Measuring Hoop Rigidity
Why at a certain tournament were there so few triples or so many sextuples?
We have no quantification about the state of croquet lawns in the past when, say, John Solomon started to produce triple peels in competitions. Why had they not been done before? Were the conditions before that time so difficult as to defeat precision croquet? There is now a movement to quantify the difficulty of conditions at an event.
The concept of measuring lawn speeds, which was introduced in 2004, appears to have proved useful. Knowing that a triple was done on a 9 plummer lawn does not raise an eyebrow whereas on a 14 plummer lawn it is an achievement.
The next target is to quantifying the ease, or otherwise, of running a hoop (its 'runability'). This will depend on the following factors among others:
Some of the above we can control, e.g. gape, others we cannot control on the day, such as the construction of the hoop and ground conditions.
Measurements are underway (2013) by the Croquet Association Equipment Committee and World Croquet Federation (currently through Prof Alan Pidcock and Martin French) to investigate the overall runability of different hoops. They have used ramps (Photo 1) to launch balls reproducibly and accurately at hoops, changing the approach angle and offset from centre. Their ramp, 40cm high, however currently only offers a modest range of ball velocities (see table later). Additionally they have used a 'Pidcock Peeling Plank' (Photo 2) which uses a simple action to peel a ball through a hoop with a croquet stroke at variable angles and offsets.
This paper however focusses on measuring a compound effect - the hoop rigidity. Rigidity will depend on a number of the above variables (ground, hoop mass, anchorage and weight distribution). It is well known that rigid hoops are more challenging than loose ones. The rigidity can be more significant than the gape. Early artificial courts were made with hoops set in concrete which proved extremely difficult to run. On grass courts however top level croquet players have no problem in running hoops which are 1/64th” wider than the ball in average conditions.
Three methods to measure hoop rigidity are discussed below ranging from the elaborate to the inexpensive. In essence a hoop is struck and then the movement of the hoop is monitored.
For repeatable measurements the hoop needs to be struck with a standard amount of energy in a repeatable position - the 'standard blow'. We could for example either hinge a weight on an arm and drop it from a known height to strike the hoop, or say roll a ball down a ramp, again from a known height, to strike the hoop.
Hall Effect Sensor
The first method tried used a Hall Effect Sensor. A Hall sensor measures the strength of a magnetic field. By attaching a magnet to the crown of a hoop and placing a fixed Hall detector nearby, the movement of the crown with respect to the Hall sensor changes the measured magnetic field strength which is translated into an electrical signal.
The electrical signal is not linear with the displacement, however we can still get the frequencies of vibration of the hoop. This was mocked up on the bench. The Hall sensor’s signal was fed to a digital storage oscilloscope and the signal downloaded to a computer. It was then fed into Excel and the amplitude-time signal transformed into a frequency-phase plot using Excel's in-built Fast Fourier Transform function.
The system worked, however the detector had to be critically placed a few millimetres from the magnet to avoid it saturating during the movement of the hoop. Secondly the signal was not linear with displacement, hence the true displacement was not readily available. Finally the digital storage oscilloscope and its mains supply were impractical for outdoor use! Basically too fiddly.
Modern smartphones have in-built accelerometers. Hence strapping a phone with an appropriate 'app' to a hoop and delivering the standard blow should produce a measure of the displacement of the hoop. The apps ‘iSeismometer’ (recommended; iPhone, Android, Windows Phone and HTML5 versions, free, does frequency analysis and will email a CSV to you), ‘Vibration’ (iPhone £25), and ‘Seismometer FREE’ (iPhone free) are examples.
This may prove a viable measurement system with a suitable clamp to attach the phone to the hoop (mocked up on right). Given that the data is collected in all three axes, the orientation of the phone is not critical. The data can be transferred to a computer and analysed. Alternatively a specialised app could be written. A Fourier transform of the displacement versus time will give an indication of power and the frequencies & magnitudes of oscillations. All that I currently needed is a smartphone!
The solutions offered above are either complex or expensive. The marble gauge described here is intended to be a cheap simple reproducible device for quantifying hoop rigidity. Unfortunately the version below failed to give reproducible results but is included as food for thought.
