Frequency, Musical Instruments, Expansion
While solving for the frequency and pitch of a simple
harmonic motion equation involving musical instruments, is it
possible to account for the way temperature will cause the metal of
the instrument to expand or contract?
Yes, Kevin, it is somewhat analyzable,
especially if metals are the dominant structural substances.
Metal can be modeled as having a simple temperature coefficient of length,
and no hysteresis (history dependence),
and spring constant independent of temperature (I think).
So it is your friend in your attempts to model a musical instrument.
Woods and plastics are a bit mushier and less predictable.
If you have metal strings on a guitar with a totally wooden neck,
the wood will expand and contract more than the metal strings,
and will change with humidity too,
and even it's history (whether it is going from longer to shorter or vice-versa)
will affect it's present length.
That said, I think the thick wooden neck of a guitar
has more total stiffness than all the thin metal strings taken together,
and the neck may also contain a metal stiffener.
So the strings will be forced to take on whatever new length the neck changes to.
Next, the change in their tension will depend on that length change and
the elastic coefficient of the metal strings,
also called spring constant, their force-change per unit length-change.
And finally, we get to where the pitch depends of the tension and built-in mass
of the strings.
Beware of screw-joints or any joint that can slip in response to large slow forces.
This is possible, but the effects to the speed of sound through the air
within the instrument can be more important. However, the main
contribution of size relates to thermodynamic expansion:
dL=L_0(1+dT). Multiply the change of temperature by the constant
called the linear expansion coefficient (i.e. alpha). Each material has
its own value. Multiply the CHANGE of temperature (usually in Celsius
or in Kelvins) by this constant. For brass, the expansion coefficient
is 19x10^-6 inverse Kelvins. If you increase the temperature of a
length of brass by one Kelvin (or one Celsius degree), the length will
multiply by 1.0000019. If the brass rod starts out at a length of ONE
meter, then it will lengthen by 0.0000019 meters for every Celsius
degree temperature increase. If the rod is not one meter long to begin
with, multiply the length by 1.0000019.
Dr. Ken Mellendorf
Illinois Central College
The condition of expansion and contraction as a result of vibration can
and for the most part be minute in nature, measured by sensitive instruments.
The equations you want to keep in mind are from classical mechanics,.....also
review Brownian motion, the concept of friction and temperature as well.
Professor Przekop, Physicist
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Update: June 2012