# Research into 2D Dynamics and Control of Small Oscillations of a Cross-Beam during Transportation by Two Overhead Cranes

Authors: Alexander V. Perig; Alexander N. Stadnik; Alexander A. Kostikov; Sergey V. Podlesny
Journal: Shock and Vibration
 (2017)

---

## Abstract

A new mathematical model of a 3DOF 2D mechanical system “transported cross-beam, two moving bridge cranes” has been proposed. Small system oscillations have been derived through the introduction of Lagrange equations. The numerical estimation of 3DOF system motion has been carried out with equation-based Modelica language. The present article uses the Lagrange method and numerical and optimization methods, realized with JModelica.org and Optimica freeware. The absolute swaying of the cross-beam with respect to the displacement of the two moving bridge cranes was estimated. The phase portraits of the 3DOF system for linear and angular coordinates were presented. An open loop optimal control problem was posed for the motion of the bridge cranes. A “bang-bang” control strategy was implemented for the derivation of an optimal control solution, which enables the travel of two bridge cranes at a prescribed distance for minimum time and minimum swaying of a heavy cross-beam. The derived results of the numerical simulation can be easily practically realized by crane operators with good agreement with simple engineering estimations. The proposed control strategy enables synchronous motion of two bridge cranes with a cross-beam that practically solves the posed problem of unwanted excessive oscillations of a heavy cross-beam during transportation.


---

## Body

## 1. Background and Introduction

Current civil engineering technologies often assume a modular concept for the assembly of buildings and other large structures. Individual modules are manufactured at specialized plants and plant departments with further transportation of modular sections to a storage warehouse. Further transportation routes of individual modules lead from the warehouse to the erecting yard at the construction site. Individual modules and modular sections are many times very heavy and bulky objects. Construction plant operation requires simultaneous usage of several cranes, working together at the various stages of transportation of these heavy modules [1, 2].An example of modular unit construction would be the erection of the tower of a wind-powered generator with several modular sections, utilizing two boom cranes working together. This practical case of the two-crane assembly of a wind machine tower requires 3D turning and tumbling of cylindrical modular sections from a horizontal to a vertical position. In contrast to this example the present article (Figures1, 2, 3, and 6) is focused on the solution of a problem of horizontal transportation of a finished heavy steel beam from a fabrication shop to a storage warehouse.Figure 1
Typical plant photos of heavy cross-beam transportation with two overhead cranes.Figure 2
The computational scheme for transportation of heavy cross-beam DE by two bridge cranesA and B.Figure 3
The computational scheme of the initial position of mechanical system during transportation of heavy cross-beam DE by two bridge cranesA and B.A shipbuilding example of collaborative working of two or more cranes would be the problem of transportation of a heavy bulk marine engine weighing in the range of 3–3000 tons to the installation site.Other examples in civil engineering include the lifting-and-handling problems of modular building elements of construction, bays of a building, ceiling and floor slab panels, and spandrel walls. Construction problems of roofs, floors, and domes of roofed stadiums and shopping centers require preliminary manufacturing of the heavy bulk trussed frame modules with further transportation to building sites.It is important to note that attempts to transport heavy bulk stores with a single crane are very undesirable because of the high probability of torsional oscillations of a heavy load. Moreover, usage of two cranes working together for heavy load transportation is a truly natural way for handling displacements in civil and mechanical engineering as nature itself has supplied humans with two hands and two legs for more efficient performance of lifting-and-handling operations.Overhead crane dynamics has been studied in the research efforts of such scientists as Abdel-Rahman et al. (2003) [1], Deen Ali et al. (2005) [2], Arena et al. (2015) [3], Cartmell et al. (1998) [4], Castelli et al. (2014) [5], Cha et al. (2010) [6], Goodwin (1997) [7], Huang et al. (2015) [8], Lahouar et al. (2009) [9], Pigani and Gallina (2014) [10], Sawodny et al. (2002) [11], Smoczek (2014) [12], Zi et al. (2008) [13].Abdel-Rahman et al. (2003) have discussed mechanical design schemes and mathematical models of gantry cranes and bidirectional gantry cranes with translational displacements of payloads [1]. However payload transportation by two overhead cranes with cross-beam usage is not addressed in [1].Arena et al. (2015) have proposed several 3D kinematic and dynamic models of container cranes with constant distances between fixed points of supporting rigid and elastic cables [3]. However some disadvantage of Arena’s model is associated with a prescribed imposition of geometric constraints on the distances between fixed points for supporting cables. A 2D kinematic model of the present article generalizes Arena’s model in the vertical plane [3].Castelli et al. (2014) have studied a kinetostatic model of a Cartesian cable-suspended robot [5]. A kinematic model of Castelli et al. (2014) is focused on the transportation of a payload using double cables with variable lengths, which provide translational linear motion of the payload without obstacles. There is an analogy between a swaying Cartesian cable-suspended robot (Castelli, 2014) and the swaying cross-beam ED in our problem. However, during transportation the payload may have additional extraneous transverse oscillations. Moreover, successful payload transportation requires a synchronized change of lengths of both supporting cables.Cha et al. (2010) have studied extra-heavy cargo lifting by two floating cranes with a barge ship connected between the two cranes [6].Huang et al. (2015) have proposed a new double-pendulum mechanical model, describing the transportation of a large payload with distributed mass, which is attached to one trolley by one suspension cable and two rigging cables [8]. Huang et al. (2015) have suppressed payload vibration during transportation by control of trolley acceleration [8]. Mechanical disadvantage of Huang’s model is a usage of only one suspension cable. However there are possible strong engineering benefits of using two suspension cables without rigging cables, which is beyond the scope of Huang’s research [8].Kostikov et al. (2016) and Perig et al. (2014) have studied cargo transportation by crane with one slewing pivot point ([14–17]).Analysis of references [1–18] clearly shows that the dynamics of transportation of the cross-beam by two overhead cranes is not fully addressed in available works [1–18]. This confirms the actuality, relevance, and industrial significance (Figure 1) of the present research.


