Vibration membrane exercice physics

[louisnach]

Homework Statement

We have a square membrane and a circular membrane, both fixed from the edges. They are made of the same material with a surface mass σ , and their surface tension is T. We want to tune the lowest eigenfrequency (the fundamental frequency) of both membranes to the same frequency f0
What would the area of each of the two membranes be in this case?

Calculate the first five partials of both membranes.

Comment on the relationship between the partials. •

Is it possible to tune the membranes harmonically? If so, how could this be achieved?

Homework Equations

The Attempt at a Solution

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The first question is easy by using equation of eigenfrequency of a circular f = Xm,n/(2*pi*radius)

where x is the value of the nth non-trivial zero of the mth–order Bessel function, so we equalize this with f0 with X0,1=2.4048 and find the radius

Same way with eigenfrequency of square with size a: w = c*pi*root(m²/a²+n²/a²) with m=n=1But i don't understand what means a "partial" and "tune the membrane harmonically"

Does anyone have an idea ?

Best wishes for 2017,

 

[mark726]

Hello,

Thank you for your question. I would like to offer some insights on your post.

Firstly, when the fundamental frequency (f0) is equal, the area of each membrane can be calculated using the equations you provided. This is because the fundamental frequency is directly proportional to the area of the membrane.

Secondly, the partials refer to the different modes of vibration that a membrane can have. These modes are characterized by different frequencies, known as partial frequencies. The first five partials can be calculated using the equations you provided, by changing the values of m and n.

Thirdly, the relationship between the partials can provide information about the shape and properties of the membrane. For example, the ratio of the second partial frequency to the fundamental frequency can give the shape factor of the membrane, which can be used to determine the mode of vibration.

Lastly, it is possible to tune the membranes harmonically by adjusting the tension or mass of the membranes. This can be achieved by changing the surface tension or adding weights to the membranes. However, it is important to note that this may affect the relationship between the partials and the overall behavior of the membranes.

I hope this helps clarify your doubts. Best of luck with your calculations and experiments! Happy New Year!