- Email: [email protected]
Dynamics of a viscous ball rolling down on a rigid staircase
ARTICLE IN PRESS
Physica A 374 (2007) 524–532 www.elsevier.com/locate/physa
Dynamics of a viscous ball rolling down on a rigid staircase Hua Yana,b, Qingfan Shia,b, Decai Huanga, Gang Suna, a
Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China Department of Physics, Beijing Institute of Technology, Beijing 100081, China
b
Received 14 March 2006; received in revised form 23 June 2006 Available online 14 August 2006
Abstract We theoretically investigate the motion of a ball rolling down on a periodical staircase. Our research is restricted in the case of completely inelastic collision when the ball falls down on the surface of the stairs. The ball is accelerated when it rolls cross the edge of the stair, while it is decelerated when it rolls on the horizontal surface due to the rolling friction. The competition between them causes two different regimes according to the parameters of the system. One is the steady moving regime in which the ball keeps moving forever, and the other is the still regime in which the ball finally stops after rolling on a finite number of stairs. The diagram of these two regimes is given in the reduced parameter space. The tendency that smaller scale of the staircase can keep moving on smaller inclined angle is found. r 2006 Elsevier B.V. All rights reserved. Keywords: Friction; Granular materials; Granular solids; Mechanical contacts
1. Introduction The flow of granular materials has been extensively investigated due to not only its importance in various industrial processes but also the nonlinear and dissipative behavior of the materials in flow processes [1–3]. In particular, the understanding of the dynamics of the dense volcanic granular flow is important for predicting and protecting the natural disasters such as landslides of mountain bodies and mud-rock flows, etc. However, despite intensive efforts in the field, the problem is far from being completely understood [4]. Many questions are still unknown at present. In order to approach the dense volcanic granular flow, it is important to find the basic physical mechanisms that would simply explain the main behaviors of the flow. For this purpose, the research on a simpler problem of the dynamics of a single ball interacting with a set of boundaries was been carried out both theoretically and experimentally in recent years [5–10]. This seemingly simple problem, in which the most complicated multibody interactions are ignored, can show different regimes in parameter space resulting from the competition between the factors of acceleration and deceleration. These regimes are dominated by several basic physical mechanisms and have some key characters that are similar to the dense volcanic granular flow. Especially, Alonso et al. have investigated the dynamics of a single ball rolling down on a staircase. They treat the Corresponding author.
E-mail address: [email protected] (G. Sun). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.07.037
ARTICLE IN PRESS H. Yan et al. / Physica A 374 (2007) 524–532 vi
525
N ω
mg
vi+1
f
L y
H o
x
Fig. 1. Schematic of our system. A viscous spherical ball with radius R is rolling down on a rigid staircase. The stairs are with uniform height H and width L.
problem by introducing two parameters of energy increase due to gravity and energy dissipation in rough surface [10]. In this paper, we further investigate the system of a single ball rolling down a staircase but without unknown parameters. The simplified model we considered is that a single ball rolls down on a regularly spaced staircase under gravity as depicted in Fig. 1. The mass and radius of the ball are m and R, respectively. It is noticeable that the outer edges of the stairs are rounded by a small circle with radius d to eliminate the singular geometry in the stairs, though the final results are obtained in the limiting case of d ! 0. The height and length of each stair are H and L, respectively. The collision between the ball and the horizontal surface of the stair is assumed to be completely inelastic, which means that the vertical momentum would be dissipated completely when the ball falls down on it. The sliding friction coefficient is set to be infinite to avoid sliding, and rolling friction is described by rolling friction coefficient krol as proposed by Brilliantov and Poschel [11,12]. After all these parameters settle down, two different contributions to the motion of the ball come forth. One is the acceleration in horizontal direction when the ball rolls across the edge of the stair, and the other is the deceleration when the ball rolls on the horizontal surface of the stairs due to rolling friction. The competition between these two contributions results in different physical regimes. The model is also inspired by the real systems in nature and contains no unknown parameters, though it is greatly simplified for being solved analytically. We hope that our results can be regarded as a first step towards the understanding of real systems.
2. Acceleration by rolling cross the edge To solve the problem, we choose the position just above the edge of the stair as a starting point of each stair. The horizontal velocity of the ball at this position is the only quantity specifying the state of motion, because the vertical velocity of the ball will dissipate completely when it falls down on the surface of the stairs and the no-sliding condition will give a unique relation between horizontal velocity and the angular velocity. From this starting point, the ball will be accelerated when it falls around the edge. We firstly consider the acceleration process in detail (see Fig. 1). The Newton equations in normal and tangential directions are N mg cos f ¼ m
v2 , Rþd
(1)
ARTICLE IN PRESS H. Yan et al. / Physica A 374 (2007) 524–532
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and mg sin f f ¼ m_v,
(2)
respectively. And the rotational equation is _ Rf ¼ I o.
