310
8 Thermoelastic Vibrations of Timoshenko Microbeams
N
∂ w
∂ x
+ k 3
∂ w
∂ x
+ ψ
−k 4
∂
3 w
∂ x 3 −
∂
2
ψ
∂ x 2
x=0, L
= 0
or
δ w| x=0, L = 0,
k 4
∂
2 w
∂ x 2 −
∂ ψ
∂ x
x=0, L
= 0
or
δ
∂ w
∂ x
− ψ
x=0, L
= 0,
(8.39)
k 2
∂ ψ
∂ x
− M
T
x=0, L
= 0
or
δ ψ| x=0, L = 0.
Let us express N and M
T by temperature θ 1 (x, t) and θ 2 (x, t) as follows:
N =
k 1
L
L
0
1
2
∂ w
∂ x
2
dx −
Eα
L
G 1
L
0
θ 1 (x, t) dx, M
T
= EαG 3 θ 2 (x, t) .
(8.40)
As a result, we obtain the closed system of equations (8.32), (8.33) and (8.38)–
(8.40) for the nonlinear analysis of TED of the Timoshenko microbeams.
Our goal is to derive a non-dimensional counterpart form of the governing PDEs.
For this purpose, we introduce the following non-dimensional parameters
¯
x =
x
L
, ¯
z =
z
h
, ¯
w =
w
h
,
¯
ψ =
L
h
ψ, ω 0 =
E I
ρ AL 4 ,
¯
t = ω 0 t, τ 0 =
h
2
D
, τ = ω 0 τ 0 , ¯
θ =
θ
T 0
, ¯
R = R/αT 0 ,
P =
AL
2
I
, H =
k s G AL
2
E I
, F =
1
4
l
2 G A
E I
,
¯
G 1 =
G 1
A
,
¯
G 2 =
G 2 h
2
A
,
¯
G 3 =
G 3
Ah
, ¯
G 4 =
G 4 h
A
.
(8.41)
Formulas (8.40) yield
N = E A ¯
N ,
¯
N =
h
L
2
1
0
1
2
∂ ¯
w
∂ ¯
x
2
d ¯
x − α T 0 ¯
G 1
1
0
¯
θ 1 (x, t) dx,
M =
E I h
L 2
¯
M,
¯
M =
∂ ¯
ψ
∂ ¯
x
− α T 0 P ¯
G 3 ¯
θ 2 (x, t) ,
(8.42)
and hence Eqs. (8.32) and (8.33) take the following non-dimensional form (bars over
non-dimensional quantities are omitted):
∂ θ 1 (x, t)
∂ t
+
R
G 1
h
L
2 ∂
∂ t
1
2
∂ w
∂ x
2
+
1
τ
G 2
G 1
θ 1 (x, t) = 0,
(8.43)