Brachistochrone on a surface with Coulomb friction

International Journal of Non-Linear Mechanics 43 (2008) 437 – 450 www.elsevier.com/locate/nlm

Brachistochrone on a surface with Coulomb friction ˇ c a,∗ , Miroslav Veskovi´c b Vukman Covi´ a Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11000 Belgrade, Serbia and Montenegro b Faculty of Mechanical Engineering, University of Kragujevac, Dositejeva 19, 36000 Kraljevo, Serbia and Montenegro

Received 5 November 2006; received in revised form 9 November 2007; accepted 1 February 2008

Abstract The problem of brachistochronic motion of a particle on a surface with the simultaneous action of gravity and Coulomb friction has been solved. Analytical solutions of the problem in special cases have been found. In the case of a cylindrical surface, it is shown that the solutions found includes, as a special case, all well-known results which refer to the brachistochronic motion of a particle. The results are illustrated with a series of concrete examples. 䉷 2008 Elsevier Ltd. All rights reserved. Keywords: Brachistochrone on a surface; Coulomb friction; Piecewise optimality

1. Introduction The first significant result which generalizes Bernoulli’s case of brachistochronic motion [1,2] was obtained by Euler, who considered such a motion of a particle in a resisting medium (cf. [3, p. 241]). Further results can be found in papers Pennachietti [4] and Mc Connel [5]. Two positions from Mc Connel [5] referred to the lines of action of the active potential forces as well as to the reactions of the constraints, are extended in Stojanovi´c [6]. The results of the quoted papers are generalized ˇ c and Lukaˇcevi´c [8]. in Djuki´c [7] and Covi´ The brachistochronic motion of a conservative nonholonomic mechanical system was considered in Djuki´c [9]. A mechanical system whose brachistochronic motion is subject to linear non-homogeneous constraints was considered in ˇ c [10], and the general case of non-holonomic Zekovi´c and Covi´ systems with non-linear constraints is solved in Zekovi´c [11]. The generalization of the main result in Bernoulli’s classical brachistochrone problem to a mechanical system, which determines the form of the trajectory of a particle, is realized ˇ c and Lukaˇcevi´c [12] and Covi´ ˇ c et al. [13]. A special in Covi´ case of a multibody system with motion limited by external ˇ c and Veskovi´c [14]. constraints is considered in Covi´ ∗ Corresponding author. Tel./fax: +381 11 3246382.

ˇ c). E-mail address: [email protected] (V. Covi´ 0020-7462/$ - see front matter 䉷 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2008.02.004

The mentioned Euler problem was, in special cases, solved in the analytical form in Vratanar and Saje [15]. The motion of a particle under the simultaneous influence of gravity and Coulomb friction was treated firstly in Oonok [16] where differential equations of motion obtained by a numerical procedure were solved, then in van der Heijden [17] and van der Heijden and Diepstraten [18] where approximate differential equations of brachistochronic motion of a point were solved. The first correct solution of this problem, expressed in elementary functions, was obtained in the paper by Ashby et al. [19]. This problem was also considered in Lipp [20], Wensrich [21], and Hayen [22]. The brachistochronic motion of a particle on a smooth surface was investigated in Djuki´c [23]. This paper solves the problem of brachistochronic motion of a particle on a surface with the simultaneous action of gravity and Coulomb friction. 2. Formulation of the problem Consider the motion of a particle M of unit mass in the stationary field of Newtonian forces F = F(r, v),

(1)

where v denotes the absolute velocity of M, and r its radius vector with respect to the inertial Cartesian rectangular coordinate system Oxyz. Let this motion be subject to the action of a real

ˇ c, M. Veskovi´c / International Journal of Non-Linear Mechanics 43 (2008) 437 – 450 V. Covi´

438

stationary unilateral constraint having the form f (x, z, y) 0, which, if the coordinates of the particle belong to the domain (): ():

f (x, y, z) = 0,

(2)

acts on it by the reaction R=N+F , where its ideal component N is directed along the unit normal n of the surface (2) at the point M has the form N = Nn, while the force of Coulomb friction has the well-known form F = −|N |v/|v|, where  is the coefficient of friction. Assume that n is directed to the part of space in which the particle moves after leaving the domain (). In this case, in the interval of time in which particle moves on (2), N > 0 holds. The motions of a particle in which it does not leave (2) will be further consider. Let the position of the particle M on the surface (2) be determined by the generalized coordinates q = (q 1 , q 2 ), which are related to Cartesian coordinates of M by the transformation x = x(q),

y = y(q),

z = z(q) ∧ f (x(q), y(q), z(q)) ≡ 0. (3)

The radius vector of M is expressed in the form r=r(q 1 , q 2 ), the velocity v and acceleration a of M being1 v=

jr  q˙ , jq 

a=

jr  j2 r q ¨ + q˙  q˙  . jq  jq  jq 

The principle of work and energy then reads   d 1 v · v = (F + F ) · v, dt 2

(4)

(5)

(7)

is obtained. Note that the sign in (6) must be in accordance with the chosen orientation of n, mentioned above. The first term on the right-hand side of the last expression represents the second quadratic form (cf. [24]) of the surface (2) b (q)q˙  q˙  = n ·

jq  jq 

q˙  q˙  ,

b = b .

(9)

where v denotes the projection of the velocity of the particle on the direction of the unit tangent vector of the particle trajectory, while Q are generalized forces of the field (1). The motion which satisfies the following conditions is further considered: • During the motion from the initial position M0 ∈ () to the final one M1 ∈ (), the particle M is acted on, besides quoted forces, by the control force u, which does not change the law (9). • Among all possible trajectories of the particle completely belonging to (), the one for which the time of the mentioned motion has minimal value is chosen. The first condition obviously leads to the conclusion that the power of the control force is equal to zero, i.e. that u · v = 0, → u q˙  = 0,

(10)

where u is the generalized control force. The second condition leads to the conclusion that the motion considered is brachistochronic and that its differential equations are the Eulerian equations corresponding to the constrained variational problem t1

dt → inf ,

(11)

0

wherefrom the demanded relation

j2 r

˙ q˙  + N (q, q)v ˙ = 0, v v˙ − Q (q, q)



and its explicit form demands that N be expressed as a function of coordinates and velocities. To do that, one has to project the differential equation of motion of the particle on the unit normal n defined by      jr jr jr jr   (6) n=± × 2  jq 1 × jq 2  , → n = n(q), jq 1 jq

˙ N = b (q)q˙  q˙  − F · n, → N = N (q, q)

Finally, the explicit form of (5) reads

(8)

1 The indices ,  take the values from 1 to 2. Repeated indices denote summation.

in which the relation (9) and v 2 − a q˙  q˙  = 0,

(12)

where a = jr/jq  · jr/jq  , are constraints. If the mentioned brachistochronic motion is admissible, it is obvious that in the interval (0, t1 ) holds v > 0. Let the particle considered, in the given position M0 (q0 ) has a velocity of known intensity v = v0 . Let the final position M1 (q1 ) of the particle be prescribed, too, while the time t1 of the brachistochronic motion is to be determined. Having in mind these demands (11) gets the form 

t1

F dt → inf ,

0

t = 0: t = t1 :

q 1 = q01 , q

1

= q11 ,

q 2 = q02 , q

2

v = v0 ,

= q12 ,

(13)

where the integrand is given by the relation ˙ q˙  + N (q, q)v] ˙ F = 1 + [v v˙ − Q (q, q) 2   + [v − a (q)q˙ q˙ ]

(14)

in which  and  are Lagrange multipliers corresponding to the constraints (9) and (12).

