Frequency for Vibration Modes of a Square Membrane

[Johnny122]

So the equation to obtain the frequency of the modes of a square membrane is something like

ω m,n = ∏ [(m/a)^2 + (n/b)^2]^(1/2)

This equation can be used to get the frequency for Modes such as (2,1) and (1,2). How do I get the frequency for such modes as (2,1)+(1,2) and (2,1)-(1,2) ? Picture attached shows the modes.

 

[AlephZero]

If it is a square plate, the frequencies for (2,1) and (1,2) are the same, and the others in your picture are linear combinations of them, also at the same frequency.

For a rectangular plate with $a \ne b$, the (2,1) and (1,2) frequencies are different and the "(2,1)+(1,2) and (2,1)-(1,2) modes" don't exist.

 

[Johnny122]

Let's say that the frequency for (2,1) = x and (1,2) = y , so by linear combination, do you mean something like a x + b y = z where a and b are constants? And z would be the frequency for (2+1)+(1,2) or (2-1)-(1,2) ?

 

[AlephZero]

Let's say that the frequency for (2,1) = x and (1,2) = y

It's a square plate, so a = b in your formula for the frequencies. So x = y.

In any structure that has two (or more) modes with the same frequency, and combination of the modes is also a "mode" with the same frequency.