(3)
In the equations, v is the absolute value of the velocity of the ball and it is a function of the angle f between the y-axis and the line connecting the center of the ball and the edge (see Fig. 1). N is the normal force acting on the ball by the edge (see Fig. 1) and f is the sliding friction acting on the ball. o is the angular velocity of the ball to its mass center. Under the no-sliding condition, we have the following two confinement equations _ þ dÞ ¼ oR. The initial velocity is v0 . The moment of inertia of the ball is denoted by v ¼ oR and fðR I ¼ 25 mR2 . Using Newton equation in tangential direction and the rotational equation, we can get an equation of f € ¼ f
mgR2 sin f. ðmR þ IÞðR þ dÞ 2
pffiffiffiffiffiffi Integrating the equation and introducing a dimensionless parameter u ¼ v= gR, we have 10 d 1þ ð1 cos fÞ. u2 ¼ u20 þ 7 R
(4)
(5)
It is noticeable that this equation can also be obtained by energy conservation laws, because there is no sliding occurs during the whole process. The ball will depart from the edge of the corner of the stair at a certain angle fap . This angle and the velocity at this time can be derived from the Newtonian equation in normal direction with condition N ¼ 0. After taking the limit d=R ! 0, we have, fap ¼ arccos
7u20 þ 10 , 17
(6)
and uap
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7u20 þ 10 . ¼ 17
(7)
When the ball leaves the edge of the stair, the acceleration process is completed. It is noticeable that u0 must be less than 1, otherwise the ball will depart from the edge at the initial time with horizontally parabolic flight. In this case, there is no acceleration effect in horizontal direction, the velocity of the ball must turn back to u0 o1 owing to the deceleration effect of rolling friction when it rolls on the horizontal surfaces of stairs. The horizontal and vertical components of uap now are expressed by 2 3=2 7u0 þ 10 ap ux ¼ uap cos fap ¼ , (8) 17 and uap y
¼ uap sin fap
2 1=2 " 2 2 #1=2 7u0 þ 10 7u0 þ 10 ¼ 1 . 17 17
(9)
When the ball falls down on the horizontal surface, the vertical velocity will be dissipated completely, while the horizontal velocity remains to move it forward. The angular velocity also remains in the value when it leaves the corner, i.e., oap ¼ vap =R. The no-sliding condition does not hold for this horizontal velocity and angular velocity when it just falls down on the surface of the next stair. However, the ball will tend to nosliding condition very quickly and the distance of the sliding is negligible, as the sliding friction coefficient was supposed to be infinite in our model. The final no-sliding velocity can be easily obtained by ordinary
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0.6
0.5
u′-u0
0.4
0.3
0.2
0.1
0.0 0.0
0.2
0.4
0.6
0.8
1.0
u0
Fig. 2. The variation of the acceleration function with the initial dimensionless velocity u0 .
mechanical equations, the result is sffiffiffiffi 5 ap 2 R 0 u ¼ ux þ oap 7 7 g 2 3=2 1=2 5 7u0 þ 10 2 7u20 þ 10 ¼ þ 7 17 7 17 1=2 5 2 7u20 þ 10 7u20 þ 10 þ ¼ . 7 5 17 17 According to Eq. (10), we define an acceleration function by 1=2 5 2 7u20 þ 10 7u20 þ 10 þ f a ¼ u0 u0 ¼ u0 . 7 5 17 17
ð10Þ
(11)
The acceleration function is only a function of u0 and is plotted in Fig. 2. We can find that the function monotonically decreases from its maximum value at u0 ¼ 0 to zero at u0 ¼ 1. The function is concave between ½0; 1, which is the interesting range in our problem. This concave property is important for illustrating the unique equilibrium point in the later. We finally emphasize that Eq. (11) is correct only for the condition that the ball would not touch the horizontal surface before it leaves the corner, which is expressed by H4Rð1 cos fap Þ. 3. Deceleration by rolling on the horizontal surface The ball will be decelerated when it rolls on the horizontal surface due to rolling friction. We treat this problem by adapting the moment of the friction proposed by Brilliantov and Poschel [11,12], i.e., M ¼ krol vN ¼ krol vmg. The dynamic equations are f ¼ mv_,
ð12Þ
2 fR M ¼ mR_v. 5
ð13Þ
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pffiffiffiffiffiffi These equations give an exponential decay velocity, i.e., vðtÞ ¼ v0 expðð5krol g=7RÞtÞ, where v0 ¼ u0 gR is the initial velocity of the ball when it starts rolling on the horizontal surface of the staircase. Accordingly, as time tends to infinite the rolling distance of the ball is finite, i.e., 7Rv0 =5krol g. For a rolling distance L0 shorter than the value (otherwise the ball will stop in the midway), the velocity of the ball is rffiffiffiffi 5krol g 0 00 0 u ¼u (14) L, R 7R where u00 is the velocity of the ball after passing through the distance L0 . In our system, the ball will roll from the landing point to the end of the stair if it has enough speed. The horizontal distance that the ball passes through before landing can be calculated by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2H 0 2 DL ¼ R sin fap þ Ruap , (15) uap ðuap y Þ þ x y þ R where H 0 ¼ H Rð1 cos fap Þ. Similarly, H 0 40 must be satisfied to ensure the ball does not touch the horizontal surface before it leaves the corner. DL is related to u0 and H=R. We plot DL as a function of u0 for several fixed H=R in Fig. 3. We can see that DL is a monotonic increase function with concave curvature for all the cases. The rolling distance on the horizontal part of the stair is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2H 0 2 0 ap ap L ¼ L DL ¼ L R sin fap Rux uy þ ðuap . (16) y Þ þ R By Eqs. (14) and (16), we can define a deceleration function as rffiffiffiffi 5krol g 0 0 00 fd ¼ u u ¼ L. R 7R
(17)
According to the discussion of DL, we can see that f d is a monotonic decrease function with convex curvature. 5 H/R=10.0 H/R=5.0 4 H/R=3.0 H/R=1.0
ΔL
3
2
1
0 0.0
0.2
0.4
0.6
0.8
1.0
u0
Fig. 3. The horizontally distance that the ball pass through before landing as a function of the initial dimensionless velocity u0 for several fixed H=R.