ˇ c, M. Veskovi´c / International Journal of Non-Linear Mechanics 43 (2008) 437 – 450 V. Covi´

⎡

3. General case Eulerian differential equations corresponding to the problem (13) have the form ˙ − N − 2v = 0, v

C = const.,

  jF jF F− = 0, v˙ − q˙ jv˙ jq˙ (t=t1 )

(16)

(18)

(19)

(20)

in virtue of which the transversality condition (18) reads (21)

The general solution of equations (15), (16), (9) and (12) contains maximum five independent constants of integration. By eliminating the velocity v and multiplier  from (9) and (16), by using (12) and (15), a system of differential equations, which is autonomous with respect to q and  is obtained d dt



 1   a q˙ q˙ − Q q˙  + N a q˙  q˙  = 0, 2

(23)

The general solution of this system is of the form q  = q  (C1 , C2,..., C5, t),

 = (C1 , C2,..., C5, t).

(24)

It is obvious that, by using these relations the remaining solutions  = (C1 , C2,..., C5, t)

(25)

are obtained from (12) and (15). The constants of integration C1 , C2,..., C5 , as well as the time t1 of the brachistochronic motion, are determined by the help of the following algebraic equations (cf. (12), (13) and (20)): q  (C1 , C2,..., C5 , t0 ) = q0 ,

q  (C1 , C2,..., C5 , t1 ) = q1 , v(C1 , C2,..., C5 , t0 ) = v0 .

(26)

Consider first the case when the stationary field of Newtonian forces (1) is potential. Then Q = −j /jq  , where = (q) is the potential energy of the field of Newtonian forces. The differential equations of motion (9), (12), (15) and (16) then take the form ˙ + N v = 0, v 2 − a q˙  q˙  = 0, v v˙ + ˙ − N − 2v = 0, v d dt

(t=t1 )

[v 2 ](t=t1 ) = − 21 .



3.1. The case of potential forces

 = 0, → (t=t1 ) = 0,

⎟ ⎥ ⎠ a q˙ ⎦

1 ⎜˙  ⎟ ja   q˙ q˙ = 0. ⎠ ⎝ − N  2 a q˙  q˙  jq

(C1 , C2,..., C5 , t1 ) = 0,



jF jv˙

+

v = v(C1 , C2,..., C5, t),

Taking into consideration that the velocity is not prescribed in the final position of the particle, there are also the endconditions 

a

q˙  q˙ 

(17)

obtains the form   jQ   jN   q˙ q˙ +  N −  q˙ v + 2v 2 = −1. jq˙  jq˙



jQ  jN + q ˙ −  a q˙  q˙    jq jq ⎞ ⎛

which, by the condition of transversality at the right end-point



⎜ − ⎝˙ − N 

One obtains the complete set of differential equations of motion by adding constraints (9) and (12) to Eqs. (15) and (16). This is, obviously, the system of five differential equations, where (9), (12) and (15) are equations of the first order, while the two equations (16) are of the second order. The differential equations (15) and (16) have the first integral jF jF v˙ −  q˙  = C, jv˙ jq˙

  jQ  jN d ⎢   q˙ +  a q˙ q˙ ⎣ −Q − dt jq˙  jq˙  ⎛ ⎞ ⎤

(15)

    jQ  d jN  q˙ + v  − 2a q˙  −Q − dt jq˙  jq˙   jQ  ja jN + q˙ − v  +   q˙  q˙  = 0. jq  jq jq

F−

439

(22)

 j jN jN  + ˙  − v  − 2a q ˙   jq˙ jq jq ja   +   q˙ q˙ = 0. jq 

v

(27)

The end-condition (20) and the condition of transversality (21) remain unchanged, while the first integral (19) gets the form   jN  (28)  N −  q˙ v + 2v 2 = −1. jq˙ 3.1.1. Motion on a sphere; an analogy with the case of viscous friction Consider the motion of a particle inside of a rough sphere of radius R, taking that F ≡ 0. Let the position of the particle be defined (Fig. 1) by the spherical coordinates = R, , ϑ.

440

ˇ c, M. Veskovi´c / International Journal of Non-Linear Mechanics 43 (2008) 437 – 450 V. Covi´

or, in Cartesian coordinates (x, y, z) Ax + By − z = 0,

(35)

and obviously represents a plane which includes the origin O of the coordinates. By the initial position of the particle (M0 ), and the final one (M1 ), the Eq. (34) is given in the form (cot ϑ0 sin 1 − cot ϑ1 sin 0 ) sin ϑ cos  + (cot ϑ1 cos 0 − cot ϑ0 cos 1 ) sin ϑ sin  − sin(1 − 0 ) cos ϑ = 0.

(36)

As the plane (36) includes M0 and M1 , the trajectory of the particle is obviously the great circle of the sphere, i.e. the geodesic on the sphere. The velocity of the particle on the geodesic is obtained by solving the differential equation v˙ +

 2 v = 0. R

The solution of this equation reads v=

Fig. 1.

At the initial position, the velocity of the particle is v0 . The initial position of the particle, as well as the final one, is prescribed: M0 : t = t0 = 0 → = R,  = 0 , ϑ = ϑ0 , M1 : t = t1 → = R,  = 1 , ϑ = ϑ1 .

v = v0 , (29)

The second relation in (27) in the case considered takes the form v 2 − R 2 sin2 ϑ ˙ 2 − R 2 ϑ˙ 2 = 0.

(30)

As F ≡ 0, there results, in accordance with (7) N = b q˙  q˙  = R(sin2 ϑ ˙ 2 + ϑ˙ 2 ), → b q˙  q˙  =

v2 . R

(31)

Consider variational problem (13) with (see (9), (30) and (31))    F = 1 +  v˙ + v 2 + (v 2 − R 2 sin2 ϑ ˙ 2 − R 2 ϑ˙ 2 ). (32) R The Eulerian equations of the variational problem considered are  ˙ − 2 v − 2v = 0, R

d ˙ = 0, (2R 2 sin2 ϑ) dt

d ˙ − 2R 2 sin ϑ cos ϑ (2R 2 ϑ) ˙ 2 = 0. dt

(33)

The last two equations in (33) are reduce to the form which corresponds to the free linear oscillator d2 w + w = 0, d2 where w = cot ϑ. The solution of the last equation reads A sin ϑ cos  + B sin ϑ sin  − cos ϑ = 0,

(34)

