Chapter 9

The Vibrations of Drumheads and Soundboards

Glockenspiel Bar

Experiment by striking the bar at various places Where is the fundamental the loudest? Where is it weakest?

We should expect the fundamental to be basically one-half wavelength with antinodes at the ends, since they are free.

Position of Nodes

Touching a node makes no change in the sound (Why?)Notice that the felt support for the bar is at a node.To get at other modes of vibration, recall that each mode adds a node. Modes 2 and 3 might look like…

Mode 2

Confirmed by tapping predicted nodes and antinodes

Add one-half wavelength to fundamental

Presumed Mode 3

Touching two fingers on either side of center should kill modes 1 and 2, leaving mode 3.Tapping at the center should produce 3096 Hz of mode 3.Mode 3 is not excited by these simple tests

Mode Three

There are two perpendicular nodal linesMode 3 is a twisting mode

Finding Modes

Motion on one side of a node is opposite from the other side of the node. Tapping at the node does nothing to stimulate that mode.Tapping near antinode gives maximum stimulation of that mode.

Mode Shapes

Length Modes

Width Modes

Mode 1

Mode 2

Estimating Mode Frequency

By direct measurement upper octave bars are 70% the length of lower octave.What frequency do we get by cutting a bar in two? Note that 0.7 * 0.7 0.5 Each 0.7 is an octave so we have two octaves Glockenspiel has length 2.5 times width

Width frequencies are about six times length frequencies.

Mode Families

Length and width modes are different families.Mixed modes can exist.

(2,1) mode

Wooden Plates

Wooden plates have a grain or preferred direction. Stiffness is much lower against the

grain than with it. Wood can flex better at the grain boundaries.

Frequencies of the width modes are decreased compared to the uniform plate.

f S

Mode 3 revisited

For wooden plates where l is about 3 times w (or uniform plates where l = w) this is the lowest frequency.Important in violin making

Classes of Plates Free Edge – antinodes always appear at the edges Glockenspiel Cymbals, gongs, bells Tuning forks

Clamped Edge – ends are merging into nodes rather slowly Soundboards of pianos and harpsichords

Hinged Edge – ends come more rapidly into nodes Violin family (purfling)

First Four Modes of Guitar

First Four Modes of Rectangular Wood

Clamped vs. Hinged Edge

More bending at the edges of a clamped plate produces higher frequency modes than the hinged edge.Frequency differences between clamped and hinged are less important for the higher modes.

1-D Example

Purfling

Thin hardwood inlaid strips in violins give the edge a hinge-like quality.If the violin hasn’t been played in awhile, the purfling gets stiff. Loud playing of chromatic scales can

loosen it up again

Violin Parts

Membrane Thickness

Variations in the thickness of a membrane can alter the natural frequencies it produces.

Drumhead is to a Circular PlateAs

Flexible String is to a Bar

Analogy

Flexible string and drumhead don’t have much stiffness They need to be

stretched at the edges to produce tension.

Drumhead under tension acts like a plate with hinged edges.

Normal Modes of a Vibrating Membrane

Normal Modes

Frequency Comparison

Mode Drumhead Plate1 1.000 1.0002 1.593 2.0923 2.135 3.4274 2.295 3.9105 2.653 6.067

Notice that the mode frequencies are much farther apart for the plate

Frequency Comparison

0

1

2

3

4

5

6

7

0 2 4 6

Mode Number

Fre

qu

ency

(n

orm

aliz

ed)

Drumhead

Plate

Drumhead/Plate Comparisons

Above mode 5 the plate has nearly a constant interval between mode frequencies. (Straight line graph)Interval for the drumhead grows smaller at higher modes. Graph turns almost horizontal.

Tuning a Plate – a Model

Adding mass will decrease the frequencyAdd small amounts of mass to the plate Positioned near a node has no effect on that

mode Positioned near an antinode has maximum

effect on that mode Rayleigh found…

fractional change in frequency 2 X the fractional change in mass

Also several lumps of wax should have the same effect as the sum of their individual effects.

f = constant* S M

Rayleigh’s Condition in Symbols

M

M

f

f

2

f = change in frequency

M = change in mass

Example

A plate of iron has a diameter of 10 cm and a thickness of 0.025 cm and is clamped around the rim.Mode one has a frequency of 250 HzThe volume is 2V = d / 4t = 1.96 cm3

Using the density of iron(7.658 grams/ cm3) the mass is 15 g.

Adding Mass

Place .5 grams of wax at the center Antinode for mode 1

By Rayleigh m

m

f

f

2

Fractional change of frequency =-2(.5gm/15 gm) = -0.067

Mode 1 has its frequency changed by 250*.067 = -16.7 Hz and is now 233.3 Hz (just above A3#). Note decrease

Mode Frequency Differences

Mode 2 has a frequency of 2.092 times mode one frequency or 523 Hz (C5) Frequency difference before wax was

applied 523 – 250 = 273 Hz

The wax does not affect mode two since the center of the plate is a mode two node New frequency difference after wax is

523 – 233.33 = 289.7 Hz

Moving the Added Mass

Move wax to midway between center and edge Here mode 2 has an antinode Now mode 2 has its frequency decreased by

6.7 % to 488 Hz

Mode 1 also affected at this position of the wax, but only 1% since this is not an antinode (makes frequency 247.5 Hz) Frequency difference is 240.5 Hz Much less of a change by moving the mass.

Fixing the Frequency Difference

Trial and error could be used to find a position where the frequency difference between the first two modes is one octave (here, 250 Hz).

Effect of Thinning the Plate

Changing the plate thickness affects the plate stiffness Since f (S/M)½, thinning the plate

decreases the mass (raising the frequency) M means f

Thinning the plate also lowers the stiffness (lowering the frequency) S means f

Trade-off

Rayleigh finds that the change caused by stiffness in one direction is about three times the effect caused by mass in the other direction.f/f 4 * ()

Building a Sounding PlateThe craftsman finds the places where he can add wax to get the frequencies he wants.Wax adds mass without affecting stiffness. The change in stiffness dominates in the other

direction

Cut away wood at the positions of the wax. The amount of wood mass removed is half the

mass of the wax.

Note: these ideas don’t apply to membranes (drumheads). Adding mass to those raises the frequency.

Building a Sounding Plate

Kettledrums

Calfskin or Plastic Membrane

Hemispherical Copper Shell

Mode Ratios (as before)

Mode Drumhead Plate1 1.000 1.0002 1.593 2.0923 2.135 3.4274 2.295 3.9105 2.653 6.067

Why aren’t these ratios whole numbers?

Deviations from Whole Integer Mode Ratios

The shell itself is a trapped volume of air Normal play mode is to strike about half way from center to edge, thus enhancing mode 2 But even striking near the center

gives very little mode 1 The reason is the vent hole that

tends to damp mode 1

Mode Component Ratios Component Ratio Component Ratio

P 1.000 U 2.494Q 1.502 V 2.800R 1.742 W 2.852S 2.000 X 2.979T 2.245 Y 3.462

Careful tuning can get S exactly twice P, and X is not far off

Also note that Q and X form an harmonic sequence fX 2fQ

Other Sequences

Recall that your ear will assign the fundamental, even if it is not there, provided that an harmonic sequence is present.For a fundamental of C2 = 65.4 Hz fP = 130.8 Hz = 2*C2 fQ = 196.72 Hz = 3*C2 fs = 261.6 Hz = 4*C2 fU = 326.22 Hz = 4.99 C2 fX = 389.65 Hz = 5.96 C2