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Equation of motion (5) when using (4), (8) in matrix form
σ
= C
ε
−
D
T,
ε
= BU
,
Can be transformed into the following form:
LU
−
GΘ +
λ
MV = 0
.
(11)
Here
L
,
G
,
M
— element matrices, and
V
— vector of rate values at
mesh points.
Elements of matrices in (8)–(11) are calculated in the following way:
L
e
=
Z
Ω
B
T
CB
dr dz,
G
e
=
Z
Ω
B
T
DB
dr dz,
M
e
=
Z
Ω
ρ
NN
T
dr dz.
Matrices
C
,
D
,
B
depend on physical properties of the material and
the form of interpolation functions
C =
E
(1 +
ν
)(1
−
2
ν
)
1
−
ν ν ν
0
0
0
ν
1
−
ν ν
0
0
0
ν ν
1
−
ν
0
0
0
0 0 0 1
/
2
−
ν
0
0
0 0 0 0 1
/
2
−
ν
0
0 0 0 0
0 1
/
2
−
ν
,
D =
Eα
(1
−
2
ν
)
[1 1 1 0 0 0]
T
,
B = [B
1
. . .
B
m
]
,
B
i
=
∂
N
i
∂r
0
0
N
i
r
n
N
i
r
0
0
0
∂
N
i
∂z
−
nN
i
2
r
1
2
∂
N
i
∂r
−
N
i
r
0
0
1
2
∂
N
i
∂z
n
N
i
2
r
1
2
∂
N
i
∂z
0
1
2
∂
N
i
∂r
,
(
i
= 1
, . . . , m
)
.
ISSN 0236-3933. HERALD of the BMSTU. Series Instrument Engineering. 2015. No. 2
33