In essence it is a plate with different diameter holes drilled through it which is clipped horizontally to the top of a hoop. Marbles of a standard diameter are placed in the holes. The wider the hole is the deeper the marble sits in the hole (see photograph below) and the more difficult it is to dislodge. When the hoop is struck with a ‘standard blow’ one or more marbles will be shaken from the holes. The rigidity of the hoop is read by seeing which marbles remain. It would be improved by the addition of a spirit level element to ensure the plate was absolutely horizontal.
The device was field tested for reproducibility and range.
Field tests were disappointing. A thick fibreglass plate was fitted across the gape of the hoop (using magnets) and balls lauched perpendicularly at the centre of the plate. The primary failure was that the balls would fall off in any sequence - not according to to the socket diameter. Hence the 9mm and 13mm ball could be displaced with the others staying in place. Clearly there is scope for using larger holes and heavier marbles, but 13mm is the maximum hole size easily acheived in plastic with home DIY tools.
If a selection of marbles and hole sizes can be found then one concept design is shown on the right. The plate would be fixed to the hoop uprights with clips or magnets with its base at approximately ground level. A pendulumn is integrated into the design and strikes the plate at the same height as a ball strikes the hoop. The pendulumn can be varied from a hard heavy material to, say, a bag of sand.
Provided the same materials and dimensions are used in the finalised gauge then results can be compared across venues.
Appendix 1 - Ball Energy
How hard should the 'standard blow' be? People run hoops gently or by hitting the ball as hard as they can, perhaps with the idea that the hoop will deflect and allow the passage of the ball. There were reports of ball speeds being measured with a radar gun at a hoopball tournament in Egypt.
Robert Fulford recalls:
The aim however is to use conditions which are typical of ball-hoop interactions. Unfortunately simple mechanics cannot be used to model the situation since the ball collision is an impulse.
The table below gives (initial) ball velocities in a number of units and the amount of energy carried by the ball. (Extended from that in 'Lawn Speeds').
The comparison to the plummer rating of the lawn gives an idea of how hard the ball was hit. The table indicates that to use a ramp or pendulumn to load a ball with an amount of energy equivalent to the hardest shots would require a ramp or pendulumn 13-23m off the ground. Clearly the mass being dropped or rolled can be increased (doubling the mass halves the height) but to get a realistic rejection of a ball hitting a hoop, a ball is the best candidate.
The coefficient of restitution of the pendulumn weight material will indictate how much energy is partitioned to the hoop and ground during the collision. Clearly a powerball will rebound well carrying away much of its initial energy, whereas a bag of sand will deliver most of its energy into the hoop. For Martin French's 40cm ramp (above) the ball is not moving at all fast (<6.8mph).
Appendix 2 - Marble Variation
Smaller marbles were chosen simply because it is easier to drill holes up to 13mm (~½") at home; larger size drills are not common. Whilst ball bearings could have been used they are neither as cheap nor ubiquitous as marbles. I paid 10p each for marbles. The reason to have the hole sizes in the plate alternate about the plate centre instead of decrease from say left to right, was to minimise errors if one side of the plate was displaced more than the other.
The reproducibility of the dimensions of marbles will affect the performance of the marble gauge.
Marbles come in a range of standard diameters - the white ones (photo above) were ~14mm, clear ones with a candy twist were ~16mm. There is a diameter variation of ±0.2mm even within a single specimen. The weights of the white marbles ranged from 3.39 - 3.74gm (7.3%) and the clear ones from 4.86 - 5.50gm (12%). This suggests that it could be necessary to pre-select the marbles.
To lift a marble so it is free of the hole takes the energy m.g.d where m is the marble mass, g the gravitational constant and d is the depth the base of the marble which initially lies below the plane of the hole. We can use the relationship between chords of a circle to work out the relationship between the radius of the hole rh and the distance, d, that the marble needs to be lifted given the radius of the marble rm.
If two chords intersect inside a circle, then the product of the segments of one chord equals the product of the segments of the other chord.
Whilst it is easy to calculate the minimum energy to lift a marble out of various diameter holes, the marbles are mainly being displaced from the marble gauge by side-to-side shaking.
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