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8X cos 8Z AB+D angular motions angular rates angular velocity angular-momentum vector center of mass characteristic equation control command damp the angular damp to one-half damper axes damper rotor damper system damper-axis system defined as shown device was assumed dL dL equation 23 equations 30 equations of motion Euler angles external torque acting five degrees ft-lb gimbal angle gimbal axis histories of q/q0 identity matrix IDS(p implies stability kinetic energy Lagrangian formulation Lagrangian function Langley Research Center Laplace transform left-hand side moments of inertia MOTIONS OF SPINNING NUMERICAL EXAMPLES one-half amplitude oscillation pencillike vehicle principal damper-axis principal vehicle-axis coordinate proportional radians per sec radians/sec rate gyro real root real values rjq0 plotted Routh-Hurwitz stability criterion servo control side of equation spherical damper spherical inertia distribution spin rate SPINNING SPACE VEHICLES tion toroidal vehicle VEHICLES By JERROLD vehicles with constant X-axis Z-axes zero