Dynamic parameters define a rigid body’s reaction to external forces. While their importance for a sword’s behaviour is known since the 19th century [1–3], many data sets of original swords, replicas and training weapons include mass and the centre of mass, but lack a third parameter such as moment of inertia, radius of gyration, or corresponding centres of oscillation/percussion*. A third parameter, however, is required to calculate a rigid object’s response. Several setups for the measurement of the moment of inertia have been commercially available for decades; but among HEMA practitioners and sword researchers, the most widespread method for the assessment of a third parameter is the so‐called *waggle test* [5, 6] which was introduced in a point‐up variant in the early 21st century [7], and is now more commonly used point‐down [8]. The waggle test gives the position of the centre of oscillation corresponding to any given axis, which might explain why many attempts to interpret the dynamic properties of a sword focus on the positions of centres of oscillation.

Le Chevalier explained the proper execution of the waggle test. [8] He also examined the effect of the imposed oscillation period on the precision of the test. In this article, overall precision and trueness of the waggle test will be assessed on the basis of experimental data.

## Setup and Methods

For the experiment, two wooden broomsticks were sawed to cylinders with the length , radius and mass . The calculated first principal moment of inertia is thus . The participants of the experiment were instructed to mark the centre of mass using a rubber band, mark the point at which they wanted to hold the stick and then find and mark the corresponding centre of oscillation following the instructions given in Le Chevalier’s introduction to the waggle test [8]. Additionally, they were instructed to turn the broomstick upside down, repeat the waggle test using the found centre of oscillation and check if they would now find the original point as the corresponding centre of oscillation. The product of the broomstick’s mass and the distances of the corresponding centres of oscillation from the centre of mass is the moment of inertia about the centre of mass. For each measurement it should be equivalent to the calculated one.

One part of the experiment was conducted at the DDHF‐Trainertage, an educational event for HEMA trainers, where eight data sets were obtained. The second part was conducted at the club Tremonia Fechten. The participants were beginners, experienced practitioners and one of the trainers. This part provided seven data sets.

Additionally, five data sets of swords and rapiers from the 16th and 17th century were obtained from the GEEhW, which analyses period swords, in particular the swords’ morphological properties. These data sets include mass, the position of the centre of mass and two pairs of corresponding centres of oscillation measured with the waggle test. [9] The moment of inertia is thus overdetermined. While we cannot calculate the swords’ true moments of inertia, we can compare the results from each of the pairs of corresponding centres of oscillation in order to determine the precision of the measurements.

## Results and Discussion

The broomstick experiment gives an arithmetic mean of the measured principal moment of inertia with the standard deviation . The coefficient of variation is thus and the relative bias is . See Fig. 1 for a graphic representation. The results from the trainers group (DDHF, measurements 1 through 8) are similar in terms of trueness and precision to those of the mixed group (Tremonia Fechten, measurements 9 through 15).

The GEEhW guidelines suggest measuring the centres of oscillation corresponding to both ends of the sword grip, so we can calculate the moment of inertia about the centre of mass from each centre of oscillation. As the moment of inertia about a given axis does not change in a rigid body, the difference between the results indicates the precision of the waggle test. The results are shown in Fig. 2. The calculated moments of inertia differ from their respective arithmetic means by 56 % to 95 %, the mean relative difference is 73 %.

Typical simple measurements of distances involve errors in the low percent range. Most kitchen scales work within even lower error margins. So, the limiting factor for an accurate determination of the dynamic parameters of swords is the waggle test.

In two separate experiments, participants with fairly accurate measurements were told that their data was incorrect, and they were asked to reproduce results which were described as being exact, but actually were farther off than their own respective measurements. In both cases, the participants were able to reproduce the given wrong values as correct. If this susceptibility to suggestion is a common effect, particular caution should be exercised if measurements were made with an expectation in mind, e.g., that a specific type of sword is supposed to have a certain position of a centre of oscillation.

## Conclusion and Outlook

While we do see some waggle test results which are fairly close to the calculated moment of inertia, the overall data indicate that results from waggle tests should be taken with caution. This observation is supported by the degree of mismatch between the moments of inertia from different pairs of corresponding centres of oscillation that could be found in waggle test data from period swords and rapiers. It is therefore suggested to replace the waggle test with a more accurate method.

I am currently collaborating with constructor, HEMA practitioner and bladesmith Patrick Schröter on a rigid gravity pendulum for measuring the moment of inertia of swords with sufficient accuracy. The device will be simple to use. First measurements with a preliminary setup have a relative bias of 4 % and a relative standard deviation of 4 %.

## Acknowledgement

My fellow fencers at Tremonia Fechten, the fellow trainers at the DDHF and Tilman Wanke at the GEEhW kindly produced or provided the data used in this article. Their support is gratefully acknowledged. I would also like to thank Jan Hoffmann, who proofread this article for nothing more than getting mentioned in the acknowledgements.

