Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 1, pp. 147-159, Warsaw 2007

OPTIMIZATION OF THE WORKING CYCLE OF HARBOUR

CRANES

Josif Vukovic

Ugljesa Bugaric

Dusan Glisic

Dusan Petrovic

Faculty of Mechanical Engineering, University of Belgrade, Serbia and Montenegro

e-mail: ubugaric@mas.bg.ac.yu

The paper presents one of the possible ways optimization of motion of
the harbour crane grab minimization of the working (unloading) cycle,
energy consumption and material dissipation during the grab dischar-
ging. The optimization procedure of the working cycle is divided into
two phases. Firstly, it is optimization of the cargo and grabmotion and,
secondly, determination ofmotion of the cranemechanisms upon the ob-
tained optimal parameters of cargo and grab trajectory. The developed
mathematical model enables direct application of the optimal control
theorymethods, i.e. amethod of optimization of cargo and grabmotion
by making use of Pontryagins maximum principle. All relevant expres-
sions are derived analytically.

Key words: optimization, working cycle, harbour crane, maximum
principle

1. Introduction

Designing of complex transport systems presents a challenge for the designer,
from the point of view of device selection, facilities layout and appropriate so-
ftware for program-controlled systems. Analysis and selection of a solution for
the given design task of complex systemswith different levels of links between
the elements andmutual influences, whichmay be deterministic or stochastic,
could be found only by application of amodeling process. An approach to the
system as amethod which gives the best results and amodel of the investiga-
ting medium which contributes to the observation of the complex reality are
used.



148 J. Vukovic et al.

The unloading of bulk cargo presents organization of different activities,
connected with control and handling of the material flow from a vessel to the
transport or storage system, which provides best service conditions of vessels
with minimization of costs.

Unloading devices present knot points of unloading terminals, and in the
mostnumberof cases, bottle necks, so their functioning is thebasicprerequisite
for the optimal work of the whole unloading system.

The unloading (working) cycle of an harbour crane grab device consist of:
material grabbing from a vessel, grab and cargo transfer from the vessel to
the receiving hopper, grab discharging and empty grab return transfer from
the receiving hopper to the vessel. Full automation of the unloading process of
harbour crane facilities with the grab, is possible but very expensive. On the
other hand the crane operator could not repeat the optimal unloading cycle
in a longer time period. The only practical feasible solution is to introduce
a half-automatic unloading cycle which consists of a manual part, where the
crane operator controls grab motion, and of an automatic part in which a
computer controls the grabmoving according to the given algorithm.

The manual part of the half-automatic unloading cycle consists of lowe-
ring of the empty grab to the material surface in the vessel, from one of the
three points of the end of the automatic part of the unloading cycle (Fig.1),
material grabbing and grab hoisting with cargo to one of the three points of
the beginning of the automatic part of the unloading cycle. The automatic
part of the half-automatic unloading cycle consists of grab transfer from one
of the three points of the beginning of the automatic part of the unloading
cycle to the receiving hopper, grab discharging and empty grab return from
the hopper to the one of the three possible points of the end of the automatic
part of the unloading cycle. The position of the three points, which presents
the beginning/end of the automatic part of the half-automatic unloading cyc-
le is virtual and depends on given geometry of the system, river water level,
material level in the vessel, etc. (Oyler, 1977).

2. The mathematical model of an harbour crane with moving

grab

Figure 1 shows a simplified harbour crane and cargomoving scheme onwhich
the mathematical model is based. The assumption is that the rope in the
initial time is in the vertical position with a defined initial length and the
grab position could be one of the three possible. This assumption corresponds
to the time immediately before the beginning of the automatic part of the
unloading cycle.



