On Brachistochronic Motion of a
Multibody System with Two Degrees of
Freedom with Real Constraints
D. Djurić
Assistant-Professor
University of Belgrade
Faculty of Mechanical Engineering
The paper considers a case of brachistrochronic motion of the mechanical
system in the field of conservative forces, subject to the action of
constraints with Coulomb friction. In this case an analogy is made between
the two approaches of solving this problem of the mechanical system with
two degrees of freedom. The mathematical model used to compute the
brachistohrone in this special case of the multibody system with two
deegres of freedom is based on varational calculus. The complete analogy
is made with a solution in relation to material point.
Keywords : Brachistochronic motion, Coulomb friction.
1. INTRODUCTION
The Bernoulli’s case of brachistochronic motion of a
particle (cf. [1]) was for the first time extended by Euler
in [2] . Euler considered motion of a particle in resisting
medium in which the force of resistance depends only
on velocity of the particle.
The case of brachistochronic motion of a particle in
the field of gravity subject to the action of Coulomb
friction solved by Ashby in [3] presents a special case in
which the power of friction forces is described as linear
function in relation to generalized velocities and
accelerations. The brachistochronic motion of a particle
along a rough surface was treated by Čović and
Vesković in [4] where the power of friction forces is
given as the sum of two functions, one in relation to
generalized coordinates and velocities and the other
function which is linear in relation to accelerations and
whose coefficients depend on generalized coordinates
and velocities.
The Bernoulli’s brachistochrone problem was
extended to the system of rigid bodies by Čović and
Vesković in [5].
The brachistochronic motion of a mechanical system
with two degrees of freedom subject to the action of
constraints with Coulomb friction presented by Djuric
in [6] use the form of the power of friction forces given
in [4] but in case when the power of friction forces,
potential and kinetic energy does not depend on
generalized coordinates. In the paper [7] Čović and
Vesković showed complete analogy between the
brachistochronic motion of a mechanical system with
two degrees of freedom and the brachistochronic motion
of a particle under friction forces where the power of
friction forces is expressed as function in relation to
generalized velocities and accelerations, in a more
general form than in [3] and [4] .
In the paper [8] Šalinić, Obradović, and Mitrović
considered the case of brachistochronic motion of
Received: April 2013, Accepted: May 2013
Correspondence to: Dragutin Djurić
Faculty of Mechanical Engineering,
Kraljice Marije 16, 11120 Belgrade 35, Serbia
E-mail: [email protected]
doi:10.5937/fmet1502173D
© Faculty of Mechanical Engineering, Belgrade. All rights reserved
mechanical system with two degrees of freedom in
which ideal bilateral constraints and one unilateral
constrain with Coulomb friction are imposed on the
system.
In this paper, the choice of the functions of
generalized velocities in expression for power of
Coulomb friction shows that in case of mechanical
system with two degrees of freedom of motion the
complete parameterization of differential equations of
motions is similar to Ashby’s frictional brachistochrone
in relation to material particle. However introduced
functions are different than in the papers mentioned
above and this leads to new differential equations of
brachistochronic motion which are solved.
2. FORMULATION OF THE PROBLEM
We consider, in this paper, the motion of a mechanical
system in a stationary field of potential forces with
potential    (q ) , subject to the action of real
constraints. The configuration of the system is defined
by
the
set
of
Lagrangian
coordinates
1 2
n
q  ( q , q ,..., q ), to which correspond the
1 2
n
generalized velocities q  ( q , q ,..., q ) Lagrangian
function of the system has the form ([2])
L ( q , q )  T ( q , q )   ( q )
(1)
where T is kinetic energy of the system
T 
1
 
a ( q ) q q ,  ,   1, 2,..., n.
2 
(2)
Differential equations of motions of the mechanical
system have the well known form (cf.[9])
d L
L

