Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 41, 4, pp. 853-872, Warsaw 2003 DYNAMIC ANALYSIS OF THE LOAD HOISTING PROCESS Juliusz Grabski Jarosław Strzałko Faculty of Mechanical Engineering, Technical University of Łódź e-mail: julgrabs@p.lodz.pl Proposals of two models of a load lifted by cranes have been discus- sed. Special attention has been paid to the phenomena occurring before detachment of the load from the ground. Stick-slip motion of the load on the foundation has been observed. One of two models presented – the model in the form of a rigid body resting on an elastic foundation has turned out to be more advantageous for the analysis of the lifting process. The results of numerical computations have revealed a pheno- menon of solution bifurcation that occurs at a slight change in values of parameters of the model. Key words: cranes, 3D model of load, dynamics of lifted load, load hoisting 1. Introduction The load hoisting process is an interesting problem in dynamics of bodies. The positioning of a load and anti-sway prevention are examples of practical applications. The task that consists in computer simulation of load motion during its lifting from the ground in the situation when the rope direction is not vertical canbe of significance fromthepractical point of view.Undernormal operating conditionsof a crane, thiswayof liftinga load isnotproper,however it can take place under certain circumstances. For instance, for rescue trucks equipped with cranes that are used to lift auto hulks out of ditches, this kind of work is not exceptionally rare. When there is a necessity to shift a load with respect to the ground or if the vertical direction of the rope cannot be maintained, the lifting process 854 J.Grabski, J.Strzałko becomes complex. In such cases, frictional effects, rotations of the load, and load impacts due to collisions with the groundmay occur. The consideration of these phenomenamakes the algorithm and the load dynamics analysismore complex. Thus, a rigid body seems to be a suitable loadmodel for such a task (cf. Strzałko and Grabski, 1998). In the literature devoted to lifting and carrying loads, themost often mo- del of the lifted load is a particle (e.g. Borkowski et al., 1996, Jedliński and Grzesikiewicz, 1990; Strzałko andGrabski, 1995; Hakala and Sorsa, 1998;Ma- czyński and Kościelny, 1999; Tomczyk and Cink, 1999; Miyata et al., 2001; etc.), and sometimes a 2D rigid body (cf. Grabski 1998). A three-dimensional model of a crane and a particle model of a load has been analysed byWojna- rowski et al. (1990), Lee (1998). In the papers byGrabski andStrzałko (1997), Grabski (1998), Strzałko (1998), a 3D body was the model of the lifted load, and in those cases mechanical parameters (e.g. rotation inertia of the body, positions of the bodymass centre and the rope fixture) of the lifted load could be considered. In the present study, two models of the load have been compared. They differ as far as the body-ground contact modelling is concerned. In the first case, a direct contact between the rigid body and the rigid foundation has been assumed. In the second case, it has been assumed that the rigid body contacts the elastic foundation. Linearly elastic models of bodies, which are employed in contact problems (e.g. Konderla, 1993;Wriggers and Zavarise, 1993), describe well local pheno- mena in the contact region of flexible solids. These phenomena can serve as a clue for thedistribution of surface forces (ground reactions) acting on the load. In the analysis of load motion, especially finite motions, the determination of local deformations of the body and the foundation is not important. The relationships presented in this paper, which describe the initial period of the crane operation, i.e. rope tensioning, stick-slip motion of the load with respect to the ground, themoment of detachment andmotion of the load after its detachment, concern systems in which the lifted load is modelled as a 3D rigid body. In the problem solutions known to the authors, these issues are usually neglected. The results of numerical calculations prove quantitative and qualitative agreement between the results obtained for both themodels. However, in the analysis of the lifting process, the model of the load in the form a rigid body resting on an elastic foundation