Symmetric Matrix Eigenvalue Problems - Numerical Methods in Engineering with MATLAB - page 353

= λ = λ

convert the eigenvalue problem Ax Bx to the standard form Hz z .What is

the relationship between x and z ?
2. Convert the eigenvalue problem Ax

= λ Bx , where

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

4 − 1 0 2 − 1 0

A = − 1 4 − 1 B = − 12 − 1

0 − 1 4 0 − 11

to the standard form.
3. An eigenvalue of the probleminProb. 2 is roughly 2.5. Use the inverse power
methodwith eigenvalueshifting to compute thiseigenvaluetofour decimal
places.Start with x

1 00 T

= . Hint : two iterations shouldbesufficient.

4. The stress matrix at apoint is

⎡
⎣

⎤
⎦

150 − 600

S = − 60 1200 MPa

0 0 80

Compute the principalstresses (eigenvalues of S ).

5.

L L

θ 2

θ 1

k

m

2 m

The two pendulums areconnectedbyaspring which is undeformedwhen the
pendulums are vertical. The equationsofmotion of the system can be shown
to be

mL ¨

kL ( θ 2 − θ 1 ) − mg θ 1 = θ 1

2 mL ¨ − kL ( θ 2 − θ 1 ) − 2 mg θ 2 = θ 2

where θ 2 are the angular displacements and k is the spring stiffness.

Determine the circular frequencies of vibration and the relative amplitudes
of the angular displacements. Use m

θ 1 and

= 0 . 25 kg, k = 20N/m, L = 0 . 75 m and

g = 9 . 80665 m/s 2 .