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We have to point out that the deceleration function may be less than zero for some cases. This happens when LoDL, i.e., the ball will jump over the horizontal surface of the stair. In this case, the ball may collide on the corner of the stairs and cause unpredictable behavior. We shall exclude the situation from the rest of this paper, and only restrict our attention to the case of f d ju0 ¼1 40. 4. Equilibrium state and phase diagram When the acceleration equals the deceleration, the velocity of the ball will not change. From previous discussion on the acceleration and deceleration functions, we know that the acceleration function has no other parameters while the deceleration function has some. According to the parameters in deceleration function, two different type of solutions can be found. Firstly, when f d ju0 ¼0 4f a ju0 ¼0 , there is no solution in the range [1, 0], because the acceleration function is concave and that of the deceleration function is convex (as shown in Fig. 4(a)). In this case, the deceleration is larger than the acceleration for all u0 , so the ball will finally stop at a certain position. In contrary, when f d ju0 ¼0 of a ju0 ¼0 , there is a unique solution (as shown in Fig. 4(b)). The deceleration is larger than the acceleration when u0 is larger than the solution, while the deceleration is smaller than the acceleration when u0 is smaller than the solution. This means that dimensionless velocity of getting out of the stair will decrease when u0 larger than the solution, while it will increase when u0 smaller than the solution. Thus, this solution corresponds to a stable equilibrium state. Considering both acceleration and deceleration processes, we can obtain an iteration function, uiþ1 ¼ F ðui Þ ¼ ui þ f a ðui Þ f d ðui Þ,
(18)
where ui is the dimensionless velocity at the starting point of ith stair. Fig. 5 is an illustration of how to use this iteration function to show the time evolution of the ball rolling down on the staircase. As shown in Fig. 5, the 0.8 (a) 0.6
0.4
0.2
0.0 (b) 0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
u0 Fig. 4. Schematic view of (a) the case having no solution when f d ju0 ¼0 4f a ju0 ¼0 and (b) that having a unique solution when f d ju0 ¼0 of a ju0 ¼0 . The solid and dashed lines are for acceleration and deceleration function, respectively.
ARTICLE IN PRESS H. Yan et al. / Physica A 374 (2007) 524–532
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(a) u0
0.8
ui+1
0.6 0.4 0.2 0.0
(b) 0.8 u0 ui+1
0.6 u*
0.4 0.2 u0′ 0.0 0.0
0.2
0.4
0.6
0.8
1.0
ui
Fig. 5. The illustration of iteration mechanism of Eq. (18) for the case of (a) F jui ¼0 40 and (b) F jui ¼0 o0. The final steady state will be that with a positive velocity u or with zero velocity, respectively.
