CONSIDERATIONS OF VARIOUS MOVING LOAD MODELS IN STRUCTURAL DYNAMICS OF LARGE GANTRY CRANES

Transcription

1 11 th International Conference on Vibration Problem Z. Dimitrovová et al. (ed.) Libon, Portugal, 9-12 September 2013 CONSIDERATIONS OF VARIOUS MOVING LOAD MODELS IN STRUCTURAL DYNAMICS OF LARGE GANTRY CRANES Gašić M. Vlada* 1, Nenad Đ. Zrnić 1, Milorad Đ. Milovančević 1 1 Facult of Mechanical Engineering, Univerit of Belgrade vgaic@ma.bg.ac.r*, nzrnic@ma.bg.ac.r, mmilovancevic@ma.bg.ac.r Keword: Gantr Crane, Moving Load, Structural Dnamic, Vibration. Abtract. The paper deal with the moving load problem within the tructural dnamic anali of large gantr crane a high-performance machine. The emphai i given on combined method approach, i.e. finite element method and analtical potulation, for obtaining the mathematical model of crane. Moving trolle i conidered throughout everal model: moving force, moving ma, moving ocillator and moving ocillator with winging object. Each model ha characteritic which determine the repone of the crane tructure, along with it dnamic propertie. The title problem i olved b calculating the forced vibration repone of the two-dimenional framework with time-dependent propert matrice and ubjected to an equivalent moving load. Improving the moving load model increae complexit a well for potulating a for obtaining the olution from overall crane model. Comparative preentation of model i hown here with concluion that lead to an appropriate wa of model election prior to crane problem potulation.

2 1 INTRODUCTION Moving load problem i ver important topic in tructural dnamic. In the beginning, it wa related with the deign of railwa bridge and highwa tructure which howed additional vibration effect from the vehicle movement. Thi problem triggered the reearch into the moving load problem, initiall with Stoke and Timohenko. Man paper are preented in excellent monograph b Frba [1], which decribe baic potulation of moving load problem and their analtical olution. Irrepective of the man viewpoint, moving load problem methodolog ha two ide: analtical approach and finite element approach. Analtical approach are limited to imple cae of tructure (uch a beam) and baic tpe of load. Cloed-form olution for the governing equation i hard to find, even for the imple cae, and involve intenive mathematic. That i wh finite element approach improved the tudie in moving load problem with wide range of complex tructure and load model. However, it demand certain numerical integration cheme in the time-domain anali which can be intenive computational proce. Over the ear, moving load problem have gained interet in the field of machine and mechanical tructure due to the fact that working parameter are increaing. Tpical tructure under a moving load in mechanical engineering are bridge crane, gantr crane, unloading bridge, tower crane, cablewa, guidewa, and container crane. Here, the quaide container crane (QCC) and rail mounted gantr container crane (RMG) can be pointed out in moving load anali becaue of the high peed of trolle, large lifting capacitie and overall tructural dimenion. Thi paper tudie the dnamic repone of the RMG container crane ubjected to variou tpe of moving load-model upon the tructural deign of trolle. The inpiration i gained with the fact that container tranport i increaing earl average 8% and that up-to-date RMG crane can have pan to 50 m with trolle peed of 3 m/, which place it in highperformance machine. Thu, mathematical model are needed for moving load anali at crane, particularl becaue it i ver difficult and expenive in practice to do an experimental reearch on a real-ize crane. 2 MOVING LOAD MODELS The main ditinction in moving load problem i modelling of the vehicle/trolle tem. Firt and baic approach i moving force model (MFM). Fundamental potulation i model of tructure a imple beam ubjected to contant force (weight of vehicle-m g) moving with contant peed v, [2]. Moving force model i ea to ue and bring attention in contemporar tudie a well [3]. Moving ma model (MMM) include inertial effect of the vehicle/trolle, and in fundamental potulation, for vertical diplacement of the beam-w(x,t), can be preented a w w w( vt, t) A EI [ m ] ( ) 2 4 g m x vt 2 t x t which clearl indicate the tructure-trolle interaction. Hence, the peed of moving ma, tructure flexibilit and the ma ratio of moving paload and tructure are important factor that contribute to creating the interaction force [4]. The interaction force i highl non-linear in nature in it local and convective derivative and change it poition and magnitude with time, which i uuall repreented with Dirac delta function (δ), a given in Eq. (1). MMM arie with comlexit when variable peed of moving load i included [5]. However, even for (1) 2