Optimization of the working cycle of harbour cycle 149

Fig. 1. A simplified scheme of an harbour crane

The generalized coordinates are: ϕ – angle of the jib, θ – angle of the
lever – luffing. The remaining denotations used in the mathematical model
are: m1 –mass of the jib, m2 –mass of the lever-luffing, m – grab and cargo
mass, l1 – length of the jib, l2 – length of the lever-luffing, ψ – rope angle,
xk – distance between the vessel and hopper, zk – height difference between
the vessel and hopper, MA – drivingmoment acting on the jib, MB – driving
moment acting on the lever-luffing, F – force in the rope, lc1 – distance
between thepoint A and the center of gravity of the jib, lc2 –distancebetween
the point B and the center of gravity of the lever-luffing, JA – moment of
inertia of the jib with respect to the axis through the point A, Jc2 –moment
of inertia of the lever-luffing with respect to the axis through the center of
gravity of the lever-luffing.

It is assumed that the centers of gravity of the jib and lever-luffing lie
on straight lines between the points A and B, and B and C, respectively.
The drivingmoments MA and MB are reduced to the points A and B. The
obtaining of real drivingmoments requires decomposition of thewhole driving
structure of the cranewhich is not the same for all harbour cranes (depends on
amanufacturer), and is not a subject of this work. The forces in the rope that
connects the lever-luffing and structure of the harbour crane are taken into
consideration by reducing the real drivingmoments to the points A and B.

The optimization procedure of the working cycle will be divided into two
phases in the following analysis. Firstly, cargo and grab motion will be opti-



150 J. Vukovic et al.

mized and, secondly, the crane drivingmoments will be determined upon the
obtained optimal parameters of the cargo and grab trajectory. According to
the thus found motion of the harbour crane, mechanisms and movement of
the grab and cargo will be observed separately.

Differential equations which describe motion of the crane mechanisms
(Fig.1) are

(JA+m2l
2
1)ϕ̈− [m2l1lc2 sin(ϕ−θ)]θ̈+[m2l1lc2cos(ϕ−θ)]θ̇

2 =

= MA−m1glc1cosϕ−m2gl1cosϕ−Fl1cos(ψ−ϕ)
(2.1)

−[m2l1lc2 sin(ϕ−θ)]ϕ̈+(Jc2+m2l2c2)θ̈− [m2l1lc2 sin(ϕ−θ)]ϕ̇2 =
= MB +m2glc2 sinθ+Fl2 sin(ψ−θ)

Fig. 2. Forces acting on the grab and cargo

Motion of the grab and cargo will be analyzed in the coordinate system
xOz (Fig.2). At the beginning of motion, the grab and cargo are placed at
the point O. In that case, differential equations which describemotion of the
grab and cargo are

mẍ = F sinψ mz̈ = F cosψ−mg
F

m
= S

ẍ = S sinψ z̈ = S cosψ−g
(2.2)

Grab and cargo, for the time interval known in advance [0, tc]:

— from the initial state, t =0

x(0)= 0 ẋ(0)= 0

z(0)= 0 ż(0)= 0
(2.3)

— should came to the ending state, t = tc

x(tc)= xk ẋ(tc)= 0

z(tc)= zk ż(tc)= 0
(2.4)



Optimization of the working cycle of harbour cycle 151

with a limitation that the grab and cargo should pass through the point
(xk/2,zk) and after that continue to move horizontaly, i.e.

x(τ)=
xk
2

z(τ)= zk z(τ ¬ t ¬ tc)= zk (2.5)

where the time instant τ is not known in advance.
If such functions ψ(t),S(t) > 0 can be found, together with the following

conditions

ψ(0)= 0 ψ̇(0)= 0 S(0)= g

ψ(tc)= 0 ψ̇(tc)= 0 S(tc)= g
(2.6)

so that appropriate solutions to equations (2.2) fulfill conditions (2.3), (2.4)
and (2.5), the whole system can be controlled.
By increasing the order of differential equations (2.2), those equations can

be written as

...
x = Ṡ sinψ+Sψ̇cosψ

...
z = Ṡ cosψ−Sψ̇ sinψ (2.7)

and conditions (2.6) can be written as

ẍ(0)= 0
...
x(0)= 0 z̈(0)= 0

ẍ(tc)= 0
...
x(tc)= 0 z̈(tc)= 0

(2.8)

