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Journal of Sound and Vibration (1999) 223(4), 645657Article No. jsvi.1999.2154, available online at onTIPPING LOADS OF MOBILE CRANES WITHFLEXIBLE BOOMSS. KILICASLAN, T. BALKAN ANDS. K. IDERDepartment of Mechanical Engineering, Middle East Technical University,06531 Ankara, Turkey(Received 23 July 1997, and in nal form 4 January 1999)In this study the characteristics of a mobile crane are obtained by using aexible multibody dynamics approach, for the determination of safe loads toprevent tipping of a mobile crane. Only the boom of the crane is assumed to beexible since it is the only element that has considerable deections inapplications. The coupled rigid and elastic motions of the crane are formulatedand software is developed in order to carry out the dynamic analysis. Thevariation of piston force with respect to boom angular position for dierentboom motion times are simulated, load curves are generated and the results arecompared with the experimental results obtained from a 10 t mobile crane.# 1999 Academic Press1. INTRODUCTIONCranes as mechanical systems are in general closed-chain mechanisms withexible members. In the problem of determination of safe loads which as afunction of the boom angular position, the solution of the dynamic equations isnecessary.There are few studies related to the dynamics and control of mobile cranes forvarious applications. In almost all of these studies the body exibility is nottaken into consideration. A dynamic model for the control of a exible rotarycrane which carries out three kinds of motion (rotation, load hoisting and boomhoisting) simultaneously is derived by Sato and Sakawa 1. Only the jointbetween the boom and the jib is assumed to be exible. The goal is to transfer aload to a desired place in such a way that at the end of the transfer the swing ofthe load decays as quickly as possible. The application of a hook load and safeload indicator and limiter for mobile cranes is presented by Balkan where themicroprocessor-based control system for the determination of current hook loadis based on oil pressure and boom angle 2.In this paper, mobile crane characteristics are determined by using exiblemultibody analysis. Kinematics and equations of motion of the exiblemultibody system are derived. Software has been developed to carry outdynamic analysis of the crane. In the exible dynamic analysis, the coupled rigidand elastic motion of the system is formulated by using absolute co-ordinates0022460X/99/24064513 $30.00/0# 1999 Academic Press646S. KILICASLANET AL.and modal variables 3, 4. Then, joint connections and prescribed motions areimposed as constraint equations. The exible body is modelled by the niteelement method and modal variables are used as the elastic variables by utilizingmodal transformation.The variations of the piston force with respect to the boom angular positionsare analyzed for different boom motion times to illustrate the effect of exibilityby using the developed software. Load curves are generated for various boommotion times and compared to those of the manufacturer.2. MODELLING OF THE CRANESince there is experimental work on the tipping load control of a COLESMobile 930 crane, for the application of the developed software, the structure ofthe above mentioned crane and its parameters are used 5. However, the methodof analysis can easily be applied to similar types of cranes with simplemodications.In general, mobile cranes are operated under blocked conditions by thevertical jacks. The load is attached to the hook and the boom is hoisted. Sincethe excessive raising of the load is dangerous, the height of the load is controlledby lengthening the rope. During hoisting, lowering and transportation of theload, the crane is not rotated, due to some restrictions such as very huge and/orheavy loads, space problems, etc.In every angular position of the boom, there is a maximum load above whichtipping might probably occur. Since the angular position of the boom changesonly during the up and down motion of the load, which is actually a planarmotion, modelling and analysis are carried out in two dimensions.Figure 1. Schematic representation of the test crane.MOBILE CRANES(5)Cn(5)647n(5)1n(2)n2GAn1Body 1DBody 3Body 512n(1)Body 2n(1)12n(2)n(3)(3)1(1)(2)n(3)1BBody 4aA02n(4)(4)On(4)12B0Figure 2. Kinematic model of the test crane. Dimensions in mm: A0G 2000; GC 17 500; A0A5823; A0D 5850; AD 565; BD 3455; OB02350; OA0805.Schematic representation of the test crane is shown in Figure 1 and thekinematic model of the test crane which can be represented by ve bodies isshown in Figure 2.Cross-section, material properties and dimensions of each body are obtainedfrom the technical data sheet and measured directly from the test crane. Thecross-section of Body 1 is a hollow polygon of thickness, t, as shown in Figure 3.The cross-section dimensions increase from A0to G and decrease from G to Clinearly, and the dimensions at sections A0, G and C are shown in Figure 3.Body 2 is a cylindrical rod 25 mm in diameter and Body 3 is a piston 180 mm ine0c0A0GCf0t=20e0453c0200f0120160160440207120120Figure 3. Cross-section of Body 1. Dimensions in mm.648S. KILICASLANET AL.diameter with spool thickness of 20 mm. Body 4 is a cylinder with innerdiameter of 230 mm, outer diameter of 246 mm and length of 3440 mm.Additionally, modulus of elasticity and mass density of the Body 1 are taken as200 GPa and 5750 kg/m3, respectively. Mass density of other bodies are taken as7850 kg/m3.When dimensions (lengths and cross-sections) and elastic properties of thebodies of the crane are considered, it is sufcient to take only the Body 1 (theboom) as exible. In