ES480 3D Dynamic and Gyroscopic Motion 1 Problem 1 The slender bar of length l and mass m is rigidly attached to a shaft which is free to turn with constant angular velocity in the two bearings as indicated. The y and y axes point into the page at the mass center C. Determine the moment about the mass center C supplied by the shaft to disk. Answer: 2 2 sin cos 12 Cy ml M In addition, (1) evaluate the reaction forces in the bearings A and B, (2) judge whether the proper dynamic balance should be made in the sense M or N. Problem 2 The uniform slender bar of mass per unit length is freely pivoted about the y-axis at the clevis, which rotates about the fixed vertical z-axis with a constant angular velocity . Determine the steady-state angle assumed by the bar, length b is greater than length c. Answer: 2 2 1 2 3 3 3 sin 2 g b c b c Problem 3 The large satellite-tracking antenna has a moment of inertia I about its z-axis of symmetry and a moment of inertia OI about eac
h of the x-axis and y-axis. Determine the angular acceleration of the antenna about the vertical Z-axis caused by a torque M applied about Z by the drive mechanism for a given orientation . Answer: 2 2cos sinO M I I Problem 1 Problem 2 Problem 3 ES480 3D Dynamic and Gyroscopic Motion 2 Problem 4 The disk is pinning about the slight axel and the whole assembly precesses about the vertical axis with constant rate 32 rad/s. If the axel is observed to remain horizontal, find the magnitude and the direction of the spin of the disk. Answer. 30.7 /rad s about +z axis Problem 5 The slender rod OA of mass m and length l rotates freely about a horizontal axis through O. The pivot at O and attached shaft rotate with a constant angular velocity about the vertical z-axis. Write the expression for the angle assumed by the rod. What minimum value must attain before the rod will assume other than a vertical position. Answer. 1 2 3 cos 2 g l , min 3 2 g l Problem 4 Problem 5