g
KW
m
21
1
3.9
1
0
d
eg
CJ
k
g
The r
oot
s
of t
h
e al
ge
brai
c E
q
. (
1
2)
were c
a
l
c
ul
at
ed usi
n
g
a com
b
i
n
at
i
on of B
i
rge
-
Vi
t
a
m
e
t
hod a
n
d
Newt
on
-R
ap
hs
on
m
e
t
hod.
In
t
h
e p
r
ese
n
t
cas
e sim
p
l
e
B
i
rge
-
Vi
t
a
m
e
t
hod
d
o
es
not
w
o
r
k
f
o
r
fi
n
d
i
n
g t
h
e r
oot
of
t
h
e al
geb
r
ai
c equat
i
o
n.
Aft
e
r
obt
ai
ni
ng t
h
e
r
oot
s o
f
t
h
e al
g
e
brai
c eq
uat
i
o
n usi
n
g B
i
rge
-
Vi
t
a
m
e
t
hod, t
h
e ro
ot
s
are corrected for the desi
red
accuracy
usi
n
g the Newton-R
aphson m
e
thod
. This com
b
ination
has ove
r
com
e
th
e d
i
fficu
lties in
find
ing
th
e roo
t
s o
f
t
h
e alg
e
braic equ
a
tio
n
s
o
f
t
h
e govern
i
n
g
eq
u
a
ti
on
s. To
v
a
lid
at
e th
e
prese
n
t a
n
alysis a com
p
arative study is
pres
ented i
n
Ta
bl
e.1
fo
r d
i
fferen
t
v
a
lu
es
of th
ickn
ess to inn
e
r
rad
i
u
s
rat
i
o
(
h
/
b
=
0
.
1
,
0.
2,
0.
3) a
n
d
cent
e
r a
ngl
e
00
0
30
,
6
0
,
90
of a cylindrica
l
panel in t
h
e
abse
nce of the
r
m
a
l
an
d ro
tatio
n
a
l
effect.
A co
m
p
arison
is m
a
d
e
b
e
tween
t
h
e
no
n d
i
m
e
n
s
io
n
a
l freq
u
e
n
c
ies
of th
erm
a
l
l
y in
su
lated
and i
s
ot
he
rm
al
m
odes o
f
vi
b
r
at
i
on o
f
a rot
a
t
i
ng an
d n
o
n
ro
tatin
g
cylin
drical sh
ell with
resp
ect to
d
i
fferen
t
ro
tation
a
l speed
in Tab
l
.2 and Tab
l
e.3
,
resp
ectiv
ely. Fro
m
Table.2 a
n
d Ta
ble.3 it is clear that as t
h
e rot
a
tional
spee
d increa
se
s, the non dimensional
frequencies are als
o
increases i
n
bo
th
ro
tating
and
non
ro
tatin
g
cases.
As th
e ro
tation
of th
e cylind
r
ical sh
ell increases, the
coupling effect of
various
i
n
teracting fields also
i
n
creases
res
u
l
t
i
ng i
n
hi
g
h
er
f
r
e
que
ncy
.
Tabl
e 1.T
h
e
l
o
west
nat
u
ral
fr
eque
ncy
of
Zi
n
c
cy
l
i
ndri
cal
pan
e
l with resp
ect to
th
ick
n
e
ss
to
inn
e
r rad
i
u
s
ratio
.
Ta
ble 1.
C
o
m
p
ari
s
o
n
bet
w
een
t
h
e n
on
di
m
e
nsi
onal
f
r
e
que
n
c
i
e
s of R
o
t
a
t
i
ng an
d
No
n
-
R
o
t
a
t
i
ng t
h
e
r
m
o
-el
a
st
i
c
cylin
d
r
ical sh
ell for th
erm
a
ll
y in
su
lated bou
nd
ary
i
n
t
h
e
fi
rst
t
h
ree
m
odes o
f
vi
brat
i
o
n.
h/b
(
)
Re
f[2
3]
Re
f[2
4]
P
r
e
s
e
n
t
0.
1
30
60
90
0.
7207
0.
8262
0.
9697
0.
7207
0.
8257
0.
9680
0.
7190
0.
8192
0.
9533
0.
2
30
1.
3448
1.
3429
1.
3325
60
1.
3118
1.
3055
1.
1990
0.