3 the cae when more accurate reult are obtained with MMM [6], the conequence of neglecting thi interaction ma ometime be minor even for the extreme up-to-date QCC [7]. Moving ocillator model (MOM) aume that paload i attached to a ma of moving vehicle through a pring. The fundamental analtical potulation i given in [8]. With high value of the pring tiffne the MOM can be reduce with MMM in mot cae [9]. Figure 1: a) RMG container crane, b) Moving force (MFM), c) Moving ma (MMM), d) Moving ocillator (MOM), e) Moving pendulum (MPM), f) Moving ocillator with pendulum (MOPM). In tructural dnamic of crane, the ver important moving load problem i approach with model of moving trolle hoiting a winging object or moving pendulum problem (MPM). The literature dealing with thi problem i rare and onl few reearche can be found [10,11]. In ome tudie [12], it i ued equivalent moving ma matrix which reduce the MPM to MMM. It i hown that influence of the winging angle of the paload i big on horizontal repone of the tructure. The upgrade of the MPM i the model of trolle with upended hoit with winging object which can be named a moving ocillator with pendulum (MOPM). Compared with previou model, thi model i the mot complex for gaining the governing equation and obtaining the olution [13]. The goal of thi paper i to preent comparative overview of the variou model uage at gant crane dnamic and their characteritic that can lead to appropriate modelling. The combined method-finite element and analtical method i adopted to gain the mathematical model [14]. Prior to gantr crane dnamic, it can be concluded that ever reearch ha to include following: (i) Both the horizontal and the vertical repone of the tructure, (ii) Real ccle of trolle movement with emphai on acceleration/deceleration period, (iii) Lowet level of approximation when trolle inertial effect included. 3

4 3 MODEL FORMULATION The approach ued here i a combined finite element and analtical method for obtaining tranvere and longitudinal vibration of a gantr crane tem ubjected to a moving load. The general approach in moving load problem at crane i alo ued here, thu the tem of the gantr crane (Figure 1.a) i divided into two part: the framework (tructure) and the moving tem. The framework i a planar (2D) dicrete model conited of top beam with length L, pier leg with height H and heer leg with height h. The dicretization of the framework (Figure 2.a) i done b uing FEM, with plane-frame element, [15]. The top beam i divided in 10 identical element and each leg b 2 element. Hence, framework ha 41 DOF' (with extraction of the retrained diplacement from upport) forming the tructure diplacement vector U. Figure 2: a) FE model of the gantr crane tructure, b) Equivalent nodal force of the element, c) Speed pattern of the moving load, d) Moving load model - MMM, MOM and MOPM, e) Contact point diplacement. The global poition of the moving tem on the top beam, Figure 2.a, i aumed to be known and defined b coordinate x m (t). Here, the acceleration/deceleration i alo included in calculation becaue of the trolle trapezoidal peed pattern, Figure 2.c. It i aumed, b model of the gantr crane tem, that a loading i mmetricall ditributed on the top beam rail() and furthermore that relationhip between the framework and the moving tem can be implified into one moving load P(t), with projection in two-dimenional direction P x (t) and P (t), Figure 2.b. The moving tem i conidered a MMM, MOM and MOPM, Figure 2.d, while MFM i ued a a pecial cae of MMM when inertial effect from moving ma are neglected. The ma of the moving tem i conited of ma of trolle (m 1 ), hoit ma (m 2 ) and paload ma (m 3 ). Accordint to ued model, additional DOF' are -vertical diplacement of ocillator and φ- winging angle of the paload. It i aumed that trolle and tructure are alwa in contact and that trolle i moving on the mooth urface on the top beam. 4 EQUATION OF MOTION FOR THE SYSTEM The equation of motion for a multiple degree of freedom tem i repreented a follow Mq Cq Kq F() t (2) 4

