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Technical
Moment of Inertia in Designing the Heads of Croquet Mallets

By Guy Towlson

Some questions have come up on the Nottingham croquet list relating to the importance of different designs of croquet mallet-heads and the effects of mass distribution on their moments of inertia.

It is suggested that a mallet with a higher moment of inertia (MI, symbol I), about the axis of the shaft, will be more stable during the swing, and so can be more accurate, since small asymmetries in the player's swing will have less of an effect on the direction in which the mallet points. Also such a mallet turns less when shots are hit off-centre.

The MI of a mallet-head can be changed by altering its weight distribution, without needing to change its total weight. This is commonly done in two ways; by making the head longer, and by moving some of the head's weight from the centre, towards the faces, a technique that is called 'peripheral weighting'. In 1999, I designed and made a mallet-head specifically to be have a high MI, by using both of these methods. What I arrived at had an MI that was more than double that of my previous 'conventional' head (one that is simply a block of constant density material, such as a singe block of wood).

What is presented below is derived from a tidied and edited copy of my workings from when I was designing that mallet. It is intended as a guide to the derivations that are needed to calculate the MIs of both conventional homogeneous mallet-heads and peripherally weighted ones of a simple design. There is little other quantitative information available about mallet designs, and the author hopes that this article may prove useful to those who are making or buying a mallet.

Introduction to moment of inertia

The mallet's moment of inertia, about the axis of the shaft, is a measure of the difficulty that is experienced in causing the mallet to rotate about its shaft.

Definition of moment of inertia

Suppose there is a small point-like mass, m, and it is rotated around an axis at a distance, radius r, then the moment of inertia is defined as:

I = m r 2

As one might well expect, the moment of inertia is directly proportional to the mass being accelerated (imagine doubling the mass - it will require twice as much effort to accelerate it in the same way). Also note how the MI of an object is proportional to the square of its distance from the axis of rotation. All other moments of inertia can be calculated from this one definition.

The two point-mass head

So, if a head of weight (M) 1.0kg and length (L) 30.5cm (12") were made of two point-like weights at the ends of a negligibly light 'body', this would have a MI (about the axis of the shaft) of:

Ipm = M L2 / 4 = 233 kg cm2

The 'two point-mass' head is useful: as will become apparant subsequently, for mallets of ordinary size, the finite width of the head only makes a small contribution to the head's MI, and so the MI of the two point-mass head serves as a good guide to the maximum possible MI for a head of chosen mass and length.

However, most useful masses, such as mallet-heads, are not point-like masses, but of finite shapes and densities - the moments of inertia of such shapes are standard results and discussed below, starting with that of a 'thin' (1D) rod.

'Thin' uniform rods

If the head above were instead made of a very thin rod of uniform line-weight (mass per unit length), then the MI would be:

I = M L2 / 12 = 77.5 kg cm2

Again the MI is proportional to the square of the length of the head. However, just by comparing this result with the previous one, the importance of weight distribution can be seen: by bringing in the weight from the ends of the head, to be uniformly distributed along its length, the MI is reduced by a factor of three.

'Conventional' mallets

By 'conventional', it is meant, of uniform density, as per a solid wood mallet-head. Many simple wooden mallets have small correction weights added to bring the whole head to the desired weight, but for small additions they continue to approximate reasonably well to this description.

'Conventional' square-sectioned mallets

Naturally a mallet-head is not 'thin', but has a quite definite width (W) and height (H). When rotating the head about the axis of the shaft only its width, and not its height, affects the MI. In plan-view the 'conventional' head's homogeneous weight distribution is the same as that of a similarly sized rectangle of uniform weight distribution. (It is assumed that the mallet is of rectangular cross-section: some expressions for calculating the MI of round heads are given in the next section.)

Iconv = M ( L2 + W2 ) / 12

(This result can most easily be obtained from the perpendicular axis theorem)
Alternatively, in terms of the density, p, this can be written:

Iconv = pLW2 ( L2 + W2 ) / 12

Since heads are generally much longer than they are wide by a factor of over four, it can be seen that the contribution to the MI due to the finite width of the mallet is quite small in comparison to the contribution from its length (the order of a few percent).

Example 1 - Short headed square-sectioned mallets

An ordinary short-headed mallet, of head mass approximately 1.0kg, total length 9.25" (23.5cm), width 6.0cm, height 6.5cm, and density 1.09 g/cm3:

Ishort = 49.0 kg cm2

(Incidentally, although sticklers for consistent SI (CGS), might prefer the measurements to be expressed in kg (g) and m (cm), but I have used kg and cm for the MIs since it gives 'conveniently sized' values!)

