CC BY
32
80 BecmHUK CaMry — EcmecmeeHHOHaynuan cepun. 2007. №2(52)
MEXAHMKA
YAK 534.1
ON DETERMINATION OF LINEAR FREQUENCIES OF BENDING VIBRATIONS OF PIEZOELECTRIC SHELLS AND PLATES BY EXACT AND AVERAGED TREATMENT
© 2007 A.G.Bagdoev, A.V. Vardanyan, S.V. Vardanyan1
In this paper the derivation and numerical solution of disspersion relations for frequencies of free bending vibrations for piezoelectric cylindrical thin shells with longitudinal polarization and plates with normal polarization is made. Solution is done by exact space treatment and by Kirchhoff hypothesis. Comparision of obtained tables shows that frequencies by exact and based on Kirchhoff hypothesis are quite different.
Introduction
The bending vibrations of magnetoelastic shells and plates by averaged treatment based on classic theory are considered in [1—5]. By the new space treatment at first developed for elastic plates in [6], the magnetoelstic vibrations of plates and shells are considered in [7-9]. The dispersion relation for Lamb waves in piezoelastic strip by exact treatment is obtained in [10], [11], where are obtained numerically five modes of mentioned waves, but is not made carefully investigation of solution of transcendent dispersion equation corresponding to law of relation of frequency from wave number for bending waves for thin plates. The above mentioned investigation is carried out analytically in [7-9] for magnetoelastic plates and it is shown that almost for all cases the results obtained by exact solution are distinguished essentially from averaged solution based on Kirchhoff hypothesis, which formerly give excellent results for elastic plates [6]. In present paper by space treatment of [6-11] are determined analytically and numerically the frequencies of free bending vibrations of the piezoelectric cylindrical shell with longitudinal polarisation and the comparison with averaged treatment is carried out. Besides the same investigation for piezoelectric plate with transverse polarization is carried out and are made and
1 Bagdoev Alexander Georgevich (bagdoev@mechins.sci.am), Vardanyan Anna Vanikovna (avardanyan@mechins.sci.am), Vardanyan Sedrak Vanikovich (sedrak@mechins.sci.am) Institute of Mechanics of NAS RA, Marshal Baghramyan 24 b, Yerevan, 0019, Armenia.
compared calculations of frequencies by space and averaged treatment. As for shell as for plate it is shown that Kirchhoff hypotesis for determination of free bending vibrations frequencies for piezoelectric is not applicable.
1. Statement of problem and solution for cylindrical shell
Let us consider the infinite cylindrical shell of small thickness 2h and radius of middle section R made from piezoelectric with elastic constants C11, C12, C13, C44 and piezomodulus e^\, 633, e\5 [10]. For the case of axial polarization of shell choosing coordinate along axis of cylinder and as radial coordinate one can write the stresses and electrical induction components in shell
[10] as
dUr Ur duz
arr - (-11—-----1- C12---1- C13—----1- eji—,
dr r dz dz
IdUr Ur\ dUz
ozz - C13 "T—1------+ C33——1- 633 —,
( dr r j dz dz
(dUr dUz \
Orz - C44 —------1- —— + £15 —,
\dz dr! dr (11)
dur ur duz dcj) ^ ' >
% - Li2^— + Cn— + C13— +631 — , dr r dz dz
(dUr dUz \
Dr - ei5 ——1- -7— ~ eii"S->
( dz dr j dr
(dUr Ur\ dUz
“a- + — + e33 "T-------£33 -T-,
dr r j dz dz
where Ur, Uz are displacements components, ^— potential of electrical field, Eh, £33 are dielectric permeabilities.
Then equations of motion and induction yield [10]
d2Ur 1 dUr Ur\ d2Ur ,d2Uz
Cn ~ty + --r----------j + ^44~aY + ^13 +
dr2 r dr r2 dz2 drdz
. . d2§ d2Ur
+ (g3i + eis) —— = p-
(1.2)
drdz dt2
^ , d2uz 1 duz\ ^ d2uz n d2ur 1 dur\
44 TT + + 33 TT + ('13 + ^44) 7T7T + +
dr2 r dr) dz2 \drdz r dz j
d2§ 1 \ d2§ d2Uz
+e'5^ + 7») + enl^ = p W
d2Uz 1 dUz\ d2Uz N /d2Ur 1 dUr
z z z r r
ei5 + -~r~ + ^33“TT + (ei5 + e3l) 7r~T" +
dr2 r dr dz2 \drdz r dz
jd2<b 1 d2<b
“eil|a^ + 7aFrC33aJ= ’
where is density. The exact particular solution of (1.2), (1.3) for propagated along axis plane wave as in corresponding piezoelectric plate [11] and in mag-netoelastic shell and plate [7-9] can be looked for in space treatment in form
= rvJ, j = 12,3,
Ur = AJI1 (?,) elkz-m + AJK1 (?,) elkz-m + c.c.,
X 1 J V 1 (1.4)
Uz = BjIo (?j) elkz—ltot + BJKo (J elkz—ltot + c.c.,
$ = ^ jIo Jkz—m + jo (J eikz—ltot + c.c.,
where Io, 1, Ko, 1 are Bessel functions of imagine argument on is carried out summation from 1 to 3. On account relations
I0 (?) = I1 (?), ko (?) = -K1 (?),
+ \h (?) = h (?), + \k, (?) = -Ko (?) (1'5)
d? ? d? ?
