Sensitivity of Eigenproblems

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Sensitivity of Eigenproblems - PowerPoint PPT Presentation

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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Presentation Transcript
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