FLEXURAL VIBRATIONS OF PIEZOELECTRIC
QUARTZ BARS AND PLATES*
I. INTRODUCTION
1. Applications of Piezoelectrically-Controlled Oscillators.-One of
the important applications of piezoelectricity consists in the use of
vibrating quartz crystals for the stabilization of high frequency oscil-
lators. Nearly all radio broadcasting stations use piezoelectrically-
controlled oscillators in order to enable simultaneous radio transmis-
sion of a great number of stations and to keep each of them within
+ 50 cycles'of the allotted frequency channel. The frequency of os-
cillators thus controlled is so constant that the variations amount to
less than one part in a hundred thousand. The range of stabilized
frequencies used in broadcasting and relay communication covers 550
to 21 520 kc. There are, however, many applications of low-frequency
resonators, filters and oscillators in the range from 1 to 100 kc.,
whose stabilization would be useful. Very little has been done to ex-
tend piezoelectric control for the lower frequencies.
2. Difficulties Encountered in Stabilizing Low-Frequency Oscilla-
tors.-The main difficulty consisted in the necessity of using very large
crystals. In the usual method of controlling the frequency the inter-
action of an oscillator with the longitudinal vibration of a piezoelectri-
cally-excited quartz bar is utilized. The stability of frequency is ob-
tained by choosing the length dimension b of the vibrating crystal
as shown in Fig. 2 so that its natural frequency fb will be precisely
equal to the required constant frequency. The natural frequency fb of
a vibrating bar is known to depend on the velocity of sound propa-
gation v, so that fb = v/2b. Therefore, the lower the frequency re-
quired, the greater the length of the bar has to be made. The limit is
set by the size of crystals obtainable at reasonable cost. If, for in-
stance, the required frequency fb should be 10 000 cycles per second,
the length would have to be 27 cm., and for fb = 1000 cycles, the
length would have to be 270 cm. Quartz crystals sufficiently large to
make bars of such dimensions are not available.
In order to reduce the dimensions of the bars, Harrisont suggested
the use of flexural vibrations. Preliminary experiments disclosed, how-
ever, that below 50 kc. the usual circuits did not yield complete piezo-
*The results of this investigation were presented in a paper before the annual meeting of
the Illinois State Academy of Science, May 1, 1936 at Quincy. An abstract was published in
the Trans. Illinois State Academy of Science. Vol. 29, 1936, No. 2, pp. 225-227.
tJ. R. Harrison, Proc. I.R.E. Vol. 15, p. 1041-1927.
ILLINOIS ENGINEERING EXPERIMENT STATION
electric control, and that the frequency of the oscillator was not suf-
ficiently independent of its circuit constants.
For the development of efficient, low frequency, piezoelectrically-
stabilized oscillators, it was first of all necessary to obtain knowledge
of the natural frequencies of flexurally-vibrating quartz bars, and
how these frequencies depend on the dimensions of the crystal. Be-
sides the investigation of Harrison the only references found in litera-
ture* relate to crystals of small height compared to their length. They
were used in vacuum tubes for the excitation of glow discharges and
served as visible indicators of resonance. An experimental study
showed that such crystals are hardly adaptable for sustained power-
ful oscillations of constant frequency and amplitude as required for
power oscillators. The formulae given by the authors proved to be
approximations applicable only for a narrow range of crystal di-
mensions, and could not serve as a basis for the calculation required
for the design of oscillators.
3. Object of Investigation.-To find, experimentally, the precise
relation between natural frequency and the geometrical dimensions of
quartz bars and plates sustained in flexural vibrations by piezoelectric
excitation was the object of this investigation. Its further aim was to
derive an expression which could be used for calculating the dimen-
sions of quartz resonators in connection with low-frequency oscillators
and filters.
4. Acknowledgments.-This investigation has been carried on as
part of the work of the Engineering Experiment Station under the
general administrative direction of DEAN M. L. ENGER, Director of the
Engineering Experiment Station, and PROF. ELLERY B. PAINE, Head
of the Electrical Engineering Department of the College of Engineer-
ing. Acknowledgment is made of the able service rendered by W. W.
BROOKS, Research Graduate Assistant, and to J. A. STEWART, Research
Graduate Assistant, for his assistance in the final stage of the in-
vestigation.
II. FLEXURAL VIBRATIONS OF Y-CUT QUARTZ CRYSTALS
5. Method of Cutting Quartz Bars.-A Brazilian quartz crystal of
unusual size was first sliced into thin parallel slabs. The method of
cutting into slices is shown in Fig. 1. The apparatus consisted of a
sheet iron cutting wheel D, 0.16 cm. (1/r in.) thick and 19.6 cm. (8 in.)
*E. Giebe and A. Scheibe, Zeitschr. fuer Hochfr. Vol. 35, p. 165-1930.
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
FIG. 1. APPARATUS FOR CUTTING QUARTZ BARS
in diameter, which was driven by an electric motor B, and rotated at
164 r.p.m. The circumference of the wheel moved with a speed of
about 0.5 m. per sec. through a trough C filled with a mixture of finely
granulated No. 50 carborundum and water. A small amount of
powdered soap was added to increase the cutting efficiency.
The original crystal stock A, which measured 18 cm. in the direc-
tion of the optical axis and 14 cm. in the direction of one of the
electrical axes, was cemented to a wooden base E. The latter was
then clamped near one end of a flat iron lever F, which was attached
by means of a hinge G to the frame H of the apparatus. The other
end of the lever carried a weight I, movable along a rod J. By shifting
this weight the pressure of the crystal against the cutting wheel could
be adjusted so that the force exerted was about 6 kg.
Slabs of quartz S (see Fig. 2) were thus obtained which required
more or less time to cut, depending upon their area. Their dimensions
a, b, and t were respectively 5 to 15 cm. in the direction of the optical
O-Z axis, about 14 cm. in the direction of the electrical O-X axis, and
0.2 to 0.3 cm. in the direction of the mechanical O-Y axis.
The rough slabs S were then trued up on an iron lapping wheel K,
ILLINOIS ENGINEERING EXPERIMENT STATION
Fia. 2. ORIENTATION OF MAIN DIMENSIONS OF QUARTZ BARS
CUT FROM A CRYSTAL
which could be driven by the same motor B (see Fig. 1). When all the
rough cutting marks made by the iron wheel were thus ground out, the
slab was placed on glass plates and ground flat on one face by using
finer and finer grades (from No. 200 to No. 500) of carborundum and
a succession of grinding glass plates. In order to obtain glass plates of
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
FIG. 3. TOOL FOR GRINDING SERIES OF BARS TO PRECISE HEIGHT
satisfactory flatness, they were ground one against the other. A suc-
cession of plates was used also in this process. Whenever one side of
the crystal slab showed a uniformly cut surface after the application
of a very fine grade of abrasive (No. 500), it was judged to be flat.