In thatway, the task ofmotion control of the grab and cargo can be stated
in a following form

xIV = ux z
IV = uz (2.9)

and

x(0)= 0 ẋ(0)= 0 ẍ(0)= 0
...
x(0)= 0

z(0)= 0 ż(0)= 0 z̈(0)= 0

x(tc)= xk ẋ(tc)= 0 ẍ(tc)= 0
...
x(tc)= 0

z(tc)= zk ż(tc)= 0 z̈(tc)= 0

x(τ)=
xk
2

z(τ)= zk z(τ ¬ t ¬ tc)= zk

(2.10)

where ux and uz are allowed values of control which belong to an open set.
The beginning condition for

...
z is not set in order to ensure movement in

the z direction at the beginning of themovement, while the ending condition
for

...
z is automatically fulfilled due to the transverse condition.
According to (2.2) and (2.7), equations (2.9) and conditions (2.10) are

equivalent to equations (2.2) and conditions (2.3)-(2.6).



152 J. Vukovic et al.

3. Optimal motion of the grab and cargo

The main objectives of the optimization process are the minimal working
(unloading) cycle, minimal rope inclination angle, minimal dissipation of the
material and, therefore, minimal expense of energy needed for motion of an
harbour crane.

By introducing new variables yi (i =1,2, . . . ,8), system (2.9) and condi-
tions (2.10) can be written in a following form

ẏ1 = y2 ẏ2 = y3 ẏ3 = y4 ẏ4 = ux

ẏ5 = y6 ẏ6 = y7 ẏ7 = y8 ẏ8 = uz
(3.1)

and

y1(0)= 0 y2(0)= 0 y3(0)= 0 y4(0)= 0

y5(0)= 0 y6(0)= 0 y7(0)= 0
y1(tc)= xk y2(tc)= 0 y3(tc)= 0 y4(tc)= 0

y5(tc)= zk y6(tc)= 0 y7(tc)= 0

y1(τ)=
xk
2

y5(τ)= zk y5(τ ¬ t ¬ tc)= zk

(3.2)

which allows direct application of Pontryaginsmaximumprinciple. The values
ux and uz are control values in the x and z direction (Bugaric et al., 2004;
Sage andWhite, 1977; Zrnic et al., 1995).

During the grab and cargo transfer from a vessel to hopper and vice versa
the minimal rope inclination angle as well as no more than one oscillation of
the grab and cargo are demarded. Beside that, changes in the rope load as a
result of the grab and cargo transfer should be reduced to minimum. In that
sense, condition of optimality (3.3) presents good enoughmeasure of behavior
of those values

J =

tc
∫

0

1

2
(y23 +y

2
4 +u

2
x+y

2
8) dt → inf (3.3)

Differential equations (3.1) and conditions (3.2) together with condition of
optimality (3.3) formulate the task of optimal control.

In otherwords, on the basis of equations (2.2), it can be concluded that the
rope inclination and its angular velocity have greater influence on movement
in the x direction, i.e. on values y3, y4 and ux, while a change in the rope load
has greater influence onmovement in the z direction, i.e. on the value y8. So,
the minimal value of (3.3) fulfills the demands and represents an optimality
criterion for the discussed problem. It also provides that the values of control



Optimization of the working cycle of harbour cycle 153

and rope inclination angle do not become too big, ensuresminimal number of
oscillations, continuity of forces in the rope, uniformwork, etc.
The problem defined by relations (3.1)-(3.3) is reduced to a form which

makes possible direct application of themaximumprinciple. For these reasons,
considering (3.1) and (3.3), the following function is established

H =−
1

2
(y23 +y

2
4 +u

2
x+y

2
8)+

8
∑

i=1

λiyi+1 (3.4)

where the values λi satisfy a system of differential equations

λ̇i =−
∂H

∂yi
i =1, . . . ,8 (3.5)

and

λ̇1 =0 λ̇2 =−λ1 λ̇3 = y3−λ2 λ̇4 = y4−λ3

λ̇5 =0 λ̇6 =−λ5 λ̇7 =−λ6 λ̇8 = y8−λ7
(3.6)

According to the theorem of the maximum principle, function (3.4) has
the maximal value for the optimal solution. According to the condition of
extremum

