162
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
λ
2 ∂
∂ x
J 2
∂ϕ
∂ x
− α J 4
∂ϕ
∂ x
+
∂
2 w
∂ x 2
− λ
4
(A 0 − 3α A 2 )
ϕ +
∂ w
∂ x
+
+3α
λ
4
(A 2 − 3α A 4 )
ϕ +
∂w
∂ x
− 3α B 2 λ
4
ϕ +
∂w
∂ x
+
+
λ
2
4
∂
∂ x
B 0
∂ϕ
∂ x
−
∂
2 w
∂ x 2
− 3α B 2
∂ϕ
∂ x
+
∂
2 w
∂ x 2
−
(6.111)
−αλ
2 ∂
∂ x
(J 4 − α J 6 )
∂ϕ
∂ x
− α J 6
∂ 2 w
∂ x 2 +
3
4
B 2
∂ϕ
∂ x
−
∂ 2 w
∂ x 2
− 3α B 4
∂ϕ
∂ x
+
∂ 2 w
∂ x 2
+
λ
2
2
(C 0 − 3αC 2 ) =
∂
2 u
∂t 2 + (I 2 − α I 4 )
∂
2
ϕ
∂t 2 − α (I 4 − α I 6 )
∂
2
ϕ
∂t 2 +
∂
3 w
∂ x∂t 2
,
λ
2 ∂
∂ x
(A 0 − 3α 2 )
ϕ +
w
∂ x
+
∂
∂ x
J 0
∂u
∂ x
+
1
2
∂w
∂ x
2
∂w
∂ x
+
+
1
4
∂
∂ x
∂
∂ x
B 0
∂ϕ
∂ x
−
∂
2 w
∂ x 2
− 3α B 2
ϕ +
∂
2 w
∂ x 2
+
+α
∂
2
∂ x 2
(J 4 − α J 6 )
∂ϕ
∂ x
− α J 6
∂
2 w
∂ x 2
−
−3λ
2
α
∂
∂ x
(A 2 − 3α A 4 )
ϕ +
∂ w
∂ x
− 3α
B 2
∂w
∂ x
+ ϕ
−
−
3
4
∂
∂ x
B 2
∂ϕ
∂ x
−
∂
2 w
∂ x 2
− 3α B 4
∂ϕ
∂ x
+
∂
2 w
∂ x 2
+
(6.112)
+
1
2
∂
∂ x
(C 0 + 3αC 2 ) + q = ε
∂w
∂t
+ I 0
∂ 2 w
∂t 2 + α I 4
∂ 3 ϕ
∂ x∂t 2 − α 2 I 6
∂ 4 w
∂ x 2 ∂t 2 +
∂ 3 ϕ
∂ x∂t 2
,
where in Eq. (6.112) the term ε∂w/∂t is introduced which follows idea of the set-up
method and guarantees damping of vibrations.
The boundary conditions for the homogeneous beam take the following form:
J 0
∂u
∂ x
+
1
2
∂w
∂ x
2
− N
T
x=1
x=0
= ¯
N or u|
x=1
x=0 = ¯
u,
(6.113)