The other side was then repeatedly ground and checked by a microm-
eter caliper until it proved to be parallel to the first side.
From such slabs of uniform thickness bars B (see Fig. 2) were
cut by means of a hacksaw frame into which had been placed a smooth
copper strip. The quartz slab was cemented to a metal backing.
Guide pins fixed in a supporting base served to keep the copper strip
aligned during the cutting operation. After this operation the edges of
the bars were rough and in most cases they were not exactly parallel,
nor were the angles exactly square. In order to avoid the lengthy
work of finishing each of a series of bars individually to a precise
height and symmetry, a specially-machined iron tool was used, as
shown in Fig. 3. It consisted of an iron block, into which a groove
was carefully machined to be parallel and square with the ends.
The width W of the groove was sufficient to accommodate six quartz
bars q. For each required height of the quartz bars, a pair of such tools
was made, of which one was slightly different in the dimension H from
the other by ten thousandths of an inch. This allowed the use of the
tool with the deeper groove for the finishing of one set of edges and
then, by removing the bars and placing them into the shallower trough
of the other tool, with the finished side in the bottom of the groove, the
other edges could be finished. A number of such pairs of tools, of vari-
ous dimensions, were prepared to serve for the finishing of bars of
different lengths. In Fig. 1 three of such tools L are shown placed on
the shelf below the lapping wheel K. The left tool could be used for
ILLINOIS ENGINEERING EXPERIMENT STATION
FIG. 4. COLLECTION OF QUARTZ BARS AND PLATES
both of the operations. It was provided with two grooves of different
depth, one cut into the upper, the other into the lower face of the block.
By allowing the ends of the quartz bars to protrude beyond the
squared end of the tool, the ends of the bars could be ground to be-
come parallel and squared. The grinding action of the carborundum
and water on a glass plate had little effect on the tool, as the iron
member, loaded with carborundum, ground the glass, but did not
change its dimension or shape. In this way all of the crystals were
finished more quickly and more exactly than could have been done
singly, without the use of the tool. Even in connection with single
crystals this method of finishing was found advantageous.
6. Material Investigated.-The experimental study started with
five bars. In the course of the investigation the number of quartz bars
and plates increased gradually. In order to verify certain conclusions
for as large a range of dimensions as possible more crystals of inter-
mediate sizes were found necessary. Finally, a collection of twenty-
five Y-cut and five X-cut bars and plates was available. Twenty-one
of them were photographed (see Fig. 4) together with a large quartz
crystal similar to one from which most of the specimens originated.
The Y-cut crystals were classified into four groups, each including
crystals of various lengths, but of constant height. The height (a in
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
Fig. 2), coincident with the optical axis, was, for the four groups,
about 0.75, 1.00, 1.5, and 1.8 cm., respectively. The thickness (t in
Fig. 2), coincident with the mechanical axis, varied from 0.14 to 0.27
cm. It was chosen approximately the same for each of the groups, with
the exception of the second group, which contained crystals of two
thicknesses. The length (b in Fig. 2), coincident with the electrical axis,
varied from 0.7 to 13 cm. Among the specimens there were five in the
collection (Nos. 1, 2, 3, 4, and 5) which were cut from another quartz
rock, and differed from the rest in appearance because they were highly
polished. All other bars were cut from one single quartz rock. Their
surface had a dull and smooth appearance produced by grinding with
the finest abrasive. Some of them were etched with hydrofluoric acid.
7. Crystal Mountings.-A number of four-electrode crystal holders
mounted on supports made of bakelite were used, each adaptable for
the particular range of bar dimensions. The space between the elec-
trodes was adjusted to be about 0.02 cm. larger than the thickness of
the quartz bar investigated, so that the latter could freely slide on the
bakelite base within the electrodes. For the purpose of stabilizing the
vibrations each bar had a loop of silk thread tightened around its
nodal plane. In Fig. 4 these supporting silk loops appear as two white
lines on each of the quartz bars. The distance of the two nodal planes
from the nearest end faces of the bars was, for the fundamental mode
of flexural vibrations, d = 0.224b, b denoting the total length of the
bar. This distance d for each of the bars investigated was included in
column 6 of Table 1.
8. Wavemeters.-For radio frequency ranges, precision wavemeters
of the General Radio Company's Type 224 were used. For lower fre-
quencies from 2 to 50 kc. similar tuning circuits were applied, which
consisted of variable air condensers and inductance coils coupled to
aperiodic circuits. The latter contained copper oxide rectifying units
connected with d-c. indicating instruments.
Low-frequency wavemeters applied for flexural vibrations were
calibrated by absolute measurements. For this purpose an electro-
magnetically driven tuning fork, whose natural frequency was 992
cycles per second, was used in connection with a two-stage amplifier
to drive a synchronous clock. The latter indicated its number of revo-
lutions within a given interval of time, checked by the standard time
as broadcasted by the Arlington Radio Station. The output of the
first amplifier was also fed into another one-stage amplifier whose grid
excitation was made abnormally large for the purpose of producing
ILLINOIS ENGINEERING EXPERIMENT STATION
Osci/lator I C,
W o ve-e-,--
C/rculf 7T i
(a)-Fo~r Freq'ueAnc,4s Ah&ove /5 Ac-.
(b)-For Freqzuenc/rs Be/ow /S1 c.
FIG. 5. METHODS FOR DETERMINATION OF NATURAL FREQUENCIES OF
FLEXURAL VIBRATIONS OF QUARTZ BARS
harmonics of the fundamental frequency of the tuning fork. These
harmonics supported by the tuned output circuit of the amplifier were
used for the calibration of the wavemeter. For frequencies above 10 kc.,
more stages of amplification were used, each stage being in connection
with a tuned circuit of a higher range of harmonic frequencies.
9. Methods of Frequency Determination.-Two methods were used
for the determination of natural frequencies of bars and plates vibrat-
ing flexurally.
For the frequencies above 15 kc., the first method represented
diagrammatically in Fig. 5a was used. An oscillator circuit a whose fre-
quency was variable by means of a condenser C, was driven by a
thermionic tube. The inductance L1 served to couple a with circuit b.