one can from (1.2)-(1.5) obtain
2
—iAjiCuvj — C44k2 + pto2^ + (C13 + C44) VjkBj+
+ (e31 + e15) Vjk§J = 0,
AjVjik (C13 + C44) + (C44V^ — C33k2 + pto2) Bj+ (1.6)
^ J-(e15V^ — e33k2} = 0,
(e31 + e15) AjV jik + Bj^v2 — e33k2) + ^ J (—eUV2 + 633k2) = 0.
The equation for V2 is distinguished from equation of [11] for plate with normal polarization and yields
pto2 2 C44 C13 + C44 C33
yJ = lJk’ 7^12 = v ’ r = ^1’ ---------------r------- = ^2’ r~ = ^4’
Cuk2 C11 C11 C11
Cu = 1
C„ 2’ (1.7)
e33 e15 2 E11 ,7 (e31 + e15)
M-5 =-----------, ^6 =-------------, ^7 = —, k{ = —-------------------,
e31 + e15 e31 + e15 E33 C11E33
k2 = , k3 = ^.5k\, det Hflj-jl = 0,
where
«11 = — X2 + ^1 — V2, «12 = «21 = M^Xj, a13 = k(KJ,
«22 = ^X2 — ^4 + V2, (1.8)
«23 = k2 (^X2 — ^5) , «31 = XJ, «32 = ^X2 — M-5, «33 = 1 — ^X2.
For elastic case when ^5 = ^ = k2 = k2 = k3 = 0 (1.7), (1.8) have two roots
X0 3, and for piezoelectric one must seek solution of (1.7) starting from values
0 0 1 (e31 + e15) ^j
of "h.2 = X3 = X~, Xi = —. Denoting -------------------------—-------- = qpy one can obtain from
M-7 C11
(1.6)
and [11]
where
iAj«11 + Bj«12 + 9 j«13 = 0,
iAj«21 + Bj«22 + 9 J«23 = 0, (1.9)
iAj«31 + B j«32 + 9 j«33 = 0
iAj = ajUj, Bj = pjUj, 9j = YjUj, (1.10)
(Xj) = «12«23 — «13 «22, P j (Xj)
= «12«23 — «13«22, P j (X= «21 «13 — «11 «23, X A = «11 «22 — «2
(1.11)
12
For Aj, —Bj, —9j one obtains the same (1.11) equation expressed by Uj. Then (1.4) gives
Ur = —iaJUJI1 (?^eikz—itot — iaJUjK1 (?^eikz—itot + c.c., Uz = pjUjIo (?j) eikz—itot — PjUjKo (?j) eikz—itot + c.c.,
(e31^15H = yjUjIo (%) elkz~imt - yjU'Ko (%) elkz~m + c.c., (L12)
(e31 + e15) i
—r--------$ = <P>
C11
where is carried out summation on from 1 to 3. Using (1.1), (1.5) and (1.12) one obtains
°rr _ _n TJ \ V /fc \jkz-imt _ ^.12 „ TT \ 1 (^■/) Jkz-imt
ike - c
+
11 C11 ?j
C
_P ft TT 7~ (’£ \ Jkz-mt n TTf\ i^f ("t \ Jkz-mt
c 11
+
- ^-pyt/;X0 (^)eite-itof+
C11 j ?j C11 j
e31 !/,y,£^,*0 (§,)**”“ +c.c„
e31 + e15 V ' e31 + e15
O
44
rz T T 7 \ Jkz—itot . 1/,*TTf^(<il \ Jkz—itot .
= rijUjl\ j e + rijUjK\ (§yJ e + c.c.,
kC
* , 13 \ , ^6C11,
» . = « + (> /. + J,
C44
t(e°;ei!)=w
f‘ = - E‘ (1.13)
(e31 + e15)2
The boundary conditions on shell surfaces as in case of piezoelectric plates give [11]
orr (R ± h, z) = 0, orz (R ± h, z) = 0,
C11 (1.14)
9 (R ± h, z) = 9 (R ± h, z), Dr (R ± h, z) = Dr (R ± h, z)
e31 + e15
The dimensionless potential qp(r, z, t) in dielectric out of shell satisfy the equa-d29 1 dcp d2p
tion —— H------------1---— = 0 and one can look for solution outside of shell in
dr2 r dr dz2
form
(9 = (j)+eikz itotK0 (kr) + c.c., r > R + h, (9 = (—eikz—itotI0 (kr) + c.c., r < R — h.