∂H

∂ux
=0

∂H

∂uz
=0 (3.7)

the controls in the x and z direction are obtained

−ux+λ4 =0 ⇒ ux = λ4
(3.8)

λ8 =0 ⇒ λ̇8 =0 ⇒ y8 = λ7

The following transverse conditions should be added to conditions (3.2)

λ8(0)= 0 λ8(tc)= 0

what is trivially fulfilled in (3.8).
The structureof systemsof differential equations (3.1) and (3.6) shows that

the optimization of grab and cargo movement in the x and z directions can
be done separately. The system of differential equations for the optimization
grab and cargo movement in the x direction has the following form

ẏ1 = y2 ẏ2 = y3 ẏ3 = y4 ẏ4 = λ4

λ̇1 =0 λ̇2 =−λ1 λ̇3 = y3−λ2 λ̇4 = y4−λ3
(3.9)

Boundary conditions are:



154 J. Vukovic et al.

— for t =0

y1(0)= y2(0)= y3(0)= y4(0)=0 (3.10)

— for t = tc

y1(tc)= xk y2(tc)= y3(tc)= y4(tc)= 0 (3.11)

The system of differential equations for the optimization of grab and cargo
movement in the z direction has the following form

ẏ5 = y6 ẏ6 = y7 ẏ7 = y8 ẏ8 =−λ6

λ̇5 =0 λ̇6 =−λ5 λ̇7 =−λ6 λ̇8 =0
(3.12)

Boundary conditions are:

— for t =0

y5(0)= y6(0)= y7(0)= λ8(0)= 0 (3.13)

— for t = τ

y5(τ)= zk y6(τ)= y7(τ)= λ8(τ)= 0 (3.14)

— for τ ¬ t ¬ tc

y5(τ)= zk y6(t)= y7(t)= λ8(t)= 0 (3.15)

Each of differential equations (3.9) and (3.12) defined that way, with con-
ditions (3.10), (3.11) and (3.13)-(3.15) presents a two-point boundary value
problem. Due to the configuration of differential equations (3.9) and (3.12),
each of them can be solved analytically (Sage andWhite, 1977).

4. Analytical solutions

According to systems of differential equations (3.1) and (3.9), following rela-
tions can be established: (movement in the x direction)

ux = λ4 λ1 = L1 λ2 =−L1t+L2

λ3 = y2+
1

2
L1t
2−L2t+L3

λ4 = y3−y1−
1

6
L1t
3+
1

2
L2t
2−L3t+L4

ẏ4 = ẏ2−y1 =−
1

6
L1t
3+
1

2
L2t
2−L3t+L4



Optimization of the working cycle of harbour cycle 155

Finally, differential equations (3.9) can be reduced to one fourth-order
differential equation

yIV1 − ÿ1+y1 =−
1

6
L1t
3+
1

2
L2t
2−L3t+L4 (4.1)

where L1, L2, L3, L4 are arbitrary constants.
A solution to the previous differential equation has the following form

y1 = x =
(

A1e
√

3t

2 +B1e
−

√

3t

2

)

cos
t

2
+
(

C1e
√

3t

2 +D1e
−

√

3t

2

)

sin
t

2
+

+E1t
3+F1t

2+G1t+H1

Differentiating theprevious expressionby tyields expressions for y2, y3, y4

y2 = ẋ =
[√
3
(

−B1+A1e
√

3t
)

cos
t

2
+
(

D1+C1e
√

3t
)

cos
t

2
+

−
(

B1+A1e
√

3t
)

sin
t

2
+
√
3
(

−D1+C1e
√

3t
)

sin
t

2

]

2e−
√

3t

2 +

+3E1t
2+2F1t+G1

y3 = ẍ =
[(

B1+A1e
√

3t
)

cos
t

2
+
√
3
(

−D1+C1e
√

3t
)

cos
t

2
+

+
√
3
(

B1−A1e
√

3t
)

sin
t

2
+
(

D1+C1e
√

3t
)

sin
t

2

]

