Документ взят из кэша поисковой машины. Адрес оригинального документа : http://ics.org.ru/upload/iblock/d40/126-galton-board_ru.pdf Дата изменения: Wed Oct 28 19:30:55 2015 Дата индексирования: Sun Apr 10 00:00:09 2016 Кодировка:

V. V. KOZLOV
Department of Mechanics and Mathematics Moscow State University, Vorob'ievy Gory 119899, Moscow, Russia

M. Yu. MITROFANOVA
E-mail: mitrofanov@breitmeier.de

GALTON BOARD
Received October 1, 2003

DOI: 10.1070/RD2003v008n04ABEH000255

In this paper, we present results of simulations of a model of the Galton board for various degrees of elasticity of the ball-to-nail collision.

1. Intro duction
The Galton b oard is an upright b oard with evenly spaced nails driven into its upp er half. The nails are arranged in staggered order. The lower half of the b oard is divided with vertical slats into a numb er of narrow rectangular slots. From the front, the whole installation is covered with a glass cover. In the middle of the upp er edge, there is a funnel in which balls can b e p oured, the diameter of the balls b eing much smaller than the distance b etween the nails. The funnel is lo cated precisely ab ove the central nail of the second row, i. e. the ball, if p erfectly centered, would fall vertically and directly onto the upp ermost p oint of this nail's surface (Fig. 1). Theoretically, the ball would rep eatedly b ounce off this nail's upp ermost p oint. Obviously, such a motion of the ball is unstable. In fact, due to unavoidable inaccuracy in the b oard's p ositioning and imp ossibility to completely exclude the lateral comp onent, no matter how small, of the ball's velo city, each ball, generally sp eaking, would meet the nail somewhat obliquely. The ball would then deviate from the vertical line and, after having collided with many other nails, fall into one of the slots. If the exp eriment is run with a large numb er of balls, dropp ed one after the other, then the following results are obtained: the balls are distributed evenly to the left and to the right of the central compartment (left and right deviations are equiprobable). Besides, the balls would more rarely fall into the leftmost and rightFig. 1 most compartments, for large deviations are more rare to app ear than small ones. However, despite the presence of nails and all the imp erfections in the construction, the ma jority of the balls will agglomerate in the central compartment as this provides the smallest deviation. The numb er of balls in the compartments would approximately corresp ond to the Gaussian law of errors. In the earlier exp eriments with the Galton b oard the funnel was filled with p ellets or millet grains.
Mathematics Sub ject Classification 37A50, 70F45, 70F35, 82C22



REGULAR AND CHAOTIC DYNAMICS, V. 8,

4, 2003

431


V. V. KOZLOV, M. Yu. MITROFANOVA

2. Statement of problem
In Galton b oard exp eriments ball-to-nail impacts have always b een inelastic. In this pap er, we present results of simulations of a mo del of the Galton b oard for various degrees of elasticity of the ball-to-nail collision. We mo del the ball as a mass p oint. Hence, the ball's motion can b e regarded as the motion of a mass p oint in a vertical plane under the action of gravity accompanied with multiple collisions with the nails. These collisions are characterized by the co efficient of restitution e, which affects only the normal comp onent of the ball's velo city after the impact. The co efficient of restitution is the first parameter of the problem. It can vary from 0 to 1. A value of e = 1 corresp onds to absolutely elastic impact for which the ball's energy do es not change. The other extreme case, e = 0, corresp onds to absolutely inelastic impact: the ball "sticks" to the nail. The nail's radius R is the second parameter of the problem. Since the ball leaves the funnel and falls onto the nail centrally, but p ossibly with some small departure to the left or to the right, we adopt that the first drop of the ball ob eys the Gaussian law. On the other hand, if the balls distribute uniformly over the funnel's op ening, then their distribution over the rectangular compartments will b e far from normal (Fig. 2). This distribution resembles the arcsine law . Incidentally, according to Paul Levy, the distribution of time intervals over which a Brownian particle is lo cated on the p ositive semi-axis, has a similar form. This observation is, probably, not just a coincidence. The p oint is that a particle in Brownian motion exp eriences a large numb er of random collisions with molecules of the surrounding fluid (in our case, the "molecules" are regularly placed and fixed).

