Dragutin Djurić
Research Assistant
University of Belgade
Faculty of Mechanical Engineering
On Brachistochronic Motion of a
Multibody System with Real Constraints
The paper considers a case of brachistochronic motion of the mechanical
system in the field of conservative forces, subject to the action of
constraints with Coulomb friction. In the special case, an analogy is made
between Ashby’s brachistochrone and the brachistochrone of the
mechanical system with two degrees of freedom.
Keywords: Brachistochronic motion, Coulomb friction
1. FORMULATION OF THE PROBLEM
We are considering the motion of a mechanical system
in a stationary field of potential forces where Π (q )
= Π (q ) is the system’s potential energy. Let this motion
be subject to the action of real constraints1.
The configuration of the system is defined by the set of
Lagrangian coordinates q = (q1 , q 2 ,..., q n ) , to which
correspond
the
generalized
velocities
q = (q 1 , q 2 ,..., q n ) . Lagrangian function of the system
has the form ([2])
L ( q , q ) = T ( q , q ) − Π ( q ) ,
(1)
where T is kinetic energy of the system 2
T =
1
aαβ ( q ) q α q β .
2
t1
I = ∫ dt → inf . ,
we will consider brachistochronic motion. A problem of
brachistrochonic motion will be solved by variational
calculus.
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 uα qα = 0 , we shall formulate variational
problem as constrained with constraint which represents
the principle of work and energy
− Pµ = 0 ,
⇒ T + Π
T = P µ − Π
(6)
where power of generalized forces of Coulomb friction
has the form
P µ = Qαµ q α ,
(3)
(7)
so that relation (5) becomes
where Qαµ are generalized forces of Coulomb friction
and uα are generalized control forces.
Let us assume that initial position of the system is
defined by the set of given coordinates q0α at moment
t = t0 , which is a set in advance, where it was at rest
and let the final position is defined by the set of
coordinates q1α at moment t = t1 , which is unknown.
The time the system needs to move from initial to final
position is determined by the relation
t1
I = ∫ dt
(5)
0
(2)
Differential equations of motions of the mechanical
system have a well-known form
d ∂L
∂L
−
= Qαµ + uα ,
α
dt ∂q
∂q α
If we assume that the system moves from initial to final
configuration along one definite trajectory for which Eq.
(4) has a minimum value
(4)
t1
I1 = ∫ Fdt → inf . ,
where
(
Constraints with Coulomb friction.
The indices take the following values: α , β , γ , π = 1, 2, ..., n
)
− Pµ .
F (λ , q , q , q) = 1 + λ T + Π
(9)
2. GENERAL PART
Let us consider a general case of brachistochronic
motion presented in [5], in which Eqs. (7) has the form
P µ = ψ (q , q ) + ϕβ (q , q ) q β ,
0
1
(8)
0
(10)
assuming that
2
Received: Decembar 2007, Accepted: Decembar 2007
Correspondence to: Dragutin Djurić
Faculty of Mechanical Engineering,
Kraljice Marije 16, 11120 Belgrade 35, Serbia
E-mail: [email protected]
© Faculty of Mechanical Engineering, Belgrade. All rights reserved
ψ (q , q ), ϕ β (q , q ) ∈ C 2
(11)
holds. Let us now examine a case of brachistochronic
motion in which functions (11) have the following form
ψ (q , q ) = −bα (q ) q α , ϕβ (q , q ) = −dαβ (q ) q α . (12)
FME Transactions (2007) 35, 183-187
183
In order to avoid second-order functional let us
introduce the following constraints in terms of
variational calculus
q α − uα = 0, 2T − aαβ uα u β = 0 ,
(13)
and the integrand of functional (8) gets the form1 (cf.(6),
(10) and (12))

∂Π α ~ ~ β 
F = 1 + λ  T +
u −ψ − ϕ β u  +


∂q α


α β
+ θ 2T − aαβ u u + σ α (q α − uα ).
