206
6 Free Vibration of Multiple Degrees of Freedom System
Further, the following relation follows from the stiffness property of the spring.
N
L
i = N
R
i − 1 = k i (x i − x i − 1 )
(6.61)
Equations (6.60) and (6.61) can be rearranged in the following form
x i = x i − 1 +
N
R
i − 1
k i
N
L
i = (0) x i − 1 + N
R
i − 1
(6.62)
Writing Eq. (6.62) in matrix form, we get
x 1
N
L
i
=
1
1
k 1
0 1
x i − 1
N
R
i −1
(6.63)
or
x
N
L
i
=
1
1
k 1
0 1
x
N
R
i − 1
or
{Z }
L
i = [T ] i {Z }
R
i − 1
(6.64)
[T ] i matrix relates the state vector {Z }
L
i with {Z }
R
i − 1 which is known as field transfer
matrix or simply as field matrix.
The mass is considered to be rigid. Therefore, the displacement of the mass to the
left is equal to that to the right.
x
R
i = x
L
i
(6.65)
The spring forces to the left and right of the mass are N
L
i and N
R
i , and there is an
inertia force m i p
2 x i acting. Therefore,
N
R
i = N
L
i − m 1 p
2 x i
(6.66)
It may be noted that while deriving Eq. (6.66), advantage has been taken of the
fact that ¨
x = − p
2 x. Equations (6.65) and (6.66) can be written in matrix notations
as follows:
x
N
R
i
=
1
0
− m i p
2 1
L
i
x
N
L
i
(6.67)