Benchmark conditions for objective racquet comparisons
In order to compare racquets objectively, five benchmark conditions have been assumed. The assumptions in the benchmark conditions eliminate the variables of stringing. Strings quickly lose their tension, and these premium racquets do not come pre-strung, so to compare them as to their performance characteristics, using the formulas developed on this site, we simply assumed that the combination of string tension, ball newness, head size, and stiffness was such that they all had the same coefficient of restitution (bounce): 0.85. This coefficient of restitution figure is typical for impacts at the center of the string. See H. Brody, The physics of tennis, III. The ball-racquet interaction
Am. J. Phys. 65, 981 (1997). The assumed value for dwell time is now 10 milliseconds, rather than 4, following electrical contact lab results from Roland Sommer. String tension, ball newness, head size, and frame stiffness do not make a difference because they are all comprised in the coefficient of restitution and the dwell time. Frame stiffness (flex) is, however, accounted for in the macro criteria that are a weighted composite of the results under these benchmark conditions.
The First Benchmark Condition is the 70 mph [31.29 m/s] return of a shot received at 20 mph [-8.94 m/s] (velocities measured along the line from the player to the net, away from the player being positive); duration of impact (dwell time) 0.010 second; coefficient of restitution of the racquet/ball system 0.85; ball mass 0.057 kg.; axis of rotation 7 cm from the handle end; and impact at 16 cm from the racquet tip and on the centerline. The ball speeds postulated are in accord with recent evidence that shows about a 50% loss in speed after the bounce.
The Second Benchmark Condition is a 110 mph [49.17 m/s] serve; duration of impact (dwell time) 0.010 seconds; coefficient of restitution of the racquet/ball system 0.85; ball mass 0.057 kg.; axis of rotation 5 cm from the handle end; and impact at 16 cm from the racquet tip and on the centerline.
The Third Benchmark Condition is a 40 mph [17.88 m/s] volley received at 40 mph; duration of impact (dwell time) 0.010 seconds; coefficient of restitution of the racquet/ball system 0.85; ball mass 0.057 kg.; axis of rotation 7 cm from the handle end; and impact at 16 cm from the racquet tip and on the centerline.
The Fourth Benchmark Condition is a 50 mph [22.35 m/s] return of a serve received at 50 mph; duration of impact (dwell time) 0.010 seconds; coefficient of restitution of the racquet/ball system 0.85; ball mass 0.057 kg.; axis of rotation 7 cm from the handle end; and impact at 16 cm from the racquet tip and on the centerline.
The Fifth Benchmark Condition is a 70 mph [31.29 m/s] second serve; duration of impact (dwell time) 0.010 seconds; coefficient of restitution of the racquet/ball system 0.85; ball mass 0.057 kg.; axis of rotation 5 cm from the handle end; and impact at 16 cm from the racquet tip and on the centerline.
| \(A_x\) | = | 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 |