The latter consisted of a coupling inductance L2, the quartz bar q
·
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
mounted in a four-electrode holder, and a copper oxide rectifier output
meter d. A wavemeter w was loosely coupled with circuit b. The
quartz bar acted as an electromechanical resonator, which absorbed
energy from the circuit a at a rate which depended on how close its
frequency (varied by varying the capacitance Ci) approached the
fixed natural frequency of the vibrating bar q. The polarity of the four
electrodes placed parallel to the electrical axis X of the bar was such
that the direction of the electric field across the upper pair of electrodes
was opposite to the field direction in the lower pair of electrodes. Due
to the converse effect the piezoelectrically-stressed quartz bar was thus
subjected to compression along the upper part whenever tension was
produced along the lower part, and vice versa, in accordance with the
phase of alternating potentials induced by the oscillator at the two
pairs of electrodes. Under these conditions the bar q vibrated flexur-
ally in the plane determined by the electrical axis X, and the optical
axis Z. At resonance the bar with its circuit b responded vigorously,
and the output meter indicated accordingly a sharply-defined pulse p.
Thus, by tuning with the condenser C1, its resonance setting-point
could be found, and the frequency determined by adjusting the variable
condenser Cs of the wavemeter circuit w. The reaction of the latter
upon the crystal circuit b influenced the current through the output
meter, which served also as resonance indicator for the wavemeter.
Whenever greater sensitiveness was required, an aperiodic circuit with
a current indicator in series was coupled to the wavemeter.
For the determination of frequencies of flexurally-vibrating bars,
whose natural frequency was below 15 kc., the second method (repre-
sented in Fig. 5b) proved more sensitive. A similar oscillating circuit
a was used for the excitation of the circuit b. In the latter, however, a
balancing bridge e was inserted. The resistance R' and the quartz bar
q with its four-electrode holder formed the piezoelectric branch of the
bridge, and the resistance R" with a variable capacitance C" formed
the compensating branch of the bridge. The purpose of the bridge was
to balance at the points g and h that part of the potential across the
quartz crystal q which was independent of the piezoelectric reaction,
and which was caused by the excitation of the circuit b at the points
i and j by the oscillator a. The bridge could be balanced roughly by
adjusting the condenser C1 of the circuit a so that it was off by about
1 per cent from the resonance frequency fo of the bar q. Then a finer
adjustment was made by varying the condenser C" until the input of
the three-stage amplifier f was not energized at that frequency. The
balance was regarded as established with sufficient accuracy when the
ILLINOIS ENGINEERING EXPERIMENT STATION
1066
166
46
26
6
o
Lenq,- of Bar "b''" /i cm.
FIG. 6. FUNDAMENTAL FREQUENCIES OF PIEZOELECTRIC QUARTZ BARS
IN FLEXURAL AND LENGTH VIBRATIONS
output meter indicated less than 0.5 volt. Under these conditions
precise frequency determinations could be made by adjusting the
condenser C1 for a setting at which a sudden increase to several volts
in the deflection of the output meter d was observed.
Besides the fundamental flexural vibrations, the two modes of
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
longitudinal length and thickness vibrations of the bars were also
investigated.
For the determination of the natural frequencies of longitudinal
vibrations the quartz bars were placed in two-electrode holders; other-
wise the method resembled that shown in Fig. 5a. Whenever the
inductive coupling of the circuits a and b was not sufficient for the
excitation of the bars, the coil L2 was removed and the circuit b con-
nected directly to the coil L1. For the detection of frequencies other
than fundamental it was found necessary to insert (see Fig. 5a) in the
circuit b an amplifier in place of the meter d, and to connect the latter
to the output terminals of the amplifier.
10. Results of Measurements.-When the measured fundamental
frequencies were plotted against the length b of the quartz bars, the
curves, I, II, IV, and L, shown in Fig. 6, were obtained. Curves I,
II, and IV represent the natural frequencies fp of the flexural mode of
vibrations of three groups of bars which differed from each other by
the magnitude of the a dimension. All bars of group I had a height of
0.75 cm., the bars of group II had a height of 1.06 cm., and the bars of
group III a height of 1.8 cm. The respective number marked on each
bar is indicated on the curves.
In contrast to the flexural vibration, the longitudinal vibrations in
the direction of the length of these three groups of bars gave fre-
quencies fb which could be represented by a single curve L. The
results of measurements confirmed the known relation that the natural
frequency of bars vibrating longitudinally in the length direction is
independent of the height a of the bar, and is a linear function of the
length b of the bar, namely,
fb = Ab E1 (1)
b 2b6 p
where the average frequency constant, Ab, was found to be 2.7 X 105
per cm.-1 Also, for the frequency ft of the longitudinal thickness vibra-
tion, the measurements were in accord with the relation
At
ft = -- (2)
t
and gave an average value for At = 1.96 X 105 per cm."1
As to the flexural vibrations of the bars, it was found that their
measured natural frequencies f, did not correspond to the known rela-
tion
ff = A--
b2
ILLINOIS ENGINEERING EXPERIMENT STATION
TABLE 1
DATA OF MEASUREMENTS AND CALCULATIONS FOR TWENTY-FIVE Y-CUT QUARTZ
BARS AND PLATES
.4
4)
z
Dimensions
t
cm.
0.176
0.176
0.151
0.175
0.27
0.27
0.26
0.261
0.166
0.166
0.149
0.168
0.149
0.26
0.227
0.261
0.226
0.261
0.228
0.262
0.164
0.144
0.150
0.150
0.1504
-I
b
cm.
3
3.680
5.553
6.64
9.205
0.721
1.103
1.583
2.213
2.823
3.631
3.714
4.346
5.058
6.23
7.902
7.992
9.424
10.830
11.636
12.724
9.251
5.171
6.328
7.556
9.649
a
cm.
4
0.750
0.750
0.745
0.745
1.058
1.056
1.055
1.057
1.061
1.061
1.062
1.061
1.059
1.06
1.059
1.057
1.060
1.057
1.067
1.055
1.498
1.806
1.806
1.806
1.802
a
o
a/b
0.204
0.135
0.112
0.081
1.467
0.957
0.666
0.468
0.376
0.292
0.286
0.244
0.209
0.17
0.134
0.132
0.1121
0.0971
0.091(
0.083
0.162
0.349
0.285
0.239
0.187
r
a
d
Cl
I
d
cm.
0.825
1.243
1.488
2.063
0.167
0.247
0.355
0.496
0.632
0.814
0.832
0.974
1.132
1.396
1.77
1.790
2.115
2.43
2.61
2.85
2.074
1.16
1.418
1.69
2.162
Longitudinal Vibrations
.a
0
A
4)4)
ft
kc.
