ω
1
ð Þ
0 ¼ 1:3198 Â 10
10 rad/s. Below the first resonance, there is a relatively small but
finite response which is the static response in the low-frequency limit. The top of the
first peak of the curve with Q ¼ 100 is 4.79 Â 10
À11 m which has been cut off in the
figure for a better global view. When Q ¼ 200, 500, and 5000, the corresponding
peak values of the first resonance are 6.35, 7.72 and 8.74 Â 10
À11 m, respectively,
increasing monotonically because of less damping. When Q is very large, the
amplitude at the first resonance becomes less sensitive to Q because in this case
the dissipation due to semiconduction becomes more influential than material
damping. (b) shows the phase angle defined by θ ¼ tan
À1 [Im{u 3 (L )}/
Re {u 3 (L )}]. At resonances there is a phase jump of π. The jumps at ω=ω
1
ð Þ
0 close
to 2 and 4 are caused by Re{u 3 (L )} ¼ 0 and do not represent resonances. For an
elastic rod, the resonance frequencies are exactly odd integral multiples of ω
1
ð Þ
0 . For a
piezoelectric dielectric rod, the three frequencies are higher because of the piezoelectric stiffening effect. If the piezoelectric rod is also semiconducting, the mobile
charges screen the polarization charges, reduce the piezoelectric stiffening effect,
and lower the frequencies. Therefore ω
(n) are somewhat higher than ω
n
ð Þ
0 .
Next we examine the displacement fields at resonances (modes). Strictly speaking, vibration modes are defined by the eigenvalue problem of a free vibration
analysis. They can also be felt from our forced vibration analysis by examining
the displacement fields at resonances as follows. Based on the complex notation
used, the physical field of the mechanical displacement can be obtained from the real
part of the complex displacement through
Re u 3 exp iω t
ð Þ
f
g¼ Re u
R
3 þ iu
I
3
À
Á
cos ω t þ i sin ωt
ð
Þ
È
É
¼ u
R
3 cos ω t À u
I
3 sin ωt,
u
R
3 ¼ Re u 3
f g, u
I
3 ¼ Im u 3
f g,
ð3:148Þ
which depends on both the real and imaginary parts of the complex displacement.
When Q ¼ 100, in Fig. 3.19, we plot both the real and imaginary parts of the axial
mechanical displacement at the three resonance frequencies. The displacement
vanishes at the fixed left end as dictated by the boundary condition. At the first
resonance, the displacement does not have nodal points (zeros) except at the fixed
left end. At the second and the third resonances, the displacement has one and two
nodal points, respectively, in addition to the left end. These are typical behaviors of
modes at resonances. Corresponding to the relatively high peak at the first resonance
shown in Fig. 3.18a, the displacement at the first resonance is relatively large.
As a mechanical-to-electrical energy converter, the output power and efficiency
are of fundamental interest. The mechanical input power P 1 at the right end of the rod
and the electrical output power P 2 on Z can be calculated from
3.11 Harmonic Vibration
65
1
ð Þ
0 ¼ 1:3198 Â 10
10 rad/s. Below the first resonance, there is a relatively small but
finite response which is the static response in the low-frequency limit. The top of the
first peak of the curve with Q ¼ 100 is 4.79 Â 10
À11 m which has been cut off in the
figure for a better global view. When Q ¼ 200, 500, and 5000, the corresponding
peak values of the first resonance are 6.35, 7.72 and 8.74 Â 10
À11 m, respectively,
increasing monotonically because of less damping. When Q is very large, the
amplitude at the first resonance becomes less sensitive to Q because in this case
the dissipation due to semiconduction becomes more influential than material
damping. (b) shows the phase angle defined by θ ¼ tan
À1 [Im{u 3 (L )}/
Re {u 3 (L )}]. At resonances there is a phase jump of π. The jumps at ω=ω
1
ð Þ
0 close
to 2 and 4 are caused by Re{u 3 (L )} ¼ 0 and do not represent resonances. For an
elastic rod, the resonance frequencies are exactly odd integral multiples of ω
1
ð Þ
0 . For a
piezoelectric dielectric rod, the three frequencies are higher because of the piezoelectric stiffening effect. If the piezoelectric rod is also semiconducting, the mobile
charges screen the polarization charges, reduce the piezoelectric stiffening effect,
and lower the frequencies. Therefore ω
(n) are somewhat higher than ω
n
ð Þ
0 .
Next we examine the displacement fields at resonances (modes). Strictly speaking, vibration modes are defined by the eigenvalue problem of a free vibration
analysis. They can also be felt from our forced vibration analysis by examining
the displacement fields at resonances as follows. Based on the complex notation
used, the physical field of the mechanical displacement can be obtained from the real
part of the complex displacement through
Re u 3 exp iω t
ð Þ
f
g¼ Re u
R
3 þ iu
I
3
À
Á
cos ω t þ i sin ωt
ð
Þ
È
É
¼ u
R
3 cos ω t À u
I
3 sin ωt,
u
R
3 ¼ Re u 3
f g, u
I
3 ¼ Im u 3
f g,
ð3:148Þ
which depends on both the real and imaginary parts of the complex displacement.
When Q ¼ 100, in Fig. 3.19, we plot both the real and imaginary parts of the axial
mechanical displacement at the three resonance frequencies. The displacement
vanishes at the fixed left end as dictated by the boundary condition. At the first
resonance, the displacement does not have nodal points (zeros) except at the fixed
left end. At the second and the third resonances, the displacement has one and two
nodal points, respectively, in addition to the left end. These are typical behaviors of
modes at resonances. Corresponding to the relatively high peak at the first resonance
shown in Fig. 3.18a, the displacement at the first resonance is relatively large.
As a mechanical-to-electrical energy converter, the output power and efficiency
are of fundamental interest. The mechanical input power P 1 at the right end of the rod
and the electrical output power P 2 on Z can be calculated from
3.11 Harmonic Vibration
65