80
Вестник СамГУ — Естественнонаучная серия. 2007. №2(52)
МЕХАНИКА
УДК 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.
On determination of linear frequencies of bending vibrations. . .
81
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 e31 , e33 , e15 [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
∂ur
ur
∂uz
∂φ
+ C12 + C13
+ e31 ,
∂r
r
∂z
∂z
∂uz
∂φ
∂ur ur
+
+ C33
+ e33 ,
σzz = C13
∂r
r
∂z
∂z
∂φ
∂ur ∂uz
+
+ e15 ,
σrz = C44
∂z
∂r
∂r
∂ur
ur
∂uz
∂φ
+ C11 + C13
+ e31 ,
σϑϑ = C12
∂z
∂z
r
∂r
∂φ
∂ur ∂uz
+
− ε11 ,
Dr = e15
∂r
∂r
∂z
∂uz
∂φ
∂ur ur
+
+ e33
− ε33 ,
Dz = e31
∂r
r
∂z
∂z
where ur , uz are displacements components, φ– potential of electrical
ε11 , ε33 are dielectric permeabilities.
Then equations of motion and induction yield [10]
2
∂2 uz
∂2 ur
∂ ur 1 ∂ur ur
(C
)
−
+
+
+
+
C
+
C
C11
44
13
44
r ∂r
∂r∂z
∂r2
r2
∂z2
∂2 ur
∂2 φ
=ρ 2 ,
+ (e31 + e15 )
∂r∂z
∂t
2
∂2 ur 1 ∂ur
∂2 uz
∂ uz 1 ∂uz
(C
)
+
C
+
+
+
+
+
C
C44
33
13
44
r ∂r
∂r∂z r ∂z
∂r2
∂z2
2
∂2 uz
∂2 φ
∂ φ 1 ∂φ
+
e
+
=
ρ
,
+e15
33
r ∂r
∂r2
∂z2
∂t2
2
2
∂ ur 1 ∂ur
∂2 uz
∂ uz 1 ∂uz
+ e33 2 + (e15 + e31 )
+
−
+
e15
r ∂r
∂r∂z r ∂z
∂r2
∂z
2
∂2 φ
∂ φ 1 ∂φ
−
ε
+
= 0,
−ε11
33
r ∂r
∂r2
∂z2
σrr = C11
(1.1)
field,
(1.2)
(1.3)
82
A.G. Bagdoev, A.V. Vardanyan, S.V. Vardanyan
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 magnetoelastic shell and plate [7–9] can be looked for in space treatment in form
ξ j = rν j , j = 1, 2, 3,
ur = A j I1 ξ j eikz−iωt + Aj K1 ξ j eikz−iωt + c.c.,
uz = B j I0 ξ j eikz−iωt + Bj K0 ξ j eikz−iωt + c.c.,
φ = φ j I0 ξ j eikz−iωt + φj K0 ξ j eikz−iωt + c.c.,
(1.4)
where I0,1 , K0,1 are Bessel functions of imagine argument on is carried out
summation from 1 to 3. On account relations
I0 (ξ) = I1 (ξ) , K0 (ξ) = −K1 (ξ) ,
dI1 (ξ) 1
dK1 (ξ) 1
+ I1 (ξ) = I0 (ξ) ,
+ K1 (ξ) = −K0 (ξ)
dξ
ξ
dξ
ξ
(1.5)
one can from (1.2)–(1.5) obtain
−iA j C11 ν2j − C44 k2 + ρω2 + (C13 + C44 ) ν j kB j +
+ (e31 + e15 ) ν j kφ j = 0,
A j ν j ik (C13 + C44 ) + C44 ν2j − C33 k2 + ρω2 B j +
φ j e15 ν2j − e33 k2 = 0,
(e31 + e15 ) A j ν j ik + B j e15 ν2j − e33 k2 + φ j −ε11 ν2j + ε33 k2 = 0.
(1.6)
The equation for ν2j is distinguished from equation of [11] for plate with normal
polarization and yields
ν j = λ j k,
ρω2
C44
C13 + C44
C33
= ν2 ,
= µ1 ,
= µ2 ,
= µ4 ,
2
C
C
C
C11 k
11
11
11
C12 1
= ,
C11 2
e33
e15
ε11 2 (e31 + e15 )2
, µ6 =
, µ27 =
, k =
,
µ5 =
e31 + e15
e31 + e15
ε33 1
C11 ε33
k2 = µ6 k12 , k3 = µ5 k12 , det ai j = 0,
(1.7)
where
a11 = −λ2j + µ1 − ν2 , a12 = a21 = µ2 λ j , a13 = k12 λ j ,
a22 = µ1 λ2j − µ4 + ν2 ,
a23 = k12 µ6 λ2j − µ5 , a31 = λ j , a32 = µ6 λ2j − µ5 , a33 = 1 − µ27 λ2j .