ball will finally tend to a steady moving state or a still state according to the value F jui ¼0 . When the initial velocity is positive, the ball approaches asymptotically to a positive velocity u if F jui ¼0 40, i.e., f d ju0 ¼0 of a ju0 ¼0 , while it will stop at some place if F jui ¼0 o0, i.e., f d ju0 ¼0 4f a ju0 ¼0 . It is apparent that u is the solution of f d ¼ f a , which exists uniquely for the condition that F jui ¼0 40, as discussed in detail in the beginning of this section. The system we considered contains four independent parameters ðkrol ; R; L; yÞ. In the parameter space, there are two special regimes in which our previous formulae may be not satisfied. One corresponds to that of the ball jumping over the horizontal surface of the stair at high speed (u0 ¼ 1), and the other corresponds to that of the ball touching the horizontal surface before it leaves the corner at very low speed (u0 ¼ 0þ ). These two regimes do not depend on the parameter krol , and are expressed by DLju0 ¼1 oL,
(19)
H4Rð1 cos fap Þju0 ¼0 ,
(20)
and
respectively. If we measure the length of the stair L by the radius of the ball R, the previous equations can be rewritten by only two parameters, l ¼ L=R and tan y ¼ H=L. The detailed formulae are lo2 tan y,
(21)
and 7=17 , (22) tan y respectively. The two regimes are shadowed by slash and back slash lines, respectively, in the parameter space y–l (see Fig. 6). lo
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25 μ = 0.05 μ = 0.1 μ = 0.2
20
μ = 0.3
l
15
10
5
0 0
15
30
45
60
75
90
θ (degree)
Fig. 6. The boundaries of the steady moving regime and still regime in y–l parameter space for a series of fixed m. The regime below the boundary corresponds to that of the steady moving state, while that above the boundary to that of the still state. The regimes that the ball will jump over the horizontal surface of the stair and touch the horizontal surface before it leaves the corner is shadowed by slash and back slash lines, respectively.
In other parameter space, our formulae are satisfied and there exist two regimes, one of which is where the final state of the ball is a steady moving state and the other is where the ball will stop at end. The boundaries of the steady moving regime and the still regime are determined by f a ju0 ¼0 ¼ f d ju0 ¼0 . After some trivial deducing qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and introducing a parameter m ¼ gk2rol =R, which is a parameter only dependent on the properties of the ball, we have, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 7c1 c3 7 2 tan y ¼ , (23) l 1 ð10=17Þ þ c4 þ 2c2 l 5m 2l 17l where 2 10 10 1=2 þ , 5 17 17 3 10 ap 2 c2 ¼ ðux Þ ju0 ¼0 ¼ , 17 " 2 # 10 10 ap 2 1 c3 ¼ ðuy Þ ju0 ¼0 ¼ , 17 17 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 10 10 ap ap 1 . c4 ¼ ux uy ju0 ¼0 ¼ 17 17 c1 ¼ f a ju0 ¼0 ¼
ð24Þ
Eq. (23) means that the boundary is determined only by three reduced parameters, i.e., l, y and m. We plot the boundaries in y–l space for a series of fixed m in Fig. 6. The regime below the line corresponds to that of the steady moving state, while the regime above the line to that of the still state. From Fig. 6, we can find the following basic physical properties. Firstly, from the monotonic increasing curve of the boundary for fixed m, we find that the larger scale of the staircase, i.e., larger l, has larger repose
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angle of the staircase, which means that the ball is prone to stop for larger scale staircases. For different m, our results show that the regime of steady moving state increases as m decreases. According to the definition of m, pffiffiffi parameter m is proportional to krol and g. Thus, the previous results mean that the regime of steady moving state will become wider by decreasing the rolling friction coefficient krol or decreasing the gravitational acceleration g. It is straightforward that the decreasing of the rolling friction coefficient will increase the regime of the steady moving state because it only reduces the effect of rolling friction. However, the changes of the regime on g is not straightforward, because the gravitational acceleration affects both the acceleration and deceleration processes. 5. Conclusion We have analytically solved the problem of a ball rolling down on a periodical staircase. In our model, the rolling process is considered in detail. To eliminate the singular geometry of the stairs, the outer edges of the stairs are first rounded by a small circle, and then we take the limit of the circle to be a point for the final result. To solve the problem analytically, we have also restricted that the collision between the ball and the horizontal surface of the stair is completely inelastic and the sliding friction coefficient is infinite. Despite all these simplifications, the result contains the most important physical components, i.e., the acceleration process when the ball rolls across the edge of the stair and the deceleration process when it rolls on the horizontal surface due to the rolling friction. Our solution shows that the competition between them causes two different regimes. One is the steady moving regime and the other is the still regime. The boundary of the two regimes is determined by three reduced parameters. Our results also show that smaller scale of the staircase can keep the ball moving on smaller inclined angle. Acknowledgments We wish to thank Prof. K.Q. Lu for fruitful discussions. This work is supported by the National Key Program for Basic Research and the Chinese National Science Foundation project no. 10374111. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
D. Bideau, A. Hansen (Eds.), Disorder and Granular Media, North-Holland, Amsterdam, 1993. A. Metha (Ed.), Granular Matter: An Interdisciplinary Approach, Springer, Heidelberg, 1994. H.J. Herrmann, J.P. Hovi, S. Luding (Eds.), Physics of Dry Granular Media, Kluwer Academic Publishers, Dordrecht, 1997. M. Bursik, A. Patra, E.B. Pitman, C. Nichita, J.L. Macias, R. Saucedo, O. Girina, Rep. Prog. Phys. 68 (2005) 271. A. Valance, D. Bideau, Phys. Rev. E 57 (1998) 1886. G.G. Batrouni, S. Dippel, L. Samson, Phys. Rev. E 53 (1996) 6496. C. Henrique, M.A. Aguirre, A. Calvo, I. Ippolito, S. Pippel, G.G. Batrouni, D. Bideau, Phys. Rev. E 57 (1998) 4743. B. Tadic, Phys. Rev. E 57 (1998) 4375. L. Quartier, B. Andreotti, S. Douady, A. Daerr, Phys. Rev. E 62 (2000) 8299. J.J. Alonso, J.P. Hovi, H.J. Herrmann, Phys. Rev. E 58 (2000) 672. N.V. Brilliantov, T. Poschel, Eur. Phys. Lett. 42 (1998) 511. N.V. Brilliantov, T. Poschel, Eur. Phys. J. B 12 (1999) 299.
