Simplify three equations and write them in a matrix form?

Questions
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1

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I want simply last three equation interms of Amn, Bmn, Cmn and also want to write in a matrix form.

Attaching herewith the expected solution image.

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2

Please post code, not images, or attach a notebook with the code.

3

I am attaching herewith the code.

research work.nb (7.29 KB)

4

Attaching herewith the code

research work.nb (7.29 KB)

5

Without any constraints on the various constants in the equations, one solution is

{Amn -> 0, 
 Bmn -> -((Dqx ?xmn Csc[(n ? y)/b] Sec[(m ? x)/a])/(h ? ?mn^2)), 
 Cmn -> -((Dqy ?ymn Csc[(m ? x)/a] Sec[(n ? y)/b])/(h ? ?mn^2))}

I do not see how these relate to the matrix you have.

6

Thank you sir for your reply. i am having three equation which consists of three terms namely Amn, Bmn and Cmn which needs to be sorted out and write in matrix form.

In all three equation sin and cos terms are common through out the equation and thus can be eliminated and that is why sin and cos terms are not mentioned in the given matrix.!

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First i want to simplify those three equation by removing common terms( Sin and Cos terms).
Second I want to convert those simplified equation in terms of matrix form with Amn, Bmn and Cmn in 3*1 matrix.

[3 x 3 matrix ][ 3 x 1 matrix ]=0.

7

Can anybody solve the above equation???

8

What was wrong with the result posted by Rohit Namjoshi ?

9

Did you mean this? (See notebook)

But it won’t give Matrix.{Amn,Bmn,Cmn} ==0, because there are terms containing Dqx ans Dqy

But look at

pp = cM.{Amn, Bmn, Cmn} - {0, Dqx \[Psi]xmn, Dqy \[Psi]ymn};
pp[[1]] - e1 // Simplify
pp[[2]] - e2 // Simplify
pp[[3]] - e3 // Simplify

research work.nb (23 KB)

10

Sir thank you for reply
I got all the coefficient of Amn, Bmn and Cmn, but now I want to get rid of x and y terms from all three equation. I mean to say want to get rid of Sin and Cos terms which contains x and y variables i.e. Sin(mpix/a), Sin(npiy/b), Cos(mpix/a), Cos(npiy/b) from all three equation. Kindly help me out.

11

Can you use this method to get what you need?

{4,Sin[m Pi x/a], 2 Sin[n Pi y/b],m*Cos[m Pi x/a], Pi b Cos[n Pi y/b]}/.{Sin[__]->1,Cos[__]->1}

which replaces each Sin and Cos with 1 and returns

{4, 1, 2, m, b Pi}
12

What do you mean by “get rid of”? For me Sin and Cos contain essential information, and they have rarely the same value. You could try

cM /. Sin[ m_ (Pi x_)/a_] :> s[m/a] /. Cos[ m_ (Pi x_)/a_] :> c[m/a]

See the notebook attached (at the very end)

Did you mean that?

research work.nb (31 KB)

13

Sir
In the present work value of x and y is unknown so somehow i want to eliminate term containing x and y variable from all three equation. I am finding free vibration frequency of a plate using the assumed solution but the problem is x and y is unknown and so i want equation without x and y term.

14

Vin,

I showed you how to remove the Sin and Cos terms using Assuming on this other thread BUT you must be certain they are not zero (or if they are zero that it does not matter to your solution).

https://community.wolfram.com/groups/-/m/t/1856471?p_p_auth=L9dN9FO4

Did you try that solution?

15

  1. Rohit’s solution is perfectly correct.

  2. Amn is zero, Bnm and Cmn are functions of x and y! So they have different values for different values of x and y. I don’t see how one could avoid that

3)Again, what do you mean by “get rid of”?

  1. Are you sure that your model is correct? Looking at the plots at the end of the notebook it seems they are somewhat strange.

research work.nb (2.81 MB)