(1.15)
— d<f
For Dr = e— one obtains dr
—Dr = -<b+ee,kz mtK\ (hr) + c.c., r > R + h, k
\br = qb_ee'fe-'“f/i (kr) + c.c., r < R-h. k
The first line equations (1.14) give four equations
■jW (5.7) + W" Ov)+
(1.16)
C K1 (?±) C
+a,U’l\,K[ (<•*) + -2.ajUfo-±±L + (?;) - (1.17)
11 ?j 11
' + (1-| '
— (1 — UjYjIo (?j±) + (1 — UjYjKo (?j±) = 0,
n*Ujh (?±) + nUjK1 (?±) = 0, ?± = (R ± h) kXj,
where is carried out summation on from 1 to 3. The last conditions in (1.14) and (1.12), (1.15) yield.
YjUjIo (?+) — YjUjKo (?+) = (+Ko {(R + h)k},
YjUjIo (?") — YjUjKo (?—) = (—Io {(R — h)k},
( ) ( ) C fpth (67) + t'jU'jK, (SJ) = “ *+*, № + ««,
( ) ( ) C
fpth (%)+>;u’Kt (57) = -fe[ "eis)A-^ № -»«•
or excluding of (j —, (j +
y Uh (?J)-y uk (?;)
(e31 + e15)2
6C11
Y UA (?-) — Y UK (?-)
K'J (?;)+t; uk (?;))
K0{(R + h)k} Kx {(R + h)k}’
(e31 + e15)2
eC
11
fcJ (?;)+t;uK (?;))
_X]I0{(R-h)k} h{(R-h)kY
(1.18)
where is carried out summation on from 1 to 3. Equations (1.17), (1.18) relate all Uj, Ujj by homogeneous linear system, where determinant equation is
n+ n+ n+ M+ M+ M++
n— n; n; M— M2— M3—
P1+1 P+ P3+ n+ n+ n+
P1- P2- P3- n— n- n-
N+ N+ N++ A+ A++ A++
N— N2- N3- A1- A2— A3—
= 0,
(1.19)
C1
I1 (?±)
C
C12 1 j C13
UJ ~ aJ J 1 (^/) c^aJ
“(1 — ^) YjI0 (?±),
K>(?*) . c
My = ajkjK[ (?j) + ^a,l
C12 Cn
rv
+ (1 — ^) yjKo (?j±) , P±=J (?±), nj±=n;K1 (?±),
N+ = v T - (g31+g!5)2 j Yy0l^J Ecn K\ {k(R + h)}
J1 (?-)
11
(1.20)
Ko {k (R + h)},
AJ = —Y ,K0 (?;)
a; = j (?;)
(e31 + e15)
2t
eC11 I1 {k (R — h)}
(e31 + e15)
Io {k (R — h)},
2t
K1 (?j+)
eC11 K1 {k (R + h)}
Ko {k (R + h)},
(e31 + e15)
2t
;K> (?;)
-Io {k (R — h)}.
eC11 I1 {k (R — h)}
Where there is not summation by j.
We must carry out calculations for piezoelectric case (1.8), (1.19). For all values
+
of constants for BaTio3 are as follows
1 5 i
M-l = ~, M-2 = T> M"4 = 1,
3 6
^ [L5=2, [16 = 7:, M-7 = 1, — = 10, (L21)
Cii 2 2 e
Jfc? = —, « = 4, 50, 100.
1 300
Placing X.1,2,3 (v) from (1.7) in (2.4), (1.19), one must solve dispersion equation for small values of kh , i.e. for h = 0.1 cm, k = 0.1, 0.2, 0.3, 0.4, 0.5 1/cm,
R = 103 cm and obtain tables of v = v (k) or CD = k+\-----------------------v(&). Results are
v p
brought in table 1.
Table 1
h = 0.1, R = 103
k2 - — 1 300 k
0.1 0.2 0.3 0.4 0.5
n=4 0.0145 0.0102 0.0083 0.0072 0.0064
«=50 0.0299 0.02118 0.0173 0.0149 0.0134
«=100 0.0366 0.0259 0.0211 0.0183 0.0164
2. The case of elastic cylindrical shell
For elastic shell one must put
e3i = 0, £33 = 0, ei5 = 0, 9 = 0. (2.1)
and take place (1.4) for ur, uz. (1.7) yields
aii«22 - a22 = 0, (2.2)
where atk are done by (1.8) and there are two roots X^. The relations (1.11)
yield
yield (2.3)
aj (Xj) = -«i3«22, Pj (Xj) = -ai2aB, j = i, 2.
Then one has equations (1.17) on boundary of shell in which yj and n* are given by (1.13).