## 2. Research Actuality and Prime Novelty of Research

The aim of the present research is the dynamic description of the transportation of a heavy cross-beam by two overhead cranes (Figures1, 2, 3, and 6).The objective of this research is a 3DOF 2D guided mechanical system “cross-beam ED, cables AE and BD, moving bridge crane A, moving bridge crane B.”The subject of this research is the study of small oscillations of a heavy cross-beam, induced by the guided motion of two overhead cranes.The prime novelty of the present research is the introduction of a new mathematical model of a 3DOF 2D mechanical system “cross-beam, two moving bridge cranes” with further dynamic and optimization analysis of small system oscillations and numerical evaluation of guided system motion in the first approach.The main contribution of the present original research article to the field of engineering transportation is as follows. The 2D problem of horizontal transportation of a heavy steel structure by two bridge cranes, operating at different horizontal levels with different length hoisting cables, has been addressed.The theoretical novelty of the research is the formulation of differential equations for the motion of a 2D mechanical system “cross-beam, two moving bridge cranes,” determination of natural frequencies and periods of oscillations, and the numerical derivation of laws of system motion for coherent initial conditions. An open loop optimal control problem was formulated and solved with JModelica.org and Optimica freeware. The posed optimal control problem numerical solution provides minimization of transportation time for bodies of the mechanical system together with minimal swaying of the heavy bulk load.The practical sphere of engineering applications of the present transportation problem is in lifting-and-handling machinery and civil engineering.The plant photo in Figure1 shows the two bridge cranes with crane operators’ cabins, which move along the different horizontal levels. The different lengths of supporting cables l1 and l2 for holding and carriage of heavy cross-beam by two bridge cranes are clearly observable in Figures 1, 2, 3, and 6.Figure1 contains an experimental plant photo of a cross-beam being transported with two independently moving cranes and provides a schematic illustration of the cross-beam inclination angles. The straight inclined line within the cross-beam, which passes through the fixing points of conjunction of the carrying cables’ roping to the cross-beam, forms a slope angle γ0 to the horizon. As a matter of engineering convenience, the present article introduces three interconnected angles γ, γ0, and δ, which uniquely determine the angle of rotation of the cross-beam. The angle γ in Figure 2 is the current angle of rotation of a cross-beam ED with respect to the horizontal x, where cross-beam ED displacement occurs along the horizontal x direction. Factually the angle γ in Figure 2 is a kind of incidence angle, which characterizes the current angle of a cross-beam ED with the direction of the velocity vector of the horizontal displacements of the cranes. It will be shown further that the law of the change of angle γ will be a harmonic one with respect to a certain position of static equilibrium of cross-beam ED, which is characterized with angle γ0 (Figures 1 and 3). The new angle δ in Figure 2 will be introduced as an angle of rotation of a cross-beam ED with respect to the angle γ0 (Figures 1 and 3) according to the formula δ=γ-γ0. Therefore the introduction of angle δ provides a way to select the harmonic component δ of angle γ of cross-beam ED rotation. It is necessary to note that angular velocity of cross-beam ED rotation, which is necessary for calculation of cross-beam kinetic energy, will be determined in the same way using δ and γ angles, as ωED=dδ/dt=dγ/dt. It will be shown later that both δ and γ angles are second-order infinitesimal quantities with respect to the small angle α of the deviation of cable 1 from the vertical in Figure 2. Therefore it is possible to consider that the cross-beam ED makes a translational movement with ωED=dδ/dt=dγ/dt≈0.