    Q  R ,

dt q
q
(3)

where Q are generalized forces of Coulomb
friction and R are generalized control forces. Let us
assume that initial position of the system is defined by
FME Transactions (2015) 43, 173-178
173

the set of given coordinates q0 at moment t  t0 ,
which is set in advance, where it was at rest and let final

position is defined by the set of coordinates q1 at


q  u  0,
t
I   01dt  inf
(5)
we will consider brachistochronic motion. If we now
introduce Bernoulli's condition's (cf.[3]), i.e. the
conditions which do not disturb the principle of work
and energy subject to the action of control forces in
virtue of Ri q i  0 , we formulate variational problem as
constrained with constraint which represents the
principle of work and energy
 
T  P  

  P   0,
T  
(6)
where power of generalized forces of Coulomb friction
has the form
P

 
 Q q ,

(14)

~    b u , ~  ( q u )   d u ,
( q u )
(15)
F
q
q

 

(10)
and let us assume now that functions  ( q , q ) and
 ( q , q ) have the form

 ( q , q )  b (q ) q  ,
 ( q , q )   d ( q ) q  ,
 
apply
(17)
This transformation leads to a new integrand of
functional (8)


F   1   T   c    b    d     



   q






 ,
(18)
where
1
T         , c  c k ,
2
b  b k , d  d k k ,

(19)


   k .

Euler's equations for (18) are given in [6].
4. MECHANICAL SYSTEM WITH TWO DEGREES OF
FREEDOM
Let us consider a special case of motion of mechanical
system with two degrees of freedom. Assuming that
condition (16) is further satisfied and having in mind
(17)
1
1 1
1 2
q  k1 q  k 2 q ,
(11)
q
2
2 1
2 2
 k1 q  k 2 q ,
(20)
where
where the following
2
 ( q , q ),   ( q , q )  C ,
(12)
holds.
Our aim is to obtain differential equations of
brachistochrinic motion of the system. For that reason,
we are able to minimize the integral (4) but in order to
avoid second order functional we introduce the
following constraints in terms of varational calculus
174 ▪ VOL. 43, No 2, 2015
shall

q   ,
  ( q , q )    ( q , q ) q ,
we
where  is Kroneker delta simbol.
3. GENERAL PART

(16)
k   const .,    a k k ,
 k q ,

P
  0,
are further satisfied (cf.[6]),
transformation to coordinates
(9)
Let us consider a case of brachistochronic motion
presented in [7] in which the power of forces of
Coulomb friction (cf. (7)) has the form:

and where    (t ) ,    (t ) and      (t ) are
Lagrange's multipliers.
Assuming that conditions
where





   ( q   u  )   2T  a u  u  .
  2T     
  P .
F (  , q , q , q )  1   T  
(13)
where
so that relation (5) becomes
(8)
 0,

(7)
t
I1  01 Fdt  inf .,
u

 
  
F  1    T 
 u    u

q

(4)
If we assume that the system moves from initial to final
configuration along one definite trajectory for which
Eqs. (4) has minimum value
 
and the integrand of functional (9) gets the form
moment t  t1 , which is unknown. The time the system
needs to move from initial to final position is
determined by relation
t
I  01 dt .
2T  a u
1
1
1
1
k1 
, k2 
,
a
b
a  a11  2 a12  a22 ,
2
1
2
1
k1  k1 , k 2  s k 2 ,
a a
s   11 12 ,
a12  a 22
(21)
2
b  a11  2 s a12  s a22 , a12  a22  0.
Taking into account (21), kinetic energy of the
system considered (cf. (2)) can be written in the form
FME Transactions
1
T 
2
2
V ,
V
2
2
2
 q1  q 2 .
(22)
where
4
Potential energy (cf. (16)) get the form


 1
 2
  c1 q  c 2 q ,
(23)
1

c2 
(c1  s c2 ).
b

 i
 i j
 bi q  d ij q q ,
i , j  1, 2,
(25)

b1 
a

2

2
3
5
(32)
 
 
r1  b1  c1 , r2  b2  c2,




r3  d12  d21, r4  d11  d22,



r5  1  d11, r6  3  d11  4 d22.
Formulating Euler's equations for (31) in relation to q1 ,
q2
, V i z , we get (cf.(19))