this case, other bodies are assumed to be rigid.The following assumptions are considered in the analysis of the crane.1. The mass of the hydraulic oil is included in the mass of the cylinder(Body 4). Varying mass of the cylinder due to varying amounts of hydraulic oilinside it is taken into consideration.2. Hydraulic oil is assumed incompressible.3. The hook load is considered as a point mass and connected to the end ofthe boom with a rope which is taken as a rigid rod. This rope is free for planarrotation about point C. This assumption is valid as long as the oscillations of therod about the vertical position are small and the rod remains in tension. Theseconditions are satised for normal operation speeds and hook loads.4. The structural damping of the boom is taken into account by assumingRayleigh damping.5. The distance between the load and the base is assumed to be kept constantby varying the length of the rope during the up and down motion of the crane.3. DYNAMIC EQUATIONSLet nkrepresent a body reference frame relative to which the deformation ofBody k is dened and n represent a xed frame. Let xkrepresent the position ofthe origin Q of nkin n, and okbe the angular velocity of Body k.Using the nite element method, the deformation displacement vector ukiof anarbitrary point P in element i of Body k is wherefkiis the element shape function matrix transformed tonk,Bkiis theelement connectivity Boolean matrix andakis the vector of body nodalvariables.The velocity of P is written aswhere qkiis the position vector from Q to P in nkincluding deformation, qkiisthe skew symmetric matrix ofqki,Tkis the co-ordinate transformation fromnkto n, o kTkiTok,wkis modal transformation used to reduce the elastic degreesof freedom, andZkis the vector of body modal variables. Equation (2) can beexpressed asMOBILE CRANES649The joint connections and prescribed motions in the system of Ninterconnected bodies are represented by kinematic constraint equationsexpressed at velocity level aswhere y is the system generalized speed vector given byCis the constraint Jacobian matrix which can be formed by the velocityinuence coefcient matrices and g indicates the prescribed velocities.Kanes equations are used to determine the equations of motion of the systemaswhere lis the vector of constraint forces, M is the generalized mass matrix, Q,Fs,FdandFare vectors of Coriolis forces, elastic forces, damping forces andapplied forces, respectively and650v0v(t)0S. KILICASLANET AL.t1t2t3Figure 4. Velocity prole with cycloidal acceleration and deceleration.whereKkis the structural stiffness matrix andDkis the structural Rayleighdamping matrix of Body k. In the simulations, the weights of the structuralmass and stiffness matrices used in formingDkcorrespond to a 2% dampingratio.When the space dependent terms in equations (8) and (9) are separated, a setof time invariant matrices are obtained 3, 4. These mass properties areevaluated once in advance. Equation (6) and the derivative of equation (4)represent linear equations for the accelerations y and the constraint forces l. Theaccelerations obtained from these equations are numerically integrated by usinga variable step, variable order predictorcorrector algorithm to obtain the timehistory of the generalized speeds and generalized co-ordinates.The boundary conditions used for the description of the deformation of Body1 are that for axial deformation A0is xed, and for bending A0is hinged and Ais xed. The rst axial mode, the rst bending mode of part A0A and the rsttwo bending modes of part AC of Body 1 are taken as the modal co-ordinatessince the higher modes are observed negligible. Therefore the generalized speedvector of the system isMOBILE CRANES651The boom is driven by a hydraulic actuator which is controlled by theoperator. In general, throughout the motion, the hydraulic actuator is drivenwith constant velocityv0so that the boom and piston oscillations are kept to aminimum level. Moreover, to avoid impact loading, the actuator velocity isincreased from zero to v0at the beginning of the motion and decreased from v0to zero at the end of the motion which can be assumed cycloidal in time. Thisdesired velocity prole is shown in Figure 4 and can be expressed as follows.If the pivots of Bodies 1 and 2 were at different points, the system would be astructure. The system is moveable owing to the special dimension obtained dueto the concurrency of the pivots. Thus, the constraint equations written for Body1 and Body 2 are linearly dependent. For this reason, one of the constraintequations is dropped to remove the linear dependency.652600400200020S. KILICASLANET AL.ExperimentalSimulation406080Boom angular position (degree)Figure 5. Piston force with respect to boom angular position (324 kN hook load and 30 sboom upward motion).4. COMPUTER SIMULATION OF THE CRANE CHARACTERISTICS ANDCOMPARISON WITH THE EXPERIMENTAL RESULTSSoftware has been developed for the analysis of the test crane. In thissoftware, one can take any number of nite elements and modal variables forBody 1.Experimental studies have been carried out by Balkan for the working rangeof the boom in which the boom was moved in 30 s 2. This speed was selected inorder to minimize the effect of exibility. In that study, the pressure in thehydraulic actuator and the angular positions of the boom were measured. Theoscillations in the pressure resulting from the boom oscillations are ltered out800400020406080Boom angular position (degree)Figure 6. Piston force with respect to boom angular position (324 kN hook load and 10 sboom upward motion). Piston force (kN) Piston force (kN)0.000.080.16MOBILE CRANESNode 381265320406080Boom angular position (degree)Figure 7. Transverse deections of nodes 3, 8 and 13 with respect to boom angular position(324 kN hook load and 30 s boom upward motion).in the control system, hence they are not seen in the measured data. The testcrane was moved with a 324 kN hook load in the upward direction, and thevariations of the pressures in the hydraulic actuator with respect to the boomangular positions are obtained for the 30 s motion of the boom. Therefore, thevariations of the piston force with respect to the boom angular positions for the30 s boom upward motion can be calculated for the 324 kN hook load.The variations of the piston force with respect to the boom angular positionsfor the 324 kN hook load are simulated for the 30 s boom upward motion byusing the computer code and given in Figure 5.Experimental results for the 30 s motion of the boom are also shown in Figure5. The data do not include piston acceleration and deceleration intervals.Moreover, since the boom oscillations are ltered out, they are not seen in theNode 30.0080.08130.1620406080Boom angular position (degree)Figure 8. Transverse deections of nodes 3, 8 and 13 with respect to boom angular position(324 kN hook load and 10 s boom upward motion). Transverse deflection (m) Transverse deflection (m)6540.00S. KILICASLANET AL.0.080.160246810048Time (s)Frequency (Hz)Figure 9. (a) Time response of transverse deection of node 13. (b) FFT of transverse deec-tion of node 13.gure. It is seen from the gure that simulation and experimental results for the30 s boom motion are close to each other.Similarly, the variations of the piston force with respect to the boom angularpositions for the 324 kN hook load are simulated for the 10 s boom upwardmotion by using the computer code in order to make the effect of exibilitymore signicant as shown in Figure 6.In the simulations, the boom is discretized by 12 nite elements. Two of themare taken on A0G where the cross-sectional area is increasing from A0to Glinearly and ten of them are taken on GC where the cross-sectional area isdecreasing from G to C linearly. Damping is included for Body 1 by using a 2%damping ratio for the rst two modes. It is assumed that the rst 15 s is usedfor the acceleration and the last 15 s is used for the deceleration of the boom forthe 30 s boom motion. In the case of 10 s boom motion, acceleration anddeceleration intervals are assumed to be 1 s.A0FA01B0FA02FB01FB02B1A1FAAFCACA2B2BFBFigure 10. Free body diagram of the crane chassis. Dimensions in m: A 550; C 337; A1257;A2460; B1177; B2225. Tranverse deflection (m) Arbitrary units16012080400MOBILE CRANES655610Radius (m)1418 Load (kN)656S. KILICASLANET AL.The magnitudes at small frequencies correspond to the trend due to theexcitation of the system. As the boom moves upwards, it goes towards thevertical position causing the boom transverse deections to decrease. Thefrequency due to the load oscillations also falls into this frequency range. Thefrequencies over 15 Hz are due to the boom oscillations at its naturalfrequencies. The variation of the natural frequency with time is a characteristicof multibody systems and results in a chirp signal as seen in Figure 9(a).5. SIMULATION OF THE LIFTING CAPACITY ON THE HOOKTipping simulation is performed in the blocked condition of the crane to seewhen tipping occurs as the boom moves in the upward and downwarddirections. When one of the reaction forces coming from the ground to thevertical jacks becomes zero, tipping occurs. Using the free body diagram of thecrane chassis, shown in Figure 10, equation (21) is written for the tipping caseaswhere FA01, FA02and FB01, FB02 are components of the reaction forces exerted bythe boom and the cylinder on the crane chassis;FAandFBare the reactionforces exerted by the ground to the jacks and FCis the body force of the cranechassis.When FAis smaller than or equal to zero, tipping condition occurs. For the30 s and 10 s boom motions, the boom angular positions where FAbecomes zeroare determined by using the developed software for different hook loads. Thesimulation results for the test crane are given in Figure 11. The allowable loadspecied by the manufacturer of the test crane is also shown in Figure 11 whereradius is dened as the horizontal distance from the vertical axis of rotation ofthe crane to the tip of the boom at the tipping position, calculated asIt can be seen from Figure 11 that when the boom motion time is decreased,the allowable load for the same radius decreases. Although there is noinformation about the conditions such as boom motion time while the allowableload data are being obtained, the plot of the allowable load is very similar to30 s boom motion time simulation. In addition to this, it is noted by themanufacturer that these allowable load data should be used with a safety factorof 1-5.6. CONCLUSIONIn this study, the mobile crane characteristics are determined by using exiblemultibody analysis. In order to achieve this goal software has been developedwhich is capable of carrying out dynamic analysis of the crane.The coupled rigid and elastic motions of the system are formulated by usingabsolute co-ordinates and modal variables 3, 4. Then, joint connections andprescribed motions are imposed as constraint equations. The exible body isMOBILE CRANES657modelled by the nite element method and the modal variables are used as theelastic variables by utilizing modal transformation.The variations of piston force with respect to the boom angular positions for324 kN hook load are simulated for both 30 s and 10 s boom upward motionsby using the computer code for the velocity prole with cycloidal accelerationand deceler
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