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.

 

[sardar]

I can provide a response to your question about obtaining the frequency for combined modes of a square membrane. The equation you have provided is known as the modal frequency equation for a square membrane, and it is used to calculate the natural frequencies of the membrane for different modes.

In order to calculate the frequency for combined modes such as (2,1)+(1,2) and (2,1)-(1,2), we need to consider the superposition principle. This principle states that the response of a system to multiple inputs is equal to the sum of the individual responses to each input.

In the case of a square membrane, the modes (2,1) and (1,2) can be considered as two separate inputs. Therefore, the frequency for the combined mode (2,1)+(1,2) can be calculated by adding the frequencies of the individual modes (2,1) and (1,2). Similarly, the frequency for the combined mode (2,1)-(1,2) can be calculated by subtracting the frequency of mode (1,2) from the frequency of mode (2,1).

In mathematical terms, the frequency for the combined mode (2,1)+(1,2) can be represented as ω(2,1)+(1,2) = ω(2,1) + ω(1,2), where ω(2,1) and ω(1,2) are the frequencies of modes (2,1) and (1,2) respectively. Similarly, the frequency for the combined mode (2,1)-(1,2) can be represented as ω(2,1)-(1,2) = ω(2,1) - ω(1,2).

It is important to note that the modal frequencies for a square membrane are dependent on the dimensions of the membrane, represented by a and b in the equation you have provided. Therefore, the frequencies for combined modes will also be affected by the dimensions of the membrane.

In summary, to calculate the frequency for combined modes of a square membrane, we can use the superposition principle and add or subtract the frequencies of the individual modes.