中英文文獻翻譯-載荷擺動引起的塔式起重機的動力響應,中英文,文獻,翻譯,載荷,擺動,引起,引發(fā),塔式起重機,動力,響應 F. Ju a , Y.S. Choo a, * , F.S. Cui b motions is analyzed using the proposed formulations and computational method. It is found that the dynamic responses of the tower crane are dominated by both the first few natural frequencies of crane structure and the pendu- resulting in a coupled dynamics problem of the vibration of the crane structure and the pendulum motion of the payload. This kind of pendulum-induced vibration may cause instability or serious damage to the * Corresponding author. Tel.: +65 6874 1312/2994; fax: +65 6779 1635. E-mail address: cvecysnus.edu.sg (Y.S. Choo). International Journal of Solids and Structures 43 (2006) 376389 0020-7683/$ - see front matter C211 2005 Elsevier Ltd. All rights reserved. lum motion of the payload. The dynamic amplification factors generally increase with the increase of the initial pendu- lum angle and the changes are just slightly nonlinear for the planar pendulum motion. C211 2005 Elsevier Ltd. All rights reserved. Keywords: Tower crane; Finite element; Pendulum; Structural dynamics 1. Introduction Tower cranes are used to lift and move heavy payloads in the construction of high-rise buildings. Under internal or external excitations, the payload always has a tendency to oscillate about its vertical position, a Centre for Oshore Research and Engineering, Department of Civil Engineering, National University of Singapore, No. 1 Engineering Drive 2, Singapore 117576, Singapore b Institute of High Performance Computing, 1 Science Park Road, #01-01, The Capricorn, Science Park II, Singapore 117528, Singapore Received 18 October 2004; received in revised form 23 March 2005 Available online 23 May 2005 Abstract Dynamic response of tower cranes coupled with the pendulum motions of the payload is studied in this paper. A simple perturbation scheme and the assumption of small pendulum angle are applied to simplify the governing equa- tion. The tower crane is modeled by the finite element method, while the pendulum motion is represented as rigid-body kinetics. Integrated governing equations for the coupled dynamics problem are derived based on LagrangeC213s equations including the dissipation function. Dynamics of a real lung crane model with the spherical and planar pendulum Dynamic response of tower crane induced by the pendulum motion of the payload doi:10.1016/j.ijsolstr.2005.03.078 cranesystem.Drivenbytheneedsofcranedesignandcontrol,eortsarebeenmadebyresearcherstounder- stand the physical nature and engineering implication of the dynamics of crane systems including the pay- load. A recent review on crane dynamics, modeling and control is given by Abdel-Rahman et al. (2003). Studies related to the dynamics and control of cranes have been mostly based on simplified models of crane structures. Chin et al. (2001) modeled a boom crane as a spherical pendulum and a rigid system with two degrees-of-freedom and assumed that the motion of the platform influences, but is not influenced by the swinging of the payload. The crane structure was also considered as rigid bodies with or without dis- crete springs in the studies carried by Towarek (1998), Kiliceaslan et al. (1999) and Ghigliazza and Holmes (2002). Oguamanam et al. (2001) studied the dynamics of overhead crane system where a beam model was used to represent the flexibility of crane structures. Generally speaking, these simplifications on the mod- eling of crane structures are reasonable when the dynamics of the pendulum motion of the payload is the main concern. However, when the stress and dynamic response of crane structures are of interest, a detailed modeling of crane structures is certainly needed. A tower crane is a complex structural system con- sisting of space frames, cable system and lumped mass and, therefore, finite element analysis appears to be an appropriate approach in this case. Ju and Choo (2002, 2005a) studied the natural vibration and dynamic response of the tower