Relative Motion Analysis using Rotating Axes - Acceleration

The acceleration of point B is found by differentiating the velocity equation with respect to time. It consists of the linear acceleration of point A from a fixed frame, plus the cross product of angular acceleration with the relative position vector rBA, plus the cross product of angular velocity with the rate of change of rBA, and finally the time derivative of angular velocity effects from the rotating frame.

The cross product of angular acceleration with the position vector rBA represents the tangential acceleration component of point B due to changes in the angular velocity of the rotating frame. This term captures how the rotational motion of the body affects the acceleration of point B relative to point A.

The rotating frame of reference simplifies the analysis of complex motion by decomposing it into manageable components. It allows separation of the linear acceleration of point A, the relative acceleration of point B within the rotating frame, and the effects caused by the frame's rotation, making the motion easier to understand and calculate.

The distributive property of vector products is used to expand the term representing the cross product of angular velocity and the rate of change of position vector rBA. This expansion simplifies the expression and allows the final acceleration equation to be written in a more manageable form for calculations and analysis.

The first two terms—the linear acceleration of point A and the cross product of angular acceleration with rBA—together represent the acceleration of point B as measured within the rotating frame of reference. These terms capture how point B accelerates relative to the rotating coordinate system attached to the rigid body.

The last two terms represent the effects caused by the rotating frame itself. The cross product of angular velocity with the rate of change of rBA and the time derivative of angular velocity effects together account for how the frame's rotation influences the absolute acceleration of point B in the fixed reference frame.

The acceleration equation is derived by differentiating the velocity equation with respect to time. Since velocity combines the absolute velocity of point A, the relative velocity in the rotating frame, and angular velocity effects, time differentiation of each component yields the corresponding acceleration terms that comprise the complete acceleration equation.