|
Part 2
Ron Larham
Abstract: In Part 1 of this paper I described the validation problem that had arisen in connection with a paper on modeling the speed of the domino effect wave. I also described my method of measuring the wave speed using DEMON-like analysis of a sound recording of the progressive collapse of a regular domino array. In this paper I present some further background material on the domino effect and my experimental results. I also discuss my data and other data from the literature in connection with the validity of two simple models of the wave speed of the domino effect.
Some Theory
McLachlan et al. [5] conclude from dimensional analysis that the limiting wave speed V for thin dominoes satisfies:
for some function G. For thin dominoes is the same as:
Where H is the height of the dominoes, d the gap between adjacent dominoes, and L the distance between equivalent points on neighboring dominoes (that is the pitch of the domino array). McLachlan et al [5] do not give the details of their derivation of this so I have provided one in the appendix to this paper. The proposed model of Efthimiou and Johnson [1] aims to determine what the form of the function G 1 must be (see Fig. 2.1 for a diagram showing the significance of the variables).
Notes: There are additional invisible arguments in functions G and G 1 as the normalized speed may also depend on the dimensionless constants relevant to the system, in this case these include the coefficient of friction between dominoes, and the coefficient of restitution for inter domino impacts. Efthimiou and Johnson make particular assumptions, which mean that none of these dimensionless parameter appear explicitly in their model.
Figure 2.1. Diagram of domino array showing the dimensions relevant to domino wave speed.
Results
Figure 2.2 shows the results of my measurements, together with the measurements from references [3] and [5]. The two curves are the predictions from the models in references [1] and [6]. Both of these models assume that only two dominoes are involved in driving the wave at any instant, which is a simplification as it is obvious from observing a wave that a number of dominoes are interacting at any one time, leaning over onto one another in sliding contact. In general, the more complex models of references [3] and [4] give better agreement with experiment with appropriate choice coefficient of friction, but even so we can see that Bank's [6] predictions are not very far from the mark.
All of the experimental results are in reasonably good agreement considering that the dominoes used in the experiments are of materials with different coefficients of restitution and sliding friction (as well as different radius of curvature along their edges).
It is difficult to see the variability in the experientially determined wave speeds in Fig. 2.2 due to the vertical scale required to show the model predictions. The experimental data shown in Fig. 2.2 together with that from McLachlan et al. are shown in Fig. 2.3 with a vertical scale more suited to showing this data. In this plot the variation of wave speed with spacing parameter can be seen. For most of the data sets there appears to be a weak dependence of normalized wave speed on the spacing parameter h/H. The data from McLachlan et al. show what appears to be a stronger dependence on the spacing parameter apparently rising as the spacing parameter becomes small. Given the other data and the description by McLachlan et al. of their experimental method (hand timing the collapse of an array of ~100 dominoes). this is possibly a misleading trend. This will be discussed further in a subsequent paper.
Figure 2.2. + and O are the present author's measurements, £ and ´ are Strong and Shu's [3] data.
Figure 2.3. + and ´ are the present authors measurements, £ and ¯ are Strong and Shu's [3] data, and D and Ñ are data from McLachlan et al.
Discussion
It has been shown that the speed of a domino wave can be measured from a sound recording of the collapse of a domino array. This required no new equipment as all the apparatus required is already present in my home, and presumably a significant proportion of homes in the developed world. The measurements are in reasonable agreement with previously published measurements (with differently proportioned dominoes made from different materials). The clustering of these measurements seems to support the claim that to a hand-waving approximation the normalized wave speed depends upon only the dimensionless array spacing parameter d / H . References [3] and [4] explain the residual variation in wave speed using detailed models of the physics including the coefficients of restitution and sliding friction between dominoes.
It might be of interest to attempt to measure the start transient for a domino wave. This could be done by timing the time for the wave to propagate through arrays of 10, 20, 30, 40 . . . dominoes. Then the transient would manifest itself as a departure from a linear relation between time for the array to collapse and number of dominoes. For this some standard method of toppling the first domino will be needed. Currently I am thinking of something like the boot from the game "Mouse Trap," which is essentially a pendulum released from a standard position kicking off the first domino. This could also use the sound recorder/microphone set up for data collection, but the data analysis would consist of timing the first impact and the last and taking the difference of the two times.
The treatment of errors in the positioning of the dominoes in an array is not discussed here nor in any of the references. It may be interesting to investigate the effect of errors on mean wave speed and the widest spacing of dominoes that will support a wave. Van Leeuwen [4] quotes a maximum normalized gap between dominoes for wave propagation of ~0.87.
References
[1] C. J. Efthimiou, M. D. Johnson, Domino Waves, SIAM Review 49, 2007, 111-120.
[2] W. J. Stronge, The domino effect: a wave of destabilizing collisions in a periodic array, Proceedings of the Royal Socity of London A 409, 1987, 199-208.
[3] W. J. Stronge, D. Shu, The Domino Effect: Successive Destabilisation by Cooperative Neighbours, Proceedings of the Royal Socity of London A 418, 1988, 155-163.
[4] J. M. J van Leeuwen, The Domino Effect, arXiv:physics/0401018v1 (2004).
[5] B. G. McLachlan, G. Beaupre, A. B. Cox, L. Gore, SIAM Review 25, 1983, 403-404.
[6] R. B. Banks, Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics, Princeton University Press, 1998.
The object of dimensional analysis is to seek information about the relationship between the variables in a physical situation in a dimensionally homogeneous fashion. That is the equations relating the variables must be dimensionally correct.
The way we go about this is to list the relevant variables that have dimensions (there may be dimensionless variables but they do not enter into our analysis). In the case of domino waves the list includes g the acceleration due to gravity, V the wave speed, H the domino height, and d the interval between the dominoes (we could have also had L the domino array pitch but this is just d plus the width of the dominoes, but we are going to make the assumption of "thin" dominoes here so d and L are equivalent variables for our purposes). We wish to learn something about the wave speed, so we want to write and equation where a dimensionless group containing V appears on the left hand side and some function of the other dimensionless groups appears on the left.
The dimensions of the variables we are considering are:
 ,  ,  , 
where [L] denotes dimensions of length, and [T] denotes dimensions of time.
Now it is clear that we want one of our groups to be a non-dimensional wave velocity, this group has to be:
and the other group/s to not contain V, which leaves us with some power of d / H. For simplicity we take this to be the first power of this ratio, so we have:
But we are at liberty to redefine our non-dimensional wave velocity by multiplying through by any power of d / H , so we can change our relation above to:
Which is the form given in McLachlan et al [5]. 
|