crane system due to the acceleration or deceleration of the payload, where detailed finite element modeling of crane system was employed. The motion of the payload in their study was con- fined to the vertical direction without any swing or pendulum motion. The objective of this study is to derive integrated finite element formulations to analyze the dynamics of machine system. Fig. 1 schematically shows a simplified tower crane model and its deformed shape caused F. Ju et al. / International Journal of Solids and Structures 43 (2006) 376389 377 by the pendulum motion of the payload. As shown in the figure, the angle of the payload swing away from x y z O A Jib Mast B C B u B v B w P L P J L M H the tower crane coupled with the pendulum motion of the payload. 2. Theory and formulations A real tower crane is a complex space structure including base, mast, jib, balance beam structures and Fig. 1. Schematic geometry of deformed tower crane structure with payload pendulum. the vertical u. For For pure complex P reference B in as u B 2.1. Kinetic where the position equation of the payload. The where 378 F. Ju et al. / International Journal of Solids and Structures 43 (2006) 376389 Based on the finite element discretization, the kinetic and potential energies of tower crane structure can be respectively expressed as T C 1 2 f _ Dg T MC138f _ Dg 1 2 _ D r _u B _v B _w B 8 : 9 = ; T M rr M ru M rv M rw M ur m uu m uv m uw M vr m vu m vv m vw M wr m wu m wv m ww 2 6 6 6 4 3 7 7 7 5 _ D r _u B _v B _w B 8 : 9 = ; 4a and U C 1 2 fDg T KC138fDg 1 2 D r u B v B 8 : 9 = ; T K rr K ru K rv K rw K ur k uu k uv k uw K vr k vu k vv k vw 2 6 6 6 4 3 7 7 7 5 D r u B v B 8 : 9 = ; 4b m P and g are the mass of the payload and the acceleration of gravity, respectively. C2 coshsinu_v B _ h 2L P sinhcosu_v B _u 2L P sinh _w B _ hg3a and U P m P gH M C0 L P cosh w B 3b T P 1=2m P v P C1 v P 1=2m P f_u 2 B _v 2 B _w 2 B L 2 P _ h 2 L 2 P sin 2 h _u 2 2L P coshcosu_u B _ h C0 2L P sinhsinu_u B _u 2L P Here the flexibility of the pendulum wire is neglected and L P is therefore a constant. The kinetic energy and the potential energy of the payload can then be derived as velocity vector of the payload P can be obtained by the time derivative of r P as v P _u B L P coshcosu C1 _ h C0 L P sinhsinu C1 _uC1i _v B L P coshsinu C1 _ h L P sinhcosu C1 _uC1j _w B L P sinh C1 _ hC1k 2 tip displacements (u B , v B and w B ), the eect of the elastic deformation of the crane structure is included in i, j and k are unit vectors along the x-, y-, and z-axis, respectively. It can be seen that, by including the The position vector of payload P as shown in Fig. 1 can be expressed as r P L J u B L P sinhcosuC1i v B L P sinhsinuC1j H M w B C0 L P coshC1k 1 system is set at the base centre of the tower crane. The elastic displacements of the jib tip point the three directions, where the pendulum of the payload is connected to the jib structure, are denoted , v B and w B , respectively. energy, potential energy and dissipation function Critical parameters including the height of the mast (H m ), length of the jib (L J ) and the length of the pendulum (L ) are also indicated in Fig. 1. For the convenience of the kinetic analysis, the origin of the line is denoted as h, while the angle of the payload rotating about the vertical line is denoted as the pure planar pendulum motion of the payload, h is a function of time but u remains unchanged. spherical pendulum motion of the payload, u is a function of time and h is a constant. For the motion of the payload, u and h are time dependent. w B K wr k wu k wv k ww w B and veloci of the C C C C where The total structure where The LagrangeC213s equation including dissipation function is d dt oL o_q r C0 oL oq r oF o_q r 0 7 where q r and _q r are general coordinates and general velocities of the system. For the present problem, D r , u B , v B , w B , h and u are the general coordinates, and _ D r ; _u B ; _v B ; _w B ; _ h and _u are the corresponding