(1.8)
For elastic case when µ5 = µ6 = k12 = k2 = k3 = 0 (1.7), (1.8) have two roots
λ02,3 , and for piezoelectric one must seek solution of (1.7) starting from values
On determination of linear frequencies of bending vibrations. . .
of λ2 = λ02 , λ3 = λ03 , λ1 =
83
(e31 + e15 ) φ j
1
. Denoting
= ϕ j one can obtain from
µ7
C11
(1.6)
iA j a11 + B j a12 + ϕ j a13 = 0,
iA j a21 + B j a22 + ϕ j a23 = 0,
(1.9)
iA j a31 + B j a32 + ϕ j a33 = 0
and [11]
where
iA j = α j U j , B j = β j U j , ϕ j = γ j U j ,
(1.10)
α j λ j = a12 a23 − a13 a22 , β j λ j = a21 a13 − a11 a23 ,
γ j λ j = a11 a22 − a212 .
(1.11)
For Aj , −Bj, −ϕj one obtains the same (1.11) equation expressed by U j .
Then (1.4) gives
ur = −iα j U j I1 ξ j eikz−iωt − iα j U j K1 ξ j eikz−iωt + c.c.,
uz = β j U j I0 ξ j eikz−iωt − β j U j K0 ξ j eikz−iωt + c.c.,
(e31 + e15 ) φ
= γ j U j I0 ξ j eikz−iωt − γ j U j K0 ξ j eikz−iωt + c.c.,
C11
(e31 + e15 )
φ = ϕ,
C11
(1.12)
where is carried out summation on from 1 to 3. Using (1.1), (1.5) and (1.12)
one obtains
I
ξj
1
C
σrr
12
= −α j U j λ j I1 ξ j eikz−iωt −
α jU jλ j
eikz−iωt +
ikC11
C11
ξj
C13
β j U j I0 ξ j eikz−iωt − α j U j λ j K1 ξ j eikz−iωt −
+
C11
K1 ξ j
C13
C12
α j U j λ j
eikz−iωt −
β j U j K0 ξ j eikz−iωt +
−
C11
ξj
C11
e31
e31
ikz−iωt
U j γ j I0 ξ j e
−
U j γ j K0 ξ j eikz−iωt + c.c.,
+
e31 + e15
e31 + e15
σrz
= n∗j U j I1 ξ j eikz−iωt + n∗j U j K1 ξ j eikz−iωt + c.c.,
kC44
µ6C11
λ jγ j,
n∗j = α j + β j λ j +
C44
Dr
= t∗j U j I1 ξ j eikz−iωt + t∗j U j K1 ξ j eikz−iωt + c.c.,
k (e31 + e15 )
t∗j = µ6 α j + µ6 β j λ j −
ε11C11
(e31 + e15 )2
γ jλ j.
(1.13)
84
A.G. Bagdoev, A.V. Vardanyan, S.V. Vardanyan
The boundary conditions on shell surfaces as in case of piezoelectric plates
give [11]
σrr (R ± h, z) = 0, σrz (R ± h, z) = 0,
ϕ (R ± h, z) = ϕ̄ (R ± h, z) , Dr (R ± h, z) = D̄r (R ± h, z)
C11
.
e31 + e15
(1.14)
The dimensionless potential ϕ̄ (r, z, t) in dielectric out of shell satisfy the equa∂2 ϕ̄ 1 ∂ϕ̄ ∂2 ϕ̄
+ 2 = 0 and one can look for solution outside of shell in
+
tion
r ∂r
∂r2
∂z
form
ϕ̄ = φ̄+ eikz−iωt K0 (kr) + c.c., r > R + h,
(1.15)
ϕ̄ = φ̄− eikz−iωt I0 (kr) + c.c., r < R − h.
For D̄r = ε
∂ϕ̄
one obtains
∂r
1
D̄r = −φ̄+ εeikz−iωt K1 (kr) + c.c., r > R + h,
k
1
D̄r = φ̄− εeikz−iωt I1 (kr) + c.c., r < R − h.
k
(1.16)
The first line equations (1.14) give four equations
I1 ξ±j
C12
C13
α jU jλ j ± −
β j U j I0 ξ±j +
α j U j λ j I1 ξ±j +
C11
ξj
C
11
±
K1 ξ j
C12
C13
α jU jλ j
+
β j U j K0 ξ±j −
+α j U j λ j K1 ξ±j +
±
C
ξ
C
11
11
j
±
− 1 − µ6 U j γ j I0 ξ j + 1 − µ6 U j γ j K0 ξ±j = 0,
n∗j U j I1 ξ±j + n∗j U j K1 ξ±j = 0, ξ±j = (R ± h) kλ j ,
(1.17)
where is carried out summation on from 1 to 3. The last conditions in (1.14)
and (1.12), (1.15) yield.
γ j U j I0 ξ+j − γ j U j K0 ξ+j = φ̄+ K0 {(R + h) k} ,
γ j U j I0 ξ−j − γ j U j K0 ξ−j = φ̄− I0 {(R − h) k} ,
t∗j U j I1 ξ+j + t∗j U j K1 ξ+j =
C11
φ̄+ K1 {(R + h) k} ,
(e31 + e15 )2
C11
φ̄− I1 {(R − h) k} ,
t∗j U j I1 ξ−j + t∗j U j K1 ξ−j = −
(e31 + e15 )2
On determination of linear frequencies of bending vibrations. . .
