15.10:
Relative Motion Analysis using Rotating Axes-Problem Solving
The crane's telescopic boom rotates with an angular velocity of 0.04 rad/s and angular acceleration of 0.02 rad/s2.
Simultaneously the boom extends with a constant speed of 5 m/s, measured relative to point C. At the given instant, the distance between points C and D is 60 meters.
Determine the magnitudes of the velocity and acceleration of point D at the same instant.
Here, point D is translating and rotating with respect to point C, so relative motion analysis can be done using a rotating frame of reference.
At the instant, the linear velocity and the linear acceleration of point C are zero. The angular velocity and angular acceleration of point D are along the negative z-axis.
Writing the relative velocity equation for point D in the rotating frame of reference, and substituting the known quantities, gives the magnitude of the velocity of point D.
Similarly, using the relative acceleration equation in a rotating frame and substituting the known quantities gives the acceleration of point D.
15.10:
Relative Motion Analysis using Rotating Axes-Problem Solving
Consider a crane whose telescopic boom rotates with an angular velocity of 0.04 rad/s and angular acceleration of 0.02 rad/s2. Along with the rotation, the boom also extends linearly with a uniform speed of 5 m/s. The extension of the boom is measured at point D, which is measured with respect to the fixed point C on the other end of the boom. For the given instant, the distance between points C and D is 60 meters.
Here, in order to determine the magnitude of velocity and acceleration for point D, relative motion analysis is used. For the given situation, point D is translating and rotating with respect to point C, so the relative motion analysis can be done using a rotating frame of reference. At the considered instant, the linear velocity and linear acceleration of point C are zero. At the same time, the angular velocity and angular acceleration of point D are along the negative z-axis as the boom is rotating in the clockwise direction.
The relative velocity and relative acceleration equations for the rotating frame of reference for point D are used to calculate the magnitudes of linear velocity and linear acceleration of point D.
Substituting the known values, magnitudes of linear velocity and linear acceleration of point D are found to be 5.55 m/s and 1.4 m/s2, respectively.