1149
1144
1292
1119
876
864
747.7
749.0
1145
1145
1282
1145
1282
753.3
870
870
768
870
768
1180
1338
1281
1281
1276
x
4)
2
4)
A
I
4)
A5
X 105
2.06
2.04
1.95
1.96
1.99
1.96
1.94
1.96
1.90
1.90
1.91
1.92
1.91
1.96
1.97
1.97
2.01
1.98
2.02
1.94
1.93
1.98
1.98
1.91
d
o-
ft
kc1
II
kc.
72.95
48.51
40.8
29.45
384
245.2
171.8
123.4
93.0
73.49
72.60
61.40
52.70
43.6
34.19
33.88
28.90
25.02
23.30
21.40
29.25
52.14
42.59
35.78
28.01
0
I"-
S
4s
see
P)
sec.
10
X 10-
1.37
2.06
2.45
3.39
0.261
0.408
0.585
0.81
1.075
1.36
1.38
1.65
1.90
2.29
2.92
2.95
3.46
4.00
4.30
4.67
3.42
1.92
2.35
2.79
3.57
x
"
03
g
Ab
11
X 105
2.68
2.70
2.72
2.71
2.77
2.707
2.72
2.73
2.63
2.66
2.70
2.67
2.67
2.71
2.69
2.70
2.72
2.71
2.71
2.71
2.7
2.70
2.70
2.70
2.70
Frequencies calculated according to this relation gave values from 2
to 75 per cent larger than the measured ones. The frequency factor
A, was found to be a function of the ratio a/b of the bar. For crystals
whose lengths were over ten times larger than their heights the
measured fundamental frequency was in close agreement with this
relation. For crystals with a ratio a/b = 0.5 the discrepancy amounted
to 40 per cent, and for a square plate, a/b = 1, the discrepancy reached
60 per cent.
~
--
---
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
TABLE 1.-Concluded
DATA OF MEASUREMENTS AND CALCULATIONS FOR TWENTY-FIVE Y-CUT QUARTZ
BARS AND PLATES
Flexural Vibrations
SI
fr
kc.
12
27.60
12.91
9.06
4.82
305.9
212.6
130.1
81.10
55.60
36.60
35.30
26.70
20.45
14.04
8.97
8.76
6.433
4.863
4.253
3.585
9.12
29.10
20.90
15.40
9.785
13
X 10-'
2.005
1.89
1.86
1.83
6.71
4.09
3.23
2.60
2.39
2.175
2.18
2.105
2.016
1.94
1.89
1.88
1.86
1.85
1.845
1.84
1.92
2.32
2.16
2.055
1.977
14
X 10-i
2.01
1.89
1.866
1.83
6.71
4.10
3.30
2.60
2.39
2.18
2.18
2.07
2.08
1.94
1.89
1.88
1.86
1.85
1.85
1.84
1.94
2.31
2.16
2.01
1.97
--
x I
A/B
15
X 106
0.498
0.529
0.538
0.546
0.149
0.244
0.310
0.385
0.418
0.459
0.459
0.474
0.495
0.514
0.529
0.532
0.538
0.541
0.541
0.543
0.521
0.431
0.463
0.488
0.505
>4
4) e
0C'
0i
460
7;
16
X 10I
0.500
0.528
0.536
0.543
0.158
0.230
0.303
0.377
0.420
0.460
0.463
0.482
0.498
0.514
0.529
0.530
0.535
0.539
0.540
0.542
0.518
0.432
0.46
0.485
0.508
0.508
0
46
B
17
1.103
1.046
1.03
1.017
3.50
2.404
1.822
1.464
1.315
1.20
1.192
1.144
1.108
1.073
1.045
1.046
1.032
1.024
1.021
1.018
1.07
1.276
1.185
1.139
1.087
6
0
a
per cent
20
+0.36
-0.58
-0.22
-0.84
+4.85
-6.83
-1.92
-1.82
+0.54
+1.08
+0.98
+1.33
+0.63
0.0
-0.22
-0.57
-0.67
-0.27
-0.71
-1.27
-0.99
+0.34
+0.48
-0.46
+0.51
--
The results of measurements and calculations, systematically
arranged, are included in Table 1. The table consists of three main
groups of columns: dimensions (columns 2-6), longitudinal vibrations
(columns 7-11), and flexural vibrations (columns 12-23). The speci-
mens investigated are divided into four groups, I, II, III, IV. Each
group contains quartz bars and plates of a definite height a.
Column 1 indicates the numbers by which the specimens were
marked in the consecutive order in which they were cut.
18
27.71
12.84
9.04
4.78
321.5
199.0
127.6
79.65
55.90
37.00
35.65
27.06
20.58
14.04
8.95
8.71
6.39
4.85
4.25
3.54
9.03
29.20
20.98
15.33
9.84
-0.04
+15.6
-13.6
-2.50
-1.45
+0.30
+0.35
0.3
+0.13
0.0
-0.02
-0.05
-0.017
-0.002
-0.04
-015.6
-13.6
-2.50
+0.40
+0.35
+0.36
+0.13
0.0
-0.02
-0.05
-0.043
-0.013
-0.003
-0.041
-0.09
+0.10
+0.08
-0.07
+0.056
46
10.50
24.08
16.20
4.27
3.750
31.23
14.66
11.45
..3
.M.
...
11.45
-a
0
0o
per cent
22
12.72
-25.04
11.4
-17.1
10.28
-4.60
-7.32
4.87
-17.05
'6
cs
o
fpb
23
X 10'
1.020
0.713
0.600
0.440
2.32
2.20
2.01
1.765
1.580
1.343
1.324
1.176
1.041
0.874
0.709
0.700
0.602
0.526
0.495
0.450
0.839
1.508
1.317
1.160
0.950
ILLINOIS ENGINEERING EXPERIMENT STATION
Columns 2, 3, and 4 give in centimeters the bar dimensions of
thickness t, length b, and height a, respectively.
Column 5 gives the ratio of height to length, a/b.
Column 6 indicates the distance, d = 0.224b, from both ends of the
bars, at which nodal zones were formed when the bars vibrated flexur-
ally.
Column 7 enumerates the measured natural frequencies ft of thick-
ness vibrations.
Column 8 shows the frequency constant for thickness vibrations,
At - ftt, calculated from the data of columns 2 and 7.
Column 9 enumerates the measured natural frequencies fb of the
length vibrations.
