Predicting string behavior

[Richard Moody]

Imagine the string clamped on one end with vice grips and dangling
vertically.  That would be one case of no tension. 
If the string is held horizontaly clamped at each end with no tension, the
degree of slackness should still affect freq. I imagine. Imagine the vice
grips are 20mm apart but the wire is 420 mm long.   What freq?  Now the
vice grips are 42mm apart, what freq now?  Imagine the vice grips 420 mm
apart, what is the tension? what is the freq?  Suppose the vice grips are
421 mm apart, what is the tension, and the freq?  and so on for the vice
grips 430mm apart what are the measurements there?. 

`How does this model compare with your model? Regarding your model, how
did you come up with your numbers for no tension? 

Please for the "metric illiterate" could you include American lengths and
wire guages? 

you wrote
	> If the fundamental is A1 and the second partial is A2,
> Cents deviation =A2-2*A1
> It that correct?

That is the diff of freqs. Cents has to be figured from the diff of freqs.
The theoretical diff between A1 and a second partial A2 is one octave, or
1200 cents. The fundamental is usually expressed in cps, (cycles per
second) The 2nd partial is usually measured at close to 2*A1 but in cps
not cents. If the freq of A1 is known and the freq of A2 is measured then
the diff in cents can be figured by.this formula from Niklas Eliasson

>>>>>	 1200*(log(F/f)/log(2)) =  cents 
> 
> F = first frequency
> f =  second frequency.
> 
> Next  how would one solve for  f  or F  for any  value of cents     
 
Just put it

F=f*2^(cents/1200)

or

f=F*2^(-cents/1200)

(note the minus-sign in the second formula!)

It all goes back to the fact that an octave is divided into 12 equal
parts and each of those parts (semitones) is divided into 100 parts.

Conclusion - an octave is divided into 1200 equal parts, and each
part is called a "cent".


Be in touch if you get into any problems!!! I am always glad to help!


Niklas E,
pianotech, Linkoping
> From: Niklas Eliasson <e96nikel@isy.liu.se><<<<<<<

Doug wrote
	> 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.

Yes I am intersted esp in what is considered a good or poor bridge
termination. 


Richard Moody 

----------
> From: Doug Richards <Doug.Richards@quantum.com>
> To: 'pianotech' <pianotech@ptg.org>
> Subject: Predicting string behavior
> Date: Friday, August 28, 1998 4:00 AM
> 
> 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).