(
where
ψ α
)
α
q1 = k11 q1 + k12 q 2 , q 2 = k12 q1 + k22 q 2 ,
(14)
Assuming that conditions
∂q
= 0,
∂d βα
∂q
= 0,
γ
(16)
q
kγα = const.,
δ γπ = aαβ kγα kπβ ,
(18)
(19)
holds. This transformation leads to a new integrand of
functional (8)
(
)
+θ ( 2T * − δγπ ω γ ω π ) + σ γ* ( q γ − ω γ )
*
F * = 1 + λ T * + cγ* ω γ + bγ* ω γ + dπγ
ωπ ω γ +
(20)
1
T * = δγπ ω γ ωπ , cγ* = cα kγα ,
2
*
bγ* = bα kγα , dπγ
= dαβ kπα kγβ ,
(21)
, σ γ = σ α kγα .
*
Formulating Euler's equations for (20) we get (cf. (21))
λ − 2 θ = 0,
*
*
)
*
λ δ γπ + dπγ
ωπ +
(
)
*
*
+ λ[ dπγ
− dγπ
w π − cγ* − bγ* ] + Cγ* = 0.
1
1
T = V 2 , V 2 = q12 + q22 .
2
(25)
Π * = c1* q1 + c2* q 2 ,
(26)
λ = λ (t ) , θ = θ (t ) , σ α = σ α (t ) are Lagrange's
multipliers.
2
δ γπ is Kroneker delta symbol.
184 ▪ VOL. 35, No 4, 2007
1
c1* =
a
(c1 + c2 ), c2* =
1
b
(c1 + s c2 ) .
(27)
Power of generalized forces of Coulomb friction (cf.
(10) obtain the form (cf. (23))
P µ * = −bi* q i − dij* q i q j , i, j = 1, 2,
(28)
where (cf. (21))
1
(b1 + b2 ), b2* =
1
(b1 + s b2 ),
a
b
1
*
d11
= ( d11 + d12 + d 21 + d 22 ) ,
a
1
*
[d11 + d 21 + s (d12 + d 22 )],
d12 =
ab
*
d 21
=
1
ab
(29)
[d11 + d12 + s (d 21 + d 22 )],
1
*
d 22
= [d11 + s (d12 + d 21 ) + s 2 d 22 ].
b
Let us introduce natural parameter ε = ε (t ) (cf. [1] and
[4]) by substitution
*
σ γ = Cγ , Cγ = const.,
(
(24)
Taking into account (23), kinetic energy of the system
considered (cf. (2)) can be written in the form
b1* =
where
q =ω
,
where (cf. (21), (24))
= kγα qγ ,
γ
b
Potential energy (cf. (17)) gets the form
where2
γ
1
a +a
a = a11 + 2 a12 + a22 , s = − 11 12 ,
a12 + a22
(17)
we shall apply transfomation to coordinates
α
(23)
b = a11 + 2 s a12 + s 2 a22 , a12 + a22 ≠ 0.
aαβ = const., Π = cα q α , cα = const.,
γ
a
, k12 =
k12 = k11 , k22 = s k12 ,
are further satisfied, which lead to the existance of the
following conditions
∂bα
1
k11 =
α
∂F
= 0,
∂q α
Let us consider a special case of motion of mechanical
system with two degrees of freedom. Assuming that
condition (17) is further satisfied and having in mind
(19) relations (18) get the form
Where
= −bα u , ϕ β α α = − dαβ u . (15)
( q =u )
( q = uα )
3. MECHANICAL SYSTEM WITH TWO DEGREES OF
FREEDOM
(22)
q1 = V cos ε ,
q 2 = V sin ε .
(30)
Eliminating the velocities q1 i q 2 by (30), the principle
of work and energy (6) has the form
ψ + ϕ V + ρ V ε = 0 ,
(31)
wherefrom integrand (9) gets the folowing form
FME Transactions
F = 1 + λ ( ψ + ϕ V + ρ V ε ) +
(
)
(
+ σ1* q1 − V cos ε + σ 2* q 2 − V sin ε
)
λ =
(32)
where
*
+ r3 (cos ε )2 −
ρ = − d 21
V=
r3
sin 2ε ,
2
r4
sin 2ε ,
2
(33)
r1 = b1* + c1* , r2 = b2* + c2* ,
*
*
*
*
r3 = d12
, r4 = d11
.
