216
7 Mathematical Models of Functionally Graded Beams in Temperature Field
U =
1
2
L
0
k 1
u , x +
1
2
w , x
2
2
+ k 2
ψ , x
2 +
+ k 3
ψ + w , x
2 + k 4
ψ , x − w , xx
2
dx ,
(7.39)
K =
1
2
L
0
m 0
u , t
2 + 2Q u , t ψ , t + ˜
I
ψ , t
2 + m 0
w , t
2
dx,
(7.40)
where
k 1 =
A
E (˜ z) d A, k 2 =
A
E (˜ z) z
2 d A =
A
E (˜ z) (˜ z − ˜
z c )
2 d A ,
k 3 = k s
A
μ (˜ z) d A, k 4 =
1
8
A
β (˜ z) d A =
1
4
A
μ (˜ z) l
2
(˜ z) d A,
m 0 =
A
ρ (˜ z) d A, Q =
A
ρ (˜ z) z d A =
A
ρ (˜ z) (˜ z − ˜
z c ) d A,
˜
I =
A
ρ (˜ z) z
2 d A =
A
ρ (˜ z) (˜ z − ˜
z c )
2 d A .
(7.41)
Hamilton’s principle yields
δ
t 2
t 1
(K − U + W ) d t = 0.
(7.42)
The carried-out variation with respect to u, w and ψ, integration by parts, and
setting the terms standing by δu, δw and δψ to be equal to zero yield the following
equations of motion:
k 1
u , x +
1
2
w , x
2
, x
+ G = m 0 u , tt + Q ψ , tt ,
k 2 ψ , x + k 4
ψ , x − w , xx
, x
− k 3
ψ + w , x
+
C
2
= Q u , tt + ˜
I ψ , tt , (7.43)
k 1
u , x +
1
2
w , x
2
w , x + k 3
ψ + w , x
, x
+
+
k 4
ψ , x − w , xx
, xx
+ q +
C , x
2
= m 0 w tt ,