5 / 10
where
E
— Young’s modulus;
α
— heat conductivity coefficient;
T
0
—
temperature of the environment (initial temperature);
е
— expansion
deformation;
ν
— Poisson’s ratio.
For
e
, the following relation is true
e
=
ε
x
+
ε
y
+
ε
z
=
∇ ∙
u
.
(8)
Let us try to look gor a solution of equation for
Δ
T
and
U
in the form
Δ
T
=
T
−
T
0
=
e
λt
Θ
,
u =
e
λt
U
,
v = ˙u =
e
λt
V
,
(9)
where
λ
=
iω
+
δ
.
It should be noted that
Θ
,
U
,
V
depend on coordinates and the number
of vibrations tone
n
only. In simulating, only the second mode shape was
considered (Fig. 1,
c
).
Using Galerkin’s standard scheme/pattern for the finite element method
[14] we obtain from (6)–(8) a matrix equiation for temperature
Θ
and
displacements
U
in mesh points:
(K +
λ
H)Θ +
λ
FU = 0;
(10)
here
K
,
H
,
F
— matrices whose elements are composed from shape
functions
N
with number of mesh points
m
.
The axisymmetric resonator’s geometry assumes the transition to
cylindrical coordinates
r
,
θ
,
z
. Then (6) will take on the form
ρC
P
∂T
∂t
−
k
∂
2
T
∂r
2
+
1
r
∂T
∂r
+
1
r
2
∂
2
T
∂θ
2
+
∂
2
T
∂z
2
=
−
EαT
0
(1
−
2
ν
)
∂e
∂t
,
while for element matrices from (10) the following relations are true:
K
e
=
Z
Ω
k
∂
N
∂r
∂
N
T
∂r
+
∂
N
∂z
∂
N
T
∂z
+
n
2
r
2
NN
T
drdz,
H
e
=
Z
Ω
C
V
NN
T
drdz,
F = [F
1
. . .
F
m
]
,
F
i
=
EαT
0
1
−
2
ν
N
∂
N
i
∂r
+
N
i
r
−
n
N
i
2
r
+
1
2
∂
N
i
∂z
n
N
i
r
+
1
2
∂
N
i
∂r
−
N
i
r
+
1
2
∂
N
i
∂z
∂
N
i
∂z
−
n
N
i
2
r
+
1
2
∂
N
i
∂z
T
,
(
i
= 1
, . . . , m
)
.
32
ISSN 0236-3933. HERALD of the BMSTU. Series Instrument Engineering. 2015. No. 2