realized for a hydraulic mobile crane. In addition to the structural elements, the mathematical modelling for

Modern mobile cranes, that have been built till today, have oft a maximal lifting capacityof 3000 tons and incorporate long booms. Crane structure and drive system must be safe,functionary and as light as possible. For economic and time reasons it is impossible to build

Mechanism and Machine Theory 38 (2003) 14891508*Corresponding author. Tel.: +86 2423900601.hydraulic drive- and control system is described. The crane rotating simulation for arbitrary working

conditions has been carried out. As a result, a more exact representation of dynamic behaviour, not only for

the crane structure, but also for the drive system is achieved.

2003 Elsevier Ltd. All rights reserved.

Keywords: Hydraulic mobile rotary crane; Flexible multibody; Drive and control system; Dynamic responses

1. IntroductionDynamic responses of hydraulic mobile cranewith consideration of the drive system

Guangfu Sun a,*, Michael Kleeberger b

a Department of Trac and Mechanical Engineering, Shenyang Architectural and Civil Engineering University,

Shenyang Hunnan, Shenyang, 110168, Chinab Institute of Material Handling, Material Flow and Logistics Technical University of Munich, Germany

Received 13 August 2002; accepted 28 March 2003

Abstract

The dynamic behaviour of mobile cranes is determined not only by the steel structures and the external

loads but also by the drive- and control systems. In todays dynamic calculation of mobile cranes, the drivesystems are modelled through the method of kinematic forcing or by measurements for outputs of the

drive system. To improve this situation, a new method for dynamic calculation of mobile cranes has been

developed. In this method, the exible multibody model of the structure will be coupled with the model ofthe drive system. In that way the elastic deformation, the rigid body motion of structures and the dynamic

behaviour of the drive system can be determined in an integrated model. The calculation method has been

www.elsevier.com/locate/mechmtE-mail address: guangfu_sun@yahoo.com (G. Sun).

0094-114X/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0094-114X(03)00099-5

1490 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508prototypes for great cranes. Therefore, it is desirable to determinate the crane dynamic responseswith the theoretical calculation.Modern mobile cranes include the drive and the control systems. Control systems send the

feedback signals from the mechanical structure to the drive systems. In general, they are closed-chain mechanisms with exible members [1].Rotation, load and boom hoisting are fundamental motions of the mobile crane. During

transfer of the load as well as at the end of the motion process, the motor drive forces, the structureinertia forces, the wind forces and the load inertia forces can result in substantial, undesired os-cillations in crane. The structure inertia forces and the load inertia forces can be evaluated withnumerical methods, such as the nite element method. However, the drive forces are dicult todescribe. During start-up and braking the output forces of the drive system signicantly uctuate.To reduce the speed variations during start-up and braking the controlled motor must producetorque other than constant [2,3], which in turn aects the performance of the crane.Several published articles on the dynamic responses of mobile crane are available in the open

literature. In the mid-seventies Peeken et al. [4] have studied the dynamic forces of a mobile craneduring rotation of the boom, using very few degrees of freedom for the dynamic equations andvery simply spring-mass system for the crane structure. Later Maczynski et al. [5] studied the loadswing of a mobile crane with a four mass-model for the crane structure. Posiadala et al. [6] haveresearched the lifted load motion with consideration for the change of rotating, booming and loadhoisting. However, only the kinematics were studied. Later the inuence of the exibility of thesupport system on the load motion investigated by the same author [7]. Recently, Kilicaslan et al.[1] have studied the characteristics of a mobile crane using a exible multibody dynamics ap-proach. Towarek [16] has concentrated the inuence of exible soil foundation on the dynamicstability of the boom crane. The drive forces, however, in all of those studies were presented byusing so called the method of kinematic forcing [6] with assumed velocities or accelerations. Inpractice this assumption could not comply with the motion during start-up and braking.A detailed and accurate model of a mobile crane can be achieved with the nite element

method. Using non-linear nite element theory Guunthner and Kleeberger [9] studied the dynamicresponses of lattice mobile cranes. About 2754 beam elements and 80 truss elements were used formodelling of the lattice-boom structure. On this basis a ecient software for mobile crane cal-culationNODYA has been developed. However, the inuences of the drive systems must bedetermined by measuring on hoisting of the load [10], or rotating of the crane [11]. This is neitherecient nor convenient for computer simulation of arbitrary crane motions.Studies on the problem of control for the dynamic response of rotary crane are also available.

