160
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
C 0
2
+ λ
2 A 0
ϕ +
∂ w
∂ x
+
1
4λ 2 k
2 B 0
∂w
∂ x
+
1
4
∂
∂ x
B 0
∂ϕ
∂ x
−
∂
2 w
∂ x 2
+
+
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
+ J 1
∂ϕ
∂ x
∂w
∂ x
−N
T
∂w
∂ x
x=1
x=0
= ¯
Q
(6.102)
or
w|
x=1
x=0 = ¯
w ,
∂ϕ
∂ x
−
∂
2 w
∂ x 2
x=1
x=0
= 0
or
∂w
∂ x
x=1
x=0
=
____
∂w
∂ x
.
(6.103)
The initial conditions remain the same as in the case of the Sheremetev–Pelekh
beam (6.96).
Equations of beam motion based on the Bernoulli–Euler hypotheses are yielded
by relations (6.97)–(6.103), and they have the following form:
λ
2 ∂
∂ x
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
− J 1
∂
2 w
∂ x 2
+
+ f − λ
2 ∂ N
T
∂ x
= I 0
∂
2 u
∂t 2 − I 1
∂
3 w
∂t 2 ∂ x
,
(6.104)
∂
2
∂ x 2
J 1
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
− J 2
∂
2 w
∂ x 2
−
∂
2
∂ x 2
B 0
∂
2 w
∂ x 2
−
− k
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
− J 1
∂
2 w
∂ x 2
+
+
∂
∂ x
J 0
∂u
∂ x
+ kw +
1
2
∂w
∂ x
2
− J 1
∂
2 w
∂ x 2
∂w
∂ x
+
+
1
λ 2
k
2
4
∂
∂ x
B 0
∂w
∂ x
+
∂ C 0
∂ x
+ k N
T
−
∂
∂ x
N
T ∂w
∂ x
−
∂
2 M
T
∂ x 2 + q =
= I 0
∂
2 w
∂ t 2 +
I 1
λ 2
∂
3 u
∂ t 2 ∂ x
−
I 2
λ 2
∂
4 w
∂ t 2 ∂ x 2 .
(6.105)