262
7 Forced Vibration Analysis of Multiple Degrees of Freedom System
In mode superposition method, which is also termed as normal mode method,
the equations are uncoupled with the help of normal coordinates [1–5]. But before
we start to evaluate the response, the results of natural frequencies and mode shapes
must be available. The displacements are expressed as a linear function of mode
shapes [same as Eq. (6.26)].
{x} = []{ξ }
(7.2)
Substituting {x} and { ¨
x} from Eq. (7.2) into Eq. (7.1), and premultiplying by []
T ,
we get
[]
T [M][]
¨
ξ
+ []
T [K ][]{ξ } = []
T
{F(t)}
(7.3)
Combining Eqs. (6.29) and (7.3), we get
[]
T [M][](
¨
ξ
+
p
2
{ξ }) = []
T
{F(t)}
(7.4)
If [M] is a diagonal matrix, then Eq. (7.4) will result in n uncoupled equations.
The rth equation is given by
¨
ξ r + p
2
r ξ r =
¯
f r
¯
m r
(7.5)
where
¯
f r =
φ
(r )
T {F(t)} =
n
i=1
φ
(r )
i F i (t)
(7.6)
and
¯
m r =
φ
(r )
T [M]
φ
(r )
=
n
i=1
m i
φ
(r )
i
2
(7.7)
Equation (7.5) is identical to the SDF system forced vibration equation. The
solution of this equation is [see Eq. (3.69)]
ξ r =
1
p r
t
o
¯
f r
¯
m r
sin p r (t − τ )dτ
(7.8)