Rv 0 , v0 t + R

so that obviously limt→∞ v = 0, wherefrom it follows that the brachistochronic motion considered is a segment of the asymptotic motion with respect to the velocity. Consider the following analogy. In Vratanar, Saje [15] the brachistochronic motion of the particle in the fields of homogeneous gravitation and viscous friction forces was considered. The viscous friction force, in this paper, was taken in the form Fw = −f (v)v/v where f = f (v) is a continuous positive function of the particle velocity. Note that in absence of gravity in the case f (v) = (/R)v 2 , during the motion of the particle inside a smooth sphere of the radius R, all the solutions obtained for the considered case of Coulomb friction remain valid in the case of viscous friction mentioned, under the condition that the initial and the final positions in these two cases coincide. 3.1.2. Brachistochronic motion of a particle on a rough inclined plane Consider the brachistochronic motion of a particle in the homogeneous gravitational field on a rough plane inclined at angle  to the horizontal. The coefficient of Coulomb friction on the inclined plane is  ( < tan ). The particle starts from the position M0 , where it was at rest. The final position M1 of the particle is determined by the orthogonal Cartesian coordinates x =x1 and y =y1 (y1 > 0) with respect to the coordinate system xM 0 y, with the axis M0 y directed down the inclined plane and parallel to the line of maximal inclination to the horizontal. The potential energy of the particle is = −g1 y, while N , having in mind that all the coefficients of the second quadratic form in the case considered are equal to zero, is (cf. (7)) N =g2 . The quantities g1 and g2 in the preceding expressions have the values g1 = g sin , g2 = g cos , where g denotes the intensity of acceleration of gravity. With these results, the differential equations (27) obtain the form v v˙ − g1 y˙ + g2 v = 0,

v 2 − x˙ 2 − y˙ 2 = 0,

ˇ c, M. Veskovi´c / International Journal of Non-Linear Mechanics 43 (2008) 437 – 450 V. Covi´

˙ − g2 − 2v = 0, v

leads to the result

d (−2x) ˙ = 0, dt

d (−2y˙ − g1 ) = 0. dt

(37)

The last two equations results in the integral (C2 + g1 )x˙ − C1 y˙ = 0,

C1 , C2 = const.,

C1 y˙ − C2 x˙ . g1 x˙

cos2 dx , = −K d [1 − R sin( − 1 )]3 dy sin cos = −K , d [1 − R sin( − 1 )]3

(38)

x= −

The boundary condition at the right end-point (cf. (20)) reads

×

[C1 y˙ − C2 x] ˙ (t=t1 ) = 0,

y=

or, taking into account the integral which results from the fourth equation in (37):

which leads to the result g1 cos v= . C2 g2 cos + C1 (g1 − g2 sin )

(40)

(41)

(42)

(43)

and, further, in the form d g1 sin − g2 = , dt dv/d

(44)

R=

g2 , g1 cos 1

C3 , C4 = const.,

a = 8R 2 (3R 2 cos 2 1 − R 2 − 1), a0 = 8(1 − R 2 )2 sin 2 1 ,

b = −24R 4 sin 2 1 ,

b0 = 8(1 − R 2 )2 cos 2 1 ,

b1 = −4R 3 (2 + R 2 ) sin 2 1 , a2 = −16R(1 − R 2 )2 sin 2 1 ,

C1 g12 cos2 dx , =− d [C2 g2 cos + C1 (g1 − g2 sin )]3

(45)

C1 g12 sin cos dy =− , d [C2 g2 cos + C1 (g1 − g2 sin )]3

(46)

which further, by using (39) transformed in accordance with (41) into the form (47)

2 The quantity , obviously, represents the angle of inclination between the tangent on the particle trajectory and the axis M0 x. As the particle trajectory for  = 0 becomes a classical brachistochrone, one takes, as in Ashby et al. [19] and Hayen [22], that the trajectories for an arbitrary coefficient of friction from the interval (0, 1) are convex in the same way as the trajectories of a classical brachistochrone.

b2 = −16R(1 − R 2 )2 cos 2 1 ,

a3 = 2R 2 [(2 − 5R 2 ) cos 2 1 + 3R 2 ], b3 = −2R 2 (2 − 5R 2 ) sin 2 1 .

(50)

By presenting the general solution (49) of the differential equations (48) in the form x = K( 1 , ) + C3 ,

which, on account of (42), gives Eqs. (41) the form

C2 = C1 tan (t=t1 ) → C2 = C1 tan 1 ,

16(R 3 − R)2  b0 +b1 cos( − 1 )+b2 sin( − 1 )+b3 sin(2 −2 1 ) × [R sin( − 1 )−1]2   R + tan (( − 1 )/2) + C4 , +b arctan (49) √ 1 − R2

a1 = 4R 3 ((2 + R 2 ) cos 2 1 + R 2 − 1),

Having in mind (41), the first equation in (37) can be written as v˙ − g1 sin + g2 = 0,

[R sin( − 1 )−1]2  R + tan (( − 1 )/2) + C3 , +a arctan √ 1 − R2 K

where

Eqs. (38) and (40) can be simplified by the substitution2 y˙ = v sin ,

− R)2 a0 +a1 cos( − 1 )+a2 sin( − 1 )+a3 sin(2 −2 1 ) 

1 + g2 v + 2v 2 = 0,

x˙ = v cos ,

K 16(R 3



(39)

and this expression, together with the transversality condition at the right end-point, gives the relation (21) which will not be used in further considerations. The first integral (28) of the Eulerian differential equations in this case reads

C1 v 2 1 + g2 v − = 0. x˙

(48)

where K = 1/(g1 C12 ), R = g2 /(g1 cos 1 ). The solutions of the differential equations (48) read

wherefrom =

441

y = K ( 1 , ) + C4 ,

(51)

where the meaning of the functions =( 1 , ) and = ( 1 , ) is obvious, the following is obtained: x = K[( 1 , ) − ( 1 , = 0 )], y = K[ ( 1 , ) − ( 1 , = 0 )],

(52)

where 0 = (t=0) . The last relations lead to the first condition for determining 1 and 0 in the form 1 ( 1 , 0 ) = y1 [( 1 , = 1 ) − ( 1 , = 0 )] − x1 [ ( 1 , = 1 ) − ( 1 , = 0 )] = 0.

(53)

The second condition is obtained by the next considerations. Namely, by the given value v0 , in accordance with (42) and (47), one obtains 2  cos 0 v02 = Kg 1 = K( 1 , = 0 ), (54) 1 − R sin( 0 − 1 )

442

ˇ c, M. Veskovi´c / International Journal of Non-Linear Mechanics 43 (2008) 437 – 450 V. Covi´

Fig. 3. Fig. 2.

wherefrom, by using one of the relations in (52) with its value in the final position of the particle, for instance the first of them, one obtains the second condition for determining 1 and 0 2 ( 1 , 0 ) = v02 [( 1 , = 1 ) − ( 1 , = 0 )] − x1 ( 1 , = 0 ) = 0.

