82
4 Multi-Degree of Freedom (MDOF) Systems
0
5
10
15
20
25
30
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
t
Φ
i q
i (t)
0
5
10
15
20
25
30
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
x
i (t)
Fig. 4.6 Modal responses 1 q 1 (t) and 2 q 2 (t) (solid and dashed lines) for ν = 3 and the
corresponding system displacements {x i (t)} (left and right panels)
4.4.1.3 Solution Uniqueness
We have seen that the modal shapes can be determined up to a multiplicative
constant so that it makes sense to question the validity of the modal analysis method
discussed in this section. It is necessary to show that the displacement vector x(t)
is unique in the sense that it does not depend on the particular scaling of the modal
shapes used in analysis. The following arguments show that we obtained the same
displacement vector x(t) irrespective of modal shape scaling.
Consider the representation of x(t) in Eq. 4.19, which implies that the modal
coordinates are the solutions of Eq. 4.26 with initial conditions given by, e.g.,
Eq. 4.31. Suppose that a different scaling is used for the modal shapes, i.e., the
original modes { i } are replaced with {e i i }, where {e i } are arbitrary scaling
constants. Accordingly, the representation of the solution x(t) in Eq. 4.19, denoted
temporarily by y(t), becomes
y(t) =
n
i=1
e i i
p i (t) = e p(t),
(4.33)
where e is an (n, n)-diagonal matrix with non-zero entries {e i } and p(t) is an ndimensional vector. The modal coordinates p(t) are the solutions of (see Eq. 4.22)
e
T m
e
¨
p(t) +
e
T c
e
˙
p(t) +
e
T k
e
p(t) =
e
T f(t),
(4.34)
which gives (see Eq. 4.24)
diag{e
2
i ˜
m i } ¨
p(t) + diag{e
2
i ˜
c i } ˙
p(t) + diag{e
2
i
˜
k i } p(t) = e
T
T f(t),
(4.35)