Fig. 2

Accordingly, the problem's third parameter is the variance of the distribution of the balls over the funnel's op ening. Thus, we intro duce three parameters for the problem: 1) the co efficients of restitution e, 2) the nail's radius R, 3) the variance 0 of the normal distribution of the first ball-tonail impact. It is also necessary to cho ose the dimensions of the mo del b oard. The geometry of the b oard should meet the two requirements: · the balls should not reach the vertical b oundaries of the Galton b oard; · each ball should exp erience at least several collisions with the nails b efore it gets into one of the rectangular compartments. Figure 3 (a­d) shows the balls' distribution over the rectangular compartments for different dimensions of the mo del b oard, namely, 50 в 50 (a), 100 в 100 (b), 200 в 200 (c), and 400 в 300 (d). In these cases, the three parameters of the problem are: e = 0.8, R = 0.7, and 0 = 0.05. As we can see, the distribution of the balls lo oks similarly for all the sp ecified dimensions of the b oard, but in


432

REGULAR AND CHAOTIC DYNAMICS, V. 8,

4, 2003


GALTON BOARD

a

b

c
Fig. 3

d

the case of 50 в 50 (Fig. 3a) the balls do reach the vertical b oundaries of the mo del Galton b oard. For the 100 в 100 b oard, the balls no longer reach the b oundaries (Fig. 3b). Further enlargement of the mo del b oard's dimensions (its length and its height) do es not affect the pattern of the balls' distribution over the compartments (Figs. 3c, d), but greatly increases the computation time. Thus, the dimensions of the mo del Galton b oard can b e set to 100 в 100 without any loss of quality. So, we are going to investigate the prop erties of the balls' distribution over the compartments of the Galton b oard and the dep endency of the variance of this distribution on the three sp ecified parameters.

3. Mathematical mo del
The metho d of investigation consists in simulating the motion of the balls (mass p oints) and taking into account their collisions with obstacles (the nails) for different values of the three sp ecified parameters of the problem. On the Galton b oard, we intro duce an orthogonal co ordinate system O xy in the following way: the axis O x is directed horizontally and passes through the upp er b oundaries of the rectangular compartments, in which the falling balls are to b e collected (for brevity, from this p oint on, we will say compartments instead of rectangular compartments). The axis O y is directed vertically and go es through the center of the nail that a ball is to hit first. The length of the b oard is taken large enough for the balls not to reach its vertical b oundaries (as was sp ecified earlier). Thus, a pair x, y represents the p osition of a ball in the plane of the Galton b oard. The fall of the ball is describ ed with a set of two ordinary differential equations: x = 0, Ё y = -g , Ё (3.1)

where g is the gravitational acceleration. Since the ball falls from the funnel and onto the first nail under gravity, the velo city of the ball at the p oint of the first impact is v0 = 2g (h0 - R sin 0 ), where h0 is the distance b etween the funnel's op ening and the center of the first nail, R is the nail's radius, while 0 is the angle b etween the axis O x and the radius drawn to the p oint where the ball hits the nail (Fig. 4). We will investigate the further motion of the ball according to the following plan: 1. Intro duce a co ordinate system fixed to the nail: its origin is at the ball-to-nail impact p oint, and its axes are the tangent and the normal to the nail's surface at this p oint. Thus, with resp ect


REGULAR AND CHAOTIC DYNAMICS, V. 8,

4, 2003

433


V. V. KOZLOV, M. Yu. MITROFANOVA

Fig. 4

to this co ordinate system, the velo city of the particle at the first impact p oint has the following n comp onents: v0 = v0 cos 0 , v0 = -v0 sin 0 .
2. After the impact with the nail, the velo city comp onents will change and take the form: v 1 = n n = v0 cos 0 , v1 = -ev0 = ev0 sin 0 , where e is the co efficient of restitution.

