topic angular momentum kinetic energy 3d equations of motion precessio

Question

Topic: Angular Momentum, Kinetic Energy/3D Equations of Motion/Precession, Stability 5. The quad rotor drone is modelled as shown with a total mass of M=20 m and radius of gyration about the centre of mass O of d/2 and includes four rotor blade systems, each of mass m and diameter d. The rotor blade systems provide thrust in the+z directions that increases with the magnitude of the spin rate p. Assume no friction between the drone body and each rotor.Assume each drone rotor blade (note 2 blades per rotor) maybe modelled as a uniform thin rod with mass moment of inertia about its centre of mass I com-m d²/12.The rotors are spun up to speeds of p, p+4p and p-4p (in rad/s)relative to the drone body in the directions as shown above using motor torques acting between the rotor and body. The drone body is initially on the ground during spin up. (a) Calculate the angular momentum components and rotational kinetic energy of the drone after it is spun up to the steady speeds shown and reached a steady attitude (ie all other angular velocities are zero). (b) Explain why it is not necessary to hold the body of the drone fixed to the ground while it is being spun up. (c) Explain why the drone lifts off the ground, tilts and moves forward towards the -X direction (d) The tilting forward occurs at an angular rate of Qj rad/s (where 2 is -ve). Assuming this occurs at the steady rotor speeds,calculate the gyroscopic moments on each of the arms of the drone and the total on the drone. (e) Predict the free precession rate and direction of one of the spinning rotor systems as a function of the spin rate p if it breaks free of the drone body and is disturbed slightly with respect to the vertical/gravity.Predict and explain whether this motion is stable or not.