# Analysis of movement of the BOP crane under sea weaving conditions

Abstract

In the paper, mathematical models for dynamic analysis of a BOP crane un-der sea weaving conditions are presented. The BOP crane is a kind of gantry crane. It is installed on drilling platforms and used for transportation of the Blowout Preventor (BOP). The most important features characterising its dynamics are: motion of the crane base caused by sea weaving, clearance in the supporting system (between the support and rails), impacts of the load into guides and a significant weight of the load. In order to investigate dy-namics of the system, its mathematical model taking into consideration all these features has been formulated. Equations of motions have been derived using homogenous transformations. In order to improve numerical effecti-veness of the model, the equations have been transformed to an explicit form. The input in the drive of the travel system has been modelled in two ways: the kinematic input via a spring-damping element and the force input. Exemplary results of numerical calculations are presented.

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Available from: Andrzej Maczyński, Dec 23, 2014JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

48, 3, pp. 677-701, Warsaw 2010

ANALYSIS OF MOVEMENT OF THE BOP CRANE UNDER

SEA WEAVING CONDITIONS

Andrzej Urbaś

Marek Szczotka

Andrzej Maczyński

University of Bielsko-Biala, Faculty of Management and Computer Science, Bielsko-Biała, Poland

e-mail: aurbas@ath.bielsko.pl; mszczotka@ath.bielsko.pl; amaczynski@ath.bielsko.pl

In the paper, mathematical models for dynamic analysis of a BOP crane un-

der sea weaving conditions are presented. The BOP crane is a kind of gantry

crane. It is installed on drilling platforms and used for transportation of the

Blowout Preventor (BOP). The most important features characterising its

dynamics are: motion of the crane base caused by sea weaving, clearance in

the supporting system (between the support and rails), impacts of the load

into guides and a signiﬁcant weight of the load. In order to investigate dy-

namics of the system, its mathematical model taking into consideration all

these features has been formulated. Equations of motions have been derived

using homogenous transformations. In order to improve numerical eﬀecti-

veness of the model, the equations have been transformed to an explicit

form. The input in the drive of the travel system has been modelled in two

ways: the kinematic input via a spring-damping element and the force input.

Exemplary results of numerical calculations are presented.

Key words: modelling, dynamic analysis, BOP crane

1. Introduction

The exploitation of undersea oil and natural gas pools is one of the fastest gro-

wing ﬁelds of economy. In view of speciﬁc environmental conditions, technical

instruments used in this branch of industry need to fulﬁl speciﬁc exceptional

requirements. These instruments are called the oﬀshore equipment, while the

whole section of engineering concerning this ﬁeld is known as the oﬀshore tech-

nology. The motion of a base of oﬀshore instruments is the most characteristic

feature of the oﬀshore engineering. A vital problem faced by researchers who

wish to perform calculations of oﬀshore equipment dynamics is how to describe

678 A. Urbaś et al.

this motion. The principal factor here is the sea waving which is a very complex

phenomenon. For analysis of certain oﬀshore equipment, particularly cranes,

an assumption is often made that the ship moves only in the vertical plane

coincident with the transversal symmetry axis of its deck. Sinusoidal waves of

angular frequency 0.56 and 0.74 rad s−1and height 1 m propagating along the

ship transversal axis were assumed in Das and Das (2005). In Balachandran et

al. (1999) two kinds of functions describing motion of the on-board crane jib

sheave have been used. They were harmonic and pseudo-harmonic functions.

Masoud (2000) assumed swaying and surging oscillations as well as heaving,

pitching and rolling of a ship with a crane installed on board. Calculations were

based on empirically measured data from Fossen (1994) covering swaying and

surging oscillations and heaving motion of a chosen point of the ship (referen-

ce point). Also Driscoll et al. (2000) used measured vertical dislocations of an

A-frame to investigate the model of a cage suspended at considerable depths.

The additional loads caused by wind, impact of ﬂoe or ice ﬁeld, hoarfrost, sea

currents and many others usually not occurring in the land technology also

need to be taken into account (Handbook of Oﬀshore Engineering, 2005).

Cranes are an important kind of oﬀshore devices. There are many publi-

cations concerning their dynamics and control. Ellermann and Kreuzer (2003)

investigated the inﬂuence of a mooring system on the dynamics of a crane.

Jordan and Bustamante (2007), Maczyński and Wojciech (2008) analysed the

taut-slack phenomenon.

Motion of oﬀshore crane bases cause the load to signiﬁcantly sway even

when the crane does not execute any working movements. There are papers

concerning methods of stabilisation of the load positioning for oﬀshore cranes

– Cosstick (1996), Birkeland (1998), Pedrazzi and Barbieri (1998), Li and

Balachandran (2001), Maczyński and Wojciech (2007). More bibliographical

information concerning oﬀshore cranes and motion of their bases can be found

in Maczyński (2006).

One of the types of oﬀshore cranes is a BOP crane. The construction of

Protea from Gdańsk is presented in Fig.1a. It is a gantry crane installed

on a drilling platform designed to transport a system of valves named BOP

(Blowout Preventor). The BOP is used to block an uncontrolled outﬂow of

oil or natural gas from a wellbore at the seabed. After drilling the wellbore,

the BOP is put inside it and afterwards the risers are being connected to the

BOP. The risers drain oﬀ oil or gas into suitable tanks. In view of the plug

task, weight of the BOP reaches hundreds of tons. During the transportation

process (during the travel of a gantry crane) the BOP is protected by a system

of guides presented in Fig.1b.

