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The Galerkin process can be regarded as a ‘least-squares’ fit of the assumed
solution eqn 7.22 to the exact solution of eqn 7.11.
Duncan has also shown6 that the Galerkin and Rayleigh–Ritz methods, and therefore
the Lagrange method, are all equivalent in linear problems, but notes that the Galerkin
method can be extended to non-linear and non-conservative problems.
The application of the Galerkin method differs from the other two in that the
integrals to be calculated are those derived from the terms in the differential equation
7.11, instead of those to be found in eqns 7.88 and 7.90. This means that in the
Galerkin method the integrand of the elastic terms involves differential coefficients
of the stiffness, and this may be a disadvantage if the stiffness undergoes sudden
changes.
Figure 7.8 shows the rigid mode and the first bending mode when the first four
functions
1
0
–1
0.5 1
S1(x), λ1 = 1
S2(x), λ2 = 2.68
Fig. 7.8 Rigid and first bending modes and frequencies of a uniform beam
r/R
Structural dynamics of elastic blades 251
γ = x, γ
i = (i + 2)(i + 3)xi+1/6 – i(i + 3)xi+2/3 + i(i + 1)xi+3/6 i = 2, 3, 4
are used to calculate the mode shapes and frequencies of a uniform beam.
7.2.2.2 Method of lumped parameters
We now consider two methods of analysis which are described as ‘lumped-parameter’
methods. In this technique the continuous blade is represented by a number of discrete
segments, so that the partial differential equation of blade bending is replaced by a
set of simultaneous ordinary differential equations.
The two methods to be described are
(a) the Myklestad method,
(b) the dynamic finite element method.
(a) The Myklestad method
This is a development of the Holzer7 method which was originally used for the
calculation of torsional oscillations but was later extended and modified by Myklestad
for the calculation of lateral beam vibrations.
Consider the deflected blade, Fig. 7.9 divided into a number of concentrated
masses between which the elastic properties remain unaltered. The nth element and
the forces acting on it are shown in Fig. 7.10.
With reference to Figs 7.11 and 7.12 we also define the following elastic coefficients
in relation to the application of a unit force and of a unit moment.
Assuming the elastic properties are constant over the element, the four coefficients
can be expressed as
u l EI
u l EI
l EI
Mn ln EI
Fn n n
Mn n n
Fn n n
n
= /6( )
= /2( )
= /2( )
= /( )
3
2
v 2
v
We now imagine the blade to be forced to vibrate with harmonic motion of frequency
ω. This adds an inertia force mn+1ω2Zn+1 to the element, and the equilibrium of the
element leads to the following relations:
Gn+1 = Gn + mn+1Ω2rn+1 (7.23)
Z
rn
rn+1 αn+1
mn αn
Zn+1
Z
Z1
m1
r
Gn+1
Zn+1
Sn+1
ln
Zn
Gn
αn
Sn
αn+1
mn+1
Mn+1
Fig. 7.9 Distributed-mass representation of helicopter
blade (Myklestad)
Fig. 7.10 Forces on blade element (Myklestad
method)
252 Bramwell’s Helicopter Dynamics
Sn+1 = Sn – mn+1ω2Zn+1 (7.24)
Mn+1 = Mn – Snln + Gn(Zn – Zn+1) (7.25)
αn+1 = αn(1 + GnvFn) – SnvFn + MnvMn (7.26)
Zn+1 = Zn – (ln + uFnGn)αn + uFnSn – uMnMn (7.27)
We obtain at once from eqn 7.23
Gn m r
i
n
= i i
=1
Σ Ω2 (7.28)
and eliminating Zn – Zn+1 between eqns 7.25 and 7.27 gives
Mn+1 = Mn(1 + uMnGn) – Sn(ln + uFnGn) + Gn(ln + uFnGn) (7.29)
For a given value of ω, these recurrence relations enable us to calculate S, M, α
and Z at any point along the blade in terms of the corresponding values at one end.
Thus, for example, the boundary conditions at the blade tip are
S1 = m1ω2; M1 = 0; α1 = α10, say; Z1 = 1
It may seem that the first boundary condition contradicts the requirement of zero
shear force at the tip, but it must be remembered that the mass of the last element is
concentrated at the tip so that it really represents an average condition for the element.
The last two boundary conditions are arbitrary and merely define the scale of the
blade displacement. It can easily be seen that, at any given station, S, M, α, and Z
must be of the form
Sn = cn + dnα10 (7.30)
Mn = en + fnα10 (7.31)
αn = gn + hnα10 (7.32)
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Bramwell’s Helicopter Dynamics(126)