Note that it has been approximated that the mallet faces are of the same density as the 'body' material. Also the contribution due to the mallet shaft has been ignored. Both of these approximations are discussed in the appendix.

Example 2 - Long headed square-sectioned mallets

In contrast, more expensive mallets are often much longer. If material of the same density were used to make another mallet with a 12" (30.5cm) head length, with the same 1.0kg mass, and 5.0 cm in width and 6.0cm in height:

Ilong = 79.6 kg cm2

which is 62% larger, for just an extra 30% length.

These two examples highlight the way in which an increase in head length causes a disproportionately larger increase in MI (approximately as the square of the length). Note how the MIs for these mallets are only slightly above the MIs for comparable 'thin' rods (by only 2.7% for the 'Long' head). Also it is shown that the MI of the 'Long' head is only slightly above one third (34%) of the MI of the comparable 'two point-mass' head - revealing that for anyone seeking a high MI, there is vast scope for improvement even on the 'Long', homogeneous, 'conventional' head.

'Conventional' round headed mallets

The MI of a 'conventional' round mallet-head:

IRnd = Mw ( L2 + 3a2 ) / 12

where 'a' is the radius of the head (half the diameter).
Alternatively, in terms of the density, p, this can be written:

IRnd = ( pi pLA2 / 12 ) ( L2 + 3a2 )

(where pi = 3.14159).

Peripherally weighted mallets

A further way of increasing the MI of a mallet head is to use materials of different densities in its construction, so as generate a non-uniform mass-distribution (in plan view), which seeks, at least partially, to emulate the 'two point-mass head', by pushing some of the weight out to the ends. This redistribution of mass away from the axis of the shaft is generally known as 'peripheral weighting', although thinking of it as end-weighting is more useful, since peripheral can suggest that the weight has been put to the sides, where as it is considerably more important to ensure that it is as close to the ends a possible. Typically this is done by forming most of the body of the mallet-head from a low density material, and then by making up the balance of the mass with small high density weights at the ends of the head.

One simple method, which independently John Airey and I both chose, is to use low density wood to make the main body of the head, and then to attach blocks of brass to the ends of the wood, and the faces are then attached to the metal (this is also the method used by Alan Pidcock in Manor House's '2000' model).

Another design is to drill cylindrical holes into the body of the head, and fill them with lead weights (this is used by John Hobbs in his peripherally weighted mallets).

Yet a further possibility is to remove even more of the weight from the body of the head, by forming it from a light tube, allowing almost all of the head's weight to be positioned in the brass weights (the method used in Manor House's '2001' model).

Square-sectioned peripheral weights and end-faces

The MI of the whole head is determined by adding together the separate MIs of the main body, the peripheral weights and the faces. The MI of the main body is given by the same expression as for the 'conventional' head. The MI of the weights is given by:

Ipw = (ppwW2 / 12) . { L2 ( L22 + W2 ) - L1 ( L12 + W2 ) }

where L2 is the separation of the outside faces of the brass, and L1 the separation of the inside faces. This same expression is also used for the MI of the end-faces (since they are effectively peripheral weights too...), substituting the density and separations of the faces for those of the weights. (This expression can be obtained either by use of the parallel axis theorem or by subtracting the MI of a shorter object from the centre of a longer one.)

Example 3 - Long-headed mallet, with square-sectioned peripheral weights

Head mass, M = 1.0kg; head width, W = 5.0cm and square in section. The head is made from a central section of wood, of length L1 = 26.0cm, and density p1 = 0.8283 g/cm3, capped at each end by identical brass blocks, of thickness 1.0cm and of density p2 = 8.3 g/cm3, and weight 207.5g each. These blocks are in turn capped by end-faces, of thickness 1.25cm and of the same density as the wood. So, in the above expression, the separations to use for the brass are L2br = 28.0cm, and L1br = 26.0cm, and those for the end-faces are L2efc = 30.5cm, and L1efc = 28.0cm.