Analysis of movement of the BOP crane... 679

Fig. 1. (a) BOP crane, (b) guide system

The clearance between the load and the guide system equals a few cen-

timeters. The weight of the presented crane is 200 T, hoisting capacity 550 T

and height about 30 m. The BOP cranes are hardly ever a topic of scientiﬁc

papers.

The analysis of a travel system is an interesting and important problem

concerning the dynamics of a BOP crane. The crane is supported on rails

and its motion is realised by means of a rack and a toothed wheel – Fig. 2a.

The maximum velocity of travel of the crane is equal to 3m/min. Due to

the movement of the platform deck caused by sea weaving and wind forces,

protection systems are used. These systems limit the movement of the crane

in the vertical and horizontal direction, perpendicular to the longitudinal axis

of the rails. This task is particularly realised by an anti-lift system presented

in Fig. 2b.

When the drive system of the crane is being designed, the dynamic forces

should be taken into consideration despite the insigniﬁcant velocity of crane

travel. The dynamic forces can be critical for some reasons:

•signiﬁcant weight of the moving crane,

•general motion of the deck caused by sea weaving,

•clearance in the anti-lift system,

•additional dynamic forces generated by the load impact against the gu-

ides.

680 A. Urbaś et al.

Fig. 2. (a) Rack travel system, (b) anti-lift system

In the paper, preliminary results of the dynamic analysis of the drive system

of the BOP crane are presented. In the mathematical model, all the aforemen-

tioned features are taken into consideration.

2. The dynamic model of a BOP crane

The mathematical model of the system has been formulated to enable dyna-

mic analysis of the BOP crane, see Urbaś and Wojciech (2008, 2009). In these

papers, general relations concerning the equations of crane motion are presen-

ted. In our work, we have derived formulas describing Lagrange operators in

an explicit form. It is important due to convenience of numerical calculations.

Furthermore, we have evolved more accurate models of the drivers of the travel

system of the BOP crane.

The schema of the model of the BOP crane together with more important

coordinate systems is presented in Fig. 3.

The following basic assumptions for modelling are established:

•motion of the base (system {D}) is known and described by functions

x(D)=x(D)(t)y(D)=y(D)(t)z(D)=z(D)(t)

ψ(D)=ψ(D)(t)θ(D)=θ(D)(t)ϕ(D)=ϕ(D)(t)(2.1)

•structure of the crane (frame) is treated as a rigid body – it should be

noticed that the construction of the BOP crane is a kind of combination

of two A-frames; the A-frame has been a subject of many analyses pre-

sented by Fałat (2004) which proved that the inﬂuence of ﬂexibility of

Analysis of movement of the BOP crane... 681

Fig. 3. Schema of the BOP crane model

the frame on dynamics of the whole system (on motion of the load) is

slight,

•load is a rigid body of a rectangular shape,

•load is suspended on two ropes – their ﬂexibility and damping are taken

into account,

•load can touch the guides only along its edges,

•clearance and ﬂexibility between the load and guides are taken into con-

sideration,

•frame is mounted ﬂexibly to the deck and, additionally, in the c

Y(D)

direction a clearance can occur,

•input in the drive system has been modelled in two ways: kinematic

input via a spring-damping element and force input,

•wind force can be taken into consideration,

•homogenous transformations are used to describe the system geometry

(Wittbrodt et al., 2006).

682 A. Urbaś et al.

Both the load (system {L}in Fig. 3) and the frame (system {F}) have

6 degrees of freedom with respect to the deck (system {D}). So, the model

has 12 degrees of freedom and the vector of generalised coordinates has the

following form

q="q(F)

q(L)#(2.2)

where

q(F)= [x(F), y(F), z(F), ψ(F), θ(F), ϕ(F)]⊤

q(L)= [x(L), y(L), z(L), ψ(L), θ(L), ϕ(L)]⊤

Equations of motion of the system have been derived from Lagrange equations

of the second kind

d

dt

∂E

∂˙qk

−∂E

∂qk

+∂V

∂qk

+∂D

∂˙qk

=Qkk= 1,...,12 (2.3)

where Eis the kinetic energy of the system, V– potential energy, D– function

of energy dissipation, Qk– non-potential generalised force corresponding to

the coordinate k,qk– element of the vector q.

In the next sections, the features of the BOP crane that potentially have

bigger inﬂuence on the dynamics of the drive system are described in greater

detail.

2.1. Motion of the base of the BOP crane – the system {D}

It has been mentioned that motion of the base (deck of the platform), that

means motion of the system {D}, with respect to the deck has been assumed

as known. It is described by pseudo-harmonic functions

y(D)

i=

n(D)

i

X

j=1

A(D)

i,j sin(ω(D)

i,j t+ϕ(D)

i,j )i= 1,...,6 (2.4)

where A(D)

i,j ,ω(D)

i,j ,ϕ(D)

i,j denote the amplitude, angular frequency and phase

angle of the input, respectively, n(D)

i– number of harmonics in the series.

The application of homogenous transformations (Wittbrodt et al., 2006)

allows one to convert the position vector of the point deﬁned in the system {D}

to system {·} according to relation

r{·}

P=T(D)r{D}

P(2.5)

Analysis of movement of the BOP crane... 683

where r{·}

p= [xp, yp, zp,1]⊤is the position vector of the point Pin the

system {·},r{D}

P= [x{D}

P, y{D}

P, z{D}

P,1]⊤– position vector of the point P

in the system {D},T(D)– matrix of homogenous transformation from the

system {D}to the system {·}.

