6.3 Free Undamped Vibration Analysis of MDF Systems
187
p
is a scalar of dimension T
− 1 ,
{φ } is a non-dimensional vector, such that
{φ}
T
= {φ 1 , φ 2 , φ 3 , · · · , φ n }
Therefore,
{ ¨
x } = − ap
2 e
i pt
{φ }
(6.7)
Substitution of the values of { ¨
x } and {x } from Eqs. (6.6) and (6.7) into Eq. (6.5)
yields
− p
2
[M] {φ } + [K ] {φ } = {O }
(6.8)
or
[M] {φ } − λ [K ] {φ } = {O }
(6.9)
where λ =
1
p 2 .
or
( [M] − λ [K ] ) {φ } = {O }
(6.10)
Equation (6.9) will have a non-trivial solution only if the determinant corresponding to ( [M] − λ [K ] ) vanishes, that is
| [M] − λ [K ] | = 0
(6.11)
The determinant is known as the frequency determinant. The solution of Eq. (6.11)
will give n values of λ or 1/p
2 , where n is the number of degrees of freedom.
p 1 , p 2 , ..., p r , ..., p n are called natural frequencies of the system. Corresponding
to each value of p r , a vector {φ
(r )
} can be evaluated. These are variously termed as
natural mode, normal mode or principal mode of vibration.
Equation (6.9) is rewritten in terms of a typical eigenvalue problem. Premultiplying both sides of Eq. (6.9) by [K ]
− 1 yields
[K ]
− 1
[M] {φ } = λ [I ] {φ }
(6.12)
or
[D] {φ } = λ [I ] {φ }
(6.13)