85
or excluding of φ̄− , φ̄+
γ j U j I0 ξ+j − γ j U j K0 ξ+j =
& K0 {(R + h) k}
(e31 + e15 )2 % ∗
t j U j I1 ξ+j + t∗j U j K1 ξ+j
,
εC11
K1 {(R + h) k}
γ j U j I0 ξ−j − γ j U j K0 ξ−j =
=−
=−
(1.18)
& I0 {(R − h) k}
(e31 + e15 )2 % ∗
t j U j I1 ξ−j + t∗j U j K1 ξ−j
,
εC11
I1 {(R − h) k}
where is carried out summation on from 1 to 3. Equations (1.17), (1.18) relate
all U j , U j by homogeneous linear system, where determinant equation is
Π+1
Π−1
P+1
P−1
N+1
N−1
Π+2
Π−2
P+2
P−2
N+2
N−2
Π+3 M+1 M+2 M+3 Π−3 M−1 M−2 M−3 P+3 Ω+1 Ω+2 Ω+3 = 0,
P−3 Ω−1 Ω−2 Ω−3 N+3 Λ+1 Λ+2 Λ+3 N−3 Λ−1 Λ−2 Λ−3 I1 ξ±j
C12
C13
α jλ j ± −
β j I0 ξ±j −
Π±j = α j λ j I1 ξ±j +
C
ξj
C11
11
±
− 1 − µ6 γ j I0 ξ j ,
K1 ξ±j
C12
C13
α jλ j
+
β
K
ξ±j +
M±j = α j λ j K1 ξ±j +
j
0
C11
ξ±j
C11
+ 1 − µ6 γ j K0 ξ±j ,
(1.19)
(1.20)
P±j = n∗j I1 ξ±j , Ω±j = n∗j K1 ξ±j ,
(e31 + e15 )2 t∗j I1 ξ+j
K0 {k (R + h)} ,
N+j = γ j I0 ξ+j −
εC11
K1 {k (R
+
h)}
(e31 + e15 )2 t∗j I1 ξ−j
−
I0 {k (R − h)} ,
N j = γ j I0 ξ−j +
εC11
I1 {k (R − h)} (e31 + e15 )2 t∗j K1 ξ+j
K0 {k (R + h)} ,
Λ+j = −γ j K0 ξ+j −
εC11
K1 {k (R
+
h)}
(e31 + e15 )2 t∗j K1 ξ−j
−
I0 {k (R − h)} .
Λ j = −γ j K0 ξ−j +
εC11
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
86
A.G. Bagdoev, A.V. Vardanyan, S.V. Vardanyan
of constants for BaTio3 are as follows
1
5
µ1 = , µ2 = , µ4 = 1,
3
6
3
ε33
C12 1
= 10,
= , µ5 = 2, µ6 = , µ7 = 1,
C11 2
2
ε
n
, n = 4, 50, 100.
k12 =
300
(1.21)
Placing λ1,2,3 (ν) 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 ν = ν (k) or ω = k
C11
ν (k) . Results are
ρ
brought in table 1.
Table 1
n
300
n=4
n=50
n=100
k12 =
0.1
0.0145
0.0299
0.0366
h = 0.1, R = 103
k
0.2
0.3
0.0102
0.02118
0.0259
0.0083
0.0173
0.0211
0.4
0.5
0.0072
0.0149
0.0183
0.0064
0.0134
0.0164
2. The case of elastic cylindrical shell
For elastic shell one must put
e31 = 0, e33 = 0, e15 = 0, ϕ = 0.
(2.1)
and take place (1.4) for ur , uz . (1.7) yields
a11 a22 − a212 = 0,
(2.2)
where aik are done by (1.8) and there are two roots λ01,2 . The relations (1.11)
yield
(2.3)
α j λ j = −a13 a22 , β j λ j = −a12 a13 , j = 1, 2.
Then one has equations (1.17) on boundary of shell in which γ j and n∗j 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
+
Π1 Π+2 M+1 M+2 Π− Π− M− M− 2
1
2
+1
(2.4)
+
+
+ = 0,
P1 P2 Ω1 Ω2 P− P− Ω− Ω− 1
2
1
2
where Π±1,2 , M±1,2 , P±1,2 , Ω±1,2 are done in (1.20), where µ5 = 0, µ6 = 0.
On determination of linear frequencies of bending vibrations. . .
87
3. Solution for cylindrical shell based on Kirchhoff hypothesis
For comparison with results of Kirchhoff hypothesis for piezoelectric shells
one can assume that
σrz ≈ 0, σrr ≈ 0, ur = A sin kz, ϕ = φ0 (r) cos kz.
(3.1)
where multiplier e−iωt is omitted.
Then one obtains using (1.1)
∂ur e15 ∂ϕ
∂uz
=−
−
,
∂r
∂z
C44 ∂r
∂ur e15
−
ϕ + u (z) ,
uz = (R − r)
∂z
C44
C12 ur C13 ∂uz e31 ∂ϕ
∂ur
=−
−
−
.