Matrix T(D)can be presented as the product of six matrices where each

of them is a function of one variable dependent on time

T(D)(t) = T(D)

1T(D)

2T(D)

3T(D)

6T(D)

5T(D)

4(2.6)

and according to Fig. 3

T(D)

1=T(D)

1(x(D)) =

1 0 0 x(D)

0 1 0 0

0 0 1 0

0 0 0 1

T(D)

2=T(D)

2(y(D)) =

1 0 0 0

0 1 0 y(D)

0 0 1 0

0 0 0 1

T(D)

3=T(D)

3(z(D)) =

1 0 0 0

0 1 0 0

0 0 1 z(D)

0 0 0 1

T(D)

4=T(D)

4(ϕ(D)) =

1 0 0 0

0cϕ(D)−sϕ(D)0

0sϕ(D)cϕ(D)0

0 0 0 1

T(D)

5=T(D)

5(θ(D)) =

cθ(D)0sθ(D)0

0 1 0 0

−sθ(D)0cθ(D)0

0 0 0 1

T(D)

6=T(D)

6(ψ(D)) =

cψ(D)−sψ(D)0 0

sψ(D)cψ(D)0 0

0 0 1 0

0 0 0 1

684 A. Urbaś et al.

and

x(D)=x(D)(t) = y(D)

1y(D)=y(D)(t) = y(D)

2

z(D)=z(D)(t) = y(D)

3ϕ(D)=ϕ(D)(t) = y(D)

4

θ(D)=θ(D)(t) = y(D)

5ψ(D)=ψ(D)(t) = y(D)

6

sα = sin α cα = cos α

The order of rotations included in the matrix T(D)is agreeable with Euler

angles ZY X .

2.2. Kinetic and potential energy of the frame and load

The application of equations (2.3) requires the deﬁnition of relations de-

scribing kinetic and potential energy of bodies composing the analysed system

– in this case the energy of the frame and the load. One of the ways of calcu-

lation of kinetic energy of a rigid body is the usage of formula (Wittbrodt et

al., 2006)

E=1

2Z

m

tr( ˙

r˙

r⊤)dm =1

2tr {˙

TH ˙

T⊤}(2.7)

where ris the position vector of the point Pin the system {·},m– mass

of the rigid body, T– transformation matrix to the system {·},H– pseudo-

inertia matrix the form of which is inter alia given in Wittbrodt et al. (2006).

Introducing the notion of Lagrange operator

εk=d

dt

∂E

∂˙qk

−∂E

∂qk

(2.8)

where kis the number of the generalised coordinate, and using the transfor-

mation presented by Wittbrodt et al. (2006), one can obtain a short form

εk= tr{TkH¨

T⊤}=

n

X

i=1

ak,i ¨qi+

n

X

i=1

n

X

j=1

tr{TkHT⊤

i,j }˙qi˙qj(2.9)

where

Tk=∂T

∂qk

Ti,j =∂2T

∂qi∂qj

ak,i = tr {TkHT⊤

i}

The notation (2.9) has been used in many publications, e.g. Maczyński and

Wojciech (2003), Adamiec-Wójcik et al. (2008). However, as far as the eﬃ-

ciency of numerical calculations is concerned, the notation is not the most

Analysis of movement of the BOP crane... 685

proﬁtable. It requires repeated multiplication of matrices of 4 ×4 dimension

and then calculation of the trace of the resultant matrices. In this work, the

authors decided to derive formulae describing Lagrange operators in an expli-

cit form. An important feature of a rotational matrix, that is its orthogonality,

has been used.

If one denotes the homogenous transformation matrix from the frame sys-

tem {F}to the deck system {D}as e

T(F)and from the load system {L}

as e

T(L), the transformation matrices from the frame system and from the

load system to the system {·} can be calculated as

T(F)=T(D)e

T(F)T(L)=T(D)e

T(L)(2.10)

Next, the transformations have a universal character and they are not depen-

dent on the local coordinate system. Therefore, instead of equations (2.10) we

will use one general formula

T=T(D)e

T(2.11)

Time derivatives of the transformation matrix Tare

˙

T=˙

T(D)e

T+T(D)˙

e

T¨

T=˙

T(D)e

T+ 2 ˙

T(D)˙

e

T+˙

T(D)¨

e

T(2.12)

so, relation (2.9) can be presented in the following form

εk= trnT(D)e

TkH¨

T(D)e

T+ 2 ˙

T(D)˙

e

T+T(D)¨

e

T⊤o=

= tr T(D)⊤T(D)e

TkH¨

e

T

⊤

| {z }

εk,2

+ tr¨

T(D)⊤

T(D)e

TkH

e

T⊤

| {z }

εk,0

+ (2.13)

+2 tr˙

T(D)⊤

T(D)e

TkH˙

e

T

⊤

| {z }

εk,1

Below the manner of calculation of the components εk,2,εk,0,εk,1is presented.