In (1.17) unknown functions are . The determinant equation will give as in (1.19) for first four lines the same form without third and sixth columns
n+ n+ M+ M+
n- n- M- M2-
P+ P+ n+ n+
P- P- n- n-
are done in (1.20)
= 0,
where n±2, M±2, P±2, ^±2 are done in (1.20), where ^5 = 0, ^ = 0.
3. Solution for cylindrical shell based on Kirchhoff hypothesis
For comparison with results of Kirchhoff hypothesis for piezoelectric shells one can assume that
arz ~ 0, arr ~ 0, ur = A sin kz, 9 = ^0(r)cos kz. (3.1)
where multiplier e~imt is omitted.
Then one obtains using (1.1)
duz dur ei5 89 dr dz C44 dr ’
uz = (R-r)^ - ^-qp + u(z), (3.2)
dz C44
dur Ci2 ur Ci3 duz e3i d9
dr C\\ r Cn dz Cn dz'
Equations of motion are
dorr dorz orr — d2ur
dr dz r P dt2 ’
d°rz dozz orz d uz
+ — + — = p-
(3.3)
dr dz r dt2
From (1.1), (3.2) one obtains
ur Ci2Ci3\ duz
a’: = 7[c'3-—) + i;
dy j e3iCi3\
+ dz V33 Cn ’
C2
i3
L33 -
Cii
+
ur
0» = —
r
C2
i2
C11 - JT-Cii
duz ( C12C13 ,
+ c”-—l+
(3.4)
e33Ci2\
dz V31 Cn )'
Integrating (3.3) on r from R -h to R + h , using that on r = R ± h, orr = 0, orz = = 0 ,and multiplying of second equation (3.3) by R - h and integrating, one obtains equations
R+h R+h
dQ i P , d2u,
/Onadr = pd Ur2h, Q = f arzdr, m dt J
R-h R-h
R++h (3.5)
d Ozzdr R+h
R-h n dM C
-------^------= 0, — = Q, M = I (r-R) ozzdr,
R-h
where small terms it one obtains
d2uz
Q
as well as — in third equation are neglected, and from dt2 R
du
dz
Ci3 i -
i2
ii
ei3
£31
Cn
£il
C 44
C33 -
C2
i3
C
ii
R
C33 -
C2
C
C33 -
C2
-k(0 sin kz.
C
Substituting (3.1), (3.2), (3.6) in last equation (1.3) one obtains
3 kA kA
£33 k A(r - R) + e 31-£33—C13
rR
i
i2
ii
4>o + ~y - vo^o =
C33 -
C2
- 1'
C
C
'11
44
+ eii
where
ei5
£33 + ^33-pr- +
C44
e33
e3i
C33 -
C2
i3
C
C\2 ~ e\5~
C
ii
ii
C
44
C33 -
i5
C
+ eii
44
(3.6)
(3.7)
(3.8)
To simplify (3.7) one can assume that in terms with piezoelectric effects one can neglect terms with — and one obtains equations
R
A." 2 j. e33PA(r-R)
^0 - V0^0 = ---2-------•
e
(3.9)
i5
C
+ eii
44
For solution of (3.9) one obtains
(0 (r) = Cichv0 (r - R) + C2shv0 (r - R) - %A (r - R).
(3.10)
For solution out of shell for potential 9 one obtains
r > R + h, 9 = cos kzK0 (kr) (j)+e ,mt + c.c., r < R - h, 9 = cos kzI0 (kr) (-e-,mt + c.c.
For induction in shell Dr in (1.1) one obtain
Dr = -(0 (r)
2
e
i5
C
+ eii
44
cos kz + c.c.
u
r
+
2
2
2
— dffi
From continuity conditions for r = R + h of qp = qp , Dr = — one obtains
dr
Ci = 0, C2 =
Vo
xA
cj)o = —shv0 (r-R)- %A (r-R), <4>0 Vo
From (3.4), (3.5), (3.6), (3.12) one obtains
2/ m2
—v0(r-R) .
(3.12)
(3.13)
M = C33 -
R+h
J' Oftftdr =
R-h
C2 ^ ^13
C11 sin kz
sin kzk2 A—
- IC13 -
R
C13C12 1
1 -
C2
- 1'
12
C2
11
C11 A2h-
1
C11 R
—A smkz2hC\T,-
12
11
C33 -
C2
^13
Cn
(3.14)
where is used that function ^0 is add with respect to r — R, and values of highly order smallness on are dropped out. Substituting of (3.14) in (3.5) one obtains
2
pm2 =
C33 -
C2
^13
Cn
k4— + — 3 R2
C11 -
C13 - C13
12
11
2 \
C33 -
C2
- 1 •
C
(3.15)
and using also values (1.21)
1
2 1
= -k2^ + —-4 3R2
(3.16)
which in the main order coincides with dispersion relation for elastic anisotropic shell. The numerical results by (3.16) are given in table 2 calculated by Kirchhoff hypothesis
Table 2
k 0.1 0.2 0.3 0.4 0.5
0.000957427 0.00216025 0.00457347 0.00804156 0.0125266
The comparison of table 1 and table 2 shows that the results by space treatment are quite different form those obtained by hypothesis.