3
90
30
60
90
1.
3015
1.
9803
1.
8362
1.
6937
1.
2901
1.
9706
1.
8099
1.
6552
1.
2877
1.
9690
1.
8135
1.
6743
Rotating
Non-Rotating
n
=
1
n
=2
n
=3
n
=1
n
=2
n
=
3
0.
1
0.
1033
0.
1159
0.
1462
0.
0899
0.
1059
0.
1259
0.
3
0.
3721
0.
4821
0.
5250
0.
2897
0.
2707
0.
3779
0.
5
0.
5285
0.
6221
0.
6614
0.
5406
0.
5241
0.
6327
0.
7
0.
9898
0.
9053
0.
7999
0.
7840
0.
9005
0.
8945
1.
0
1.
3144
1.
3728
1.
4663
1.
1353
1.
2064
1.
3977
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
252
-88
14
IJA
A
S
Vol
.
2
,
N
o
.
1,
M
a
rc
h
20
1
3
:
4
1
–
50
48
Ta
ble 2.
C
o
m
p
ari
s
on
bet
w
ee
n t
h
e
n
o
n
di
m
e
nsi
o
nal
f
r
e
que
nc
ies of R
o
tatin
g and
No
n-R
o
tatin
g
t
h
erm
o
-elastic
cy
l
i
ndri
cal
s
h
el
l
fo
r i
s
ot
herm
al
bo
u
nda
ry
in th
e
first three m
o
d
e
s
o
f
v
i
bratio
n.
A di
s
p
er
si
o
n
c
u
r
v
e i
s
d
r
a
w
n
bet
w
ee
n t
h
e
no
n
-
di
m
e
nsi
o
n
a
l
wave
num
ber ve
rs
us di
m
e
nsi
onl
es
s
pha
se vel
o
ci
t
y
i
n
case
of
r
o
t
a
t
i
ng an
d
n
o
n
-
r
ot
at
i
n
g t
h
ermall
y in
su
lated
cylin
drical sh
ell with
respect to
diffe
re
nt thic
k
n
ess
pa
ram
e
ter
s
*0
.
1
,
0
.
2
5
,
0
.
5
tb
a
R
for t
h
erm
a
ll
y in
su
lated and
iso
t
h
e
rm
al b
o
und
aries
i
s
sho
w
n i
n
Fi
g.
1 an
d Fi
g
.
2 res
p
ect
i
v
el
y
.
The sol
i
d
line curves c
o
rrespond to
ro
t
a
tin
g
th
erm
o
elastic
cylin
d
r
ical sh
ell an
d th
e do
tted
lin
e curv
es t
o
that of
no
n-ro
tatin
g sh
ell. Fro
m
th
e Fi
g
s
.1 and
2
,
it is
o
b
serv
ed
t
h
at
t
h
e
n
o
n
-
di
m
e
nsi
onal
p
h
a
s
e vel
o
ci
t
y
dec
r
eases
rapi
dl
y
to
becom
e
linear at
higher val
u
es
of wave
num
ber
for
b
o
t
h
th
ermally in
su
lated
an
d iso
t
h
e
rm
al
cases.Th
e
ph
ase v
e
l
o
city o
f
lo
wer
v
a
lu
e
of
*
t
in
case
o
f
no
n
ro
tating
sh
ell i
s
o
b
serv
ed
to
i
n
crease
fro
m
z
e
ro
wav
e
number a
n
d bec
o
me stable at
hi
g
h
er val
u
es o
f
wave
n
u
m
b
e
r
for
bo
th
th
e th
erm
a
l b
o
und
aries. Th
e
phase
velocity at highe
r
value of
*
t
attain
qu
ite larg
e
v
a
lu
es
at
t
h
e vani
s
h
i
n
g
wave
num
ber
and a
r
e n
o
n
-
di
spersi
ve d
u
e t
o
r
o
t
a
t
i
on.
Wh
en t
h
e t
h
i
c
k
n
e
ss param
e
t
e
r of t
h
e
cy
l
i
ndri
cal
pa
nel
i
s
i
n
creased, t
h
e di
m
e
nsi
onl
ess
pha
se
v
e
lo
city is d
ecreases fo
r both
ro
tating
and
no
n-
ro
tating
cylin
drical sh
ell.
Fi
g.
1.
