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Sensitivity of EigenproblemsPowerPoint Presentation

Sensitivity of Eigenproblems

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Sensitivity of Eigenproblems

- Review of properties of vibration and buckling modes. What is nice about them?
- Sensitivities of eigenvalues are really cheap!
- Sensitivities of eigevectors.
- Why bother getting them?
- Think of where you want your car to have the least vibrations

The eigenproblem

- Common notation for vibration and buckling
- For vibration M is mass matrix, for buckling it is geometric stiffness matrix.
- Usually W=M
- u is vibration or buckling mode, and is the square of the frequency of buckling load
- What are the properties of K and M?
- What do we know about the eigenvalues and eigenvectors?

Derivatives of eigenvalues

- Differentiate:
- Pre multiply by :
- What is the physical meaning?
- Why is it cheap to calculate?

Problems eigenvalue sensitivity

- How you would apply the physical interpretation of the derivatives of eigenvalues to raising or lowering the frequency of a cantilever beam?
- Check this by using the beam in the semi-analytical problem, assuming that it has a cross-section of 4.5”x2”, and is made of steel with density of 0.3 lb/in3. Compare the effect of halving the height of the first and last of the 10 elements. Check the frequency of the original beam against a formula from a textbook or web.

Eigenvector derivatives

- Collecting equations
- Difficult to solve because top-left matrix is singular. Why is it?
- Textbook explains Nelson’s method, which uses intermediate step of setting one components of the eigenvector to 1.

Spring-mass example

- Fig. 7.3.1
- Stiffness and mass matrices (all springs and masses initially equal to one.
- Solution of eigenproblem

Derivative w.r.t k

- Derivatives of matrices
- Derivative of eigenvalue
- See in textbook derivative of eigenvector
- Do those pass sanity checks?

Eigenvectors are not always unique

- When can we expect two different vibration modes with the same frequency?
- Why does optimization with frequency constraints likely to lead to repeated eigenvalues?
- Vibration modes are orthogonal when eigenvalues are distinct, but any combination of modes corresponding to the same frequency is also a vibration mode!

Example 7.3.2

- Problem definition and solution
- Eigenvectors for x=0
- Eigenvectors for y=0

- At x=y=0 eigenvalues are the same and eigenvectors are discontinuous

Deriviatives of repeated eigenvalues

- Assume m repeated eigenvectors
- To find eigenvalue derivatives need to solve a second eigenvalue problem!

Calculation of derivatives w.r.t x

- At x=y=0 any vector is an eigenvector.
- Similarly get

Why are these derivatives of limited value

- What happens if we try to use them for dy=2dx=2dt?

Problems (optional)

- Explain in 50 words or less why derivatives of vibration frequencies are relatively cheaper than derivatives of stresses
- When eigenvalues coalesce, they are not differentiable even though we can still use Nelson’s method to calculate derivatives. How can you reconcile the two statements?
- Why is the accuracy of lower frequencies (and their derivatives) better than that of higher frequencies?

Source: Smithsonian Institution

Number: 2004-57325

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