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Ren et al. / J Zhejiang Univ Sci A 2008 9(1):26-31 26 Dynamic response analysis of a moored crane-ship with a flexible boom * Hui-li REN , Xue-lin WANG , Yu-jin HU, Cheng-gang LI (School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China) E-mail: ; w_ Received June 12, 2007; revision accepted Aug. 5, 2007; published online Dec. 14, 2007 Abstract: The dynamic response of moored crane-ship is studied. Governing equations for the dynamic response of a crane-ship coupled with the pendulum motion of the payload are derived based on Lagranges equations. The boom is modeled based on finite element method, while the payload is modeled as a planar pendulum of point mass. The dynamic response was studied using numerical method. The calculation results show that the large-amplitude responses occur at wave periods near the natural period of the payload. Load swing angle is smaller for crane-ship with flexible boom, in comparison with rigid boom. The ship surge mo- tions have large vibrations for crane-ship with flexible boom, which were not observed for a rigid boom. The analysis identifies the significance of key parameters and reveals how the system design can be adjusted to avoid critical conditions. Key words: Dynamic response, Moored crane-ship, Finite element method, Rigid-flexible coupling dynamic model doi:10.1631/jzus.A071308 Document code: A CLC number: U615.35 INTRODUCTION Floating cranes play an important role in the offshore projects. However, the sea wave shakes the crane-ship and may excite its large movement. For heavy duty lifting, the operations of crane-ship in waves, even as the sea is relatively quiet, are often restricted by the excessive motions of the crane load. In some cases, the dynamical behavior of floating cranes is to be considered as critical with respect to the amplitudes of the motion of the ship or the load. A small disturbance of the system can cause the colli- sion between the load and the ship or other objects. In addition, the amplitude of the motion of the ship has to stay small in order to achieve the required precisely positioning and to avoid damages to the mooring system. The investigation of crane-ship dynamics has been of interest in a large number of recent researches (Dong and Han, 1993; Masoud et al., 2004; Eller- mann, 2005). A rigid massless cable and massive point load were used to model the crane load system, and the results of the computer simulation were veri- fied experimentally using a three degree-of-freedom (DOF) ship-motion simulation platform (Henry et al., 2001). The method of multiple scales is used to ana- lyze the dynamics of a cable-suspended load. The results show that a parametric excitation at twice the natural frequency leads to sudden jumps in the re- sponse as the cable is unreeled (Chin et al., 2001). Cargo pendulation control of an elastic ship-mounted crane was concerned using the Maryland Rigging system. The dynamics of the crane is described by a multi-model problem depending on the current cable length and boom luff angle. A variable-gain observer and a variable-gain controller are designed. Simula- tion and experimental results showed that the ex- pressed control strategy has a significant effect on suppressing the vibrations for different operating conditions and payload masses (Al-Sweiti and Sf- fker, 2007). As the first approach for evaluating the dynam- Journal of Zhejiang University SCIENCE A ISSN 1673-565X (Print); ISSN 1862-1775 (Online) E-mail: Corresponding author * Project supported by the National Natural Science Foundation of China (No. 50675077) and the Research Fund for the Doctoral Program of Higher Education of China (No. 20050487047) Ren et al. / J Zhejiang Univ Sci A 2008 9(1):26-31 27 ics of a crane-ship, a linear theoretical model is used. The linear differential equation of motion is derived under the assumption of small amplitudes, and does not consider occurring restoring forces. A dynamic model was established using multi-body dynamics methods and the dynamics of the crane was analyzed (Chen et al., 2002). The effects of cable reeling and unreeling on cargo pendulations were studied with the boom crane modeled as a planar pendulum and the ship as a rigid body. The results show nonlinear be- havior (Kral et al., 1996). In almost all of these studies, the flexibility of the boom is not taken into consideration. This may be reasonable for