Our first conclusion is that our observed frequencies corresponded to our theoretical predictions of k-values.  We did notice that our calculations yielded different values for the ratio r/(E*I).  As the length increased, this ratio decreased.  Upon further researching, we re-discovered that I was not a constant, but was a function of the length of the wire.  As a result, the ratio r/(E*I) should decrease as the length of the rod increases.

    When we first started investigating the solutions to the fourth-order differential equation, we noticed that at high k-values, all the k's were solutions.  A very dull-looking graph showing this can be seen below.

We also noticed during our experiment, that after seeing a couple of overtones, the rod appeared not to vibrate anymore.  At the time, we thought that the limited amplitude on the function generator-amplifier was responsible for this lack of visible oscillation.  However, we found another cause upon further investigation.  It turns out that Young's modulus is only valid for a limited portion of the elasticity curve.  In the graph below of Load versus Elongation, the line from O to A is the region in which Young's modulus holds.  In fact, the slope of the line for each material is Young's modulus.  After the system goes beyond point A, which is known as the elastic limit, Young's modulus no longer holds and the material doesn't fully recover (much like a spring that has been stretched too far).  

As a result, the graphical solutions only hold at lower values of k (and hence frequency)  where the load is still not that great.  Looking back on our results, we believe that our data points were obtained while still in this range.  However, we did notice that when the wires were removed from the apparatus, they were slightly more bent than they were when they entered the apparatus.

    As for the linearity of the nodal position as a function of the length of the wire, we were not able to find any theoretical information to back up our results, and there was not sufficient time to go about discovering the theory.  In the future, more work should be done in regards to this aspect.

    The wire did not behave in accordance with Chladni's law.  The frequency was observed to be proportional to almost the cube of the number of nodes, as opposed to the square.

     On the other hand, the hoop of wire behaved very much in accordance with Chladni's law.   A more detailed explanation can be found in the hoop of wire page.

 

 

References:

  1. Chalmers, Bruce.  Physical Metallurgy.  John Wiley and Sons, Inc.  New York, NY. 1959. p.82, 215.
  2. Armstrong, R.L. and J.D. King.  Mechanics, Wave, and Thermal Physics.  Prentice Hall, Inc.  Englewood Cliffs, NJ. 1970. p.296, 451-453.

See also references in theory section

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