Modos de Un Timbal Con Tension en El Borde No Uniforme

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Proceedings of Meetings on Acoustics

Volume 5, 2008 http://asa.aip.org

156th Meeting

Acoustical Society of AmericaMiami, Florida

10 - 14 November 2008

Session 3aMU: Musical Acoustics

3aMU5. Drum tuning: an experimental analysis of membrane modes under non-uniform

tension

Randy Worland*

*Corresponding authors address: Physics, University of Puget Sound, 1500 N. Warner, Tacoma, WA 98416-1031,

[email protected]

Results of an experimental study of normal mode vibrations in single-headed musical drums under non-uniform tension

are presented. Although uniform tension is often assumed in theoretical treatments, in practice the musical drum only

approximates this condition, even after careful tuning by the drummer. This study investigates the behavior of normal

mode shapes and frequencies under non-uniform tension, as they relate to the tuning process. In particular, the role of the

(1,1) mode is described. Experimental results include electronic speckle pattern interferometry (ESPI) images of modal

shapes along with the associated frequencies. A finite element model is used for comparison with the experimental results.

Published by the Acoustical Society of America through the American Institute of Physics

Randy Worland

2009 Acoustical Society of America [DOI: 10.1121/1.3138888]

Received 16 Mar 2009; published 2 May 2009

Proceedings of Meetings on Acoustics, Vol. 5, 035001 (2009) Page 1


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1. Introduction

While many discussions of drumhead physics assume uniform tension, drummers know

that this is at best an approximation. Tuning real drums is a challenging process thatusually starts with the need to tune a single head to itself (what timpanists refer to as

clearing the head). In this paper we discuss the way in which drummers tune a single

head and relate this procedure to the physics of the normal modes on a circular membrane.

2. Ideal membrane theory

Theoretical results for the ideal circular membrane under uniform tension are well

known.1-3

The normal mode shapes are given by:

mn (r, ) ~Jm (kmnr) {cos (m), sin (m)}, (1)

where is the displacement as a function of radial and azimuthal coordinates, r and.The Jm are Bessel functions of order m. These modes exhibit nodal lines consisting ofdiameters and concentric circles, based on the circular symmetry of the drumhead. Figure

1 shows the lowest twelve normal mode shapes, along with the (m,n) notation indicating

the number of nodal diameters and circles respectively. Any normal mode containing atleast one nodal diameter (m > 0) is two-fold degenerate

4as seen in the sine and cosine

solutions of Equation 1. This degeneracy may be lifted by a variety of perturbations,including non-uniformities in the applied tension. The role of this degeneracy in drum

tuning is the focus of this paper. For simplicity, only the tuning of a single head to itself is

considered, without regard to the overall pitch of the drum.

Figure 1. The first 12 modes of an ideal membrane, with the (m,n) designation and thefrequency relative to the (0,1) fundamental also shown.

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2. Drum tuning practice

Drums such as tom toms and snare drums contain membranes (typically Mylar) that arestretched over the bearing edges of a cylindrical drum shell by a mechanical arrangement

consisting of a counter hoop and threaded tension rods screwed into symmetrically placedtuning lugs (see Figure 2). Tightening a tension rod stretches the membrane and generally

raises the pitch of the drum. Drums of the type considered here usually have six, eight, or

ten tuning lugs.

Figure 2. A 10 diameter tom tom with six tuning lugs. The membrane is lightly damped

at the center while tapping near each tension rod. Tension adjustments are made using adrum key, seen here on the tension rod nearest to the drum stick.

Drumheads are generally tuned by tapping near each tuning lug and adjusting the tensionrods with a drum key in an attempt to make each tap sound the same. Drummers often

place the drum on a flat surface and/or mute the drumhead lightly at the center. Bothactions will minimize the response of the fundamental mode (0,1). The flat surface also

damps the lower head, as these drums usually contain two heads (batter and resonant).