5 where M, C, K are the overall ma, damping and tiffne matrice of the tem, repectivel; qq,, qare, repectivel, acceleration, velocit and diplacement vector for the tem and F(t) i the external force vector. Apart from the tructural diplacement vector U, the overall diplacement vector include the coordinate and φ, which depend on the ued load model. 4.1 Structural tiffne, ma and damping matrix The equation of motion for a framework (tructural tem) i repreented a follow M U C U K U P() t (3) t t t where M t, C t, K t are the ma, damping and tiffne matrice of the tructural tem; U, U, U are the repective acceleration, velocit and diplacement vector for the tructural tem and P(t) i the external force vector acting upon the tructure. According to the hown FE model of the framework, the tiffne matrix can be obtained b aembling all the element tiffne matrice [16] up to forming the quare matrix K t that correpond with 41 DOF' of the tructure. Similarl, the ma matrix of the framework M t can be obtained. A uuall practiced in FE dnamic anali of MDOF tem, the damping matrix i contructed b uing the theor of Raleigh damping in following form [17] C a M b K (4) t t t with contant a and b that correpond to firt two frequencie and their damping ratio (ξ). 4.2 Equivalent nodal force and external load vector The tructural external force vector P(t) take the form [17] P () t [ f () t f () t f () t f () t f () t f () t.. 0 0] (5) where (), 1 6 i T f t i, are the equivalent nodal force of the element, determined b f1 () t N1( x) Px () t, f4 () t N4( x) Px () t f2 () t N2( x) P () t, f3 () t N3( x) P () t (6) f5 () t N5( x) P () t, f6 () t N6( x) P () t Noting that l i the element length and x i the ditance along the element to the point of the application of the force (Figure 2.b), the relative ditance i given b x / l, thu, the hape function, N i = N i (x) = N i (ξ) (i =1-6), take the following form N1 1, N4, N , N3 l( 2 ), N5 3 2, N6 l( )(7) The trolle move from left end of the beam, at time t = 0, to the right end with defined peed pattern and it poition on the top beam at time t i known - x m (t). The element number on which the moving force are applied, at an time t (t 0), i xm () t IntegerPart[ ] 1 (8) l The nodal force can be calculated in term of the global poition, b xm () t (-1) l (9) l 5

6 According to the Eq. (6) one ma repreent the element nodal force vector a where T T { f ( t)} N P N P (10) N x [ N1 0 0 N4 0 0], N2 N3 N5 N6 Adjutment with total DOF' give the following matrice N X x x N [0 0 ] (11) [0 0 N 0], N [0 0 N 0] (12) x Thee matrice are with non-zero element which correpond to the DOF' of element where the moving load i located at, while other element are zero. The ubmatrice N x and N are calculated b (9,8,7). One ma ee that thee matrice engage onl 6 value which move along in the (12), a moving load change the poition on the top beam. The axial diplacement at an location within the finite element, w x = w x (x,t), and tranveral diplacement at an location within the finite element w = w (x,t), Figure 2.e, can be preented in matrix form a wx X Y N U, w N Y U (13) Second derivate of the expreion (13) can be preented [13], in following form w x N X U '' 2 ' ', w NYU x NYU x 2 NYU x NYU (14) where the upercript ('), ('') are repreenting the firt and econd derivate of expreion (7) with repect to x, while x i the velocit and x i the acceleration of the moving tem. The tructural external force vector take the matrix form T T P() t N P N P (15) X which can be ued, in combination with (3), for forced vibration of the framework. The interaction force between the tructure and the moving load model are, in general, x, x i, x Y P f( w, w, m,, ) (16) which i detailed preented in [13] for MMM, MOM, MOPM. The additional equation for ocillator and pendulum are obtained according to the Second Newton' law. Combination of thee equation with Eq. (3-16) up to the form of the Eq. (2) give the equation of motion for the ued tem. The general procedure i to divide the total time τ, needed for moving tem to travel from the left end to the right end of the top beam, into p tep with a time interval Δt and to calculate all the given matrice for each time tep r (r =1-p). 5 NUMERICAL RESULTS AND DISCUSSION Dnamic behaviour of the gantr crane ubjected to variou moving load i obtained b olution of the Eq. (1). Original in-houe oftware i created to olve the title problem with direct integration method baed on the Newmark algorithm [18]. The maximal time interval Δt i 5. The gravitational acceleration g i taken to be 9.81 m/ 2. The crane tructure i made of teel with denit 7850 kg/m 3 and modulu of elaticit Pa. Initial model include tructural damping with ξ=ξ 1 =ξ 2 =0.5 %. Geometric characteritic of the gantr crane are L=40 m and H=h=15 m. Element propertie are: A n =0.09 m 2, I n =0.041 m 4 (n=1-10), A 11 =0.085 m 2, I 11 =0.036 m 4, A 12 =0.07 m 2, 6