short crane booms and small pay- load-to-ship mass ratios, but for long crane booms and large payload-to-ship mass ratios, the influence of the flexibility of the crane boom cannot be ignored any more. To the authors best knowledge, until recently there is no report on the modeling of such system. The study aims to present a rigid-flexible coupling dy- namic model for the prediction of the dynamics of a moored crane-ship and its payload. The prediction is the foundation of the dynamical design and an accu- rate residual working life assessment. It is also a basis for the control of the vibrations of the system. MODEL DESCRIPTION The schematic system under investigation is shown in Fig.1. When dimensions and elastic prop- erties of the crane-ship body are considered, it is suf- ficient to take only the boom as flexible. The fol- lowing assumptions are used in the analysis of the crane-ship: (1) The motion is only in the vertical plane. The load is regarded as a point mass, and can have pen- dulum motion in the plane, but without twisting. (2) The rope is taken as a rigid rod. This as- sumption is valid as long as the oscillations of the load in the vertical direction are small and the rope remains in tension. (3) The structural damping of the system is not taken into account, because cranes are typically lightly damped. Todd et al.(1997) reported that the damping of a ship mounted crane is from 0.1% to 0.5% of the critical damping. As depicted in Fig.1, a coordinate system OXY is fixed to the ground and denoted as “Earth-Fixed”, which is taken to be the inertial frame of reference. The other system O 0 X 0 Y 0 is fixed to and, hence, moves with the ship, which is denoted as “Body- Fixed”. J is the rotary inertia of the ship, m ship and m p are the masses of the ship and load, respectively. The elastic displacements of the boom tip point B are denoted as u and w. The critical parameters including the boom length L b , the luff angle of the boom , the length of the rope L, the displacement in surge direc- tion x, the pitch angle , the heave motion y and the swing angle of the load are also indicated in Fig.1. MATHEMATICAL MODEL External forces acting on the crane-ship In the model, different external forces have to be considered (Ellermann et al., 2002), the hydrostatic force T bwwshippm 0, , ( ) ,gA y m m gh = +f (1) with the density of water w , the cross-sectional area of the ship at the still water level A w and the meta- centric height h m . The mooring line forces, which are approxi- mated by a third order polynomial 3T m12 3 |, 0 0,cx c x x cx= f (2) where c 1 , c 2 , c 3 are the linear, quadratic, and cubic characteristic coefficients of the mooring system. Forces due to viscous drag T dwd | | / 2, 0, 0cWT x x=f (3) X 0 Y 0 O 0 Y O u w P B L L b X Fig.1 Moored crane-ship model Ren et al. / J Zhejiang Univ Sci A 2008 9(1):26-31 28 are proportional to the density of water w , the em- pirical drag coefficient c d , the width of the ship W and the draught T. The frequency-dependent wave excitation forces, which can be split into a periodically changing part and the constant drift forces, can be modeled as: 2 ri d wr i ri ( cos( ) sin( ) ( cos( ) sin( ) , (cos() sin() xx yy A ktktAp Ak t k t Ak t k t + = f (4) with the wave amplitude A, the real and imaginary parts of the frequency dependent coefficient k rj and k ij (j=x,y,), and the coefficient of the drift force p d . The forces are abbreviated by T 123 b m d w , , .FFF=+F ffff (5) Dynamic equation of crane-ship The position vector of payload P as shown in Fig.1 can be expressed as p (cos() sin) (sin() cos), j j xL uL yL wL =+ + + + ri j (6) where i and j are unit vectors along the X- and Y-axes, respectively. It can be seen that, by including the boom tip displacement u and w, the effect of the elastic deformation of the boom is included in the position equation of the payload. It is assumed that the luff angle of the boom and the length of the payload pendulum are constant. Based on Eq.