Q ,3 ¼ ρA€ v,
M ,3 À Q ¼ ρI €
ψ,
D
1
ð Þ
3,3 À D
0
ð Þ
2 ¼ qI Àn
1
ð Þ
,
J
n 1
ð Þ
3,3 À J
n 0
ð Þ
2
¼ qI _
n
1
ð Þ ,
ð4:45Þ
Q ¼ c 44 A v ,3 þ ψ
ð
Þþe 15 Aϕ
1
ð Þ ,
M ¼ c 33 Iψ ,3 þ e 33 Iϕ
1
ð Þ
,3 ,
ð4:46Þ
D
0
ð Þ
2 ¼ e 15 A v ,3 þ ψ
ð
ÞÀε 11 Aϕ
1
ð Þ ,
D
1
ð Þ
3 ¼ e 33 Iψ ,3 À ε 33 Iϕ
1
ð Þ
,3 ,
ð4:47Þ
J
n 0
ð Þ
2
¼ Àqn 0 μ
n
11 Aϕ
1
ð Þ
þ qD
n
11 An
1
ð Þ ,
J
n 1
ð Þ
3
¼ Àqn 0 μ
n
33 Iϕ
1
ð Þ
,3 þ qD
n
33 In
1
ð Þ
,3 ,
ð4:48Þ
c 44 A v ,33 þ ψ ,3
À
Á þ e 15 Aϕ
1
ð Þ
,3 ¼ ρA€ v,
c 33 Iψ ,33 þ e 33 Iϕ
1
ð Þ
,33 À c 44 A v ,3 þ ψ
ð
ÞÀe 15 Aϕ
1
ð Þ
¼ ρI €
ψ,
e 33 Iψ ,33 À ε 33 Iϕ
1
ð Þ
,33 À e 15 A v ,3 þ ψ
ð
Þþε 11 Aϕ
1
ð Þ
¼ qI Àn
1
ð Þ
,
Àqn 0 μ
n
33 Iϕ
1
ð Þ
,33 þ qD
n
33 In
1
ð Þ
,33 þ qn 0 μ
n
11 Aϕ
1
ð Þ
À qD
n
11 An
1
ð Þ
¼ qI _
n
1
ð Þ
:
ð4:49Þ
The boundary conditions are
v 0
ð Þ ¼ 0, ψ 0
ð Þ ¼ 0, M L
ð Þ ¼ 0, Q L
ð Þ ¼ f y t
ð Þ ¼ F exp iωt
ð Þ,
ð4:50Þ
D
1
ð Þ
3 0
ð Þ ¼ 0, J
n 1
ð Þ
3
0
ð Þ ¼ 0, D
1
ð Þ
3 L
ð Þ ¼ 0, J
n 1
ð Þ
3
L
ð Þ ¼ 0,
ð4:51Þ
where F and ω are real constants and i is the imaginary unit. We use the usual
complex notation with
v, ψ, ϕ
1
ð Þ , n
1
ð Þ
n
o
¼ Re V, Ψ, Φ, N
f
gexp iωt
ð Þ
f
g :
ð4:52Þ
Then Eq. (4.49) reduces to the following four ordinary differential equations for V,
Ψ, Φ, and N:
x1, x
x2, y
L
a
c
x3, z
fy(t)
Fixed
end
Shear force
Fig. 4.4 A ZnO beam
under a time-harmonic shear
force
4.4 Time-Harmonic Bending
99