Eigenvalue problem -- Elastic deformation of a membrane

In summary, using the correct eigenvalues and eigenvectors, each eigenvalue will give the amount of stretch or compression in the direction of the associated eigenvector. That should give you some idea of how the circle will be deformed due to the transformation.
  • #1

Homework Statement


An elastic membrane in the x1x2-plane with boundary circle x1^2 + x2^2 = 1 is stretched so that point P(x1,x2) goes over into point Q(y1,y2) such that y = Ax with A = 3/2* [2 1 ; 1 2] find the principal directions and the corresponding factors of extension or contraction of the elastic deformation. Sketch the shape of the deformed membrane.

Homework Equations


det(A-lambda*I) = 0
(A-lambda*I)x = 0

The Attempt at a Solution


Using det(A-lambda*I) = 0
I got

lambda = 4.5
lambda = 0

Using (A-lambda*I)x = 0
I got eigenvectors

x2*[1;1]
x2*[-.5;1]

What do I do next?
 
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  • #2
shreddinglicks said:

Homework Statement


An elastic membrane in the x1x2-plane with boundary circle x1^2 + x2^2 = 1 is stretched so that point P(x1,x2) goes over into point Q(y1,y2) such that y = Ax with A = 3/2* [2 1 ; 1 2] find the principal directions and the corresponding factors of extension or contraction of the elastic deformation. Sketch the shape of the deformed membrane.

Homework Equations


det(A-lambda*I) = 0
(A-lambda*I)x = 0

The Attempt at a Solution


Using det(A-lambda*I) = 0
I got

lambda = 4.5
lambda = 0
Check your algebra. Both eigenvalues are incorrect.
shreddinglicks said:
Using (A-lambda*I)x = 0
I got eigenvectors

x2*[1;1]
x2*[-.5;1]
The first eigenvector is correct, although it appears that you got it by accident, not on purpose. The second one isn't correct.
shreddinglicks said:
What do I do next?
After you get the correct eigenvalues and eigenvectors, each eigenvalue will give the amount of stretch or compression in the direction of the associated eigenvector. That should give you some idea of how the circle will be deformed due to the transformation
 
  • #3
Yes, it is incorrect. I recalculated.

lambda = 1.5 and 4.5

eigenvectors are:
x2*[-1 1]
x2*[1 1]

I'm confused about the next part. Am I plugging the eigenvalues into x2? Then Am I plugging the vectors into the circle equation x1^2 + x2^2 = 1?
 
  • #4
shreddinglicks said:
Yes, it is incorrect. I recalculated.

lambda = 1.5 and 4.5
Those still aren't right. The eigenvalues I get are both integer values. It would be helpful if you showed more of your work, such as the characteristic equation you're working with.

It's a good idea to check your work. If ##\lambda_1## is an eigenvalue with associated eigenvector ##\vec{x_1}##, then ##A\vec{x_1} = \lambda_1\vec{x_1}## should be a true statement. IOW, applying the matrix to an eigenvector should result in a scalar multiple of that eigenvector -- a vector that is longer or shorter or possibly pointing the opposite direction (if the eigenvalue is negative).

I have checked both of my eigenvalues and eigenvectors as described above.
shreddinglicks said:
eigenvectors are:
x2*[-1 1]
x2*[1 1]
These are the eigenvectors I get, but I have no idea how you're getting them, as your eigenvalues are both wrong.
shreddinglicks said:
I'm confused about the next part. Am I plugging the eigenvalues into x2? Then Am I plugging the vectors into the circle equation x1^2 + x2^2 = 1?
The two directions of your eigenvectors define four points on the untransformed circle. Using the correct eigenvalues, the transformed points from the circle will be in the same direction as the eigenvectors, but pushed out or sucked in. The transformation should deform the circle into a rotated ellipse.
 
  • #5
I checked my eigenvalues on MATLAB to be sure.

upload_2018-9-12_10-37-33.png

The hand calculation is also attached.
 

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  • #7
OK, my mistake. I wasn't using that factor of 3/2 in my calculations. Since both of your eigenvalues are positive, the points on the circle in the directions of your eigenvectors will be stretched out by factors of 1.5 and 4.5.
 
  • #8
I want to understand this. So I have two answers.

One where

x2*[-1 1]' where x2 = 1.5

and the other answer is

x2*[1 1]' where x2 = 4.5

I plug each of these vectors into the circle equation and then plot?
 
  • #9
The line in the direction of the vector <-1, 1> intersects the unit circle at ##(\frac{-\sqrt 2} 2, \frac{\sqrt 2} 2)## and at ##(\frac{\sqrt 2} 2, \frac{-\sqrt 2} 2)##. The transformation A will stretch these points out by a factor of 1.5 away from the origin. What are the coordinates of these two transformed points?
Do the same computations for the other eigenvector/eigenvalue pair.
Is that clear?
 
  • #10
Makes perfect sense. I just didn't think of the vectors as having an infinitely long line traveling through its direction.
 

1. What is the eigenvalue problem in the context of elastic deformation of a membrane?

The eigenvalue problem in this context refers to finding the values (eigenvalues) and corresponding functions (eigenvectors) that satisfy the governing equations of motion for a deforming membrane. These values and functions provide important information about the behavior and stability of the membrane under different loading conditions.

2. How is the eigenvalue problem solved for elastic deformation of a membrane?

The eigenvalue problem is typically solved using numerical methods, such as finite element analysis, which involves dividing the membrane into small elements and solving for the eigenvalues and eigenvectors at each element. Analytical solutions can also be obtained for simple membrane geometries and boundary conditions.

3. What factors influence the eigenvalues for elastic deformation of a membrane?

The eigenvalues are influenced by various factors such as the material properties of the membrane (e.g. elasticity, thickness), the boundary conditions (e.g. fixed or free edges), and the type of loading (e.g. uniform tension or point load). Changes in these factors can result in different eigenvalues and eigenvectors, indicating different deformation patterns and modes of vibration.

4. Can the eigenvalues be used to predict the behavior of a deforming membrane?

Yes, the eigenvalues and eigenvectors provide valuable information for predicting the behavior of a membrane under different loading conditions. They can be used to determine the critical loads at which the membrane will buckle or undergo other types of instability, as well as the resulting deformation patterns and mode shapes.

5. Are there any limitations to using the eigenvalue approach for analyzing elastic deformation of a membrane?

One limitation is that the eigenvalue approach assumes linear elastic behavior, which may not be accurate for highly deformed or nonlinear materials. Additionally, the results obtained from the eigenvalue analysis are only valid for the specific geometry and loading conditions used in the analysis and may not be applicable to other scenarios.

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