Sato et al. [14], derived a control law so that the transfer a load to a desired position will takeplace that at the end of the transfer of the swing of the load decays as soon as possible. Gustafsson[15] described a feedback control system for a rotary crane to move a cargo without oscillationsand correctly align the cargo at the nal position. However, only rigid bodies and elastic jointbetween the boom and the jib in those studies were considered. The dynamic response of thecrane, for this reason, will be global.To improve this situation, a new method for dynamic calculation of mobile cranes will be

presented in this paper. In this method, the exible multibody model of the steel structure will becoupled with the model of the drive systems. In that way the elastic deformation, the rigid body

motion of the structure and the dynamic behaviour of the drive system can be determined with

one integrated model. In this paper this method will be called complete dynamic calculation fordriven mechanism.On the basis of exible multibody theory and the Lagrangian equations, the system equations

for complete dynamic calculation will be established. The drive- and control system will be de-scribed as dierential equations. The complete system leads to a non-linear system of dierentialequations. The calculation method has been realized for a hydraulic mobile crane. In addition tothe structural elements, the mathematical modelling of hydraulic drive- and control systems isdecried. The simulations of crane rotations for arbitrary working conditions will be carried out.As result, a more exact representation of dynamic behaviour not only for the crane structure, butalso for the drive system will be achieved. Based on the results of these simulations the inuencesof the accelerations, velocities during start-up and braking of crane motions will be discussed.

2. The principle of complete dynamic calculation for mechanism

The principle of complete dynamic calculations, which is based on integrated model [12] fromexible multibody model of the mechanism and the mathematical model of the drive system isshowoutpand d

G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508 1491following sections.

2.1. Modelling of the driven mechanism system

Using Lagranges equation, the motion of bodies in the mechanism is given by

d

dtoTo _qq

T oT

oq

T oC

oq

Tk Qin Qa Qd 1n in Fig. 1. The input is the desired value such as position or velocity of the mechanism. Theut is the dynamic response of the complete system, which consists of the mechanism systemrive system including control. The mathematical models of the two systems are given in theFig. 1. The principle of complete dynamic calculation for mechanism.

where T is total kinetic energy, q is the vector of the generalized coordinate,Qin is the vector of theinternal forces, Qa is the vector of the applied external forces, Qd is the vector of the drive forcesand k is the vector of Lagrange multiplies.

draulic, the drive forces in general can be expressed through following algebraic equation

Qd s

moti

reference xxi fxx1i;xx2i;xx3ig and a global inertia reference x fx1; x2; x3g are selected. Theposit

1492 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508Euler-angle, determinate the location and direction of the body reference. The motion of body i isgiven by the following Lagranges equation

1 In this paper, bold characters are used to represent vectors expressed in the global reference system, and over-barbold cion vector Ri and the rotation vector Hi, which is usually described with Euler-parameter orsystems consisting of interconnected rigid and deformable bodies, can be greatly enhancedthrough dynamic simulation, provided the deformation eect is incorporated into the rigid mo-tion.In the exible multibody dynamics, the large rigid body motion is usually described in the

global inertia reference system. On the other hand, the body deformation is given in the bodyreference system [17]. For an arbitrary body i in the system, which is shown in Fig. 2, a body

T T 1on. The design and performance analysis of cranes, which can be modeled as multibody3. Flexible multibody formulation

Crane as driven mechanism, their bodies mostly undergo large translational and rotationalimultaneously.Qd fdz; t 4The equations of motion for the complete system will be achieved through the combination of Eq.(1)(4). The fundamental problem of complete dynamic calculation for mechanism is, withconsideration of the input and the boundary conditions, to solve all of the time valuables q, k, z,The constrained conditions of the bodies can be written in the following vector equation

Cq; t 0 2

2.2. Modelling of the drive system with control

The drive system has a signicant inuence on the dynamic response of the driven mechanismand should be included in the dynamic model. Electronic and hydraulic drive systems with controlare usually used in mobile cranes. In general the drive system with control can be described usingthe following rst order explicit dierential equation

_zz Fz; q; _qq; t 3where z is the state space vector of the drive system, q is the feedback vector from the drivenmechanism.According to the corresponding physical law for the drive system, such as electronic or hy-haracters are used for vectors that are dened in the body reference system.