Column 10 gives the period, Tb = 1/fb, calculated from the data in
column 9.
Column 11 gives the frequency factor, Ab = fbt, for length vibra-
tions calculated from the data in columns 3 and 9.
Column 12 enumerates the measured natural frequency f' of the
flexural vibrations.
The remaining columns will be discussed in Sections 10 and 13.
11. Comparison with Theory.-The theory of lateral vibrations
of bars, as given by Rayleigh for isotropic material, in its simplest
form leads to a differential equation of the fourth order
02y iKE l4y
+ = 0 (4)
at2 p Ox4
where y* is the transversal displacement of the bar at any point x
along the length axis, K the radius of gyration of the bar's cross section,
E Young's modulus of. elasticity, and p the density of the bar's
material. The ratio E/p is the square of the velocity of propagation
of longitudinal vibrations. For a bar free at both ends with the
a2y 03y
boundary conditions = 0 - = 0 Equation (4) yields the
Ox2 aOx
very well known relation for the natural frequencies of bars
m2 K E
IF= 2-2 b V (5)
in which b is the length of the bar in centimeters, and m depends on
the mode of vibrations, K = 1, 2, 3, . . . . so that
m = (K + 1/2)r (6)
*In the particular case of the Y-cut quartz bars investigated the displacement y is directed
along the optical axis marked O-Z in Fig. 2. The dimension of the bar in this direction is de-
noted a.
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
For the fundamental vibration of a bar with an area ab of rectan-
gular cross section, K = 1, m2 = 22.37 and K = a/"/ 2, the theo-
retical frequency of flexural vibrations according to Equation (5)
assumes a form similar to that of Equation (3), namely,
fE-x a(7)
f = 1.028 E1b (7)
p b
The values calculated from this formula do not agree at all with
the measured values obtained from the experimental study of crystals,
as enumerated in column 12 of Table 1. As mentioned in Section 10,
the deviations become considerable for a/b > 0.1, and reach 75 per
cent for a crystal plate with a/b = 1.5.
The differential equation which served as a basis for the derivation
of Equation (7) evidently does not take into account all the physical
phenomena which enter into the process of flexural vibrations of quartz
bars. It was assumed that the displacements of each cross section along
a vibrating bar are strictly normal to the X-Y plane (see Fig. 2).
Actually, however, rotational to and fro motions of the cross sections
about axes perpendicular to the plane of vibration, O-X, had to be
taken into consideration, in addition to the motion of translation. A
corresponding term
K2a4y
QOx2t2
added by Rayleigh on the left side of Equation (4) produced a more
complete differential equation.
a2y K2E 04y K2 04y
Y + = 0 (8)
at2 p Ox4 Ox2at2
Rayleigh* showed, for the case of a bar free at one end, how the intro-
duction of rotational energy influences the expression (7) for the fre-
quency. The required correction for a bar free at both ends, as given
by Goens,t consists of a factor by which Young's modulus of elasticity
E1 must be decreased, namely,
E1
KE2 (9)
1 + a(m)-
b2
The physical meaning of this correction has to do with the realiza-
tion that the assumed velocity of propagation, vi =-- E, which is
*Lord Rayleigh, Theory of Sound, second ed., Vol. 1, p. 294, 1894.
tGoens, E. Ann. der Phys. Vol. 11, p. 649, 1931.
ILLINOIS ENGINEERING EXPERIMENT STATION
true for pure longitudinal waves, applies only for infinitely thin bars.
With increasing cross sectional dimensions, a different velocity of
propagation, namely, that of transversal waves comes more and more
into play. The resultant velocity is then
SE2 El
pv = (10)
S 1 (m) b
and depends on the frequency of vibrations, because the function
a(m) depends upon the mode of vibration. For the fundamental
flexural vibration, a(m) = 49.48. By inserting this value for a(m) and
a2/12 for K2 in Equation (9), the modulus of elasticity becomes
E1
E = (11)
1 + 4.12(a/b)2
When this correction was used in connection with Equation (7) for
the calculation of the frequency of the quartz bars the disagreement
with measured values decreased, but did not disappear altogether.
Especially for the larger values of a/b, discrepancies of about +7 to
+11 per cent resulted in spite of Rayleigh's correction.
A notable contribution to the theory of flexural vibrations was
made by Timoshenko,* who extended still further the differential
equation, by considering the effect of shearing forces in addition to
the translatory and rotary motions of the cross-sectional elements.
Correspondingly, he inserted two more terms with the modulus of
shearing G and a coefficient ý to account for the non-uniform distribu-
tion of shearing forces throughout each cross section of the vibrating
bar. He derived the following differential equation:
K2Y E 0y ( E\ 04 pK2 4Y
+Y + - 1+ -) + = 0 (12)
Ot2 p Ox4 G aX2t2 G at4
A complete solution of this equation was given by Goens.t How-
ever, its application for the calculation of natural frequencies was
found to be very involved. Therefore Goens' simplified solution, in
form of a correction factor for the modulus of elasticity,
E1
E = (13)
K2 / El \ K2/ £El
1 + 2mp(m)- 3 - (-- + m2(m)-- 1 + ý--
b \ G b) \ G
*Timoshenko, S.P. Phil. Mag. Vol. 41, 1921 p. 744; Vol. 43, 1922 p. 125.
5See footnote p. 19.
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
was applied for checking the results of frequency measurements made
on 25 bars and plates enumerated in column 12 of Table 1.
For the fundamental mode of flexural vibrations of a bar of
rectangular cross section, the specific values of m, p(m), K, and ý are
as follows:
m = 4.73 p(m) = 0.9825 K2 = a2/12
m2 = 22.373 p2(m) = 0.9653 ý = 1.2
As to the ratio Ei/G, reliable data for crystalline quartz are not
available, because it depends on the Poisson's ratio X, which is too
small for dependable measurements. Considering that in the known
relation
E/G = 2(1 + X) (14)
X is practically equal to zero for quartz, the value 2 for the ratio
EI/G was used. When these specific values were inserted in Equa-
tion (13), the modified modulus
E1
E3 = (15)
1 + 6.55(a/b)2
was obtained. The frequency of the quartz bars was then checked
by Equation (7) in which E3 was substituted for E1.
The deviations of values thus calculated from measured values of
fundamental frequencies were in most cases negative, and indicated
that the values calculated according to the Timoshenko-Goens cor-
rection were too small, while the positive deviations of the corre-
sponding Rayleigh correction gave values which were too large com-
pared with measured frequencies.