7 I 12 =0.024 m 4 and A n =0.048 m 2, I n =0.01 m 4 (n=13-14). The moving tem conit of paload of 52 t, trolle of 3 t and hoit of 5 t, thu m =60 t. Initial pring tiffne in moving tem i k=10 9 N/m, while rope length i L u =12 m. The tem move with 2 peed pattern, where pattern v 1 aume v r =3 m/, a u =a k =0.6 m/ 2, which correpond to characteritic of nowada tem of trolle, while the pattern v 2 aume v r =5 m/, a u =a k =1.25 m/ 2 which i extreme aumption, but expected in the cloe future. Firt, the MMM model i analed. A expected from phical intuition, the bigget value of vertical diplacement are for node 6, i.e. the middle point. Horizontal diplacement for all the top beam point are the ame, becaue of high axial tiffne of the element. Figure 4 how the reult for both the peed pattern. The repone are higher for pattern v 2. Thi influence i ver ignificant for horizontal diplacement U X1 where value reach the maximum of 5.41 cm in the deceleration period, while maximum value with pattern v 1 i 4.1 cm, Figure 3. The difference i much maller for vertical diplacement of middle point, Figure 4.b. Maximum value occur when trolle i at midpan, and for v 1 i 5.2 cm and 5.4 cm for pattern v 2. a) Horizontaldiplacement, node 1 m v1 v Normalized time t t Normalized time t t b) Figure 3: a) Horizontal diplacement U X1, b) Vertical diplacement U Y6 ; v 1,v 2. When inertial effect of the moving ma are neglected, MFM i analzed and comparion of reult how big difference in value for horizontal diplacement, Figure 4.a. MFM implifie the mathematical algorithm but when MMM i included one ma calculate the frequencie of the whole tem at each time tep. The reult for firt 3 frequencie are hown at Figure 4.b. It i obviou that frequencie are dependent on poition of ma moving on the top beam. Vertical diplacement, node 6 m v v 1 2 a) Horizontaldiplacement, node 1 m Moving force Moving ma Time t b) f 1 f f 3 Time t Figure 4: a) MMM/MFM Horizontal diplacement Ux1, v 2, b) MMM firt 3 frequencie The vertical and horizontal diplacement of node 6, for MOM model with peed pattern v 2, are preented in Figure 5.a. When compared with value from Figure 3, for peed pattern v 2, one can find negligible difference becaue the pring tiffne of k=10 9 N/m i high Frequencie firt 3 Hz 7