## 3. Kinematics of the Mechanical System: Geometric Constraints, Coordinates, and Velocities

The mechanical model of the present problem (Figure1) is shown in Figures 2, 3, and 6. Dynamic equations for the motion of the mechanical system will be derived using the Lagrange equations [14, 16–18] for generalized coordinates x1, x2, and α. So it is necessary to determine kinetic and potential energies of the mechanical system as functions of our chosen generalized coordinates. For this purpose we have to calculate the coordinates of bodies A, B, and material point C3 of the cross-beam body DE.The Cartesian coordinates of pointB, which coincides with the point C2, are as follows:(1)xB=acos⁡γ0+x2;zB=bxB=x1+l1·sinα+a·cosγ-l2·sinβ;zB=-l1·cosα+a·sinγ+l2·cosβ.System (1) of transcendental equations, which determine the Cartesian coordinates of point B, contains variables x1; x2; and α. It is necessary to solve system (1) with respect to the unknown angles β and γ, that is, to derive β=fβ(x1,x2,α) and γ=fγ(x1,x2,α) for γ=γ0+δ, where the current angle δ is counted off from the initial inclination angle γ0 in the direction of cross-beam ED motion in Figures 2 and 3. The simplest way to solve system (1) with respect to the β, γ, and δ is usage of a small angle assumption (Appendix A–D).In order to calculate the initial inclination angleγ0 (Figures 1 and 3) of cross-beam ED with a horizon x it is possible to introduce an additional geometric parameter Δz0=a·sin⁡(γ0) in Figure 3 by substitution of the initial time moment t0 = 0 (s) into the second lower equations of system (1), which yields the following expression zB=b=-1·l1·1+a·sin⁡(γ0)+l2·1, and Δz0=a·sin⁡(γ0)=b+l1-l2 or the sinus of γ0 angle is sin⁡(γ0)=(b+l1-l2)/a=Δz0/a (Figure 3). These reasons confirm alternative results in formulae (A.2)–(A.4).Equation (A.12) establishes a simplified linear dependence between swing angles α and β. So one gets from (A.12) that the small angle β in Figure 2 can be estimated on the basis of (A.12) in the following way:(2)β≈1l2·x1-x2+l1α.Equation (2) determines the swing angle β as the simplified linear function of the variables x1, x2, α; that is, β=fβ(x1,x2,α).The rate of change of the swing angleβ, that is, the small angular velocity ωβ=(dβ/dt), can be estimated on the basis of (2) in the following way:(3)dβdt≈1l2·dx1dt-dx2dt+l1dαdt.Equation (B.9) determines the small inclination angle δ to be a simplified linear function of the variables x1, x2, α; that is, δ=fδ(x1,x2,α). Algebraic expression (B.9) accurately proves the correctness of statement (A.9) concerning the second order of infinitely small δ (infinitesimality of δ=O(α2)) for a small inclination angle δ, which determines the current inclination of the cross-beam ED in Figure 2. Equation (B.9) allows the development of an alternative approximate expression for the small angle γ for cross-beam ED inclination in Figure 2:(4)γ=γ0+δ≈γ0+12·a·l2·cos⁡γ0x1-x2+l1α2-l1·l2·α2.The derived alternative equations (B.9) and (4) provide a way for the estimation of the value of the angular velocity ωδ=(dδ/dt)=ωγ=(dγ/dt) by the following expression:(5)dγdt=dδdt≈1a·l2·cos⁡γ0x1-x2+l1α·dx1dt-dx2dt+l1dαdt-l1·l2·α·dαdt.Algebraic expression (5) also proves the correctness of statement (A.9) concerning the second order of infinitely small ωδ=ωγ (infinitesimality of ωδ=(dδ/dt)=ωγ=(dγ/dt)=O(α2)) for the angular velocity of the cross-beam ED in Figure 2.After simplifications (C.4) yields that (6)zC3≈12·l1·α2+14·l2x1-x2+l1α2-l1·l2·α2+12·b-l1-l2.