2e−
√

3t

2 +

+6E1t+2F1

y4 =
...
x =
[

D1cos
t

2
+C1e

√

3tcos
t

2
−A1e

√

3t sin
t

2
−B1 sin

t

2

]

e−
√

3t

2 +6E1

where A1,B1,C1,D1,E1,F1,G1,H1 are constantswhichare tobedetermined
upon boundary conditions (2.3) and (2.4).
For movement in the z direction, according to differential equations (3.1)

and (3.12), following the relations can be established

uz =−λ6 λ5 = L5 λ6 =−L5t+L6
ẏ8 = λ̇7 ẏ8 =−λ6 ẏ8 =L5t−L6

where L5, L6 are arbitrary constants.
Substituting (L5,−L6) with (A2,B2), the required expressions for move-

ment in the z direction are obtained as

yIV5 = z
IV = A2t+B2

...
y5 =

...
z =
1

2
A2t
2+B2t+C2

ÿ5 = z̈ =
1

6
A2t
3+
1

2
B2t
2+C2t+D2

ẏ5 = ż =
1

24
A2t
4+
1

6
B2t
3+
1

2
C2t
2+D2t+E2

y5 = z =
1

120
A2t
5+
1

24
B2t
4+
1

6
C2t
3+
1

2
D2t
2+E2t+F2



156 J. Vukovic et al.

where A2, B2, C2, D2, E2, F2 are constants which are to be determined upon
boundary conditions (3.2).
Directly from differential equations (2.2), expressions for ψ and S are

obtained as

ψ =arctan
ẍ

z̈ +g
S =
√

ẍ2+(z̈2+g)2

Figure 3 show results of the grab and cargo optimization process in time.
Those results are: changes of the grab and cargo velocity in the x direction
(ẋ, Fig.3a), changes of the acceleration in the x direction (ẍ, Fig.3d), chan-
ges of the grab and cargo velocity in the z direction (ż, Fig.3b), changes of
the acceleration in the z direction (z̈, Fig.3d), changes of the rope inclina-
tion angle (ψ, Fig.3e), changes of the angular velocity of the grab and cargo
(ψ̇, Fig.3f), changes of forces in the rope (F/m, i.e. S, Fig.3g), optimal path
of the grab (Fig.3h), changes of the displacement x (Fig.3i) and z (Fig.3j).
Parameters, uponwhich the results shown inFig.3havebeenobtained, are:

distance between the vessel and hopper in the x direction xk =9m, distance
between the vessel and hopper in the z direction zk =8m, tc =20s – time,
known in advance, needed for obtaining one half of the automatic part of the
half-automatic unloading cycle, i.e. grab transfer from the vessel to hopper
or vice versa (determined upon maximal allowed velocities and accelerations
in the x and z directions (Bugaric and Petrovic, 2002)) and τ = x−1xk/2 –
time needed for grab and cargo transfer to one half of the distance between
the vessel and hopper, i.e. z(τ ¬ t ¬ tc)= zk.

5. Motion of the crane mechanisms

On the basis of the previous concept of cargo movement, a link between the
cargo and crane peakmovements can be established as

l1cosϕ+ l2 sinθ+ ξ sinψ+x−xk−a =0

l1 sinϕ− l2cosθ− ξcosψ−z +zk+ b =0

Realising that this is a redundant system, we can deem that the rope
length transition ξ(t), or something else, is a prominent time function, which
generally depends on structural characteristics of the crane. Changes of the
rope length ξ(t) in time should be determined upon real characteristics of
driving mechanisms for a specific type of an harbour crane, depending on a
manufacturer.Aproblemtobe resolvednow ishowthedirect task of dynamics
and unknown moments MA and MB can be determined from differential
equations (2.1) on the basis of the obtained parameters of cargo motion.