The tuning lugs are adjusted in an attempt to minimize perceived frequency differences

and beats. The tuning process is iterative, as adjustment at a particular lug changes thetension at all perimeter points.

3. The role of the (1,1) mode

The fundamental mode (0,1) is generally the loudest mode produced when playing a drum,particularly when striking near the center. However, this mode is not degenerate and thus

does not split into two frequencies in response to a perturbation. Adjusting a given tension

rod will raise or lower the fundamental frequency, but will not provide the drummer withinformation regarding the uniformity of the tension.

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The (1,1) mode is the lowest degenerate mode and typically the most prominent of themodes that can be split. Under ideal circumstances, the (1,1) mode can occur with a single

nodal diameter in any orientation, (linear combinations of two orthogonal states).

Depending on the excitation (e.g. location where the drumhead is struck), the nodaldiameter may be re-oriented, but the frequency of the mode will not change.

If the drumhead is slightly out of tune with itself, two orthogonal (1,1) modes will result,

with slightly different frequencies as shown in Figure 3. The figure shows arbitrarily

chosen vertical and horizontal nodal lines, with the former taken to have the higherfrequency (f+), and the latter the lower frequency (f-). Tapping near the perimeter of the

drumhead excites the modes consisting of nodal diameters preferentially, and particularlythe prominent (1,1) mode. We see in Fig. 3(b) that tapping near the top or bottom lugs will

preferentially excitef-, while tapping near the left or right will excite mainlyf+. Tapping

along the diagonals will cause both modes to be heard, producing audible beats. Thus, by

tapping around the perimeter, the drummer hears higher and lower pitches and beats atvarious locations if the drum is out of tune.

Listening to the (1,1) ModeListening to the (1,1) Mode

X

X

(1,1) f+ (1,1) f-

When drum is slightly out of tune, tapping on thenodal line of one mode excites primarily the other(1,1) mode.

Tapping between nodal lines causes bothfrequencies to be heard.

a) Slightly perturbed modes: b) Tapping (X) near each lug:

X

Hear f- Hear f+

Hear f+ and f-

Figure 3. In (a) a slightly non-uniform tension has lifted the degeneracy of the (1,1) mode,

producing two modes of slightly different frequency: f+ andf-. In this example theorthogonal nodal lines are arbitrarily assumed to be vertical and horizontal. Part (b) shows

that we may excite either or both of these modes depending on where the drumhead isstruck.

Correcting the tuning is not simply a matter of tightening the lugs where the pitch is low.

Due to the two-dimensional nature of the membrane, tightening one lug will cause an

increase in tension at points other than just the region directly across the correspondingdiameter. A further complication is the high degree of friction between the membrane and

the bearing edge, which makes the tuning process undetermined and not reproducible with

the same sequence of tension rod adjustments. When we tighten one lug we are pulling the

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membrane slightly over the bearing edge at that point, but across the drum there may be acombination of slippage over the bearing edge and stretching of the membrane. This com-

bination is not always the same tightening a rod turn and then loosening the same rod

turn will not in general return the membrane to the original state of applied tension(radial force per unit length) around the perimeter or stress throughout the surface.

4. ESPI experimental data

To study the effect of various states of applied tension, electronic speckle pattern interfer-ometry (ESPI) was used to image the mode shapes on the drumhead and identify the corre-

sponding frequencies. The ESPI system is based on a design due to Thomas Moore.5,6

Drums driven acoustically, using a speaker connected to a function generator. Sinusoidal

frequencies were scanned manually to locate the modal images.

It is easier to detune a drum than to tune it. In the lab, it proved best to start with a tuneddrum and then systematically tighten or loosen a single lug in an attempt to see a gradualdeviation in the mode shapes and frequencies. The hope is that by understanding the

detuning process we can infer something about the physics of what happens when a

drummer tunes a drum.