∂r
C11 r
C11 ∂z C11 ∂z
(3.2)
Equations of motion are
∂2 ur
∂σrr ∂σrz σrr − σϑϑ
+
+
=ρ 2 ,
∂r
∂z
r
∂t
2
∂ uz
∂σrz ∂σzz σrz
+
+
=ρ 2 ,
∂r
∂z
r
∂t
From (1.1), (3.2) one obtains
⎛
⎞
2 ⎟
C13
ur
C12C13
∂uz ⎜⎜⎜⎜
⎟⎟⎟
C13 −
+
σzz =
⎜⎝C33 −
⎟+
C11
∂z
C11 ⎠
r
e31C13
∂ϕ
e33 −
,
+
∂z
C11
⎛
2 ⎞
C12
⎟⎟⎟ ∂uz
ur ⎜⎜⎜
C12C13
⎜⎝C11 −
⎟⎠ +
C13 −
+
σϑϑ =
r
C
∂z
C
11
11
e33C12
∂ϕ
e31 −
.
+
∂z
C11
(3.3)
(3.4)
Integrating (3.3) on r from R−h to R+h , using that on r = R±h, σrr = 0, σrz =
= 0 ,and multiplying of second equation (3.3) by R − h and integrating, one
obtains equations
∂Q 1
−
∂z R
∂
R+h
R+h
R+h
∂2 ur
σϑϑ dr = ρ 2 2h, Q =
σrz dr,
∂t
R−h
σzz dr
R−h
∂z
R−h
R+h
∂M
(r − R) σzz dr,
= 0,
= Q, M =
∂z
R−h
(3.5)
88
A.G. Bagdoev, A.V. Vardanyan, S.V. Vardanyan
where small terms
it one obtains
∂2 uz
Q
in third equation are neglected, and from
as well as
2
R
∂t
ur
∂u
=−
∂z
R
C13
C12
1−
C11
2
C13
C33 −
C11
+
e31
e15
e13 −
−
C11 C44
⎛
⎞
⎜⎜⎜
C2 ⎟
⎜⎜⎝C33 − 13 ⎟⎟⎟⎟⎠
C11
2
C13
C33 −
C11
kφ0 sin kz.
(3.6)
Substituting (3.1), (3.2), (3.6) in last equation (1.3) one obtains
kA
kA
− e33 C13
e33 k3 A (r − R) + e31
r
R
φ
0 +
φ0
− ν20 φ0 =
r
e215
C44
C12
C11
2
C13
C33 −
C11
1−
ε33 + e33
e15
+
C44
ν20 = k2
(3.7)
+ ε11
where
e33 −
,
C33 −
e31
C12 − e15
C11
C2
C33 − 13
C11
2
C13
C11
C44
e215
.
(3.8)
+ ε11
C44
To simplify (3.7) one can assume that in terms with piezoelectric effects one
1
and one obtains equations
can neglect terms with
R
2
φ
0 − ν0 φ0 =
e33 k3 A (r − R)
.
e215
+ ε11
C44
(3.9)
For solution of (3.9) one obtains
φ0 (r) = C1 chν0 (r − R) + C2 shν0 (r − R) − χA (r − R) .
(3.10)
For solution out of shell for potential ϕ̄ one obtains
r > R + h, ϕ̄ = cos kzK0 (kr) φ̄+ e−iωt + c.c.,
r < R − h, ϕ̄ = cos kzI0 (kr) φ̄− e−iωt + c.c.
For induction in shell Dr in (1.1) one obtain
⎛ 2
⎞
⎜⎜⎜ e15
⎟⎟
⎜
+ ε11 ⎟⎟⎟⎠ cos kz + c.c.
Dr = −φ0 (r) ⎜⎝
C44
(3.11)
89
On determination of linear frequencies of bending vibrations. . .
From continuity conditions for r = R ± h of ϕ = ϕ̄ , Dr =
C1 = 0, C2 =
φ0 =
∂ϕ
one obtains
∂r
χA
,
ν0
(3.12)
χA
χA 2
ν (r − R)2 .
shν0 (r − R) − χA (r − R) , φ0 ≈
ν0
6 0
(3.13)
From (3.4), (3.5), (3.6), (3.12) one obtains
⎞
⎛
2 ⎟
3
⎜⎜⎜
C13
⎟⎟⎟⎟ sin kzk2 A 2h ,
M = ⎜⎜⎝C33 −
⎠
C11
3
R+h
⎛
2 ⎞
C12
⎟⎟
sin kz ⎜⎜⎜
⎜
σϑϑ dr =
⎝1 − 2 ⎟⎟⎠ C11 A2h−
R
C11
(3.14)
R−h
C12
1−
C13C12 1
C11
A sin kz2hC13
− C13 −
C11
R
C2
C33 − 13
C11
where is used that function φ0 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 ⎞
⎜⎜⎜
C12 ⎟⎟⎟
⎛
⎞
⎜⎜⎜
⎟⎟
C13 − C13
2 ⎟
2
2
⎜⎜⎜
C13
C12
⎜⎜⎜
C11 ⎟⎟⎟⎟
1
h
⎟
⎟
2
4
⎟⎟ k
⎟⎟⎟
+ 2 ⎜⎜⎜C11 −
−
ρω = ⎜⎜⎝C33 −
2
⎟⎟⎟
C11 ⎠ 3
C11
R ⎜⎜⎜
C13
⎜⎝
⎟⎠
C33 −
C11
(3.15)
and using also values (1.21),
1
2 1
ν2 = k2 h2 +
4
3 R2
(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
k
0.1
0.000957427
0.2
0.00216025
0.3
0.00457347
0.4
0.00804156
Table 2
0.5
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.