## 4. Dynamics of the Mechanical System: Potential and Kinetic Energies, Generalized Forces, and Lagrange Equations

The potential energy for the mechanical system in Figure2 can be calculated as(7)Π=m3g·zC3.Substitution of approximated expression (6) into (7) yields(8)Π≈m3g·12·l1·α2+14·l2x1-x2+l1α2-l1·l2·α2+12·b-l1-l2.Estimation of potential energy enables calculation of the generalized forcesQx1, Qx2, and Qα, which can be calculated as partial derivatives of potential energy Π with respect to the general coordinates x1, x2, and α (Appendix E). Substitution of (E.2), (E.4), and (E.9) into a one-column matrix yields the following vector of generalized forces:(9)Q≈m3g·12·l2·-x1+x2-l1αx1-x2+l1α-l1·l2+l12·α-x1·l1+x2·l1.Simplification of (F.7) results in the following approximated expression:(10)T≈12·m1+m3·dx1dt2+12·m2·dx2dt2+12·m3·l12·dαdt2+m3·l1·dx1dt·dαdt.The matrix form of the left-hand sides of Lagrange equations (G.1), (G.7), and (G.13) or (G.5), (G.11), and (G.17) yields the following matrix expression:(11)ddt∂T∂q˙-∂T∂q≈m1+m30m3·l10m20m3·l10m3·l12·d2x1dt2d2x2dt2d2αdt2-0.The vector form of Lagrange equations is as follows:(12)ddt∂T∂q˙-∂T∂q=Q,where the left-hand side of (12) is determined by matrix product (11) and the right-hand side of (12) is determined by the vector-column (9).Substitution of (11) and (9) into (12) results in the following matrix form of Lagrange equations for the motion of the system in Figures 2 and 3:(13)m1+m30m3·l10m20m3·l10m3·l12·d2x1dt2d2x2dt2d2αdt2≈m3g·12·l2·-x1+x2-l1αx1-x2+l1α-l1·l2+l12·α-x1·l1+x2·l1.Derived equation (G.6), (G.12), (G.18), or (13) allows the determination of natural periods of system oscillations, natural frequencies of mechanical system, and deriving motion equations for each of the three bodies of studied mechanical system for different initial conditions in Figures 1–3. The numerical solution of Lagrange equation (G.6), (G.12), (G.18), or (13) enables the determination of motion patterns and formulating applied engineering recommendations concerning motion of the system.