(6), the velocity vector of the payload P can be obtained by the time derivative of r p as pb b (sin()cos) (cos()sin). xu L L yw L L =+ + + + Vi j (7) So the kinetic energy of the payload can then be derived as pppp 222222 p 22 b /2 22 2cos2sin2cos 2 sin 2 sin( ) b Tm mx u y w L xu yw xL yL uL wL L xL = =+ + + VV bb b 2 sin( ) 2 sin( )cos 2 cos( ) 2 cos( ) 2 cos( )sin / 2. (8) uL L L yL wL LL + + + The potential energy of the payload pp b (sin() cos).UmgyL wL =+ + (9) The kinetic energy and the potential energy of the ship can be respectively expressed as 222 ship ship ()/,TJmxy=+ + (10) ship ship .Umgy= (11) Based on the finite element discretization, the kinetic and potential energy of boom can be respec- tively expressed as T rrrrurwr T bu wr wu ww 11 , 22 Tumu wmmw = UMMMU UMU M M (12) T rrrrurwr T bu wr wu ww 11 = , 22 Uuku wkkw UKKKU UKU K K (13) where M and K are global mass and stiffness matrices of the boom, U and U are displacement and velocity vectors of the flexible boom; (, )uw and (, )uw are the nodal displacement and velocities of the boom tip point B. U r and r U are vectors of displacements and velocities for the rest degrees of freedom of the boom structure. The Lagrangian function of the system can be expressed as T rrrrurwr 2 ur uu uw wr wu ww 22 222 2 22 ship p 22 bb 11 ()/2 2 22cos2sin2cos 2sin 2 L TU u m m u J wm w mxy mxuyw L xu yw x L y L u L wL L xL = + + + + + UMMMU M M b sin( ) 2 sin(uL + Ren et al. / J Zhejiang Univ Sci A 2008 9(1):26-31 29 bb T rrrrurwr ur uu uw ship wr wu ww pb ) 2 sin( )cos 2 cos( ) 2cos()2 cos()sin/2 1 2 (sin() cos). (14) LL yL wL L L ukkumgy wk w mg y L w L + + + + + + + + UKKKU K K The Lagranges equation is d , d j jj LL Q tq q = (15) where q j and j q are general coordinates and general velocities of the system. Q j is general force of the system. Substituting Eqs.(5) and (14) into Eq.(15) gives rr p p 2 pb 2 b 2 p 2 bb 0 0 (sin sin( ) cos( ) cos ) sinos cos( ) sin( ) muu mw w mL L LxL mgy L L LL + + =+ + + UU MK 00 0 0 0 0 , (16) 2 ship p p p p 2 bb1 () cos sin sin( ) cos( ) , (17) mmxmumL mL LLF + + += 2 ship p p p p 2 ship p b b 2 () sin cos ()cos()sin(), mmymwmL mL mmgL L F + + + + + += (18) pb b 2 2 pb 3 (sin()sin() sin( )cos sin( )sin cos( ) cos( ) cos( )sin cos( )cos ) cos( ) , JmLLx u LL yw L LmgF + + + + + = (19) b 2 bb 2 b cos sin cos sin sin( )cos cos( )cos cos( )sin sin( )sin sin 0. (20) Lx y u w L LL Lg + + + + + = As in several other investigations, the attention is focused on the horizontal surge motion. In this special case, equations of motion of the system can be re- duced to rr p p 2 p 2 p 0 0 ( sin cos ) , (21) (sincos) muu mw w mL L x mg L L + = UU MK 00 0 0 0 0 2 ship p p p p 1 () cos sin,mmxmumL mL F+ = (22) cos cos sin sin 0.Lx u w g + += (23) If the boom structure is assumed to be rigid, the deformation of the boom vanishes in Eqs.(21)(23). In this case, the equations of motion of the system can be reduced to 2 ship p p p 1 ()cos sin,mmxmL mL F+ = (24) cos sin 0.Lx g + += (25) SIMULATION RESULTS AND DISCUSSION The dynamic response of the crane-ship is investigated in time domain based on a Newmark method and an iterative approach (Ju et al., 2006). The type of finite element used in modeling the crane boom is space frame element. The basic parameter values of the ship used for the analysis are given as follows: c 1 =21200 N/m, c 2 =9440 N/m 2 , c 3 =13.82 N/m 3 , k r =5540 N/m, k i =426000 N/m, c d =0.2, p d =15800 Nm 2 , m p =200000 kg, m ship =1920000 kg, B=25 m, L b =60 m, T=1.69 m, =1000 kg/m 3 , =60, A=1.2 m. Influence of wave excitation frequency Consider the effect of different wave excitation frequencies on the load-swing angles. The cable length is 30 m, which corresponds to a load natural frequency of 0.626 Hz. The wave frequencies are 0.313 Hz, 0.626 Hz, and 0.8138 Hz. The dynamic responses of load-swing angles are shown in Fig.2. Ren et al. / J Zhejiang Univ Sci A 2008 9(1):26-31 30 The results show that the load-swing amplitudes depend on the wave excitation frequency. As the ex- citation frequency approaches the natural frequency of the load, the amplitudes of the swing angles increase, and the amplitude of the load oscillation was max =50. Influence of flexibility of the boom Consider the effect of a rigid boom and a flexible boom on the dynamic response of crane-ship. The cable length is 45 m, the excitation wave amplitude is 1.2 m and the wave frequency is 0.626 Hz. The dy- namic responses of load-swing angles of crane-ship with the rigid and flexible boom are demonstrated in Fig.3a. The load-swing angle is smaller for the crane-ship with flexible boom as compared with the crane-ship with a rigid boom. The load vibrations during the swing process can be seen from the load swing angle curves in Fig.3a. The surge motions of a crane-ship with a rigid and a flexible boom are demonstrated in Fig.3b. It can be seen that there is small