Physica A 374 (2007) 524–532 www.elsevier.com/locate/physa
Dynamics of a viscous ball rolling down on a rigid staircase Hua Yana,b, Qingfan Shia,b, Decai Huanga, Gang Suna, a
Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China Department of Physics, Beijing Institute of Technology, Beijing 100081, China
b
Received 14 March 2006; received in revised form 23 June 2006 Available online 14 August 2006
Abstract We theoretically investigate the motion of a ball rolling down on a periodical staircase. Our research is restricted in the case of completely inelastic collision when the ball falls down on the surface of the stairs. The ball is accelerated when it rolls cross the edge of the stair, while it is decelerated when it rolls on the horizontal surface due to the rolling friction. The competition between them causes two different regimes according to the parameters of the system. One is the steady moving regime in which the ball keeps moving forever, and the other is the still regime in which the ball finally stops after rolling on a finite number of stairs. The diagram of these two regimes is given in the reduced parameter space. The tendency that smaller scale of the staircase can keep moving on smaller inclined angle is found. r 2006 Elsevier B.V. All rights reserved. Keywords: Friction; Granular materials; Granular solids; Mechanical contacts
1. Introduction The flow of granular materials has been extensively investigated due to not only its importance in various industrial processes but also the nonlinear and dissipative behavior of the materials in flow processes [1–3]. In particular, the understanding of the dynamics of the dense volcanic granular flow is important for predicting and protecting the natural disasters such as landslides of mountain bodies and mud-rock flows, etc. However, despite intensive efforts in the field, the problem is far from being completely understood [4]. Many questions are still unknown at present. In order to approach the dense volcanic granular flow, it is important to find the basic physical mechanisms that would simply explain the main behaviors of the flow. For this purpose, the research on a simpler problem of the dynamics of a single ball interacting with a set of boundaries was been carried out both theoretically and experimentally in recent years [5–10]. This seemingly simple problem, in which the most complicated multibody interactions are ignored, can show different regimes in parameter space resulting from the competition between the factors of acceleration and deceleration. These regimes are dominated by several basic physical mechanisms and have some key characters that are similar to the dense volcanic granular flow. Especially, Alonso et al. have investigated the dynamics of a single ball rolling down on a staircase. They treat the Corresponding author.
E-mail address: [email protected] (G. Sun). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.07.037
ARTICLE IN PRESS H. Yan et al. / Physica A 374 (2007) 524–532 vi
525
N ω
mg
vi+1
f
L y
H o
x
Fig. 1. Schematic of our system. A viscous spherical ball with radius R is rolling down on a rigid staircase. The stairs are with uniform height H and width L.
problem by introducing two parameters of energy increase due to gravity and energy dissipation in rough surface [10]. In this paper, we further investigate the system of a single ball rolling down a staircase but without unknown parameters. The simplified model we considered is that a single ball rolls down on a regularly spaced staircase under gravity as depicted in Fig. 1. The mass and radius of the ball are m and R, respectively. It is noticeable that the outer edges of the stairs are rounded by a small circle with radius d to eliminate the singular geometry in the stairs, though the final results are obtained in the limiting case of d ! 0. The height and length of each stair are H and L, respectively. The collision between the ball and the horizontal surface of the stair is assumed to be completely inelastic, which means that the vertical momentum would be dissipated completely when the ball falls down on it. The sliding friction coefficient is set to be infinite to avoid sliding, and rolling friction is described by rolling friction coefficient krol as proposed by Brilliantov and Poschel [11,12]. After all these parameters settle down, two different contributions to the motion of the ball come forth. One is the acceleration in horizontal direction when the ball rolls across the edge of the stair, and the other is the deceleration when the ball rolls on the horizontal surface of the stairs due to rolling friction. The competition between these two contributions results in different physical regimes. The model is also inspired by the real systems in nature and contains no unknown parameters, though it is greatly simplified for being solved analytically. We hope that our results can be regarded as a first step towards the understanding of real systems.
