312
8 Thermoelastic Vibrations of Timoshenko Microbeams
which is substituted into (8.42). Therefore
N =
⎛
⎝ 1 −
2iω α T 0 R
1
τ
G 2
G 1
− iω
h
L
2
⎞
⎠
1
0
1
2
∂ w
∂ x
2
dx = S ω
1
0
1
2
∂ w
∂ x
2
dx,
(8.51)
where
S ω =
⎛
⎝ 1 −
2iω α T 0 R
1
τ
G 2
G 1
− iω
h
L
2
⎞
⎠ .
(8.52)
Function S ω can be presented in the following form:
S ω =
⎛
⎜
⎜
⎝ 1 +
2ω
2
α T 0 R
1
τ 2
G 2
G 1
2 + ω 2
h
L
2
− i
2ω α T 0 R
1
τ
G 2
G 1
1
τ 2
G 2
G 1
2 + ω 2
h
L
2
⎞
⎟
⎟
⎠ ,
Re [S ω ] = 1 +
2ω
2
α T 0 R
1
τ 2
G 2
G 1
2 + ω 2
h
L
2
,
Im [S ω ] = −
2ω α T 0 R
1
τ
G 2
G 1
1
τ 2
G 2
G 1
2 + ω 2
h
L
2
.
(8.53)
We proceed in the same way with Eqs. (8.42) and (8.44) to get
G 4
G 3
− iω
θ 2 (x) =
iω R
PG 3
∂ ψ
∂ x
.
(8.54)
Hence
θ 2 (x) =
iω R
PG 3
1
G 4
G 3
− iω
∂ ψ
∂ x
(8.55)
is substituted into (8.42) to yield
M =
⎡
⎣ 1 −
iω αT 0 R
G 4
G 3
− iω
⎤
⎦
∂ ψ
∂ x
= D ω
∂ ψ
∂ x
,
(8.56)
where
D ω =
⎡
⎣ 1 −
iω αT 0 R
G 4
G 3
− iω
⎤
⎦ .
(8.57)