The blue arrows in figure represent the angular velocity and angular acceleration of the link it is associated with.

The purple arrows represent the tangential acceleration and velocity of the point at the end of each link.

The red arrows represent the normal acceleration of the link it is associated with.

The green arrow represents the acceleration and velocity of the body as a whole. In fact the green arrow can represent point A as it is the only point that isn’t rotating or moving in a curvilinear motion. The green arrow is straight indicating that the body is moving in a rectilinear motion.

In order to bring in relative motion analysis let’s say that we have recorded the motion of the runner’s body on camera, so we know the velocity and acceleration directions and magnitudes of the body or of point A. Let’s also say that we know the angles of each link with respect to the x axis and the length of each link.  Since we have the camera we can measure the change in angles and we know the change in time.

Given all this information we can calculate the velocity and acceleration of all the other points on the body.

These equations will prove to be helpful:

ω = dθ/dt,       v = ω*r,           where r is the length of the link
Tricker & Tricker, The Science of Movement. 215

                                                                                                                at = α*r,           an = ω²*r,         a = (at² + an²)^1/2
 

 

For point B:
Velocity

vB = vA + vB/A, where vB/A is the relative velocity of “B with respect to A”

The relative motion is circular and the magnitude vB/A = ω*rB/A

vB = vA + ω*rB/A,           

We know the change in angle and time so we know the angular velocity and we just use the equation to get the velocity of point B. Note: The equation would have to be used for the x and y direction and be used in the following equation:        v =  (vx² + vy²)^1/2

Acceleration
aB = aA + (aB/A)t + (aB/A)n,        where (aB/A)t  is the relative tangential acceleration component of “B with respect to A.”

                                                   and (aB/A)n  is the relative normal acceleration component of “B with respect to A.”

 aB = aA + α*rB/A + ω²*rB/A

The angular acceleration can be found out by looking at two frames captured by the camera before the frame we are looking at to obtain the initial and final angular velocity to get the angular acceleration. Note: The equation would have to be used for the x and y direction and be used in the following equation:        a = (ax² + ay²)^1/2

From all this we can get the velocity and acceleration of every point by relative motion analysis however the angular velocity and angular acceleration values we would get would not be accurate as the angular acceleration is not constant. We can only get an approximation of the angular velocity and acceleration.

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Sideview of Runner

Frontview of Runner

Relative Motion

Bibliography