298
7 Forced Vibration Analysis of Multiple Degrees of Freedom System
As such ¯
m r = 1, obtained by adjustment of [] matrix. However, ¯
f r quantities are
to be calculated.
In the frequency response method, the solution is supposed to be complex and
so is the load. The load is therefore extended to a complex load vector to obtain a
simple solution of the differential equation. The dynamic response of the structure
will then be given by the real part of the solution.
The harmonic load in d.o.f. r is F r (t) which is
F r (t) = F r cos(ωt + α r )
(7.88)
To facilitate the solution of the differential equation, the load vector is expanded
into a complex quantity:
F r (t) = F r [cos(ωt + α r ) + i sin(ωt + α r )]
(7.89)
which is expressed as
F r (t) = F r e
i(ωt+α r )
(7.90)
Equation (7.90) can be expressed as follows
F r (t) =
F
(r )
R + i F
(r )
I
e
iωt
(7.91)
where F
(r )
R is the real part and F
(r )
I is the imaginary part.
Therefore, from Eqs. (7.6) and (7.91), we get
¯
f r =
⎡
⎣
n
j=1
φ
(r )
j F r (t)
⎤
⎦
or,
¯
f r =
⎡
⎣
n
j=1
φ
(r )
j F
(r )
R + i
n
j=1
φ
(r )
j F
(r )
I
⎤
⎦ e
iωt
(7.92)
In Eq. (7.92), we indicate the real terms by ¯
f
(r )
R and imaginary terms by ¯
f
(r )
I and
the resultant complex vector of Eq. (7.92) is given by
¯
f r =
¯
f
(r )
R + i ¯
f
(r )
I
e
iωt
= ¯
f
∗
r e
iωt
(7.93)