3. Then the ball will move in a parab ola. To find its path, we solve the equations (1) with the following initial values: x (0) = x0 , y (0) = y0 , x (0) = v1 cos 0 , y (0) = v1 sin 0 , where (x0 , y0 ) n are the co ordinates of the ball at the time it hits the nail, v 1 = (v1 )2 + (v1 )2 , while 0 is the angle b etween the axis O x and the velo city vector v 1 . Thus, the ball's path is the following parab ola: g y=- 2 (x - x0 )2 + (x - x0 ) tan 0 + y0 . (3.2) 2v1 cos2 0 The p ortion of the parab ola the ball will take is determined by the direction of the ball's velo city vector after its impact with the nail. 4. From (1) we find the velo city with which the ball will approach the next nail. Let (x 1 , y1 ) b e the co ordinates of the p oint of the next ball-to-nail impact. Then the velo city of the ball on the surface of this nail has the following comp onents:
x v2 = v1 cos 0 ,

g (x1 - x0 ) y v2 = - v cos + v1 sin 0 . 1 0

Then another collision o ccurs, and again the ball's motion is calculated according to the pro cedure describ ed ab ove. The whole op eration is rep eated until the ball crosses the axis O x. As so on as the ball's path crosses the axis O x, we find the intersection p oint and thus determine the compartment the ball falls into. One of the most imp ortant things ab out this mo del is to find the nail that the ball is going to hit next. To that end, consider the p erp endiculars to the ball's path which go through the nails' centers. Such p erp endiculars are describ ed with linear equations: x - xn +
434

-

g (xn - x v cos2
2

imp

) + tan

(y - yn ) = 0,
4, 2003


(3.3)

REGULAR AND CHAOTIC DYNAMICS, V. 8,


GALTON BOARD

where (ximp , yimp ) are the co ordinates of the previous ball-to-nail impact, (x n , yn ) are the co ordinates of the path's p oint through which a p erp endicular is drawn, v is the magnitude of the ball's velo city after the previous impact, and is the angle b etween the axis O x and the velo city vector v . Since our goal is to find p erp endiculars through the nails' centers, we insert the co ordinates of the center of one of the nails (xc , yc ) into (3). From this equation, we find a pair (x n , yn ) which meets the following requirements: · the distance b etween the nail's center and the p oint (x n , yn ) is smaller than the nail's radius; · the absolute value of the difference b etween the abscissa of the previous impact p oint and the abscissa of the path's p oint, through which the p erp endicular is drawn, is as small as p ossible. The first requirement is to ensure that the ball's path meets the nail, i. e. the co ordinates of the next impact p oint (x, y ) can b e found. These co ordinates satisfy the following set of equations: (x - xc )2 + (y - yc )2 = R2 , where (xc , yc ) are the co ordinates of the nail's center, R is the nail's radius, and (x imp , yimp ) are the co ordinates of the previous impact p oint. The second requirement is to take the impacts in their sequence. This follows from the parametric form of the ball's path. We solve the equations (1) with the following initial values: x (0) = x imp , y (0) = yimp , x (0) = v cos , y (0) = v sin , where (x imp , yimp ) are the co ordinates of the previous impact p oint, v is the magnitude of the ball's velo city after the previous impact, while is the angle b etween the axis O x and the velo city vector v . The result is the parametric form of the ball's path after it has hit the nail: x = v t cos + x
imp

y = - 2v 2 cos2 (x - x

g

imp

)2 + (x - x

imp

) tan + y

imp

,

,

g y = - t2 + v t sin + y 2

imp

.

