Predicting string behavior

[Doug Richards]

Hi all,

I couldn't sleep tonight so thought I would put in my two cents for the year
(I'm a long time lurker).

With all the discussion on string inharmonicity, I finally decided to finish
a little study using the modeling tools I use to predict structural
properties of disk drives (my day job).

I decided to start very simple.
Steel string, 420mm long and 0.8mm diameter.
The string ends are constrained like a cantilever beam (it would be
something like somehow welding both ends into huge steel blocks)

Case #1 is with no tension on the string.
I included this case to show that the upper bending modes of the string do
not relate to an even number multiple of the fundamental.
                           FREQUENCY           CENTS DEVIATION
FUNDAMENTAL      14.12378                         (0)
2nd PARTIAL           45.77155                      (17.52)
3rd PARTIAL            95.50265                      (67.26)
4th PARTIAL          163.32415                     (135.08)
5th PARTIAL          249.24185                     (220.99)


Case #2 applies enough tension to "tune" the string to very near 440 Hz.
                           FREQUENCY           CENTS DEVIATION
FUNDAMENTAL       440.00518                         (0)
2nd PARTIAL          880.55096                       (0.54)
3rd PARTIAL         1322.17697                       (2.16)
4th PARTIAL         1765.42088                       (5.40)
5th PARTIAL         2210.81749                      (10.79)
If you curve fit the CENTS DEVIATION, the best fit is a 3rd order polynomial
equation.
y = 0.0896x^3 + 0.0023x^2 - 0.0924x


One thing I wanted to verify is that the Cents deviation is calculated
correctly.  
If the fundamental is A1 and the second partial is A2,
Cents deviation =A2-2*A1
It that correct?

I have been able to make avi files of the animated mode shapes, but they are
~5Mb for each frequency.
If anyone is interested in what they look like .......

If anyone is following this, next time I could simulate bending the wire
over a bridge and look at how good and poor bridge termination affects
string frequencies.

Comments?????????


C. Douglas Richards                                                 Quantum
Corporation
Mechanical Engineer
408.894.4592
Finite Element Specialist
doug.richards@qntm.com
DPSG Mechanical Design Group	                500 McCarthy Blvd, Milpitas
CA

Oh yeah, also PTG member in the Santa Clara Chapter (in case it matters).