Assuming that the rotation angles of the frame and the load are small, the

matrix e

Tcan be written as

e

T=

1−ψ θ x

ψ1−ϕ y

−θ ϕ 1z

0 0 0 1

(2.14)

and moreover

e

T=I+

6

X

j=1

Djqj(2.15)

686 A. Urbaś et al.

where qjare suitable elements of vectors q(F)or q(L), and matrices Djcan

be deﬁned:

— for j= 1,2,3

Dj="0aj

00#(2.16)

where

a1=

1

0

0

a2=

0

1

0

a3=

0

0

1

— for j= 4,5,6

Dj="Rj0

00#(2.17)

where

R4=

0 0 0

0 0 −1

0 1 0

R5=

0 0 1

0 0 0

−1 0 0

R6=

0−1 0

100

000

From (2.15), the following relationships occur

˙

e

T=

6

X

j=1

Dj˙qj

¨

e

T=

6

X

j=1

Dj¨qj

e

Tk=∂

e

T

∂qk

=Dj(2.18)

If one uses denotations

T(D)="Φ0S0

01#˙

T(D)="Φ1S1

00#¨

T(D)="Φ2S2

00#

(2.19)

then

T(D)⊤T(D)="Φ⊤

00

S⊤

01#"Φ0S0

01#="Φ⊤

0Φ0Φ⊤

0S0

S⊤

0Φ0S⊤

0S0+ 1#="IΦ⊤

0S0

S⊤

0Φ0S⊤

0S0+ 1#

˙

T(D)⊤

T(D)="Φ⊤

10

S⊤

10#"Φ0S0

01#="Φ⊤

1Φ0Φ⊤

1S0

S⊤

1Φ0S⊤

1S0#(2.20)

¨

T(D)⊤T(D)="Φ⊤

20

S⊤

20#"Φ0S0

01#="Φ⊤

2Φ0Φ⊤

2S0

S⊤

2Φ0S⊤

2S0#

The relation Φ⊤

0Φ0=Iis a consequence of orthogonality of the rotation

matrix Φ0. The matrix His deﬁned in the coordinate system where axes of

the system are central axes of inertia of the body, so

Analysis of movement of the BOP crane... 687

H="J0

0m#(2.21)

where Jij (ij = 1,2,3) are elements of the matrix J={Jij }deﬁned by

formulae presented inter alia in Wittbrodt et al. (2006).

2.2.1. Determination of εk,2components

For k= 1,2,3

Using (2.15), (2.18) and (2.21), after executing proper multiplications, one

obtains

e

TkH¨

e

T

⊤="0ak

00#"J0

0m#6

X

j=1

D⊤

j¨qj=

3

X

j=1

¨qj"maka⊤

j0

00#(2.22)

and next, taking into account (2.20) and (2.22)

εk,2=

3

X

j=1

¨qjtr (" IΦ⊤

0S0

S⊤

0Φ0S⊤

0S0+ 1#"maka⊤

j0

00#)=

3

X

j=1

¨qjtr{maka⊤

j}=m¨qk

(2.23)

For k= 4,5,6

In this case, again based on (2.15), (2.18) and (2.21), one can calculate

e

TkH¨

e

T

⊤="Rk0

00#"J0

0m#6

X

j=1

D⊤

j¨qj=

6

X

j=4 "RkJR⊤

j0

00#¨qj(2.24)

and then

εk,2= tr

"IΦ⊤

0S0

S⊤

0Φ0S⊤

0S0+ 1#6

X

j=4 "RkJR⊤

j0

00#¨qj

=

(2.25)

=

6

X

j=4

tr{RkJR⊤

j}¨qj=

6

X

j=4

tr{R⊤

jRkJ}¨qj

Finally, after making suitable multiplications R⊤

jRk, one obtains

ε4,2= (J11 +J22)¨q4−J32 ¨q5−J31 ¨q6

ε5,2=−J23 ¨q4+ (J11 +J33)¨q5−J21 ¨q6(2.26)

ε6,2=−J13 ¨q4−J12 ¨q5+ (J22 +J33 )¨q6

688 A. Urbaś et al.

2.2.2. Determination of εk,0components

Taking into consideration (2.15), (2.18), (2.21) and (2.20), and repeating

analogical calculations like above, one can deﬁne:

— for k= 1,2,3

εk,0=m(S⊤

2Φ0)k+

3

X

j=1

qjm(Φ⊤

2Φ0)j,k (2.27)

— for k= 4,5,6

εk,0= tr {Φ⊤

2Φ0RkJ}+

6

X

j=4

qjtr{Φ⊤

2Φ0RkJR⊤

j}(2.28)

2.2.3. Determination of εk,1components

Proceeding analogically as for the components εk,2and εk,0, one obtains:

— for k= 1,2,3

εk,1=

3

X

j=1

˙qjm(Φ⊤

1Φ0)j,k (2.29)

— for k= 4,5,6

εk,1=

6

X

j=4

˙qjtr (Φ⊤

1Φ0RkJR⊤

j)j,k (2.30)

The implementation of relationships (2.23), (2.26), (2.27), (2.28), (2.29) and

(2.30) in a computer program describing Lagrange operators in explicit forms

instead of general form (2.9), allows one to extremely reduce the calculation

time.

The derivatives of the potential energy are gravity forces acting on the

element of mass mand they can be presented in the form of a vector

∂Vg

∂q= [mgt(D)

31 , mgt(D)

32 , mgt(D)

33 ,0,0,0]⊤(2.31)

where qis the vector of coordinates of the frame or the load (deﬁned in (2.2)),

respectively, m– mass of the frame or the load, t(D)

31 ,t(D)

33 ,t(D)

33 – corresponding

elements of the third row of the matrix T(D).

