Eigenvalues. Berlin Chen Department of Computer Science & Information Engineering National Taiwan Normal University

Transcription

1 Eigenvalues Berlin Chen Department of Computer Science & Information Engineering National Taiwan Normal University Reference:. Applied Numerical Methods with MATLAB for Engineers, Chapter 3 & Teaching material

2 Chapter Objectives Understanding the mathematical definition of eigenvalues and eigenvectors Understanding the physical interpretation of eigenvalues and eigenvectors within the context of engineering systems that vibrate or oscillate Knowing how to implement the polynomial method Knowing how to implement the power method to evaluate the largest and smallest eigenvalues and their respective eigenvectors Knowing how to use and interpret MATLAB s eig function NM Berlin Chen 2

3 Dynamics of Three Coupled Bungee Jumpers in Time V =200m/s V 0 =00m/s V 3 =00m/s Initial Conditions (set the jumpers initial positions to the equilibrium values) Is there an underlying (latent) pattern??? NM Berlin Chen 3

4 Mathematics (/2) Up until now, heterogeneous systems: [A] {x} = {b} Have a unique solution when equation are linearly independent (i.e., A has a nonzero determinant) What about homogeneous systems? [A] {x} = 0 At face value, it has the trivial solution: {x} = 0 Is there another way of formulating the system so that the solution would be meaningful (nontrivial)??? NM Berlin Chen 4

5 Mathematics (2/2) What about a homogeneous system like: (a ) x + a 2 x 2 + a 3 x 3 = 0 a 2 x + (a 22 ) x 2 + a 23 x 3 = 0 a 3 x + a 32 x 2 + (a 33 ) x 3 = 0 Or, in matrix form [[A] {x} = 0 For this case, there could be a value of that makes the equations equal zero. This is called an eigenvalue For non-trivial solutions to be possible [A] = 0 Expend the determinant yields a polynomial in, called the characteristic polynomial The roots of the polynomial are eigenvalues of A NM Berlin Chen 5

6 A Two-Equation Case NM Berlin Chen 6

7 The Polynomial Method (/3) Example 3. NM Berlin Chen 7

8 The Polynomial Method (2/3) Example 3. NM Berlin Chen 8

9 The Polynomial Method (3/3) The eigenvectors provide the ratios of the unknowns representing the solution. Example 3. NM Berlin Chen 9

10 MATLAB Built-in Functions NM Berlin Chen 0

11 Physical Background: Oscillations or Vibrations of Mass-Spring Systems NM Berlin Chen

12 Model with Force Balances (AKA: F = ma) Collect terms: m d 2 x dt 2 = k x + k(x 2 x ) m 2 d 2 x 2 dt 2 = k(x 2 x ) kx 2 m d 2 x dt 2 k ( 2x + x 2 ) = 0 m 2 d 2 x 2 dt 2 k (x 2x 2 ) = 0 NM Berlin Chen 2

13 Assume a Sinusoidal Solution (/2) Based on vibration theory x i = X i sin ( t) where angular frequency amplitude Differentiate twice: x i = X i 2 sin ( t) T p Substitute back into system and collect terms NM Berlin Chen 3

14 Assume a Sinusoidal Solution (2/2) k m X k m 2 X + k m 2 k m X 2 X 2 Given: m = m 2 = 40 kg; k = 200 N/m (0 2 ) X 5 X 2 = 0 5 X + (0 2 ) X 2 = 0 This is now a homogeneous system where the eigenvalue represents the square of the angular frequency ( ) NM Berlin Chen 4

15 Solution: The Polynomial Method X = X 2 0 Evaluate the determinant to yield a polynomial = ( 2 ) 2 2 The two roots of this "characteristic polynomial" are the system's eigenvalues: 2 = 5 5 or Hz 2.36 Hz NM Berlin Chen 5

16 2 = 5 /s 2 Interpretation = /s T p = 2 /2.236 = 2.8 s (0 2 ) X = /s T p = 2 /3.373 =.62 s 5 X 2 = 0 5 X + (0 2 ) X 2 = 0 (0 ) X 5 X 2 = 0 (0 ) X 5 X 2 = 0 5 X + (0 ) X 2 = 0 5 X + (0 ) X 2 = 0 X 5 X 2 = 0 5 X + X 2 = 0 X = X 2 2 = 5 /s 2 X 5 X 2 = 0 5 X X 2 = 0 X = X 2 V = eigenvector V = eigenvector NM Berlin Chen 6

17 Principle Modes of Vibration NM Berlin Chen 7

18 The Power Method Iterative method to compute the largest eigenvalue and its associated eigenvector Simple Algorithm: [[A] [I]]]{x} = 0 [A]{x} = {x} function [eval, evect] = powereig(a,es,maxit) n=length(a); evect=ones(n,);eval=;iter=0;ea=00; %initialize while() evalold=eval; %save old eigenvalue value evect=a*evect; %determine eigenvector as [A]*{x) eval=max(abs(evect)); %determine new eigenvalue evect=evect./eval; %normalize eigenvector to eigenvalue iter=iter+; if eval~=0, ea = abs((eval-evalold)/eval)*00; end if ea<=es iter >= maxit,break,end end NM Berlin Chen 8

19 The Power Method: An Example (/3) First iteration: Initial guesses of X s that have all element equal to Second iteration: = = 20 0 Normalize the right-hand side vector to make the large element equal to = = 40 a = % = 50% NM Berlin Chen 9

20 The Power Method: An Example (2/3) Third iteration: = 80 = a = 00% = 50% 80 Fourth iteration: = = ) a = 00% = 24% 70 NM Berlin Chen 20

21 The Power Method: An Example (3/3) Fifth iteration: = = a = % = 2.08% The process can be continued to determine the largest eigenvalue (= ) with the associated eigenvector [ ] Note that the smallest eigenvalue and its associated eigenvector can be determined by applying the power method to the inverse of A NM Berlin Chen 2

22 Determining Eigenvalues & Eigenvectors with MATLAB >> A = [0-5;-5 0] eigenvector A = >> [v,lambda] = eig(a) eigenvector v = lambda = eigenvector eigenvalue eigenvalue NM Berlin Chen 22