P. Schmidt
4/11/02
Introduction
Chimes are musical instruments consisting of suspended bars, tubes, or rods which are caused to sound by striking. The fundamental frequency varies as the inverse square of the length. Overtones are non-harmonic; i.e, not integer multiples of the fundamental. This note provides details of some of the math that describes how chimes function.
Fundamental frequency vs. length of chimes
The fundamental frequency of bar free at both ends is given by
f = ( 1.133 p / l2 ) sqrt( Q K2 / r )
where
l = length of bar (cm)
r = density (gm/cm3)
Q = Young's Modulus of Elasticity (dynes/cm2)
K = radius of gyration (cm)
For a solid circular cross-section
K = 0.5 r where r is the radius (cm)
For a hollow circular cross-section
K = 0.5 sqrt( ro2 + ri2 ) where ro and ri are the outer and inner radii (cm)
Reference:
Music, Physics and EngineeringHarry F. Olson
Dover Publications, New York, 1967 (second edition)
With the speed of sound in the medium
v = sqrt( Q / r )
the fundamental frequency can also be written as
f = 1.133 p K v / l2
The characteristics of some representative materials (from the "Handbook of Chemistry and Physics") are
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(gm/cm3) |
(dynes/cm2) |
(cm/sec) |
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As an example, let's calculate the length of 0.5" aluminum rod required for each note of the chromatic scale, with equal temperament, over the octave from C6 to C7. For reference, C4 is middle C, and A4 is concert A = 440 Hz.
Under equal temperament, the frequencies of adjacent semitones in the chromatic scale are in the ratio of the twelfth root of two, or 1.0595 .
Taking the first equation in this note, and solving for length in terms of frequency
l = sqrt[ ( 1.133 p / f ) sqrt( Q K2 / r ) ]
The radius of gyration for a solid rod of 0.5" diameter (0.25" radius) is
K = 0.5 r = 0.5 x 0.25 (in) x 2.54 (cm/in) = 0.3175 (cm)
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(cm) |
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The tones (fundamental and overtones)
and their node positions (also from "Music, Physics and Engineering") are
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For the least effect on the sound of the fundamental, the chime should be hung at the node point of the fundamental frequency, or 22.42% of the length from one end.
Having had the opportunity recently to look at the metallophone instruments (gangsa) of a Balinese gamelan ensemble, I found that the bronze bars were in fact mounted at two points approximately 25% from each end!
Math Description: Overtones, Nodes, etc.
These relationships and numbers can
be derived from an analysis of transverse vibration of an elastic bar;
this is described by a fourth-order differential equation. The general
solution is a linear combination of sine, cosine, hyperbolic sine and hyperbolic
cosine. The boundary conditions at both ends of a free bar are that shear
(third derivative) and bending moment (second derivative) are zero.
The equation for the normal modes of vibration then becomes
cosh(b l) cos(b l) - 1 = 0 (shown on the two plots below)
which has solutions
b l = 3.0112 (p/2) , 5 (p/2) , 7 (p/2) , .....
for the Fundamental, 1st
Overtone, 2nd Overtone, ....
cos(b l) = 1 / cosh(b l)
because, as cosh(b
l) increases exponentially to very large values, cos(b
l) rapidly approaches zero.
The actual vibration frequencies
f vary directly with (b
l)2 according to
f = [ (b
l)2 / 2p ] sqrt( Q K2 / r
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As an example, the ratio of frequencies
of the first overtone to the fundamental is then (5 / 3.0112)2
= 2.76 , as in the table above. Note that this also means that the overtones
are non-harmonic; i.e., not integer multiples of the fundamental frequency.
The extent to which the modes of
vibration are non-sinusoidal is measured by the relative size of the hyperbolic
component compared to the sinusoidal, as summarized in the table below:
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relative to cos() |
relative to sin() |
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Ratio |
= -sin(b l/2)/sinh(b l/2) |
= sin(b l/2)/sinh(b l/2) |
Note that, from symmetry considerations,
the vibration modes are alternating even and odd functions: cos(b
x) and cosh(b x)
for even modes, and sin(b
x) and sinh(b x)
for odd modes, respectively. Here x is the coordinate along the bar with
the origin at its center.
For example, the spatial displacement of the fundamental, and of the first two overtones, is represented by
y = Ao(t) [ cos(b x) - 0.133 cosh(b x) ]
y = A1(t) [ sin(b x) - 0.0278 sinh(b x) ]
y = A2(t) [ cos(b x) + 0.0058 cosh(b x) ]
where Ai(t) is the amplitude,
a function of time.
The actual wave forms of the fundamental
and the first two overtones, showing the vibration nodes, are shown in
the plot below. The nodes occur where the wave forms cross the horizontal
axis (i.e., Amplitude = 0). The maximum amplitude has been arbitrarily
set to 2 for each tone.
Some informative Internet web sites
on musical chimes are:
http://www.geocities.com/cllsj/
http://www.metalwebnews.com/howto/wchime/wchime.html
http://home.fuse.net/engineering/Chimes.htm
http://faculty.millikin.edu/~jaskill.nsm.faculty.mu/webpages.html
(Site no longer active; Prof. Askill has other problems)
http://ccrma-www.stanford.edu/CCRMA/Courses/150/percussion.html