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Doublet splitting in the circular plate
Journal
of Sound and
Vibration (1984) 95(3), 389-395
DOUBLET
SPLITTING IN THE CIRCULAR
J. P. MURPHY,
R. PERRIN
PLATE
AND T. CHARNLEY
Department of Physics, Loughborough University of Technology, Loughborough LE 11 3 TU, England (Received 5 May 1983, and in revised form 12 October 1983)
It is pointed out that a previously derived “selection rule” concerning the splitting of degenerate doublets in bells when subjected to perturbations with certain symmetry properties, is equally valid for other systems with symmetry group C,, or Dmh. Experimental confirmation of the predictions of the rule is presented for the case of the centrally clamped flat circular plate with free edges.
1. INTRODUCTION It was shown by Perk and Charnley [l] using group theoretical techniques that, because they have symmetry group C,, ideal bells have normal modes which must have nodal patterns consisting of n circles parallel to the rim and 2m equally spaced meridians where n, m > 0. Further have nodal between
each mode with m > 0 belongs
patterns
which are identical
those of the other. Modes
for them. The absolute
locations
equivalent.
In practice
imposing
conditions,
or imperfections
according
to Rayleigh’s
perturbation
Principle
[2].
able. The effects of such a perturbation those for different
but usually doublets;
cause slight differences Although
the frequencies
properties
are concerned,
the “system”
conditions
plate rigidly clamped
supply
between
patterns:
group
that, so far as symmetry
not only the physical Thus,
(3) to
members.
of the bell, they are equally
one remembers
includes
a
locations
connection
for any system with symmetry
to which it is subjected.
by
do so
and unpredict-
in the nodal
within the context
(or indeed the higher group Dmh) provided
metallurgy
the imperfections
of the doublet
minor adjustments,
all azimuths locations
(1) to fix the absolute
(2) to cause slight distortions
to a few obvious
also the boundary
case
and/or
bell, but complicated
are as follows:
the above results were deduced
valid, subject C,,
in the geometry
In the latter
because
fixes the absolute
in such a way that there is no obvious
between
whose members
of one being midway
are indeterminate
either the clapper
which should be small, for a well-cast
of the meridians,
doublet
with m = 0 are singlets since no such partner can exist
of these meridians
are dynamically initial
to a degenerate
apart from the meridians
system itself but
for example,
as its centre but free at the edges will have normal
a flat circular modes with all
the above properties apart from the obvious modification that the n nodal circles lie in the plane of the plate and the 2m nodal meridians become m nodal diameters. Other systems with essentially the same properties, provided suitable boundary imposed, include the right circular cylinder, the cone and the circular constant shape of cross-section).
conditions ring (with
are any
It was shown by Perrin [3], again using group theoretical techniques, that if a perturbation is applied to an ideal bell which breaks some but not all of its non-trivial symmetries, then some,
but not necessarily
all, of the doublets
are split. Those
which are split have
the absolute locations of their nodal meridians fixed: otherwise these remain indeterminate. If the perturbation consists of some basic element (e.g., a cherub for a church bell!) 389 @ 1984 Academic Press Inc. (London) Limited 0022--460X/84/150389 +07 $03.00/O
390
J. P. MURPHY.
R.
PERRIN
AND
T. CHARNLEY
which is repeated r times around the bell, always at the same distance from the rim, so that the symmetry group of the perturbation is C:, or perhaps C,,, then splitting occurs only for those doublets whose m values satisfy the selection rule mlr=$p,
wherep=
1,2,3 ,....
