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damping, κΩ is the natural undamped frequency, and lb = I/Mb rg.
The displacement of the centre of gravity of the rotor in the y direction has already
been found to be
y r bk
b
rg g =0 k k
–1
= ( / ) Σ ξ cos ψ (9.30)
so that the equation of motion of the airframe and chassis is
( + b) = – b rg – ( + b) c – 2 ( + )
2 2
M bM ˙y˙ bM ˙y˙ M bM κΩy κcΩδc M bMb y
= g b =0 cos – ( + )
–1
b c
r M M bM 2 2y
k
b
k k Σξ ψ κ Ω
– 2κcΩδc (M + Mb) ˙y (9.35)
ψk
ξk
y˙˙
y˙˙dm
r
Fig. 9.21 Inertial force on blade due to fuselage motion
Aeroelastic and aeromechanical behaviour 347
where δc is the damping coefficient of the chassis and κcΩ is its undamped natural
frequency, M being the effective mass of the fuselage in the chassis mode.
The periodic terms in eqns 9.34 and 9.35 can be removed by using the Coleman
co-ordinates, Appendix A.3. Let
η = – (2/ ) ξ sin ψ
=0
–1
b
k
b
k k Σ
and
ζ = – (2/ ) ξ cos ψ
=0
–1
b
k
b
k k Σ
from which we get
Σ Σ k
b
k k k
b
b k k b
=0
–1
12
=0
–1
12
ξ˙ sin ψ = (Ωζ – η˙ ); ξ˙ cos ψ = – (Ωη – ζ˙)
Σ k
b
k k b
=0
–1
12
ξ˙˙ sin ψ = – (η˙˙ – 2Ωζ˙ – Ω2η)
Σ k
b
=0 k
–1
12
ξ˙˙ cos ψ = – (ζ˙˙ + 2Ωη – Ω2ζ)
Then, multiplying eqn 9.34 by sin ψk and summing over all the blades, we get
η˙˙ + 2Ωδη˙ + (κ2 – 1)Ω2η – 2Ωζ˙ – 2Ω2δζ = 0 (9.36)
Repeating the process with cos ψk gives
ζ˙˙ + 2 δζ˙ + (κ2 – 1) ζ + 2 η˙ + 2 δη – ˙˙/ = 0
Ω Ω2 Ω Ω2 y lb (9.37)
and eqn 9.35 can be written
˙y˙ + 2 cy˙ + c y – r ˙˙ = 0
2
Ωδ κΩ2 μgζ (9.38)
where μ = /( + ) 12
bMb M bMb is the ‘mass ratio’, i.e. the ratio of half the blade mass
to the total mass.
Equations 9.36, 9.37 and 9.38 are three simultaneous differential equations with
constant coefficients in the variables η, ζ, and y. The solutions of these equations can
be obtained in the same way as the stability equations of Chapter 5 by assuming that
η = η0eλt, ζ = ζ0 eλt, y = y0 eλt
Substituting in the equations and expanding the determinant of the co-efficients
leads to a frequency equation of the form
Aλ6 + Bλ5 + Cλ4 + Dλ3 + Eλ2 + Fλ + G = 0 (9.39)
where the coefficients are functions of the non-dimensional chassis and blade natural
frequencies κcΩ and κΩ. Unfortunately, the above sextic cannot be solved in general
terms and, even when solved numerically, the many parameters involved make
348 Bramwell’s Helicopter Dynamics
interpretation difficult. One method of obtaining useful results is to note that the
regions of instability will be bounded by curves representing undamped (neutral)
oscillations. These can be found by assuming a solution to eqn 9.39 in the form
λ = iω, where ω is real. Inserting this solution into the sextic and equating real
and imaginary parts leads to the equations
Aω6 – Cω4 + Eω2 – G = 0 (9.40)
Bω4 – Dω2 + F = 0 (9.41)
in which A = 1.
To solve these equations we take a range of values of κc, calculate the values of a,
c, and e (which are functions of κc) and solve the biquadratic equation 9.41 for ω.
These values of ω are inserted into the left-hand side of eqn 9.40 which is then
plotted against ω. The values of ω which give zero values to the left-hand side of eqn
9.40 are solutions of eqns 9.40 and 9.41. We therefore have the values of the chassis
frequency at which undamped oscillations occur. Coleman and Feingold11 and Price12
have presented these boundaries in the form of charts which enable the ranges of
rotor speed, if any, for which the oscillations are unstable to be found.
Although the method described above gives a complete solution to the equations,
　
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