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coefficients, and to solve them we put*
u = u0eλτ, w = w0eλτ, θ = θ0eλτ
where u0, w0, θ0, and λ are constants.
Substituting in eqns 5.11, 5.12, and 5.13 and cancelling throughout by eλτ we
obtain
(λ – xu) u0 – xww0 + wcθ0 cos τc = 0 (5.14)
– zuu0 + (λ – zw) w0 – ( ˆVλ – wc sin τc)θ0 = 0 (5.15)
– 0 – ( + )0 + ( – ) = 0
2
muu λmw˙ mww λ mqλ θ 0 (5.16)
For non-trivial and consistent values of u0, w0, θ0 it is necessary for the determinant
of the coefficients in eqns 5.14, 5.15, and 5.16 to be zero, i.e. for
– – cos
– – –( – sin )
– – ( + ) –
= 0
c c
c c
2
λ τ
λ λ τ
λ λ λ
x x w
z z V w
m m m m
u w
u w
u w w q
ˆ
˙
Expanding this determinant leads to the characteristic equation
A1λ4 + B1cλ3 + C1λ2 + D1λ + E1 = 0 (5.17)
where
A1 = 1
B1c = N1 – mq – Vˆmw † ˙ (5.18)
C1 = P1 – N1mq – Q1mw˙ – Vmw ˆ (5.19)
D1 = S1mu – P1mq – R1mw˙ – Qmw (5.20)
E1 = T1mu – R1mw (5.21)
and
N1 = – xu – zw (5.22)
* Care must be taken not to confuse θ0 here with the collective pitch angle; from this point down
to eqn (5.16) θ0 refers to the maximum amplitude of pitch θ of the whole aircraft.
†B1c as a coefficient of the characteristic equation is distinct from B1, the longitudinal cyclic pitch
angle.
Flight dynamics and control 145
P1 = xuzw – xwzu (5.23)
Q1 = –Vˆxu – wc sin τc (5.24)
R1 = – wc(zu cos τc – xu sin τc) (5.25)
S w Vxw 1 = c cos τc – ˆ (5.26)
T1 = – wc(zw cos τc – xw sin τc) (5.27)
the suffix 1 denoting longitudinal coefficients.
Equation 5.17 will have four roots λ1, λ2, λ3, λ4 which may be either real or
complex. Thus the general solution for u can be written
u = c1e + c2e + c3 e + c4e
λ1τ λ2τ λ3τ λ4τ (5.28)
where the constants c1, c2, c3, and c4 can be determined from the initial conditions.
Solutions for w and θ can be written in the same way.
Since we are concerned here only with the stability of the motion, we need consider
only the values of λ.
When λ is real and positive, eλτ increases without limit and the motion is known
as a divergence, Fig. 5.3.
When λ is real and negative, eλτ decreases steadily to zero and the motion is
known as a subsidence, Fig. 5.3.
When λ is complex it can be written as
λ = λre ± iλim
since the complex roots always appear as conjugate pairs.
The mode of motion corresponding to this pair of roots can be expressed as
u = k1e sin im + k2e cos im
λreτ λ τ λreτ λ τ (5.29)
If λre is positive, the motion is a divergent oscillation, Fig. 5.4; if λre is negative,
the motion is a convergent or damped oscillation.
The rate at which these motions subside or diverge is determined from the real
values or real parts of λ. It is usual to express this rate as the time to halve or to
double the amplitude of the motion. It can easily be seen that the time to half
amplitude Τh is given by
λ positive
(divergence)
λ negative
(subsidence)
τ
eλτ
Fig. 5.3 Divergence and subsidence
146 Bramwell’s Helicopter Dynamics
eλreτ
Damped oscillation (λre negative)
Divergent oscillation (λre positive)
Fig. 5.4 Damped and divergent oscillations
Th = ln 2tˆ/(–λ) = 0.693tˆ/(–λ) (5.30)
when λ is real and negative or
Th = 0.693tˆ/(–λ re ) (5.31)
when λ is complex and λre is negative.
The time to double amplitude, Td, is given by
Td = 0.693tˆ/λ (5.32)
for real and positive λ
or Td = 0.693tˆ/λ re (5.33)
when λ is complex and λre positive.
From eqn 5.29 it can be seen that the periodic time, T, is given by
λimτ = λimT/tˆ = 2π
i.e. T = 2πtˆ/λim (5.34)
τ
τ
Flight dynamics and control 147
　
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