wherTh

scrib

wherrefer

withand uutransAc

positcleargener

With

G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508 1493d

dtoTio _qqi

oTioqi

oCoqi

k Qini Qai Qdi 5

T T T

Fig. 2. A body with rigid and elastic motion.e i 1; 2; . . . ; n and n is the total number of the mechanism members.e deformable position vector of the arbitrary point p in the element j of body i can be de-ed as

rpij Ri upij 6e Ri is a set of Cartesian coordinate that dene the location of the origin Oi of the bodyence. Clearly,

upij uoij ueij Aiuuoij uueij 7uoij is the position of point p in the unreformed state; u

eij is the elastic displacement vector; uu

oij

eij are the coordinate of the vectors u

oij and u

eij with respect to the body reference xxi; Ai is the

formation matrix, which is a matrix function of Hi.cording to the principle of the nite element method, uuoij and uu

eij can be interpolated by nodal

ion coordinate in the undeformed state qqoi and the nodal displacement qqei , respectively. It is

that rpij can be determined by Ri, Hi and qqei from Eqs. (6) and (7). Therefore we dene the

alized coordinate qi of the body i as

qi RiT HiT qqei Th iT

8this denition the vector r

pij can be expressed as the function of the generalized coordinate qi

rpij rpijqi 9

culat

withTh

place

respe _

Th

wher

UQ

c

i cavecto

wher

Subs

1494 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508generalized coordinate q qT1 ; qT2 ; . . . ; qTnT, the new variable u as the system generalized velocityandQini Qei Qci 20tituting from (11), (16), (17) and (20) into Lagranges equation (5), introducing the system

Di _qqi; qi Ti DiTi _qqi;TiqiTi 19that are total stiness matrix and total damping matrix of body i, respectively.The total internal nodal force vector is given bye

Kiqi TTi KiTiqiTi 18TQei TTi Qe

i TTi KiTiqiTiqi Kiqiqi 16Qci TTi Q

c

i TTi DiTi _qqi;TiqiTi _qqi Di _qqi; qi _qqi 17sing this transformation matrix the nodal elastic force vectorQi and the damping force vectorn be transferred to the generalized elastic force vector Qei and the generalized damping forcer Qci associated with the generalized coordinate Ri, Hi and qq

ei asTi 0 0 I 15with Ti is the transformation matrix.

eei i

qqei Tiqi 14ct to qqi and qqqqi , Ki and Di are the stiness matrix and damping matrix, respectively.e nodal displacements qqe can be written in terms of the generalized coordinate q of body i asQe

i Kiqqei qqei 12Q

c

i Di _qqqqei ; qqei _qqqqei 13where Q

e

i and Qc

i are the nodal elastic force vector and the nodal damping force vector withe eMiqi is the total mass matrix of the body i.e elastic nodal forces and the nodal damping forces can be described by the nodal dis-ment vector qqei and the nodal velocity vector _qqqq

ei asj1 j1ed using the sum of the all of elements

Ti Xne

Tij 12

_qqTiXne

Mijqi" #

_qqi 1

2_qqTi Miqi _qqi 11where qij is the mass density, Vij is the element volume and Mijqi is the mass matrix, which ingeneral not constant, but is dependent on qi. The total kinetic energy of the body i can be cal-ij2 Vij

ij ij ij ij 2 iij i iThe kinetic energy of the element j is given by

T 1Z

q _rrp T _rrp dV 1 _qqTM q _qq 10combining with the kinematic constraints between adjacent bodies, yield

_qq u 21

Mq _uu Du; qu Kqq oCoq

Tk Qa Qd Qv 22

Cq; t 0 23where Mq is the system mass matrix, Du; q and Kq are system damping and stiness matrix.In Eq. (22) Qv is given by

Qv _MMqu 12

ooq

uTMqu T

24

which represents the gyroscopic and Coriolis force components.

The mathematical model of the hydrostatic closed-loop system, the controller and the regulatorunit that are often used in mobile cranes, will be described in the following sections.