general veloci C0 1 2 r u B v B w B : = ; rr ru rv rw K ur k uu k uv k uw K vr k vu k vv k vw K wr k wu k wv k ww 6 6 6 4 7 7 7 5 r u B v B w B : = ; C0 m P gH m C0 L P cosh w B 6 2L P sinh _w B hg D 8 9 T K K K K 2 3 D 8 9 2L P coshsinu_v B _ h 2L P sinhcosu_v B _u _ A 2 m P f_u B _v B _w B L P h L P sin h _u 2L P coshcosu_u B _ h C0 2L P sinhsinu_u B _u B B B B C C C C L T C0 U 2 B _v B _w B : ; ur uu uv uw M vr m vu m vv m vw M wr m wu m wv m ww 6 6 4 7 7 5 B _v B _w B : ; 1 2 2 2 2 _ 2 2 2 2 0 1 2.2. Integrated finite element formulations of governing equations Based on Eqs. (5), the Lagrangian of the system can be expressed as 1 _ D r _u 8 = T M rr M ru M rv M rw M m m m 2 6 3 7 _ D r _u 8 = F C 1 2 f _ Dg T CC138f _ Dg 1 2 D r _u B _v B _w B : = ; rr ru rv rw C ur c uu c uv c uw C vr c vu c vv c vw C wr c wu c wv c ww 6 6 6 4 7 7 7 5 D r _u B _v B _w B : = ; 4c C is the damping matrix. Obviously, D D r T u B v B w B C138 T ; _ D _ D r T _u B _v B _w B C138 T ; D D r T u B v B w B C138 T 4d kinetic energy, potential energy and dissipation function of the crane system including crane and the pendulum of the payload are thus can be expressed as T T C T P U U C U P F F C 5 T P , U P , T C , U C and F C are given by Eqs. (3a,b)and (4a,b,c), respectively. Similarly, the Rayleigh dissipation function can be expressed as _ 8 9T2 3 _ 8 9 ties of the jib tip point B, while D r and _ D r are vectors of displacements and velocities for the rest degrees-of-freedom of the crane structure. Here M and K are the global mass and stiness matrices of the tower crane structure; D and _ D are dis- placement and velocity vectors of the whole system; (u B , v B , w B ) and _u B ; _v B ; _w B are the nodal displacements F. Ju et al. / International Journal of Solids and Structures 43 (2006) 376389 379 ties. Substituti m P L P C0coshcosu h sinhcosu _ h 2 2coshsinu _ h _u sinhsinuu sinhcosu _u 2 = Eqs. coupled 2.3. Simplificat For Eqs. It tions, 2.4. A If dierent L P hu 2L P _ h _u 0 10b Eqs. C0m P L P hh h C0m P g L P h C0 L P h _u 2 cosuu B sinuv B hw B gh 0 9b L P hu 2L P _ h _u C0 sinuu B cosuv B 0 9c can be seen that, even with the assumption of small pendulum angle, the simplified dierential equa- Eqs. (9a)(9c) are still nonlinear for the coupled structural dynamics with the pendulum motion. special case with rigid structure assumption the crane structure is assumed to be rigid, Eqs. (8a)(8c) will degenerate to the following nonlinear ial equations L P h C0 L P h _u 2 gh 0 10a MC138 0 0 m P 0 0 00m P 6 4 7 5 B C A v B w B : ; CC138 _v B _w B : ; KC138 v B w B : ; 0 m P L P C0cosu h hcosu _ h 2 2sinu _ h _u hsinuu hcosu _u 2 m P L P C0sinu h hsinu _ h 2 C0 2cosu _ h _u C0 hcosuu hsinu _u 2 _ 2 8 : 9 = ; 9a C0m P L P sinhh coshh C0m P g L P h C0 L P sinhcosh _u 2 coshcosuu B coshsinuv B sinhw B gsinh 0 8b L P sinhu 2L P cosh _ h _u C0 sinuu B cosuv B 0 8c (8a)(8c) are the integrated finite element formulations for structural dynamics of the tower crane with the pendulum motion of the payload. ion for small pendulum angle small angular displacement, sin h can be replaced by h and cos h by 1 (Nelson and Olsson, 1986), and (8a)(8c) are then simplified to 0000 0 m P 00 2 6 6 3 7 7 0 B B 1 C C D r u B 8 = _ D r _u B 8 = D r u B 8 = m P L P C0coshsinu h sinhsinu _ h 2 C0 2coshcosu _ h _u C0 sinhcosuu sinhsinu _u 2 _ 2 : ; 8a MC138 0 m P 00 0 0 m P 0 0 00m P 6 6 4 7 7 5 B B C C A u B v B w B : ; CC138 _u B _v B _w B : ; KC138 u B v B w B : ; 0 8 9 ng Eqs. (4c) and (6) into Eq. (7) gives 0000 2 6 3 7 0 B 1 C D r 8 = _ D r 8 = D r 8 = 380 F. Ju et al. / International Journal of Solids and Structures 43 (2006) 376389 (10a) and (10b) can also be obtained directly from NewtonC213s Second Law of Motion. Furtherm hj t=0 C0L P _u 2 g 0 with u 012a 3. Approximation for numerical computation Katriel pendulum structure, deform The _u x 0 _e u t u e u t 13b Here, angular velocity and acceleration, derived from Eq. (13a), are where e u (t) is small perturbation. It can be