Dr Ian Plummer
What is required is a convenient and cheap method of testing the bounce (related to the coefficient of restitution) of croquet balls. It should be accurate, reproducible and preferably absolute. If not absolute then there should be an inexpensive means of calibrating the measurement.
The coefficient of restitution C, is the ratio of the difference in velocities before and after collision.
C = vf /vi, where vi is the initial velocity (difference) and vf is the final velocity (difference).
The impact force at which the ball is tested should lie within those encountered in average strokes played in croquet. The test should allow the bounce from a specific region of a ball to be recorded so that variation over the surface can be determined. Finally there should be no damage to the ball.
Basics
In the methods conceived so far energy must be provided to the ball or a proofing surface which then collide. The parameters at collision or the results thereafter are recorded and analysed to yield a 'bounce height'. The source of energy conventionally has been potential energy derived from allowing a weight to fall a measured distance under gravity. This has the advantage that it is cheap, simple and reproducible. The height is a measure of Kinetic Energy, which is related to the square of the velocity.
Ek = 1/2( m.v2 )
Methods to date give the coefficient of restitution in terms of a bounce height under standard conditions: for a ball dropped from 60" it should bounce to a height between 30-45". When dropped from hi = 60" it has a total initial energy Ei of:
m.g.hi = Ei = 1/2( m.vi2)
m = mass of ball, g is accelleration due to gravity (9.8 m/s) and h the height. After the collision (bounce) it reaches a new height hf:
C = sqrt( hf / hi )
Hence a rebound range of 50-75% yields a coefficient restitution of sqrt(0.50) to sqrt(0.75) [= 0.707 ... 0.866]. The reduction in bounce height is a consequence of energy losses on collision, due to deformation and heating of the ball and energy transfer to the metal plate.
Methods
Steel Plate Drop Rig
The steel plate rig is described elsewhere. It consists of a 1" steel plate set solidly in concrete onto which a ball is dropped and the rebound height is recorded. This has been found to be sensitive to the details of the steel to concrete bond, the size of the concrete block and the seating of the block on the ground. Variations of 5% have been documented.
Calibrated Steel Plate Rig
As indicated above the problem with the steel plate rig is that each rig will have its own variaion due to its construction. One method of obviating this is to calaibrate the rig. It is not viable to send calibrated balls around the world hence a convenient calibration standard is required. Suggestions to date have been to use a large steel ball bearing or a child's large marble (solid glass sphere). This would be dropped from a standard height and its rebound height used to calibrate the rig. It then produces a coefficent by which to modify the measured results on that rig.
Pendulum Impact
In
order to remove the problems of the concrete block from the measurement, a
pendulum method is proposed here. A ball attached to the end of a piece of
string is allowed to fall in an arc and hit a metal plate similarly suspended
from the same pivot point (see diagram). The initial energy given to the ball
is a function of the height from which it is dropped. The moving ball strikes
the stationary plate and both rebound. The energy of each is a function of
the height that they rebound to. Three measurements are needed, together with
the mass of the ball and plate: the initial and final height of the ball and
the rebound height of the plate.
The mass multiplied by the height times a constant (g - the gravitational constant) gives the energy of that object. In a perfect system, with no losses, the initial energy of the ball due to its height would be partitioned into the steel plate and ball.
mb*hib * g = (mb*hfb + mp*hp)*g + L
where
| mb | mass of ball |
| hib | initial height from which ball is dropped relative to bottom of arc |
| g | graviational constant (9.81 ms-1) |
| hfb | final maximum height of ball after collision relative to bottom of arc |
| mp | mass of plate |
| hp | final maximum height of plate after collision relative to bottom of arc |
| L | energy losses |
As an initial approximation the energy loss is due to the coefficient of restitution, C. This yields:
More accurately the right-hand side is equal to the energy losses divided by g. Addition energy losses, which should not be incorporated into the coefficient of restitution, include the air resistance of the two bodies and any normal modes (vibrations) excited in the plate. The design of the rig should be such that the plate and ball gain no angular momentum in the collision. In practice all that is required is the masses of the ball and plate and a measurement of the drop height and the heights reached by the ball on the plate on rebound.
Practical details
To
reproduce the same impact collisions as used in the conventional steel plate
test the ball velocity can be matched by allowing it to fall (in an arc) through
the same vertical distance as for the plate test. Thus the ball centre-pivot
distance should be 60". The mass of the plate is a user variable. If too small
it is difficult to measure the rebound (or indeed follow through) of the ball,
and if too large then the deflection of the plate is too small to measure accurately.
A mass in the order of 0.5 - 1.5 times the mass of the ball (454gm, 16oz) would
be appropriate.
The measurement to the centre of the ball (equivalent to its centre of mass) from the pivot should be 60". Similarly the same measurement is used for the plate. The ball must strike the plate at its centre of mass otherwise torques (rotation) will be introduced. These will make the deflection difficult to measure and abstract energy. It is therefore proposed that the plate is suspended from two strings lying in the same plane as the pivot and centre of mass of the plate at right angles to the motion of the ball. (See diagram). Similarly the ball can be in a cradle supported by two similar strings. The masses of the strings should be negligible in comparison to the plate or ball masses. The cradle will allow different part s of the surface of the test ball to strike the plate allowing the variation of bounce with position to be measured.
When measuring deflections it is the height of the centres of masses which needs to be recorded.
Thanks to Nick Furze for spotting a missing sqrt.