2.3. Model of the support of BOP crane

It has been assumed that the frame of the BOP crane is supported ﬂexibly

in four points denoted as P(k)(k= 1,2,3,4). The crane is moving on a dedi-

cated rail system in the direction parallel to c

X(D)axis – Fig. 4. Additionally,

a constructional clearance can occur in the c

Y(D)direction.

Analysis of movement of the BOP crane... 689

Fig. 4. Flexible connection of the frame to the deck

The reaction force, i.e. the reaction force of the base on the frame, is

depicted by the vector

F(F)

P(k)= [F(F,x)

P(k), F (F,y)

P(k), F (F,z )

P(k)]⊤(2.32)

The F(F,z)

P(k)component can be calculated as

F(F,z)

P(k)=F(F,z)

S,P (k)+F(F,z)

T, P (k)(2.33)

where F(F,z)

S,P (k)is the stiﬀness force and F(F,z )

T, P (k)the damping force.

The stiﬀness and damping forces are determined by relations

F(F,z)

S,P (k)=−cz

P(k)δz

P(k)∆zP(k)F(F,z)

T, P (k)=−bz

P(k)δz

P(k)∆˙zP(k)(2.34)

where

δz

P(k)=(1 when ∆zP(k)<0

0 when ∆zP(k)0

and ∆zP(k)=z(D)

P(k)−z(D,0)

P(k), where z(D,0)

P(k)= 0, and z(D)

P(k)is the zcoordinate

of the point P(k)in the system {D},∆˙zP(k)= ˙z(D)

P(k),cz

P(k),bz

P(k)– stiﬀness

and damping coeﬃcients of the connection in the b

Z(D)direction, respectively.

In the case of the component F(F,y)

P(k), the possibility of occurrence of cle-

arance in the anti-lift system is taken into account. To model the clearance,

two spring-damping elements acting in the c

Y(D)direction are introduced.

One is the type Relement and the second – type L. They are shown in Fig. 4.

690 A. Urbaś et al.

The characteristics of force in the spring-damping elements are presented in

Fig. 5. They are not linear because of computer implementation (avoidance of

discontinuous derivative of the force).

Fig. 5. Characteristics of the force Fin the spring-damping element (∆– clearance,

d– deﬂection of the element), (a) type R, (b) type L

The function describing characteristics given in Fig. 5 can be deﬁned as:

— for the element type R

F=

c(d−∆) when d > a∆

FRwhen 0 ¬d¬a∆

0 when d < 0

(2.35)

— for the element type L

F=

0 when d > 0

FLwhen −a∆ ¬d¬0

c(d+∆) when d¬ −a∆

(2.36)

It has been assumed that the functions FRand FLhave the form

F(R,L)=αd2eβd (2.37)

which guarantees the fulﬁlment of the following conditions

F(R,L)(0) = 0 F(R,L)′(0) = 0 (2.38)

The parameters α,βfrom (2.37) can be obtained by using conditions:

— for the element type R

FR(a∆) = c(a−1)∆=FR

0(2.39)

FR′(a∆) = c=F′

0

Analysis of movement of the BOP crane... 691

— for the element type L

FL(−a∆) = −c(a−1)∆=FL

0(2.40)

FL′(−a∆) = c=F′

0

Finally, the component F(F,y)

P(k)from (2.32) can be presented as:

— for the element type R

F(F,y)

P(k)=

−cy,R

P(k)(∆yP(k)−∆y,R

P(k))−by,R

P(k)∆˙yP(k)when ∆yP(k)> a∆y,R

P(k)

FRwhen 0 ¬∆yP(k)¬a∆y,R

P(k)

0 when ∆yP(k)<0

(2.41)

— for the element type L

F(F,y)

P(k)=

−cy,L

P(k)(∆yP(k)−∆y,L

P(k))−by,L

P(k)∆˙yP(k)when ∆yP(k)>0

FLwhen −a∆y,L

P(k)¬∆yP(k)¬0

0 when ∆yP(k)¬ −a∆y,L

P(k)

(2.42)

where ∆yP(k)=y(D)

P(k)−y(D,0)

P(k),∆˙yP(k)= ˙y(D)

P(k),cy,L

P(k),cy,R

P(k),by,L

P(k),by,R

P(k)is

the stiﬀness and damping coeﬃcients of the connection in the c

Y(D)direction,

respectively.

The component F(F,x)

P(k)from (2.32) can be expressed by the formula

F(F,x)

P(k)=−sgn(υ(D,x)

P(k))S(F,x)

P(k)(F(F,y)

P(k), F (F,z )

P(k)) (2.43)

where S(F,x)

P(k)is the resisting force caused by rolling or sliding friction, υ(D,x)

P(k)–

component xof the velocity of the point P(k)in the coordinate system {D}.

After calculating suitable coordinates and velocity of points of the support,

the generalised force of ﬂexible connection of the frame and deck can be written

as

Q(F)

P(k)=U(F)

P(k)

⊤F(F)

P(k)(2.44)

where

U(F)

P(k)=

1 0 0 −y(F)

P(k)z(F)

P(k)0

0 1 0 x(F)

P(k)0−z(F)

P(k)

0 0 1 0 −x(F)

P(k)y(F)

P(k)

692 A. Urbaś et al.

Generalising relation (2.44) to four supports, one can obtain

Q(F)

p=

4

X

k=1

Q(F)

P(k)=

4

X

k=1

U(F)

P(k)

⊤F(F)

P(k)(2.45)

2.4. Modelling of the clearance between the load and guides

The guides have been replaced by spring-damping elements (sde) with a

clearance (sde E(k,p)) that limited the movement of the load in the c

X(D)and

c

Y(D)directions, see Fig. 6. It has been assumed that the load can contact with

guides only along its edges and the number of spring-damping elements can

be diﬀerent for each edge. The manner of calculation of stiﬀness and damping

forces coming from each side is analogical to that one presented in Section 2.3.