Vari
at
i
o
n
o
f
w
a
ve
n
u
m
b
er
verses
p
h
ase
vel
o
ci
t
y
wit
h
d
i
fferen
t
t*
for
th
erm
a
ll
y in
su
l
a
ted
Zin
c
shell.
The c
o
m
p
ari
s
on
o
f
Fi
g
.
1 a
nd
Fi
g.
2 s
h
o
w
s t
h
at
t
h
e
n
o
n
-
di
m
e
nsi
onal
phase
vel
o
ci
t
y
decrease
s
ex
pon
en
tially fo
r sm
aller wave n
u
m
b
e
r in
case o
f
th
erm
a
ll
y
in
su
lated
an
d
iso
t
h
e
rm
al b
o
u
n
d
a
ries fo
r all v
a
lue
of
t
*
,b
ut
t
h
e
case o
f
hi
g
h
e
r
wa
ve
n
u
m
b
er t
h
e
n
o
n
-
di
m
e
nsi
o
nal
pha
se
vel
o
ci
t
y
i
s
st
eady
a
n
d
sl
o
w
fo
r al
l
v
a
lu
es of t*.
Fi
g.
2.
Vari
at
i
o
n
o
f
w
a
ve
n
u
m
b
er
verses
p
h
ase
vel
o
ci
t
y
wi
t
h
di
ffe
re
nt
*
t
fo
r is
othe
rm
al Zinc shell.
Rotating
Non-Rotating
n=1 n=2
n=3
n=1 n=2
n=3
0.
1
0.
1026
0.
1215
0.
1413
0.
0741
0.
1078
0.
1214
0.
3
0.
4443
0.
4549
0.
5245
0.
2243
0.
3550
0.
4247
0.
5
0.
6077
0.
7075
0.
7378
0.
5922
0.
7071
0.
7077
0.
7
0.
9196
0.
8200
0.
9044
0.
9094
0.
9909
0.
9909
1.
0
1.
4149
1.
4256
1.
4644
1.
4142
1.
4156
1.
4142
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
AA
S I
S
SN
:
225
2-8
8
1
4
Di
spersi
o
n
of
Ther
mo
El
a
s
t
i
c
Waves
i
n
a
Ro
t
a
t
i
ng C
y
l
i
n
dri
c
al
P
anel
(
R
.
S
e
l
v
am
ani
)
49
7.
CO
NCL
USI
O
N
The t
h
ree
di
m
e
nsi
o
nal
di
s
p
e
r
si
o
n
anal
y
s
i
s
of a
h
o
m
ogen
e
ou
s i
s
ot
r
o
pi
c rot
a
t
i
n
g cy
l
i
ndri
cal
pa
nel
su
bj
ected
to the tractio
n free
b
oun
d
a
ry co
nditio
n
s
h
a
s
b
e
en
con
s
id
ered for th
is
p
a
p
e
r. Fo
r th
is
p
r
ob
lem
,
th
e
go
ve
rni
ng e
q
u
a
t
i
ons
of t
h
ree
di
m
e
nsi
onal
l
i
n
ear t
h
e
r
m
o
el
ast
i
c
i
t
y
have
been
em
pl
oy
ed an
d s
o
l
v
e
d
by
t
h
e
B
e
ssel
fu
nct
i
o
n s
o
l
u
t
i
o
n
wi
t
h
c
o
m
p
l
e
x ar
g
u
m
e
nt
. The e
f
f
ect
of t
h
e
wav
e
n
u
m
b
er o
n
t
h
e
phas
e
vel
o
c
i
t
y
of a
closed Zinc
cylindrical s
h
ell is inve
stigated
and the
res
u
lts
are
presente
d
as dis
p
ersi
on c
u
rves. T
h
e
rota
tional
spee
d a
n
d
di
f
f
ere
n
t
t
h
e
r
m
a
l b
o
u
n
d
ari
e
s
i
n
fl
uence
t
h
e
wave
pr
opa
ga
t
i
on c
h
aract
e
r
i
s
t
i
c
s. I
n
a
d
di
t
i
on,
a
com
p
arative study is m
a
de between the rot
a
ting and n
on
rot
a
t
i
n
g cy
l
i
n
d
r
i
cal
shel
l
and
t
h
e fre
que
ncy
chan
ge
i
s
obse
r
ve
d t
o
be hi
g
h
est
f
o
r t
h
e r
o
t
a
t
i
ng cas
e. Al
so
, a com
p
ari
s
on
of t
h
e
no
n di
m
e
nsi
o
n
a
l
freq
u
enci
es
fo
r t
h
e
di
ffe
re
nt
t
h
i
c
k
n
ess t
o
i
n
ner
ra
di
us
rat
i
o
of cy
l
i
ndri
cal
p
a
n
e
l
with
ou
t th
ermal an
d
ro
tational effects sh
ows well
ag
reem
en
t with
tho
s
e of ex
istin
g
literature.
