6.13 Matrix Deflation Procedure
229
This is a much better convergence than in (a), as indicated above. Continuing in
this way for a few more iterations, the quantities involved in second normal mode
may be determined.
6.14 Rayleigh’s Method
Rayleigh’s method is an approximate method to determine the fundamental frequency
of a MDF system. The concept can be extended easily to continuous systems, as
has been shown in a later chapter. For practical problems, the knowledge of the
fundamental frequency of a system is always of great interest.
Let [K] and [Ml be the stiffness and mass matrices, respectively, of a MDF system.
From Eq. (6.6), we can write
{x} = Ae
i pt
{φ}
(6.99)
The maximum kinetic and potential energies can be written as
T max =
1
2
{ ˙
x}
T
max [M] { ˙
x} max
(6.100)
U max =
1
2
{x}
T
max [K ] {x} max
(6.101)
Substituting values of {x} and { ˙
x} from Eq. (6.99) into Eqs. (6.100) and (6.101),
we get
T max =
1
2
p
2 A
2
{φ}
T
[M] {φ}
(6.102)
U max =
1
2
A
2
{φ}
T
[K ] {φ}
(6.103)
Equating the maximum energies of the system, we get
p
2
=
{φ}
T
[K ] {φ}
{φ} T [M] {φ}
(6.104)
The above method is referred to as Rayleigh’s method. As it is always possible to
make a fairly reasonable estimate of the fundamental mode shape. Rayleigh’s method
gives a good idea about the fundamental frequency. The method is upper bound.
Example 6.12 Determine by Rayleigh’s method, the fundamental frequency of the
system shown in Fig. 6.17.