306
8 Thermoelastic Vibrations of Timoshenko Microbeams
∂
∂ x
k 1
∂ u
∂ x
+
1
2
∂ w
∂ x
2
− N
T
+ G (x, t) = ρ A
∂
2 u
∂ t 2 ,
∂
∂ x
k 1
∂ u
∂ x
+
1
2
∂ w
∂ x
2
− N
T
∂ w
∂ x
+ k 3
∂
2 w
∂ x 2 +
∂ ψ
∂ x
−
− k 4
∂
4 w
∂ x 4 −
∂
3
ψ
∂ x 3
+ F(x, t) +
1
2
∂ C
∂ x
= ρ A
∂
2 w
∂ t 2 ,
k 2
∂
2
ψ
∂ x 2 − k 3
∂ w
∂ x
+ ψ
− k 4
∂
3 w
∂ x 3 −
∂
2
ψ
∂ x 2
−
∂ M
T
∂ x
+
C
2
= ρ I
∂
2
ψ
∂ t 2 ,
(8.19)
and the following boundary conditions
k 1
∂ u
∂ x
+
1
2
∂ w
∂ x
2
− N
T
− ¯
N
x=0, L
= 0
or
δ u| x=0, L = 0 ,
k 1
∂ u
∂ x
+
1
2
∂ w
∂ x
2
− N
T
∂ w
∂ x
+ k 3
∂ w
∂ x
+ ψ
−
−k 4
∂
3 w
∂ x 3 −
∂
2
ψ
∂ x 2
+
1
2
C − ¯
V
x=0, L
= 0
or
δ w| x=0, L = 0,
k 4
∂
2 w
∂ x 2 −
∂ ψ
∂ x
− ¯
M m
x=0, L
= 0
or
δ
∂ w
∂ x
− ψ
x=0, L
= 0, (8.20)
k 2
∂ ψ
∂ x
− M
T
− ¯
M σ
x=0, L
= 0
or
δ ψ| x=0, L = 0.
8.3.3 Coupled Problem of Thermoelasticity for Timoshenko
Beams
Assuming small temperature variation, a coupled heat transfer equation for a thermoelastic body can be recast to the following form [98]:
λ θ , ii = ρ c v
∂ θ
∂ t
+
Eα T 0
(1 − 2ν)
∂ ε ii
∂ t
,
(8.21)
where λ—heat transfer coefficient; c v —heat capacity of the unit material volume.
Recall that the heat transfer equation is coupled with beam deformation by heat
production through compression/extension of the beam. It is assumed that the volume