1568 Fivd Iremos a5 Reviewed

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International Review of Modelling and Simulations (IREMOS), Vol. xx, n. x

Manuscript received March 2009, revised March 2009 Copyright 2009 Praise Worthy Prize S.r.l. - All rights reserved

Mathematical Model for the Kinematics

of a Continuous Ship Unloader

Filip Van den Abeele1, Any Noca2, Marc Van Steelant3

Abstract Unlike traditional portal cranes, a Continuous Ship Unloader (CSU) is not hamperedby an intermittent offloading mechanism. The digging foot, consisting of an endless chain with

offloading buckets, allows reaching very high capacities when unloading large Panamax vessels

carrying iron ore or coal. However, severe problems were reported with the hydraulic control

system of the CSU during service. Measurements of cylinder displacements and hydraulic

pressures revealed that the digging foot fails to follow the imposed positions. A profound

kinematic analysis, presented in this paper, sheds a brighter light on the evolution of the chain

length during the transition from idle state to bypass. The aberrations in the calculated chain sag

can explain why some imposed positions cannot be obtained by the hydraulic control system.

Copyright 2009 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords:Kinematics, Continuous Ship Unloader, Digging Foot, Chain Length

Nomenclature

Ci Cylinder i

(Ci) Control signal for cylinder i

(Ci) Encoder values for cylinder iPp Hydraulic pressure at the side of the piston

Pr Hydraulic pressure at the side of the rodMb Bucket mass

Mtot Total mass

F Force

t Time

t Time interval

I. Continuous Ship Unloaderand Digging Foot

Unlike traditional cranes, a Continuous Ship

Unloader (CSU) is not hampered by an intermittentoffloading mechanism. Thanks to its ingenious digging

foot, consisting of an endless chain with offloading

buckets, a CSU has a maximum capacity of up to 3 000

tonnes/hour (for iron ore[1]) and 2 300 tonnes/hour (for

coal [2]) whereas conventional portal cranes can only

reach capacities up to 850 tonnes/hour. Hence,

Continuous Ship Unloaders allow increasing the

productivity of a harbour site [3]. The digging foot of

the Continuous Ship Unloader is schematically shown

on Fig. 1.

Fig. 1: Schematic of the CSU digging foot

The digging foot is actuated by four hydrauliccylinders (like shown on Fig. 2), allowing ample

versatility to easily unload and clean large ocean-going

vessels. The cylinders C1L and C1R are mechanically

coupled, and can rotate the entire digging foot. The

position of these coupled cylinders, denoted C1 in our

analysis, influences both the inclination of the leg,

and the rotation of the toe. The hydraulic cylinder C2connects the leg with the toe, and its position alters the

tension that is imposed on the chain of buckets. The

fourth cylinder C3 can tilt the toe of the digging foot,

hence modifying the rotation angle .


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F. Van den Abeele, A. Noca, M. Van Steelant

Copyright 2009 Praise Worthy Prize S.r.l. - All rights reserved International Review of Modelling and Simulations, Vol. xx,

n. x

Fig. 2: Hydraulic cylinders actuating the digging foot

When judiciously steering these hydraulic cylinders,

the digging foot can unload virtually any type of vessel,

following the sequence [4] presented in Table I.

TABLEI

OPERATIONAL SEQUENCE TO UNLOAD A VESSEL

The control algorithm of the digging foot has to

ensure that the chain sag S, like defined in Fig. 3, stays

within an acceptable range, i.e. 250 mm S 350 mm.The chain will slip if the sag is too high, while

catastrophic failure of the structure might occur when

the chain sag is too low.

During service, however, serious problems with the

hydraulic system actuating the digging foot were

reported [5]. When maneuvering the digging foot,

unacceptable chain sags occurred (as seen on Table II),

and some imposed positions could not even be obtained.

In this paper, a mathematical model for the

kinematics of the continuous ship unloader is derived to

calculate the different positions of the digging foot

during unloading, and the corresponding chain sag. This

kinematic model provides an input for a root cause

analysis to address the operational concerns.

