For the mass and stiffness matrix below one eigenvalue natur

For the mass and stiffness matrix below one eigenvalue (natural frequency squared) is omega^2 = 1. Determine its corresponding eigenvector [M] = [3 0 0 0 3 0 0 0 3] [K] = [6 2 -5 2 6 -5 -5 -5 13]

Solution

The eigenvalue 1 corresponds to - w^2

matrix equation for the eigenvector K= (u,v,w) is

AK = 1.K

A = {3 2 -5}

{2 3 -5}

{-5 -5 10}

forming the three eqns with eigenvalue 1,

2u +2v -5w =0

2u +2v -5w=0

-5u -5v+9w =0

from the any of the 1st two, and the third, it is seen that w=0

so u+v =0 => u=-v => (1,-1,0) is a m egenvector, and similarly (-1,1,0 ) is an eigenvector.