+ d 21
− d 22
Formulating Euler's equations for (32) in relation to q1 ,
q2 , V i ε , we get (cf. (21))
σ1* = 0 → σ1* = C1* = const.,
σ 2* = 0 → σ 2* = C2* = const.,
ϑ2 + (ϕ ′ − ρ ) λ ε + ϕ λ = 0,
(ϑ − ρ λ ) V + λ[V (ϕ ′ − ρ ) + ψ ′] = 0,
(34)
where (cf. (33))3
(35)
ε 0 = arctan[
(37)
λ=−
(38)
As the integrand F does not contain t explicitely,
Euler's equations (34) have the first integral
1 − ϑ2 V + λψ = C , C = const.
(39)
Eliminating V by (31), Euler's equation (cf. (34)) in
relation to ε gets the form
(
)
V ϕ ϑ1 − ρ λ + λ [(ψ + V ρ ε )( ρ − ϕ ′ ) + ϕψ ′] = 0, (40)
wherefrom we get
3
*
C1
and
*
C2
are Lagrange’s multipliers corresponding to the
constraints (30).
FME Transactions
*
r2 d12
)
].
*
− r1 (1 + d 22
)
(43)
ϑ2 ρ + ϑ1ϕ
,
ϑ1ϕψ + ϑ2 [(2 ρ − ϕ ′)ψ + ϕψ ′]
(44)
*
d * + κ (1 + d 22
ρ (ε1 ) + ϕ (ε1 ) tan ε1
)
(45)
= C1* 12 1
*
*
ρ (ε1 ) − ϕ (ε1 ) tan ε1
1 + d11 + d 21κ1
In order to obtain the equations of motion of the system
considered (cf. (33), (35)) taking into account (45) and
(46)
the differential equations (30) can be written in the form
dq1
1 V (V ρ + ϕV ′) cos ε ,
=−
dε
(C1* ) 2 ψ
dq 2
1 V (V ρ + ϕV ′) sin ε ,
=−
dε
(C1* ) 2 ψ
(47)
where (cf. (35) , (42), (45))
V =
wherefrom we get
1
(42)
If we apply the same procedure to (39), having in mind
(42), we shall get the Lagrange's multiplier
[1 − ϑ2 V + λψ ](t = t ) = 0 .
(36)
1
Taking into consideration that V and ε is not
prescribed in the final position of the system, we have
also the end-conditions
λ(t =t ) = λ1 = 0.
(
*
*
−r2 1 + d11
+ r1 d 21
ψ ′′ = −ψ , ϕ ′′=2ρ ′, ρ ′′ = −2 ρ ′,
ϑ1′ = ϑ2 , ϑ2′ = −ϑ1 ,
The condition of transversality at the right end-point
gets the form
∂F
∂F
]
= 0, [ ](t = t ) = 0,
1
∂V (t = t1)
∂ε
ψ ( ρ − ϕ ′ ) + ϕψ ′
,
ϑ1 ϕψ + ϑ2 [( 2 ρ − ϕ ′ )ψ + ϕψ ′]
wherefrom in a case that the system in the start position
has a velocity of known intensity V (t0 ) = 0 we shall get
(cf. (33)) a value of the natural parameter at moment
t = t0
C2* = C1*
ϑ1 = C1* sin ε − C2* cos ε ,
[
λ
[(ψ + V ρ ε )( ρ − ϕ ′ ) + ϕψ ′]}. (41)
Vϕ
wherefrom taking into account (38), we obtain
1
ϑ2 = C1* cos ε + C2* sin ε ,
ψ ′ = r2 cos ε − r1 sin ε ,
ϕ ′ = r3 cos 2ε − r4 sin 2ε ,
ρ ′ = −r4 cos 2ε − r3 sin 2ε .