Fig. 3: Definition of chain sag

TABLEII


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UNACCEPTABLE CHAIN SAG

S > 350 mm

II. Measurements during Transitionfrom Idle State to Bypass

In order to identify the shortcomings of the hydraulic

system, measurements were performed on the digging

foot with and without bucket chain. To encompass the

full range of possible positions, the transition from idle

state (= 0, = 0) to bypass (= 90, = 90) isstudied. Table III shows the subsequent positions of the

hydraulic cylinders during this transition.

TABLEIII

TRANSITION FROM IDLE STATE TO BYPASS

=0,=0

=45,=45

=90,=90

For each trial, the control signals ((Ci), steering thecylinders valve, see Fig. 4) and the corresponding

encoder values ((Ci), reflecting the displacement of thepiston) are reported. At the same time, the hydraulicpressures {Pp, Pr} in the cylinders are monitored as

well.

Fig. 4: Proportional valve controlling the hydraulic cylinder


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II.1. Digging Foot without Bucket ChainAn overview of the control signals (Ci) for the

different cylinders, and their corresponding

displacements (Ci) is summarized in Table IV. The

control signal (C1) = -100% at t = 5s, requiringcylinder C1 to be pulled in at full force. Indeed, the

encoder value decreases correspondingly from 3 000

mm to 200 mm during the course of this transition. The

similar graph for cylinder C2 already reveals a possible

problem: although the control signal (C2) = -100% att = 5s, the position of the cylinder remains unchanged.

Apparently, the hydraulic system cannot cope with this

request .The control signal for cylinder C3 reaches a

maximum of (C3) = 70%, indicating that the bypassposition cannot be obtained, and the transition stops at

= 90, = 67.

Table V compares the hydraulic pressure Pp at the

piston with the pressure Pr at the side of the rod as afunction of the control signal (Ci) that is fed into thevalve of cylinder Ci (i= 1,2 or 3). It is remarkable to

note that, although (C1) = -100%, urging cylinder C1to pull in, the pressure at the piston increases from

Pp = 3 bar to Pp = 50 bar. The cylinder is drawn in

nonetheless, thanks to the considerable increase of

pressure at the side of the rod (from Pr = 45 bar to

Pr = 165 bar), but it is clear that this control scheme

does not yield the most economic use of hydraulic

power. Yet again, the observations for cylinder C2 are

quite odd: although there is a considerable pressure

gradient over the piston, the cylinder does not move. A

similar peculiarity is reported for cylinder C3: the

encoder values increase (meaning that the cylinder C3

becomes longer) despite the fact that Pr >> Pp. Once

more, this reveals some non conformity in the hydraulic

control system of the digging foot.

II.2. Digging Foot with Bucket ChainThe measurements for the digging foot with a chain

of 50 empty buckets, each with a mass of Mb = 312 kg,

are summarized in Table VI. The control signals (Ci)

and the corresponding encoder values (Ci) are similar

to the situation without bucket chain.

TABLEIV

CONTROL SIGNALS AND ENCODER VALUES (WITHOUT CHAIN)

(C1)AND (C1)

(C2)AND (C2)

(C3)AND (C3)

For cylinder C2, a control signal (C2) = -100%does not lead to the expected displacement. Note that

there is a time delay of 30s in the control signal

(C3). The corresponding hydraulic pressures areshown in Table VII. The evolution of the pressure

profiles is comparable with Table V, but due to the

additional mass (Mtot = 50 Mb = 13.6 tonnes), the

magnitude is somewhat higher.

In Fig. 5, the required forces

p p r rF P A P A= (1)

with the notations of Fig. 4, are compared for the

measurements with (solid line) and without (dotted line)

bucket chain.


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TABLEV

CONTROL SIGNALS AND PRESSURE VALUES (WITHOUT CHAIN)

(C1)AND P(C1)

(C2)AND P(C2)

(C3)AND P(C3)

For the simulation with (empty) buckets, the required

forces are higher, and the total transition time is longer.

Fig. 5: Hydraulic forces on the different cylinders

TABLEVI

CONTROL SIGNALS AND ENCODER VALUES (WITH CHAIN)

(C1)AND (C1)

(C2)AND (C2)

(C3)AND (C3)

The biggest difference is seen for cylinder C1, where

the additional force requirement mounts up to F = 200

kN, and the time delay is t = 30s.