1  0  1  C1  const.,
where (cf. (19))
1
4
1  r1  (2 r2  r1 z) z  (4 r2  r1 z) z  (2 r2  r1 z) z
(24)
Power of generalized forces of Coulomb friction (cf.
(10)) obtain the form (cf. (20))
P
6
1  2 d12 (1  z )  4 r4 z (1  z )  4 (d12  2 d21) z
where (cf. (19), (21))
1

c1 
( c1  c2 ),
a
2 2
1  2 r3(1  z )z  r6 (1  z ) z  r5 (1  z ),

b2 
(b1  b2 ),
 2  0   2  C2  const.,
 2    1  ( 1  1 ) z   0,
s
(b1  1 b2 ),
s
b
2
1



where

d12 
s
1
[ d11  d 21  1 ( d12  d 22 )],
s
ab
2

d 21 
s
[ d11  d12  1 ( d 21  d 22 )],
s
ab
2
1
1   C1
(26)
s
s
1 ( d  d )  ( 1 ) 2 d ],
12
21
22
s
s
2
2
4z
2 (1  z )
 C2
,
2
2
(1  z )
1 z2
1 z2
2z
 C2
,
1  z2
1  z2
1  2 r6 (1  2 z 2 ) z  2 r3 (1  5 z 4 )  6 r5 z 5 ,
1  2 r2 (1  6 z 2  5 z 4 )  2 r1 (1  2 z 2  3 z 4 ) z.
(34)
 2  C1
The condition of transversality at the right end-point
gets the form
s  a11  a12 ,
1
s  a12  a22 ,
2
1
2
we are able to eliminate the velocities q and q by the
following relations
1
q  V f1 ,
2
q  V f 2 ,
(27)
where
2
2
q1  q 2 .
V 
(28)
The choice of functions f1 i f 2 gives the different
form of the principle of the work and energy (6).
Furthermore we introduce the functions with
parameter z  z (t )
f1 

V (1   1 )  [V 1   1  1 ]  0,
1

d11  d11  d12  d 21  d 22 ,
a
1

d 22  [ d11 
b
(33)
1 z
2
1 z
2
,
f2 
2z
1 z
2
,
(29)
wherefrom the principle of the work and energy (6) has
the form
1   1 V  1 V z  0
(30)
(35)
[1   1   2 V ](t t )  0,
1
and Euler's equations (33) have the first integral
1   1   2 V  C ,
(36)
C  0.
Taking into consideration that V and z is not
prescribed in the final position of the system, the endconditions have the following form
[
F
F
 0, [ ] (t t )  0,
]
V (t t1)
z
1
(37)
wherefrom we get
(t t )  1  0.
1
(38)
Eliminating V by (30), from Euler's equation (cf.
(33)) in relation to z we are able to eliminate  from
Euler's equation (cf. (36)) in relation to V , wherefrom
we get (cf. (36))
V
1 1  1 ( 1  1 )
,
1 11   2 [1 1  1 ( 2 1  1 )]
(39)
and integrand (9) gets the following form
F  1   (1   1 V  1 V z ) 
 2
 1
 1 ( q  Vf )   2 ( q  Vf )  0
1
2
FME Transactions

