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It has been seen that the rotor blade, whether supposed rigid or flexible, can be
treated as a spring–mass–dashpot, or second order, system when the blade displacement
from equilibrium is small. Further, in steady flight, the loads forcing the blade are
periodic and can be expressed in the form of a Fourier series. The blade motion can
therefore be derived from the response of a second order system to a harmonic
forcing function and it is useful to present the main results in this Appendix.
The differential equation of the motion can be represented typically as
m˙x˙ + cx˙ + kx = F0 cos ωt (A.4.1)
where m is the mass of the system, c is the viscous damping coefficient, k is the
spring stiffness, and F0 is the amplitude of the applied force.
When F0 = 0 we have the case of free motion; when the damping is zero the free
motion is sinusoidal with natural angular frequency ωn given by ωn = √(k/m).
Let us write the critical damping coefficient (just suppresses oscillatory motion)
ccrit = 2 √(k/m) = 2k/ωn and define a non-dimensional damping factor ζ by ζ = c/ccrit.
The equation of motion, eqn A.4.1, can then be written
˙x˙ + 2ζωnx˙ + ωn2 x = (F0 /m) cos ω t
The solution of this equation is known to be
x = x0 cos (ω t – ε) (A.4.2)
where x0 = F0/ √[(k – mω2)2 + c2ω2]
and ε = tan–1 [cω/(k – mω2)]
Defining a frequency ratio ω˜ by ω˜ = ω/ωn, x0 and ε can be written
x0 = F0 /[k √{(1 – ω˜ 2)2 + 4ζ2ω˜ 2}] (A.4.3)
and
ε = tan–1 [2ζω˜ /(1 – ω˜ 2)]
Appendices 369
Now F0/k is the static deflection, i.e. the displacement of the system under a steady
load F0. Denoting this deflection by xst we can represent the amplitude x0 of the
oscillating displacement as the static deflection multiplied by a magnification factor
μ , where
μ = x0/xst = 1/√[(1 – ω˜ 2)2 + 4ζ2ω˜ 2] (A.4.4)
Thus the response of the system can be completely expressed by the quantities μ
and ε as functions of the frequency ratio ω˜ and the damping ratio ζ, these quantities
being shown plotted in Figs A.4.1 and A.4.2.
It can be seen from the solution of eqn A.4.2 that the response frequency is always
0.5 1.0 1.5 2.0 2.5 3.0
180°
135°
90°
45°
0
0.1
0.25
0.5
1
2
4
4
2
0.5
1
0.1
ω˜
ε
0.25
ξ = 0.01
Fig. A.4.2 Phase relationship between response of second order system and forcing function
5
4
3
2
1
0
0.5 1.0 1.5 2.0 2.5
0.1
0.15
0.25
0.5
1
4
ξ = 0
ξ = 0
μ
ω˜
2
Fig. A.4.1 Amplitude of second order system to harmonic forcing function
370 Bramwell’s Helicopter Dynamics
the same as the forcing frequency. When ω˜ = 1, i.e. when ω = ωn, the system is said
to be forced at its resonant frequency. It might be thought that the response amplitude
is greatest at this frequency, but examination of Fig. A.4.1 shows that the maximum
amplitude occurs at rather less than the resonant frequency. In fact, the aerodynamic
damping factor of a flapping blade is typically of the order ζ = 0.4, from which we
see that the maximum amplitude occurs at ω˜ = 0.85. However, at resonance the
phase angle is always 90° however great the damping.
Index
Rayleigh–Ritz method, 246
Southwell formula, 248
flapwise bending modes and frequencies,
238
intermodal coupling, 262
lagwise bending modes and frequencies,
257
forced response equation, 269
torsional modes and frequencies, 260
Biot–Savart law, 198
Blade deflections measured in flight, 276
Blade element theory:
forward flight, 92
vertical flight, 46
Blades, bending, see bending of blades
Blades, dynamic design, 294
Blade–vortex interactions (BVI), 219–221
Boundary layer on rotating blade:
forward flight, 235
hovering flight, 233
Circulation:
effect of finite number of blades,
62
Coleman co-ordinates, 367
Coleman et al:
ground resonance, 342
induced velocity distribution, 79
Coning angle, equation for, 107
Control derivatives, 180–183
Control response, 180–192
to cyclic pitch, 185
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