1. Eigenvalues Appendix A
Chapter 22
Eigenvalues
Appendix A

2. Non-homogeneous system
Eigenvalue Problems
Engineering Problems involving vibrations, elasticity, oscillating systems, etc.,
Determine the eigenvalues for n homogenous linear equations in n unknowns
Non-homogeneous system
homogeneous system

3. Mathematical Background
For nontrivial solutions ==>
Characteristic polynomial: det( ) = fn( )
The root of fn( ) = 0 are the solutions for the eigenvalues

4. Equilibrium positions
Mass-Spring System
Equilibrium positions

5. Mass-Spring System Homogeneous system
Find the eigenvalues  from det[ ] = 0

6. Polynomial Method m1 = m2 = 40 kg, k = 200 N/m
Characteristic equation det[ ] = 0
Two eigenvalues  = 3.873s1 or s 1
Period Tp = 2/ = 1.62 s or 2.81 s

7. Principal Modes of Vibration
Tp = 1.62 s
X1 = X2
Tp = 2.81 s
X1 = X2

8. Buckling of Column Axially loaded column Buckling modes
M – bending moment
E – modulus of elasticity
I – moment of inertia
Curvature:

9. Buckling of Axially Loaded Column
Eigenvalue problem
Buckling loads
Fundamental mode: n = 1
Euler formula

10. Buckling Modes

11. Polynomial Method ODE Finite-difference method
Characteristic equation: (2n)th-order polynomial
Which
Scheme?
Order of
Errors?

12. Polynomial Method One interior node (h = L/2)
Two interior nodes (h = L/3)

13. Polynomial Method
Three interior nodes (h = L/4)

14. Power Method Power method for finding eigenvalues
Start with an initial guess for x
Calculate w = Ax
Largest value (magnitude) in w is the estimate of eigenvalue
Get next x by rescaling w (to avoid the computation of very large matrix An )
Continue until converged
Power method also gives you eigenvectors

15. k is the dominant eigenvalue
Power Method
Start with initial guess z = x0
rescaling
k is the dominant eigenvalue

16. Power Method
For large number of iterations,  should converge to the largest eigenvalue
The normalization make the right hand side converges to  , rather than n

17. eigenvalue eigenvector
Example: Power Method
Consider
Assume all eigenvalues are equally important, since we don’t know which one is dominant
Start with
eigenvalue eigenvector

18. Example
Current estimate for largest eigenvalue is 21 Rescale w by eigenvalue to get new x
Check Convergence (Norm < tol?)
Norm

19. Update the estimated eigenvector and repeat
New estimate for largest eigenvalue is Rescale w by eigenvalue to get new x
Norm

20. Convergence criterion -- Norm (or relative error) < tol
Example
One more iteration
Norm
Convergence criterion -- Norm (or relative error) < tol

21. Example: Power Method

22. Script file: Power_eig.m

23. MATLAB Example: Power Method
» [z,m] = Power_eig(A,100,0.001);
it m z(1) z(2) z(3) z(4) z(5)
error =
8.3175e-004
» z
z =
0.9199
0.7085
1.0000
» m
m =
» x=eig(A)
x =
0.6686
MATLAB Example:
Power Method
eigenvector
eigenvalue
MATLAB function

24. MATLAB’s Methods e = eig(A) gives eigenvalues of A [V, D] = eig(A)
eigenvectors in V(:,k)
eigenvalues = Dii (diagonal matrix D)
[V, D] = eig(A, B) (more general eigenvalue problems) (Ax = Bx)
AV = BVD

25. Inverse Power Method Power method give the largest eigenvalue
Inverse Power method gives the smallest
*Eigenvalues of B = A-1 are inverse of eigenvalues of A (i.e.,  = 1/)
So one could use power method on w = Bx to get largest eigenvalue of B - smallest of A
Calculating B is wasteful - instead use

26. Inverse Power Method Basic power method gives the dominant eigenvalue
Inverse power method gives the smallest eigenvalue

27. Script file for Inverse Power Method Use LU_factor and LU_solve

28. Smallest eigenvalue [L] [U] [B] = [L][U] eigenvector eigenvalue
» max_it=100; tol=0.001;
» [z,m] = InvPower(A,max_it,tol);
L =
U =
B =
it = 1 4
» z
z =
0.1163
1.0000
» m
m =
0.6686
» x=eig(A)
x =
eigenvector
eigenvalue
MATLAB function
[L]
[U]
[B] = [L][U]
Smallest eigenvalue

29. CVEN Homework No. 15
Finish the HW but do not hand in. I will post the solution on the net.