(1)
It should be noted that this statement of the rule is simpler than that originally given by Perrin, although its content remains unchanged, and that for a given value of m there is a maximum value of r for which splitting can occur, namely 2m. In a real bell there are usually imperfection perturbations, as mentioned above, producing distortions and splitting over and above those due to the ornaments but, unless something has gone wrong during the casting process, they should be relatively small. It should be emphasized that the derivation of equation (1) does not depend upon any assumption about the perturbing elements being small. Since the derivation of the equation (1) depends only upon the symmetry properties of the original system (including boundary conditions) and of the perturbation, the same result should hold for any system with symmetry group C,, (or Dm,) subjected to perturbations with symmetry groups C, or C,,. The only system for which detailed tests of the selection rule have been made is the thin circular ring of rectangular cross-section subjected to perturbations of identical small cylinders placed at the vertices of regular polygons inscribed in the ring. Charnley and Perrin [4] showed that the rule is satisfied by this system, to within experimental error limits, by both axial and inextensional radial modes with m = 3, 4 and 5 using r = 10 (such modes occur only for m 2 2). The study was limited to a small range 1,2,..., of m values because of the severe practical difficulty of working with thin rings in this context. In the present paper results are presented of a similar experimental study in which a centrally clamped flat uniform circular plate with free edges was used. The system is much easier to handle experimentally than the ring, so it has been possible to investigate a much wider range of m values as well as to check that the second quantum number n has no influence on the validity of the rule. 2. THE UNPERTURBED
PLATE
A flat circular steel plate of diameter (25 1-Ok 0.1) mm, uniform thickness (1 .OO* 0.01) mm and mass (385.5 f 0.5) g was selected for study. It was clamped at its centre by means of a screw going into a massive steel base. The plate was kept horizontal and was driven by a signal from an oscillator (Briiel and Kjaer (B & K) Type 1022) input via a magnetic transducer (B 8z K Type MM0002). Pick-up was usually with a second such transducer feeding a low-noise amplifier whose output was fed to a twin-beam oscilloscope and to a logarithmic recorder (B & K Type 2305) which was also used to drive the sweep facility on the oscillator when this was required. Sometimes it was necessary to replace the pick-up transducer by two accelerometers (B & K Type 4344) each feeding via a pre-amplifier (B & K Type 2625) to a channel of the oscilloscope. One of these was fixed to the plate either near the drive point or diametrically opposite to it, and was used as a phase reference. The other was held against the plate by hand. Approximate frequencies of all the plate’s normal modes up to about 9 kHz were found by making frequency sweeps, with a variety of drive and pick-up positions. Each mode was then investigated in detail in order to ascertain its frequency more accurately, identify its n and m values and pair it up with its doublet partner, if it had m > 0. The experimental techniques required to do this depended upon the closeness to degeneracy of the pair and the amplitude of vibration which could be produced. In some cases a large enough
DOUBLET
SPLI’ITING
IN THE
CIRCULAR
PLATE
391
amplitude of vibration could be obtained to produce a Chladni pattern by sprinkling a fine sand on the vibrating plate. It was then simply a matter of counting diameters and circles in the sand pattern to obtain m and n except in a few cases where “accidental” degeneracy occurred [5]: a little ingenuity was then required to unravel the component modes. In the other cases m and n could still be found by use of the two accelerometers mentioned above: each time the roving accelerometer crossed a nodal diameter or circle an extra phase change of 7~ appeared between its output and that of the fixed reference accelerometer. This technique was at least as reliable as the use of Chladni patterns, but was tedious and relatively time-consuming. In cases of true degeneracy between doublet members the location of the driving transducer in the azimuthal sense was irrelevant, an antinodal diameter always automatically locating itself at the transducer’s position. The radial location, however, had to be selected with care to be at or close to an antinodal circle in order to maximize plate response. If the driving transducer was moved azimuthally then the entire nodal pattern rotated with it as expected. The location of the pick-up transducer was not vital but it was desirable to keep it at or close to another circle/diameter antinode. In cases where the doublet was split, due to imperfections in the plate, it was necessary to locate the driving transducer at a circle/diameter antinode for one component of the doublet by a process of trial and error. The drive frequency was then adjusted to give maximum response and the value read off the timer-counter. The driver was then relocated at a nodal diameter/antinodal circle for this mode, and hence at an antinodal diameter/circle for its partner, and the oscillator retuned in order to bring the partner to resonance and hence obtain its frequency accurately. In a few cases the splitting was too small to allow the two components to be tuned-in separately. The best that could be done then was to identify m and n by looking at one mode and then to measure the frequency difference between the pair by examining beats on decay.