90
A.G. Bagdoev, A.V. Vardanyan, S.V. Vardanyan
4. Calculations of frequencies by exact solution
for piezoelectric plate
For piezoelectric strip equations of motion
∂2 u x
∂2 uz
∂2 ϕ
∂ux
2
+
= 0,
+
µ
+
e
u
+
µ
1
x
2
∂x∂z ∂x∂z
∂x2
∂z2
∂2 uz
∂2 uz
∂2 uz
∂2 ϕ
∂2 ϕ
+ µ1 2 + µ4 2 + e2 uz + µ5 2 + µ6 2 = 0,
µ2
∂x∂z
∂x
∂z
∂z
∂x
2u
2u
2u
2ϕ
2ϕ
∂
∂
∂
∂
∂
x
z
z
− k12
+ k2 2 + k3 2 + µ27 2 + 2 = 0.
∂x∂z
∂x
∂z
∂x
∂z
(4.1)
For considered antisymmetric problem one has
ux (x, z) = U j shλ j pz cos px,
uz (x, z) = V j chλ j pz sin px,
(4.2)
ϕ (x, z) = φchλ j pz sin px,
where is carried out on j summation from 1 to 3, Substituting of (4.2) in (4.1)
for U j , V j , φ one obtain homogeneous system, where determinant
(4.3)
det ai j = 0
determining λ,
a11 = 1 − µ1 λ2 − ν2 , a12 = −µ2 λ, a21 = a12 , a13 = −λ,
a22 = −µ1 + µ4 λ2 + ν2 , a23 = µ5 λ2 − µ6 , a31 = k12 λ,
a32 = k2 − k3 λ2 , a33 = λ2 − µ27 .
One can write (4.2) in form [11]
ux (x, z) = α j shλ j pzU j cos px,
uz (x, z) = β j chλ j pzU j sin px,
(4.4)
ϕ (x, z) = γ j chλ j pzU j sin px,
where is carried out summation on j from to 3,
α j λ j = a12 a23 − a13 a22 ,
β j λ j = a21 a13 − a11 a23 ,
γ j λ j = a11 a22 − a212 ,
potential of electric field ϕ̄ in region out of plate |z| > h can be written as
ϕ̄ = φ̄e∓pz sin px,
(4.5)
On determination of linear frequencies of bending vibrations. . .
which satisfy the equation
[11]
91
∂2 ϕ̄ ∂2 ϕ̄
+
= 0. The stress components in plate are
∂x2 ∂z2
σ xx = C11 pt∗j U j shλ j pz sin px,
σzz = C33 pm∗j U j shλ j pz sin px,
(4.6)
σ xz = C44 pn∗j U j chλ j pz cos px,
where there is summation on j from 1 to 3,
t∗j = −α j + µ2 − µ1 β j λ j + 1 − µ6 γ j λ j ,
µ6
C13
γ j , µ8 =
,
µ1
C33
µ
m∗j = −µ8 α j + µ54 γ j λ j + β j λ j .
n∗j = α j λ j + β j +
Here there is no summation.
Boundary conditions σzz (x, ±h) = 0, σ xz (x, ±h) = 0 are satisfied by
U j = ∆ j U0 , ∆1 = m2 n3 − m3 n2
∆2 = m3 n1 − m1 n3 , ∆1 = m1 n2 − m2 n1 ,
(4.7)
m j = m∗j shλ j ph, n j = n∗j chλ j ph.
From (4.4)-(4.7) and conditions z = ±h, ϕ = ϕ̄ one obtains
e−ph φ̄ = γ∗j ∆ j U0 , γ∗j = γ j chλ j ph.
Using also conditions of continuity z component of induction z = ±h ,
∂ϕ̄
,
Dz = D̄z , z = ±h, D̄z = S
∂z
Dz = pq j ∆ j shλ j pzU0 sin px,
(4.8)
(4.9)
D̄z = ±pγ∗j ∆ j U0 e∓z sin px.
One obtains the dispersion equation
S
R1 ,
R21 (p, ν) = 0, R21 = R2 −
S 33
⎫ ⎧ ∗ ⎫
⎧
⎪
⎪
γ ⎪
R ⎪
⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎪ j ⎪
⎪
⎪ 1 ⎪
⎬
⎨
⎬
⎨
=
∆ j,
⎪
⎪
⎪
⎪
∗
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭ ⎪
⎩ qj ⎪
⎭
⎩ R2 ⎪
where is carried out summation on j from 1 to 3,
γ∗j = γ j chλ j ph, q∗j = q j shλ j ph,
q j = −γ j λ j − e31 µ9 α j + e33 µ9 β j λ j ,
µ9 =
e31 + e15
,
C11 S 33
S 33 is dielectric constant for plate
S
S 33
< 1.
(4.10)
92
A.G. Bagdoev, A.V. Vardanyan, S.V. Vardanyan
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
∂σ xx ∂σ xz
∂2 u x
+
=ρ 2 ,
∂x
∂z
∂t
∂2 uz
∂σ xz ∂σzz
(5.1)
+
=ρ 2 ,
∂x
∂z
∂t
∂D x ∂Dz
+
= 0.
