c 44 A V ,33 þ Ψ ,3
ð
Þþe 15 AΦ ,3 þ ρω
2 AV ¼ 0,
c 33 IΨ ,33 þ e 33 IΦ ,33 À c 44 A V ,3 þ Ψ
ð
ÞÀe 15 AΦ þ ρω
2 IΨ ¼ 0,
e 33 IΨ ,33 À ε 33 IΦ ,33 À e 15 A V ,3 þ Ψ
ð
Þþε 11 AΦ þ qIN ¼ 0,
Àqn 0 μ
n
33 IΦ ,33 þ qD
n
33 IN ,33 þ qn 0 μ
n
11 AΦ À qD
n
11 AN À iωqIN ¼ 0:
ð4:53Þ
We look for a solution to Eq. (4.53) in the following form:
V, Ψ, Φ, N
f
g¼ C 1 , C 2 , C 3 , C 4
f
g exp λx 3
ð Þ:
ð4:54Þ
The substitution of Eq. (4.54) into Eq. (4.53) results in four linear homogeneous
algebraic equations for C 1 –C 4 . For nontrivial solutions, the determinant of the
coefficient matrix of the equations has to vanish, which leads to a polynomial
equation of degree eight for λ. Let the eight roots of this equation be λ
(m)
, and the
corresponding nontrivial solution of C 1 –C 4 be C
m
ð Þ
1 À C
m
ð Þ
4 . Since Eq. (4.53) is
homogeneous, only the ratios among C
m
ð Þ
1 À C
m
ð Þ
4
can be determined. Then the
general solution to Eq. (4.53) can be written as
V
Ψ
Φ
N
8
> > > <
> > > :
9
> > > =
> > > ;
¼
X 8
m¼1
G
m
ð Þ
C
m
ð Þ
1
C
m
ð Þ
2
C
m
ð Þ
3
C
m
ð Þ
4
8
> > > > > <
> > > > > :
9
> > > > > =
> > > > > ;
exp λ
m
ð Þ x 3
,
ð4:55Þ
where G
(m) are undetermined constants. Substituting Eq. (4.55) into Eqs. (4.50) and
(4.51), we obtain eight linear algebraic equations for G
(m)
, which are solved on a
computer using MATLAB.
As an example, consider a circular ZnO beam with L ¼ 600 nm and a ¼ 25 nm.
n 0 ¼ N
þ
D ¼ 10
23 m
À3 . Material damping is included by allowing the elastic constants to assume complex values which can describe viscous damping in the
material. In our numerical calculation, the real elastic constant c pq are replaced by
c pq (1 + iQ
À1 ), where Q is the quality factor of the material, a large, real and positive
number. We fix Q to be 10
2 in our calculation. In the static bending analysis in the
previous section, end forces greater than or equal to 0.02 nN were used. In this
section, since we are interested in the behavior of the beam near resonances where a
small driving force can produce significant motions, in our calculation below we
reduce the end force to F ¼ 0.002 nN so that even at resonances all of the fields are
small, and the linear theory used for small deformation is still valid.
We plot the modulus (absolute value) of the complex deflection at the right end
versus the driving frequency in Fig. 4.5. In the frequency range shown, there are
three resonances at ω (1) ¼ 6.1570 Â 10
8 rad/s, ω (2) ¼ 3.7651 Â 10
9
rad/s, and
ω (3) ¼ 1.0171 Â 10
10
rad/s where the right end displacement is large. The first
peak is relatively more pronounced. The second and the third peaks are barely visible
100
4 Bending of Beams
ð
Þþe 15 AΦ ,3 þ ρω
2 AV ¼ 0,
c 33 IΨ ,33 þ e 33 IΦ ,33 À c 44 A V ,3 þ Ψ
ð
ÞÀe 15 AΦ þ ρω
2 IΨ ¼ 0,
e 33 IΨ ,33 À ε 33 IΦ ,33 À e 15 A V ,3 þ Ψ
ð
Þþε 11 AΦ þ qIN ¼ 0,
Àqn 0 μ
n
33 IΦ ,33 þ qD
n
33 IN ,33 þ qn 0 μ
n
11 AΦ À qD
n
11 AN À iωqIN ¼ 0:
ð4:53Þ
We look for a solution to Eq. (4.53) in the following form:
V, Ψ, Φ, N
f
g¼ C 1 , C 2 , C 3 , C 4
f
g exp λx 3
ð Þ:
ð4:54Þ
The substitution of Eq. (4.54) into Eq. (4.53) results in four linear homogeneous
algebraic equations for C 1 –C 4 . For nontrivial solutions, the determinant of the
coefficient matrix of the equations has to vanish, which leads to a polynomial
equation of degree eight for λ. Let the eight roots of this equation be λ
(m)
, and the
corresponding nontrivial solution of C 1 –C 4 be C
m
ð Þ
1 À C
m
ð Þ
4 . Since Eq. (4.53) is
homogeneous, only the ratios among C
m
ð Þ
1 À C
m
ð Þ
4
can be determined. Then the
general solution to Eq. (4.53) can be written as
V
Ψ
Φ
N
8
> > > <
> > > :
9
> > > =
> > > ;
¼
X 8
m¼1
G
m
ð Þ
C
m
ð Þ
1
C
m
ð Þ
2
C
m
ð Þ
3
C
m
ð Þ
4
8
> > > > > <
> > > > > :
9
> > > > > =
> > > > > ;
exp λ
m
ð Þ x 3
,
ð4:55Þ
where G
(m) are undetermined constants. Substituting Eq. (4.55) into Eqs. (4.50) and
(4.51), we obtain eight linear algebraic equations for G
(m)
, which are solved on a
computer using MATLAB.
As an example, consider a circular ZnO beam with L ¼ 600 nm and a ¼ 25 nm.
n 0 ¼ N
þ
D ¼ 10
23 m
À3 . Material damping is included by allowing the elastic constants to assume complex values which can describe viscous damping in the
material. In our numerical calculation, the real elastic constant c pq are replaced by
c pq (1 + iQ
À1 ), where Q is the quality factor of the material, a large, real and positive
number. We fix Q to be 10
2 in our calculation. In the static bending analysis in the
previous section, end forces greater than or equal to 0.02 nN were used. In this
section, since we are interested in the behavior of the beam near resonances where a
small driving force can produce significant motions, in our calculation below we
reduce the end force to F ¼ 0.002 nN so that even at resonances all of the fields are
small, and the linear theory used for small deformation is still valid.
We plot the modulus (absolute value) of the complex deflection at the right end
versus the driving frequency in Fig. 4.5. In the frequency range shown, there are
three resonances at ω (1) ¼ 6.1570 Â 10
8 rad/s, ω (2) ¼ 3.7651 Â 10
9
rad/s, and
ω (3) ¼ 1.0171 Â 10
10
rad/s where the right end displacement is large. The first
peak is relatively more pronounced. The second and the third peaks are barely visible
100
4 Bending of Beams