15.9:
Relative Motion Analysis using Rotating Axes – Acceleration
Consider member AB undergoing linear motion and simultaneous rotation about point A. The velocity of point B is expressed as the sum of the velocity of point A and the relative velocity of point B in the rotating frame and the angular velocity effects caused by the rotating frame.
Differentiating with respect to time gives the acceleration of point B. The first term is the linear acceleration of point A measured from a fixed frame.
The second term is the cross product of the angular acceleration with the position vector rBA.
The third term is the cross product of angular velocity and the rate of change of position vector rBA. This term can be expanded using a distributive property of vector product.
The last term is the time derivative of angular velocity effects caused by the rotating frame of reference.
Here, the first two terms denotes the acceleration of point B in the rotating frame of reference. The last two terms can be simplified to get the final equation for the acceleration of point B.
15.9:
Relative Motion Analysis using Rotating Axes – Acceleration
Consider a component AB undergoing a linear motion. Along with a linear motion, point B also rotates around point A. To comprehend this complex movement, position vectors for both points A and B are established using a stationary reference frame. The absolute velocity of point B is determined by adding the absolute velocity of point A, the relative velocity of point B in the rotating frame, and the effects caused by the angular velocity within the rotating frame.
Time differentiation is employed in order to understand the acceleration of point B. The first term derived from this process represents the linear acceleration of point A, gauged from a stationary frame. The second term is identified as the cross-product of the angular acceleration and the position vector determining the relative position of point B with respect to point A, rBA. Thereafter, the third term presents itself as the cross product of angular velocity and the rate of change of the position vector rBA. This term can be further expanded using the distributive property of vector products. The final term is the time derivative of the effects of angular velocity caused by the rotating frame of reference.
In this scenario, it is important to note that the initial two terms signify the acceleration of point B within the rotating frame of reference. Meanwhile, the last two terms can be simplified to derive the final equation for the acceleration of point B, thus providing a comprehensive understanding of its movement.