The MI of the wood uses the same expression as before, for a conventional head:

Iwd = p1 L1W2 ( L12 + W2 ) / 12 = 31.45kg cm2

The MI of the brass blocks and the end-faces:

Ibrss = 76.53 kg cm2
Ief = 11.19 kg cm2

So, the MI of the whole head:

Itotal = Iwd + Ibrss + Ief = 119 kg cm2

The MI of this peripherally weighted mallet-head is approximately 50% more than that of a conventional mallet head (79.6) of the same length and width. Note also that this is nearly two and a half times larger than that the MI of a typical short (9.25") conventional head!

Cylindrical peripheral weights and end-faces

Cylindrical peripheral weights of lead or brass can be inserted into holes drilled in the wood near the faces. These cylinders can be mounted with their axes vertically, horizontally and transverse to the length of the mallet, or horizontally and along it.

For calculating the MIs of two identical cylindrical weights, of total mass Mw, radius a, length b, and that are fitted symmetrically with their centres separated by L", the following expressions are useful.

When fitted horizontally (weights that are mounted along the axis of the head (on-axis) have the same MI as those fitted transversely):

Ipwh = Mw ( 3a2 + b2 + 3L"2 ) / 12

and when fitted vertically:

Ipwv = Mw ( 2a2 + L"2 ) / 4

It is assumed in these expressions that the weights are fitted with their centres coinciding with the central axis of the head. Of the three, the on-axis orientation is the most aesthetically pleasing peripheral weighting option, since it allows the weights to be descretely hidden under the faces, and is the method used by John Hobbs in his mallets.

Example 4 - Long-headed square-sectioned mallet, with on-axis cylindrical peripheral weights

For this example John Hobbs has kindly provided me with the design of Reg Bamford's mallet, which he made. Reg is shown playing with this mallet on the front cover of the CA's September 2001 Croquet Gazette, on his way to winning The 2001 Lincoln WCF World Championships.

The total head weight is 1050g, and the dimensions are 30.5 x 4.8 x 5.2cm (LxWxH). The body is made of wood of length 28.0cm, density 0.762 g/cm3 and weight 501g, and is capped at each end by identical end-faces of thickness 1.25cm and mass 42g. The balance of the weight is made up by two identical cylindrical peripheral lead weights that are mounted along the central axis of the head, one just under each end-face. These weights are of length 4.65cm, and density 11.35 g/cm3.

The MIs of the peripheral weights, wood (with the holes for the weights already drilled) and the end-faces:

Ipwh = 64.6 kg cm2
Iwd = 31.3 kg cm2
Ief = 18.2 kg cm2

giving a total MI:

Itotal = 114.1 kg cm2

Tube-sectioned peripherally weighted mallets

By hollowing-out the body of a mallet head, it is possible redistribute the removed weight into the peripheral weights, and in that way generate an extremely high MI. The most simple design for achieving this is to form the body of the mallet from a hollowed-out cylinder, a tube. In the extreme, as the tube is made lighter, the total MI of the head approaches that of a two-point mass.

The MI of a tube can be found by calculating the MI it would have if it were a solid cylinder (as above), and then subtracting from it the MI of the smaller hollowed-out part:

Itube = ( pi pLtube / 12 ) ( a22 - a12 ) ( L2 + 3a22 + 3a12 )

Where a2 is the radius of the outside of the cylinder, and a1 that of the cylindrical hole. The MI of the horizontal (on-axis) cylindrical peripheral weights is given by the expression above.

Example 5 - Tube-sectioned mallet, with cylindrical peripheral weights

Such a mallet, could be designed in this way: Total head mass, M = 1.0kg; head length L = 30.50cm and diameter 2a2 = 6.00cm. Body/tube; length Ltube = 25.52cm, density ptube = 1.00g/cm3, internal diameter 2a1 = 5.00cm and mass Mtube = 222.73g. The tube is capped by peripheral brass weights of the same diameter, each of thickness b = 1.56cm and density pbrss = 8.30g/cm3, and mass Mbrss/ 2 = 366.02g. Each end-face of thickness b = 0.80cm, the same density as the tube, and of mass Mef/ 2 = 22.62g.

The MIs of the tube, periperal weights and end-faces:

Itube = 13.19 kg cm2
Ibrss = 138.59 kg cm2
Ief = 10.10 kg cm2

giving as the total MI of the head:

Itotal = 161.9 kg cm2

This MI is 70% of that of the comparable two-point mass head (232.6 kg cm2).

