Meriam, Kraige & Bolton — Introduction to Three-Dimensional Dynamics of Rigid Bodies: Problem 7_109

7/109 The earth-scanning satellite is in a circular orbit of period τ. The angular velocity of the satellite about its \(y\)- or pitch-axis is \(\omega = 2\pi/\tau\), and the angular rates about the \(x\)- and \(z\)-axes are zero. Thus, the \(x\)-axis of the satellite always points to the center of the earth. The satellite has a reaction-wheel attitude-control system consisting of the three wheels shown, each of which may be variably torqued by its individual motor. The angular rate \(\Omega_z\) of the \(z\)-wheel relative to the satellite is \(\Omega_0\) at time \(t = 0\), and the \(x\)- and \(y\)-wheels are at rest relative to the satellite at \(t = 0\). Determine the axial torques \(M_x, M_y\), and \(M_z\) which must be exerted by the motors on the shafts of their respective wheels in order that the angular velocity \(\vec{\omega}\) of the satellite will remain constant. The moment of inertia of each reaction wheel about its axis is \(I\). The \(x\) and \(z\) reaction-wheel speeds are harmonic functions of the time with a period equal to that of the orbit. Plot the variations of the torques and the relative wheel speeds \(\Omega_x, \Omega_y\), and \(\Omega_z\) as functions of the time during one orbit period. (Hint: The torque to accelerate the \(x\)-wheel equals the reaction of the gyroscopic moment on the \(z\)-wheel, and vice versa.)