Further, consider the set of two brachistochrones having the same initial and final values of the parameter , i.e. the brachistochrones for which 0 = ∗0 , 1 = ∗1 . According to (52), in the final position M1 (x∗1 , y∗1 ) of the particles M (x∗ , y∗ ), one obtains

(55)

x∗1 = K∗ [( ∗1 , ) − ( ∗1 , = ∗0 )],

It is not difficult to demonstrate that in this case the final value of satisfies the following condition:

y∗1 = K∗ [ ( ∗1 , ) − ( ∗1 , = ∗0 )],

− arccos(g2 /g1 ) < 1 < 0 ∧ 1 = arccos(g2 /g1 ).

(56)

Finally, the constant of integration K is determined by (cf. (52) and (54)) x1 K= ( 1 , = 1 ) − ( 1 , = 0 ) v02 y1 = . =

( 1 , = 1 ) − ( 1 , = 0 ) ( 1 , = 0 )

(57)

In the case when v0 =0 according to (54), it follows that 0 =/2. With that solution, the value 1 is determined from (53). One obtains the finite equations of brachistochronic motion of the particle from (52) by means of the relation t =t ( ), which can be determined by the next considerations. In accordance with (42) and (44), it is obtained that dt D =− , d [1 − R sin( − 1 )]2

(58)

where D = 1/(C1 g1 ). The solution of (58), taking into account that t( = 0 ) =0 holds, reads    D R − tan(( − 1 )/2) 2 t= 2 1 − R arctan √ 2 2 (1 − R ) 1 − R2  R − tan(( 0 − 1 )/2) − arctan √ 2   1−R cos( 0 − 1 ) cos( − 1) +R(1−R 2 ) − . 1−R sin( − 1 ) 1−R sin( 0 − 1 ) (59) The time t1 of the particle motion is determined by the value of the function (59) for = 1 . The brachistochrones on the inclined plane, with v0 = 0, x1 = 10, y1 = 2, g1 = 8, g2 = 6 (it is taken that g = 10), and with the coefficients of friction  = 0.15 and  = 0.22 (from lowest to highest), are shown in Fig. 2.

 = 1, 2,

(60)

where K∗ denotes the constant of integration in the considered case. From (60) results the expression

( ∗1 , = ∗1 ) − ( ∗1 , = ∗0 ) y∗1 = , ∗ x1 ( ∗1 , = ∗1 ) − ( ∗1 , = ∗0 )

(61)

wherefrom one concludes that geometric loci of the particle final positions on the brachistochrones having the same value ∗0 of the parameter in the initial position and the same value ∗1 of the parameter in the final position, is a straight line (l ∗ ) (O ∈ (l ∗ )) with the coefficient of direction determined by the right-hand side of (61). It is important to remark that in the case of brachistochrones which satisfy the condition v0 = 0, one concludes that the geometric loci of the final positions of the particles at the brachistochrones, having the same value ∗1 of the parameter , is a straight line (l ∗ ) (O ∈ (l ∗ ))  

( ∗1 , = ∗1 ) − ∗1 , = /2 ∗  x . y∗ = (62) ( ∗1 , = ∗1 ) −  ∗1 , = /2  The set of brachistochrones with v0 = 0,  = 0.2, g1 /g2 = 4/3, 1 = −/4 is presented in Fig. 3. The question of existence of a brachistochrone with the initial velocity v0 =0, with respect to the given coefficient of Coulomb friction , as well as to the inclination of the inclined plane (g1 , g2 ) and the coordinates of the final position (x1 , y1 ) of the particle, can be solved by using the following relation: lim

1 →¯ 1

y1

( 1 , = 1 ) − ( 1 , = −/2) = lim 1 →¯ 1 ( 1 , = 1 ) − ( 1 , = −/2) x1 g2 = . 2 g1 − 2 g22

This result leads to the conclusion that the brachistochrone in the case mentioned exists if the condition y1 > g2 x1 / g12 − 2 g22 holds.

ˇ c, M. Veskovi´c / International Journal of Non-Linear Mechanics 43 (2008) 437 – 450 V. Covi´

443

3.1.3. On the piecewise optimality of the trajectory In addition, it will be demonstrated that concerning the optimality of the solution (52), there exists an essential difference in comparison with the Bernoulli’s brachistochrone. In order to do that, one has to start from the Bernoulli brachistochrone ( = 0) in the form xb = Kb lim [( 1 , ) − ( 1 , = 0 )], R→0

yb = Kb lim [ ( 1 , ) − ( 1 , = 0 )], R→0

Kb = const.,

(63)

wherefrom, taking the values =( 1 , ) and = ( 1 , ) from (49), is obtained Kb (2 0 − 2 + sin 2 0 − sin 2 ), 4 Kb yb = (cos 2 − cos 2 0 ). 4 xb =

(64)

Now, return to the brachistochrone with Coulomb friction and choose the points M0 (0, 0) and M1 (x1 , y1 ) as the initial and final positions of the particle, respectively. The initial velocity of particle is v = v0 . In accordance with (52) and (57) there result the finite equations of the brachistochrone (b) x1 ( 0 , 1 , ), ( 0 , 1 , = 1 ) y1 y= ( 0 , 1 , ), ( 0 , 1 , = 1 )

x=

(65)

( 0 , 1 , ) = ( 1 , ) − ( 1 , = 0 ), ( 0 , 1 , ) = ( 1 , ) − ( 1 , = 0 ). Further, choose the values 00 and 11 of parameter so that (66)

holds, and determine the coordinates of M˜ 1 (x11 , y11 ) ∈ (b) and M˜ 0 (x00 , y00 ) ∈ (b) which correspond to those parameters: x11 =x1

( 0 , 1 , = 11 ) , ( 0 , 1 , = 1 )

y11 =y1

( 0 , 1 , = 11 ) , ( 0 , 1 , = 1 )

(67)

x00 =x1

( 0 , 1 , = 00 ) , ( 0 , 1 , = 1 )

y00 =y1

( 0 , 1 , = 00 ) . ( 0 , 1 , = 1 )

(68)

It is taken that M˜ 1 is the final position of the particle on its new brachistochronic motion. The initial position and the initial velocity of that motion corresponding to the point M˜ 0 (x00 , y00 ) ∈ (b). In the orthogonal coordinate system x˜ M˜ 0 y˜ for which M˜ 0 x M ˜ 0 x holds, the finite parametric equations of this brachistochronic motion have, in accordance with (60), the form ˜ 0 , ˜1 , ), x˜ = K(˜

Finally, taking into account (67), it is obtained the parametric ˜ equations of the brachistochrone (b) x˜ = x1 y˜ = y1

[( 0 , 1 , = 11 ) − ( 0 , 1 , = 00 )](˜ 0 , ˜1 , ) , ( 0 , 1 , = 1 )(˜ 0 , ˜1 , = ˜1 ) [( 0 , 1 , = 11 ) − ( 0 , 1 , = 00 )](˜ 0 , ˜1 , ) . ( 0 , 1 , = 1 )(˜ 0 , ˜1 , = ˜1 ) (69)

It is assumed that the brachistochrone (b) is piecewise optimal, i.e. that the particle, in its new brachistochronic motion, ˜ ⊂ (b). That asmoves from its initial position M˜ 0 so that (b) sumption leads to the following identities: x( ) − x( = 00 ) ≡ x( ), ˜

y( ) − y( = 00 ) ≡ y( ), ˜

(70)

the first3 of which, after differentiating with respect to , results in

where

( 1 < 11 < 00  0 ) ∧ 00 , 11 = arccos p

Fig. 4.