## 5. First Computational Example of Numerical Integration of Derived Lagrange Equations for the Studied Mechanical System with JModelica.org Freeware

Both Figure3 and formulae (2)-(3) yield that at the initial time moment t0 = 0 (s), (2) results in β0≈(x10-x20+l1·(α0))/l2=0. The time derivative of this expression provides an analogical algebraic equation for initial velocities ωβ(0)=d(β0)/dt≈(V10-V20+l1·((d(α0))/dt))/l2 of system in Figures 1–3, which results in the following equations: l2·ωβ0=V10-V20+l1·ωα(0) and ωα(0)=d(α0)/dt = (V20-V10+l2·ωβ(0))/l1 = (VB0-VA0+l2·ωβ(0))/l1. The numerical JModelica.org-enhanced solution of the posed problem in Figures 4 and 5 of the present article was derived in the typical engineering case V20=VB0≠0 of right-hand side motion of the crane B along x direction, when the system motion started from the system equilibrium position with zero-initial-values of three generalized initial coordinates x10=x20=(α0)=0. Assumption of V10=VA0=0 (m/s) and V20=VB0≠0 yields ωβ(0)=d(β0)/dt≈(l1·ωα(0)-V20)/l2. Additional assumption ωα(0)=0 results in ωβ(0)=d(β0)/dt≈(-1·V20)/l2 = (-1·VB0)/l2. The derivation of numerical solutions of inverse dynamic problems for mechanical system in Figures 1–3 requires the assignment of different initial values of system coordinates and velocities, which satisfy the abovementioned constraint equations for initial coordinates β0≈(x10-x20+l1·(α0))/l2=0 and initial velocities ωα(0)=d(α0)/dt = (V20-V10+l2·ωβ(0))/l1=(VB0-VA0+l2·ωβ(0))/l1. These questions concerning the influence of initial conditions were properly addressed by the authors in the present research and one of the typical cases is shown in numerical plots in Figures 4 and 5.Figure 4
JModelica.org-derived numerical plots for linear coordinatesx1=x1(t); x2=x2(t) (a); and x2-x1=x2(t)-x1(t) (b), computed for system (13) with nonzero initial velocities VB(0)=0.5 (m/s) and ωβ(0)≈-0.083 (rad/s).
(a) (b)Figure 5
JModelica.org-derived numerical plots for angular coordinatesα=α(t) (a); β=β(t) (b); δ=δ(t) (c); γ=γ(t) (d); ωγ = ωγ(t) = (d(γ(t))/dt) (e); and ωγ = ωγ(γ); that is, (d(γ(t))/dt)=f(γ) (f), computed for system (13) with nonzero initial velocities VB(0)=0.5 (m/s) and ωβ(0)≈-0.083 (rad/s).
(a) (b) (c) (d) (e) (f)Figure 6
The computational scheme for transportation of heavy cross-beam DE by two guided bridge cranesA and B with applied control forces to carts A and B with forces’ magnitudes F1=u1(t) and F2=u2(t).It is useful to get the numerical solution of system (13) of Lagrange equations (G.6), (G.12), and (G.18) with freeware code JModelica.org [14, 15, 19] for the following numerical values of parameters of the mechanical system in Figure 2: gravity acceleration is g = 9.81 (m/s2); masses of movable carts A and B are m1 = m2 = 1000 (kg); mass of the cross-beam ED is m3 = 10000 (kg); length of the left cable AE is l1 = 6 (m); length of the right cable BD is l2 = 5 (m); vertical distance between the carts is b = 2 (m); length of the cross-beam ED is a = 4 (m).The initial conditions for system (13) of Lagrange equations (G.6), (G.12), and (G.18) are as follows: it is possible to assume that at the initial time moment t = 0 (s) the right upper cart B in Figure 2 moves rectilinearly to the right; that is, initial coordinate of the left cart A is x1(0)=0 (m); initial velocity of the left cart A is VA(0)=(d(x1(0))/dt)=0 (m/s); initial coordinate of the right cart B is x2(0)=0 (m); initial velocity of the right cart B, which moves to the right, is V20=VB(0) = (d(x2(0))/dt)=0.5 (m/s) ≠ 0; initial swing angle of the left rope AE is α(0) = 0(rad); initial angular velocity of the left cable AE is ωα(0)=(d(α(0))/dt)=0 (rad/s). The abovementioned constraint equations yield that ωβ(0)=d(β0)/dt≈(-1·V20)/l2=((-1)·VB0)/l2 = -1·0.5 (m/s)/6 (m) ≈ –0.083 (rad/s). The corresponding computational results are shown in Figures 4 and 5.Computational Figures4 and 5 show the increase of carts’ coordinates with respect to motion time t. The computational curves in Figures 4 and 5 also show the additional harmonic oscillations. Computational dependencies of the angular coordinates in Figure 5 are harmonic curves. The computational amplitudes in Figure 5 confirm the correctness of small angle assumption in the present article.Computational Figure5(f) shows an unstable focus at the phase plane (γ(t),d(γ(t))/dt) that means a certain increase in amplitude during cross-beam oscillations. However it is possible to neglect some increase of γ amplitude by taking into account the fact that both γ(t) and (d(γ(t))/dt) are second-order infinitesimal quantities. So from an engineering standpoint some small increase in γ(t) during a cross-beam transportation is negotiable and not essential. Moreover Figure 5(f) shows the necessity of application of additional damping devices for suppression of a γ(t) oscillations.