The question suggested itself whether the deviations might have
been caused by assuming, incorrectly, the Poisson ratio for quartz to
be X = 0. Inspection of Equation (13) showed, however, that for
X > 0 the deviations become still larger. By assuming X < 0 the
deviations become smaller, and approach zero for X = -0.54. Such
a large negative value for X is contrary to what is known of the me-
chanical properties of quartz.
It may be inferred that the Timoshenko differential equation
applies only to pure mechanical vibrations in isotropic materials,
while in piezoelectric vibrations of quartz bars additional electro-
mechanical phenomena have to be considered. Dielectric stresses,
caused by the electric potential gradient set up in quartz by the
voltage applied to the electrodes, and also the distribution of space
charges within the materials, may require additional terms.
ILLINOIS ENGINEERING EXPERIMENT STATION
The derivation of a differential equation which could take into
account the electrostrictive phenomena within a lattice of a flexurally
vibrating crystalline quartz bar is a problem in itself.*
12. Derivation of Design Formula.-For purposes of designing low-
frequency resonators for self-sustained stabilized oscillators, it was
found sufficient to derive design formulae on the basis of the experi-
mental data obtained from measurements on the 25 quartz bars enu-
merated in Table 1. Search in literature has revealed two empirical
formulae; one was given by Harrisont and the other by Giebe and
Scheibe.$ But natural frequencies calculated on the basis of these
formulae gave discrepancies even larger than those obtained by apply-
ing Rayleigh's (Equation 11) and Timoshenko-Goens' (Equation 15)
corrections. For example, a bar with a comparatively small ratio
a/b = 0.209 (bar No. 5 in Table 1), when calculated according to the
Harrison or Giebe-Scheibe formula, gave discrepancies from the meas-
ured values amounting to, respectively, +28.8 per cent and -3.18 per
cent. The deviations assumed rapidly increasing values with increasing
ratios, a/b. For a/b = 0.376 (bar No. 1 in Table 1), the corresponding
values reached +75.2 per cent and -10.4 per cent.
It was therefore necessary to obtain new formulae.
The procedure in deriving the frequency formula was as follows:
The distribution of the deviations indicated that the general char-
acter of Equations (11) and (15) might not be at fault. It was there-
fore assumed that these equations hold for piezoelectric quartz, and
that only the elastic modulus and the numerical value in the term con-
nected with a/b (marked in what follows by D) is influenced by a
factor as yet unknown. Denoting in Equation (7)
1.028 EI/p = A (16)
and in Equations (11) and (15)
1 + D(a/b)2 = B2 (17)
the fundamental natural frequency of a flexurally-vibrating quartz bar
may be expressed in the form
A a/b
fF = (18)
B b
*A further contribution to the theory of flexural vibration of bars was made by W. P.
Mason in the Journal of the Acoustical Society of America, Vol. VI, p. 246, 1935. This article
came to our attention when the present publication was in press.
tSee footnote p. 5.
tSee footnote p. 6.
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS 23
18
1./4
08
0.4
0.Z
/4
17
/6, \~
4o
Cu
'3
Curve IZY /
/ -x x /,
r3
'V" V)
56
54
sz
5.so
4.8
46
44
4.2
dn
Ilr)
0 0.05 0./0 0 a. O.z O56 Aa a035 04a
Rat/o v
FIG. 7. CURVES FOR FREQUENCY FACTORS FOR FLEXURAL VIBRATIONS
OF PIEZOELECTRIC QUARTZ BARS
From the frequency f! and the geometrical dimensions b and a of 25
bars which were determined by precise measurements (columns 3, 4,
and 12 in Table 1) the ratio
A f'b
A - (19)
B a/b
was calculated, and the results plotted against a/b. In Fig. 7, the
curve V is shown for values of a/b limited to the useful range 0 to 0.4
so as to avoid crowding of the points marked for each bar. Curve VI
in Fig. 8 gives the A/B ratios for the complete range of a/b from
0 to 1.4 investigated.
Considering that for a/b = 0 the frequency factors will be B = 1
and A/B = A, it follows that the numerical value of the factor A may
be obtained by extrapolating the full-line curve V until it intersects
the A/B axis. It was thus determined that A = 5.52 X 105. Due to the
uncertainty connected with extrapolation, the numerical value of A
was checked experimentally from the data of measurements of natural
frequency fb and the frequency factor Ab of the longitudinally-vibrat-
ing bars (columns 9 and 11 of Table 1). From Equations (1) and (16)
/ A_ 5.52+/
%=^/
""
~
r
-. 0/1 Cclrv<e ,ILK .
'U
08
- --- -·- -·-
-- --- -·
4.0
ILLINOIS ENGINEERING EXPERIMENT STATION
I
Rai/o
FIG. 8. CURVES FOR FREQUENCY FACTORS FOR FLEXURAL VIBRATIONS
OF PIEZOELECTRIC QUARTZ BARS
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
the ratio between the frequency factor A for flexural vibration and
the factor Ab for longitudinal length vibration was
A 1.028 Ei/p
-- , or A = 2.056Ab (20)
Ab 1/2V Ei/p
With the average values of Ab = 2.7 X 105, the factor A = 5.55 X 105
was obtained. From the two values for A, the former was chosen be-
cause it proved to give closer agreement when used for the calculation
of flexurally-vibrating bars. This constant factor A is indicated in
Fig. 7 by a line drawn parallel to the a/b axis for the ordinate
A/B = 5.52 X 105.
In order to obtain the frequency factor B as a function of a/b, it
was only necessary to divide for each bar the value of A by the cor-
responding value of A/B (column 15, Table 1). The result was curve
VII in Fig. 8. That the mathematical expression of the curve was in
accordance with Equation (17), where
B = V 1 + D(a/b)2
was verified by checking the numerical value of the coefficient
B2 - 1
D = (-- (21)
(a/b)2
for each of the specimens from the experimental values of B and a/b.
This coefficient was found to be a constant for all values of a/b, and
the corresponding B values represented by the curve VII in Fig. 8.
