Power and Stringbed Stiffness
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Elastic Energy and Ball Velocity
When the ball and racquet collide, the energy of the collision is divided between kinetic energy (energy of motion) and elastic energy (energy stored in deformation) in the ball, strings and racquet.
A serve presents the simplest case. Throughout impact, the racquet continues to move forward. It does not come to a complete stop. In other words, some energy of forward motion remains in the racquet, and the strings never get any of it.
The amount of energy that stays in the motion of the racquet and that gets converted into elastic energy in the strings and ball depends primarily on the mass of the racquet. How this happens is complicated. So, by clamping the racquet head, we will eliminate the complications (essentially by making the mass infinite) because then all the impact energy will be converted into elastic energy. We then will be able to look just at the elastic energy in the strings and ball. We want to do this because we can affect the share of impact energy that the ball and strings each receive by altering string tension, and as a result, the power, feel, control and comfort of the hit.
How much of the total elastic energy goes into the strings and how much goes into the ball? The energy that gets converted into elastic energy is divided between the ball and the strings. The percentage share of each will affect the speed of the ball leaving the strings. The more energy that can be directed into the strings, the faster will be the ejection speed.
The energy share is determined by the relative stiffness of the ball and strings. Whichever one is the softest will get a bigger share of the elastic energy. When the ball and strings are equally stiff (which is a fair approximation under many circumstances), then they each get half of the elastic energy. In an alternative scenario, if the strings are half as stiff as the ball, they will get two thirds of the energy and the ball will get one third.
What matters at this point is how much of the strings' and ball's respective energy shares will be lost. The answer is important because you want to channel the elastic energy into its most efficient storage location to get the most back.
How much of the string's share of elastic energy is lost? Experiments have shown that all strings give back about 95% of the energy they take in, no matter what. That means all strings lose 5% of the energy that goes into stretching them. The key concept above is that all strings give back 95% of "the energy they take in." Different strings at the same tension do take in different amounts of energy depending on their stiffness compared to that of the ball. And the same string at different tensions takes in different amounts of energy. But of that energy, all strings will lose 5% and return 95% .
How much of the ball's share of elastic energy is lost?
The rules stipulate the numbers here. All balls when dropped from a height of 100 inches must bounce between 53 and 58 inches (we'll use 55 as the average). That means that the ball loses 45% of its impact energy during the bounce.
What is the final velocity and energy of the ball? The Energy Share Calculator makes the answer to this simple by clamping the racquet so we don't have to worry about the effect of the collision on the racquet. With a clamped racquet, all the collision energy goes into elastic energy and we can then very clearly see how changing string tension affects power. The surprising answer is some, but not much.
Drag the stringbed stiffness slider to "1 times as stiff as the ball." This means that the ball and stringbed are the same stiffness. This is a good approximation with most nylon strings at tensions around 60 pounds. Drag the drop height slider to 100 inches. This is convenient because it is the height at which balls are tested. You will see from the output numbers that, in this case, the ball and strings share the energy equally (50% of the total), but the ball loses 45% of its share and the strings lose 5% of their share. The result is that the ball rebounds from the strings at 87% of the speed it came in with (velocity out / velocity in). This percentage is known as the coefficient of restitution (COR), and it measures the quality of the bounce.
Now, if we lower the stringbed stiffness by 50% (an amount far greater than the normal 5-10% that would be considered in real playing conditions) by moving the slider until it says "0.5 times as stiff as the ball," then the COR is 0.9. We only gained three percent. If we drop stringbed stiffness to 0.8, COR will only go up to 0.88.
The lesson is this: string tension affects power a little, but not much. However, it significantly affects feel, comfort, and control. When instructing students on the effects of string tension on performance, you should concentrate of these latter three features, not on power.
Note: These relationships are discussed in much greater detail in The Physics and Technology of Tennis.