%% 2.086 RECITATION 8 - Eigenvalues and Eigenvectors
% Spring 2013 - created by J Thangavelautham
%% Eigen Value Problems
%An Eigenvalue problem can be presented in the following form
% A.x = lambda.x
% (A-lambda.I).x = 0 (rearrange)
% A-lambda.I = 0
%Example A = [ 0 -1; 2 3]
%Eigenvalues: lambda1 = 1, lambda2 = 2
%Eigenvectors: v1 = [-1 1]', v2 = [1 -2]'
%% Example 1: Numerical Example in Matlab
A = [ 0 -1; 2 3]
%Using eig command
[v,d]=eig(A)
diag(d) % Eigenvalues
v(:,1) % columns are eigen vectors normalized to 1
v(:,2)
%% Example 2: Coupled Mass Spring System
%
% Consider a spring-mass system (see illustration below),
% that consists of a spring of stiffness K connected to a
% rigid wall on end and Mass 1 on the other end. Mass 1 is connected to a
% second spring with stiffness Km. The other end of the spring with stiffness
% Km is connected to Mass 2. Mass 2 is in turn connected to a second spring
% with stiffness K that attaches to a rigid wall.
%
%
% || K Km K ||
% ||\/\/\/\O/\/\/\O/\/\/\/||
% || Mass1 Mass2 ||
% Wall M1 M2 Wall
%
% Motion of system defined by these system of equations:
%
% M1*x1''= -K*x1 + Km*(x2-x1) (1)
% M2*x2''= -K*x2 + Km*(x1-x2) (2)
%
% The stiffness matrix of the system is as follows
% K_matrix = [K + Km -Km; -Km K + Km]
%
% Presume solution of the form x = x0*exp(i*omega*t) and substitute into 1
% and 2
%
% After simplification it becomes
%
% [ K + Km - M1*omega^2 -Km ][ x1(t) ] = 0
% [ -Km K + Km - M2*omega^2 ][ x2(2) ]
%By setting lambda as M*omega^2, one can present this as an eigenvalue
%problem.
% Consider specific values for M1 = 1; M2 = 1, K = 2; Km = 1;
%
%
% A is the K_matrix and we solve for lambda which is M*omega^2
A = [3 -1; -1 3]
M1 = 1;
M2 = 1;
MM = zeros(2,2);
MM(1,1) = M1;
MM(2,2) = M2;
% MM is the mass matrix.
[v,d]=eig(A,MM)
sqrt(diag(d)) %omega 1 and omega 2, the eigenfrequencies of the system.
v(:,1) %oscillatory modes
v(:,2)
%% Example 3: Consider M1 and M2 to be different, where M1 = 1; M2 = 1.5, K = 2; Km = 1;
%
%
% A is the K_matrix and we solve for lambda which is M*omega^2
A = [3 -1; -1 3]
M1 = 1;
M2 = 1.5;
MM = zeros(2,2);
MM(1,1) = M1;
MM(2,2) = M2;
% MM is the mass matrix.
[v,d]=eig(A,MM)
sqrt(diag(d)) %omega 1 and omega 2, the eigenfrequencies of the system.
v(:,1) %oscillatory modes
v(:,2)