436
11 Vibration of Ship and Aircraft as a Beam
First of Eq. (11.26) be multiplied by m 1 , second by m 2 and so on. Similarly, first
of Eq. (11.27) be multiplied by m 1 s 1 , the second by m 2 s 2 and so on. The results of
these two equations when added lead to
n
i=0
m i y i +
n
i=0
m i s i α i −
n
i=0
m i y 0 −
n
i=0
m i x i
θ 0 −
n
i=0
m i s i
α 0 = ω
2
[A 1 y 1 + · · · + A n y n + B 1 α 1 + · · · + B n α n ]
(11.30)
where
A n =
n
i=1
(m i m n a in + m i m n s i s n b in )
(11.31)
and
B n =
n
i=1
(m i m n s n a in + I n m i s i b in )
(11.32)
Equation (11.30) is rewritten using the first relation of Eq. (11.29) as
−My 0 − Pθ 0 − Qα 0 = ω
2
[A 1 y 1 + · · · + A n y n + B 1 α 1 + · · · + B n α n ] (11.33)
where M =
m i , P =
m i x i and Q =
m i s i
Similarly, the first of Eq. (11.26) is multiplied by m 1 s 1 , second by m 2 s 2 and so
on. In the same manner, Eq. (11.27) is multiplied by I 1 , second by I 2 and so on. The
results of these two equations when added yield
n
i=0
m i s i y i +
n
i=0
¯
I i α i −
n
i=0
m i s i
y 0 −
n
i=0
m i s i x i
θ 0 −
n
i=0
¯
I α 0 = ω
2
[C 1 y 1 + · · · + C n y n + D 1 α 1 + · · · + D n α n ] (11.34)
where
C n =
n
i=0
(m i m n s i a in + m n s n I i b in )
(11.35)
and
D n =
n
i=1
(m i m n s i s n a in + I i I n b in )
(11.36)