The Double Pendulum In Tennis (Rod Cross, 2011)

Source file: The_Double_Pendulum_In_Tennis.pdf.txt

Domain: biomechanics / serve-mechanics

Tier: reports/tier-1-high-value-gaps

Used in artifact: artifacts/reports/tier-1-high-value-gaps.md

Abstract

Cross models the forearm and racket as a double pendulum. Three-stage timing analysis: Stage 1 (0–0.05 s) upper arm; Stage 2 (0.05–0.103 s) forearm+racket 1700 °/s; Stage 3 (0.103–0.123 s) racket up to 6000 °/s = 1000 rpm.

Key points

Forearm slows down while the racket speeds up — energy transfer between segments.
Three-stage timing model with measured values from 300-fps slow-motion film.
Wrist torque max ~30 N·m; player generates ~20 ft-lbs of wrist force.
Forearm and racket remain at right angles for ~80% of the swing.
Whether the racket rotates the wrist or vice versa depends on racket head speed.

Raw dump (first ~1000 words)

The Double Pendulum in Tennis about:reader?url=http://twu.tennis-warehouse.com/learning_center/doub... twu.tennis-warehouse.com The Double Pendulum in Tennis 27-34 ph�t 1. INTRODUCTION Swinging a racquet is a process that is not easy to describe in words since the actions of all the various body segments are quite complicated. However, if we focus just on the forearm and the racquet then the task is a little easier, especially since these two segments by themselves act like a double pendulum. A double pendulum is just two single pendulums joined end to end. To illustrate the point, consider the two film clips shown below where the action of the arm and racquet in a serve is compared with a simple mechanical double pendulum. Both were filmed in slow motion, at 300 fps, to observe the action more clearly. The pendulum version is upside-down, but the actions are very similar. Pendulum Motion of the Serve 1 trong 24 8:33 SA, 08/03/2021 The Double Pendulum in Tennis about:reader?url=http://twu.tennis-warehouse.com/learning_center/doub... 0:00 / 0:01 Movie Screen 1. -- Double Pendulum Motion in the serve. (Note: Movies are best viewed frame by frame using keyboard arrow keys or movie controls.) Double Pendulum Demonstration Choose Movie 2 trong 24 8:33 SA, 08/03/2021 The Double Pendulum in Tennis about:reader?url=http://twu.tennis-warehouse.com/learning_center/doub... 0:00 / 0:04 Movie Screen 2. -- The difference in racquet and forearm speed between a light "racquet" and a heavy racquet. (Note: Movies are best viewed frame-by-frame using keyboard arrow keys or movie controls.) The server struck the ball at 100 mph so the racquet swung much faster than the mechanical pendulum. The mechanical pendulum was constructed using a sawn-off baseball bat as a forearm and a long wood dowel as a racquet, connected by an artificial wrist joint consisting of two eye hooks held together loosely with a small bolt. There is no wrist or elbow action in the mechanical pendulum, and gravity alone caused the pendulum to rotate. Even so, the similarity with a real serve is remarkable. With a few extra modifications to the pendulum, it could be made to behave just like a real serve. It needs only a bit of elbow action and a bit of wrist action to behave in a more realistic manner. 3 trong 24 8:33 SA, 08/03/2021 The Double Pendulum in Tennis about:reader?url=http://twu.tennis-warehouse.com/learning_center/doub... Several things stand out. One is that the forearm slows down while the racquet speeds up. Another is that the racquet rotates through a larger angle than the forearm. The forearm and racquet start out at right angles and end up almost in line by the time the racquet strikes the ball. In a real serve, the forearm and racquet remain at right angles for about 80% of the swing time. In the pendulum swing, the dowel drops straight down before it starts to swing around, because there is no wrist to lock the dowel at right angles to the forearm. We could ask several questions about this. Is it actually necessary to lock the wrist at the start of a serve? Does the wrist rotate the racquet at the end of the swing or is it the other way around? That is, does the racquet rotate the wrist? Would it help if the forearm didn't slow down? These sorts of questions are easily answered by looking at the mechanics of a pendulum, especially when we add some artificial elbow and wrist action. The actions of a double pendulum have been studied for many years, especially in relation to the swing of a golf club. The physics of swinging a tennis racquet has not been studied in nearly as much detail. Part of the problem is that there are many different ways of swinging a racquet. A golfer has a simpler task since the ball just sits there waiting to be struck and the golfer needs only to hit the ball toward the hole. Except when serving, a tennis player needs to chase down the ball and then hit it from a range of different heights to various parts of the court, either using a forehand or a backhand or a volley or a half-volley or a smash. Golfers only ever hit the ball with backspin, but tennis players can hit the ball with topspin or backspin or side spin or any other spin they like. In tennis, the wrist is not just a passive hinge, like it is in a 4 trong 24 8:33 SA, 08/03/2021 The Double Pendulum in Tennis about:reader?url=http://twu.tennis-warehouse.com/learning_center/doub... mechanical double pendulum, but plays an active role in controlling the racquet. Similarly, the elbow joint is not passive, but plays an active role in controlling motion of the forearm. The beauty and the complication of this arrangement is that motion of the forearm affects motion of the racquet, and motion of the racquet affects motion of the forearm, so the two cannot be controlled separately or independently. The results are sometimes unexpected but are consistent with the way that a double pendulum behaves. If the wrist is completely limp, then the player has no direct control over the racquet and the racquet will just swing where it wants to swing. In general, the player rotates the forearm at a controlled speed, using muscles in the upper arm, and uses the wrist as an extra aid to control exactly where the racquet needs to go. If the wrist is locked then the forearm and the racquet behave as a single, solid object and both rotate at the same speed. That is, if the forearm rotates through say 30 degrees, then so does the racquet. The rotation speeds (measured in degrees per second) of the forearm and the racquet are the same even though the actual speed of the racquet (measured in mph or feet per second) is greater than the actual speed of the forearm. The complex interaction between the forearm and the racquet is nicely described in terms of the physics of a double pendulum. Players don't worry about that since they know from experience what happens to the racquet when