## 6. Dynamics of the Guided Mechanical System

The formulation of the optimal control problem for guided motion of bridge cranesA and B requires the application of two independent external control forces to moving carts A and B with magnitudes F1=u1(t) [N] and F2=u2(t) [N] of variable forces in Figure 6.Substitution of (I.9), (I.14), and (I.19) into the one-column matrix yields the following vector of generalized forces for the guided mechanical system:(14)Qδ≈m3g·12·l2·-x1+x2-l1αx1-x2+l1α-l1·l2+l12·α-x1·l1+x2·l1+F1F20.Substitution of (11) and (14) into (12) results in the following matrix form of the Lagrange equations for the motion of the guided mechanical system in Figure 6:(15)m1+m30m3·l10m20m3·l10m3·l12·d2x1dt2d2x2dt2d2αdt2≈m3g·12·l2·-x1+x2-l1αx1-x2+l1α-l1·l2+l12·α-x1·l1+x2·l1+F1F20,where F1=u1(t) [N] and F2=u2(t) [N] in Figure 6.Matrix equation (15) shows that the addition of control into system (15) can be done through the introduction of control forces F1=u1(t) [N] and F2=u2(t) [N] in the form of the additional vector components u1t;u2t;0T, added to the right-hand side of equation (15).Formulation of the governing system of ordinary differential equations (J.4), (J.5), and (J.6) or (15) for motion of the guided mechanical system in Figure 6 allows us to pose the following optimal control problem.It is necessary to find the control functionsu1(t) and u2(t) for control forces F1=u1(t) [N] and F2=u2(t) [N] in Figure 6, which provide minimization of the objective function (16)J=tf+∫0tfαt2dt,where α(t) [rad] is the variable angle of the cable AE with the vertical axis z and tf [s] is the time of travel of the bridge cranes A and B of the prescribed desired distance S.Minimization of the objective functionJ (16) has to be done for the following constraints:(17)ddtx1t=x1pt;(18)ddtx2t=x2pt;(19)ddtαt=αpt;(20)m1+m3·ddtx1pt+m3·l1·ddtαpt≈u1t-m3g·12·l2·x1t-x2t+l1·αt;(21)m2·ddtx2pt≈u2t+m3g·12·l2·x1t-x2t+l1·αt;(22)m3·l1·ddtx1pt+m3·l12·ddtαpt≈m3g·12·l2·-l1·l2+l12·αt-x1t·l1+x2t·l1;(23)x10=0;x1p0=0;x20=0;x2p0=0;α0=0;αp0=0;(24)x1tf=S;x1ptf=0;x2tf=S;x2ptf=0;αtf=0;αptf=0;(25)u1t≤u1max;u2t≤u2max.Introduction of controlu1t;u2t;0T into equations (J.4), (J.5), (J.6), and (15) provides possibility of deriving an optimal control in dependence from the form of a minimized functional J (16) with equal weight coefficients.