Drums were tuned primarily by ear, as described above in Section 2. A spectrum analyzerwas also used to confirm that the tuning procedure did tend to minimize the frequency

splitting of the degenerate modes, particularly the (1,1) mode. Mode shapes (as seen with

ESPI) also appeared more symmetrical as the tuning was improved. The symmetry of thecircular nodal line of the (0,2) mode is a particularly good indicator of tension

uniformity.7,8 Perfect tuning, which we assume would be characterized by symmetricmode shapes and the absence of frequency splitting, was never achieved. Any variation in

the membrane material, roundness of the bearing edge, etc. would likely make this idealimpossible to attain.

As an example of an approximately tuned drum, Figure 4 shows several of the lower

modes of a snare drum batter head9

(with its bottom head damped). The drum was tuned

well enough that only a single (1,1) mode was found, but the (2,1), (3,1), and (4,1) modesall appear in pairs, indicating that their degeneracy has been lifted. In each of these pairs

the two sets of nodal lines are rotated by /2m with respect to each other (i.e. the nodaldiameters of one mode lie halfway between those of the other mode). Some frequency

splitting (ranging in magnitude from 4-8 Hz) is seen in each case. Variations in the modeshapes from the ideal symmetric shapes of Figure 1 are also seen. We will show that these

deviations from ideal symmetric mode shapes need not be directly related to the frequency

splitting.

Figure 5(a) shows a tom tom, initially tuned such that for the (1,1) modes, f = 2 Hz.From this initial state, ESPI observations were made as one lug (see arrow in Figure 5(b))

was loosened in quarter turn increments of the drum key. Figure 5(b) shows the firstchange - the modes have rotated slightly counterclockwise such that one mode aligns with

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the perturbation while the orthogonal mode rotates to an angle perpendicular to the pertur-

bation axis. The frequency splitting has also increased to f= 5 Hz.

(0,1) 258 Hz (1,1) 278 Hz (2,1) 399 Hz (2,1) 407 Hz

f= 8 Hz

(3,1) 513 Hz (3,1) 521 Hz (4,1) 627 Hz (4,1) 631 Hz

f= 8 Hz f= 4 Hz

Figure 4. Electronic speckle pattern images showing normal modes of a reasonably welltuned 14 snare drum with 10 lugs. The nodal regions appear as white areas in these

images. The (2,1), (3,1), and (4,1) modes appear in pairs with relatively small frequencydifferences.

Figure 5(c) shows the situation after 1 turns of the drum key. At this stage a new feature

is seen. The mode that is not pointed at the perturbing lug is now curved around the region

of lower tension. In addition, the frequency splitting has increased to f= 16 Hz. The

curvature of the lower frequency mode indicates that the lower tension is not carriedsymmetrically across the drumhead diameter from the perturbed lug. (If the result ofloosening this lug was simply to create a slow axis and a fast axis at right angles to each

other there would be no tendency for a nodal line to curve one way or the other.)

To investigate this idea further, the opposing lug (across the diameter) was loosened turn

(see Figure 5(d)). Note that this has the effect of straightening out the previously curved

nodal line. This also dramatically increases the frequency splitting, to f= 28 Hz. It is

apparent that loosening the opposing lug has made the decrease in tension more symmetricacross the drum (i.e. a straighter nodal line) and has also enhanced the difference between

the fast and slow axes on the membrane. We conclude that varying the tension in themanner described in Figure 5 can split the (1,1) modes and cause one of them to curve

but the two effects are not directly correlated with one another.

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(a) (b)

213 Hz 215 Hz 207 Hz 212 Hz

f= 2 H z f= 5 Hz

(c) (d)

175 Hz 191 Hz 156 Hz 184 Hz

f= 16 Hz f= 28 Hz

Figure 5. The (1,1) modes of a 12 tom tom with six tuning lugs. Arrows indicate tuning

lugs that were loosened: (a) approximately tuned, (b) loosened turn, (c) loosened 1turns, (d) opposite lug loosened turn.