8 enough to compl with moving ma model. Here, it i convenient to find the value of vertical diplacement of the beam over the point of contact -w. With decreae of tiffne to k=10 7 N/m, one ma ee light change of diplacement in relation to w, Figure 5.b. Thu, up to thee value there i no ignificant influence in dnamic repone of tructure. Diplacement - Node 6 m U X6 U Y6 w, m w a) Time b) Time Figure 5: MOM, a) Vertical and horizontal diplacement-node 6, b) Diplacement, k=10 7 N/m; v 2. Finall, the MOPM model i ued. One can find that increae of acceleration ha high influence on the winging angle of the paload, which hould bring attention in deign. The winging angle reache the value of 14.3 o, Figure 6.a, in thi cae. Moreover, additional inertial effect due to winging have influence on horizontal diplacement of the tructure, Figure 6.b, while vertical diplacement are imilar to the value with previoul preented model Angle j rad v 1 v2 Horizontaldiplacement-Node 6 m v1 v 2 a) Normalized time t t b) Normalized time t t 6 CONCLUSIONS Figure 6: MOPM, a) Swinging angle φ, b) Horizontal diplacement-node 6; v 1,v 2. The reult from Figure 4.a how that MFM i le accurate then MMM, conidering horizontal diplacement of the tructure. However, the influence of moving load acceleration/deceleration i included with MFM which approve thi approach in ome cae. MMM include inertial effect of the moving ma, with additional effect due to convective derivative a a reult, which aure potulation of the problem a in analtical cae with Eq. (1). Main advantage i poibilit to adjut the algorithm to calculate frequencie of the tem with moving ma included, Figure 4.b. Thi i important at crane becaue the paload i uuall heavier then the tructure, therefore calculation of tructural frequencie with MFM are not dealing with atifactor reult. MOM can analze the influence of the flexibilit of the trolle tructure. However, there i no ignificant difference from MMM with realitic crane parameter, Figure 5.a. Thi i in correlation with reult from all the model that how ver mall influence on vertical diplacement of the tructure. 8

9 The MOPM i the mot complex model a for potulation of governing equation a for obtaining the reult. In thi cae, CPU time required to obtain the olution i much greater but thi hould be the cot to pa if additional parameter are included. The influence of paload winging i ver important at gantr crane, epeciall when expected to achieve high performance for the trolle peed pattern, Figure 6. From the given model, author would like to give emphai on MMM and MOPM. According to previoul tated, MMM can tand for initial approach in moving load problem at gantr crane, while MOPM i extended approach for more atifactor reult in cae where ant-wa tem are not developed. Acknowledgement Thi work i a contribution to the Minitr of Education, Science and Technological Development of Republic of Serbia funded project TR REFERENCES [1] L. Frba, Vibration of olid and tructure under moving load. 3 rd edition, Thoma Telford, [2] M. Olon, On the fundamental moving load problem. Journal of Sound and Vibration, 14 5 (2), , [3] A.V. Peterev, B. Yang and L.A. Bergman, Reviiting the moving force problem. Journal of Sound and Vibration, 261, 75-91, [4] H.P. Lee, Dnamic Repone of a Beam with a moving Ma. Journal of Sound and Vibration, 191 (2), , [5] G.T. Michalto, Dnamic Behaviour of a Single-Span Beam Subjected to Load Moving with Variable Speed. Journal of Sound and Vibration, 258 (2), , [6] V. Gašić, N. Znić and M. Rakin, Conideration of a Moving Ma Effect on Dnamic Behaviour of a Jib Crane Structure. Tehnički Vjenik-Technical Gazette, 19 (1), , [7] N. Zrnić, V. Gašić, A. Obradović and S. Bošnjak, Appropriate modeling of dnamic behavior of quaide container crane boom under a moving trolle. J. Naprtek, J. Horacek, M. Okrouhlik, B. Marvalova, F. Verhult and J. Sawicki ed. 10th International Conference on Vibration Problem, Springer Proceeding in Phic 139, ICOVP 2011, 81-86, [8] A.V. Peterev, L.A. Bergman, C.A. Tan, T.-C. Tao and B. Yang, On amptotic of the olution of the moving ocillator problem. Journal of Sound and Vibration, 260, , [9] A.V. Peterev, L.A. Bergman, C.A. Tan, T.-C. Tao and B. Yang, Some recent reult in moving load problem with application to highwa bridge. Proc Int. Conf. Motion Vib Control, 6, Vol.1, K20-K27, [10] D.C.D. Oguamanam, J.S. Hanen, G.R. Heppler, G.R, Dnamic of a three-dimenional overhead crane tem. Journal of Sound and Vibration, 242(3), ,

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