spline quad

*Eigenvalue and Eigenvector MethodChapter 27.2Iterative Special importance in science & engineering( Vibration sys, elasticity, oscillatory)Provide useful information about its properties

Eigenvalues represent the natural frequencies of a system http://numericalmethods.eng.usf.edu*Eigenvectors represent the mode of these vibrationVibration

http://numericalmethods.eng.usf.edu

http://numericalmethods.eng.usf.edu*Why it is important to identify these natural frequencies ?

when the system is subjected to periodic external loads(forces) at or near these frequencies, resonance can cause the response(motion) of the structure to be amplified, potentially leading to failure of the component.


Mechanics of materials:

the principle stresses are the eigenvalues of the stress matrix, and the principle directions are the directions of the associated eigenvectors. In quantum mechanics, eigenvalues are especially important. http://numericalmethods.eng.usf.edu*


*Power MethodClassification of Power Method : 2 typesPower method for highest or largest eigenvalue (PM_H(L) E)

2. Power method for lowest or smallest eigenvalue ( PM-L(S)E)

When can the power method be used?Largest eigenvalue is desiredMust be real*Must not change the Matrix

*The vector [x] is normalized at each step

by dividing the elements of the vectorby the value of the largest element. Concept

*This makes the largest element of thevector equal to 1.

Because of this scaling/normalization at each step the power method yields the eigenvalue & associated eigenvector simultaneously.

*Power method for largest Eigenvalue (LE)The mathematical formula of PM_LE is

*PM-LE ProcedureWrite the initial eigenvector; The vector can be any nonzero vector

2. Execute the First iteration; It-1;

3. Put the largest common of y (1) as (1) the first eigenvalue

*PM-LE Procedure cont4. Find ( the first eigenvector) by dividing the component of y (1) by (1);

Repeat step(2) to execute iteration-2;

It-2;

*PM-LE Procedure6. Repeat step (3) to find (2)7. Repeat step(4) to find X(2) ;

8.Evalute the first error E1;

9. Repeat step(5) to execute the third iteration; It-3

10. Repeat step(6) to find (3)

*PM-LE Procedure11. Repeat step (7) to find

12. Evaluate the second error E2 ;

*PM-LE Procedure

13. Compare E2 with E1 ;

If E2 < E1 then stop the procedure and put as the largest eigenvalue

b) If E2 > E1 then continue to the next iteration and stop when En < En-1

*Find the dominant eigenvalue for the matrix A;EXAMPLESolution1.2.Iteration #1i =0

*3.4.5.Iteration #2;i =1Cont

*6.7.Cont

*9.8.Iteration #3;10.i =2 11.

*12.13.Since E3< E2 then stop and the largest,Eigenvalue is and the solutionis the associated eigenvector

PM-LE Procedure

The last multiplicative factor (scalar) is the largest eigenvalue, and the normalized vector is the associated eigenvector. http://numericalmethods.eng.usf.edu*


http://numericalmethods.eng.usf.edu*Exercise 1Determine the largest eigenvalue of the following matrix:


*

spline quad

Documents