3. THE PERTURBATlONS In order to produce perturbations with the required symmetry properties a number of identical small permanent magnets were attached to the plate at appropriate points. The magnets were in the form of rectangular parallelepipeds with two sides equal to 6 mm and the third to 20 mm. Their average mass was (3*40* 0.01) g and they were attached by one of their square faces by exploiting their ferromagnetic attraction to the steel plate. They were placed at the vertices of a series of regular polygons inscribed in circles
(a) Figure 1. Relative orientations of nodal diameters and perturbation is permitted. (a) n =O, m = 3, r = 3; (b) n = 0, m = 6, r = 4.
(b) polygons
for some cases where splitting
392
J. P. MURPHY.
,
c
0
/ -.,
I )
K. PERRIN
AND
I-. CHARNLEY
I /
1
/
2
I
3
/
4
5 6
I
7
1700
‘IrO: 1720
0
(”
* / 1
x\,
I
x
I
* 1 2
* 1 3
j__________:
/
4
::
1
1 5
I
1
6
*
x
I
7
Figure 2. Frequencies of selected doublet members as functions of the number of perturbing loads r. The straight line fits are made to those cases where the selection rule permits splitting. These are indicated by an asterisk immediately above the r axis. (a) n = 0, m = 2; (b) n = 0, m = 3; CC) n =O, m = 4; Cd) n=O, m=S: (e) n=O, m=6: (f) n=l, m=3; (g) n=2, m=3.
0
l\
1780
1740
-
I800
P F
394
J. P. MURPHY,
R. PERRIN
AND
T. CHARNLEY
concentric with the plate whose radii were selected so that they lay at or close to an antinodal circle for the pair of degenerate modes under study. This was to maximize the influence of the perturbation. It was found that when the amplitude of vibration became large the magnets tended to “chatter” on the plate and to move about. In order to avoid this, double-sided Sellotape was used to hold them in position. In those cases where the unperturbed doublet was truly degenerate the orientation of the polygon could be arbitrary because (a) if it split the doublet then the perturbed nodal diameters would align themselves in a way determined by the polygon’s orientation, and (b) if it did not split them then the location of the diameters remained indeterminate. However, in those cases where the “unperturbed” doublet was slightly split by the imperfections there was a problem because, in order to allow for this, the polygon had to be orientated in such a way that its position relative to the perturbed nodal diameters was the same as it would have been had the imperfections been absent. In practice this simply meant making sure that at least one of the vertices of the polygon was aligned with one of the nodal diameters of one of the “unperturbed” split pair. In order to understand the last statement, and gain some insight into the physics underlying the selection rule, it is helpful to consider some m and r combinations for which splitting is permitted, take some others for which it is not, and “play” with the relative orientations of the rth order polygon and the m equally spaced diameters. What one finds is that in cases where splitting is permitted it is always possible to line up the polygon so that every vertex lies on a nodal diameter (and hence on an antinodal diameter of the partner mode). Examples are shown in Figures l(a) and (b). In cases where splitting is forbidden by the rule, such an alignment is an impossibility.