ismore suitable. Theapplication of thismodel leads to a simpler andmore effective calculation algorithm,which results from aconstantnumberofdegrees of freedominthe systemduringthewholeprocess Dynamic analysis of the load hoisting process 855 of analysis. The model in which a rigid body contacts a rigid foundation is characterised by a time-varying number of degrees of freedom of the load that dependson themotionphase (thenumberof degrees of freedomduring shifting the load with respect to the ground is different from the number of degrees of freedomduring rotation of the loadwith respect to the edge or the corner, and it is still different at the moment when there is no contact with the ground). An interesting phenomenon that occurred during the numerical computa- tions is bifurcation of the solution. Themodel of the load under consideration is a non-linear multidimensional system with dry friction. For systems of this sort, a slight change in the value of one parameter can result in bifurcation of the solution in some ranges of values of parameters (cf. Kurnik, 1997). Fric- tion coefficient value between the load and the ground has been chosen as an active (bifurcation) parameter. The results presented in this paper illustrate evolution of the solution at a slight change in the bifurcation parameter. 2. First model – a rigid body on a rigid foundation The first model to be considered is a rigid body placed on an ideally rigid foundation. Fig. 1. Model of the load – characteristic points, systems of co-ordinates Reaction forces between the groundand the rigid body can bepresented in various manners. In Fig.2, the normal components of the foundation reaction have been presented separately, and the friction forces are presented aside. 856 J.Grabski, J.Strzałko Fig. 2. Equivalent free-body diagrams of the load: (a) on the assumption of a continuous distribution of the reaction forces; (b) for forces concentrated in corners; (c) in the case of the reduced loading Dynamic analysis of the load hoisting process 857 A distribution of the foundation reaction components is presented for the following cases, subsequently: a) normal forces n(x,y) and tangent forces t(x,y) distributed on the sur- face in a continuous way (in Fig.2a a distribution of these forces along the body edge is shown only); b) concentrated forces (ND,TD,NE,TE,NF ,TF ,NH,TH) applied to the body corners (Fig.2b); c) reduced concentrated loading acting in a given point R of the surface of the contact betweenbodies (the tangent force T and thenormal reaction force N) and the couple (T and −T) with the moment value denoted by MT (Fig.2c). 2.1. Analysis of the equilibrium of the load The presented ways of an introduction force reaction of the ground are equivalent statically, if the following is assumed: – for the model with distributed forces (Fig.2a) – a linear change in the values of normal reaction forces with respect to the ground, i.e. n(x,y)= a0+a1x+a2y (2.1) (xyz is the body local co-ordinate system) and the corresponding values of tangent forces t(x,y)¬ µn(x,y) (2.2) – for the model with forces concentrated in corners (Fig.2b) – a linear relationship that describes the values of normal reactions components Ni(xi,yi)= A0+A1xi+A2yi i =D,E,F,H (2.3) and the friction forces applied in the same points, whose values satisfy the following conditions Ti ¬ µNi i = D,E,F,H (2.4) – for themodel in the form of the resultant forces T and N (Fig.2c) and themoment MT that follows from the reduction of friction forces to the point R T ¬ µN (2.5) 858 J.Grabski, J.Strzałko Thedirections and senses of the friction forces are definedby the directions of possible velocities. The following relations hold N = ND+NE +NF +NH = a ∫ 0 b ∫ 0 n(x,y) dxdy Tx = TDx+TEx+TFx+THx = a ∫ 0 b ∫ 0 tx(x,y) dxdy Ty = TDy +TEy+TFy+THy = a ∫ 0 b ∫ 0 ty(x,y) dxdy (2.6) T = √ T2x +T 2 y MT = TDxyR−TDyxR+TExyR+TEy(xE −xR)−TFx(yF −yR)+ +TFy(xF −xR)−THx(yH −yR)−THyxR = = a ∫ 0 b ∫ 0 [tx(y−yR)+ ty(x−xR)] dxdy The forcesdistributed continuously aredescribedby thequantities nD,nE, nF and nH, which denote the values of pressure per unit area in individual corners. An exemplary relation describing nD has the following form nD = 7ab(G−Sz)+6b(SzxB −GxC −SxzB)+6a(SzyB −GyC −SyzB) a2b2 (2.7) The values of forces concentrated in corners (NE, NF and NH) can