ρ
{ϑ1 +
If we now eliminate λ in Euler's equation (cf. (34)) in
relation to V by (41) we shall get
ψ = r1 cos ε + r2 sin ε ,
*
+ r4 (cos ε ) 2 +
ϕ = 1 + d 22
1
ψ ( ρ − ϕ ′) + ϕψ ′
ϑ3ϕψ + ϑ4 (2 ρψ −ψϕ ′ + ϕψ ′)
{ {
(48)
V = ψϑ3 ψ  2( ρ − ϕ ′) 2 − ϕ (ϕ + ρ ′)  +


ϕψ ′(3ρ − 2ϕ ′)
}+
ϑ4 {[ψ (ϕ + ρ ′) + ψ ′ρ ] (ψρ ′ − ρψ ′) −
}}
2ψ 2 ρψ (ϕ − ρ ′)
{ϑ3ϕψ + ϑ4 [ψ (2 ρ − ϕ ′) + ϕψ ′]}−2 ,
ϑ3 = sin ε − cos ε
= sin ε − cos ε
(49)
ρ (ε1 ) + ϕ (ε1 ) tan ε1
=
ρ (ε1 ) − ϕ (ε1 ) tan ε1
*
*
+ κ1 (1 + d 22
)
d12
*
*
1 + d11
+ d 21
κ1
,
VOL. 35, No 4, 2007 ▪ 185
ϑ4 = cos ε − sin ε
ρ (ε1 ) + ϕ (ε1 ) tan ε1
=
ρ (ε1 ) − ϕ (ε1 ) tan ε1
= cos ε − sin ε
*
*
d12
+ κ1 (1 + d 22
)
The differential equations of motion of the system
considered (Fig. 1) have the form
.
α11q1 + α12 q 2 = Q1µ −
(50)
*
*
1 + d11
+ d 21
κ1
The solutions of the differential equations (47) has the
form
q1 = K (Φ − Φ 0 ), q 2 = K (Ψ − Ψ 0 ),
(51)
α 22 q 2 + α12 q1 = Q2µ −
∂Π
∂q 1
+ u1 ,
∂Π
∂q 2
(56)
+ u2,
where
where (cf. (48), (49))
V
Φ (ε1 , ε ) = ∫ (V ρ + ϕV ′) cos ε ds
Q1µ = − µ2 (m + M ) g − mq 2 tan α ,
V
Ψ (ε1 , ε ) = ∫ (V ρ + ϕV ′) sin ε ds
∂Π
Q2µ = − µ1m( g − q1 tan α ),
ψ
ψ
(52)
Φ 0 = Φ (ε1 , ε 0 ), Ψ 0 = Ψ (ε1 , ε 0 ),
K=
1
(C1* ) 2
∂q 2
= c2 , c2 = − mg tan α .
∆(ε1 ) = q1 (Ψ1 − Ψ 0 ) − q1 (Φ1 − Φ 0 ) = 0 ,
(53)
holds where (cf. (52))
Φ1 = Φ (ε1 , ε = ε1 )
(54)
we get a value of the natural parameter ε1 at moment
t1 , constant K from
q1
q2
,
=
Φ1 − Φ 0 Ψ1 − Ψ 0
(55)
ϕ1 = d11q 1 − d 21q 2 ,
ϕ2 = d12 q 1 − d 22 q 2 ,
b1 = g µ2 (m + M ), b2 = gmµ1 ,
d12 = − mµ2 tan α , d 21 = − mµ1 tan α ,
If we now introduce relation (23), kinetic energy (25)
and potential energy (26) of the system have the same
form , where (cf. (27))
gm tan α
EXAMPLE. Let us consider motion of the mechanical
system which consists of two prismatic rigid bodies and
moves in homogeneous field of gravity. The
configuration of the system is defined by the set of
a = M + m(tan α ) 2 ,
coordinates q = ( q 1 , q 2 ) . The system starts from the
position defined by coordinates
q (t0 ) = q
q 1 (t0 ) = q (10) and
, where it was at rest. Final position is set
by q 1 (t1 ) = q (11) and q 2 (t1 ) = q (21) . The coefficient of
Coulomb friction on the rough inclined side (at angle
α t 0 to horizontal) of prism P1 is µ1 . The coefficient of
friction on rough horizontal plane is µ2 , (Fig 1.).
a
gms tan α
b
,
(60)
Power of the generalized force is presented by (28)
where (cf. (29) and (60))
b1* =
b2* =
*
d 21
g ( m µ1 + m µ2 + M µ2 )
,
a
g ( m s µ1 + m µ2 + M µ2 )
=−
*
=−
d12
*
=−
d 22
186 ▪ VOL. 35, No 4, 2007
c2∗ = −
,
b = M + m − 2ms + s 2 m1 + (tan α ) 2 ,
M
s=−
.
m(tan α ) 2
*
=−
d11
Figure 1
(59)
d11 = d 22 = 0.
c1∗ = −
(20)
(58)
ψ = −b1q 1 − b2 q 2 ,
C1* (cf. (52)) and C2* (cf. (45)).