## 7. Second Computational Example of a Numerical Solution of Derived Governing Equations for the Studied Guided Mechanical System with JModelica.org and Optimica Freeware

It is possible to achieve a numerical solution of system (16)–(25) for minimized functional (16), governing equations (17)–(22), initial conditions (23), terminal conditions (24), and limitations on the control forces (25) with module Optimica of freeware code JModelica.org [14, 15, 19] for the following numerical values of parameters of the mechanical system in Figure 6: gravity acceleration is g = 9.81 (m/s2); masses of movable carts A and B are m1 = m2 = 1000 (kg); mass of the cross-beam ED is m3 = 10000 (kg); length of the left rope AE is l1 = 6 (m); length of the right rope BD is l2=5 (m); vertical distance between the carts is b=2 (m); length of the cross-beam ED is a=4 (m); prescribed desired transportation distance for bridge cranes A and B is S=40 (m); the maximum value of the first control force is u1max=100 (N); and the maximum value of the second control force is u2max=100 (N). Derived plots of motion of guided system with zero initial conditions (23) are shown in Figure 7.Figure 7
JModelica.org- and Optimica-derived numerical plots for linear coordinatesx1=x1(t); x2=x2(t); and x3=x3(t) (a); control forces F1=u1(t) [N] and F2=u2(t) [N] (b); angular coordinates α=α(t) and β=β(t) (c); angular coordinate δ=δ(t) (d); synchronized velocities Vx1(t)=(d(x1(t))/dt), Vy1(t)=(d(y1(t))/dt), and VxC3(t)=(d(xC3(t))/dt) of points C1, C2, C3 during guided system motion (e); and ωγ=ωγ(γ); that is, (d(γ(t))/dt)=f(γ), obtained for synchronous motion of the mechanical system with an open loop control.
(a) (b) (c) (d) (e) (f)