5. Simulation results. The mo del Galton b oard has b een implemented as an interactive Microsoft Visual C++ application. The application offers the opp ortunity to vary every parameter of the mo del: the nail's radius, the co efficient of restitution, and the variance of the initial distribution when the ball hits the first nail; it also allows varying the numb er of dropp ed balls. The application outputs a histogram of the balls in the compartments and the variance of the final distribution of the balls over the compartments. Besides, the exp eriment's results can b e visualized. First, we get a histogram of the distribution of the balls over the compartments (Fig. 5). Form this histogram, the variance is calculated using the following well-known formulas:
n

x= Ї
i=1

N xi i , N

n

=
i=1

2

(xi - x) Ї

2

Ni , N compartment, n numb er of balls curve using the with the mo del

where x is the mathematical exp ectation, x i is the co ordinate of the center of the i th Ї is the numb er of compartments, N is the numb er of dropp ed balls, N i is the final accumulated in the i th compartment, and is the variance of the distribution. Second, using the variance obtained, we plot the normal distribution (Gaussian) 2 1 Ї2 Gauss formula f (x) = e-(x-x) /(2 ) . Next, we compare the theoretical curve
2

curve (Fig. 6). The mo del curve is plotted with squares, while the theoretical curve is plotted as a solid line. The results given in this pap er were obtained for 100 000 dropp ed balls. This numb er is optimal from the viewp oint of the result's accuracy and the pro cessing time needed for the exp eriment.


REGULAR AND CHAOTIC DYNAMICS, V. 8,

4, 2003

435


V. V. KOZLOV, M. Yu. MITROFANOVA

Fig. 5

Fig. 6

Figures 7 (a­d) show the histograms of the balls' distribution over the compartments for 1 000 (a), 10 000 (b), 100 000 (c), and 1 000 000 (d) of dropp ed balls. These results were obtained with the following value of the parameters: e = 1, R = 0.1, and 0 = 0.04. We can see that the histograms shown in Figs. 7c and d are, for all practical purp oses, identical. The accuracy of the results can also b e judged by the figures given in Table 1. These figures are the values of the variance of the final distribution for 10 000, 100 000 and 1 000 000 dropp ed balls in eight series of simulations.

a

b

c
Fig. 7

d

Table 1. e = 1, R = 0.1, and 0 = 0.04

N 10000 100000 1000000

1 7.3820 7.4066 7.4028

2 7.3381 7.3817 7.3988

3 7.3367 7.3841 7.3901

4 7.4097 7.3938 7.3917

5 7.4266 7.3779 7.4066

6 7.4201 7.4387 7.3958

7 7.3159 7.4063 7.3902

8 7.3559 7.3713 7.4104

Let us first consider the case where the balls are distributed uniformly on the width of the funnel's op ening. Instead of the normal distribution of the balls over the compartments (as it might


436

REGULAR AND CHAOTIC DYNAMICS, V. 8,

4, 2003


GALTON BOARD

b e exp ected), we get a somewhat unusual distribution with p eripheral p eaks and two distinctive gaps near the center (Fig. 2). These gaps are lo cated symmetrically with resp ect to the vertical axis through the funnel's center. For this case, the 200 в 100 mo del Galton b oard was taken, otherwise the balls reach its vertical b oundaries. More precisely, Fig. 2 corresp onds to the case of absolutely elastic impact (e = 1) and R = 0.1.