seen that an additional term, e u (t), is added to Eq. (12b) to get Eq. (13a). e u (t) is employed to account for the flexibility of the structure. u x 0 t u 0 e u t 13a in this study that the approximated solution for the spherical pendulum with flexible structure is of the form and this solution is not valid for the flexible crane structure. Based on the premise that the elastic ation of the crane structure is small compared to the pendulum motion of the payload, it is assumed 3.1. Flexible crane structure with spherical pendulum Eq. (12b) gives the solution of the spherical pendulum of the payload based on the assumption of rigid dynamics of tower cranes. , 2002). Two special cases of the motions of the payload, namely spherical pendulum and planar as discussed in the previous sections, are considered in this paper to study the coupled structural Solving the coupled equations for structural dynamics and the pendulum motion may involve the com- plex nonlinear dynamical phenomena of bifurcation and chaos (Chin et al., 2001; Schwartz et al., 1999; And the solution of Eq. (12a) is u x 0 t u 0 12b where x 0 is given in Eq. (11c). It can be seen that, with the rigid structure assumption, the spherical pen- dulum of the payload is a uniform rotation with the angular speed equal to the natural frequency of the corresponding linear planar pendulum. where x 0 is the natural frequency of the pendulum given by x 0 g=L P p 11c Similarly, if the spherical motion of the payload is assumed (Ghigliazza and Holmes, 2002; Markeyev, 1998), with h = h 0 , _ h h 0 and uj t=0 = u 0 , the equation governing the motion of the spherical pendulum of the payload is obtained as h h 0 cosx 0 t 11b L P h gh 0. 11a Eq. (11a) is the well known equation of the linear planar pendulum (Nelson and Olsson, 1986) and its solution is ore, if the movement of payload is confined to a vertical plane with u = u 0 , _u u 0 and = h 0 , Eq. (10a) reduces to F. Ju et al. / International Journal of Solids and Structures 43 (2006) 376389 381 like e u (t), _e u t and e u t are also assumed to be small perturbation terms. Substituti 8 9 8 9 8 9 8 9 It decoup report of the the pen Similarly, be the where Here Substituti MC1386 7B C CC138 KC138 382 F. Ju et al. / International Journal of Solids and Structures 43 (2006) 376389 0 0 m P 0 4 5 A v B : ; _v B : ; v B : ; based on Eq. (11b), the solution for the planar pendulum with flexible structure is assumed to following form h h 0 cosx 0 te h t15a e h (t) is a small perturbation term. The angular velocity and acceleration are _ h C0 x 0 h 0 sinx 0 t_e h t h C0 x 2 0 h 0 cosx 0 te h t 15b _e h t and e h t are also small perturbations. ng Eqs. (15a) and (15b) into Eqs. (9a)(9c) yields 0000 0 m P 00 2 6 6 3 7 7 0 B B 1 C C D r u B 8 = _ D r _u B 8 = D r u B 8 = 3.2. Flexible crane structure with planar pendulum is interesting to notice that the governing equation for the dynamic response of the tower crane is led from the equation of the spherical pendulum motion of the payload. This phenomenon was also ed in Oguamanam et al. (2001). The right-hand side of Eq. (14d) indicates that the dynamic response crane structure is proportional to the amplitude of the pendulum angle h 0 , regardless of the length of dulum when the payload experiences the motion of the spherical pendulum. MC138 D r u B v B w B : = ; CC138 _ D r _u B _v B _w B : = ; KC138 D r u B v B w B : = ; m P w B gh 0 0 cosx 0 t u 0 sinx 0 t u 0 C01 : = ; 14d 0 0 L P h 0 e u sinx 0 t u 0 u B C0 cosx 0 t u 0 v B 14c where all product terms of e u (t), _e u t and e u t are dropped. Eq. (14a) can be further simplified to: P 2x 0 h 0 _e u h 0 w B cosx 0 t u u B sinx 0 t u v B 14b m P v B h 0 sinx 0 t u 0 w B C138m P h 0 gsinx 0 t u 0 C0m g : ; 0 m P u B h 0 cosx 0 t u 0 w B C138m P h 0 gcosx 0 t u 0 8 = 14a MC138 0 0 m P 0 0 00m P 6 4 7 5 B C A v B w B : ; CC138 _v B _w B : ; KC138 v B w B : ; ng Eqs. (13a) and (13b) into Eqs. (9a)(9c) gives 0000 0 m P 00 2 6 6 3 7 7 0 B B 1 C C D r u B 8 = _ D r _u B 8 = D r u B 8 = 0 00m P w B _w B w B Eq. (16c) serves as a constraint for the tip acceleration and u B v B C1 cosu 0 =sinu 0 (for sinu 0 6 0) or A height mostly lump Choo 4.1. Natural To vibrat natural deform bending pled with tures real model of Potain lung tower crane is used for numerical analysis in this study, where the total of the crane is 68.3 metres. The types of the finite elements used in modeling the crane structures are space frame and truss elements. The counter weight, equipment and the payload are modeled by mass and the lung and hoisting cables are modeled by cable-pulley element proposed by Ju and (2005b). modes and frequencies of the tower crane structure study the eect of pendulum parameters on the coupled vibration of the tower cranes, the natural ion of the tower crane structures without the payload is firstly analyzed. Fig. 2 gives the first four mode and frequencies of the tower crane. It can be seen that the first mode is dominated by the ation of the jib structure, while the second and the third modes are predominantly the complex patterns of the whole crane structure. Twist of the jib structure is found in the fourth mode, cou- the bending of the mast and the jib. Detailed discussion on the natural vibration of crane struc- v B u B C1 sinu 0 =cosu 0 (for cosu 0 6 0) will be substituted into Eq. (16b) to impose such constraint. The computational scheme for solving Eqs. (16a)(16c) is based on Newmark method (Bathe and Wilson, 1976; Nickel, 1971) and iterative approach (Kelley, 1995; Das and Sargand, 1999). The computa- tional procedures with a time interval of Dt can be summarized as: Step 1. Given D (t) , _ D t and D t , get D (t + Dt) , _ D tDt and D tDt from Eq. (16a) while neglecting e h , _e h and e h ; Step 2. Substitute u tDt B ,v tDt B and w tDt B into Eq. (16b), get e tDt h , _e tDt h and e tDt h ; Step 3. Substitute e tDt h , _e tDt h and e tDt h into Eq. (16a), get new D (t+Dt) , _ D tD and D tDt ; Step 4. Check the convergence of u tDt B , v tDt B and w tDt B , go back to Step 2 if convergence condition is not satisfied. 4. Numerical results and discussions 0 m P cosu 0 C0L P e h C0 2x 0 h 2 0 L P cosx 0 tsinx 0 t_e h 2gh 2 0 sin 2 x 0 tcosu 0 e h gh 0 cosx 0 tgh 3 0 sin 2 x 0 tcosx 0 tC138 m P sinu 0 C0L P e h C0 2x 0 h 2 0 L P cosx 0 tsinx 0 t_e h 2gh 2 0 sin 2 x 0 tcosu 0 e h gh 0 cosx 0 tgh 3 0 sin 2 x 0 tcosx 0 tC138 m P C0h 0 L P cosx 0 te h 2x 0 h 0 L P sinx 0 t_e h gh 0 cos 2 x 0 te h 2gh 2 0 cos2x 0 tC138 C0 m P g 8 : 9 = ; 16a L P e h w B ge h C0h 0 w B cosx 0 tC0u B cosu 0 C0 v B sinu 0 16b 0 u B sinu 0 C0 v B cosu 0 16c It can be seen from Eqs. (16a)(16c) that the dynamic response of tower crane is fully coupled with the pla- nar pendulum motion of the payload. F. Ju et al. / International Journal of Solids and Structures 43 (2006) 376389 383 can be found in the study by Ju and Choo (2002). 384 F. Ju et al. / International Journal of Solids and Structures 43 (2006) 376389 4.2. Dynamic response of tower crane with the spherical pendulum motion of the payload Fig. 3 shows the deformed shape of a tower crane, where the payload is experiencing the spherical pen- dulum motion with an initial pendulum angle, h 0 , of 10.0C176. The length of the pendulum, L P , is 40 m, while the weight of the payload is 2000 kg. Fig. 2. First four natural modes of a tower crane structure with bracings at the mast. F. Ju et al. / International Journal of Solids and Structures 43 (2006) 376389 385 Fig. 4a shows the factored dynamic response of the tip node B, v B (t)/(w B ) 0 and w B (t)/(w B ) 0 . Here v B (t) and w B (t) are the dynamic responses of displacements of the node B in y and z directions; (w B ) 0 is the static displacement of the node B at vertical direction without any motion of the payload. The reason for using v B (t)/(w B ) 0 instead of v B (t)/(v B ) 0 is that (v B ) 0 is very small when only a vertical load is acting at the tip node. Fig. 4b gives the spectrum of the factored power de