4. EXPERIMENTAL
PROCEDURE
First, an interesting doublet whose values of m and n were already known was selected for study. If it was already split, and the absolute locations of the nodal diameters of its members therefore fixed, then a single perturbing magnet was placed on a nodal diameter of one component, and hence at an antinodal diameter for the other, with its radial location selected to be at or close to an antinodal circle, the location of this being identical for both members. If the doublet was initially degenerate then the azimuthal location of the magnet, although not its radial one, was unimportant as the nodal patterns of the components would, according to Rayleigh’s Principle, automatically line up their nodal diameters as required. The frequencies of the split doublet members (they should always be split with a single perturbing load) were then obtained by using the same techniques as described in section 3 for the unperturbed plate with split doublets. After having obtained the one-perturbing load frequencies of the doublet, i.e., r = 1, further loads were added at the vertices of polygons of increasing order going up to an r value of at least 2m. In each case one vertex of the polygon had to be located with the same restrictions as applied to the single load case: the other vertices would then automatically have been correctly aligned as explained in section 3. This procedure was carried out for n = 0 with m = 2,3,4,5 and 6 in order to check whether or not the selection rule was obeyed by this system in the situation directly analogous to the thin ring case described by Charnley and Perrin. To study the influence of the second quantum number n, the procedure was also carried out for m = 3 with n = 0, 1 and 2. The results are shown graphically in Figures 2(a)-(g). It is not easy to estimate possible errors on the measurement of the frequencies of the split doublet members. This is because it was not the frequency measurement itself which
DOUBLET
SPLITTING IN THE CIRCULAR
PLATE
395
limited the overall experimental accuracy but rather the precision with which the various perturbing magnets could be located at the required vertices. Repeating sample sequences of the measurements indicates that an overall error estimate of perhaps 1% is realistic. 5. DISCUSSION In the light of the remarks at the end of section 3 it is clear that in those situations where the selection rule permits splitting each perturbing load is dynamically indistinguishable from its fellows. In these situations, therefore, the effect of r such loads will be identical to that of one lo&d with r times the mass located at any one of the polygon’s vertices. For this reason it is to be expected that first-order perturbation theory could be applied here in an exactly analogous fashion to the way in which it was applied to the thin ring by Charnley and Perrin [6]. This means that, provided the masses of the magnets are small compared with that of the plate, the frequencies of the components of a particular degenerate pair will fall linearly with r, for those r values permitting splitting, and that the gradient of the line for the component with a diameter antinode located at the first magnet will be very large compared with that of its partner. It can be seen from Figures 2(a)-(g) that in every case investigated this linear variation does indeed occur. In those cases where the selection rule forbids splitting the observed frequency difference is always either zero or very close to it, at least after any “unperturbed” splitting has been subtracted off. The latter adjustment is important only in the cases shown in Figures 2(a) and (b). Such splitting as then remains is always very small compared with what the interpolated first-order perturbation lines would require, and is zero to within experimental error limits if our estimate of 1% on the individual frequency values is accepted. To within experimental error limits, therefore, the selection rule was satisfied for each of the cases selected for study. 6. CONCLUSIONS The derivation by Perrin of the selection rule for the splitting of degenerate doublets in bells when subjected to perturbations with symmetry groups C, or C, is equally valid for any system with symmetry group C,, or Dmh when subject to such a perturbation appropriately applied. The validity of the rule has been demonstrated experimentally in the case of a Bat circular plate with free edges clamped at its centre. It is, therefore, unlikely that the rule would fail for other systems with the requisite symmetries. REFERENCES 1. R. PERRIN and T. CHARNLEY 1973Journal ofSound and Vibration 31,41 l-418. Group theory and the bell. 2. LORD RAYLEIGH 1894 Theory ofSound (two volumes). New York: Dover Publications, second edition, 1945 re-issue. See Volume 1, Section 198. 3. R. PERRIN 1977 Journal of Sound and Vibration 52, 307-313. A group theoretical approach to warble in ornamented bells. and R. PERRIN 1971 Acusticu 25, 240-246. Characteristic frequencies of a 4. T. CHARNLEY symmetrically loaded ring. 5. R. PERRIN, T. CHARNLEY and J. DEPONT 1983 Journal ofSound and Vibration 90, 29-49. Normal modes of the modem English church bell. 6. T. CHARNLEY and R. PERRIN 1973 Acustica 28, 139-146. Perturbation studies with a thin circular ring.