APPE
NDI
X
The param
e
ters
ij
E
i
n
Eq
.
(2
2)
a
r
e
defi
ne
d a
s
2
2
11
1
1
1
1
1
1
1
1
2
2
1
11
1
1
1
1
0
1
22
2
1
11
1
1
2
11
1
(2
)
(
(
)
/
(
)
)
(
(
)
)
(
)
/
(
1
)
(
)
/
()
()
i
L
Et
t
t
t
t
tt
t
t
R
e
Jt
J
R
J
t
JJ
d
t
J
T
t
(
A
1)
2
2
13
1
1
2
1
1
1
1
2
2
2
21
1
1
2
1
0
22
2
2
21
2
2
2
2
12
2
1
)
(
2
)
(
(
)
/
(
))
((
)
)
(
/
(
1
)
(
)
/
()
()
i
L
Et
t
t
t
t
t
tt
t
t
R
e
JJ
R
J
t
JJ
d
t
J
T
t
(A
2)
2
2
15
1
1
3
1
1
1
1
2
2
2
31
1
1
2
1
0
22
2
2
31
3
3
2
2
13
3
1
)
(
2
)
(
(
)
/
(
))
((
)
)
(
/
(1
)
(
)
/
(
)
(
)
i
L
Et
t
t
t
t
t
tt
t
t
R
e
JJ
R
J
t
JJ
d
t
J
T
t
(A
3)
22
17
1
1
1
1
1
1
1
1
1
1
11
1
1
11
(
2
)
(
(
)
(1
)
(
)
/
(1
)
(
)
/
(
)
Et
t
t
t
t
t
kk
Jk
J
k
J
k
J
k
tt
(A
4)
21
1
1
1
11
1
1
2(
/
)
(
)
(
1
)
(
)
Et
t
t
JJ
(
A
6)
23
1
1
1
21
2
2
2(
/
)
(
)
(
1
)
(
)
Et
t
t
JJ
(A
7)
25
1
1
1
31
3
3
2(
/
)
(
)
(
1
)
(
)
Et
t
t
JJ
(A
8)
2
11
1
1
1
1
22
27
1
1
1
1
1
1
(
)
()
2
(
1
)
()
/
/
(
)
Et
t
t
t
t
t
k
R
Jk
Jk
k
J
k
(
A
9)
31
1
1
1
1
1
1
1
1
(
1
)
/
()
()
L
Et
d
t
J
t
J
t
(
A
10)
3
3
2
1
21
2
1
21
(
1
)
/
()
()
L
Et
d
t
J
t
J
t
(
A
11)
35
3
1
3
1
2
1
3
1
(1
)
/
(
)
(
)
L
Et
d
t
J
t
J
t
(
A
12)
37
1
1
1
(/
)
(
)
L
Et
t
J
k
t
(A
1
3
)
(A
14
)
4
3
2
1
21
2
1
2
1
21
[
(
/
)
()
(
)
()
()
]
Ee
t
J
t
J
t
h
J
t
(A
1
5
)
4
5
3
1
31
3
1
31
31
[
(
/
)
()
(
)
()
()
]
Ee
t
J
t
J
t
h
J
t
(A16
)
47
0
E
(A
17
)
4
1
1
1
11
1
1
11
1
1
[
(
/
)
()
(
)
()
()
]
Ee
t
J
t
J
t
h
J
t
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
252
-88
14
IJA
A
S
Vol.
2
,
N
o
.