2
2
2
V
4. Calculations of frequencies by exact solution for piezoelectric plate
For piezoelectric strip equations of motion
dux d2ux 2 d2uz d29
--— + Ui---— + £ Ux + f-l?--- ---- — 0,
dx2 dz2 dxdz dxdz
d2uz d2uz d2uz 2 d2(p d2(p_
M-2T—+ M-l + + e uz + + [16TT - 0, (4.1)
dxdz dx2 dz2 dz2 dx2
(,ad2ux d2uz d2uz\ 2<92qp d2<$
i^+fc^+feu)+^ + u = 0'
For considered antisymmetric problem one has
ux (x, z) = Uj-skkj-pz cos px,
uz (x, z) = Vjchkjpz sin px, (4.2)
9 (x, z) = (chkjpz sin px,
where is carried out on j summation from 1 to 3, Substituting of (4.2) in (4.1) for Uj, Vj, ( one obtain homogeneous system, where determinant
det ||a;-j|| = 0 (4.3)
determining k,
«11 = 1 - ^k2 - V2, «12 = -M-2k «21 = «12, «13 = -k,
«22 = -^1 + ^4k2 + v2, «23 = ^5k2 - ^6, «31 = k2k,
«32 = k2 - k3k2, «33 = k2 -One can write (4.2) in form [11]
ux (x, z) = ajshkjpzUj cos px,
uz (x, z) = |3jchkjpzUj sin px, (4.4)
9 (x, z) = yjchkjpzUj sin px, where is carried out summation on j from to 3,
aj (kj) = «12«23 - «13 «22,
|3j (kj) = «21 «13 - «11 «23,
Yj (kj) = «11 «22 - «12, potential of electric field 9 in region out of plate |z| > h can be written as
9 = (j)e+pz sin px, (4.5)
d2p d2p
which satisfy the equation —- H--------------------r = 0. The stress components in plate are
J H dx2 dz2 F F
[11]
oxx = C11 pt*Ujshkjpz sin px,
ozz = C33pm* Ujshkjpz sin px, (4.6)
oxz = C44pnjUjchkjpz cos px, where there is summation on j from 1 to 3,
fj = -aj + (^2 - M-0 pjkj + 0 - Mtf) Yjkj,
* ■s C M-6 C13
n = ajlj + |3y + —YJ, [J-8 = 7—,
M-1 C33
mj = -^aj + fjjlj + fijlj.
Here there is no summation.
Boundary conditions ozz (x, ±h) = 0, axz (x, ±h) = 0 are satisfied by Uj = AjU0, A1 = m2n3 - m3n2
A2 = m3n1 - m1n3, A1 = m1n2 - m2n1, mj = mjshkjph, nj = njchkjph.
From (4.4)-(4.7) and conditions z = ±h, p = p one obtains
(4.7)
e ph(j> = y*AjUo, Y* = Yjchk}ph.
Using also conditions of continuity z component of induction z = ±h
DZ = DZ, z = ±h, DZ = S^,
Dz = pqjAjShkjpzU0 sin px,
Dz = ±pYj AjU0e+z sin px.
One obtains the dispersion equation
s
R21 (p, v) = 0, R21 = R2 ~ —Ri,
s 33
A j,
where is carried out summation on j from 1 to 3,
Yj = Yjchkjph, q* = qjshkjph,
qj = -yjkj - e31^9aj + e33jkjj e31 + e15
(4.8)
(4.9)
I Yj 1
2 V ^ qj
(4.10)
^9 =
CUS
11s 33
5”33 is dielectric constant for plate ^ < 1.
5. Piezoelectric case based on Kirchhoff hypothesis
One can obtain solution for piezoelectric plate with normal polarization based on Kirchhoff hypothesis. Equations of motion and of elastic induction are
doxx doxz d2Hx
+ = P
dx dz dt2
daxz dazz _ d2uz
dx dz ^ dt2 ’
dDx dD^ _
dx dz '
From [11] these equations can be written as
d2Ux d2Ux 2 d2Uz d2p
------ + Ui------- + £ Ux + U? ”—“ + “—” — 0,
dx2 dz2 dxdz dxdz
-
d2ux d2uz d2uz\ 2 d2p d2p
+ ^-2------o” ^-3-------T" I M*7------T ----------T" — 0,
dxdz dx2 dz2 ) dx2 dz2
C44 C13 + C44 C33 e33
- 7^> M*2 - —t;---------------------------------, M-4 - M-5 - -;-;
C11 C11 C11 e31 + e15
Comparison of (5.1), (5.3) yields
dux duz dp e31
Oxx - L11 —----------1- C13 —------1- C11 -
dz C\\S33 dx C^S33 dz
11s 33 dx C11s 33
x duz ^
(5.1)
d2uz d2uz d2uz 2 d2p d2p ^
\Xi------ + 111--- + \Xa + £ Uz + U5 + U-6-------------- = 0, id.2)
dxdz dx2 ^ dz2 ^ dz2 ^ dx2 1
e15 2 ^11 a (e31+ei5) , fx o\
^6 =-------;------, M-7 = , kt = n „----------, h = \i6lq, (5.3)
e31 + e15 s 33 C11s 33
2
7 7 2 2 “ P
k3 = \i5k\, e = ——.