a

b

c
Fig. 8

d

For a smaller value of the co efficient of restitution (e = 0.8) and an increased value of the nail's radius to R = 0.3, the balls' distribution over the compartments changes not very significantly: distinctive p eripheral p eaks are still present, while instead of two pronounced gaps we have several symmetrically lo cated small pits (Fig. 8a). With a further decrease of the co efficient of restitution the depth and structure of these pits changes. In Figs. 8b, c, and d, the histograms are shown for e = 0.6, e = 0.4, and e = 0.1, resp ectively (the nail's radius is the same, R = 0.3). If the balls are fed into the funnel according to a Gaussian law with large disp ersion 0 , then the form of the histograms will not change qualitatively. Therefore, the case of small disp ersion 0 b ecomes esp ecially interesting. Let 0 = 0.05 and R = 0.4. We are going to investigate the form of the histogram, as the co efficient of restitution e decreases from 1 to 0. Figures 9 (a­d) show the balls' distributions over the compartments for e = 1 1 (a), 0.9 (b), 0.8 (c), and 0.7 (d). We can see that the distribution in the case of absolutely elastic impact is almost Gaussian with two noticeable pits. As the co efficient of restitution decreases, the "normal" distribution gets "corrupted"; instead of the pits, distinctive gaps app ear, which b ecome deep er with a decrease of e. However, this picture holds only for e b elow a value of e 0.7. A further decrease of the co efficient of restitution makes the gaps disapp ear, and the distribution b ecomes practically indistinguishable from the Gaussian distribution (Fig. 9e). Figures 10 (a­e) show a similar series of histograms for R = 1, while the co efficient of restitution takes successively the values 0.7, 0.6, 0.3, 0.2, and 0.1. We can see that for e = 0.2 there are two gaps, while for larger and smaller values of e the distribution is, practically, Gaussian. A further increase in e results in a distribution which is very different from Gaussian. The mentioned "o ccasions" of deviation of the final distribution of the balls from a Gaussian distribution are intriguingly asso ciated with non-monotonic b ehavior of the variance as a function of two variables R and e (with 0 b eing fixed). Table 2 gives the values of for 0 = 0.05. We can see that for a fixed R , the variance a lo cal maximum. It is exactly that value of the co efficient of restitution in the vicinity of which a substantial deviation from the Gaussian distribution o ccurs.


REGULAR AND CHAOTIC DYNAMICS, V. 8,

4, 2003

437


V. V. KOZLOV, M. Yu. MITROFANOVA

a

b

c

d

e
Fig. 9

a

b

c

d

e
Fig. 10



438

REGULAR AND CHAOTIC DYNAMICS, V. 8,

4, 2003


GALTON BOARD

For example, for R = 0.4, the variance has a maximum at e 0.7; this value has already b een mentioned in connection with the analysis of the series of histograms in Fig. 9. Similarly, for R = 1, the lo cal maximum is reached at e 0.2 (as it should b e, according to Fig. 10).
Table 2. 0 = 0.05

e\R 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0.1 9.24 9.06 8.79 8.57 8.33 8.06 7.77 7.53 7.03 6.67

0.2 9.19 8.97 8.47 8.56 8.26 8.00 7.72 7.42 7.08 6.72

0.3 9.26 9.05 8.32 8.46 8.30 7.98 7.68 7.34 7.05 6.88

0.4 9.44 9.18 8.07 8.41 8.21 7.94 7.61 7.30 7.07 7.03

0.5 9.61 9.22 7.83 8.34 8.18 7.91 7.64 7.22 7.08 6.94

0.6 9.94 9.32 7.52 8.26 8.15 7.84 7.57 7.23 7.08 6.96

0.7 10.29 9.57 7.78 8.22 8.10 7.79 7.52 7.17 7.18 6.83

1 10.38 10.63 8.22 7.14 7.96 7.65 7.36 7.02 7.23 6.68

1.2 11.27 12.27 8.05 7.69 7.89 7.60 7.22 6.86 7.20 6.68

1.5 12.80 11.36 8.47 8.82 7.72 7.75 7.06 6.76 6.64 6.72

The sp ecified features of the histograms require theoretical treatment and interpretation. The problem of the gaps should b e esp ecially emphasized b ecause this problem is likely to b e most directly relevant to the famous Kirkwo o d gaps in the distribution of asteroids in the main asteroid b elt b etween Mars and Jupiter. It is well known that these gaps cannot b e satisfactorily explained with the resonance ratios of the orbital p erio ds of the ma jor planets. Meanwhile it would b e useful to investigate a simple mo del, where small planets (large asteroids) move along circular orbits, and there also is a flow of small asteroids colliding with the large ones without p erturbing their paths. In this case, the impact is not absolutely elastic (0 < e < 1). After a large numb er of collisions, one would obtain a distribution of the asteroids' flow over the semi-ma jor axes. This distribution may contain a series of gaps, as that in the case of Galton b oard. This work was partially supp orted by the grant "Principal Scientific Scho ols" (00-15-96146).



REGULAR AND CHAOTIC DYNAMICS, V. 8,

4, 2003

439