∂x
∂z
From [11] these equations can be written as
∂2 ϕ
∂2 u x
∂2 uz
∂2 u x
2
+
= 0,
+
µ
+
e
u
+
µ
1
x
2
∂x∂z ∂x∂z
∂x2
∂z2
∂2 uz
∂2 uz
∂2 uz
∂2 ϕ
∂2 ϕ
+ µ1 2 + µ4 2 + e2 uz + µ5 2 + µ6 2 = 0,
µ2
∂x∂z
∂x
∂z
∂z
∂x
2u
2u
2u
2ϕ
2ϕ
∂
∂
∂
∂
∂
x
z
z
+ k2 2 + k3 2 + µ27 2 + 2 = 0,
− k12
∂x∂z
∂x
∂z
∂x
∂z
(5.2)
C44
C13 + C44
C33
e33
, µ2 =
, µ4 =
, µ5 =
,
C11
C11
C11
e31 + e15
e15
S 11 2 (e31 + e15 )2
, µ27 =
, k =
, k2 = µ6 k12 ,
µ6 =
e31 + e15
S 33 1
C11 S 33
ω2 ρ
.
k3 = µ5 k12 , e2 =
C11
(5.3)
µ1 =
Comparison of (5.1), (5.3) yields
∂ux
∂uz
∂ϕ e31
+ C13
+ C11
,
∂x
∂z
∂z e31 + e15
∂uz
∂ux
∂ϕ
+ C13
+ C11 µ5 ,
σzz = C33
∂z
∂x
∂z
∂ϕ
∂ux ∂uz
+
+ C11 µ6 ,
σ xz = C44
∂z
∂x
∂x
∂ϕ
e31 + e15 ∂ux
e31 + e15 ∂uz
+ e31
+ e33
,
Dz = −
∂z
C11 S 33 ∂x
C11 S 33 ∂z
e31 + e15 ∂ux ∂uz
2 ∂ϕ
+ e15
+
,
D x = −µ7
∂x
C11 S 33 ∂z
∂x
σ xx = C11
ϕ is connected with electrical potential by ϕ =
hypothesis one has
(5.4)
e31 + e15
ϕ̄. Due to Kirchhoff
C11
uz ≈ uz (x) , σ xz ≈ 0, σzz ≈ 0.
(5.5)
On determination of linear frequencies of bending vibrations. . .
93
From (5.4), (5.5) one can obtain relations
∂ux ∂uz
C11 ∂ϕ
µ6 ,
+
=−
∂z
∂x
C44 ∂x
C13 ∂ux C11 ∂ux
∂uz
=−
−
,
µ5
∂z
C33 ∂x C33 ∂z
⎛
⎞
2
∂ϕ ⎜⎜⎜⎜ 2 e15 ⎟⎟⎟⎟
Dx = −
⎜µ +
⎟,
∂x ⎝ 7 C44 ⎠
⎛
⎞ e233 ⎟⎟⎟
∂ϕ ⎜⎜⎜⎜
C
e31 + e15 ∂ux
13
⎟⎟⎠ + e31 −
.
e33
Dz = −
⎜⎝1 +
∂z
C33
C33
C11 S 33 ∂x
(5.6)
One can look for uz and ϕ in form
uz = A cos px, ϕ = cos pxφ0 (z) .
(5.7)
Then (5.6) yields
C11
µ6 p sin pxφ0 (z) ,
C44
C13
C13 C11
C11
∂uz
=−
zAp2 cos px −
µ6 p2 cos pxφ0 −
µ5 cos pxφ
0,
∂z
C33
C33 C44
C33
⎛
⎞
⎜⎜
e233 ⎟⎟⎟
⎜
⎟⎟ +
Dz = − cos pxφ0 ⎜⎜⎝1 +
C33 ⎠
C13
e15
e31 + e15
e33
zA +
φ0 (z) p2 cos px.
+ e31 −
C33
C11 S 33
C44 S 33
ux = zAp sin px +
(5.8)
Substituting (5.7), (5.8) into (5.2) gives
C11
C13
µ6 φ0 + Ap2 k2 + Ap2 k3
+
C44
C33
C11
C13 C11
2 2 µ6 p2 φ0 k3 +
µ5 φ
+
0 k3 − µ7 p φ0 + φ0 = 0,
C33 C44
C33
−k12 p2 A − k12 p2
or denoting φ = Φ
Φ − ν21 Φ = Aζp2 ,
C11
C13
C13
2
2
µ7 +
µ6 k1 − k3
k12 − k2 − k3
C44
C33 2
C33
p , ζ=
.
ν21 =
C11
C11
1 + k3
µ5
1 + k3
µ5
C33
C33
(5.9)
The general solution of (5.9) yields
Φ = C1 chν1 z −
ζp2 A
.
ν21
(5.10)
94
A.G. Bagdoev, A.V. Vardanyan, S.V. Vardanyan
Substituting (5.10) in (5.8) gives
⎛
⎞
⎜⎜⎜
e233 ⎟⎟⎟
⎟⎟ shν1 z+
Dz = −C1 ν1 cos px ⎜⎜⎝1 +
C33 ⎠
e31 + e15
C13
+ e31 −
e33
zAp2 cos px+
C33
C11 S 33
⎞
⎛
⎜⎜⎜ shν1 z ζ2 pA ⎟⎟⎟ e15
C13
e33 ⎜⎝C1
− 2 z⎟⎠
p2 cos px.