2. Acceleration by rolling cross the edge To solve the problem, we choose the position just above the edge of the stair as a starting point of each stair. The horizontal velocity of the ball at this position is the only quantity specifying the state of motion, because the vertical velocity of the ball will dissipate completely when it falls down on the surface of the stairs and the no-sliding condition will give a unique relation between horizontal velocity and the angular velocity. From this starting point, the ball will be accelerated when it falls around the edge. We firstly consider the acceleration process in detail (see Fig. 1). The Newton equations in normal and tangential directions are N mg cos f ¼ m
v2 , Rþd
(1)
ARTICLE IN PRESS H. Yan et al. / Physica A 374 (2007) 524–532
526
and mg sin f f ¼ m_v,
(2)
respectively. And the rotational equation is _ Rf ¼ I o.
(3)
In the equations, v is the absolute value of the velocity of the ball and it is a function of the angle f between the y-axis and the line connecting the center of the ball and the edge (see Fig. 1). N is the normal force acting on the ball by the edge (see Fig. 1) and f is the sliding friction acting on the ball. o is the angular velocity of the ball to its mass center. Under the no-sliding condition, we have the following two confinement equations _ þ dÞ ¼ oR. The initial velocity is v0 . The moment of inertia of the ball is denoted by v ¼ oR and fðR I ¼ 25 mR2 . Using Newton equation in tangential direction and the rotational equation, we can get an equation of f € ¼ f
mgR2 sin f. ðmR þ IÞðR þ dÞ 2
pffiffiffiffiffiffi Integrating the equation and introducing a dimensionless parameter u ¼ v= gR, we have 10 d 1þ ð1 cos fÞ. u2 ¼ u20 þ 7 R
(4)
(5)
It is noticeable that this equation can also be obtained by energy conservation laws, because there is no sliding occurs during the whole process. The ball will depart from the edge of the corner of the stair at a certain angle fap . This angle and the velocity at this time can be derived from the Newtonian equation in normal direction with condition N ¼ 0. After taking the limit d=R ! 0, we have, fap ¼ arccos
7u20 þ 10 , 17
(6)
and uap
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7u20 þ 10 . ¼ 17
(7)
When the ball leaves the edge of the stair, the acceleration process is completed. It is noticeable that u0 must be less than 1, otherwise the ball will depart from the edge at the initial time with horizontally parabolic flight. In this case, there is no acceleration effect in horizontal direction, the velocity of the ball must turn back to u0 o1 owing to the deceleration effect of rolling friction when it rolls on the horizontal surfaces of stairs. The horizontal and vertical components of uap now are expressed by 2 3=2 7u0 þ 10 ap ux ¼ uap cos fap ¼ , (8) 17 and uap y
¼ uap sin fap
2 1=2 " 2 2 #1=2 7u0 þ 10 7u0 þ 10 ¼ 1 . 17 17
(9)
When the ball falls down on the horizontal surface, the vertical velocity will be dissipated completely, while the horizontal velocity remains to move it forward. The angular velocity also remains in the value when it leaves the corner, i.e., oap ¼ vap =R. The no-sliding condition does not hold for this horizontal velocity and angular velocity when it just falls down on the surface of the next stair. However, the ball will tend to nosliding condition very quickly and the distance of the sliding is negligible, as the sliding friction coefficient was supposed to be infinite in our model. The final no-sliding velocity can be easily obtained by ordinary
ARTICLE IN PRESS H. Yan et al. / Physica A 374 (2007) 524–532
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0.6
0.5
u′-u0
0.4
0.3
0.2
0.1
0.0 0.0
0.2
0.4
0.6
0.8
1.0
u0
Fig. 2. The variation of the acceleration function with the initial dimensionless velocity u0 .