˜ 0 , ˜1 , ), y˜ = K(˜

where ˜0 is the initial value, and ˜1 the final value of the parameter in this new brachistochronic motion.

(˜ 0 , ˜1 , = ˜1 )(p + tan 1 cos − sin )3 ≡ ( 00 , 1 , = 11 )(p + tan ˜1 cos − sin )3 .

(71)

This identity is obviously possible only if (˜ 0 , ˜1 , = ˜1 ) ≡ ( 00 , 1 , = 11 ), which has the consequence that ˜1 ≡ 1 and ˜0 ≡ 00 , ˜1 ≡ 11 wherefrom it follows that 11 ≡ 1 . The last result is in contradiction with the assumption (66). Therefrom the trajectory (b) is not optimal (it is not a brachistochrone) in the case when the final position of the particle is ˜ ∈ in M˜ 1 ∈ (b) ∧M˜ 1 = M1 , i.e. there holds (b) / (b). ˜ and (b) have two points in So, it is obvious that (b) common—the points M˜ 0 and M˜ 1 . In the general case, it results in the piecewise non-optimality of the brachistochrone on an inclined plane with Coulomb friction. It is also noted that, in ˜ ∈ (b) holds under the accordance with Belman’s principle, (b) ˜ condition that M1 ≡ M1 . The brachistochrone (b) which corresponds to v0 = 0,  = 0.22, M1 (10, 2), g1 /g2 = 4/3 is shown in Fig. 4. The point M˜ 0 ≡ M0 ≡ O. The point M˜ 1 , corresponding to 11 = −0.4, ˜ is chosen on that brachistochrone and the brachistochrone (b) is determined. In contrast to the brachistochrone with Coulomb friction, the Bernoulli’s brachistochrone always satisfies the conditions (70). As the parametric equations of the Bernoulli’s brachistochrone 3 The result remains unmodified if one takes the second identity (70).

444

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have the form (64), there follows b ( 0 , 1 , ) = 2 0 − 2 + sin 2 0 − sin 2 , b ( 0 , 1 , ) = cos 2 − cos 2 0 ,

while the principle of work and energy, as in the case considered Q1 ≡ 0, Q2 = g, obtains the form (0 = /R): (72)

v v˙ − g z˙ + 0 R 2 ˙ 2 v = 0.

(77)

wherefrom analogous reasoning, as in the case of the brachistochrone with Coulomb friction, results in obtaining the identity

˙ − 0 R 2 ˙ 2 + 2v = 0, v

(2˜ 0 − 2˜ 1 + sin 2˜ 0 − sin 2˜ 1 ) ≡ (2 00 − 2 11 + sin 2 00 − sin 2 11 ),

d ˙ − ) ˙ = 0, ( v  dt 0

wherefrom it follows that ˜0 ≡ 00 , ˜1 ≡ 11 . It will be shown that the last condition is always fulfilled. The values = ˜0 and = ˜1 represent the roots of the algebraic equations

These equations, together with the constraints (76) and (77), present a full set of equations for determining the law of motion of the particle. In the case considered, the first integral (28) has the form

x˜1 b (˜ 0 , ˜1 ) − y˜1 b (˜ 0 , ˜1 ) = 0, 2 v00 b (˜ 0 , ˜1 ) − g x˜1 cos2 ˜0 = 0,

1 − 0 R 2 ˙ 2 + 2v 2 = 0, (73)

where v00 is the velocity of the particle on the brachistochrone (b) in the position = 00 , i.e. in the initial position of the ˜ particle in the case of the brachistochrone (b). For the brachistochrone (b), it holds (by analogy with (67) and (68)) x˜1 b ( 00 , 11 ) − y˜1 b ( 00 , 11 ) = 0, 2 b ( 00 , 11 ) − g x˜1 cos2 00 = 0, v00

(74)

wherefrom it is obvious that the roots of Eqs. (73) fulfill the condition ˜0 ≡ 00 ∧˜ 1 ≡ 11 . In contrast to the brachistochrone with Coulomb friction, it follows that the Bernoulli’s brachis˜ ∈ (b) holds. tochrone (b) is piecewise optimal, i.e. that (b) 3.1.4. Brachistochronic motion of the particle on the inner side of a rough cylinder Let the particle move brachistochronically on the inner side of a rough rotatory cylinder of radius R, having a vertical axis. The particle starts from the given position M0 with the initial velocity v0 , and reaches the final position M1 . The active forces acting on the particle are the forces of the homogeneous field of gravity. The coefficient of Coulomb friction between the particle and the cylinder is . The position of the particle in the initial and the final moments with respect to the inertial cylindric system of coordinates O z, with the axis Oz directed vertically downwards, having in mind that the equation of the cylinder with respect to this system is = R, is determined by M0 : t = t0 = 0 → = R, M1 : t = t1 = 0 → = R,

 = 0 ,  = 1 ,

z = z 0 , v = v0 , z = z1 .

(75)

In the case considered for the second relation in (27) it is obtained that ˙ 2 − z˙ 2 = 0, v2 − R2 

(76)

while the second quadratic form (8) of the surface (q 1 =, q 2 = ˙ 2. z) becomes b q˙  q˙  = R  Taking into account that the orthogonal projection of the gravity force on the vector n is null, the intensity (7) of the ideal component of the reaction of the constraint reads N = R ˙ 2 ,

Eulerian equations of the variational problem (13) are

d (g + 2˙z) = 0. dt

(78)

(79)

while the end-condition at the right end-point (20) gives the relation between the constants of integration which appear after the integration of the last two equations in (78). If the generalized coordinates , z are now transformed by using the relations  − 0 = x/R, z − z0 = y, there arises the possibility of further considering the motion on the developed surface of the cylinder, which represents a vertical plane containing the coordinate system xM 0 y (with axis Oy directed vertically downwards) with respect to which the position of the particle is determined by the new generalized coordinates x and y. As the expressions (76), (77) and (78) do not depend explicitly on the generalized coordinates mentioned, it is possible to transform these expressions by using the relations x˙ = v cos ,

y˙ = v sin ,

(80)

which relate the projections of the particle velocity on the axes of the system xM 0 y, the intensity of this velocity and the angle made by the tangent of the particle trajectory at an arbitrary point and M0 x axis. After that by the integrating the last two equations in (78) and eliminating the multiplier  from thus obtained integrals and the integral (79) it is obtained that (g + 20 v 2 sin ) cos − C1 sin − C2 cos = 0, (cos2 − 2)0 v 3 cos + C1 v − cos = 0,

(81)

wherefrom, by (20) results t = t1 : → C2 = −kC 1 ,

v1 =

cos 1 , C1

(82)

where k = tan 1 , 1 = (t=t1 ) , v1 = v(t=t1 ) . By eliminating the multiplier  from (81) it is obtained that a0 ( ) + a1 v + a2 ( )v 2 + a3 ( )v 3 = 0,

(83)

where a0 ( ) = −g cos , a1 = C1 g, a2 ( ) = −0 sin 2 , a3 ( ) = C1 0 cos (3k−k cos 2 −sin 2 )/2. By solving (83) one obtains expressions4 for the velocity as a function of the parameter : v k = v k ( ), k = 1, 2, 3. 4 Because of their length the expressions of the root of the equation (83) are not displayed.