Its average value was D = 5.22. Accordingly, the expression for the
frequency factor B was obtained
B = V 1 + 5.22(a/b)2 (22)
In column 17 of Table 1 are given the values of B, calculated from
Equation (22) for each of the bars investigated. With two numerical
factors A and D determined, the complete expression for the funda-
mental natural frequency of flexurally-vibrating piezoelectric quartz
bars and plates follows from Equations (18), (20), and (22)
5.52 X 105 a/b
f' = (23)
V 1 + 5.22(a/b)2 b
13. Comparison of Calculated and Measured Values of Frequency.
-By calculating the ratio
A 5F 5 Y 10f
B V 1 + 5.22(a/b)2
A 5 52 X 10.2
ILLINOIS ENGINEERING EXPERIMENT STATION
and plotting against values of a/b for each bar, the broken-line curve,
VIII in Fig. 7, and the curve VI (column 16, Table 1) in Fig. 8 were
obtained. Both show the relatively good agreement with experimental
values indicated by circles. The values of frequencies f' calculated
from Equation (23) and tabulated in column 18, along with the meas-
ured values of f' shown in column 12, may serve for a quantitative
comparison of discrepancies. The deviations 8 (in per cent) are indi-
cated in column 20, Table 1. Of the 25 crystals investigated,
3 bars showed deviations from calculated values within 0 to 0.25 per cent
5 bars showed deviations from calculated values within 0.25 to 0.5 per cent
10 bars showed deviations from calculated values within 0.5 to 1.0 per cent
3 bars showed deviations from calculated values within 1.0 to 1.5 per cent
2 plates showed deviations from calculated values within 1.5 to 2.5 per cent
2 plates showed deviations from calculated values above 2.0 per cent
The classification of the crystals into bars and plates was made
on the basis of the ratio a/b of height to length in the plane of vibra-
tion. Crystals with a ratio a/b < 0.45 to 0.5 were designated as bars;
those with a ratio a/b > 0.45 to 0.5 as plates. The deviations were
Spartly positive, partly negative.
For 9 bars, calculated values were larger than measured by +0.36 to 1.33 per cent
For 13 bars, calculated values were smaller than measured by -0.22 to 1.92 per cent
For 1 bar, the calculated value was equal to the measured value.
Generally the deviations were slight for bars with dimensions of
practical significance. They may be ascribed to threefold sources:
First, errors in observation connected with methods of measurements
of frequency and length; second, the lack of homogeneity of quartz
rock, which is known to show variation not only from one crystal
specimen to another, but within different parts of the same specimen;
and third, the limitations of Equation (23), which, although derived
from theoretical considerations in conjunction with experimental data,
cannot be regarded as representing rigorously all phenomena involved
in sustained piezoelectric oscillations of quartz bars and, especially, of
plates.
14. Frequency Factors.-For the purpose of designing piezoelectric
quartz resonators it may be stated that, as the result of measurements
(see Table 1), the following average frequency factors were estab-
lished for Y-cut quartz bars:
A 5.52 X 105
For flexural vibrations, - = 5.2X
B V/1 + 5.22(a/b)2
For longitudinal length vibrations, Ab = 2.70 X 106
For longitudinal thickness vibrations, At = 1.96 X 106
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
15. Application of Formula to Typical Cases.-
(a) It may be inferred from the smallness of the deviations 8,
tabulated in column 20, Table 1, that, for all practical purposes,
Equation (23) may safely be used for the calculation of frequency
fF from given dimensions of bars as well as of plates, whose ratio a/b
is smaller than 0.75.
(b) For the case when the length b and frequency fF are given,
and it is required to calculate the height a of the crystal, it follows
directly from Equation (23) that
a= I --- = b4f,2 (25)
a A2 - Db2f \F 30.47 X 1010 - 5.22b2f (
(c) For the case when the height a and frequency fp are given,
and it is required to calculate the length b of the crystal, it follows
similarly 4A2a2
R fj Da2
b2 = -
4 2 (26)
V 6.81a4 + 30.47 X 1010(a/fp)2 - 2.61a2
(d) In the course of this investigation, it was often necessary to
calculate the dimensions a and b for a given ratio a/b and frequency
fF. The following relations were applied for this purpose:
A (a/b)2 5.52 X 105 (a/b)2
a = - (27)
B fF V1 + 5.22(a/b)2 fF
or
A a/b 5.52 X 105 a/b (
b = = (28)
B f V 1 + 5.22(a/b)2 fr
Numerical calculations were considerably simplified by the use of
curves plotted in Figs. 7 and 8. The curves, VII, VIII, and IX, give
the values of B, A/B, and A/B - a/b = bfp, respectively, for any value
of a/b. For instance, instead of calculating b from Equation (28), the
value of bfp is looked up on curve IX for a given a/b value and divided
by the given frequency f, to obtain b.
(e) Another problem presented itself when it was necessary to cut
bars of different dimensions, but of equal prescribed natural frequency.
ILLINOIS ENGINEERING EXPERIMENT STATION
Two possible cases were investigated. The first may be formulated as
follows:
If the length b of a bar be changed by multiplying by n, what must
be the factor of multiplication m for the height a, in order that the
frequency fF may remain unchanged?
The principle of similitude requires that the dimensions a and b of
one bar and the dimensions ma and nb, of the other bar of equal fre-
quency fF, be determined by an equation similar to Equation (23),
namely, A a/b
fF -
V 1 + D(a/b)2 b
A m/n - a/b (29)
- 1 + D(m/n)2(a/b)2 nb
Solving for m, the factor for the height a is obtained
n2
m = (30)
V 1 - D(a/b)2(n' - 1)
(f) Similarly, the other case may be thus formulated: If the
height a of a bar be changed by multiplying by a factor m, what must
be the factor n for the length b in order that the frequency fF may
remain unchanged? The answer is obtained by solving for n, Equa-
tion (29)
n2 -- - +m:D+ 1] + 1 (31)
2 bL4\ b b)
(g) The process of cutting and grinding quartz bars of precise
dimensions to produce the prescribed frequency involves accurate and
time-consuming work. As a result of too much grinding, the needed
dimensions may be slightly overstepped, and thus a costly specimen
may be made useless for the designed purpose. It was, therefore,
desirable to obtain formulae for the relative variation of frequency
with varying dimensions, and vice versa. The following problem pre-
sented itself:
A bar has a natural fundamental frequency fF, its dimensions a and
b may be varied by amounts expressed by the factors q and r, respec-
tively. What will be the relative factor p by which the natural
frequency will change?
The changed frequency is
A q/r. a/b
?pf = (32)
V/1 + D(q/r)\(a/b)2 rb
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
By substituting for fp the expression from Equation (18) and solving
for p, the following result is obtained:
1 + D(a/b)2 q/r (33)
1 + D(q/r)2(a/b)2 r
In case only a is to be changed, r = 1 and
1 + D(a/b)2
p = q 66(33a)
= 1 ± D(qa/b)2
Similarly, if only b is to be changed, q = 1 and
/ 1 + D(a/b) (33
p = 1/r2 1 (33b)
1 + D(a/rb)2
(h) During the final lapping of bars for a prescribed frequency,
repeated measurements of frequency must be made. In order to reduce
the number of such tests to a minimum, it is useful to calculate by
what amount both or either of the two dimensions a and b must be
changed in order to reduce the frequency just measured to the required
one. The problem may be thus formulated: Given a bar whose natural
frequency is fp and whose height and length dimensions are a and b,
respectively; it is required that its frequency be changed by a given
factor p and its length changed also by a given factor r. What must be
the factor q by which the height a must be changed?
