stoneage
Rookie
Sorry for starting another thread and sorry for the harsh title. But as I will hopefully show below, well founded. The relation MgR/I has has been kicked around as a supposedly good help to customize a racquet. I didn't find any information about why this would work, so I decided to dig a little deeper into equation. Unfortunately it turns out that it gives an infinite number of racquet configurations, most of them useless.
First a general observation: The laws of mechanics have been used fairly successfully for a couple of hundred years to describe more advanced objects than a swinging tennis racquet. The parameters you need to describe how a racquet behaves when a force is applied to it, is mass, center of mass and MOI (moment of inertia/swing weight). If you take two of these parameters and divide by the third, you don't gain any information, but you loose a lot!
If you assume that you want to keep MgR/I constant (e.g. =21) and we are playing on earth so g is constant we end up with MR/I = c where c is a constant (for example c=21/981= 0.214 if we talk kg and cm).
Further assume that we have a racquet with mass m0, center of gravity r0 and MOI I0 to which we ad a number N of point weights mi at a distance ri. We then get (MOI for a point weight is m*r^2):
Here we already see a scary thing. Any weight added at r=0 doesn't contribute to the total MR/I at all, which means that the idea of MR/I being a description of mass distribution falls apart. And anyone who thinks that adding a 1 kg lump to the handle of a racquet doesn't affect the playability raise a hand.
Further, since we want to keep c constant while adding weights we can simplify the above expression to:
This is the relation that added weights should fulfill to keep MR/I constant. Let us look at some practical cases.
1. Adding one weight means that the relation is m*r=c*mr^2
This has two solutions one is r=0 (as said before) the other is r=1/c. This means that you can add any weight you want at r=0 or r= 46.7 cm but nowhere else!
2. Two weights. The idea behind the concept is to add a weight to increase swingweight, then tweak it a little to achieve the right MgR/I. So lets add a first weight of 30g at the top of the racquet, say at 70 cm. You then end up with a relation for the second weight to fulfill:
0.03*70+m*r=c*(0.03*70*70+m*r^2) which can be simplified to
m=1/(r*(1-c*r))
If you plot the relation between m and r for the second weight you get:
This means that you have have an infinite number of solutions for r<46.7 giving very varying combination of m and r. Which in its turn leads to huge differences in mass and swingweight of racquets that fulfills MgR/I=21
The conclusion is thus:
1. MgR/I does not describe any mechanical behavior of the racquet (apart from being proportional to swing time of if you hold the racquet at the but and let it swing like a pendulum)
2. MgR/I does not describe the mass distribution
3. Using MgR/I=const leads to an infinite number of very varying solutions making it impossible to use as a design criterion.
/Sten
__________________
Test and tune you racquet with racquetTune for the iPhone
First a general observation: The laws of mechanics have been used fairly successfully for a couple of hundred years to describe more advanced objects than a swinging tennis racquet. The parameters you need to describe how a racquet behaves when a force is applied to it, is mass, center of mass and MOI (moment of inertia/swing weight). If you take two of these parameters and divide by the third, you don't gain any information, but you loose a lot!
If you assume that you want to keep MgR/I constant (e.g. =21) and we are playing on earth so g is constant we end up with MR/I = c where c is a constant (for example c=21/981= 0.214 if we talk kg and cm).
Further assume that we have a racquet with mass m0, center of gravity r0 and MOI I0 to which we ad a number N of point weights mi at a distance ri. We then get (MOI for a point weight is m*r^2):
Here we already see a scary thing. Any weight added at r=0 doesn't contribute to the total MR/I at all, which means that the idea of MR/I being a description of mass distribution falls apart. And anyone who thinks that adding a 1 kg lump to the handle of a racquet doesn't affect the playability raise a hand.
Further, since we want to keep c constant while adding weights we can simplify the above expression to:
This is the relation that added weights should fulfill to keep MR/I constant. Let us look at some practical cases.
1. Adding one weight means that the relation is m*r=c*mr^2
This has two solutions one is r=0 (as said before) the other is r=1/c. This means that you can add any weight you want at r=0 or r= 46.7 cm but nowhere else!
2. Two weights. The idea behind the concept is to add a weight to increase swingweight, then tweak it a little to achieve the right MgR/I. So lets add a first weight of 30g at the top of the racquet, say at 70 cm. You then end up with a relation for the second weight to fulfill:
0.03*70+m*r=c*(0.03*70*70+m*r^2) which can be simplified to
m=1/(r*(1-c*r))
If you plot the relation between m and r for the second weight you get:
This means that you have have an infinite number of solutions for r<46.7 giving very varying combination of m and r. Which in its turn leads to huge differences in mass and swingweight of racquets that fulfills MgR/I=21
The conclusion is thus:
1. MgR/I does not describe any mechanical behavior of the racquet (apart from being proportional to swing time of if you hold the racquet at the but and let it swing like a pendulum)
2. MgR/I does not describe the mass distribution
3. Using MgR/I=const leads to an infinite number of very varying solutions making it impossible to use as a design criterion.
/Sten
__________________
Test and tune you racquet with racquetTune for the iPhone