## 8. Discussion of Derived Results

The present problem is a new and mathematically complex research problem. Complexity of the problem is connected with the imposition of essentially nonlinear geometric constraints that cause a lot of issues and problems with the derivation of differential equations of motion.Development of the governing equations was based on the application of a small parameter linearization method. The numerical solution of the problem confirmed the correctness of applying linearization techniques. Derived simulation results were in good agreement with the basic conservation laws of classical mechanics.The linear momentum of the system at the initial timet=0 (s) is determined by the linear momentum of cart B and can be calculated as m2·VB(0) = 1000 (kg) · 0.5 (m/s) = 500 (kg·m/s). The total mass of the whole system is (m1+m2+m3) = 12000 (kg). It is possible to determine the linear velocity of the center of gravity of the mechanical system at an arbitrary time with the introduction of a theorem defining center-of-gravity motion conservation as VC=(m2·VB(0))/(m1+m2+m3)=500 (kg·m/s)/12000 (kg) ≈ 0.042 (m/s). The derived value of VC determines the average velocity for the entire system in Figure 2. So it is possible to estimate the traversed path of the carts A and B in Figure 4(a) for the moment of time τ=10 (s) with the formula S≈VC·τ≈0.042 (m/s) · 10 (s) ≈ 0.42 (m). There is good agreement of this estimated value S≈0.42 (m) with the numerical value x1τ≈0.5 (m) in Figure 4(a).It is necessary to note that, in addition to the motion due to the velocity of the center of gravity, there are additional system motions with natural frequenciesk1≈1.021 (1/s) and k2≈3.794 (1/s) (Appendix H), and the corresponding natural periods τ1=2·π/k1≈6.155 (s) and τ2=2·π/k2≈1.656 (s). These two natural periods and amplitudes are clearly observable in Figures 4 and 5.The numerical solution of an optimal control problem (16)–(25) allows determination of the control variables u1(t) and u2(t) for control forces F1=u1(t) [N] and F2=u2(t) [N] in Figure 7(b), which enable transportation of a cross-beam from the initial to the final position with minimum time and minimum swaying in Figures 7(c) and 7(d). These control variables u1(t) and u2(t) are the control forces F1=u1(t) and F2=u2(t), applied to the first and second bridge cranes A and B. Optimica-derived Figure 7(b) shows that it is possible to provide an optimal motion in the abovementioned sense by the following strategy, when at the first half of motion time the control forces u1(t) and u2(t) should be constant and positive and at the second half of motion time the control forces u1(t) and u2(t) should be negative. As a result, in the beginning, acting forces provide maximum acceleration of the bridge cranes and then deceleration. This control strategy is a “bang-bang” control method with the switching point in the time moment t∗=tf/2, where tf is the time of travel of bridge cranes from one position to another. It is shown in Figure 7(b) that during the last 2-3 seconds the control forces have negative values because it is necessary to provide slowing-down (deceleration) of the system motion. In this case the values of control forces are 0.25·max{u1(t),u2(t)}. This kind of a control strategy enables travel to the final position with near zero-values of velocities Vx1≈Vx2≈Vx3≈0 (m/s) and near zero-values of swaying angles α≈β≈0. An easy practical realization of these recommendations by bridge cranes operators is the real advantage of the found control strategy. Really, in the first half of the way, crane operators have to accelerate bridge cranes, and in the second half of the movement, crane operators have to decelerate bridge cranes. This control strategy is relatively easy for realization in manual mode using available crane control mechanisms. The proposed control strategy reduces transportation time and partially decreases intensity of the labor of crane operators.Analysis of Figure7(a) indicates that the proposed control strategy enables synchronous motion of bridge cranes with quite small and negligible oscillations of a heavy cross-beam. Analysis of Figure 7(c) shows that deviation of a cross-beam from equilibrium position in the horizontal direction does not exceed 0.002 (rad), and in the vertical direction the cross-beam deviation is much smaller and does not exceed 4·10−6 (rad).It is easy to confirm the correctness of the proposed optimal control with engineering estimation. According to the law of variation of momentum along the horizontal direction, the impulse of control forces forF1=u1(t) [N] and F2=u2(t) [N] is (u1+u2)·(tf/2)≈(100+100)·49 (N·s) ≈ 9800 (N·s). Moreover, it follows from Figure 7(e) that the linear momentum of the system to this time moment 49 s is (m1+m2+m3)·VC≈(1000+1000+10000)·0.8 (kg·m/s) ≈ 9600 (N·s). The results are approximately the same in the deceleration region. These simple engineering estimations confirm the correctness of the derived optimal control strategy.Further research directions are associated with dynamic analysis of internal forces, reactions of constraints, introduction of additional elastic-damping devices, development and analysis of other modes of optimal control, introduction of additional elastic-damping devices, and search for other natural frequencies in 3D problem.


## 9. Final Conclusions

The present problem is a truly new problem of lifting-and-handling machinery.This problem has arisen from applied engineering assignments, associated with civil engineering and transportation problems in plant departments and in field environment conditions.There are previously known studies of vertical lifting, space canting, and 3D rotation of the load up to 90 deg with two boom cranes, working together. But there is a lack of relevant and specific information concerning horizontal transportation of the load with two bridge cranes.The present original research article is focused on the development of a mathematical model, describing the horizontal transportation of the load (cross-beam) in the vertical plane.Transportation dynamics of heavy cross-beam moved by two overhead cranes was addressed with the introduction of the Lagrange equations using small angle assumptions.A 3DOF mechanical system “heavy cross-beam, two moving carts” was studied.A system of Lagrange equations for a 3DOF mechanical system was developed and numerically solved.A number of general trends and features of the heavy cross-beam motion were found using mathematical and numerical simulation techniques. Among motion features are the phenomena of insignificant horizontal swaying of a cross-beam in motion direction as well as infinitesimally small angles of cross-beam oscillations around the center of mass.One of the methodologies of the search of optimal control was proposed. An optimal control strategy with minimization of transportation time and swaying anglesα and β was developed with an application of JModelica.org and Optimica freeware.The proposed mathematical model for an optimal control system allows development of the corresponding graphs and nomographic charts for formulation of applied engineering recommendations.The posed transport engineering problem will find further development and application with the introduction of elastic-damping devices, changing hoisting cables lengths, and switching to a 3D simulation.



---

*Source: 9605657-2017-02-15.xml*