C11
13 11
dx dz dz e31 + e15
duz dux dp
®zz ~ ^33^“ + ^13^— +
dz dx dz
(dux duz\ dp
ax=-cu\~ik+ml + c'mai’ (54)
dp e31 + e15 dux e31 + e15 duz
Dz = + e3i-——,------— + e33-
2 dp e31 + e15 dux du:
D* - ~^Tx + ‘’15"c^7 (if + to
e31 + e15 _
p is connected with electrical potential by p = ——-----------------------p. Due to Kirchhoff
C11
hypothesis one has
uz ~ uz (x) , 0xz ~ 0 0zz ~ 0. (5.5)
From (5.4), (5.5) one can obtain relations
dux duz dz dx
d(f
duz C13 dux C11
£n
C44 d x ’
dz C33 dx C33
,2 \
\ly
dux dz ’
0* = -^
d x
Dz = -^ dz
^7 + # C44
„2 V
1 +
33
C
33
C13 \ e31 + e15 dux
One can look for uz and p in form
uz = A cos px, p = cos px^0 (z).
Then (5.6) yields
C
ux = zAp sin px + ~^-\ieP sin px cj>o (z), C44
du
C1
C13 C,
C1
—z -13, 2 C13 C11 2 1 C11
— = -—zAp cospx-——\i6p cospxfyo - — ^5cos^xct)0, dz C33 C33 C44 C33
Dz = - cos px§0
1 +
2
e2
33
C33 e31 + e15
I 13
+ le3i - 7^e33n n c
C33 / \ C11s 33
z^+ ei5
C44S
44s 33
^0 (zn p cos px.
Substituting (5.7), (5.8) into (5.2) gives
-klp2A - k2p2^-\i6<\>'0 + Ap2k2 +Ap2k3-Cl3
44
C
33
C C C
+ 7^7^W2q№ + ^5^3 - ^P2¥o + C =
C33 C44 C33
or denoting =0
O" - v^O = AZp2,
C
C
^7+ -^377-
C44 C33 2
kj - ^2 - li3
1 7 11
1 +^3^5 C33
The general solution of (5.9) yields
p2, Z =
13
33
1 7 CH
1 +£37^5
C33
O = C1chv1 z -
Zp2A
(5.6)
(5.7)
(5.8)
(5.9)
(5.10)
+
2
2
v
Substituting (5.10) in (5.8) gives
Dz = -C1 v1 cos px
1 +
2
33
C33 e31 + e15
shv1z+
I C13 + le3i - 7^33
C33 C11S33
zAp2 cos px+
i I3
+ le3i - 7^e33 C33
shv1z Z2 pA C i-----------------—z
v1
el5 2
p cos px.
C44S
44S33
Boundary conditions on z = h give
p = p, Dz = Dz. Where for dielectric out of plate
S
qp = C3e~p(z~h^ cos px, Dz = -C3 ——pe
-p(z-h)
S
33
cos px.
On account of (5.7), (5.8) and (5.10)—(5.13) one obtains z = h,
r b b ^plA r L\cnv\n-------- — — C3,
-C1V1
1 +
v21
2
e2
33
C
33
1 13
+ le3i - 7^e33 C33
C
£
C.
shv1h Zp2A
, , I “13 \ e31 + e15 , , 2
shv]_h + \e3i - —e33 ———hAp +
C33 ) C11s 33
v1
From (5.14) it follows that C1 shv1 h {-v1
1 +
2
e2
33
e15 2 n S
r P = ~C3—p.
