and are magnified locally in the figure. Below is the first resonance frequency where
there is a relatively small but finite response, which is the low-frequency limit.
For the deflection curve at the three resonances, it is found that the imaginary
parts are much larger than the real parts. Therefore we focus on the imaginary parts
and normalize them in Fig. 4.6. The behavior is familiar for a cantilever at resonances. In addition to the fixed left end, the three displacement distributions are with
zero, one, and two points with zero deflection (nodal points) as expected.
We are interested in the distribution of the carrier concentration perturbation
described by n
(1) x 2 or Nx 2 . Numerical results show that its imaginary part is much
larger than the real part. Therefore, only its imaginary part is shown in Fig. 4.7. The
carrier concentration perturbation becomes very large at the left end which may not
be true because it is well-known that one-dimensional theories for the bending of a
beam cannot predict behaviors at a fixed end accurately in general. For the most part
of the beam away from its ends, the one-dimensional theory can produce reliable
Fig. 4.5 Beam end
displacement versus driving
frequency showing three
resonances
0
2
4
6
x 10
-7
-1
-0.5
0
0.5
1
x 3 (m)
Im{V} (m)
ω=ω (1)
ω=ω (2)
ω=ω (3)
Fig. 4.6 Normalized
imaginary parts of the
complex deflection V(x 3 )
along the beam at
resonances
4.4 Time-Harmonic Bending
101