(31)
 2 1  1  1
1 1  1   2 [   1 1  1 (  2 1   1 )]
.
(40)
VOL. 43, No 2, 2015 ▪ 175
(1  z 2 )32 [ 2 2  1 (1  z 2 )1 ]
13 1
1  2 p11 2  p2 (1  z 2 )21  12 ,
2  p2 (1  z 2 )2 11  2 p11 (1  z 2 )1  4 p4112 ,
1  1  1 1, 1  11  1 1 ,
1  p31  2 p1 11 ,
In order to obtain the equations of motion of the
system considered (cf. (32),(34) ) the differential
equations (27) can be written in the form
dq1
  f1,
dz
dq 2
  f2 .
dz
1 
(41)
where
 [      1 (3 2  2 3 )]
 2 2 3 1
3
3 1
p1  1  2 Kz  z 2 ,
p2  2 z  K (1  z 2 ),
p3  K  Kz 4  2 z (1  z 2 ),
p4   K  3 z  3Kz 2  z 3 .
1   1 1  1 ( 2 1   1 ),
2  1  1 1 ,
1   1 1   1 1  1 ( 2 1   1 )  1 ( 2 1   1 ), (42)
2  1  1 1  1 1 ,
3  1  2  1 1  1 ,
3  1  2  1  2  1 1  1  1 1  1  1 1  1 .
Taking into account that at final position the
following relations
  z1    q  2  1  0,
(50)
holds where (cf.(48))
5. SOLUTION
i  i1  i 0 ,
Suppose that the mechanical system considered has a
velocity V (t0 )  0 at the start position then equation (39)
q 
gives
z0    
(43)
where




2
( r (1  d 22 )  d12 r2 )
1 1
.

2
( d 22 r1  r5 r2 )

C2 
(44)
(45)
where
K 
2

r5 (1  z1 )  2 d 21 z1
.
2


2 (1  d 22 ) z1  d 21 (1  z1 )
q2 
1
 2  A2

(C2 )2
in which (cf.(46))
176 ▪ VOL. 43, No 2, 2015
(52)
1
1 2
coordinates q  ( q , q ). System starts from position
1
defined by coordinates q (t 0 )  q
where it was at rest.
Final position is set by:
1
11
q (t1 )  q
10
(47)
.
The coefficient of Coulomb friction on the rough
inclined side (at angle  to horizontal) of prism P is  .
1
1
The coefficient of friction on rough vertical plane is 
2
and the coefficient of friction on rough horisontal plane
is 3 (fig 1.). Let m denote the mass of the prism P ,
1
(48)
2
20
and q (t0 )  q
and
q 2 (t1 )  q 21
where
1 ( z1, z )  1 f1 d z ,  2 ( z1, z )  1 f 2 d z ,
1
i 0 ,
Ai  qi 0 
qi 0  qi ( z0 ),
(C2 )2
i 0  i ( z1, z0 ),
i  1, 2,
i  1, 2.
Let us consider the motion of the mechanical system
which consists of three prismatic rigid bodies and
moves in homogeneous field of gravity. The
configuration of the system is defined by the set of
(46)
Integration of differential equations (41) yields to
general solutions
1
1  A1,

(C2 )2
 i
i.
q
6. EXAMPLE
Taking into account conditions (37) at the final
position of the system considered, equation (40) gives
(cf.(38))


C1  K C 2 ,
(51)
 q i  qi  q i 0 ,

constant C2 (cf.(51))
d22 r1  r5 r2
 
 q1
,
 q2
i  1, 2,
we get a value of the parameter z1 at moment t1 , and
r (1  d )  d r
  1  22 12 2 ,
q1 
(49)
m2
1
denote the mass of the prism P and let m3 denote
2
the mass of the prism P3 in a suitable system of unites.
Prisms P and P are attached for the rope. Rope passes
1
2
over drum without friction. The rotation of the drum is
not resisted by friction.
FME Transactions
1
q (t1 )  8.79
2
and q (t1 )  4.1 (at t  t ) then relation
1
(50) gives the value of the parametar z1  1.23232 .
Figure.1 An example of the mechanical system
Taking into account that the differential equations of
motion of the system considered (fig.1) have the form
 
j
aij q  Q i  i  ui ,
i , j  1, 2 ,
q
(53)
where
Figure.2 Brachistohrone of the mechanical system
considered
Q1    3 ( Mg  M 1q 2 ),