G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508 14954. Mathematical models for hydraulic drive system with control

The hydrostatic closed-loop system of the type shown in Fig. 3, which is usually used in mobilecranes, will be considered. A variable pump that is driven by a diesel engine, capable of pumpingin either direction, is directly ported to a bidirectional motor. Using this system, smooth transitionfrom forward to reverse through the zero rotating speed and full hydrostatic braking action ineither direction will be available. To prevent excessive loop pressure, cross-port relief valves arecommonly used as shown in Fig. 3. The small auxiliary supercharge pump that is usually mounteddirectly on the main pump is to provide enough ow to take care of pump and motor internalleakage. There is sometimes a need for control of motor in mobile crane, such as speed feedbacksystem with controller and regulator unit, which are also shown in Fig. 3. The controller uses theresponse signals from the mechanism and compares them with their desired values in its task ofdetermine an appropriate action fR to the regulator unit that can create the regulating value xs toadjust the pump. The signal ow is given in Fig. 4.Fig. 3. Hydraulic closed circuit system in mobile crane.

4.1. M

since

to th

1496 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508in out

with

Qout Qh Qr 27where Qr is the ow through the relief valve, which can be described as following

Qr 0 jphj6 jpsetjKrph pset jphj > jpsetj

28wherTh

withdisplTh

within out

e Kirchhos low, should be identical

Q Q 26h

dt

ChQch Ch Qh Qm 25

where Ch is the hydraulic capacity of the high-pressure side, Qh and Qm are the input ow to themotor and the output ow from the motor, respectively.Because of the neglecting of all leakage, the input ow Q and the output ow Q , accordingdp 1 1pressure line [18]. The leakage will be neglected.According to basic dierential equations of the hydraulic circuit [19], we havean auxiliary pump and relief valves are typically used to assure constant pressure in the low-For simplication it is assumed that the ow pressure p0 of the low-pressure side is constantwhile the high pressure ph of the high-pressure side is variable. This is general true in practice,odelling the hydrostatic closed-loop systemFig. 4. Signal ow.e Kr is the constant of the relief valve and pset is the prescribed limiting pressure.e pumping ow Qin from the variable pump is given by

Qin V1maxn1x1max xs 29

V1max is the maximum pumping ow of the variable pump, x1max the maximum adjustableacement, n1 is the pump rotation speed, which is assumed to be constant.e ow from the constant motor of the system can be expressed as

Qm n2V2 30n2 is the rotating speed of the motor, V2 is the motor volume.

The output torque of the motor is given by

M2 ph p0V22p

31From Eqs. (25)(30) the dierential equation of the pressure ph is given by

_pph 1Ch xsn1v1maxx1max

Qr n2V2

32

Because the motor speed n2 is a component of the generalized velocity u, substituting Eq. (28) intoEq. (32), we can write Eq. (32) in the general form as

_pph fpxs; ph; u 33The generalized driving forces, which can be obtained from the output torque of the motor in Eq.(31), can be expressed in a function form of the pressure ph as

Qd fdph 34

show

G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508 1497Using Newtons law the motion of the control valve piston is given by the following dierentialequation

mvxxv fR dv _xxv cvxv 35where mv is mass of the valve piston, dv is the damping constant of the valve, cv is the springconstant, fR is regulating force from the controller and xv is the motion of the valve piston.s a typical regulator unit system.4.2. Modelling the regulator unit system

The regulator unit system usually consists of a cylinder and a 4/3-way control valve. Fig. 5Fig. 5. Regulator unit system.

Th

with

both

withvolumTh

sprinIf

1498 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508_xx6 1m fR dvx6 cvx2_xx2 x6_xx3 Eolva Q1 Q2 Asx5

_xx4 Eolvb Q4 Q3 Asx5

_xx5 1ms x3 x4As dsx5 csx1 f0

40yields the following state equations for the regulator unit

_xx1 x5s s 0

g, ds is the damping constant of the cylinder and xs is the servo-moton.we dene the state variables

x1 xs; x2 xv; x3 pa; x4 pb; x5 _xxs; x6 _xxvmsxxs pa pbAs ds _xxs csxs f0 39where m is the piston mass, c is the spring constant, f is the initial elastic force of the cylinderAs is the area of the cylinder piston, vm is the initial system volume of the cylinder with thees of all drillings, pipelines of the unit, as the piston is at the position xs 0.e motion of the cylinder piston can be written asva vm Asxsvb vm Asxs

38ol a b

side of a and b, which are given by_ppb Eolvb Q4 Q3 As _xxs37

where E is eective bulk modulus of the oil, v and v , respectively, are the oil volumes in the_ppa olva Q1 Q2 As _xxsThe pressure rates in the both cylinder chambers can be calculated by the continuity equations