Additionally, one has to determine equivalent coeﬃcients of ﬂexibility of the

elements modelling the guides. Suitable calculations have been executed by

means of the ﬁnite elements method. They will be presented in details in the

doctoral thesis by Urbaś.

Fig. 6. Load and spring-damping elements with clearance

2.5. The drive of the travel system

The input in the drive of the travel system has been modelled in two ways:

kinematic input via a spring-damping element (ﬂexible) and force input (rigid)

– Fig. 7. It has been assumed that the drive acts in the points P(1) and P(4).

Analysis of movement of the BOP crane... 693

Fig. 7. The travel system of the crane (a) ﬂexible, (b) rigid

2.5.1. Kinematic input

In this case, the potential energy of elastic deformation and the dissipation

function of the drive system can be calculated as

V(i)

t=1

2cx

P(i)[δx

P(i)(t)−x(D)

P(i)]2D(i)

t=1

2bx

P(i)[˙

δx

P(i)(t)−˙x(D)

P(i)]2(2.46)

for i= 1,4, where δx

P(1) (t), δx

P(4) (t) is the assumed displacement (kinematic

input), cx

P(i),bx

P(i)– stiﬀness and damping coeﬃcients of the drive of the travel

system, respectively.

After determining the coordinates x(D)

P(i)as functions of elements of the

vector q(F), one should place suitable derivatives in the ﬁrst six equations of

motion of system (2.3).

2.5.2. Force input

In this case, the unknown forces F(F)

P(1) ,F(F)

P(4) and suitable constraint equ-

ations have been introduced. Generally, the forces can be placed on the left-

hand side of the equations of motion of the system, which can be written

as

A¨

q−DF=f(2.47)

where

D=

0 0

U(F)

P(1),1

⊤U(F)

P(4),1

⊤

F="F(F)

P(1)

F(F)

P(4) #

U(F)

P(1),1

⊤,U(F)

P(4),1

⊤– the ﬁrst rows of matrices from (2.44).

694 A. Urbaś et al.

In the analysed problem the constraint equations have the form

x(D)

P(1) =δ(D)

P(1) (t)x(D)

P(4) =δ(D)

P(4) (t) (2.48)

Because of the convenience of computer implementation, they can be presented

in the matrix and acceleration form

D⊤¨

q=¨

δ=

δ(D)

P(1) (t)

δ(D)

P(4) (t)

(2.49)

2.6. Energy of elastic deformation and energy dissipation in ropes

The load is suspended on two ropes, so their energy of elastic deformation

and energy dissipation can be written as

V(p)

r=

2

X

p=11

2c(p)

rδ(p)

r[∆l(p)

ApBp]2D(p)

r=

2

X

p=11

2b(p)

rδ(p)

r[∆˙

l(p)

ApBp]2

(2.50)

where c(p)

r,d(p)

rare stiﬀness and damping coeﬃcients of the rope p, respecti-

vely, ∆l(p)

ApBp– deformation of the rope p, and

δ(p)

r=

0 when ∆l(p)

ApBp¬0

1 when ∆l(p)

ApBp>0

Derivation of formulae deﬁning suitable derivatives of relations (2.50) was

presented by Urbaś and Wojciech (2008).

3. Numerical calculations

Taking into consideration all components of Lagrange equations (2.3), a system

of diﬀerential equations has been obtained

A¨

q=f(t, q,˙

q) (3.1)

where A=A(t, q) is the mass matrix.

In the case when the input in the drive of the travel system has been

modelled as the force input, equations (3.1) have to be completed by con-

straint equations (2.49) and the equations of motion have to be presented in

Analysis of movement of the BOP crane... 695

form (2.47). The fourth-order Runge-Kutta method has been used to solve the

system of equations.

Masses and geometrical parameters have been chosen based upon technical

documentation (2007). The main parameters are: mass of the frame 73955 kg,

mass of the load 550 000 kg, dimension of the load 4.8×5.5×20.3 m. Data

concerning the motion of the deck that should be taken into calculation are also

provided in Technical documentation (2007), see Table 1. In our simulations,

the operational conditions have been assumed.

Table 1. Deck motion due to waves

Condition Heading Heave Pitch Roll

[deg] [m] [rad] [rad]

Z1 0 0.1343 0.0023 0

Z2 45 0.1115 0.0008 0.0023

Z3 90 0.1140 0 0.0045

Calculations for the BOP crane that does not move on the deck have been

denoted according to Table 2. The same denotations are used in the graphs.

Table 2. Analysed load cases – gantry crane not moving

Symbol Description Clearance Deck motion

Z1-M0-C0 No clearance

in travel

system

0 Z1

Z2-M0-C0 0 Z2

Z3-M0-C0 0 Z3

Z1-M0-C1 With

clearance in

travel system

1cm Z1

Z2-M0-C1 1cm Z2

Z3-M0-C1 1cm Z3

Figure 8 presents time courses of the general coordinates ψ(L)of the load

of the BOP crane with and without clearance in the travel system.