of Sound and
Vibration (1984) 95(3), 389-395
DOUBLET
SPLITTING IN THE CIRCULAR
J. P. MURPHY,
R. PERRIN
PLATE
AND T. CHARNLEY
Department of Physics, Loughborough University of Technology, Loughborough LE 11 3 TU, England (Received 5 May 1983, and in revised form 12 October 1983)
It is pointed out that a previously derived “selection rule” concerning the splitting of degenerate doublets in bells when subjected to perturbations with certain symmetry properties, is equally valid for other systems with symmetry group C,, or Dmh. Experimental confirmation of the predictions of the rule is presented for the case of the centrally clamped flat circular plate with free edges.
1. INTRODUCTION It was shown by Perk and Charnley [l] using group theoretical techniques that, because they have symmetry group C,, ideal bells have normal modes which must have nodal patterns consisting of n circles parallel to the rim and 2m equally spaced meridians where n, m > 0. Further have nodal between
each mode with m > 0 belongs
patterns
which are identical
those of the other. Modes
for them. The absolute
locations
equivalent.
In practice
imposing
conditions,
or imperfections
according
to Rayleigh’s
perturbation
Principle
[2].
able. The effects of such a perturbation those for different
but usually doublets;
cause slight differences Although
the frequencies
properties
are concerned,
the “system”
conditions
plate rigidly clamped
supply
between
patterns:
group
that, so far as symmetry
not only the physical Thus,
(3) to
members.
of the bell, they are equally
one remembers
includes
a
locations
connection
for any system with symmetry
to which it is subjected.
by
do so
and unpredict-
in the nodal
within the context
(or indeed the higher group Dmh) provided
metallurgy
the imperfections
of the doublet
minor adjustments,
all azimuths locations
(1) to fix the absolute
(2) to cause slight distortions
to a few obvious
also the boundary
case
and/or
bell, but complicated
are as follows:
the above results were deduced
valid, subject C,,
in the geometry
In the latter
because
fixes the absolute
in such a way that there is no obvious
between
whose members
of one being midway
are indeterminate
either the clapper
which should be small, for a well-cast
of the meridians,
doublet
with m = 0 are singlets since no such partner can exist
of these meridians
are dynamically initial
to a degenerate
apart from the meridians
system itself but
for example,
as its centre but free at the edges will have normal
a flat circular modes with all
the above properties apart from the obvious modification that the n nodal circles lie in the plane of the plate and the 2m nodal meridians become m nodal diameters. Other systems with essentially the same properties, provided suitable boundary imposed, include the right circular cylinder, the cone and the circular constant shape of cross-section).
conditions ring (with
are any
It was shown by Perrin [3], again using group theoretical techniques, that if a perturbation is applied to an ideal bell which breaks some but not all of its non-trivial symmetries, then some,
but not necessarily
all, of the doublets
are split. Those
which are split have
the absolute locations of their nodal meridians fixed: otherwise these remain indeterminate. If the perturbation consists of some basic element (e.g., a cherub for a church bell!) 389 @ 1984 Academic Press Inc. (London) Limited 0022--460X/84/150389 +07 $03.00/O
390
J. P. MURPHY.
R.
PERRIN
AND
T. CHARNLEY
which is repeated r times around the bell, always at the same distance from the rim, so that the symmetry group of the perturbation is C:, or perhaps C,,, then splitting occurs only for those doublets whose m values satisfy the selection rule mlr=$p,
wherep=
1,2,3 ,....
