linear_algebra

size of A | norm error 3 | 4 |

5. Solve spring mass system: M = (1 0 0 0 2 0 0 0 4) K = (k1+k2 -k2 0
   -k2 k2+k3 -k3
   0 -k3 k3+k4) K = ( 30 -20 0 -20 40 -20 0 -20 30 ) Mx'' + Kx = 0

1x1''+ 30x1 - 20x2 = 0 2x2''- 20x1 + 40x2 - 20x3 = 0 4x3''- 20x2 + 30x3 = 0

k~ = ( 30 -14.2 0 -14.2 20.16 -7.1 0 -7.1 7.5 ) (k~ - (eigenvalue)I)*v = 0 det(k~ - (eigenvalue)I) = 0 eigenvalue = E

(30-E -14.2 0 -14.2 20.16-E -7.1 = 0 0 -7.1 7.5-E)

(30-E)(((20.16-E)(7.5-E)) + 7.1(-7.1)) + 14.2(-14.2(7.5-E))

(30-E)(100.79 + 12.66E + E^2) + -1512.3 - 201.64E 3023.7 + 379.8E + 30E^2 - 100.79E - 12.66E^2 -E^3 - 1512.3 - 201.64E 1511.4 + 77.37E + 17.34E^2 -E^3 = 0

E1 = 23.4 wn1 = 4.837m/s

of segments | largest | smallest | # of eigenvalues

5 6 10