8.16 Rayleigh–Ritz Method for Determining Natural …
357
Substituting Eq. (8.197) into Rayleigh’s quotient in Eq. (8.195), we obtain
p
2
=
L
0 E I
n
i=1 C i
d
2 φ i
dx 2
2
dx
L
0 ρ A
n
i=1 C i φ i
2 dx
=
n
i=1
n
j=1 C i C j
L
0 E I
d
2 φ i
dx 2 ·
d
2 φ j
dx 2 dx
n
i=1
n
j=1 C i C j
L
0 ρ A φ i φ j dx
(8.198)
or
p
2
=
n
i=1
n
j=1 k i j C i C j
n
i=1
n
j=1 m i j C i C j
(8.199)
where
k i j =
L
0
E I
d
2
φ i
dx 2 ·
d
2
φ j
dx 2 dx
m i j =
L
0
ρ A φ i φ j dx
The frequencies determined from Eq. (8.199) are minimum. In order to do that,
we differentiate p
2 with respect to each of these constants and put them equal to zero.
Therefore,
2
n
j=1 k i j C j
i
j m i j C i C j
−
2
n
i=1
n
j=1 k i j C i C j
n
j=1 m i j C j
n
i=1
n
j=1 m i j C i C j
= 0
(8.200)
Combining Eq. (8.200) with Eq. (8.199), we get
k i j C j − p
2
m i j C j = 0
(8.201)
Equation (8.201) can be written in matrix form as
([K ] − p
2
[M]){φ} = 0
(8.202)
Equation (8.202) represents an eigenvalue problem, which on solution yields
natural frequencies and mode shapes.
Example 8.9 Determine the first two natural frequencies of a uniform cantilever
beam by Rayleigh–Ritz method.