1,
M
a
rc
h
20
1
3
:
4
1
–
50
50
In w
h
ich
*
1
12
ta
R
t
,
*
2
12
tb
R
t
and
*
tb
a
R
is the thickness -to-m
ean radius
ratio of
t
h
e pa
nel. Ob
vio
u
sly
2,
4,
6,
8
ij
Ej
can
o
b
taine
d
by
just
replaci
n
g
m
odified B
e
ssel fu
nctio
n
of t
h
e
first kin
d
in
1,
3
,
5
,
7
ij
Ei
with the
ones
of the second
kind, respectively, while
5,
6
,
7
,
8
ij
Ei
can be
obtaine
d by
jus
t
replacin
g
1
t
in
1,
2
,
3
,
4
ij
Ei
with
2
t
.
REFERE
NC
ES
[1]
Nowa
c
k
i
W.
D
y
namical problems of th
ermo elasticity
, Noordhoff, Ley
d
en, Th
e Netherlands, 1975
.
[2]
Lord S
hulm
a
n. “A
generali
zed d
y
nam
i
cal th
eor
y
of therm
o
elas
tic
it
y”
,
Journal of Mechanics of
Physics of Solid
s
,
Vol. 15
. Pp. 299
–309, 1967
.
[3]
A.
E
Gree
n a
nd K.
A
L
i
ndsay
.
“
T
herm
o elas
t
i
c
i
t
y
”,
Journal o
f
Ela
s
ticity
, Vol. 2. Pp.1–7, 1972.
[4]
X.W
a
ng. “
T
hree
dim
e
nsional an
aly
s
is of m
u
lti la
y
e
red fu
n
c
tion
a
lly
grad
ed an
isotropic c
y
lindr
ic
al pan
e
l unde
r
therm
o
m
echan
i
cal
load
”,
Me
cha
n
ics
of
m
a
ter
i
als
, Vol. 40
. Pp. 23
5-254, 2008
.
[5]
C.B H
a
llam
an
d E. O
llerton
. “
T
herm
al s
t
res
s
e
s
in axiall
y c
onnect
ed circu
l
ar c
y
lind
e
rs
”,
Journal of Strain
Analysis,
Vol/Issue: 8(3)
. Pp. 160
-167, 1973
.
[6]
C.F
G
a
o and N
.
N
oda. “
T
herm
al
-induced int
e
rfa
cia
l
crack
ing of m
a
gneto ele
c
tro
elas
ti
c m
a
teri
als
.
”,
International
Journal of Engin
eering S
c
ien
ces
,
Vol. 42
, Pp.1347
-1360, 2000
.
[7]
W
.
Q
Chen et.al. “
P
oint tem
p
eratur
e s
o
lution
for a penn
y
-
s
h
aped cra
c
k in a
n
infinite
trans
v
ers
e
l
y
is
otropi
c
therm
o
-piezo-
e
l
a
s
tic
m
e
dium
”,
Engineering Ana
l
ysis
with
Bound
ary elements
, Vol. 29
. Pp. 524-53
2, 2005
.
[8]
M
.
Abouham
ze a
nd M
.
Shakeri. “
M
ulti obje
c
tiv
e
stacking seque
n
ce optim
iz
ation
of lam
i
nated
c
y
lindric
al pan
e
ls
using th
e g
e
netic algor
ithm
and nueral
network
”
,
Composite structures
, Vol. 81
Pp. 253-263
, 2007
.
[9]
P.Chadwick. “B
asic prosperities
of
plane harm
onic waves in a
pre stresse
d heat conducting el
astic m
a
terial”,
Journal of Thermal Stresses
, Vo
l. 2
.
Pp. 193-214
, 2002
.
[10]
J.N Sharm
a
. “Three dim
e
nsio
nal vibration analy
s
is of
homogenous transversely
isotrop
i
c
therm
o
elastic
cy
lindrical p
a
nel”,
Journal o
f
Acoustical
Society
of America
, Vol. 110. Pp. 648-65
3, 2001
.
[11]
J
.
H
P
r
evos
t and D
.
Tao “
F
inite
e
l
em
ent an
al
ys
is
of d
y
n
a
m
i
c cou
p
led th
erm
o
elas
tici
t
y
probl
em
s
w
ith rel
a
xat
i
on
tim
es
”,
J.
App
l
.
Mech.
T
r
ans
. AS
ME
, Vol. 50
. Pp
. 817–822, 1983.
[12]
M.R Eslam
i
and H Vahedi. “Coupled
therm
o
elasticity
beam
problem
s”,
AIAA Journal
, V
o
l/I
s
s
u
e: 27(5). P
p
.
