A schematic for the satellite and scientific probe for the Gravity Probe-B (GP-B)

Chapter 7, Problem 7.2

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A schematic for the satellite and scientific probe for the Gravity Probe-B (GP-B) experiment that was launched on April 30, 2004 is sketched in Fig. 7.83. Assume that the mass of the spacecraft plus helium tank, m1 , is 2000 kg and the mass of the probe, m2 , is 1000 kg. A rotor will float inside the probe and will be forced to follow the probe with a capacitive forcing mechanism. The spring constant of the coupling k is 3.2 10 6 . The viscous damping b is 4.6 10 3 . (a) Write the equations of motion for the system consisting of masses m 1 and m 2 using the inertial position variables, y1 and y2 . (b) The actual disturbance u is a micrometeorite, and the resulting motion is very small. Therefore, rewrite your equations with the scaled variables Z1 = 10 6y1 , Z2 = 10 6y2 , and v = 1000u. (c) Put the equations in state-variable form using the state x = [Z1 1 Z2 2] T , the output y = Z2 , and the input an impulse, u = 10 -3 (t) N-sec on mass m1 . (d) Using the numerical values, enter the equations of motion into MATLAB in the form and define the MATLAB system: sysGPB = ss(F,G,H,J). Plot the response of y caused by the impulse with the MATLAB command impulse(sysGPB). This is the signal the rotor must follow. (e) Use the MATLAB commands p = eig(F) to find the poles (or roots) of the system and z = tzero(F,G,H, J) to find the zeros of the system. Figure 7.83 Schematic diagram of the GP-B satellite and probe