Page 345 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

332 Introduction to the Finite Element Method Chap. 10

The equation of motion can now be written as
(o~ml ., El
V = - m i l ' l H V
By multiplying and dividing the right hand term by El/P and transferring it to the
left side, we obtain
(jo^ml , El
K + ' ^ H v = o
t l

M - A = 0
t l
where
420 £ /

A =
With the boundary conditions [ j = = 0, we need only to maintain the lower
right quadrant of the matrices. Also by remembering that the inside the matrices all
cancel in the solution for the eigenvalues and eigenvectors, we can let I = 1.0. The
final equation of motion is then

156 - 2 2 ' 12 - 6 ' ~r ' 0.4286 -0.06429 ^’2
1 ,
- 2 2 4 a( - 6 4 i E l j -0.06429 -0.02381 )J
7 ^ J V^2
0
0

This equation can be solved for the eigenvalue A by assuming a number for the
rotation parameter. If we choose il^ml^/EI = 1.0, we obtain
_
156 - 2 2 ‘ - A 12.43 -6.064
_-22 4 -6.064 4.024

(156 - 12.43A) -(2 2 - 6.064A)
= 0
-(22 - 6.064A) (4 - 4.024A)
The eigenvalues and natural frequencies from the determinant are

A, = 30.65

A, = 0.345
and the associated eigenvectors and mod
0.545 0.0807
0.749 0.615