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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