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.

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Sensitivity of eigenproblems
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
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
Derivatives of eigenvalues

  • Differentiate:

  • Pre multiply by :

  • What is the physical meaning?

  • Why is it cheap to calculate?

Problems eigenvalue sensitivity
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
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
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
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
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
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
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
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
Why are these derivatives of limited value

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

Problems optional
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