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Converted document DYNAMICS PROBLEMS

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 1 Helicopter undergoing a pull-up maneuver

Find:

The angular momentum of the rotor in the coordinate system shown in Fig. 2↓ which moves with the helicopter.
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 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) I_x = mR² = 3750kgm² I_y = I_z = (1)/(2)mR² = 1875kgm²

Next, the angular velocity about each axis is required.

(2) ω_x = 35rad ⁄ sec ω_y = (V)/(r) = 0.7rad ⁄ sec ω_z = 0

1. It is now possible to find the angular momentum.

(3) H = I_xω_xî + I_yω_yĵ + I_zω_zk̂ = 3750⋅35î + 1875⋅0.7ĵ = ⎡⎢⎢⎢⎣ 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