As a final illustration, Figure 6 shows a frame drum that was tuned (not shown) and then

had a single lug tightened, rather than loosened, by nearly two full turns. We see the

alignment of the (1,1) modes relative to the perturbation and some frequency splitting (f=

9 Hz). Furthermore, the mode perpendicular to the axis of the perturbing lug (this is thehigher frequency mode when tightening occurs) curves away from this lug. Thus, the

curvature is around the region of relatively lower tension, as it was in Figure 5c. Again,we see that the degree of curvature (quite large in this example) and the amount of

frequency splitting (rather modest here) do not appear to be directly related. In Section 5we use a finite element model to explore these results.

(1,1) 225 Hz f= 9 Hz (1,1) 234 Hz

Figure 6. ESPI images showing the (1,1) modes of a 10 frame drum with eight lugs.

Arrows indicate the tuning lug that was tightened approximately two full turns.

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5. Finite element model

Commercial finite element analysis software10

was used to model the circular drumhead

under non-uniform tension. In this model the membrane was broken up into six symmetricpie-shaped wedges (see Figure 7). The radial force at the perimeter can be specified sepa-

rately for each wedge, subject to the constraint that the net applied force must equal zero.The software then performs a pre-stress modal analysis, with normal mode shapes and

frequencies as the output. No attempt was made to reproduce the numerical frequencies of

the drums used in the lab; the goal of the model was to make qualitative comparisons ofthe mode shapes and frequency splittings in response to particular tension perturbations.

Figure 7. A screen shot of the finite element model geometry, showing the circular

membrane divided into six regions. The green arrows represent the radial forces appliedalong the six perimeter arcs.

In Figure 8 we consider the case of initially uniform tension to which some additional

radial force has been added to sector 1. This force must be compensated for to keep thedrum in equilibrium. One way to do this is by providing an equal and opposite force

across the diameter, at sector 4. Another acceptable solution is to have three equal forcesseparated by 120

o(sectors 1, 3, and 5). The general case in which sectors 3, 4, and 5 all

contribute can be broken down into a sum of the two-fold and the three-fold symmetriccases.

Figure 8. Finite element model geometry depicting the two-fold (left) and three-fold

(right) perturbation symmetries. Large black arrows indicate sectors with added tension.

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Figure 9 shows the models output for the two-fold tension perturbation (equal and

opposite forces added across the diameter). Note the alignment of the two perpendicularmodes relative to the perturbation and the resulting frequency splitting. The two-fold

perturbation does not lead to any curvature of the nodal lines. (The symmetry of thisperturbation does not allow for the curvature seen in the experimental data of Figures 5(c)

and 6.)

(1,1) 298 Hz f= 9 Hz (1,1) 307 Hz

Figure 9. Two-fold perturbation symmetry: tension is added in equal amounts across a

diameter, as indicated by the two arrows. The orthogonal (1,1) modes are aligned relativeto the perturbation and show a frequency splitting. The mode shape is not changed.

Figure 10 shows the three-fold symmetric perturbation equal forces added at sectors 1, 3,

and 5. Here we see curvature of the nodal lines but no frequency splitting. The three-fold

perturbation does not create orthogonal fast and slow axes that would couple with the (1,1)modes to produce two different frequencies. By comparing Figures 9 and 10 we see that

the frequency splitting and curvature of the (1,1) modes come from different sources, andcan be created independently. This is consistent with our previous experimental observa-

tion (e.g. Figure 6) that large frequency differences and large amounts of nodal line

curvature do not necessarily appear together.

Figure 11 shows the more general case; a combination of two-fold and three-foldsymmetric perturbations. These results look qualitatively similar to what is typically seen

in the lab: alignment with respect to the perturbation, curvature around the region of lower

tension, and frequency splitting. It appears that this case simulates qualitatively what onesees if a single lug is adjusted on an initially tuned drum. (Compare with experimental

results shown in Figures 5(c) and 6).

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(1,1) 343 Hz f= 0 Hz (1,1) 343 Hz

Figure 10. Three-fold perturbation symmetry: tension is added in equal amounts at 120o

intervals as shown by the black arrows. The (1,1) nodal lines become curved, but there is

no frequency splitting.

