Data and Conclusions


We studies four types of plates:  a thick circular plate, a thick square plate, a thin square plate, and a thin rectangular plate. Each plate had unbounded edges.

 

Thick Circular Plate With Unbounded Edges:

Cladni hypothesized that modal frequency for thin circular plates with bounded edges would follow the relation  f ~ [Graphics:chlad.txtgr3.gif] .   So, if we graph    f / fo on a log-log scale versus [Graphics:chlad.txtgr4.gif], the graph should have a slope of two.  After extensive research, Mary Waller, posited that the frequency is a function of (m+bn), where b increases from 2 to 5, as m increases. (Rossing, Thomas D. "Chladni's Law For Vibrating Plates." Am. J. Phys. 50(3), March 1982, 273.)

View Picturesof Circular Plates

Fundamental Frequency = .261 kHz

[Graphics:chlad.txtgr2.gif][Graphics:chlad.txtgr6.gif]

Slope of our best-fit line is 1.20569.  

 

Now plot Log (f/fo) versus k*Log(m+2n).

[Graphics:chlad.txtgr2.gif][Graphics:chlad.txtgr12.gif]

[Graphics:chlad.txtgr2.gif][Graphics:chlad.txtgr13.gif]

For the circular plate with unbounded edges, the values of m (diametric modes) were very difficult to determine, partly because the plate was unbounded and partly because the plate was not thin enough. So, we treated m and n the same and graphed the mode frequencies in order of occurrence as we gradually increased the frequency.

This data does not clearly reflect Chladni's law.

To get a more clear picture of how this plate matches up with Chladni's law, we picked out only the values where we knew "m" to be zero. Thus, our next graph will show only the modes with concentric circular modes as a function of (m+2n).

Plotting just the modes where m = 0 (only circular nodes appear).

Fundamental Frequency = 0.261 kHz

m = number of nodal diameters ; n = number of concentric nodal circles

[Graphics:chlad.txtgr15.gif]
[Graphics:chlad.txtgr16.gif]
[Graphics:chlad.txtgr19.gif]
[Graphics:chlad.txtgr20.gif]

Slope of our best-fit line is 0.971168.

Now plot Log (f/fo) versus k Log(m+2n).

[Graphics:chlad.txtgr24.gif]

Note: Slope of this best fit line is about 1, which means data does not reflect Chladni's law.  This data indicates that f/fo ~ (m+b*n)^k when k is around 1, not 2. This is not surprising, as Chladni's law applied to plates with fixed boundaries, unlike our experiment.

 

Thick Square Plate With Unbounded Edges:

For thick square plate, nodal lines were very difficult to observe as this situation diverges greatly from the ideal membrane theory. So, we plot modes in order of ocurence as we increased the frequency.

View Picture of Thick Square Plate

Fundamental Frequency = .186 kHz.

[Graphics:chlad.txtgr27.gif]
[Graphics:chlad.txtgr30.gif]
[Graphics:chlad.txtgr31.gif]

Slope of our best-fit line is 1.14191.

Now plot Log (f/fo) versus k* Log(m+n)

[Graphics:chlad.txtgr35.gif]

This matches theoretical prediction, but does not prove it. Since the plate is square, increasing m should have the same effect as increasing n. Thus, we can number the modes ordinally and still get a straight line. The graph shows that there is some power relation between the mode numbers and the frequency.

 

Thin Square Plate With Unbounded Edges

[Graphics:chlad.txtgr37.gif]

Modes were tough to see clearly due to abberations in the plate. Thus, we have little data.

Fundamental Frequency = .970 kHz.

[Graphics:chlad.txtgr39.gif]
[Graphics:chlad.txtgr40.gif]

Now plot Log (f/o) versus k* Log([Graphics:chlad.txtgr41.gif]+[Graphics:chlad.txtgr42.gif])

[Graphics:chlad.txtgr45.gif]
[Graphics:chlad.txtgr46.gif]

Slope of our best-fit line is 1.04386.

Now plot Log (f/fo) versus k* Log(m+n)

[Graphics:chlad.txtgr49.gif]

Comparing this with the thick plate, we see that the thick plate diverges from the ideal more than does the this plate. This data closely fits the straight line and shows that the frequency does vary as a function of ([Graphics:chlad.txtgr50.gif]+[Graphics:chlad.txtgr51.gif])^power. The slope of our line does not confirm that the power is 1/2.

Rectangular Plate With Unbounded Edges

View Picture of Rectangular Plate

Fundamental Frequency = .433 kHz

Width of plate was 1.5 * Length.

[Graphics:chlad.txtgr55.gif]
[Graphics:chlad.txtgr56.gif]
[Graphics:chlad.txtgr58.gif]
[Graphics:chlad.txtgr59.gif]

Slope of our best-fit line is 1.03424.

Now plot Log(f/fo) versus k*Log( ([Graphics:chlad.txtgr61.gif]+([Graphics:chlad.txtgr62.gif])

[Graphics:chlad.txtgr64.gif]

In truth, the data fits a straight line very well. Slope is equal to 1.03424. This shows us that the mode frequency does indeed vary as follows: f ~[Graphics:chlad.txtgr65.gif], but it does not show that the power is 1/2 as we expected. Note that the slope is the same in this case as in the case of the thin square plate.


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