ˇ c, M. Veskovi´c / International Journal of Non-Linear Mechanics 43 (2008) 437 – 450 V. Covi´

445

In accordance with (83) there results =

2r + 1  → v = 0, 2

r = 0, ±1, ±2, . . . .

(84)

By the extending of the equations of motion to the interval ∈ [(2r1 + 1)/2, 1 ],

1 > (2r1 + 1)/2,

(85)

the number r1 having the lowest absolute value in a set satisfying the second relation in (85), it can be concluded that, under the condition the brachistochrone required is possible, the function v = v( ) has, on the limits of the interval ∈ [(2r1 + 1)/2, (2r1 + 2)/2]

Fig. 5.

(86)

values equal to zero. Also, the function v = v( ) in that interval must be continuous with the continuous first derivative with respect to parameter . It will be noted that under the given boundary conditions the mentioned conditions of continuity in the interval (86) are obtained combining two functions of the set v k = v k ( ), k = 1, 2, 3 (for instance, in the case 0.25C1 30, 0 1 0.3, by combining v 1 = v 1 ( ) and v 2 = v 2 ( )). Further, from (77), one obtains the differential equation dt =

dv/d d , g sin − 0 v 2 cos2

(87)

from which there results  t = t0 + ( ) d , → t = t ( , 0 , C1 , k), 0

(88)

where the meaning of the function  = ( ) is completely determined by v k = v k ( ) and (87). The relations (87) and (80) lead to the general solution of the parametric equations of the brachistochrone required:   x= ( ) d , y = ( ) d , (89) 0

0

The whole procedure of the determination of solutions (88) and (91) is performed by numerical integration, which is the consequence of the complexity of the function v k = v k ( ). The brachistochrone (bc ) obtained by numerical integration for v0 = 0, 0 = 0.15, x1 = 5.8, y1 = 2.5 is shown in Fig. 5. The Bernoulli’s brachistochrone is drawn, for comparison, in the same figure. The way of obtaining the analytical solution of parametric equations (91) in the absence of gravity force will be shown. The analogy with such a case of the here considered motion inside a sphere would yield the conclusion that the motion of the particle inside a rough cylinder would be along a geodesic (a helicoid in O z, a straight line in M0 xy). But the next analysis will show that such analogy does not exist. In the case considered, the expression (83) for the velocity is reduced to v=K

sin , 3k − k cos 2 + sin 2

here K = 1/C1 , while the functions appearing in (90) take the form

where v(dv/d ) cos , ( ) = g sin − 0 v 2 cos2 v(dv/d ) sin ( ) = , g sin − 0 v 2 cos2

0 = (t=0) .

( ) =

3k cos + k cos 3 + 4 sin3 , 0 sin 2 (3k − k cos 2 + sin 2 )

(94)

(90)

( ) =

3k cos + k cos 3 + 4 sin3 , 0 cos2 (3k − k cos 2 + sin 2 )

(95)

(91)

wherefrom, after the substitution tan( /2) = u, the general solution of the parametric equations of the brachistochrone (cf. (89)) reads

The solution (89) can be written in the form x = x( , 0 , C1 , k),

y = y( , 0 , C1 , k),

whereas for the determination of the integration constants 0 , C1 , 1 the next algebraic relations are available x1 = x( = 1 , 0 , C1 , k = tan 1 ), y1 = y( = 1 , 0 , C1 , k = tan 1 ), v0 = v( = 0 , 0 , C1 , k = tan 1 ).

(93)

(92)

Having determined the integration constants, one obtains the finite equations of the brachistochrone x = x( (t)), y = y( (t)), in which the function = (t), found by the using of (88), where the values of the integration constants from (92) are taken into account.

1 x= 20 2 y= 0





u u0

u u0

8(u3 + u5 )−k(u2 − 1)3 (u2 + 1) u(u2 −1)(k+2u + 6ku2 −2u3 +ku4 ) (u2 +1)[−8u3 +k(u2 − 1)3 ]

(u2 −1)2 (k+2u + 6ku2 −2u3 +ku4 )

du,

du,

(96)

(97)

where, taking into account the relation u1 =tan( 1 /2), the value of the constant k is given by the expression k = 2u1 /(1 − u21 ), and where u0 = tan( 0 /2).

ˇ c, M. Veskovi´c / International Journal of Non-Linear Mechanics 43 (2008) 437 – 450 V. Covi´

446

i =

k + 2u0i + 3ku20i + 2k 2 u0i (1 − 3u20i ) + 4k 3 (1 + 3u20i ) 1 − 3u20i + 2ku0i (u20i + 3)

,

i = 1, . . . , 4, ui0 (i = 1, . . . , 4) being the roots of a polynomial of the fourth degree which is the second factor in the denominator of the integrands in (96) and (97), and u0 —the value of the parameter u in the initial position of the particle. The roots quoted have the form ! " " 2 ei 1 − 8k 1 e¯i # 2 − 4k 2 2 1 − 8k 2 u0i = + + + , 2k 2 2 k k2 k2 k2 (100) where e1 = e2 = e¯2 = e¯4 = 1, e3 = e4 = e¯1 = e¯3 = −1. The integration constants u0 and u1 are determined from the algebraic equations obtained when one puts x =x1 , y =y1 , u= u1 in (98) and (99). A sequence of brachistochrones on a cylinder in the absence of the gravity force for x1 = 2.8, y1 = 6, 0 = 0, 0.1, 0.5, 0.6 are drawn (from lowest to highest) in Fig. 6. 4. Comparison with the existing results and their generalization

Fig. 6.