## References

- [1] N.‑J. Didiez. ‘Note sur le nouveau système d’escrime pour la cavalerie’. In:
*Le Spectateur Militaire.*Vol. 17. Noirot and Anselin, 1834, pp. 461–468. - [2] A. Fehn.
*Die Fechtkunſt mit Stoß‐ und Hiebwaffen.*Hannover: Carl Rümpler, 1851. - [3] R. F. Burton.
*The Book of the Sword.*London: Chatto & Windus, 1884. - [4] V. Le Chevalier.
*History of the Centre of Percussion.*Ensis Sub Caelo. 2014‑05‑04. URL: http://blog.subcaelo.net/ensis/history-centre-percussion/ (visited on 2016‑07‑05). - [5] F. Fortner and J. Schrattenecker.
*A Comparison of Late 16th to Early 17th Century Rapiers with Modern Reproductions.*Fechtschule Klingenspiel. 2015‑08‑27. URL: http://historisches-fechten.at/wp-content/uploads/2015/08/Comparison_of_Period_Rapiers_to_Modern_Reproductions.pdf (visited on 2016‑07‑06). - [6] B. Grotkamp‐Schepers et al., eds.
*Das Schwert. Gestalt und Gedanke.*Solingen: Deutsches Klingenmuseum, 2015. - [7] G. L. Turner.
*Dynamics of Hand-Held Impact Weapons.*Version 5. Association for Renaissance Martial Arts. 2002‑07‑22. URL: http://armor.typepad.com/bastardsword/sword_dynamics.pdf (visited on 2016‑02‑02). - [8] V. Le Chevalier.
*A dynamic method for weighing swords.*Ensis Sub Caelo. 2014‑11‑15. URL: http://www.subcaelo.net/ensis/weighing/weighing.pdf (visited on 2016‑02‑02). - [9]
*User’s Manual for the Database.*Version 1.1. Gesellschaft zur Erforschung und Erprobung historischer Waffen. 2013‑02‑14. URL: http://historische-waffenkunde.de/Downloads/Benutzerhandbuch%20V%201.1.pdf (visited on 2016‑07‑05).

Hi Robert !

A while ago, a new member of our training group found out about the waggle test. When I pointed him to your article, he said that the mismatch between the mean measured value (blue) and the calculated value that you found in Fig.1 was hardly significant because the grey area almost extends to the calculated value (red). He's an engineer and probably knows alot more about math then I do, so maybe you could clear this up ?

BTW, I really like your approach of combining HEMA with physics !

Hi Daniel! Thanks for your kind words and sorry for the delayed answer! I usually get notified of comments on my articles, but this time the mail appears to have vanished on its way. Regarding your question about the significance of the difference, the short answer is: The difference is more than just hardly significant (p = 1.6E-4). The long answer is: I suspect your fellow fencer confuses confidence interval (CI) and standard deviation (SD). SD characterises the width of the distribution of the measured values around their centre, and, without any further information, does not tell you if the position of the assumed centre of the distribution is significantly different from another value. The CI is the interval in which you assume the centre of your distribution on a certain significance level, usually 5 %. The CI for the measurements used in my article is [26.1 gm²; 41.1 gm²], so the calculated moment of inertia of 51.5 gm² is way outside the CI. I hope that helps. Feel free to ask should you have any further questions!

Thanks alot for your explanations, Robert ! May I ask why you skipped this significance issue in your article ? And a question on the stndard deviation: You said that standard deviation characerises the width of the distribution. I am not sure if I really unterstand what you mean with distribution. Would you explain this a little more ? Also in Fig.1, there are some crosses which are very close to the calculated I value. Does that give a clue about the proportion of "experts" who do actually master the waggle test ?

You are very welcome! On the distribution: Assume you were measuring a certain property, say a certain sword's mass, many times. If you are measuring on a fine scale, you will not always find the same value, but some fluctuation about a central value. Values which are closer to the centre will appear more often while those which are farther off are less likely to occur. Look up "bell curve" or "Gaussian distribution" for a popular example. The bell curve can be wide, i.e. it's not too unlikely that you measure values which are further away from the centre (greater standard deviation), or it can be slim, when the measured values stack more closely around their centre (smaller standard deviation). Now, the confidence interval is where you assume the centre of the distribution (at a given confidence level, usually 95 %). If the distribution is slim (i.e. if we have a small standard deviation), we are more certain about the position of the centre, so we will have a smaller CI. Also, if we have more measurements, we are more certain about the actual shape of the distribution, which in turn allows for a better guess of the centre's position, and therefore again, a smaller CI. However, more certainty about the shape of the distribution does not necessarily decrease the SD. If you want to characterise the width of distribution (which is what I wanted to do), the SD is a good choice. If you want to charakterise the certainty of the distribution's centre, then take the CI.

Now to your next question, were the fairly accurate measurements made by experts? Now, assume you were shooting a target usin a rifle with a slightly bent barrel, such that on average your bullets hit left. Now, let's say that additionally your hands are shaking, so the bullet hits are scattered across and around the target. If you shoot often enough, however, there will be some cases where bias and scatter cancel out and you hit your target accurately. This coincidence wouldn't make you a good marksman, though. The situation in Fig. 1 is similar -- the "accurate" measurements don't stick out from the distribution around the mean value, so we can't even tell if the persons who measured these values can be considered experts or if they were just lucky.

Why did I skip the significance thing? I wanted to make the article as accessible as reasonably possible, and therefore keep statistics at a basic level. Adding statistical significance to the discussion would require an analysis of the distribution on my side, and on the side of the reader knowledge of tests and a general interest in statistics. Also, I didn't feel that statistical significance would add much weight to what can already be read from the raw values in Fig. 1 by just eyeballing the data.

I understand that you wanted to keep the article accessible. Actually I didn't really know what "significantly" means until you explained it. Well, to be honest, I'm not sure if I understand it now, but at least I see that there's more maths involved than I had expected. 😉 You wrote that in your broomstick study you couldn't find any experts. How about other researchers ? Are there any people out there on whose measurments you would rely ? I see you have 'Das Schwert. Gestalt und Gedanke' on your literature list.

Not that much verifyable waggle test data is actually published, and the data which is published unfortunately often indicates, together with accompanying explanations, that the respective researcher hasn't spent much time reading mechanics textbooks. In general, published data reflects the trend that I found in my study: More often than not, the waggle test fails. There might be more exceptions to the rule, but I know only one, Vincent Le Chevalier (the guy who made the diagrams for "Das Schwert -- Gestalt und Gedanke"). He's aware of the limitations of the waggle test and knows how to validate the data. Also, he knows how to make a pendulum, so I would trust his data.