Strength Analysis - Finite Element Analysis for Satellite Structures - page 121

If each eigenvector fg i is normalized to the mass matrix:

fg i ½fg i ¼ M r ¼ 1 ð 4 : 20 Þ

Hence:

c i ¼ fg i ½fg ð 4 : 21 Þ

The vector describing the excitation direction has the form:

fg¼ ½fg ð 4 : 22 Þ

where:

T

fg¼ D 1 D 2 D 3 ...

D j = excitation at DOF j in direction a; a may be either X, Y, Z, or rotations
about one of these axes.

2
4

3
5

100 0 ð Z Z 0 Þ Y Y 0 ð Þ

010 Z Z 0 ð Þ 0 ð X X 0 Þ

001 ð Y Y 0 Þ X X 0 ð Þ 0

½¼

000 1 0 0

000 0 1 0

000 0 0 1

X, Y, Z = global Cartesian coordinates of a point
X o , Y o , Z o = global Cartesian coordinates of point about which rotations are
done (reference point)
fg = six possible unit vectors
The effective mass for the ith mode (which is a function of excitation direction) is:

M ei ¼ fg i ½fg c i ð 4 : 23 Þ

Note from Eq. ( 4.20 ) that:

M ei ¼ c i ð 4 : 24 Þ

The sum of the effective masses of all modes (i = 1, 2, 3, …, n) is equal to the
total mass of the structure. This results in a means of determining the number of
individual modal responses necessary to accurately represent the structural
response. If the total response of the system is to be represented in terms of a small
number of modes ''p'' (where p n), and if the sum of the ''p'' effective masses is
greater than a predefined percentage of the total mass of the structure, then the
number of modes ''p'' considered in the analysis is adequate. However, if this is
not the case, then additional modes must be considered. Dynamic analysis
procedures of large MDOF systems specify that for the ''p'' modes considered in
the analysis, at least 90 % of the participating mass of the structure must be
included in the response calculations for each principal horizontal direction.