C44S33 S33
I ^13
+ e31 ~ 7; ^33
e15 p
2
+hAp2\e3i - ^e33
C33
C33) \ C33 ) C44S 33 v1
e31 + e15 e15 Zp2
S
S
33
C1chv1h -
„ C11s33
Zp2 A'
C44S
44s 33 V
2
1
1
(5.11)
(5.12)
(5.13)
(5.14)
(5.15)
C, = -
hApl\e31 - ^-e33
C33
£31_+_fl5 ei5 ^
CuS
11S33
C44S33 v2
2^ + S t,p3A
1
S33 v
shv1h < -v1
1 +
2
e2
33
C
33
e15 p
1 I3
+ le31 - 7^33 I n c
C33 C44S33 v1
d2ux
+ -^—pchvih S33
(5.16)
From (5.1), neglecting of p—— and multiplying on z, after integration on
dt2
z, one obtains
d f zoxxdz h
-h I doxz , n
-dz = 0
d x
+ /z
-h
dz
v
h
2
v
+
2
and after integrating on z by part in second term one obtains
h h
r dM r
Q = I a^dz, — = Q, M = I zaxxdz.
hh
(5.17)
One account boundary conditions oxz = oxx = 0 one obtains after integration of second equation in (5.1)
d2M dQ ^ d2uz dx2 ~ dx ~ s dt2'
(5.18)
From (5.4), (5.6) one obtains
oxx =
C11 -
C2
- 1'
C
dux Cii / ^ £33^13 \ <9qp
dx + e31 + e15 T31 C33 / dz
and substituting of (5.7), (5.8), (5.10) one obtains
°rr — C11
xp cos px +
C2
^13
C33
C3311
zA +
C\\ 215
C44 e31 + e15
Cl , lp2A
—snv\z-----------—
v1
1
e31
^33^13
e31 + e15 \ C33
Substituting (5.19) in (5.17) one obtains
(5.19)
C1v1 shv1z cos px.
M=
h
I
C11 -
C2
- 1 '
C
C44 e31 + e15
C1 h ZP2A 2
—zshv\z-----------—zr
v1 v2
-h v
+------—U31 - e3?,-^\CiVishviz\dz,
e31 + e1^ C33,
h
and on account that | zshv\zdz = ^vi^3 one obtains
2/?3 I
M = cos px < p
2
C11 -
C2
C
1
^P2 Cn ei5
V2 C44 £31 + e\5
A +
C\\ C iei5
C44 e31 + e15
+
e31 + e15
C
33
In elastic case from (5.16) one obtains C1 = 0 ,
2h3p2 M = —-—A cos px
C11 -
C2
13
C
33
Dispersion relation can be obtained from (5.18), thus one has
h2 p4
C11 -
C2
13
C
33
_£u_ti%p4 £31 + ^15 3
e31 - e33
13
33
= pm
(5.20)
X
z
+
v
+
3
Values of constants for BaTiO3 are
C11 = 1,5 * 1011 N/m2, C13 = 6,6 * 1010N/m2, C33 = 1,4 * 1011 N/m2,
C44 = 4,5 * 1010N/m2, S33 = 10-9<fy/m, e31 = -4K/m2,
e33 = UK/m2, e15 = 11K/m2.
Then
1 n.. 1 n.. n..
2
C13 1 C13 1 Cn = 3, £n _
Cn “ 2’ C33 2 C44 c33
£15 _ e31 -3, ^ e31 = -4, ^6 3 “ 2’ ^5 = 2,
n
300’
And from (5.20) one has
1
3k2
1 +-------(5.21)
1 + 4k2 V J
The results of calculations by the exact treatment, made by (4.10), are brought in (4.3) tables 3,4 and by hypothesis are done by (5.21) and are done by table 5.
Table 3
¥=10
n
k2 - — 1 300 P
0.1 0.2 0.3 0.4 0.5
«=0.1 0.660153 0.660142 0.660132 0.660121 0.660111
n=1 0.696237 0.664531 0.716595 0.716833 0.717054
n=2 0.588632 0.588632 0.588632 0.588632 0.588632i
n=3 0.594389 0.503435 0.594389 0.594389 0.594389
n=4 0.720844 0.703199 0.733194 0.73344 0.733659
n=5 0.693198 0.693535 0.693854 0.694158 0.694447
«=10 0.720122 0.720582 0.720998 0.748948 0.72173
S 33
5
Table 4
=1
kj = — 1 300 P
0.1 0.2 0.3 0.4 0.5
«=0.1 0.588 0.57 0.5888 0.5889 0.589002
«=1 0.71 0.73 0.7369 0.782 0.782102
«=2 0.58 0.588 0.58863 0.58 0.5886
«=3 0.594 0.59 0.594 0.59 0.5943
«=4 0.73 0.73 0.736 0.73 0.0.7371
«=5 0.6 0.6i 0.6053 0.605 0.6054
«=10 0.63 0.63 0.630 0.6307i 0.63072
Table 5
S33
The Kirchhoff case table —- = 10
_______________________________________________O_______________________________
k2 - — 1 300 P
0.1 0.2 0.3 0.4 0.5
«=0 0.005 0.01 0.015 0.02 0.025
«=0.1 0.00500416 0.0100083 0.0150125 0.0200166 0.0250208
«=1 0.00504095 0.0100819 0.0151229 0.0201638 0.0252048
«=2 0.00508052 0.010161 0.0152416 0.0203221 0.0254026
«=3 0.00511878 0.0102376 0.0153563 0.0204751 0.0255939
«=4 0.0051558 0.0103116 0.0154674 0.0206232 0.025779
«=5 0.00519164 0.0103833 0.0155749 0.0207666 0.0259582
«=10 0.00535504 0.0107101 0.0160651 0.0214202 0.0267752
Comparision of tables 3 and 5 shows that the solution by exact space treatment essentially is distinguished from solution obtained due to Kirchhoff hypothesis.
