Page 209 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

196 Properties of Vibrating Systems Chap. 6

Any of the modes corresponding to must be orthogonal to if it is to be
a normal mode. If all three modes are orthogonal, they are linearly independent
and can be combined to describe the free vibration resulting from any initial
condition.
The eigenvectors associated with the equal eigenvalues are orthogonal to the
remaining eigenvectors, but they may not be orthogonal to each other.
Example 6.11-1
Consider the system of Fig. 6.11-1 of a flexible beam with three lumped masses. Of
the three possible modes shown, the first two represent rigid body motion of
translation and rotation corresponding to zero frequency, and the third mode is that
of symmetric vibration of the flexible beam. With the mass matrix equal to

1 0 0
M = 0 2 0
0 0 1
the modes are easily shown to be orthogonal to each other, i.e.,

Next, multiply (¡>2 by a constant h and add it to (/>, to form a new modal vector
^12*
1 - b
M 1
1
u 1 i / \ 1 + h
i.e.,
1 0 0 1-6^

(-1 1 -1) 0 2 0 1 } ==0
0 0 1 \ + h\
Thus, the new eigenvector formed by a linear combination of (/)j and </>2 is orthogonal

I
O ■o— ..... o
m Z m m

x^ = o
o_------------------ -O'-*

ck ___
\ ^ ----
3E/
+3= 1 X=
‘O - - - Figure 6.11-1.