(1)
It should be noted that this statement of the rule is simpler than that originally given by Perrin, although its content remains unchanged, and that for a given value of m there is a maximum value of r for which splitting can occur, namely 2m. In a real bell there are usually imperfection perturbations, as mentioned above, producing distortions and splitting over and above those due to the ornaments but, unless something has gone wrong during the casting process, they should be relatively small. It should be emphasized that the derivation of equation (1) does not depend upon any assumption about the perturbing elements being small. Since the derivation of the equation (1) depends only upon the symmetry properties of the original system (including boundary conditions) and of the perturbation, the same result should hold for any system with symmetry group C,, (or Dm,) subjected to perturbations with symmetry groups C, or C,,. The only system for which detailed tests of the selection rule have been made is the thin circular ring of rectangular cross-section subjected to perturbations of identical small cylinders placed at the vertices of regular polygons inscribed in the ring. Charnley and Perrin [4] showed that the rule is satisfied by this system, to within experimental error limits, by both axial and inextensional radial modes with m = 3, 4 and 5 using r = 10 (such modes occur only for m 2 2). The study was limited to a small range 1,2,..., of m values because of the severe practical difficulty of working with thin rings in this context. In the present paper results are presented of a similar experimental study in which a centrally clamped flat uniform circular plate with free edges was used. The system is much easier to handle experimentally than the ring, so it has been possible to investigate a much wider range of m values as well as to check that the second quantum number n has no influence on the validity of the rule. 2. THE UNPERTURBED
PLATE
A flat circular steel plate of diameter (25 1-Ok 0.1) mm, uniform thickness (1 .OO* 0.01) mm and mass (385.5 f 0.5) g was selected for study. It was clamped at its centre by means of a screw going into a massive steel base. The plate was kept horizontal and was driven by a signal from an oscillator (Briiel and Kjaer (B & K) Type 1022) input via a magnetic transducer (B 8z K Type MM0002). Pick-up was usually with a second such transducer feeding a low-noise amplifier whose output was fed to a twin-beam oscilloscope and to a logarithmic recorder (B & K Type 2305) which was also used to drive the sweep facility on the oscillator when this was required. Sometimes it was necessary to replace the pick-up transducer by two accelerometers (B & K Type 4344) each feeding via a pre-amplifier (B & K Type 2625) to a channel of the oscilloscope. One of these was fixed to the plate either near the drive point or diametrically opposite to it, and was used as a phase reference. The other was held against the plate by hand. Approximate frequencies of all the plate’s normal modes up to about 9 kHz were found by making frequency sweeps, with a variety of drive and pick-up positions. Each mode was then investigated in detail in order to ascertain its frequency more accurately, identify its n and m values and pair it up with its doublet partner, if it had m > 0. The experimental techniques required to do this depended upon the closeness to degeneracy of the pair and the amplitude of vibration which could be produced. In some cases a large enough
DOUBLET
SPLI’ITING
IN THE
CIRCULAR
PLATE
391
amplitude of vibration could be obtained to produce a Chladni pattern by sprinkling a fine sand on the vibrating plate. It was then simply a matter of counting diameters and circles in the sand pattern to obtain m and n except in a few cases where “accidental” degeneracy occurred [5]: a little ingenuity was then required to unravel the component modes. In the other cases m and n could still be found by use of the two accelerometers mentioned above: each time the roving accelerometer crossed a nodal diameter or circle an extra phase change of 7~ appeared between its output and that of the fixed reference accelerometer. This technique was at least as reliable as the use of Chladni patterns, but was tedious and relatively time-consuming. In cases of true degeneracy between doublet members the location of the driving transducer in the azimuthal sense was irrelevant, an antinodal diameter always automatically locating itself at the transducer’s position. The radial location, however, had to be selected with care to be at or close to an antinodal circle in order to maximize plate response. If the driving transducer was moved azimuthally then the entire nodal pattern rotated with it as expected. The location of the pick-up transducer was not vital but it was desirable to keep it at or close to another circle/diameter antinode. In cases where the doublet was split, due to imperfections in the plate, it was necessary to locate the driving transducer at a circle/diameter antinode for one component of the doublet by a process of trial and error. The drive frequency was then adjusted to give maximum response and the value read off the timer-counter. The driver was then relocated at a nodal diameter/antinodal circle for this mode, and hence at an antinodal diameter/circle for its partner, and the oscillator retuned in order to bring the partner to resonance and hence obtain its frequency accurately. In a few cases the splitting was too small to allow the two components to be tuned-in separately. The best that could be done then was to identify m and n by looking at one mode and then to measure the frequency difference between the pair by examining beats on decay.
