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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