be determined from formulas similar to that one for ND ND = 3ab(G−Sz)+2b(SzxB −GxC −SxzB)+2a(SzyB −GyC −SyzB) 4ab (2.8) On the basis of the equations of equilibrium, the co-ordinates (xR,yR) of the point R, through which the resultant reaction force N = G−Sz comes through, and the values of the friction force T = √ S2x+S 2 y as well as their moment MT can be determined. During the analysis of the system, the state of equilibrium is controlled. The system under analysis is in the state of equilibrium for Dynamic analysis of the load hoisting process 859 N > 0 T < µN |MT | < max i |Tx(yi−yR)−Ty(xi−xR)| i = D,E,F,H (2.9) 0 < xR < a 0 < yR < b If the above-mentioned conditions of the equilibrium are not satisfied, a transition to the next calculation phase takes place, namely to solving the equations of motion of the model under analysis. 2.2. Equations of motion of the system The motion of the body is analysed by means of the Newton-Euler equ- ations that describe general motion of the body (displacements of the mass center and rotations) mpC = ∑ P ďKC dt +ω×KC = ∑ MC (2.10) These equations in the matrix form are presented as follows m    ẌC ŸC Z̈C    =     SX SY SZ     +     TX TY N −G     (2.11)     Jξ −Jξη −Jξζ −Jξη Jη −Jηζ −Jξζ −Jηζ Jζ         ω̇ξ ω̇η ω̇ζ     +     0 −ωζ ωη ωζ 0 −ωξ −ωη ωξ 0     · ·     Jξ −Jξη −Jξζ −Jξη Jη −Jηζ −Jξζ −Jηζ −Jζ         ωξ ωη ωζ     =     0 −ζB ηB ζB 0 −ξB −ηB ξB 0         Sξ Sη Sζ     + +     0 −ζQ ηQ ζQ 0 −ξQ −ηQ ξQ 0         Tξ Tη Tζ     +     0 −ζR ηR ζR 0 −ξR −ηR ξR 0         Nξ Nη Nζ     where ẌC, ŸC, Z̈C denote the projections of the acceleration vector of the point C on the axes of the fixed coordinate system (XY Z), ω̇ξ, ω̇η, ω̇ζ are the projections of the angular acceleration vector of the body on the axes attacheed to the body (ξηζ), whereas (ξB,ηB,ζB), (ξR,ηR,ζR), (ξQ,ηQ,ζQ) 860 J.Grabski, J.Strzałko are the co-ordinates that describe the positions of B, R and Q in which the forces S,N and T are applied, respectively. The motion of the boom tip (modelled with a particle of the mass m0, which is located at the end of the boom) is described by the following vector equation m0pA = ∑ P0 (2.12) It means m0     ẌA ŸA Z̈A     =     kX∆XA kY∆YA kZ∆ZA     +     −SX −SY −(SZ +G0)     (2.13) where: kX, kY , kZ denote the equivalent stiffnesses of the boom, and ∆XA, ∆YA, ∆ZA – its deformations. Two additional equations are used in themathematical model. In the case of an elastic rope, the rope tension is obtained as S = cl [ √ (XB −XA)2+(YB −YA)2+(ZB −ZA)2− l(t) ] (2.14) where cl is the rope stiffness and l(t) the rope length. For the model of an inextensible rope, the geometrical constraint equation is introduced (XB −XA) 2+(YB −YA) 2+(ZB −ZA) 2− [l(t)]2 =0 (2.15) The number of degrees of freedomand the form of the equations ofmotion of the load undergo changes during the lifting process. Both the number of relations thatdescribeconstraints and the formof constraint equations change. The subsequent phases of motion are solved with procedures including proper relations for each phase. The conditions and equations describing onephase ofmotionwill be shown bymeans of an example of the sliding of the load on the ground. 2.3. The sliding of the load If the following conditions are fulfilled Zi =0 i = D,E,F,H T = µN N > 0 0 < xR < a 0 < yR < b (2.16) the body slides along the ground – planemotion of the body occurs. Dynamic analysis of the load hoisting process 861 The equations ofmotion are obtained on the basis of relations (2.11), after the following substitutions: ωζ = ψ̇, Z̈C =0, ωξ =0, ωη =0, ω̇ξ =0, ω̇η =0, ϑ =0, ϕ =0, (and also: Nη = Nξ =0, Nζ = N, Tζ =0). The resulting system of algebraic and differential equations should be sup- plemented with additional kinematics relationships which define the constra- ints imposed on the directions of friction forces Viξ = ẊC cosψ+ ẎC sinψ+ ψ̇ηi i = D,E,F,H (2.17) Viη = ẎC cosψ− ẊC sinψ+ ψ̇ξi i = D,E,F,H The components of friction forces, applied in individual corners, are determi- ned then as Tiξ =−µNi Viξ √ V 2 iξ +V 2 iξ i = D,E,F,H (2.18) Tiη =−µNi Viη √ V 2iη+V 2 iη i = D,E,F,H and their sum, according to (2.6)2,3, yields the resultant friction force com- ponents Tξ, Tη. The coordinates of the point Q, where the resultant friction force is applied, can be described by means of the following