mechanical equations, the result is sffiffiffiffi 5 ap 2 R 0 u ¼ ux þ oap 7 7 g 2 3=2 1=2 5 7u0 þ 10 2 7u20 þ 10 ¼ þ 7 17 7 17 1=2 5 2 7u20 þ 10 7u20 þ 10 þ ¼ . 7 5 17 17 According to Eq. (10), we define an acceleration function by 1=2 5 2 7u20 þ 10 7u20 þ 10 þ f a ¼ u0 u0 ¼ u0 . 7 5 17 17
ð10Þ
(11)
The acceleration function is only a function of u0 and is plotted in Fig. 2. We can find that the function monotonically decreases from its maximum value at u0 ¼ 0 to zero at u0 ¼ 1. The function is concave between ½0; 1, which is the interesting range in our problem. This concave property is important for illustrating the unique equilibrium point in the later. We finally emphasize that Eq. (11) is correct only for the condition that the ball would not touch the horizontal surface before it leaves the corner, which is expressed by H4Rð1 cos fap Þ. 3. Deceleration by rolling on the horizontal surface The ball will be decelerated when it rolls on the horizontal surface due to rolling friction. We treat this problem by adapting the moment of the friction proposed by Brilliantov and Poschel [11,12], i.e., M ¼ krol vN ¼ krol vmg. The dynamic equations are f ¼ mv_,
ð12Þ
2 fR M ¼ mR_v. 5
ð13Þ
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pffiffiffiffiffiffi These equations give an exponential decay velocity, i.e., vðtÞ ¼ v0 expðð5krol g=7RÞtÞ, where v0 ¼ u0 gR is the initial velocity of the ball when it starts rolling on the horizontal surface of the staircase. Accordingly, as time tends to infinite the rolling distance of the ball is finite, i.e., 7Rv0 =5krol g. For a rolling distance L0 shorter than the value (otherwise the ball will stop in the midway), the velocity of the ball is rffiffiffiffi 5krol g 0 00 0 u ¼u (14) L, R 7R where u00 is the velocity of the ball after passing through the distance L0 . In our system, the ball will roll from the landing point to the end of the stair if it has enough speed. The horizontal distance that the ball passes through before landing can be calculated by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2H 0 2 DL ¼ R sin fap þ Ruap , (15) uap ðuap y Þ þ x y þ R where H 0 ¼ H Rð1 cos fap Þ. Similarly, H 0 40 must be satisfied to ensure the ball does not touch the horizontal surface before it leaves the corner. DL is related to u0 and H=R. We plot DL as a function of u0 for several fixed H=R in Fig. 3. We can see that DL is a monotonic increase function with concave curvature for all the cases. The rolling distance on the horizontal part of the stair is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2H 0 2 0 ap ap L ¼ L DL ¼ L R sin fap Rux uy þ ðuap . (16) y Þ þ R By Eqs. (14) and (16), we can define a deceleration function as rffiffiffiffi 5krol g 0 0 00 fd ¼ u u ¼ L. R 7R
(17)
According to the discussion of DL, we can see that f d is a monotonic decrease function with convex curvature. 5 H/R=10.0 H/R=5.0 4 H/R=3.0 H/R=1.0
ΔL
3
2
1
0 0.0
0.2
0.4
0.6
0.8
1.0
u0
Fig. 3. The horizontally distance that the ball pass through before landing as a function of the initial dimensionless velocity u0 for several fixed H=R.
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We have to point out that the deceleration function may be less than zero for some cases. This happens when LoDL, i.e., the ball will jump over the horizontal surface of the stair. In this case, the ball may collide on the corner of the stairs and cause unpredictable behavior. We shall exclude the situation from the rest of this paper, and only restrict our attention to the case of f d ju0 ¼1 40. 4. Equilibrium state and phase diagram When the acceleration equals the deceleration, the velocity of the ball will not change. From previous discussion on the acceleration and deceleration functions, we know that the acceleration function has no other parameters while the deceleration function has some. According to the parameters in deceleration function, two different type of solutions can be found. Firstly, when f d ju0 ¼0 4f a ju0 ¼0 , there is no solution in the range [1, 0], because the acceleration function is concave and that of the deceleration function is convex (as shown in Fig. 4(a)). In this case, the deceleration is larger than the acceleration for all u0 , so the ball will finally stop at a certain position. In contrary, when f d ju0 ¼0 of a ju0 ¼0 , there is a unique solution (as shown in Fig. 4(b)). The deceleration is larger than the acceleration when u0 is larger than the solution, while the deceleration is smaller than the acceleration when u0 is smaller than the solution. This means that dimensionless velocity of getting out of the stair will decrease when u0 larger than the solution, while it will increase when u0 smaller than the solution. Thus, this solution corresponds to a stable equilibrium state. Considering both acceleration and deceleration processes, we can obtain an iteration function, uiþ1 ¼ F ðui Þ ¼ ui þ f a ðui Þ f d ðui Þ,
(18)
where ui is the dimensionless velocity at the starting point of ith stair. Fig. 5 is an illustration of how to use this iteration function to show the time evolution of the ball rolling down on the staircase. As shown in Fig. 5, the 0.8 (a) 0.6
0.4
0.2
0.0 (b) 0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
u0 Fig. 4. Schematic view of (a) the case having no solution when f d ju0 ¼0 4f a ju0 ¼0 and (b) that having a unique solution when f d ju0 ¼0 of a ju0 ¼0 . The solid and dashed lines are for acceleration and deceleration function, respectively.
