In the paper the dynamic analysis of a gantry crane used for transporting of BOP (BlowOut Preventer) is presented. The crane is placed on a drilling platform. Sea waves cause motion of the platform and the load. Description of such systems can be used in the design process of control systems which allows us to compensate waving. Homogenous transformations and joint coordinates are used to describe behavior of the system. Equations of motion are derived using the Lagrange equations of the second order. In the paper the results of numerical calculations are presented as well

Andrzej URBAŚ

Stanisław WOJCIECH

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TRANSPORT PROBLEMS                                                                                                               2008 PROBLEMY TRANSPORTU                                                                             Tom 3 Zeszyt 4 Część 2     Andrzej URBAŚ*, Stanisław WOJCIECH University of Bielsko-Biała Willowa 2, 43-309 Bielsko-Biała, Poland *Corresponding author. E-mails: aurbas@ath.bielsko.pl    DYNAMIC ANALYSIS OF THE GANTRY CRANE USED FOR TRANSPORTING BOP  Summary. In the paper the dynamic analysis of a gantry crane used for transporting of BOP (BlowOut Preventer) is presented. The crane is placed on a drilling platform. Sea waves cause motion of the platform and the load. Description of such systems can be used in the design process of control systems which allows us to compensate waving. Homogenous transformations and joint coordinates are used to describe behavior of the system. Equations of motion are derived using the Lagrange equations of the second order. In the paper the results of numerical calculations are presented as well.    ANALIZA DYNAMICZNA SUWNICY BRAMOWEJ PRZEZNACZONEJ DO TRANSPORTU BOP  Streszczenie.  W  pracy  przedstawiono  model  dynamiczny  suwnicy  bramowej przeznaczonej do transportu ładunku zwanego BOP (BlowOut Preventer). Suwnice tego typu  montowane  są  na  statkach  lub  platformach  wiertniczych.  Falowanie  morza powoduje  niepoŜądane  ruchy  ładunku,  wpływające  na  pracę  suwnicy.  Podjęto  zatem próbę  opisu  dynamiki  układu,  która  umoŜliwiła  zbadanie  wpływu  falowania  na zachowanie  układu.  Do  opisu  zastosowano  formalizm  transformacji  jednorodnych i współrzędne  złączowe.  Równania  ruchu  układu  wyprowadzono  stosując  równania Lagrange’a II rodzaju. Przedstawiono wyniki symulacji numerycznych.   1. INTRODUCTION  Dynamic analysis of offshore systems mounted on platforms or vessels is especially difficult because it is necessary to consider phenomena connected with sea waves [1,3,5]. The waving causes additional motion of the base (platform) and thus there are impulse forces in the system which should be taken into account in the design process not only in stress analysis but also in design of drive and control systems. In the paper homogenous transformations and joint coordinates are used to describe dynamic analysis of the gantry crane [2,6]. Equations of motion are formulated using the Lagrange equations of the second order.  2. MATHEMATICAL MODEL OF THE SYSTEM  The gantry crane (Fig. 1) is considered as a system with 12DoF (Fig. 2). The frame {F} is treated as a rigid body (6DoF) connected with platform {D} by spring-damping elements (SDE). The load is 86                                                                                                                            A. Urbaś, S. Wojciech  also treated as a rigid body with 6DoF with respect to platform {D}. The load is connected with the frame by means of two flexible ropes and additionally its motion in ) ( ˆ F X  and ) ( ˆ F Y  directions is limited by guides.    a)    b)  Fig.  1.  