Conclusion
The derivation of disspersion relation for free bending vibrations of thin piezoelastic cylindrical shells with longitudinal polarization and for plates with normal polarization by exact space treatment, proposed at first for elastic plates by V. Novatski, is given. It is done numerical solutions of obtained transcendent equations. Also the same considerations are made by treatment based on Kirchhoff hypothesis.
The table 1 corresponds to shell with radius R = 103cm calculated by space treatment, and table 2 by hypothesis, in the last in main order frequency does not depend from piezoelectric properties. The results of table 1 and table 2 are quite different. Also are constructed by space treatment tables 3, 4 for piezoelectric plates and table 5 by averaged method based on hypothesis. The tables 3 and 5 are distinguished by several times. Thus in considered problem as in magnetoelastic plates and shells in piezoelectricity Kirchhoff hypothesis not applicable.
Literature
[1] Ambartsumyan, S.A. Magnetoelastocity of thin shells and plates / S.A. Ambartsumyan G.E. Bagdasaryan, M.V. Belubekyan. - M.: Nauka, 1977. (In Russian).
[2] Ambartsumyan, S.A. Some problems of electromagnetoelasticity of plates / S.A. Ambartsumyan, M.V. Belubekyan. Yerevan State University. - 1991. - 143 p.
[3] Ambartsumyan, S.A. Electroconducting plates and shells in the magnetic field / S.A. Ambartsumyan, G.E. Bagdasaryan. - M.: Phys.-Math. Literature, 1996. - 286 p.
[4] Bagdoev, A.G. Nonlinear vibrations of plates in longitudinal magnetic field / A.G. Bagdoev L.A. Movsisyan // Izv. AN Arm SSR. Mekanika. -V. 35. - No.1. - 1982.
[5] Bagdasaryan, G.E. Vibrations and stability of magnetoelsatic system / G.E. Bagdasaryan // Yerevan State University. - 1999. - 439 p. (In Russian).
[6] Novatski, V. Elasticity / V. Novatski M.: Mir. 1975. 863 p. (In Russian)
[7] Bagdoev, A.G. The stability of nonlinear modulation waves in magnetic field for space and averaged problems / A.G. Bagdoev, S.G. Sahakyan // Izv RAS MTT. - 2001. - No.5. - P. 35-42 (In Russian)
[8] Safaryan, Yu.S. The investigation of vibrations of magnetoelastic plates in space and averaged statement / Yu.S. Safaryan // Information technologies and management. - 2001. - No.2.
[9] Bardzokas, D.I. The propagations of waves in electromagnetoelastic media / D.I. Bardzokas, B.A. Kudryavcev, N.A. Sennik. - M.: 2003. - 336 p.
[10] Bardzokas, D.I. Electroelasticity of piece-homogeneous bodies / D.I. Bardzokas, M.L. Filshtinski. Universitetskaia kniga. Sumi. - 2000. - 309 p.
[11] Bagdoev, A.G. Linear bending vibrations frequencies determination in magnetoelastic cylindrical shells / A.G. Bagdoev, A.V. Vardanyan, S.V. Vardanyan // Reports of National Academy of Sciences of Armenia. - 2006. - V.106. - No.3. - P. 227-237.
Paper received 13/X///2006. Paper accepted 13/X///2006.
ОПРЕДЕЛЕНИЕ ЛИНЕЙНЫХ ЧАСТОТ ИЗГИБНЫХ КОЛЕБАНИЙ ПЬЕЗОЭЛЕКТРИЧЕСКИХ ОБОЛОЧЕК И ПЛАСТИН ПО ТОЧНОМУ И ОСРЕДНЕННОМУ ПОДХОДАМ
© 2007 А.Г. Багдоев, А.В. Варданян, С.В. Варданян2
В работе рссматривается вывод и численное решение диссперсионных соотношений для частот изгибных свободных колебаний пьезоэлектрических цилиндрических тонких оболочек с продольной поляризацией и тонких пластин с поперечной поляризацией. Решение дается по точному пространственному подходу и по гипотезе Кирхгоффа. Сравнение полученных таблиц показывает, что частоты по точному и основанному на гипотезе Кирхгоффа подходам значительно различаются.
Поступила в редакцию 13jXT7j2006; в окончательном варианте — 13jX//j2006.
2 Багдоев Александер Георгиевич, Варданян Анна Ваниковна, Варданян Седрак Ваникович, Институт механики, Армения, Ереван, ул. Маршала Баграмяна, 24б.