662–665, 1989
.
[13]
N.N Huang and T.R Tauc
her
t
. “Therm
o elastic solution for
cr
oss-ply
cy
lindrical p
a
nels”,
Jou
r
nal of Thermal
St
re
sse
s,
Vol. 14
. Pp. 227–237, 1
991.
[14]
P.Ponnusam
y
an
d R.Selvam
ani.
“W
ave pr
opagation in gener
a
lized therm
o
elas
tic plate em
bedded
in an elastic
m
e
dium
”,
Interaction
and multiscale mechan
ics,
Vol. 5
.
Pp.13-26
, 2012
.
[15]
P.Ponnusam
y
and R.Selvam
ani. “Dispe
rsion analy
s
is of generalized m
a
gneto
therm
o
elastic waves in
a
transeversely
isotropi
c cy
li
ndric
al
pa
ne
l
”
,
Journal of
thermal stres
s
es,
Vol. 35. Pp.
1119-1142, 201
2.
[16]
R.Selvam
ani
an
d P.Ponnusam
y
.
“Dam
ping
of g
e
neralized th
erm
o
el
astic plate
in
hom
ogeneous isotropic plate”,
Materials
Physics and mechan
ics
, Vol. 14
. Pp. 64
-73, 2012
.
[17]
C.T Lo
y
and K.Y Lam
.
“Vibration of
Rotating
Thin C
y
lindr
ica
l
P
a
nel”
,
Applied
Acoustic
, Vol.4
6
. Pp. 327-343,
1995.
[18]
A.A.Bhim
raddi. “A higher order theo
r
y
fo
r free vibration analy
s
is of
circu
l
ar
c
y
lindri
c
a
l
s
h
ell
”
,
International
Journal of Solid
and Structure,
V
o
l. 20
. Pp. 623-6
30, 1984
.
[19]
X.M Zhang. “The param
e
tr
ic
an
aly
s
is
of frequen
c
y
of ro
tating lam
i
nated co
m
posite cy
lindrical
shell using wave
propagation app
r
oach”,
Computer methods in applied
mechanics and engineering,
Vol. 191
. Pp. 2027-2043
,
2002.
[20]
J.N Sharm
a
and
D. Grover
.
“B
od
y
w
a
ve p
r
opa
gation
in ro
tat
i
n
g
therm
o
e
l
as
t
i
c
m
e
dia”
,
M
echa
n
ical
Res
e
ar
ch
Communications,
Vol. 36
. Pp. 71
5-721, 2009
.
[21]
A.M Abd-Alla
and F.S Bay
one
s
.
“
E
ffect of rotation in a gen
e
rali
zed m
a
gneto
therm
o
vis
c
o elas
tic m
e
di
a”,
Advances in
Theoretical and
App
lied
Mechan
ics
,
Vol. 4
.
Pp. 15-4
2
, 2011
.
[22]
J.W
a
uer, “W
aves in rotating an
d conducting piezoelectric m
e
dia”,
Journal of
Acoustical
Society of America
,
Vol/Issue: 106(2
)
. Pp. 626-636, 1
999.
[23]
S.K Roy
c
houdh
uri and S. Mukh
opadh
y
a
y
.
“Effect of rotation an
d relaxation tim
es on plane waves in generalized
therm
o
vis
c
o
el
a
s
ticit
y”
,
IJMMS
, Vol/Issue: 23(7
)
. Pp. 497-505, 2
000.
[24]
Chen et
.a
l. “
V
i
b
ration
anal
ys
is
of
orthotropic cy
lindrical shells
with fre
e ends
b
y
the Ray
l
eigh
–Ritz m
e
thod
”,
Journal of Soun
d and Vibration
, Vol. 195. Pp. 11
7–135, 1996
.
[25]
L.I H
u
a and K
.
Y
Lam
.
“
F
requenc
y
char
ac
teris
t
i
c
s
of a thin ro
tating cy
lindrical
shell using gen
e
ra
l
di
ffe
r
e
n
ti
al
quadratur
e m
e
th
od”,
In
ternation
a
l Journal
of Mechanica
l Scien
c
es
, Vol. 40. Pp. 4
43-459, 1998
.
Evaluation Warning : The document was created with Spire.PDF for Python.