a) The  gantry  crane  used  for  transporting  BOP  –  PROTEA  Gdańsk-Olesno,  b) The  BOP  (Blowout Preventer) system  Rys. 1. a) Suwnica bramowa do transportu BOP – PROTEA Gdańsk-Olesno, b) Zestaw zaworów BOP Dynamic analysis of the gantry crane…    87    ) ( ˆ D X) ( ˆ D Y) ( ˆ L Z} {F1 B1 A2 A2 B) ( ˆ L Y) ( ˆ F Y) ( ˆ L X} {L) ( ˆ F Z) (F q) (F y) (L q) ( ˆ F X) (F j) ( ˆ D Z } {D) (L y) (D y) (D q) (L j) (D j) (F C) (L C) , , () ( ) ( ) ( ) ( FCFCFCF z y x C) , , () ( ) ( ) ( ) ( LCLCLCL z y x Cload (BOP) {L} frame {F} deck {D} ) 0 ( ˆ X) 0 ( ˆ Z} 0 {) 0 ( ˆ Yguides heading angle a) (D y) (D x) (D z  Fig. 2. Model of the gantry crane Rys. 2. Model suwnicy bramowej  It is assumed that the motion of the platform is known. Consequently the position of coordinate system {D} with respect to global coordinate systems {0} is known. The position of coordinate system {D} with respect to {0} is defined by:           ===t z zt y yt x xD DD DD D() () () ( ) () ( ) () ( ) ( the coordinates of the origin of coordinate system {D} in {0},  (1.1) 88                                                                                                                            A. Urbaś, S. Wojciech   ===tttD DD DD D() () () ( ) () ( ) () ( ) (j jq qy y the Euler angles [2] which describe any possible orientation of {D} in {0}.  (1.2) The motion of frame {F} and load {L} with respect to platform {D} are described by independent parameters which are components of the vectors:   [ ]T F F F F F F F z y x) ( ) ( ) ( ) ( ) ( ) ( ) ( j q y = q ,  (2.1)   [ ]T L L L L L L L z y x) ( ) ( ) ( ) ( ) ( ) ( ) ( j q y = q .  (2.2) It is assumed that frame {F} is connected with the platform by means of spring-damping elements with large values of stiffness and damping coefficients. Because the motion of load {L} is limited by means of guides, angles ) ( ) ( ) ( ) ( ) ( ) ( , , , , ,L L L F F F y q j y q j  are small and the transformation matrices from coordinate systems {F} and {L} to {D} can be presented in the linearised form [5,1]:  ---=1 0 0 0111~) ( ) ( ) () ( ) ( ) () ( ) ( ) () (F F FF F FF F FFzyxj qj yq yB ,  (3.1)  ---=1 0 0 0111~) ( ) ( ) () ( ) ( ) () ( ) ( ) () (L L LL L LL L LLzyxj qj yq yB .  (3.2) The transformation matrices from local coordinate systems {F} and {L} to global coordinate system {0} can be writen in the form:   ) (~ ) ( ) ( ) ( ) ( F F D F q B A B = ,  (4.1)   ) (~ ) ( ) ( ) ( ) ( L L D L q B A B = ,  (4.2) where -- ++ -=1 0 0 0) () ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) () (D D D D D DD D D D D D D D D D D D DD D D D D D D D D D D D DDz c c s c sy s c c s s c c s s s c sx s s c s c c s s s c c ctj q j q qj y j q y j y j q y q yj y j q y j y j q y q yA .   The equations of motion are derived using the Lagrange equations of the second order:   kk k k kQqDqVqEqEdtd=¶¶+¶¶+¶¶-¶¶& &,       12 , , 1K = k ,  (5) where: E – is the kinetic energy of the system, V - is the potential energy of gravity forces, D - is the dissipation of energy of the system,  k Q - are non-potential generalised forces,  k q , k q & - are generalised coordinates and velocities, respectively – the componets of the vectors (2). The kinetic energy of the system, the potential energy of gravity forces and its derivatives can be calculated as it has been shown in [5].    Dynamic analysis of the gantry crane…    89  Additionally we have to take into account:  –  the energy of spring deformation of the ropes, –  the energy of spring deformation of the guides, –  reaction forces of rigid or flexible supports. Fig. 3 presents connection of the load and ropes. The load is hoisted by means of two barrel systems. In mathematical model of the gantry crane, the hoist system is reduced to two flexible ropes.      