+ e31 −
C33
ν1
C44 S 33
ν1
(5.11)
Boundary conditions on z = h give
ϕ = ϕ̄, Dz = D̄z .
(5.12)
Where for dielectric out of plate
S
pe−p(z−h) cos px.
S 33
On account of (5.7), (5.8) and (5.10)–(5.13) one obtains
ϕ̄ = C3 e−p(z−h) cos px, D̄z = −C3
(5.13)
z = h,
ζp2 A
C1 chν1 h − 2 = C3 ,
ν1
⎛
⎞
⎜⎜⎜
e233 ⎟⎟⎟
C
e31 + e15
13
⎟⎟⎠ shν1 h + e31 −
e33
hAp2 +
−C1 ν1 ⎜⎜⎝1 +
C33
C33
C11 S 33
⎞
⎛
⎜⎜⎜ shν1 h ζp2 A ⎟⎟⎟ e15
C13
S
e33 ⎝⎜C1
− 2 h⎟⎠
p2 = −C3
p.
+ e31 −
C33
ν1
C44 S 33
S 33
ν1
From (5.14) it follows that
⎧
⎫
⎛
⎞ ⎪
⎜⎜⎜
⎪
e233 ⎟⎟⎟
⎪
C13
e15 p2 ⎪
⎨
⎬
⎜
⎟
−ν1 ⎜⎝1 +
e33
+
C1 shν1 h ⎪
⎟⎠ + e31 −
⎪
⎪
⎩
⎭
C33
C33
C44 S 33 ν1 ⎪
⎞
⎛
⎜⎜⎜ e31 + e15
C13
e15 ζp2 ⎟⎟⎟
2
⎜
⎟⎠ =
e33 ⎝
−
+hAp e31 −
C33
C11 S 33
C44 S 33 ν21
⎛
⎞
ζp2 A ⎟⎟⎟
S ⎜⎜⎜
p ⎜⎝C1 chν1 h − 2 ⎟⎠ ,
=−
S 33
ν1
⎞
⎛
⎜⎜⎜ e31 + e15
C13
e15 ζp2 ⎟⎟⎟
S ζp3 A
2
⎟⎠ +
hAp e31 −
e33 ⎜⎝
−
C33
C11 S 33
C44 S 33 ν21
S 33 ν21
⎧
.
⎫
C1 = −
⎛
⎞ ⎪
2⎪
⎜⎜⎜
⎪
⎪
e233 ⎟⎟⎟
C
p
e
S
⎨
⎬
13
15
⎟⎟ + e31 −
shν1 h ⎪
−ν ⎜⎜1 +
e33
pchν1 h
+
⎪
⎪
⎩ 1⎝
⎭ S
C ⎠
C
C S ν ⎪
33
33
From (5.1), neglecting of ρ
z, one obtains
∂
h
44 33
1
(5.14)
(5.15)
(5.16)
33
∂2 u x
and multiplying on z, after integration on
∂t2
h
zσ xx dz
−h
∂x
+
z
−h
∂σ xz
dz = 0
∂z
95
On determination of linear frequencies of bending vibrations. . .
and after integrating on z by part in second term one obtains
h
Q=
−h
∂M
= Q, M =
σ xz dz,
∂x
h
zσ xx dz.
(5.17)
−h
One account boundary conditions σ xz = σ xx = 0 one obtains after integration
of second equation in (5.1)
∂2 uz
∂2 M ∂Q
=
2sh
=
.
∂x
∂x2
∂t2
From (5.4), (5.6) one obtains
⎞
⎛
2 ⎟
⎜⎜⎜
C13
C11
e33C13 ∂ϕ
⎟⎟⎟ ∂ux
⎜
+
e31 −
σ xx = ⎜⎝C11 −
⎟
C33 ⎠ ∂x
e31 + e15
C33
∂z
and substituting of (5.7), (5.8), (5.10) one obtains
⎛
⎞
⎛
⎞⎞
2 ⎟⎛
⎜⎜⎜
C13
⎜⎜⎜ C1
ζp2 A ⎟⎟⎟⎟⎟⎟
C11 e15
⎟⎟⎟ ⎜⎜⎜
⎜
p ⎜⎝ shν1 z − 2 z⎟⎠⎟⎠ ×
σ xx = ⎜⎝C11 −
⎟ ⎜⎝zA +
C33 ⎠
C44 e31 + e15
ν1
ν1
e33C13
C11
e31 −
C1 ν1 shν1 z cos px.
