Sensitivity of Eigenproblems

1. 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

2. 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?

3. Derivatives of eigenvalues • Differentiate: • Pre multiply by : • What is the physical meaning? • Why is it cheap to calculate?

4. 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.

5. 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.

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

7. Derivative w.r.t k • Derivatives of matrices • Derivative of eigenvalue • See in textbook derivative of eigenvector • Do those pass sanity checks?

8. 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!

9. 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

10. Eigenvalues for example 7.3.2 .

11. Deriviatives of repeated eigenvalues • Assume m repeated eigenvectors • To find eigenvalue derivatives need to solve a second eigenvalue problem!

12. Calculation of derivatives w.r.t x • At x=y=0 any vector is an eigenvector. • Similarly get

13. Why are these derivatives of limited value • What happens if we try to use them for dy=2dx=2dt?

14. 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