The previous solutions explicitly read  u2 − u20 2 1 (u − u0 ) + x= − 20 k 2 2 u + (arctanh u − arctanh u0 ) + ln k u0 +



i=4 1  u − u0i i ln , u0 − u0i 2k0

(98)

and the second quadratic form of the surface reads (cf. (8)) ⎤ ⎡  2 2y  d dy ⎦ x˙ 2 , b (q)q˙  q˙  = ⎣ 2 1+ (102) dx dx

i=1



y=

2 1 1 − u − u0 + 0 k(u2 − 1) k(u20 − 1) 1 1 u2 − 1 + 2 (arctanh u − arctanh u0 ) − ln 2 2k u0 − 1 2k −



where q 1 = x, q 2 = z. Having in mind that (cf. (1) and (7)) F = −gj, the ideal component of the reaction of the surface takes the form

i=4 1  u − u0i

i ln , u0 − u0i 2k 2 0

(99)

i=1

i =

i = 1, . . . , 4,

1 − 3u20i + 2ku0i (u20i + 3)

 N=

where 2u0i (u20i − 2) + k(9u20i + 3) + 4k 2 u0i (u20i + 1)

The brachistochrone with Coulomb friction considered in Ashby et al. [19] and Hayen [22] represents a particular case of the brachistochrone on the surface considered in this paper. In order to show that, consider the motion of a particle on a rough cylindrical surface whose relation y = y(x), in the rectangular Cartesian system Oxyz (see Fig. 7), is not given in advance, and which, without loss of generality, satisfies the condition y(x = 0) = 0. Taking into account the fact that the radius vector of the particle is given by the relation r = xi + y(x)j + zk, the unit vector of the normal on the surface (cf. (6)) has the form      jr  jr jr  × jr  → n= ×  jz jx jz jx   2   dy dy 1+ (101) i n= j− dx dx

d2 y 2 x˙ + g dx 2



 1+

dy dx

2 ,

(103)

while the principle of work and energy (9) becomes , dy x+v ˙ v v+g ˙ dx



d2 y 2 x˙ +g dx 2



 1+

dy dx

2 =0,

(104)

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447

first integrals of the form 1 =C1 ,

2 =C2 ,

˙z=C3 ,

C1 , C2 , C3 =const.,

(110)

and with respect to w, u, v, they read ˙ + 2u˙ − g − 2w + C1 = 0, u ˙ −w − 2w˙ − g − 2u + C2 =0,

˙ + 2=0.

(111)

As the integrand of the functional (108) does not explicitly depend on the independent variable t, the differential equations (110) and (111) have the first integral jF jF jF jF jF jF v˙ − u˙ − w˙ − x˙ − y˙ − z˙ = C, jv˙ ju˙ jw˙ jx˙ jy˙ j˙z C = const., (112)

F−

where, by the transversality conditions at the right end-point, C = 0. The last relation explicitly reads 1 + (g y˙ + g x) ˙ − 2˙z2 − C1 y˙ − C2 x˙ = 0.

(113)

Taking into account the fact that the quantities v, u and w are not given in the final position of the particle, the natural end-conditions follow       jF jF jF =0, =0, =0, (114) jv˙ (t=t1 ) ju˙ (t=t1 ) jw˙ (t=t1 )

Fig. 7.

where  v2 = 1 +



dy dx

2  x˙ 2 + z˙ 2 .

(105)

When forming the functional for the variational problem whose Euler equations lead to the solution required, the fact that the form of the cylindrical surface (i.e. the function y =y(x)) is not given in advance, is taken into consideration. Therefore, (104) and (105) are transformed to the form, respectively:

wherefrom the condition (t=t1 ) = 1 = 0. The complete set of differential equations of the brachistochronic motion considered includes Eqs. (106) and (107) too. In the special case when the particle moves along the coordinate line z = const., the differential equations (106), (107), (110), (111) and (113), in which z˙ ≡ 0, and the natural endcondition (t=t1 ) = 0 describe the brachistochronic motion of the particle studied in Ashby et al. [19] and Hayen [22]. In that case, with the help of transformation (41), Eqs. (106), (111), and (113), respectively, take the form v˙ + g sin + g cos + v˙ = 0,

(115)

− 2v cos + 2(v˙ cos − v˙ sin ) − g − 2v sin + C1 = 0,

(116)

v v˙ + g y˙ + v(y¨ x˙ − x¨ y˙ + g x)( ˙ x˙ 2 + y˙ 2 )−1/2 = 0,

(106)

2v sin − 2(v˙ sin + v˙ cos ) − g − 2v cos + C2 = 0,

(117)

v = x˙ + y˙ + z˙ .

(107)

v(g sin + g cos )−v(C1 sin + C2 cos )+1=0.

(118)

2

2

2

2

The variational problem (11), with these observations, takes the form  t1 F dt → inf , (108) 0

where, if the constraints x˙ = u, y˙ = w are introduced, the integrand is given in the form F = 1 + [v v˙ + gw + v(wu ˙ − uw ˙ + gu)(u2 + w 2 )−1/2 ] 2 2 2 2 + (u + w + z˙ − v ) + 1 (y˙ − w) + 2 (x˙ − u) (109) in which , , 1 and 2 are Lagrangian multipliers. The Euler equations of the problem (108) with respect to x, y, z have the

The expression which represents the result of elimination of v˙ from (116) and (117), having in mind (118), leads to the result 1 − ˙ , (119) 2v 2 which reduces (116) and (117) to the form from which it is possible to eliminate the quantity v˙ + ˙ by using (115). After that elimination, one gets the relations =

= v=

C1 − C2 − (C1 + C2 ) tan , g(1 + 2 ) (1 + 2 ) cos . C2 (1 + 2 ) − (C1 − C2 ) cos 2 + (C1 + C2 ) sin 2 (120)

ˇ c, M. Veskovi´c / International Journal of Non-Linear Mechanics 43 (2008) 437 – 450 V. Covi´

448

In the case of the following transformation: C1 = B( − A), C2 = B(1 + A), A, B = const., one gets the solution B(A + tan ) , g cos , v= B(1 + 2A cos2 + 2 sin cos )

=−

(121)

which is identical with the solution in Ashby et al. [19], with the remark that the condition of transversality at the right end-point was not formulated in that paper, which resulted by the fact that the right-hand side of (121) is multiplied by an independent integration constant K. By using that condition, one obtains K = 1. The fact, which was not considered in Ashby et al. [19] and Hayen [22], that in this case the brachistochrone has all the characteristics of a brachistochrone on an inclined plane, including the difference with respect to Bernoulli’s brachistochrone as for its piecewise optimality will be further shown. That is why it is first concluded, by (t=t1 ) = 0, that A = − tan 1 , wherefrom, taking into account (121) and (41), it is obtained that dt 2(sin cos − kcos2 ) − 1 = Kt , d [2(sin cos − k cos2 ) + 1]2

(122)

2(sin cos − k cos2 ) − 1 dx cos2 , =K d [2(sin cos − k cos2 ) + 1]3

(123)

dy 2(sin cos − k cos2 ) − 1 sin cos , =K d [2(sin cos − k cos2 ) + 1]2

(124)

where k = − tan 1 ,

K=

1 gB 2

,

Kt =

1 . gB

It is obvious that the solutions of differential equations (123) and (124) can be written in the form (52), where, in this case5    + tan ( 1 , ) = a arctan e $2 j =0 [aj cos(2j ) + bj sin(2j )] + $2 , j =0 [cj cos(2j ) + dj sin(2j )]    + tan

( 1 , ) = a¯ arctan e $2 ¯ j cos(2j ) + b¯j sin(2j )] j =0 [a + $2 , (125) ¯ j =0 [c¯j cos(2j ) + dj sin(2j )] with the constants a, a, ¯ e, aj , bj , cj , dj , a¯ j , b¯j , c¯j , d¯j depending on the coefficient  of Coulomb friction and on the finite value 1 of the parameter . The explicit expressions for these constants are not given here, as they are of no importance in further analysis. 5 We give the solution which by its external form differs from the solution in Ashby et al. [19] and Hayen [22].

