Page 205 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

192 Properties of Vibrating Systems Chap. 6
6.10 NORMAL MODE SUMMATION

The forced vibration equation for the A^-DOF system
M X + C X + K X = F (6.10-1)
can be routinely solved by the digital computer. However, for systems of large
numbers of degrees of freedom, the computation can be costly. It is possible,
however, to cut down the size of the computation (or reduce the degrees of
freedom of the system) by a procedure known as the mode summation method.
Essentially, the displacement of the structure under forced excitation is approxi­
mated by the sum of a limited number of normal modes of the system multiplied
by generalized coordinates.
For example, consider a 50-story building with 50 DOF. The solution of its
undamped homogeneous equation will lead to 50 eigenvalues and 50 eigenvectors
that describe the normal modes of the structure. If we know that the excitation of
the building centers around the lower frequencies, the higher modes will not be
excited and we would be justified in assuming the forced response to be the
superposition of only a few of the lower-frequency modes; perhaps (/),(x), (/>2(^),
and may be sufficient. Then the deflection under forced excitation can be
written as
X, = (A|(x,)î7|(0 + 4>2iXj)q2U) + 4>3ix,)q^(t) (6.10-2)
or in matrix notation the position of all n floors can be expressed in terms of the
modal matrix P composed of only the three modes. (See Fig. 6.10-1.)

f^'l
<J2
< (6.10-3)
4>2(x„)
The use of the limited modal matrix then reduces the system to that equal to the
number of modes used. For example, for the 50-story building, each of the
matrices such as /C is a 50 X 50 matrix; using three normal modes, P is a 50 X 3
matrix and the product KP becomes
P^KP = {3 x 50)(50 X 50)(50 X 3) = (3 x 3) matrix
Thus, instead of solving the 50 coupled equations represented by Eq. (6.10-1), we
need only solve the three by three equations represented by

P^MPq + P^CPq + P^KPq = P^F (6.10-4)
If the damping matrix is assumed to be proportional, the preceding equations
become uncoupled, and if the force F{x,t) is separable to p(x)fit), the three
equations take the form
Q, + ^ ^ i f i O (6.10-5)