Optimization of the working cycle of harbour cycle 157

Fig. 3. Functions of the optimized parameters

6. Duration of the working cycle

Duration of the half-automatic working (unloading) cycle consist of the follo-
wing periods needed for completion of certain operations (Bugaric, 2002):



158 J. Vukovic et al.

—automatic part of the half-automatic unloading cycle

tac =2tc+ tgd =2 ·20+8=48s

where tgd is the time needed for grab discharging,
—manual part of the half-automatic unloading cycle

tmc = tgl+ tgc+ tgh+ te =(1.2÷7.2)+15+(1.2÷7.2)+5= (22.4÷34.4)s

where: tgl = (1.2÷7.2)s is the time for grab lowering from one of the three
possible points of the end of the automatic part of the unloading cycle to
the material in the vessel with a velocity of 50m/min. The lowering distance
depends on the water level and varies between 1 and 6m; tgc = 15s – time
needed for closing of the grab; tgl =(1.2÷7.2)s – time for grab hoisting from
the material in the vessel to one of the three possible points of the beginning
of the automatic part of the unloading cycle with a velocity of 50m/min. The
hoisting distance depends on the water level and varies between 1 and 6m;
te =5s – extra time needed for the crane operator to locate themost suitable
place for grabbing.
Finally, the duration of the working cycle is:

tuc = tac+ tmc =48+(22.4÷34.4)= (70.4÷82.4)s

7. Conclusions

The characteristic feature of bulk cargo is the great importance of parameters
such as fact transport expenses, manipulation and a waiting time. An unlo-
ading bulk cargo terminal works 24 hours seven days aweek during the sailing
period. The presented optimized working cycle of an harbour crane reduces
the rope inclination angle, forces in the rope and, therefore, energy required
for realisation of such operations.
It is important to underline that the developed procedure for the optimi-

zation of grab and cargo motion has universal applications, i.e. results of the
optimization process can be applied to any transport device which performs
similar tasks (unloading bridges, overhead cranes etc.).
The application of the obtained results lies in the introducing of the half

automatic unloading cycle during unloading of the bulk cargo material. In
that case, it is possible to achieve the optimal unloading cycle, to minimize
material dissipation during grab discharging, to lowering dynamic strains in
the crane and also to eliminate the influence of the human factor in realisation
of the unloading process (training of operator, weather conditions, nightwork,
etc.).



Optimization of the working cycle of harbour cycle 159

References

1. Bugaric U., Petrovic D., 2002, Modeling and simulation of specialized ri-
ver terminals for bulk cargo unloading with modeling of the elementary sub-
systems, Systems Analysis Modeling Simulation, Taylor and Frensis, 42, 10,
1455-1482

2. Bugaric U., Tosic S., Vukovic J., Brkic A., 2004,Optimization of eleva-
tor regimesofmovement,Proceedings of the 11thWorldCongress inMechanism
and Machine Science, China Machinery Press, Tianjin, China (accepted)

3. Oyler F.J., 1977, Handling of Bulk Solids at Ocean Ports; Stacking Blen-
ding Reclaiming, R.H.Wöhlbier (Edit.), Trans.Techn. Publications,Clausthal,
Germany

4. Sage A.P., White C.C., 1977, Optimum System Control, Prentice-Hall,
Eaglewood

5. Scheid F., 1968, Theory and Problems of Numerical Analysis, McGraw Hill,
NewYork

6. ZrnicD., Bugaric U., Vukovic J., 1995, The optimization ofmoving cycle
of grab by unloading bridges, 9th World Congress on the Theory of Machines
and Mechanisms, Milano, Italy, 1001-1005

Optymalizacja cyklu roboczego dźwigów portowych

Streszczenie

W pracy przedstawiono jeden z możliwych sposobów optymalizacji ruchu chwy-
taka dźwigu portowego oraz minimalizacji cyklu roboczego pod kątem ograniczania
zużycia energii oraz strat materiału podczas rozładunku. Procedurę optymalizacji
cyklu roboczego podzielono na dwie fazy. W pierwszej zoptymalizowano ruch prze-
noszonego obiektu i chwytaka, podczas gdy w drugiej wyznaczono ruch pozostałych
elementówdźwigu dla znalezionej optymalnej trajektorii obiektu i chwytaka. Sformu-
łowanymodel matematyczny układu umożliwił na bezpośrednie zastosowaniemetod
teorii optymalnego sterowania,w tymoptymalizacji z zasadymaksimumPontriagina.
Wszystkie związane wyrażenia wyprowadzono analitycznie.

Manuscript received August 3, 2006; accepted for print October 5, 2006