Fig. 3. Hoist system Rys. 3. Układ zawieszenia ładunku  The motion of the load is limited by means of four guides which positions are set up by hydraulic cylinders (Fig. 4). 90                                                                                                                            A. Urbaś, S. Wojciech   Fig. 4. Guide system Rys. 4. Układ prowadnic  It is assumed that the load (rectangular prism) can be in contact with the guides only along lines ) ( ) ( ,k k B A , where  4 , 3 , 2 , 1 = k  (Fig. 5a). The guides are modelled as spring-damping elements with backlash  ) sde () , ( p k E , which limits the motion of the load in ) ( ˆ D X  and ) ( ˆ D Y  directions (Fig. 5b). ) 1 ( A) 2 ( A) 3 ( A) 4 ( A) ( ˆ L Z) 1 ( B) 2 ( B) 4 ( B) 3 ( BC) ( ˆ L Y) ( ˆ L X Fig. 5a. Contact lines ) ( ) ( ,k k B A  between the load and the guides Rys. 5a. Linie styku ) ( ) ( ,k k B A  ładunku z prowadnicami Dynamic analysis of the gantry crane…    91   ) (k A) ( ˆ D Z} {D) (k BxEp k ) , ( DyEp k ) , ( DyEp k b ) , (yEp k c ) , (xEp k c ) , (xEp k b ) , () ( ˆ D Y) ( ˆ D X) , ( p k E b) Fig. 5b. Spring-damping element with backlash Rys. 5b. Element spręŜysto-tłumiący z luzem   Fig. 6 presents the way of connecting the gantry to the deck. The system is supported on the rigid rails. The motion in ) ( ˆ D X  direction is realized by system of 33 rollers (15 in contact). The motion in ) ( ˆ D Y  direction is limited by system of four pressure rollers on each leg.  92                                                                                                                            A. Urbaś, S. Wojciech    Fig. 6. Connecting the gantry crane to the deck Rys. 6. Posadowienie suwnicy BOP  The gantry crane is also protected by antylift system which limits the motion of the system in vertical direction () ( ˆ D Z  axis) (Fig. 7).       Fig. 7. Antylift system Rys. 7. Układ zabezpieczający  Fig. 8 presents the model of flexible connection of the frame legs with the deck () (k P  point).  Dynamic analysis of the gantry crane…    93    ) ( ˆ D Z) ( ˆ D Y) ( ˆ D X} {D) , , () ( ) ( ) ( ) () ( ) ( ) (FPFPFPkk k k z y x P) () (DPk x) () (DPk yL yPk,) ( DR yPk,) ( DzPk b ) (zPk c ) (R yPk c,) (R yPk b,) (L yPk c,) (L yPk b,) ( Fig. 8. Flexible connection of the frame to the platform Rys. 8. Podatne mocowanie ramy do pokładu  The structure of the equations of motion is different and depends on selected drive functions of the motion of the frame in ) ( ˆ D X  direction:   A. flexible drive system   f q A = & & ,  (6) B. rigid drive system  δ q Df DF q A& & & && &== -T ,  (7) where = ) () (LFqqq ,   =) 4 () 1 (xxFFF ,   A is constant matrix,   ) , , ( q q f f & t = ,   =) () (21ttddδ  is the vector describing the displacements of ) 4 ( ) 1 ( and P P  points in ) ( ˆ D X  direction. The model of dynamics of the gantry crane considered is described by 12 (eq. 6) or 14 (eq. 7) ordinary differential equations of the second order. In order to integrate the equations of motion the Runge-Kutta method with constant step-size is used. 94                                                                                                                            A. Urbaś, S. Wojciech  Having assumed  0 ; 0 º º q q & & &  in eq.6 and eq. 7 the vector q, reaction forces in ropes and in SDEs can be calculated. In this case the system is described by the set of nonlinear equations:   0 ) , ( = q f t ,  (8) The dependence f versus t is caused by expressions connected with the base motion of the deck. Equations (8) are solved using the Newton iteration method.   3. NUMERICAL CALCULATIONS  For the mathematical model of the gantry crane the computer program has been developed. The following input data are taken from the technical documentation [4]. – the mass of the frame and the load:  kg 955 73) ( =F m , kg 000 550) ( =L m , – dimensions of the load  m 3 . 