102
4 Multi-Degree of Freedom (MDOF) Systems
4.5.4 Forced Vibration
We have seen that, if the eigenvalues are distinct, the right and left eigenvectors provide basis in R 2 n since they are linearly independent (see Sect. 3.2.2). Accordingly,
the 2 n-dimensional vector z(t) can be represented at any time t by its projections
on the right or left eigenvectors, e.g.,
z(t) =
2 n
i=1
u i q i (t) = u q(t)
(4.87)
with the notation q(t) = [q 1 (t) q 2 (t) . . . q 2 n (t)] T . As previously, we refer to the
components of q(t) as modal coordinates. If the eigenvalues are not distinct, the set
of right/left eigenvectors and their corresponding generalized eigenvectors have to
be used to represent the state vector z(t).
The representations of z(t) in Eq. 4.87 and Eq. 4.77 give
u ˙
q(t) = a u q(t) + b f(t) (which becomes)
v
T u ˙
q(t) = v
T a u q(t) + v
T b f(t) (by left multiplication with v
T )
v
T
i u i ˙
q i (t) = v
T
i a u i q i (t) +
v
T b f(t)
i
, i = 1, . . . , 2 n, (by orthogonality).
The latter equation and Eq. 4.84 give
˙
q i (t) = λ i q i (t) +
v T b f(t)
i
v T
i u i
, i = 1, . . . , 2 n,
(4.88)
whose solution is (see Appendix B)
q i (t) = q i,0 e
λ i t
+
t
0
e
λ i (t−s)
v T b f(s)
i
v T
i u i
ds,
(4.89)
where {q i,0 } are the initial values of {q i (t)}, so that
z(t) =
2 n
i=1
u i q i (t) =
2 n
i=1
u i
q i,0 e
λ i t
+
t
0
e
λ i (t−s)
v T b f(s)
i
v T
i u i
ds
.
(4.90)
Note that the coefficients {λ i } and the forcing functions {
v T b f(t)
i
/
v T
i u i
of
Eq. 4.88 are complex-valued and so are the solution {q i (t)}. It can be shown by
using properties of eigenvectors and eigenvalues that the state vector z(t) has realvalued components [1]. We will only mention that the components of this vector
are real as they represent physical quantities, displacements, and velocities of the
system masses.