III. Mathematical Model for theKinematics of the Digging Foot

In order to predict the subsequent positions as a

function of the encoder values (Ci), a mathematicalmodel for the kinematics of the digging foot is

presented in Fig. 6. The model has three degrees of

freedom and 8 distinct parts, which are listed in Table

VIII. The fixed lengths are summarized in Table IX.


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TABLEVII

CONTROL SIGNALS AND PRESSURE VALUES (WITH CHAIN)

(C1)AND P(C1)

(C2)AND P(C2)

(C3)AND P(C3)

For the kinematic analysis of the mechanism shown

in Fig. 6, the Lagrange coordinates are chosen as the

angles {,,,} and the encoder values {BA = (C1),

EF = (C2), IJ = (C3)}. Each position of the mecha-nism can be expressed in terms of these coordinates [6].

The closed loop O4DCBAO2O4implies that

4 2 2 4 0O D DC CB BA AO O O+ + + + + =

(2)

or, in complex notation,

Fig. 6: Kinematic model for the digging foot

( )

( ) ( )

( ) ( )

3

24

2

2 4 4 2 0

iii

i i

x x y y

O D e DC e CB e

BA e AO e

O O i O O

+

+ +

+

+ =

(3)

which leads to the set of equations

( )

( )

( )

( )

4

2

2 4

4

2

4 2

sin sin

cos

0

cos sinsin

0

x x

y y

O D DC

CB BA AO

O O

O D DC CB BA AO

O O

+

+

=

+ + + =

(4)

in the unknowns {, }. With BA = (C1) and well-

chosen initial values for {, }, these equations can besolved numerically. The same approach is pursued for

the closed loop GHIJKLMG, which requires

0GH HI IJ JK KL LM MG+ + + + + + =

(5)


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or, in complex notation,

( )

3

2

2 0

i i i

ii

ii

GH e HI e IJ e

JK e KL e

LM e MG e

+

+ +

+ +

+ +

+ + =

(6)

giving rise to two equations

( )

( )

cos cos

sin cos

sin 0

sin sin

cos sin

cos 0

GH HI IJ JK

KL LM

MG

GH HI IJ JK

KL LM

MG

+ + +

+

=

+ +

+ + =

(7)

where IJ = (C3) is the encoder value for cylinder C3,

{, } is the solution of (4), and {, } a set of well

chosen initial values for the unknown angles and .

TABLEVIII

PARTS OF THE KINEMATIC MODEL

N Part name Mass [kg]

1 Frame

2 Cylinder C1 1 420

3 Cylinder rod C1 2 028

4 Cylinder C2 + leg DC 6 274

5 Cylinder rod C2 (heel 7 446

6 Cylinder C3 398

7 Cylinder rod C3 280

8 Toe 10 520

The mathematical model (4)-(7) allows simulating the

transition from idle state to bypass, like shown in Table

III. The encoder values, initial guesses and solutions for

the initial position and the final state are summarized in

Table X. Using the measured encoder values (Ci) as aninput for the calculations, the entire transition can be

simulated as a function of time. Table XI compares the

simulated positions of the digging foot with the

experimental data (without bucket chain) for different

time frames, showing that the kinematic model is able

to predict the subsequent positions indeed.

TABLEIX

FIXED LENGTHS

Line Length [mm]

O2x 1 678

O4y 5 125

O2A - 450

BC 4 150

CD 1 375

O4D 1 675

O4E 0

FG 3 075

FM 4 395

GH 875

HI 500

JK 1 814

KL 491

LM 3 149

MN 5 125

Taking into account the diameters of the different chain-

wheels (see Fig. 3), the effective chain length and

correspond sag S can be calculated. Figure 7 compares

the chain sag for the simulation with (solid line) and

without (dotted line) chain bucket.

TABLEX

INITIAL GUESSES AND SOLUTIONS FOR {,,,}

IDLE BYPASS

guess solution guess solution

[] 0 3.06 90 89.12

[] 0 - 1.01 67 67.02

[] - 20 - 21.55 45 46.50

[] - 90 - 91.83 - 90 - 89.73

For most of the transition between initial position

and bypass, the calculated chain sag S is nowhere

within the acceptable limits (250 mm S 350 mm). Inthe initial position, the chain sag is much too high, and

the risk of slip is significant. The situation in bypass is

even more critical: for the simulation with bucket chain,

the sag drops to S = 0 mm after only 40 s, whereas the

full transition lasts 160 s. Once the chain sag has

disappeared, tension is induced in the chain, which can

give rise to catastrophic failure if the mechanical stress

exceeds the ultimate strength. This could explain why

the digging foot fails to follow the imposed positions,

like indicated in Table IV and VI. Hence, this kinematic

analysis at hand allows identifying the root causes of

the operational problems that occurred during service.


