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Fig. 7.26. Improved accuracy can be obtained by assuming that fi varies linearly over
an interval according to the slope of the straight line connecting the values of fi – 2 and
fi–1, Fig. 7.27. This is called a first-order hold. The differential equation for this case
is
Fig. 7.25 Discrete input frequency
f
Δψ Δψ ψ
f
ψ
Fig. 7.26 Zero-order hold forcing
Structural dynamics of elastic blades 275
d
d
+ = +
–
2
2
φ 2 –1
ψ
λ φ f ψ ψ
f f
i
i i
Δ (7.101)
with the solution
φ
λ
λ ψ
λ ψ λ ψ
λψ φ λψ φ
= (1 – cos ) + λ λψ
–
–
–
sin + cos + sin 2
–1
2
–1
3
f f f f f i i i i i
i
Δ i
Δ Δ
′
(7.102)
By proceeding as before, we obtain the recurrence relationship
φi
+2 = 2φi+1 cos λΔψ – φ
i +
1 ( – 2 + ) cos –
sin
+ 2 (1 – cos ) 2 –1 +1 +1 λ
λ ψ
λ ψ
fi fi fi Δ λ ψ fi λ ψ
Δ
Δ Δ
(7.103)
Wilkinson and Shilladay have tested the recurrence relationships eqns 7.100 and
7.103 for cases with known exact solutions and have found that they are more
accurate and allow faster computation than the Runge–Kutta method.
We can also take advantage of the fact that in steady flight the aerodynamic
loading, and the consequent blade motion, is periodic so that the right hand side of
the modal equation 7.86 is expressible in the form of a Fourier series. If this is done,
the complete solution of eqn 7.86 can be written down at once, since it represents the
well known response of a second-order system forced by a series of harmonic functions,
Appendix A.4. Of course, the loading integral on the right hand side of eqn 7.86 will
not, initially, be known exactly, since it depends in a complicated way on the motion
being calculated. An integrative solution must be adopted, and the classical rigid
blade flapping can be used as a first approximation. The loading integral is then
calculated, expressed as a Fourier series, and the values of the mode displacements
φn are obtained. The values of φn and dφn/dψ are then used to obtain a second
approximation to the loading ∂F/∂x, and the process is repeated until satisfactory
convergence has been achieved. There are, of course, a number of numerical methods
for calculating the Fourier series of a set of periodic values.
If, in the method just described, the modal equation is used as it appears in
f
ψ
Fig. 7.27 First-order hold forcing
276 Bramwell’s Helicopter Dynamics
eqn 7.86, we shall encounter the difficulty of resonance16 for those modes with a
natural frequency equal or close to the rotor frequency. This can be avoided by noting
that the aerodynamic loading contains damping which, although non-linear on account
of the non-linear aerodynamic data which will undoubtedly be used, will nevertheless
be roughly proportional to dφn/dψ.
It such a term is removed from the loading integral of eqn 7.86 and is transferred
to the left hand side of the equation, we shall have the case of a harmonically forced
system with quite high linear damping, and the large amplitudes near resonant frequency
will not occur. Since dφn/dψ will not be known when transferred, the calculation of
the damping terms remaining in the integral must be based on the blade motion of the
previous approximation, as has been described for the case earlier.
7.4 Blade deflections in flight
The deflection of the rotor blade in flight can be calculated by the method of the
previous section in conjunction with the estimated blade mode shapes and frequencies
and the aerodynamic loading. These calculations can also be used to obtain the
stresses in the blade and at the hub. For illustration, however, we give below examples
of the measurement of blade deflection obtained by photographing the blade in flight.
In this experiment, a 16 mm camera taking 240 frames per second (about 60 frames
per rotor revolution) was mounted on the hub of an experimental Westland ‘Scout’
with a hingeless rotor, and was directed to look along the span of one of the blades.
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