The answer is obtained by solving Equation (32) for q; that is
pr2
q = p (34)
V 1 + D(a/b)2(1 - p2r2)
In case no change is required for b, r = 1 is to be substituted in
Equation (34), (34a)
q =p (34a)
= 1 + D(a/b)2(1 - p2)
(i) Finally, a similar case was considered, when the natural fre-
quency of a bar had to be changed by a given factor p, and also the
height a by a given factor q. It is required to calculate the factor r by
which the length b must be changed.
By solving Equation (32) for r, the following expression is
obtained:
- 2 '2 q4 ( 2 (35)
r2 - q 2 P-a)__ p2q2 + 1 +D (35)
2 b P 4 b _)
ILLINOIS ENGINEERING EXPERIMENT STATION
For the special case where the height is to remain unchanged,
q = 1 is to be substituted in Equation (35) so that
D /a\2 1 D2 a\44 (a )2]
2 b4G) +b ^ P2 ±+[ +D )] (35a)
Numerical calculations may be simplified by remembering that
1 + D (a/b)2 = B2 (see Equations (17) and (22)) where B is a factor
given by curve VII in Fig. 8.
16. Secondary Frequencies.-Of the 25 samples investigated, 10
bars showed additional natural frequencies which were observed by
merely varying the frequency of the excitation circuit without chang-
ing the four-electrode arrangement of the crystal holder, or modifying
the diagram of connection of the resonator circuit. These frequencies,
denoted by f"p, are indicated in column 21 of Table 1, and their per-
centage deviation from the normal flexural vibration is listed in column
22. The relative intensity of these vibrations was much smaller than
that of the normal flexural vibration. No definite indication was
found of the origin of the secondary frequencies. They were probably
due to a coupling effect. As a clue, which may lead to their source,
the following observations may be of interest.
The frequency data in column 21 may be divided into three groups.
In the first group, the secondary frequencies f"F are related to the
length vibrations fb by multiple numbers as follows:
Crystal No.............. 16 15 14
Ratio fb/f"p............. 3.02 2.98 6.90
These secondary frequencies may represent subharmonics of.the longi-
tudinal vibrations. In the second group, the secondary frequencies f"r
are related to the longitudinal ones by repeating nonharmonic ratios:
Crystal No.............. 8 10 11 17
Ratio fb/f" ............. 5.73 5.72 2.44 2.45
The third group of bars is characterized by a ratio between the longi-
tudinal fb and the normal flexural vibration f' which is close to a
multiple number:
Crystal No.............. 14 6 8 10
Ratio fb/f' ............. 6.1 4.07 5.15 5.98
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
ILLINOIS ENGINEERING EXPERIMENT STATION
III. FLEXURAL AND LONGITUDINAL VIBRATIONS
OF X-CUT QUARTZ CRYSTALS
17. Material and Results of Measurements.-All quartz bars inves-
tigated thus far and enumerated in Table 1 were Y-cut. Their length
dimension b was made parallel to the direction of the electrical axis
OX, (see Fig. 2). In order to extend the investigation to bars in which
the length direction formed an angle of 30 deg. with the electrical axis,
so as to coincide with a mechanical axis, five X-cut crystals (Nos. 26
to 30) were prepared from the quartz stock used for the Y-cut samples.
Because the moduli of elasticity in the direction of the two respec-
tive axes are known to differ only slightly from each other, the flexural
frequencies of the two types of crystals having equal dimensions were
not expected to differ from each other by more than 0.5 per cent.
Measurements were made, however, for the purpose of ascertaining
that the design formulae (23 to 35) may be applied for X-cut crystals.
The results of measurements of the frequency of flexural and longi-
tudinal vibrations are given in Table 2. The notations and numbering
of the columns are similar to those in Table 1. Column 18 shows that
the deviation 8 of measured values of frequency in flexural vibrations
from calculated values was less than 1 per cent. Hence it may be con-
cluded that the expression (23), with the auxiliary design formulae,
may be applied to X-cut quartz bars also.
18. Frequency Factors.-From the data in Table 2 the average
frequency factors for X-cut crystals were found to be as follows:
5.52 X 105
For flexural vibrations A/B =
V 1 + 5.22(a/b)2
For longitudinal length vibrations A6 = 2.7 X 105
For longitudinal thickness vibrations At = 2.86 X 105
Only the last factor differs from the corresponding factor for the
Y-cut bars, the first two are identical for both X-cut and Y-cut quartz
bars.
IV. CONCLUSION
19. Summary and Conclusions.-
(1) A series of 30 piezoelectric quartz bars and plates were speci-
ally cut and investigated with a view of obtaining data for the design
of low-frequency stabilized oscillators and filters.
FLEXURAL VIBRATIONS OF PIEZOELECTRIC QUARTZ BARS
(2) The fundamental natural frequency of flexurally-vibrating
Y-cut quartz bars and plates within a range of from 3.5 to 306 kc. was
determined, along with frequencies of longitudinal vibrations.
(3) For the measurement of frequency of flexurally-vibrating
quartz resonators below 15 kc. per second a sensitive bridge method
was developed.
(4) On the basis of experimental data obtained on Y-cut quartz
bars and plates, and in conjunction with theoretical considerations, a
formula (Equation 23) was derived for the calculation of the funda-
mental natural frequency of flexurally-vibrating bars and plates.
(5) The frequency formula was found to hold within 1 per cent
for quartz bars whose ratio a/b did not exceed 0.4, and within 1.5 per
cent for bars and plates whose ratio a/b did not exceed 0.75.
(6) Measurements on X-cut quartz bars were made which show
that the frequency formula may be used also for the calculation of
dimensions of this type of resonator.
(7) Further formulae (Equations 25 to 35) were derived for cal-
culating the dimensions and frequency of flexurally-vibrating quartz
bars for particular cases which occur in the course of designing and
grinding of crystals.
(8) Average frequency factors applicable for flexural and longi-
tudinal vibrations were determined for the quartz rock used in this
investigation (Sections 14 and 18).