Conclusion

By use of the examples given above, the importance and effect of both the head's length and its mass-distribution, as factors contributing to the its moment of inertia, have been shown - that increasing the length of the head, and re-distributing some of the mass close to the faces serves to increase the moment of inertia. Also some examples of typically available mallets have been shown, together with the formulae required to calculate their moments of inertia.

If, as has been suggested (and I am inclined to agree), an increased MI increases the accuracy of play, then in buying or making a 'conventional' mallet, and seeking the maximum MI, a player should look a mallet that has a long head. (But don't forget that longer mallets can be easier to catch on the ground by accident).

Similarly, with the possibility of even higher MIs, a peripherally weighted mallet could confer advantages to all levels of players in terms of directional accuracy, particularly for making accurate rushes, long roquets, and hitting-in. This could be of particular advantage to those who are new to croquet and whose development in the game can be held back by their ability to make roquets.

Acknowledgements

In the course of designing my mallet-head I had a most useful discussion with Dr Ian Plummer who had made a structured metal mallet-head a few years previously. I was also interested later to meet someone else who had also made his own high MI mallet-head, John Airey of Swindon CC. I am also grateful to John Hobbs and Alan Pidcock who provided information about their mallet designs, and to John Riches for other assistance in writing this article.

Appendix

One important consideration is that the playing styles of all mallets differ, and this is just as true of an high MI mallet as of a 'conventional' mallet. So, a new user should of course allow a suitable amount of time 'to get used to' its playing style before rushing off to any tournaments!

Using less dense wood for the body leaves the sides of the head less resistant to damage - this might be overcome by applying a veneer.

For those without access to the milling-machines or lathes required to prepare metal weights, a peripherally weighted mallet could also be constructed by combining woods of different densities.

Approximations

For simplicity, in several places the approximation was taken that the mallet faces were of the same density as that which forms the main body of the head. If the density of the faces is know then it is relatively simple to make the appropriate minor adjustment, with the algebraic expressions given in the section about peripheral weights.

Also for simplicity, the contribution to the mallet's MI that comes from the shaft was also ignored, and calculations were done only for the heads. The MI of the shaft is not a particularly significant part of that of the whole mallet, since it is of such a small diameter - for a 3lb mallet, with a 1.0kg head, if the shaft is approximated to a uniform cylinder of mass 362g, and radius 1/2" (1.27cm), an estimate of its MI can be obtained:

Ishaft = Mshaft a2 / 2 = 0.29 kg cm2

Peripherally weighted mallets and the Laws

This is what the 6th edition of The Laws of Association Croquet (in section 3.e.3) says about the mallet-head:

"HEAD The head must be rigid and may be made of any suitable materials, provided that they give no significant playing advantage over a head made entirely of wood. It must have essentially identical playing characteristics regardless of which end is used to strike the ball. Its end faces must be parallel, essentially identical and flat, though fine grooves are permitted. The edges of the faces should be of a shape or material unlikely to damage the balls and if they are bevelled they are not part of the end face."

(Kindly reproduced with permission of the CA).

In reply to a question, on the Nottingham board, about the legality of peripherally weighted mallets under this law, Stephen Mulliner wrote:

The answer lies in the use of the word "significant". This harks back to the 1890s when some mallets had one end covered by rubber which allowed amazing roll-strokes to be played. It may well be that an all-metal mallet head or one with peripheral weighting offers some advantage over one made entirely of wood, but as long as that advantage is not deemed to be significant, the mallet head is legal. You may take it that the lawmakers do not currently regard such an advantage as significant and I am sure that there are no plans to review it.

Asymmetric mallet-heads

Since first posting this article, I have been contacted by John Riches who has an asymmetric mallet, which he made some years ago. He described its swing - if you use the light end to hit the ball, and swing back with the mallet misaligned, it will act to straighten itself at the end of the backswing.

Unfortunately the weighting was rather awkward because he was not used to such a peculiar mallet, and it did not prove to give greater accuracy for rushes and hitting-in. Also splits and rolls became difficult. However he did add that, perhaps, if he had become more used to it he might have had greater success.

This mallet was, of course, experimental and not used for tournaments! Like beauty, the test for legality is in the eye of the beholder... rather than just the observer: asymmetric weighting could only be detected by a physical examination which would find the head to be unusually unbalanced with the mallet supported horizontally with freedom to turn about its shaft.

NB - any mistakes in the algebra, numbers or to disagree - please let me know - January 2002. All rights reserved.

Author: Guy Towlson
All rights reserved © 2001


Updated 28.i.16
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