(1,1) 344 Hz f= 8 Hz (1,1) 352 Hz

Figure 11. Two-fold and three-fold symmetric tension perturbations are combined, as

shown by the black arrows. Alignment, curvature and frequency splitting of the (1,1)

modes are seen in the models output.

6. Symmetry

These results for the (1,1) mode are representative of a more fundamental principle: the

frequency splitting of a particular degenerate mode depends on both the symmetry of the

mode and the symmetry of the perturbation. Group theory has been used to investigatethese symmetry relations in bells and circular plates,

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fied for the circular drum head subject to perturbations in the applied tension.

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Of particular interest here, the group theory analysis predicts that the two-fold case is the

only symmetric tension perturbation that can split the (1,1) mode. However, higher modes

(i.e. modes with largerm values) can be split by this and other, higher order, perturbations.One consequence of this is that the drum can be tuned such that the (1,1) mode does not

split, and still have higher degenerate modes that are split. This effect is seen in thetuned snare drum data of Figure 4, which shows frequency splitting for the (2,1), (3,1)

and (4,1) modes, although not for the (1,1) mode. A more detailed analysis of the mode

splitting selection rule from group theory will be the subject of a future paper.

7. Conclusion

Drum tuning practice attempts to minimize small frequency differences and beats associ-

ated primarily with the degenerate (1,1) membrane mode. In terms of the applied tension,

this mode splits only in response to a two-fold symmetric perturbation, which creates per-pendicular fast and slow axes on the membrane. Other symmetric tension perturbations(e.g. the three-fold case) may change the (1,1) mode shape, but these shape changes have

no audible effect during the tuning process. Eliminating the two-fold tension perturbation

is a challenging and iterative process for the drummer due to both the two-dimensionalnature of the drum head and the large degree of friction between the head and bearing

edge. Finally, even with the two-fold perturbation eliminated, higher degenerate modescan be split by higher order perturbations in the applied tension.

References

[1] Lord Rayleigh [J.W. Strutt], The Theory of Sound, (Dover, New York, 1945), Vol.1, Chap. 9.

[2] T.D. Rossing, and N.H. Fletcher, Principles of Vibration and Sound, 2nd ed.,

(Springer, New York, 2004), Chap. 3.

[3] D.R. Raichel, The Science and Application of Acoustics, (AIP Press, New York,

2000), Chap. 6.

[4] P.M. Morse and K. Uno, Theoretical Acoustics, (Princeton Univ. Press, Princeton,N.J., 1968), p. 212.

[5] T.R. Moore, A simple design for an electronic speckle pattern interferometer,Am. J. Phys. 73, 1380-1384 (2004).

[6] T.R. Moore, Interferometric studies of a piano soundboard, Am. J. Phys. 119,

1783-1793 (2006).

[7] T.D. Rossing, I. Bork, H. Zhao, and D.O. Fystrom, Acoustics of snare drums, J.

Acoust. Soc. Am. 92 (1), 84-94 (1992). (See p. 85)

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[8] R.S. Christian, R.E. Davis, A. Tubis, C.A. Anderson, R.I. Mills, and T.D. Rossing,Effects of air loading on timpani membrane vibrations, J. Acoust. Soc. Am. 76

(5), 1336-1345 (1984). (See p. 1341)

[9] All heads used in this study were single-ply Coated Ambassadors, manufactured

by Remo.

[10] NEi Fusion, from Noran Engineering (www.NEiNastran.com).

[11] J.P. Murphy, R. Perrin, and T. Charnley, Doublet splitting in the circular plate, J.

Sound Vib. 95, 389-395 (1984).

[12] R. Perrin and T. Charnley, Group theory and the bell, J. Sound Vib. 31, 411-418

(1973).

Randy Worland


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Modos de Un Timbal Con Tension en El Borde No Uniforme