Page 224 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 7.1 Generalized Coordinates 211

Thus, the two constraint equations are in the form
[ ^ ] M = 0 (b)
We actually have seven coordinates u-j, Ug) and two constraint
equations. Thus, the degrees of freedom of the system are 7 - 2 = 5, indicating that
of the seven coordinates, five can be chosen as generalized coordinates q.
Of the four coordinates in the constraint equation, we choose and «7 as two
of the generalized coordinates and partition Eq. (a) as

= [«]{«} + [^]{«} = 0 (C)

Thus, the superfluous coordinates u can be expressed in terms of q as
{«} = (d)

Applying the preceding procedure to Eq. (a), we have
1 0 ■- 1 0

0 0.866 -0.5 - 0.866
'1 0 '1 0 1 O'
1 “0
0.5 0.866 0.578 1
° 0.866
By supplying the remaining q, as identities, all the w’s can be expressed in terms of
the q's as
{«} = [C]{«} (e)
where the left side includes all the w’s and the right column contains only the
generalized coordinates. Thus, in our case, the seven m’s expressed in terms of the five
q's become

^«1 0 0
^3 0 0
W4 0 0.578 0 1
0 1 0 0 (0
^6 0 0 1 0 0
Un 0 0 0 1 0
0 0 0 0
In Eq. (e) or (f), matrix C is the constraint matrix relating u to q.

Example 7.1-4
In the lumped-mass models we treated earlier, n coordinates were assigned to the n
masses of the n-DO¥ system, and each coordinate was independent and qualified as a
generalized coordinate. For the flexible continuous body of infinite degrees of
freedom, an infinite number of coordinates is required. Such a body can be treated as