20 5 . 5 8 . 4 ´ ´ .   In table 1 conditions of work for BOP are presented [4]. It is assumed that the motion of the deck is described  by  heave,  pitch  and  roll  motions  which  are  written  as  follows:  0) ( ) ( ) ( = = =D D D y x y , ) / 2 sin( 3) (0) ( T t a z zD D p + = ,  ) / 2 sin( 5) ( T t aD p q = ,  ) / 2 sin( 6) ( T t aD p j =   where  m 36) (0 =D z ,  s 10 = T , 6 5 3 , , a a a  are amplitudes depending on heading and weather conditions. Table 1 Conditions of work for BOP OPERATIONAL  SURVIVAL HEADING a  heave ) (D z  3 a  pitch ) (D q  5 a  roll ) (D j  6 a  heave ) (D z  3 a  pitch ) (D q  5 a  roll ) (D j   6 a  o 0   0.1343  0.0023  0  0.4458  0.0061  0 o 45   0.1115  0.0008  0.0023  0.3521  0.0023  0.0077 o 90   0.1140  0  0.0045  0.3724  0  0.0138  Fig. 9a presents the comparison of z component of forces in ) 1 ( P  point and Fig. 9b presents the elongation of rope ) 1 (1 1B A l D . In both cases it is asumed that only heave motion of the platform and the heading angle equal to o 0 .  Dynamic analysis of the gantry crane…    95   -1.60E+06-1.59E+06-1.58E+06-1.57E+06-1.56E+06-1.55E+06-1.54E+06-1.53E+06-1.52E+060 2 4 6 8 10 12 14 16 18 20       Rz[1]_nor       Rz[1]_gr  operational survival ] N [ ) 1 (zP F ] s [ t a)  0.1260.1270.1280.1290.130.1310.1320 2 4 6 8 10 12 14 16 18 20         dl2_nor         dl2_gr] m [) 1 (1 1B A l D ] s [ t  operational survival  b)  Fig. 9. The influence of working conditions on: a) forces in connection point ) 1 ( P , b) the elongation of rope 1 Rys. 9. Wpływ warunków pracy na: a) wartość siły w podporze ) 1 ( P , b) wydłuŜenie liny 1   Fig. 10 presents z component of forces at points ) 1 ( P  and ) 4 ( P . The results have been obtained for heading angle equal to o 45  in operational conditions of work.   96                                                                                                                            A. Urbaś, S. Wojciech   -1.70E+06-1.65E+06-1.60E+06-1.55E+06-1.50E+06-1.45E+06-1.40E+060 2 4 6 8 10 12 14 16 18 20       Rz[1]       Rz[4]] N [z F ] s [ t ) 1 ( point P) 4 ( point P  Fig. 10. Reaction forces in ) 1 ( P  and ) 4 ( P  points  Rys. 10. Siły reakcji w punktach ) 1 ( P  i ) 4 ( P    The  influence  of  backlash xE p k ) , ( D   and yE p k ) , ( D   on z  component  of  forces  at  point ) 1 ( P   can  be observed in Fig. 11. It is assumed that heading angle equals to o 90 in operational conditions of work and only pitch motion of the platform is taken into account.   -1.66E+06-1.64E+06-1.62E+06-1.60E+06-1.58E+06-1.56E+06-1.54E+06-1.52E+06-1.50E+06-1.48E+06-1.46E+060 2 4 6 8 10 12 14 16 18 20       Rz[1]_luz0       Rz[1]_luz0_01] N [z F ] s [ t  0 ) , ( ) , ( = D = DyExE p k p km 01 . 0 ) , ( ) , ( = D = DyExE p k p k Fig. 11a. The influence of backlash in SDE on reaction force (component z) in point ) 1 ( P  Rys. 11a. Wpływ luzu w EST na składową z siły reakcji w punkcie ) 1 ( P  Dynamic analysis of the gantry crane…    97   -1.70E+06-1.65E+06-1.60E+06-1.55E+06-1.50E+06-1.45E+060 2 4 6 8 10 12 14 16 18 20       Rz[1]_luz0       Rz[1]_luz0_02] N [z F ] s [ t  0 ) , ( ) , ( = D = DyExE p k p km 02 . 0 ) , ( ) , ( = D = DyExE p k p k  -1.70E+06-1.65E+06-1.60E+06-1.55E+06-1.50E+06-1.45E+06-1.40E+060 2 4 6 8 10 12 14 16 18 20       Rz[1]_luz0       Rz[1]_luz0_03] N [z F ] s [ t  0 ) , ( ) , ( = D = DyExE p k p km 03 . 0 ) , ( ) , ( = D = DyExE p k p k Fig. 11b,c. The influence of backlash in SDE on reaction force (component z) in point ) 1 ( P  Rys. 11b,c. Wpływ luzu w EST na składową z siły reakcji w punkcie ) 1 ( P   Fig. 12 presents the influence of backlash xE p k ) , ( D  and yE p k ) , ( D  on the sum of forces acting between the guides and the load. It is assumed that the heading angle equals to o 90 and pitch motion of the platform in operational conditions of work is taken into account. 