ω
n
ð Þ
0 ¼
2n À 1
ð
Þπ
2L
ffiffiffiffiffiffi
c 33
ρ
r
, n ¼ 1, 2, 3, Á Á Á,
ð3:146Þ
which are the extensional resonance frequencies of the corresponding elastic dielectric rod. We also introduce
Z 0 ¼
1
iω
1
ð Þ
0 C 0
, C 0 ¼ ε 33
S
L
ð3:147Þ
as a unit for the impedance Z of the output circuit.
We calculate the open-circuit vibration characteristics when Z ¼ 1 first. The
effect of Z will be examined later. We plot the modulus (absolute value) of the
complex displacement at the right end of the rod versus the normalized driving
frequency in Fig. 3.18a for two values of Q ¼ 50 and 100. These values of Q are
relatively small for the material damping of common crystals. We exaggerate the
damping a little so that the resonances are not too sharp and are more visible in the
figure. In the frequency range shown, there are three resonances with frequencies
ω
(n)
. It can be seen that the values of ω
n
ð Þ
=ω
1
ð Þ
0
are close to 1, 3, and 5 where
Fig. 3.18 (a) ju 3 (L )j and (b) θ ¼ tan
À1
[Im{u 3 (L )}/ Re {u 3 (L )}] versus the normalized driving
frequency showing resonances. Z ¼ 1
64
3 Extension of Rods