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TABLEXI

CONTROL SIGNALS AND ENCODER VALUES (WITH CHAIN)

T =0S

T =60S

T =100S

T =130S

Fig. 7: Calculated chain sag with and without bucket chain

IV. Conclusions and RecommendationsThe kinematic analysis of the Continuous Ship

Unloader sheds a brighter light on the evolution of the

chain length during the transition of the digging foot

from idle state to bypass. Indeed, the aberrations on thecalculated chain sag can explain why some imposed

positions cannot be obtained by the hydraulic control

system. A dynamic analysis [7-8], taking into account

(reduced) velocities, accelerations and inertia effects,

should allow identifying possible solutions to these

problems.

References

[1] C.F. Alsop, G.A. Forster and F. R. Holmes, Ore UnloaderAutomation a Feasibility Study , Proc IFAC Symp pp 295-305

(1965)

[2] Y. Sassa et.al., Continuous Ship Unloader for Coke Unloading,Fuel and Energy Abstracts 77, pp. 41-46 (1998).

[3] C. S. Park and Y.D. Noh, An Interactive Port CapacityExpansion Simulation Model, Engineering Costs and Production

Economics 11(1), pp. 109-124 (1987)

[4] Continuous Ship Unloader Operating and Maintenance Manual(1999).

[5] F. Darko and W. Dierick, Continuous Ship Unloader for IronOre and Coal, Internship Report, Department Harbour,

Transport and Recycling, ArcelorMittal Gent

[6] O. Vinogradov, Fundamentals of Kinematics and Dynamics ofMachines and Mechanisms (CRC Press, 2000).

[7] A. Noca,Mathematical Model for a Continuous Ship Unloader,M. Sc. dissertation, Department of Mechanical Construction and

Production, Faculty of Engineering, Ghent University, 2006.

[8] J. Van Wittenberghe, Global Positioning System for the Digging

Foot of a Continuous Ship Unloader, M. Sc. Dissertation,

Department of Mechanical Construction and Production, Faculty

of Engineering, Ghent University, 2006.


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Authors information

1OCAS N.V., Steel Solutions and Design, J.F. Kennedylaan 3, BE-

9060 Zelzate.2Mechanical Construction and Production, Faculty of Engineering,

Ghent University, St.-Pietersnieuwstraat 41, BE-9000 Gent.3ArcelorMittal Gent, Harbour, Transport & Recycling, J.F.

Kennedylaan 51, BE-9042 Gent

Filip Van den Abeele holds a M.Sc. in

Electromechanical Engineering, and received

a PhD in Engineering Sciences from Ghent

University for his research on numerical

modeling of impact damage in composite

materials.

Filip is currently working as a simulation

expert at OCAS, which is an advanced,

market-oriented R&D lab of ArcelorMittal, the biggest steel supplier

in the world. His main research focus is pipeline engineering, and he is

actively contributing to the safe and reliable design of tubular products

for oilfield services.Dr. Van den Abeele is a member of the Professional Institute of

Pipeline Engineers, and received the Prof. De Meulemeester-Piot

award from the Royal Society of Flemish Engineers (KVIV).

Any Nocareceived his M.Sc. in Electromechanical Engineering from

Ghent University in 2006, with a master dissertation on the

mathematical modeling of a Continuous Ship Unloader.

Anys interests include kinematics and dynamics of mechanisms,

hydraulic systems and control theory.

He is currently working as a mechanical engineer at Bosch Rexroth.

Marc Van Steelant is the electrical maintenance manager at the

Department of Raw Materials, Harbour, Transport & Recycling of

ArcelorMittal Gent, an integrated steel plant in the northern part of

Belgium.

Marc has contributed to a wide variety of engineering topics,including logistics, maintenance, lead time reduction and operational

efficiency.

He is currently focusing on the implementation of the Total Productive

Maintenance (TPM) methodology at ArcelorMittal Gent.

1568 Fivd Iremos a5 Reviewed

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