The inﬂuence of clearance in the travel system on the reaction forces in

the support system (leg no. 1) is shown in Fig. 9. The deck motions Z2 and Z3

are taken into consideration.

The biggest inﬂuence of clearance in the travel system on the dynamics

of the BOP crane occurs for input Z3, so this input is taken into account for

the next calculations. The inﬂuence of clearance in the travel system on the

reaction forces will be analysed. The travel velocity is deﬁned by the relation

v=(3at2−4bt3when t¬Tr

vnwhen t > Tr

(3.2)

696 A. Urbaś et al.

Fig. 8. Inﬂuence of clearance on the roll angle of BOP

Fig. 9. Lateral reaction in leg no. 1; (a) – in sea conditions Z2, (b) – in sea

conditions Z3

where vn= 3 m/min, Tr= 6 s, a=vn/T 2

r,b=vn/(2T3

r).

The left graph in Fig. 10 shows the drive force on the ﬁrst gear (support

no. 1) for the kinematic and force input and for the case when no clearance

occurs in the system. The right graph presents the inﬂuence of clearance on

the drive force. One can notice that the clearance causes the occurrence of

signiﬁcant dynamic forces of short duration.

Required courses of drive forces acting on legs 1 and 4 realising the esta-

blished travel of the crane are presented in Fig. 11. Kinematic and force inputs

have been simulated.

Analysis of movement of the BOP crane... 697

Fig. 10. Drive force on gear no. 1 for the ﬂexible and rigid model applied (a),

inﬂuence of diﬀerent clearances on the drive force (b)

Fig. 11. Rigid and ﬂexible drive, leg no. 1 (a) and no. 4 (b)

The obtained results (values of forces) for the assumed parameters are simi-

lar, but for the kinematic input peak values are bigger. These values depend on

the stiﬀness and damping coeﬃcients taken into account during calculations.

For graphs in Fig. 12, the additional clearance of 2 cm in the supporting

system for the undrived legs (i.e. 2 and 3) has been taken into consideration.

698 A. Urbaś et al.

Fig. 12. Rigid and ﬂex drive on leg no. 1 (a) and no. 4 (b) – double size clearance in

legs 2 and 3

The obtained values of dynamic forces prove the signiﬁcant inﬂuence of

clearance on dynamic load of the drive system, the track-way and the whole

structure of the crane.

4. Summary

The mathematical models and the computer programs presented in the pa-

per enable one to execute dynamical analysis of BOP cranes mounted on the

ﬂoating base. They can be useful in calculating dynamic loads, dimensioning

bearing elements of the crane and the track-way. They enable determination

of static and dynamic loads by simulation for arbitrarily chosen sea waving

conditions.

Special attention has been paid to the inﬂuence of clearance in the drive of

the crane travel system on characteristics of forces acting in selected elements

of the structure. It can be mentioned that for a limited value of clearances

they have slight inﬂuence on the drive forces. Practically, it is not possible to

construct and maintain the track-way with ideal geometry. Dynamic analysis

Analysis of movement of the BOP crane... 699

is a good instrument for calculating forces in the system, especially when

a bigger clearance appears. It allows one to determine whether the dynamic

loads do not transgress the constructional assumptions under given sea waving

conditions.

References

1. Adamiec-Wójcik I., Maczyński A., Wojciech S., 2008, Zastosowanie me-

tody przekształceń jednorodnych w modelowaniu dynamiki urządzeń oﬀshore,

WKŁ, Warsaw

2. Balachandran B., Li Y.Y., Fang C.C., 1999, A mechanical ﬁlter concept

for control of non-linear crane-load oscillations, Journal of Sound and Vibra-

tions,228, 651-682

3. Birkeland O., 1998, Knuckle boom combination crane, 3rd International Of-

fshore Cranes Conference Oﬀshore Cranes – Speciﬁcation, Design, Refurbish-

ment, Safe Operation and Maintenance, Stavanger, Norway

4. Cosstick H., 1996, The sway today, Container Management, 42-43

5. Das S.N., Das S.K., 2005, Mathematical model for coupled roll and yaw

motions of a ﬂoating body in regular waves under resonant and non-resonant

conditions, Applied Mathematical Modelling,29, 19-34

6. Driscoll F.R., Lueck R., Nahon G.M., 2000, Development and validation

of a lumped-mass dynamics model of a deep-sea ROV system, Applied Ocean

Research,22, 169-18

7. Ellermann K., Kreuzer E., 2003, Nonlinear dynamics in the motion of

ﬂoating cranes, Multibody System Dynamics,9, 4, 377-387

8. Fałat P., 2004, Dynamic Analysis of a Sea Crane of an A-frame Type, Ph.D.

Thesis, Bielsko-Biała [in Polish]

9. Fossen T.I., 1994, Guidance and Control of Ocean Vehicles, John Wiley and

Sons, Chichester, England

10. Handbook of Oﬀshore Engineering, 2005, Edited by Chakrabarti S., Elsevier

11. Jordan M.A., Bustamante J.L., 2007, Numerical stability analysis and con-

trol of umbilical-ROV systems in one-degree-of-freedom taut-slack condition,

Nonlinear Dynamic,49, 163-191

12. Li Y.Y., Balachandran B., 2001, Analytical study of a system with a me-

chanical ﬁlter, Journal of Sound and Vibration,247, 633-653

13. Maczyński A., 2005, Positioning and Stabilization of the Load for Jib Cranes,

University of Bielsko-Biala [in Polish]