2
Q2   1m1g cos   ( 1m1 sin    2 m2 ) q ,


 0,
 gM1,
1
2
q
q
(54)
the relations (19) get the form
1

c1 
a
1

b1 
a
gs

M 1 , c1  
M1,
b
Figure.3 The graph of velocity V(z)
( gM 3  gm11 cos  ),
The graph in (Fig.2) is showing the brachistochone
2
1
q  f (q )
and graph in (Fig.3) is showing V  V (z )
(cf.(39)) of the system considered.
1

b2 
( gM 3  gsm11 cos  ),
b
7. CONCLUSION
1

d11  (  m2  2  m11 sin   3 M1 ),
a
1

d12 
[  m2  2  m11 sin   s 3 M 1 ),
ab
1

d 21 
ab
[   3 M 1  s (  m2  2  m11 sin  ),
s

d 22  [(  m2  2  m11 sin  )   3 M 1 ],
b
(55)
where
REFERENCES
M  m1  m2  m2 ,
M 1  m1 sin   m2 ,
a  M  m2  m1 (1  2 cos  ),
b  M  ( m2  m1 ) s 2  2 m1s cos  ,
s
M  m1 cos 
.
M  m3  m1 cos  (56)

If

6
,
32
,
m1 
13


,   1,   1
3
2
20
5
3
value of parameter z (t )
1 
The paper considers a case of the brachistochronic
motion of the mechanical system in the field of
conservative forces, subject to the action of constrains
with Coulomb friction. The constraint represents the
modified form of the principle of work and energy (30)
obtained in [6] and the new Euler's equations are
formed. The complite analogy is made among solution
obtained in the example considered, solution in [6] and
the solution in relation to material point in [3].
1
FME Transactions
m2  m1 sin  ,
m
m3  1 ,
10
then the relation gives the
(at t  t 0 ) z 0  0.4187046 . If
[1] Bernoulli, J., "Curvatura Radii in diaphanus non
uniformibus, solutioque problematis a se in "Acta",
1696, p 269 propositi de invenienda "Linea
Brachistochrona" etc.", Acta Eruditorum, Leipzig,
May issue, pp. 206,
[2] Pars, L., 1962. An introduction to calculus of
variations, Heinemann, London.
[3] Ashby, N. et all., Brachistochrone with Coulomb
friction, American Journal of Physics, 43, pp 902906, 1975
[4] Čović, V., Vesković, M., Brachistochrone on a
surface with Coulomb friction, International
Journal of Non-linear Mechanics Volume 43, Issue
5, June 2008, pp.437-450, 2002
VOL. 43, No 2, 2015 ▪ 177
[5] Čović, V., Vesković, M., Extension of the
Bernoulli’s case of a brachistochronic motion to the
multibody system having the form of a kinetic
chain with external constraints, European Journal of
Mechanics A/Solids, 21 (2002), pp 347-354
[6] Djurić, D., On Brachistochronic Motion of a
Multibody system with Real constraints, FME
Transaction (2007) 35, 183-187
[7] Čović, V., Vesković, M., Brachistochronic Motion
of a Multibody system with Coulomb friction,
European Journal of Mechanics A/Solids, 28
(2009), pp 882-890
[8] Šalinić, S., Obradović, A., Mitrović, Z., On
brachistochronic mechanical system with unilateral
constraints, Mechanics Research Communications,
45, (2012) 1-6
[9] Čović,
V.,
Lukačević,
M.,
1981.
On
brachistrochronic motions of non-conservative
178 ▪ VOL. 43, No 2, 2015
dynamical system, Teorijska
Mehanika, UDK 532, pp. 41-50.
I
Primenjena
БРАХИСТОХРОНО КРЕТАЊЕ МЕХАНИЧКОГ
СИСТЕМА СА РЕАЛНИМ ВЕЗАМА
Драгутин Ђурић
У овом раду разматрано је брахистохроно кретање у
специјалном случају система са два степена слободе
који се креће у пољу конзервативних сила под
деjством веза са Кулоновим трењем. Избором
функција генералисаних брзина извршена је
параметризација
диференцијалних
једначина
кретања. Формиран је нов математички модел који
је коришћен за добијање брахистохроне и
направљена је аналогија са математичким моделима
који су изложени у радовима [3], [4] , [5] и [6].
FME Transactions