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

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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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JOURNAL 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 s1and 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,
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
0(D)(D)0
0(D)(D)0
0 0 0 1
T(D)
5=T(D)
5(θ(D)) =
(D)0(D)0
0 1 0 0
(D)0(D)0
0 0 0 1
T(D)
6=T(D)
6(ψ(D)) =
(D)(D)0 0
(D)(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
= sin α = 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
∂qiqj
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
To=
= 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=
1ψ θ x
ψ1ϕ 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=
01 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)¨q4J32 ¨q5J31 ¨q6
ε5,2=J23 ¨q4+ (J11 +J33)¨q5J21 ¨q6(2.26)
ε6,2=J13 ¨q4J12 ¨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
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(a1)=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(a1)=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)0z(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¨
qDF=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
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=(3at24bt3when 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
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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-
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fshore Cranes Conference Oﬀshore Cranes – Speciﬁcation, Design, Refurbish-
ment, Safe Operation and Maintenance, Stavanger, Norway
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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
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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
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University of Bielsko-Biala [in Polish]
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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
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fshore cranes, Twelfth World Congress in Mechanism and Machine Scien-
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http://130.15.85.212/proceedings/ WorldCongress07/articles/article cd.htm]
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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
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for naval ROV handling, 13th European ADAMS Users’ Conference, Paris,
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transporting BOP, Proc. VII Conference Telematics, Logistics and Transport
Safety, 311-323
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transporting BOP, Modeling, Simulation and Control of Nonlinear Engineering
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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,
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
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