change in the amplitude of ship surge motion. The surge velocity of a crane-ship with a rigid and a flexible boom is shown in Fig.3c. It can be seen that there is a high frequency vibration during the ship surge motion for the crane-ship with a flexible boom, Angle (deg) 10 5 0 5 10 Fig.2 Load swing angle for wave excitation with dif- ferent frequency (a) 0.313 Hz; (b) 0.626 Hz; (c) 0.8138 Hz Angle (deg) 60 40 20 0 20 40 60 Time (s) Angle (deg) 0 100 200 300 400 500 6 4 2 0 2 4 6 (a) (c) (b) Flexible Rigid Time (s) Angle (deg) 0 10 20 30 40 50 15 10 5 0 5 10 15 Fig.3 Swing angle (a), surge motion (b) and surge ve- locity (c) for a crane-ship with the rigid and flexible boom Flexible Rigid Time (s) Sur g e m o tion ( m ) 0 20 40 60 80 100 10 5 0 5 10 Flexible Rigid Time (s) Sur g e velocity (m/ s ) 0 20 40 60 80 100 2 1 0 1 2 (a) (b) (c) Ren et al. / J Zhejiang Univ Sci A 2008 9(1):26-31 31 but this is not observed for the crane-ship with a rigid boom. Fig.4 shows the elastic displacement of the boom tip-point B. It is seen that at the beginning of the process, the vibration amplitude is about 300 mm and it decreases to about 50 mm at the end of the process. CONCLUSION The governing equations for the dynamic re- sponse of a crane-ship coupled with the pendulum motion of the payload are derived based on La- granges equations. The boom is modeled based on a finite element method, while the payload is modeled as a planar pendulum of point mass. The cable is assumed to be massless and inextensible. If the crane structure is assumed to be rigid, the derived equations correctly degenerate to nonlinear differential equa- tions, which exactly correspond to Newtons Law of Motion for the planar pendulum motions. Numerical studies are then carried out for a real crane-ship with planar pendulum motions of the payload. Simulations show that the load-swing amplitude depends on the wave excitation frequency. As the excitation fre- quency approaches the natural frequency of the pay- load, the amplitudes of the swing angles increase. The influence of boom flexibility on the ship surge am- plitude and load swing angle is also investigated. Large vibrations during the ship surge motion are only observed for the crane-ship with flexible boom. Crane-ship with flexible boom has a longer period of ship surge motion than that with rigid boom. But the ship surge motion amplitudes have smaller changes for the crane-ship with flexible boom in comparison with rigid boom. References Al-Sweiti, Y.M., Sffker, D., 2007. Planar cargo control of elastic ship cranes with the “maryland rigging” system. Journal of Vibration and Control, 13(3):241-267. doi:10. 1177/1077546307078097 Chen, X.J., Shen, Q., Cui, W.C., 2002. Analysis of dynamic behavior of a multi-rigid-body system with a floating base. Engineering Mechanics, 10:139-143 (in Chinese). Chin, C., Nayfeh, A.H., Abdel-Rahman, E., 2001. Nonlinear dynamics of a boom crane. Journal of Vibration and Control, 7(2):199-220. doi:10.1177/107754630100700 204 Dong, Y.Q., Han, G., 1993. Dynamic response of lifting load system of crane vessel in waves. Shipbuilding of China, 120:63-71 (in Chinese). Ellermann, K., 2005. Dynamics of a moored barge under periodic and randomly disturbed excitation. Ocean En- gineering, 32(11-12):1420-1430. doi:10.1016/j. oceaneng.2004.11.004 Ellermann, K., Kreuzer, E., Markiewicz, M., 2002. Nonlinear dynamic of floating cranes. Nonlinear Dynamics, 27(2):107-183. doi:10.1023/A:1014256405213 Henry, R.J., Masoud, Z.N., Nayfeh, A.H., Mook, D.T., 2001. Cargo pendulation reduction on ship-mounted cranes via boom-luff angle actuation. Journal of Vibration and Control, 7(8):1253-1264. doi:10.1177/107754630100 700807 Ju, F., Choo, Y.S., Cui, F.S., 2006. Dynamic response of tower crane induced by the pendulum motion of the payload. International Journal of Solids and Structures, 43(2):376-389. doi:10.1016/j.ijsolstr.2005.03.078 Kral, R., Kreuzer, E., Wilmers, C., 1996. Nonlinear oscilla- tions of a crane ship. ZAMM, 76:5-8. Masoud, Z.N., Nayfeh, A.H., Mook, D.T., 2004. Cargo pen- dulation reduction of ship-mounted cranes. Nonlinear Dynamics, 35(3):299-311. doi:10.1023/B:NODY. 0000027917.37103.bc Todd, M.D., Vohra, S.T., Leban, F., 1997. Dynamical Meas- urements of Ship Crane Load Pendulation. Oceans 97 MTS/IEEE Conference Proceedings, 2:1230-1236. Time (s) Displacement (m) 0 100 200 300 400 500 0 0.1 0.2 0.3 0.4 Fig.4 Displacement of flexible boom tip point