98                                                                                                                            A. Urbaś, S. Wojciech    -2.50E+05-2.00E+05-1.50E+05-1.00E+05-5.00E+040.00E+005.00E+041.00E+051.50E+052.00E+052.50E+050 2 4 6 8 10 12 14 16 18 20suma_Y_12_luz0suma_Y34_luz0suma_Y_12_luz0_01suma_Y34_luz_0_01∑ ] N [ ) , (yE p k F] s [ t  0 ) , ( ) , ( = D = DyExE p k p km 01 . 0 ) , ( ) , ( = D = DyExE p k p klines 1 and 2 lines 3 and 4 lines 1 and 2 0 ) , ( ) , ( = D = DyExE p k p klines 3and 4  m 01 . 0 ) , ( ) , ( = D = DyExE p k p ka  -2.50E+05-2.00E+05-1.50E+05-1.00E+05-5.00E+040.00E+005.00E+041.00E+051.50E+052.00E+052.50E+050 2 4 6 8 10 12 14 16 18 20suma_Y_12_luz0suma_Y34_luz0suma_Y12_luz_0_02suma_Y34_luz_0_02∑ ] N [ ) , (yE p k F] s [ t  0 ) , ( ) , ( = D = DyExE p k p km 02 . 0 ) , ( ) , ( = D = DyExE p k p klines1 and 2 lines 3 and 4 lines 1 and 2 0 ) , ( ) , ( = D = DyExE p k p klines 3 and 4  m 02 . 0 ) , ( ) , ( = D = DyExE p k p kb Fig. 12. The influence of backlash in SDE on the sum of forces ∑yE p k F ) , (  Rys. 12. Wpływ luzu w EST na sumaryczną wartość siły ∑yE p k F ) , (   Dynamic analysis of the gantry crane…    99   -2.50E+05-2.00E+05-1.50E+05-1.00E+05-5.00E+040.00E+005.00E+041.00E+051.50E+052.00E+052.50E+050 2 4 6 8 10 12 14 16 18 20suma_Y_12_luz0suma_Y34_luz0suma_Y12_luz_0_03suma_Y34_luz_0_03∑ ] N [ ) , (yE p k F] s [ t  0 ) , ( ) , ( = D = DyExE p k p km 03 . 0 ) , ( ) , ( = D = DyExE p k p klines 1 and 2 lines 3 and 4 lines 1 and 2 0 ) , ( ) , ( = D = DyExE p k p klines 3 and 4  m 03 . 0 ) , ( ) , ( = D = DyExE p k p k  Fig. 12. The influence of backlash in SDE on the sum of forces ∑yE p k F ) , (  Rys. 12. Wpływ luzu w EST na sumaryczną wartość siły ∑yE p k F ) , (    4. CONCLUSIONS  Numerical simulations presented prove that the dynamic analysis of off-shore systems is very important  in  the  design  process.  Impuls  forces  acting  in  the  system  should  be  included  in  stress analysis of the guides and also the frame. We are going to analyse drive systems in order to find out how  to  compensate  the  frame  and  the  load  motion  caused  by  waving.  The  numerical  calculation presented in Fig. 10 indicates that drive systems in certain conditions should work independently.   Bibliography  1.  Adamiec-Wójcik  I.,  Maczyński  A.,  Wojciech  S.:  Zastosowanie  przekształceń  jednorodnych w modelowaniu dynamiki urządzeń offshore. Wydawnictwo Komunikacji i Łączności, Warszawa, 2008. 2.  Craig  J.J.:  Wprowadzenie  do  robotyki.  Mechanika  i  sterowanie.  Wydawnictwo  Naukowo-Techniczne, Warszawa, 1995 3.  Maczyński  A.:  Pozycjonowanie  i  stabilizacja  połoŜenia  ładunku  Ŝurawi  wysięgnikowych. Wydawnictwo Akademii Techniczno-Humanistycznej, Z.14, Bielsko-Biała, 2005 4.  Technical documentation for BOP, PROTEA, Gdańsk – Olesno, 2007 5.  Urbaś A., Wojciech S.: Dynamic analysis of the gantry crane used for transporting BOP. Proc. of 9 th Conference on Dynamical Systems – Theory and Applications, Łódź, 2007. 6.  Wittbrodt E., Adamiec-Wójcik I., Wojciech S.: Dynamic of flexible multibody systems. Rigid finite element method. Springer Berlin Heidelberg New York, 2006 100                                                                                                                           A. Urbaś, S. Wojciech  The investigation presented in the paper has been supported by grant N N502 464934 founded by Polish Committee of Science.  Received 15.04.2008; accepted in revised form 07.11.2008 

Dynamic analysis of the gantry crane used for transporting BOP

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