700 A. Urbaś et al.

14. Maczyński A., Wojciech S., 2003, Dynamics of a mobile crane and opti-

misation of the slewing motion of its upper structure, Nonlinear Dynamic,32,

259-290

15. Maczyński A., Wojciech S., 2007, Stabilization of load position for of-

fshore cranes, Twelfth World Congress in Mechanism and Machine Scien-

ce, Besancon, France [electronic document] s. 1-6 [on-line (03.04.2008)

http://130.15.85.212/proceedings/ WorldCongress07/articles/article cd.htm]

16. Maczyński A., Wojciech S., 2008, Wpływ stabilizacji położenia ładunku

żurawia typu oﬀshore na zjawisko odciążania i dociążania liny nośnej, Teoria

Maszyn i Mechanizmów, Wydawnictwo Akademii Techniczno-Humanistycznej

w Bielsku-Białej, 135-143

17. Masoud Y.N., 2000, A Control System for the Reduction of Cargo Pendulation

of Ship-Mounted Cranes, Virginia Polytechnic Institute and State University,

Doctoral Thesis, Blacksburg, Virginia, USA

18. Pedrazzi C., Barbieri G., 1998, LARSC: Launch and recovery smart crane

for naval ROV handling, 13th European ADAMS Users’ Conference, Paris,

19. Technical Documentation for BOP, 2007, PROTEA, Gdańsk-Olesno

20. Urbaś A., Wojciech S., 2008, Dynamic analysis of the gantry crane used

transporting BOP, Proc. VII Conference Telematics, Logistics and Transport

Safety, 311-323

21. Urbaś A., Wojciech S., 2009, Dynamic analysis of the gantry crane used for

transporting BOP, Modeling, Simulation and Control of Nonlinear Engineering

Dynamical Systems: State-of-the-Art, Perspectives and Applications, Springer,

49-59

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xible Multibody Systems, Springer

The research has been ﬁnanced by the ministry of Science and Higher Education

(project No. N50 2 464934).

Analiza jazdy żurawia BOP w warunkach falowania morskiego

Streszczenie

W pracy przedstawiono model matematyczny umożliwiający analizę dynamiczną

żurawia BOP w warunkach falowania morskiego. Żuraw BOP stanowi rodzaj suwnicy

bramowej. Instalowany jest na platformach wydobywczych i przeznaczony do trans-

portu zespołu zaworów BOP (Blowout Preventor). Do najważniejszych czynników

Analysis of movement of the BOP crane... 701

wpływających na jego dynamikę należą: ruch podstawy wywołany falowaniem morza,

luzy występujące w połączeniu torowiska z suwnicą, uderzenia ładunku o prowadnice

oraz znaczna masa ładunku. W celu przeprowadzenia analiz dynamicznych opraco-

wano model obliczeniowy urządzenia uwzględniający powyższe czynniki. Równania

ruchu układu sformułowano przy zastosowaniu metody transformacji jednorodnych.

W celu poprawy efektywności numerycznej modelu, wykonano przekształcenia umoż-

liwiające przedstawienie równań w sposób jawny. Wymuszenie w układzie napędu

jazdy modelowano dwoma sposobami: jako wymuszenie kinematyczne poprzez ele-

ment sprężysto-tłumiący oraz jako wymuszenie siłowe. Zaprezentowano przykładowe

wyniki obliczeń numerycznych.

Manuscript received September 30, 2009; accepted for print January 25, 2010

- CitationsCitations5
- ReferencesReferences19

- [Show abstract] [Hide abstract]
**ABSTRACT:**Offshore pedestal cranes are devices installed on offshore platforms or vessels. A characteristic feature of any floating object is the significant movement caused by sea waving. These movements cause that the offshore cranes are exposed to dynamic loads reasonably higher than structures of similar operational parameters but operated on land. Therefore, they are equipped with special systems for overload reduction. One of them is the shock absorber. The paper presents a mathematical model of an offshore pedestal crane with a shock absorber. Results of numerical simulations are presented to assess the effectiveness of the shock absorber in conditions when large dynamic overloads occur. - [Show abstract] [Hide abstract]
**ABSTRACT:**In offshore pedestal cranes one may distinguish three components of considerable length: a pedestal, a boom and a frame present in some designs. It is often necessary in dynamical analyses to take into account their flexibility. A convenient and efficient method for modelling them is the rigid finite element method in a modified form. The rigid finite element method allows us to take into account the flexibility of the beam system in selected directions while introducing a relatively small number of additional degrees of freedom to the system. This paper presents a method for modelling the pedestal, the frame and the boom of an offshore column crane, treating each of these components in a slightly different way. A custom approach is applied to the pedestal, using rigid finite elements of variable length. Results of sample numeric computations are included. - [Show abstract] [Hide abstract]
**ABSTRACT:**BOP (Blow-out preventer)Gantry Crane is a crane which is used to move BOP stack saved in Drilling Platform to BOP trolley. However, such offshore plant equipment has not been developed in Korea but a lot of loyalty has to be paid to overseas companies. Thus, domestic production of it is necessary. In this research, ANSYS was used to apply into interpretation of rolling and pitching due to wave, wind load due to Drillship, and the crane's self-weight and structural interpretation was implemented. Moreover, Simulation X was used for control system design, and sub system was modeled and hydraulic system interpretation was implemented.

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