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DYNAMICS PROBLEMS

Example: Helicopter pull-up maneuver
The helicopter in Fig. 1↓ is flying at a speed of 70 m/sec, and its rotor speed is 35 rad/sec. The rotor radius, R, is 5 m. The helicopter is performing a pull-up maneuver with a radius, r, of 100 m.
figure helicopter.png
Figure 1 Helicopter undergoing a pull-up maneuver

Find:
  1. The angular momentum of the rotor in the coordinate system shown in Fig. 2↓ which moves with the helicopter.
  2. The moment needed to be applied to the rotor to keep it from tilting during the maneuver.
Note: you may assume that the rotor disk may be approximated as a thin disk with mass m = 150 kg.
figure coord_sys.png
Figure 2 Coordinate system which moves with the helicopter

Solution
The first step is to find the moment of inertia of the rotor about each of the principle axes. For a thin disk with the coordinate system shown, the moment of inertia is:
(1) Ix  = mR2 = 3750kgm2 Iy  = Iz = (1)/(2)mR2 = 1875kgm2
Next, the angular velocity about each axis is required.
(2) ωx  = 35rad ⁄ sec ωy  = (V)/(r) = 0.7rad ⁄ sec ωz  = 0
1. It is now possible to find the angular momentum.
(3) H  = Ixωx + Iyωy + Izωz  = 3750⋅35 + 1875⋅0.7  =  131, 250 1312.5 0 kgm2 ⁄ sec
2. To find the moment applied to the rotor, it is important to note that the coordinate system is rotating with constant angular velocity because the helicopter is executing a pull-up maneuver. The rotational velocity of the coordinate system, Ω, is
(4) Ω =  0 0.7 0 rad ⁄ sec
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