## A Theoretical Study of the Angular Motions of Spinning Bodies in SpaceSummary: A theoretical study was made of the angular motions of spinning bodies in space. The analysis was based on Euler's dynamic equations which were linearized and solved analytically. The results of the study are directly applicable only to spin-stabilized vehicles with constant moments of inertia and angular displacements not exceeding about 15°. Simple analytical expressions were obtained which relate angular motions to spin-rate and inertia distributions for a given disturbance. Consideration was given to the effects produced by having artificial damping in the system. The study included numerical examples and comparisons of analytical solutions with machine solutions of exact dynamic equations. The analysis indicated that angular motions are sensitive to inertia distributions. In considering a rectangular-pulse pitching moment, it was found that the residual motion was very sensitive to the time at which the moment was removed. Artificial damping due to a perfect proportional control system seemed to be more advantageous to pencil-like configurations than to disk-like configurations. |

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1-cos Qt Aberdeen Proving Ground analysis indicated Analytical expressions analytical solutions appropriate substitutions artificial damping BODIES IN SPACE body axes center circle considering a rectangular-pulse constant moments curve which looks damping ratio disk-like configurations disturbed-state circle Duhamel integrals equa Equation 51 equation 72 equations of motion equilibrium-state circle Euler angles Euler's dynamic equations ft-lb IBM solution inertia distribution inertial axes Ix<^Iy,Iz or Ix^>Iy,Iz Iy=Iz Langley Field Langley Research Center Laplace transformation linearized and solved machine solutions ment moments of inertia MOTIONS OF SPINNING motions to spin-rate pencil-like configurations principal body principal body-axis coordinate qm(t R-83 A THEORETICAL radians radians/sec rectangular-pulse pitching reference relate angular motions residual motion sensitive to inertia slug-ft2 solutions with machine solved analytically Solving equations SPACE By JERROLD spin axis spin rate spin-rate and inertia spin-stabilized spinning bodies steady-state symbol indicates TECHNICAL REPORT R-83 theoretical study torque unit step vehicles with constant yawing mo yp-6 plane