3. THE PERTURBATlONS In order to produce perturbations with the required symmetry properties a number of identical small permanent magnets were attached to the plate at appropriate points. The magnets were in the form of rectangular parallelepipeds with two sides equal to 6 mm and the third to 20 mm. Their average mass was (3*40* 0.01) g and they were attached by one of their square faces by exploiting their ferromagnetic attraction to the steel plate. They were placed at the vertices of a series of regular polygons inscribed in circles
(a) Figure 1. Relative orientations of nodal diameters and perturbation is permitted. (a) n =O, m = 3, r = 3; (b) n = 0, m = 6, r = 4.
(b) polygons
for some cases where splitting
392
J. P. MURPHY.
,
c
0
/ -.,
I )
K. PERRIN
AND
I-. CHARNLEY
I /
1
/
2
I
3
/
4
5 6
I
7
1700
‘IrO: 1720
0
(”
* / 1
x\,
I
x
I
* 1 2
* 1 3
j__________:
/
4
::
1
1 5
I
1
6
*
x
I
7
Figure 2. Frequencies of selected doublet members as functions of the number of perturbing loads r. The straight line fits are made to those cases where the selection rule permits splitting. These are indicated by an asterisk immediately above the r axis. (a) n = 0, m = 2; (b) n = 0, m = 3; CC) n =O, m = 4; Cd) n=O, m=S: (e) n=O, m=6: (f) n=l, m=3; (g) n=2, m=3.
0
l\
1780
1740
-
I800
P F
394
J. P. MURPHY,
R. PERRIN
AND
T. CHARNLEY
concentric with the plate whose radii were selected so that they lay at or close to an antinodal circle for the pair of degenerate modes under study. This was to maximize the influence of the perturbation. It was found that when the amplitude of vibration became large the magnets tended to “chatter” on the plate and to move about. In order to avoid this, double-sided Sellotape was used to hold them in position. In those cases where the unperturbed doublet was truly degenerate the orientation of the polygon could be arbitrary because (a) if it split the doublet then the perturbed nodal diameters would align themselves in a way determined by the polygon’s orientation, and (b) if it did not split them then the location of the diameters remained indeterminate. However, in those cases where the “unperturbed” doublet was slightly split by the imperfections there was a problem because, in order to allow for this, the polygon had to be orientated in such a way that its position relative to the perturbed nodal diameters was the same as it would have been had the imperfections been absent. In practice this simply meant making sure that at least one of the vertices of the polygon was aligned with one of the nodal diameters of one of the “unperturbed” split pair. In order to understand the last statement, and gain some insight into the physics underlying the selection rule, it is helpful to consider some m and r combinations for which splitting is permitted, take some others for which it is not, and “play” with the relative orientations of the rth order polygon and the m equally spaced diameters. What one finds is that in cases where splitting is permitted it is always possible to line up the polygon so that every vertex lies on a nodal diameter (and hence on an antinodal diameter of the partner mode). Examples are shown in Figures l(a) and (b). In cases where splitting is forbidden by the rule, such an alignment is an impossibility.
