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It will be seen from energy considerations (eqns 7.87, 7.89 and 7.91) that L, termed
the Lagrangian, is the difference between the kinetic and strain energies in a given
mode of motion, i.e.
L = T – U
The Rayleigh–Ritz procedure is to assume a finite series of approximation functions
Sn(x) = A1γ1(x) + A2γ2(x) + … + Aiγi + … + Anγn
where the terms γ
i satisfy the boundary conditions, and substitute the series into
eqn 7.17. Since A1, A2, …, etc. can be arbitrarily varied, the condition that L should
have stationary values is
∂
∂
∂
∂
∂
∂
∂
∂
L
A
L
A
L
A
L
1 2 i An
= = … = = … =
giving n equations for the evaluation of the terms Ai.
1
0.5
0 0.2 0.4 0.6 0.8 1.0
S1(x)
x
0.2 0.4 0.6 0.8 1.0
x
1
0.5
0
–0.5
–1
S2(x)
Fig. 7.6 Normal mode shapes
Structural dynamics of elastic blades 247
We find that this gives exactly the same set of equations as we obtained by the
Lagrange method of the previous section. It is worth mentioning, however, that the
Rayleigh–Ritz method can be applied to problems, e.g. static equilibrium, to which
the Lagrange equations would not normally be applicable.
The Rayleigh–Ritz method is a generalization of the application of Rayleigh’s
principle2 that the frequencies corresponding to the solutions of eqn 7.11 have stationary
values. This means that, if the chosen functions differ from the exact solutions Si by
small quantities of the first order, the calculated frequencies will be in error by only
small quantities of the second order. Thus, it can be expected that quite a poor
approximation to the true mode shape will give a good approximation to the frequency.
Now, when a system oscillates in a normal mode with frequency ω
i = λ
iΩ, each
part of the system oscillates in phase or in antiphase with every other part of the
system. Thus we can write for a typical displacement in a normal mode
Z = γ
i(x) sin (ω
it + ε)
= γ
i(x) sin (λ
iψ + ε)
Then the kinetic energy T of the system, as will be seen from eqn 7.87, is
T R m x = 12
2 3 d
0
1 2
Ω ∫
∂γ
∂ψ
= 12
2 2 3 cos2( + ) ( )d
0
1
λ λψ ε 2 i ΩR i ∫my x x
and from eqns 7.89 and 7.91 the potential energy of the rotating blade is
U = UB + UG
= 1
2
sin ( + )
d
d
d + 12
sin ( + )
d
d
2 d
0
1 2
2
0
1 2
R
EI
x
x R G
x i x
i
i
λ ψ ε γ λ ψ ε γ i ∫ ∫
In this simple harmonic motion the maximum kinetic energy must equal the maximum
potential energy. Thus the coefficients of cos2(λ
iψ + ε) and sin2(λ
iψ + ε) can be
equated giving
λ
γ γ
γ
i
i i
i
EI x x R G x x
R m x
2 2 0
1
2 2 2 2
0
1
2
4
0
1
2
=
(d /d ) d + (d /d ) d
d
Ω ∫ ∫
∫
(7.18)
Then, if a mode shape γ
i(x) is assumed which is reasonably near an exact mode,
eqn 7.18 gives the corresponding frequency. The use of the Rayleigh method is
usually confined to the calculation of the lowest frequency, since the corresponding
mode shape will probably be comparatively simple and a good guess can usually be
made.
Equation 7.18 has been used by Southwell3 to obtain a relationship to the non248
Bramwell’s Helicopter Dynamics
rotating frequency of the blade and its natural frequency under rotation. Equation
7.18 can be written
ω λ ω
γ
γ
i i nr
i
i
R
G x x
m x
2 2 2 2
2
0
1
2
0
1
2
= = + 1
(d /d ) d
d
Ω ⋅∫
∫
(7.19)
where ωnr is the non-rotating frequency of the blade.
Now
G m rr
R mx x
r
R
x
= d
= d
2
2 2
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Bramwell’s Helicopter Dynamics(124)