2
(57)
where
Taking into account that at final position the following
relations (cf. [6])
K=
∂Π
Relation (10) in this case have the form (cf. (57))
P µ = ψ + ϕ1q1 + ϕ2 q 2 ,
.
Ψ1 = Ψ (ε1 , ε = ε1 ),
∂q1
= 0,
b
m ( µ1 + µ2 ) tan α
,
a
m ( s µ1 + µ2 ) tan α
a b
m ( µ1 + s µ2 ) tan α
a b
,
(61)
,
,
m s ( µ1 + µ2 ) tan α
.
b
Taking into account (30), the relation (31) has the same
form, where (cf. (33), (60))
FME Transactions
ψ = g cos ε
+ g sin ε
m ( µ1 + µ2 − tan α ) + M µ2
+
a
m ( s µ1 + µ2 − s tan α ) + M µ2
b
ϕ = 1 − m sin 2 ε tan α
− s m tan α
( µ1 + µ2 ) (1 + s )
2 ab
µ1 + µ2
b
−
−
1 s
− (cos ε )2 m ( µ1 + µ2 ) tan α ( − ),
a b
( µ + µ ) (1 + s )
ρ = −m (cos ε )2 tan α 1 2
+
ab
+ m tan α
,
( s µ1 + µ2 )
ab
(62)
Figure 3
4. CONCLUSION
−
1
1 s
− m sin 2 ε tan α ( µ1 + µ2 ) ( − ).
2
a b
Let m =
32
denote the mass of the prism P1 and let
13
20
denote the mass of the prism P2 in a suitable
13
π
1
1
and µ2 =
, then
system of units. If α = , µ1 =
10
50
4
the relation (43) gives the value of natural parameter in
the initial position (at moment t = t0 ), ε 0 = −0.413771 .
M =
If q1 (t1 ) = 12 i q 2 (t1 ) = 6 (at moment t = t1 ), then the
relation (53) gives the value of the natural parameter in
the final position (at moment t = t1 ), ε1 = 0.951221 .
The graph in Fig.2 is showing the brachistochrone
q 2 = f (q1 ) of the system considered.
The changes of the power of generalized forces of
Coulomb friction with respect to ε of the system
considered are shown graphically in Fig.3.
After solving for
ε1
t1 =
∫
ε0
dε
ε
(45)
we get the time the system needs to move from the start
to the final position, t1 = 1.63737 s .
The mathematical model used to compute the
brachistochrone in this special case of the multibody
system with two degrees of freedom is based on
variational calculus. The problem is formulated as
constrained with the constraint which represents the
principle of work and energy (6), where power of
generalized forces of Coulomb friction has the modified
form obtained from [5]. The complete analogy is made
between solution obtained in an example considered and
a solution in relation to material point.
REFERENCES
[1] Ashby, N. et al.: Brachistochrone with Coulomb
friction, American Journal of Physics, 43, pp. 902906, 1975.
[2] Pars, L.: An introduction to calculus of variations,
Heinemann, London. 1962.
[3] 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.
[4] Čović, V., Vesković, M.: Brachistochrone on a
surface with Coulomb friction, International
Journal of Non-linear Mechanics, accepted paper,
2006.
[5] Čović, V., Vesković , M.: Brachistochronic motion
of a multibody system with Coulomb friction,
Minisymposia:
Mathematical
methods
im
Mechanics, 1st ICSSM-2007, Kopaonik, Serbia.
[6] Čović, V., Lukačević, M., Vesković , M.: On
brachistochronic motions, Monographical Booklets
in Applied & Computers Mathematics, PAMM,
Budapest, 2007.
БРАХИСТОХРОНО КРЕТАЊЕ МЕХАНИЧКОГ
СИСТЕМА СА РЕАЛНИМ ВЕЗАМА
Драгутин Ђурић
Figure 2
FME Transactions
У овом раду разматрано је кретање механичког
система у пољу конзервативних сила под дејством
веза са Кулоновим трењем. У специјалном случају
направљена је аналогија између
Ешбијеве
брахистохроне и брахистохроне механичког система
са два степена слободе.
VOL. 35, No 4, 2007 ▪ 187
188 ▪ VOL. 35, No 4, 2007
FME Transactions