The figure shows a space satellite Js whose absolute pointing angle 0s is contro

Question

The figure shows a space satellite Js whose absolute pointing angle 0s is controlled by varying the speed of a reaction wheel Jw. The reaction wheel can be thought of as an electric motor fixed to the satellite with a flywheel attached to it. The angular velocity of the reaction wheel relative to the satellite can be varied by changing the voltage applied the motor armature winding. Because of the reaction torque between the satellite and the motor, there is no change in total angular momentum of the combined satellite and reaction wheel. Thus, as the speed of the wheel relative to the satellite is varied, the angular velocity of the satellite with respect to an inertial reference frame must vary such that the total angular momentum remains constant. ·?7777 Treat the diagram as the physical model. Assume that the motor stator is rigidly attached to the satellite and that the motor rotor and shaft are rigidly attached to the reaction wheel. Js is the mass moment of inertia of the satellite (excluding the reaction wheel, motor rotor, and motor shaft). Jw is the mass moment of inertia of the combined reaction wheel, motor rotor, and motor shaft· 0, 1s the absolute angular position of the satellite· is the absolute angular position of the reaction wheel. ?w-a-Ows where ?w's 1s the relative angular position of the reaction wheel with respect to the satellite. There is viscous damping between the satellite motion and the reaction wheel motion represented by a viscous damping coefficient B. Neglect actuator and sensor dynamics

Explanation / Answer

given, moment o finertia of the satellite, excluding the rotating parts = Js
moment of inertia of the rotating parts = Jw

absolute angular position of satellite = thetas
absolute angular position of wheel = thetaw
relative angular position of wheel = thetaws
thetaws = thetaw - thetas

now, from conservation of angular momentum
Js*thetas' + Jw*thetaw' = 0
differnetiating
Js*thetas" + Jw*thetaw" = 0
also, torque applied by motor = M
Jw*thetaws" + B*thetaws' = M
Jw(thetaw" - thetaw') + B(thetaw' - thetas') = M
-Js(thetas" + thetas') - B*thetas'(Js/Jw + 1) = M
M + Js*thetas" + thetas'(B(Js/Jw + 1) + Js) = 0
hence there is only one variable thetas that is responsible for the dynamic response of the system
hence

a. degree of freredom of the system = 1

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