relations, thus ξQ = 1 Tη (TηDξD+TηEξE +TηFξF +TηHξH) ηQ = 1 Tξ (TξDηD+TξEηE +TξFηF +TξHηH) (2.19) ζQ =−ZC The derived equations allow for determination of the quantities being sought: XC(t), YC(t), ψ(t) and N(t), Tξ(t), Tη(t), ξN(t), ηN(t), ξQ(t), ηQ(t) during the load sliding on the ground. An example of the solution, in which the quoted relations are used, is shown in Fig.3. The motion of the load is forced by a change in the rope length l(t)= l0−vlt. The calculations have been carried out for the following data: – load dimensions: a =2.5m, b =6.0m, c =2.5m – co-ordinates of the point A of the rope xA = [3.0,4.0,10.0] – load density ρ =1000kg/m3 – coefficient of friction between the load and the ground µ =0.25. 862 J.Grabski, J.Strzałko Fig. 3. Results of simulation of the load sliding The placing of the connection point of the rope and container in the con- tainer corner causes that the load slides along the ground in the first phase of motion. In the casewhen conditions (2.16) are not satisfied, and thusmotion of the load is not plane, other solution paths are foreseen in the algorithm. A choice of the solution path is made automatically. 3. Second model – a rigid body on an elastic foundation The second model under consideration is a rigid body that contacts an elastic foundation. This contact is made by elastic and damping elements, whose directions remain vertical (parallel to the axis Z), Fig.4. It is assumed that the reaction forces of the ground are in this case con- centrated forces applied to the corners of the body ND, TD, NE, TE, NF , TF , NH, TH. For negative vertical displacements of the corners Zi < Z0, the normal forces ND, NE, NF , NH are assumed according to the relation Ni(xi,yi)= kZi+hŻi i = D,E,F,H (3.1) Dynamic analysis of the load hoisting process 863 Fig. 4. Model of the load placed on the elastic foundation whereas for positive distances of the corners from the foundation surface Zi > Z0, zero normal forces are assumed Ni(xi,yi)= 0 i = D,E,F,H (3.2) The symbol k denotes the stiffness of the elastic elements introduced, h refers to the coefficient of damping of the damping elements. The values of both the coefficients are selected bymeans of a numerical experiment. The friction forces acting in the corners of the body have the values Ti = µNi i = D,E,F,H (3.3) whereas their directions are determined by the corner velocity directions1. 3.1. Analysis of the equilibrium of the load The model equipped with springs and damping elements remains in the equilibriumonly in the case when the force the rope acts with is equal to zero. The friction forces are equal to zero under such a loading. An increase in the loading caused by the rope results in small displacements of the system and in an appearance of frictional effects. The state in which the load rests on the ground – for a non-zero loading generated by the rope – cannot be realised in such a model. When a loading 1Thezero cornervertical velocity,whichmeans Ti < µNi (i = D,E,F,H), violates the calculation procedure, as the number of degrees of freedom of the system changes in this case. The procedure indicates this situation. In the calculations carried out, such an instance did not occur. 864 J.Grabski, J.Strzałko occurs, small displacements occur as well. A proper selection of the coeffi- cients of elasticity and damping of the foundation allows for reducing these displacements to magnitudes that are insignificant from the practical point of view. 3.2. Motion analysis The application of the loadmodel in the form of a rigid bodyplaced on an elastic foundation allows the avoiding of the necessity of analysing the system with a time-varying number of degrees of freedom. Instead of the limitations imposed by analytically described constraints on the relations between certain co-ordinates, some limitations in the formof reaction forces of the groundhave been introduced. It turns out that the selection of these forces is rather simple. Such an approach simplifies considerably the process of numerical solution of the load motion issue. The number of degrees of freedom of the system and the form of equations do not change during the problem solving. Theequations ofmotionof themodelunderdiscussionhavebeengenerated by computer. Lagrange’s equations of the second kind have been employed for this