Formula: Shoulder Crunch
Shoulder Crunch (A.K.A. Shoulder Yank) is the change in the centrifugal force acting on the racquet. You will recall that the Shoulder Pull due to the racquet’s rotation was equal and opposite to the centrifugal force. Impact changes the speed of rotation, and therefore the centrifugal force. Following the definition of centripetal force, the change in centrifugal force is expressed as
\[ \frac{Mv_1^2}{R} - \frac{Mv_2^2}{R} \]
(\(M\) = racquet Mass, in kg; \(v_1\) is the linear velocity of the racquet mass center before impact, in m/s; \(v_2\) is the linear velocity of the racquet mass center after impact; \(R\) is the distance from the racquet mass center [balance point] to the shoulder, in meters).
The excess centripetal force, which continues to act regardless of the impact lessening the centrifugal force it was countering, is the Shoulder Crunch. This is easily derived knowing Shock. Shock, or the change in the racquet’s kinetic energy due to impact, is ½ Mv1² - ½ Mv2² so by multiplying Shock by \(2 \div R\) we get Shoulder Crunch. \(R\) is the distance from the shoulder to the racquet’s balance point (\(R = 0.01r + 0.61\ meters \)), so:
\[ Shoulder\ Crunch = \frac{2}{(0.01 r + 0.61)} \times Shock \]
| \(A_x\) | = | Impulse Reaction, the translational force acting at the axis of rotation due to impact, in Newtons. Note that when \(d\) = \(q\) (\(q\) is the distance from the axis of rotation to the center of percussion), the expression within the second set of parentheses becomes zero. |
| \(a\) | = | linear acceleration of the mass center, in m/s² |
| \(b\) | = | mass of the ball, in kg |
| \(c\) | = | coefficient of restitution of the racquet/ball system |
| \(d\) | = | distance from the axis of rotation to the impact point, in cm |
| \(e\) | = | the distance from the axis of rotation to the tip |
| \(F\) | = | force applied at mass center, in Newtons |
| \(I\) | = | moment of inertia (swing weight) of racquet, in kgf/cm² |
| \(I_5\) | = | moment of inertia (swing weight) of racquet at 5cm from the butt, in kgf/cm² |
| \(I_7\) | = | moment of inertia (swing weight) of racquet at 7cm from the butt, in kgf/cm² |
| \(I_{10}\) | = | moment of inertia (swing weight) of racquet at 10cm from the butt, in kgf/cm² |
| \(I_a\) | = | moment of inertia (swing weight) of racquet at distance \(a\) from the butt, in kgf/cm² |
| \(M\) | = | mass of the racquet, in kg |
| \(m\) | = | mass in kg |
| \(ω\) | = | angular velocity of racquet, in radians/s |
| \(p\) | = | linear velocity of impact point, in m/s |
| \(r\) | = | distance in cm from mass center (balance point) to axis used in the stroke |
| \(s\) | = | ball velocity, in m/s (positive is away from player) |
| \(s_1\) | = | velocity of ball before impact, in m/s |
| \(s_2\) | = | velocity of ball after impact, in m/s |
| \(T\) | = | torque at axis of rotation, in Nms |
| \(t\) | = | dwell time, or duration of impact, in seconds |
| \(v\) | = | linear velocity of the mass center, in m/s |
| \(v_1\) | = | linear velocity, just before impact, of racquet mass center, in meters/second |
| \(v_2\) | = | linear velocity, just after impact, of racquet mass center, in meters/second |