4. EXPERIMENTAL
PROCEDURE
First, an interesting doublet whose values of m and n were already known was selected for study. If it was already split, and the absolute locations of the nodal diameters of its members therefore fixed, then a single perturbing magnet was placed on a nodal diameter of one component, and hence at an antinodal diameter for the other, with its radial location selected to be at or close to an antinodal circle, the location of this being identical for both members. If the doublet was initially degenerate then the azimuthal location of the magnet, although not its radial one, was unimportant as the nodal patterns of the components would, according to Rayleigh’s Principle, automatically line up their nodal diameters as required. The frequencies of the split doublet members (they should always be split with a single perturbing load) were then obtained by using the same techniques as described in section 3 for the unperturbed plate with split doublets. After having obtained the one-perturbing load frequencies of the doublet, i.e., r = 1, further loads were added at the vertices of polygons of increasing order going up to an r value of at least 2m. In each case one vertex of the polygon had to be located with the same restrictions as applied to the single load case: the other vertices would then automatically have been correctly aligned as explained in section 3. This procedure was carried out for n = 0 with m = 2,3,4,5 and 6 in order to check whether or not the selection rule was obeyed by this system in the situation directly analogous to the thin ring case described by Charnley and Perrin. To study the influence of the second quantum number n, the procedure was also carried out for m = 3 with n = 0, 1 and 2. The results are shown graphically in Figures 2(a)-(g). It is not easy to estimate possible errors on the measurement of the frequencies of the split doublet members. This is because it was not the frequency measurement itself which
DOUBLET
SPLITTING IN THE CIRCULAR
PLATE
395
limited the overall experimental accuracy but rather the precision with which the various perturbing magnets could be located at the required vertices. Repeating sample sequences of the measurements indicates that an overall error estimate of perhaps 1% is realistic. 5. DISCUSSION In the light of the remarks at the end of section 3 it is clear that in those situations where the selection rule permits splitting each perturbing load is dynamically indistinguishable from its fellows. In these situations, therefore, the effect of r such loads will be identical to that of one lo&d with r times the mass located at any one of the polygon’s vertices. For this reason it is to be expected that first-order perturbation theory could be applied here in an exactly analogous fashion to the way in which it was applied to the thin ring by Charnley and Perrin [6]. This means that, provided the masses of the magnets are small compared with that of the plate, the frequencies of the components of a particular degenerate pair will fall linearly with r, for those r values permitting splitting, and that the gradient of the line for the component with a diameter antinode located at the first magnet will be very large compared with that of its partner. It can be seen from Figures 2(a)-(g) that in every case investigated this linear variation does indeed occur. In those cases where the selection rule forbids splitting the observed frequency difference is always either zero or very close to it, at least after any “unperturbed” splitting has been subtracted off. The latter adjustment is important only in the cases shown in Figures 2(a) and (b). Such splitting as then remains is always very small compared with what the interpolated first-order perturbation lines would require, and is zero to within experimental error limits if our estimate of 1% on the individual frequency values is accepted. To within experimental error limits, therefore, the selection rule was satisfied for each of the cases selected for study. 6. CONCLUSIONS The derivation by Perrin of the selection rule for the splitting of degenerate doublets in bells when subjected to perturbations with symmetry groups C, or C, is equally valid for any system with symmetry group C,, or Dmh when subject to such a perturbation appropriately applied. The validity of the rule has been demonstrated experimentally in the case of a Bat circular plate with free edges clamped at its centre. It is, therefore, unlikely that the rule would fail for other systems with the requisite symmetries. REFERENCES 1. R. PERRIN and T. CHARNLEY 1973Journal ofSound and Vibration 31,41 l-418. Group theory and the bell. 2. LORD RAYLEIGH 1894 Theory ofSound (two volumes). New York: Dover Publications, second edition, 1945 re-issue. See Volume 1, Section 198. 3. R. PERRIN 1977 Journal of Sound and Vibration 52, 307-313. A group theoretical approach to warble in ornamented bells. and R. PERRIN 1971 Acusticu 25, 240-246. Characteristic frequencies of a 4. T. CHARNLEY symmetrically loaded ring. 5. R. PERRIN, T. CHARNLEY and J. DEPONT 1983 Journal ofSound and Vibration 90, 29-49. Normal modes of the modem English church bell. 6. T. CHARNLEY and R. PERRIN 1973 Acustica 28, 139-146. Perturbation studies with a thin circular ring.