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(a) u0
0.8
ui+1
0.6 0.4 0.2 0.0
(b) 0.8 u0 ui+1
0.6 u*
0.4 0.2 u0′ 0.0 0.0
0.2
0.4
0.6
0.8
1.0
ui
Fig. 5. The illustration of iteration mechanism of Eq. (18) for the case of (a) F jui ¼0 40 and (b) F jui ¼0 o0. The final steady state will be that with a positive velocity u or with zero velocity, respectively.
ball will finally tend to a steady moving state or a still state according to the value F jui ¼0 . When the initial velocity is positive, the ball approaches asymptotically to a positive velocity u if F jui ¼0 40, i.e., f d ju0 ¼0 of a ju0 ¼0 , while it will stop at some place if F jui ¼0 o0, i.e., f d ju0 ¼0 4f a ju0 ¼0 . It is apparent that u is the solution of f d ¼ f a , which exists uniquely for the condition that F jui ¼0 40, as discussed in detail in the beginning of this section. The system we considered contains four independent parameters ðkrol ; R; L; yÞ. In the parameter space, there are two special regimes in which our previous formulae may be not satisfied. One corresponds to that of the ball jumping over the horizontal surface of the stair at high speed (u0 ¼ 1), and the other corresponds to that of the ball touching the horizontal surface before it leaves the corner at very low speed (u0 ¼ 0þ ). These two regimes do not depend on the parameter krol , and are expressed by DLju0 ¼1 oL,
(19)
H4Rð1 cos fap Þju0 ¼0 ,
(20)
and
respectively. If we measure the length of the stair L by the radius of the ball R, the previous equations can be rewritten by only two parameters, l ¼ L=R and tan y ¼ H=L. The detailed formulae are lo2 tan y,
(21)
and 7=17 , (22) tan y respectively. The two regimes are shadowed by slash and back slash lines, respectively, in the parameter space y–l (see Fig. 6). lo
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25 μ = 0.05 μ = 0.1 μ = 0.2
20
μ = 0.3
l
15
10
5
0 0
15
30
45
60
75
90
θ (degree)
Fig. 6. The boundaries of the steady moving regime and still regime in y–l parameter space for a series of fixed m. The regime below the boundary corresponds to that of the steady moving state, while that above the boundary to that of the still state. The regimes that the ball will jump over the horizontal surface of the stair and touch the horizontal surface before it leaves the corner is shadowed by slash and back slash lines, respectively.
In other parameter space, our formulae are satisfied and there exist two regimes, one of which is where the final state of the ball is a steady moving state and the other is where the ball will stop at end. The boundaries of the steady moving regime and the still regime are determined by f a ju0 ¼0 ¼ f d ju0 ¼0 . After some trivial deducing qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and introducing a parameter m ¼ gk2rol =R, which is a parameter only dependent on the properties of the ball, we have, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 7c1 c3 7 2 tan y ¼ , (23) l 1 ð10=17Þ þ c4 þ 2c2 l 5m 2l 17l where 2 10 10 1=2 þ , 5 17 17 3 10 ap 2 c2 ¼ ðux Þ ju0 ¼0 ¼ , 17 " 2 # 10 10 ap 2 1 c3 ¼ ðuy Þ ju0 ¼0 ¼ , 17 17 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 10 10 ap ap 1 . c4 ¼ ux uy ju0 ¼0 ¼ 17 17 c1 ¼ f a ju0 ¼0 ¼
ð24Þ
Eq. (23) means that the boundary is determined only by three reduced parameters, i.e., l, y and m. We plot the boundaries in y–l space for a series of fixed m in Fig. 6. The regime below the line corresponds to that of the steady moving state, while the regime above the line to that of the still state. From Fig. 6, we can find the following basic physical properties. Firstly, from the monotonic increasing curve of the boundary for fixed m, we find that the larger scale of the staircase, i.e., larger l, has larger repose
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angle of the staircase, which means that the ball is prone to stop for larger scale staircases. For different m, our results show that the regime of steady moving state increases as m decreases. According to the definition of m, pffiffiffi parameter m is proportional to krol and g. Thus, the previous results mean that the regime of steady moving state will become wider by decreasing the rolling friction coefficient krol or decreasing the gravitational acceleration g. It is straightforward that the decreasing of the rolling friction coefficient will increase the regime of the steady moving state because it only reduces the effect of rolling friction. However, the changes of the regime on g is not straightforward, because the gravitational acceleration affects both the acceleration and deceleration processes. 5. Conclusion We have analytically solved the problem of a ball rolling down on a periodical staircase. In our model, the rolling process is considered in detail. To eliminate the singular geometry of the stairs, the outer edges of the stairs are first rounded by a small circle, and then we take the limit of the circle to be a point for the final result. To solve the problem analytically, we have also restricted that the collision between the ball and the horizontal surface of the stair is completely inelastic and the sliding friction coefficient is infinite. Despite all these simplifications, the result contains the most important physical components, i.e., the acceleration process when the ball rolls across the edge of the stair and the deceleration process when it rolls on the horizontal surface due to the rolling friction. Our solution shows that the competition between them causes two different regimes. One is the steady moving regime and the other is the still regime. The boundary of the two regimes is determined by three reduced parameters. Our results also show that smaller scale of the staircase can keep the ball moving on smaller inclined angle. Acknowledgments We wish to thank Prof. K.Q. Lu for fruitful discussions. This work is supported by the National Key Program for Basic Research and the Chinese National Science Foundation project no. 10374111. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
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