Originally Posted by stoneage
Thanks for the clarification!
I have had two problems with MgR/I:
1. I have yet to hear an explanation why this parameter is relevant to the mechanical behavior of the racquet in tennis. MgR/I describes how the racquet swings freely in a gravitational field, i.e. like a pendulum. MgR/I=21 means that frequency of a racquet swinging from the but is 0.73 Hz. But unless you are building a cuckoo clock of the racquet I have difficulties to see the relevance. Gravity is the least of forces involved in a tennis swing. But if you have some explanation that could shed some light on this I would be most interested (no irony intended).
2. There are an infinite number of racquet configurations that fulfill MgR/I=21 most of them very strange and unplayable. So if you use only that to find an optimal racquet you could easily go wrong.
However, you say that MgR/I should be combined with a certain swingweight. And then it becomes something different entirely! By saying that MgR'/I' = 22.5 and I' = 360 to 370 you are saying that MgR' should be around 8200. And MgR'/I' (and MgR/I) has disappeared.
So your design criteria are:
1. setting MgR' (or MgR), which is the static force (moment) you feel when you hold the racquet in the handle parallel to ground.
2. Setting the moment of inertia (swingweight) which is good description of the rotational behavior
So there is obviously no need for MgR/I at all, which I think is fine since the two above criteria are better.
racquet apps for the iPhone/iPad.
You misinterpreted my post.
The MgR/I formula works for swingweights in the 300 range to 400 range.
And so does the MgR'/I' formula.
The physics of a pendulum are relevant to a tennis stroke even in the absence of gravity, because the centrifugal force is always present.
But the laws that govern the motion of a pendulum are easiest to apply by using the laws of motion for a pendulum in a vertical plane - rotating the plane of motion closer to the horizontal plane has little effect on the formula.
I have found that a groundstroke can be reduced to the swing of a double pendulum, with the frequency of the upper pendulum (the arm and torso) almost independent of the racquet specs. Adding a kg to the butt of racquet would obviously affect the speed of the upper (arm+torso) pendulum, thereby reducing the optimum MgR/I value for the lower (racquet) pendulum. Thus the formula does not "break down" as you suggest - rather, it is the assumption that the upper pendulum is unaffected by changes to the racquet mass distribution that breaks down. The principles of the formula still hold.
An easy way to see see the significance of the MgR/I formula is to start the racquet high overhead with your forehand grip, and allow the racquet to sweep through a high-to-low-to-high stroke in a mostly vertical plane (much like a golf stroke). If you keep the wrist relaxed through the stroke, the racquet will exert a moment on your hand as it passes through the hitting zone. If MgR/I is > 21.0, the moment from the racquet handle will pull forward on your index finger. If MgR/I is < 21.0, the moment from the handle will push back on base knuckle of your index finger. If MgR/I is tuned properly, there will be virtually zero moment.
Zero moment acting on your wrist joint is highly desirable, because it means you do not need to apply any forces from the wrist during your swing to keep the racquetface aimed toward the target. This increases your ability to control the ball significantly.
If MgR/I is too low, you need to apply a forward moment from the wirst to keep the racquetface aimed toward the target through the hitting zone. This is not easy, but it is doable.
Similarly, if MgR/I is too high, the racquetface will come around too fast relative to the hand, making control difficult because you would need to apply a reverse moment to the racquet in order to control the ball (or convert the excess racquetspeed into extra vertical velocity for more spin, which is not easy to do). This is more difficult than the former situation, which is why very few stock racquets (or pro customized racquets) have specs with MgR/I > 21.
The ideal situation, however, is to have zero moment required, allowing a relaxed wrist through the hitting zone. This is why, I believe, that most of the top players in the world have MgR/I values very close to 21.0, as I have shown in the data I presented.