Originally Posted by Greg Raven
I'm curious as to how your derived your formula for Effective weight (AKA Hitting weight). I have a completely different formula, which yields completely different results.
The hitting weight is mathematically derived using the momentum balance, assuming that the swing has only translational motion at impact (neglecting the rotational component of the swing).
It is highly accurate for predicting power level on groundstrokes, because groundstrokes hit with sound technique have minimal rotational motion at impact. The correction factor to account for the Serve/Groundstroke Power Ratio (SGPR) must be applied to accurately predict relative power on the serve, because the serve has a large rotational component. If two racquets have the same hitting weight, but one has a longer balance, the power level will be about the same on groundstrokes, but the frame with the longer balance will generally be more powerful for serves.
The derivation of the formula for hitting weight is as follows.
We assume the impact of the ball on the racquet causes the racquet to pivot about the wrist joint, and that the racquet is otherwise constrained by the wrist joint (i.e., we neglect momentum losses transferred into the forearm). Let’s also neglect the momentum contribution of the hand to keep this simple.
The ball’s rebound momentum is determined by balancing the moments of the momentum vectors about the wrist joint.
Let’s define variables:
m = mass of ball
v = change in velocity of ball during impact
V = change in velocity of racquet center of mass during impact
r = distance from butt to point of impact
M = mass of hand
R = distance friom butt to balance point
d = distance from butt to wrist axis of rotation (about 4cm for a forehand, or 8cm for a 2hb).
The momentum balance is given by:
M*(R – d)*V = m*(r – d)*v
M*(R – d) = m*(r – d)*(v/V)
m*(r – d) is a constant (neglecting any changes in sweet spot location due to weight distribution), and v/V is the ratio of the ball’s change in velocity to the racquet’s change in velocity. Clearly, v/V is a measure of the power level of the racquet.
So the power level of the racquet (v/V) is proportional to M*(R – d).