In the case v0 = 0, the constants 1 and 0 are determined from (53) with the values from (125), and from the relation (55), in which in this case 2  cos 0 . (126) ( 1 , = 0 )=g 2(sin 0 cos 0 − tan 1 cos2 0 )+1 The geometric locus of the final positions of points on the brachistochrones having the same value ∗0 of the parameter in the initial position, and the same value ∗1 of that parameter in the final position is, also in this case, a straight line (l ∗ ) (O ∈ (l ∗ )) whose coefficient of direction is determined by the righthand side of the relation (61) with the values corresponding to the relations (125). It is also noted, having in mind the chosen orientation of the axis Oy, that in the case v0 = 0, there results, by (121), that 0 = −/2, wherefrom one concludes that also in this case the geometric locus of the final positions of points of the brachistochrones having the same value ∗1 of the parameter in the final position, is the straight line (l ∗ ) (O ∈ (l ∗ )) with the coefficient of direction determined by (62) and (125). As for the piecewise optimality of the brachistochrone, it is shown, by considerations analogous to the one applied to the case of the inclined plane, that there is an essential difference in comparison with Bernoulli’s brachistochrone. Namely, it is shown that, even in this case, the brachistochrone in contrast to the Bernoullian one is not piecewise optimality. It will be further shown, in the case considered, that it is possible to find, for each brachistochrone with v0 = 0, the ˜ with v0 = 0, such that (b) ∈ (b) ˜ holds. In brachistochrone (b) order to do that, will be determined the brachistochrone (b) with the initial velocity v0 = v0∗ , whose initial position is at the point O(x0 =0, y0 =0), and the final one is at M1∗ (x1 =x1∗ , y1 =y1∗ ). By solving (53) and (55), with values (125) and (126), one obtains the initial and the final values of the parameter : 0 = ∗0 , 1 = ∗1 and then the value K ∗ of the constant K from (57) with values (125). In such a way, the final parametric equations of the brachistochrone (b) are obtained: x = K ∗ [( ∗1 , ) − ( ∗1 , = ∗0 )], y = K ∗ [ ( ∗1 , ) − ( ∗1 , = ∗0 )], with the velocity  cos , v = gK ∗ ∗ (1 − 2 tan 1 cos2 + 2 sin cos )

(127)

(128)

where K∗ = =

x1∗ ∗ ∗ ( 1 , = 1 ) − ( ∗1 , = ∗0 ) y1∗ . ∗ ∗

( 1 , = 1 ) − ( ∗1 , = ∗0 )

(129)

The last three relations determine the brachistochrone (b) in the interval [ ∗0 , ∗1 ], with ∗1 < ¯1 where ¯1 = arctan[(1 − 2 )/(2)]. The functions (127) and (128) are continuous for each ∗0 ∈ [−/2, ¯1 ) and ∗1 ∈ (−/2, ¯1 ), which, for =−/2, allows determination of the following quantities: y˜0 = −K ∗ [ ( ∗1 , = −/2) − ( ∗1 , = ∗0 )], x˜0 = −K ∗ [( ∗1 , = −/2) − ( ∗1 , = ∗0 )],

(130)

ˇ c, M. Veskovi´c / International Journal of Non-Linear Mechanics 43 (2008) 437 – 450 V. Covi´

449

the constraint with Coulomb friction. This makes an essential difference from the problem considered here. Finally, it can be mentioned that also in the case considered in Ashby et al. [19] and Hayen [22] the question of the existence of a brachistochrone with the initial velocity v0 = 0 can be simply solved as in the case of an inclined plane, for a given coefficient of Coulomb friction  and the coordinates of the particle (x1 , y1 ) by the following procedure: lim

1 →¯ 1 Fig. 8.

y1

( 1 , = 1 ) − ( 1 , = −/2) = lim = −, x1 1 →¯ 1 ( 1 , = 1 ) − ( 1 , = −/2)

where the limit value of the parameter is ¯1 = arctan[(1 − 2 )/2]. This result leads to the conclusion that in the case mentioned the brachistochrone exists if |y1 | > x1 which is obtained by different approaches in Ashby et al. [19] and Hayen [22].

with the help of which the translation of the coordinate system xOy (see Fig. 8, in which the case =0.2, v0 =4, x1 =7, y1 =−2, g = 10 is presented) is performed by means of the relations x˜ = x+x˜0 , y=y+ ˜ y˜0 , to the configuration x˜ O˜ y. ˜ The brachistochrone ˜ with the initial position of the particle in O, ˜ the initial (b) velocity v0 = 0, and the final position at M1∗ will be further determined. The coordinates of this point in x˜ O˜ y˜ are (cf. (129) and (130))

Acknowledgment

x˜1 = x1∗ + x˜0 → x˜1 = K ∗ [( ∗1 , = ∗1 ) − ( ∗1 , = −/2)]

References

y˜1 = y1∗ + y˜0 → y˜1 = K ∗ [ ( ∗1 , = ∗1 ) − ( ∗1 , = −/2)], (131) ˜ so that, by (127), there results for (b) cos v˜ = g K˜ , (1 − 2 tan ˜1 cos2 + 2 sin cos ) ˜ x˜ = K[(˜ 1 , ) − (˜ 1 , = −/2)], ˜ y˜ = K[ (˜ 1 , ) − (˜ 1 , = −/2)].

K˜ = const.,

(132)

The final value ˜1 of the parameter is determined in accordance with (53) from the relation y˜1 [(˜ 1 , = ˜1 ) − (˜ 1 , = −/2)] − x˜1 [ (˜ 1 , = ˜1 ) − (˜ 1 , = −/2)] = 0,

(133)

which leads to the result ˜1 = ∗1 , as one obtains, by (131), the expression y˜1 [( ∗1 , = ∗1 ) − ( ∗1 , = −/2)] − x˜1 [ ( ∗1 , = ∗1 ) − ( ∗1 , = −/2)] = 0, and its coincidence with (133). Hence there is also the result ˜ K˜ = K ∗ and, finally, the equation of brachistochrone (b) x˜ = K ∗ [( ∗1 , ) − ( ∗1 , = −/2)], y˜ = K ∗ [ ( ∗1 , ) − ( ∗1 , = −/2)].

(134)

Having in mind that x˜( = 0 ) = x˜0 , y˜( = 0 ) = y˜0 , v˜( = 0 ) = v0∗ , ˜ It is obvious that holds, it is finally concluded that (b) ∈ (b). this conclusion is valid in the case of the brachistochrone on an inclined plane, too. The problem of the brachistochrone with Coulomb friction is considered in Lipp [20] as a problem of optimal control in which the particle, for a part of its motion, is not under

This research was supported by the Ministry of Science fund, through the Grant 144019.

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