×p cos px +
e31 + e15
C33
Substituting (5.19) in (5.17) one obtains
⎞
⎛
h ⎧
2 ⎟⎛
⎪
⎪
C13
⎟⎟⎟ ⎜⎜⎜ 2 2 C11 e15 p2
⎨⎜⎜⎜⎜
M=
⎟ ⎜⎝z Ap +
⎜⎝C11 −
⎪
⎪
⎩
C33 ⎠
C44 e31 + e15
−h
C13
C11
e31 − e33
C1 ν1 shν1 z dz,
+
e31 + e15
C33
h
and on account that
−h
(5.18)
(5.19)
⎛
⎞⎞
⎜⎜⎜ C1
ζp2 A 2 ⎟⎟⎟⎟⎟⎟
⎜⎝ zshν1 z −
z ⎟⎠⎟⎠ +
ν1
ν21
2
zshν1 zdz = ν1 h3 one obtains
3
⎧ ⎛
⎞
⎞
⎞
2 ⎟ ⎛⎛
⎪
⎪
C13
C11 C1 e15 ⎟⎟⎟
ζp2 C11 e15 ⎟⎟⎟
2h3
⎟⎟ ⎜⎜⎜⎜⎜⎜
⎨ 2 ⎜⎜⎜⎜
⎟
⎟⎠ A +
⎟⎠ +
cos px ⎪
p ⎜C −
M=
⎟ ⎜⎜1 − 2
⎪
⎩ ⎝ 11 C33 ⎠ ⎝⎝
3
C44 e31 + e15
ν1 C44 e31 + e15
⎫
⎪
C11C1 ν21
ζp2 A
C13 ⎪
⎬
≈
.
e31 − e33
,
C
⎪
1
⎭
e31 + e15
C33 ⎪
ν21
In elastic case from (5.16) one obtains C1 = 0 ,
⎛
⎞
2 ⎟
⎜⎜⎜
C13
2h3 p2
⎟⎟⎟⎟ .
A cos px ⎜⎜⎝C11 −
M=
3
C33 ⎠
Dispersion relation can be obtained from (5.18), thus one has
⎛
⎞
2 ⎟
C13
C13
C11 h2 ζp4
h2 p4 ⎜⎜⎜⎜
⎟⎟⎟
e31 − e33
= ρω2 .
⎜C11 −
⎟+
3 ⎝
C33 ⎠ e31 + e15 3
C33
(5.20)
96
A.G. Bagdoev, A.V. Vardanyan, S.V. Vardanyan
Values of constants for BaTiO3 are
C11 = 1, 5 ∗ 1011 N/m2 , C13 = 6, 6 ∗ 1010 N/m2 , C33 = 1, 4 ∗ 1011 N/m2 ,
C44 = 4, 5 ∗ 1010 N/m2 , S 33 = 10−9 φ/m, e31 = −4K/m2 ,
e33 = 17K/m2 , e15 = 11K/m2 .
Then
C13 1
= ,
C11 2
e15
= −3,
e31
C13 1 C11
C11
= ,
= 3,
= 1, µ27 = 1,
C33 2 C44
C33
e33
3
n
, h = 0, 1cm, n = 1, ..., 10.,
= −4, µ6 = , µ5 = 2, k12 =
e31
2
300
And from (5.20) one has
*
+
ν=
1 2
hp
2
1+
3k12
1 + 4k12
.
(5.21)
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
S 33
= 10
S
n
=
300
n=0.1
n=1
n=2
n=3
n=4
n=5
n=10
0.1
0.2
p
0.3
0.660153
0.696237
0.588632
0.594389
0.720844
0.693198
0.720122
0.660142
0.664531
0.588632
0.503435
0.703199
0.693535
0.720582
0.660132
0.716595
0.588632
0.594389
0.733194
0.693854
0.720998
k12
0.4
0.5
0.660121
0.716833
0.588632
0.594389
0.73344
0.694158
0.748948
0.660111
0.717054
0.588632i
0.594389
0.733659
0.694447
0.72173
Table 4
S 33
=1
S
n
=
300
n=0.1
n=1
n=2
n=3
n=4
n=5
n=10
k12
p
0.1
0.2
0.3
0.4
0.5
0.588
0.71
0.58
0.594
0.73
0.6
0.63
0.57
0.73
0.588
0.59
0.73
0.6i
0.63
0.5888
0.7369
0.58863
0.594
0.736
0.6053
0.630
0.5889
0.782
0.58
0.59
0.73
0.605
0.6307i
0.589002
0.782102
0.5886
0.5943
0.0.7371
0.6054
0.63072
97
On determination of linear frequencies of bending vibrations. . .
n
300
n=0
n=0.1
n=1
n=2
n=3
n=4
n=5
n=10
S 33
The Kirchhoff case table
= 10
S
p
0.1
0.2
0.3
0.4
k12 =
0.005
0.00500416
0.00504095
0.00508052
0.00511878
0.0051558
0.00519164
0.00535504
0.01
0.0100083
0.0100819
0.010161
0.0102376
0.0103116
0.0103833
0.0107101
0.015
0.0150125
0.0151229
0.0152416
0.0153563
0.0154674
0.0155749
0.0160651
0.02
0.0200166
0.0201638
0.0203221
0.0204751
0.0206232
0.0207666
0.0214202
Table 5
0.5
0.025
0.0250208
0.0252048
0.0254026
0.0255939
0.025779
0.0259582
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 = 103 cm 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.
98
A.G. Bagdoev, A.V. Vardanyan, S.V. Vardanyan
[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/XII/2006.
Paper accepted 13/XII/2006.
On determination of linear frequencies of bending vibrations. . .
99
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 13/XII/2006;
в окончательном варианте — 13/XII/2006.
2
Багдоев Александер Георгиевич, Варданян Анна Ваниковна, Варданян Седрак
Ваникович, Институт механики, Армения, Ереван, ул. Маршала Баграмяна, 24б.