purpose. The assumed method requires only determination of the kinetic energy of the system, the generalised forces acting on the system and the initial conditions. The remaining procedures are made by computer. The kinetic energy of the system under analysis is described by the formula T = 1 2 q̇ ⊤ 1Mq̇1+ q̇ ⊤ 2R ⊤ 1 JR1q̇2+ 1 2 q̇ ⊤ 1R ⊤ MWR1q̇2 (3.4) where: q1 = [XC,YC,ZC] ⊤ and q2 = [ϕ,ψ,ϑ] ⊤ denote the generalised co- ordinates of the load;M, J are thematrices of masses andmoments of inertia of the load; R1,R stand for thematrices of transformations of the co-ordinates; W refers to the antisymmetric matrix of the co-ordinates of the mass centre of the load (of the point C) in the body co-ordinate system ξηζ. 4. Calculation results and conclusions TheMathematica andMatlab software packages were used in the analysis and calculations carried out for the models of the load under consideration. TheMathematica was used to generate the equations of motion, whereas the numerical calculations were conducted with theMatlab. The results of the calculations allow for drawing a few conclusions. Dynamic analysis of the load hoisting process 865 4.1. Comparison of the results for different models of the system In Fig.5 and Fig.6 the results of calculations for both load models are presented, i.e.: a rigid body placed on an ideally rigid foundation and a body placed on an elastic foundation. The plots include time histories of the co- ordinates describing the location of the mass centre of the load (Fig.5) and the reaction forces fromthegroundat thecontact pointwith thebody(Fig.6a) and the force acting in the rope (Fig.6b).Theobtained results for bothmodels are slightly different. The calculations were carried out for the following data: – container dimensions: a =2.5m, b =6.0m, c =2.5m – mean density of the container material ρ =1000.0kg/m3 – properties of the foundation: coefficient of friction 0.15, coefficient of elasticity k = 1.5 · 107N/m, coefficient of viscous damping h = 0.2 · 106Ns/m – in the case of the elastic foundation model – properties of the rope: coefficient of elasticity cl =1.5 ·10 7N/m, coeffi- cient of viscous damping bl =1.3 ·10 4Ns/m – hoisting speed vl =0.04m/s. Fig. 5. Comparison of the results obtained for different models of the load – the displacements of the mass centre of the load (co-ordinates x and z) 866 J.Grabski, J.Strzałko Fig. 6. Comparison of the results for different models of the load – the rope tension force S and the ground reaction force R 4.2. Calculation results for the model of a load placed on an elastic fo- undation The calculations were carried out for the following data: – container dimensions: a =2.5m, b =6.0m, c =2.5m – mean density of the container material ρ =850.0kg/m3 – properties of the foundation: coefficient of elasticity k = 1.5 ·107N/m, coefficient of viscous damping h =2.0 ·106Ns/m – properties of the rope: coefficient of elasticity cl =1.4 ·10 7N/m, coeffi- cient of viscous damping bl =0.4 ·10 6Ns/m – rope hoisting speed vl =0.3m/s. Motion of the load (the analysed container was eccentrically loaded) is presented graphically in Fig.7. InFig.8 the timehistories of friction forces in four points of the container – ground contact are shown. Differences in the time at which individual corners detach from the ground can be seen on these diagrams (the detachment of the corner from the ground is manifested by zero values of the normal force and the friction force at this point). It can be also observed that one of the corners (point F) impacts against the ground after detachment. The directions along which the friction forces act and the trajectories of four corners of the load can be seen in Fig.9. Dynamic analysis of the load hoisting process 867 Fig. 7. Results of simulations of motion of the lifted container Fig. 8. Results of simulations of motion of the lifted container – values of the friction forces 868 J.Grabski, J.Strzałko Fig. 9. Results of simulations of motion of the lifted container – directions of the friction forces Fig. 10. Comparison of time histories of the co-ordinate x for various coefficients of friction 4.3. Bifurcation of solutions The calculations carried out for various parameters of the system revealed bifurcation of solutions that appeared during these calculations. For certain ranges of the