Quote:
Originally Posted by toly
Figure 2.7. Stosur spin serve
Definition: Racquet efficient length Rel is the distance between player’s hand (point O on the Figures (2.42.7) and the ball during impact. I think Rel = 25” (63.5 cm) in the most occasions.
Definition: Arm efficient length Ael is the distance between shoulder joint and player’s hand. Since everybody have different arm size, I guess Ael = 25” (63.5 cm) as average length.
On the pictures above,RV is the radius of the arm and the racquet vertical rotation,RH – the radius of the racquet horizontal rotation (pronation).
RV = Ael + Rel × cosβ =25” × (1+ cosβ), where:
Notation: pronation angle β is the angle between long axis of the racquet and axis of the forearm/arm (Figure 2.42.7).
RH = Rel× sinβ = 25” × sinβ
RV can vary from Ael to Ael + Rel (or from 25” to 50”) because cosβ has range from 0 to 1, depending on the β magnitude. RV can never be equal to zero, because Ael or the arm efficient length is constant and equal to 25”.
RH can vary from 0 to Rel (or from 0” to 25”) because sinβ has range from 0 to 1. RH can be equal to zero and therefore linear speed would be zero! It can be very big problem for the tennis player. Maintaining the proper magnitude of the angle β before impact is absolutely crucial for pronation! On figures from 2.4 to 2.7 the best players keep β from 35° to 45°depending on the serve type.
How they are able to do that I described in step 2.2.2.
Notation: VLV  Linear speed of the racquet in the vertical plane; VLH  Linear speed of the racquet in the horizontal plane. VLV and VLH are corresponding velocities. Reminder: the linear speed = radius × angular speed. In the last formula the angular speed should be expressed in radians. The angular speeds in degrees (from Figure 2.3) were: ΩV=2°/ft, ΩH=20°/ft. In radians they are ΩV=(π/90)/ft, ΩH= (π/9)/ft.Then linear speeds in the vertical and horizontal planes can be calculated according to the followin g formulas:
VLV= RV× ΩV= 25” × (1+ cosβ) × (π/90)/ft = 25” × (1+ cosβ) × (π/90) × 300/sec
VLH= RH× ΩH= 25” × sinβ × (π/9)/ft = 25” × sinβ × (π/9) × 300/sec
The sum of the linear racquet speeds would be VLV+ VLH. The results of the calculation are presented on the Figure 2.8
Figure 2.8. Linear speeds of the racquet in vertical VLVand horizontal VLH rotations and their summation
The data on Figure 2.8 demonstrate, if the angle β ≥ 12°the linear speed of the pronation VLH begins to prevail over the linear speed of the vertical rotation VLV.
It should be noted, unfortunately, the calculated above pronation linear velocity determines mostly theoretical potential maximum. In reality, this speed may be slower even in case when the pronation angle beta has appropriate value. I’ll explain this phenomenon later, in the step 2.2.3.
Since, RH = 25” × sinβ, then we can calculate the pronation efficiency according to following formula
Pronation Efficiency = sinβ×100%.
The results of the calculation are presented on the Figure 2.9.
Figure 2.9 Pronation’s efficiency as function of the angle β
OK, it appears I found the proof! In case of the kick serve, the pronation can really provide much bigger linear speed of the racquet than others body limbs (except the wrist) altogether! But, if the pronation angle β=0°, the pronation produces nothing at all, just the proper racquet string bed orientation.
That’s why I repeat again, the best tennis players keep the pronation angle β around 30°  45° (Figure 2.4 2.7). Maintaining the proper magnitude of the angle β before impact is absolutely crucial for pronation! If the pronation angle has the proper magnitude the pronation would be the most important and effective contributor to the powerful kick serves!

Toly, this is an excellent and very thoughtful analysis. Isn't there one additional degree of freedom though, which is extension of the arm (that is, a rapid change in the beta angle as the arm is straightened)? The serve is really the most complex stroke in tennis.