parameters, a bifurcation of the solution takes place at a slight change in the value of one parameter. The results shown in Fig.10b illustrate bifurcation of the solution that occurs at a slight change in the value of the coefficient of friction between the load and the foundation. Dynamic analysis of the load hoisting process 869 Figure 11 shows the time histories of the co-ordinates y and ϕ for different values of the coefficient of friction (within the range 0.363¬ µ ¬ 0.365). For the value µ ∼=0.3635, a distinct change in the character of the time history of the solution occurs. The remaining co-ordinates behave in a similar way – the time histories of ϑ and ψ can be seen in Fig.12. Fig. 11. Bifurcation of the solution (co-ordinates y and ϕ) resulting from a change in the coefficient of friction µ Fig. 12. Bifurcation of the solution (co-ordinates ϑ and ψ) resulting from a change in the coefficient of friction µ 4.4. Conclusions For the three-dimensionalmodel of the crane and the load in the form of a body, the algorithm that aims at the analysis of the load dynamicsmust take into account: – the rope and system tensioning duringwhich the lifted load remains still 870 J.Grabski, J.Strzałko – the beginning ofmotion (it can be translation, rotation around the body edge, rotation around one of the corners) – complete detachment of the lifted load from the foundation – possibility of accidental impacts of the load against the ground – generalmotionof the lifted load (includingactive orpassive rope-affected constraints). The global behaviour (motion) of the load contacting the ground can be described in such away that the deformations of the load itself can be neglec- ted. The physical models of the load discussed in this study, it means: a rigid body located on a rigid foundation and a rigid body interactingwith a founda- tion characterised by elastic and damping properties, proposed to be used in simulations of the lifting of the load from the ground, seem to be suitable for the purpose. Themodels used are accurate enough for the analysis of motion of the load. TheMathematica package used in the above-mentioned calculations allows one to solve the task effectively, although it requires individual consideration of each phase of the lifting of the load from the ground. Its advantage lies in the possibility of obtaining many formulas in an analytical form. The application of the Matlab package offers the possibility of numerical analysis of thewhole lifting process, aswell as of the transporting and lowering the lifted load, independentof the sequence of suchphenomenaas detachment, rotations, impacts, and sliding. The calculationsmade for various values of the parameters of themodel al- lowed for determination of the change in the character of the solution (solution bifurcation) in certain regions of the assumed values of these parameters. References 1. Borkowski W., Konopka S., Prochowski L., 1996, Dynamika Maszyn Roboczych, WNTWarszawa 2. Grabski J., 1998, Analysis of engineering machine dynamics using matrix methods, Scientific Bulletin of TUL, 804 (in Polish) 3. Grabski J., Strzałko J., 1997,Automatic generation ofBoltzmann symbols for mechanical problems,Mechanics and Mechanical Engineering, 1, 1, 61-77 4. Hakala I., Sorsa T., 1998, Sway Prevention in RTG Cranes, Koncra- nes’World (http://www.kcinet.com/kciworld/spring1998) Dynamic analysis of the load hoisting process 871 5. 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Wriggers P., Zavarise G., 1993, Recent trends in computational methods for contact problems, XI Polish Conference on Computer Methods in Mecha- nics, 127-148 872 J.Grabski, J.Strzałko Analiza dynamiki procesu podnoszenia ładunku Streszczenie W pracy zostały omówione propozycje modeli przeznaczonych do wykorzystania w analizie dynamiki ładunku podnoszonegoprzez dźwignice. Szczególna uwaga zosta- ła zwrócona na zjawiska zachodzące przed oderwaniem ładunku od podłoża. Spośród dwóch przedstawionychmodeli ładunku bardziej przydatny do analizy procesu pod- noszenia okazał się model w postaci sztywnej bryły i sprężystego podłoża. Wyniki otrzymane z przeprowadzonych obliczeń numerycznych ujawniły zjawisko bifurkacji rozwiązania przy niewielkiej zmianie wartości parametrówmodelu. Manuscript received October 7, 2002; accepted for print April 15, 2003