Ee ows through the control valve are given by the turbulent equations [20] as

Q1 Bv2xv x0 jxv x0j

jps paj

psignps pa

Q2 Bv2xv x0 jxv x0j

jpaj

psignpa

Q3 Bv2xv x0 jxv x0j

jpbj

psignpb

Q4 Bv2xv x0 jxv x0j

jps pbj

psignps pb

36

x0 is the valve lapping, ps is system supply pressure and Bv is the ow coecient of the valve.v

1 2 6 s

vector x, which is used in Eq. (33), can be directly from x obtained

4.3. M

d

desirbe re

T

ow oq ou ow oq ou

The t

G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508 1499Having determined, in the preceding sections, the equations of the mechanism, the hydraulic

close5. Total system equationsfR frq; u; xi; k;Qa;Qd 48With consideration of (45)(47) and (22) the regulating force can be written in the followinggeneral formerm _uu in (47) can be calculated using Eq. (22). Clearly, it is a function of q, u, k, Qa and Qd.we have

fR Kp xd

1Tp

xi Tv _xxd

46

with

_xxd oxd _ww oxd _qq oxd _uu oxd _ww oxd u oxd _uu 47i

_xxi xd 45p

where Kp is the proportional parameter, Tp and Tv are time constants. It is clear that the error xd isa function of the observed value q, u and the desired value w.Assuming a new variable x and letfR Kp xd 1 xd dt Tv _xxd 44which can be described with following equation Z

ed value, the output of controller is the controlled signal fR, with which the regulator unit willalized (see Fig. 4). The commonly used controller in mobile cranes is the PID controller,The input value of the controller is x , which is the dierence of the response value and theodelling the controllerwhere Ts is the Boolean translation vector. Substituting (42) into (33), the pressure rate can bewritten in the general form by

_pph fpph; x; u 43xs Tsx 42Using the vector form, the state equation of the regulator unit can be written as

_xx fxx; fR 41with x x ; x ; . . . ; x T is the state vector of the regulator actuator. The component x of the stated system and the controller system one can write the total system equations as

where

correvenie

6. Nu

Mtatiothe ccraneshowTh

nismmechTh

(2) Tt

(3) A

1500 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508he boom is exible, the suspension ties and ropes are elastic, but the superstructure, coun-erweight and crane base are regarded as rigid bodies.(1) The load is regarded as a point mass.merical simulation of a mobile hydraulic rotary crane

obile cranes are widely used in industry. The fundamental motion of a rotary crane is ro-n by which the load pendulum action due to this motion is a strong dynamic phenomenon ofrane, which must be taken into account. In this section the numerical simulation of a rotarythat is driven by a closed hydraulic system with speed control will be carried out. Fig. 6s this crane with coordinate systems.e whole superstructure of the crane can be rotated about z-axis driven by a rotary mecha-. The global technical data of the crane are given in Table 1. The technical data of the rotaryanism and hydraulic close system are shown in Table 2.e following assumptions will be used:sponding dynamic responses of the total crane system can be obtained. This is very con-nt for computer simulation of crane motions.fmq; u; k;Qa;Qd Du; qu Kqq @CoqT

k Qa Qd Qv

The system equations (49) are a mixed system of dierential and algebraic equations that have tobe solved simultaneously. By solving Eq. (49) not only the responses of the mechanism q, u and k,but also the state values of the drive system x, fR, xi, ph and the output force Q

d can be simul-taneously obtained. By arbitrary input of desired values and parameters of the controller themy dydt

fy; t 49

with

y qT uT xT xi ph fR QdT kT T

fy; t

u

fmq; u; k;Qa;Qdfxx; fRxdw; q; ufpph; x; u

fR frq; u; xi; k;Qa;QdQd fdph

Cq; t

266666666664

377777777775

my diagI;Mq; I; 1; 1; 0; 0; 0ll frictional and damping forces will be neglected.

G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508 1501(4) T(5) T

Twcoordto thloaddescrselec

Theerencpensistaticrotathe oil is compressible, but the oil leakage of the hydraulic system is not considered.here are not any load hoisting or boom hoisting during the rotary motions of the crane.

o reference systems: the global inertial coordinate system x, y and z and the body referenceinate system xx, yy and zz are used. The vertical axises of the both systems z and zz are identicale rotary axis as shown in Fig. 6(b). The global coordinate R and h dene the location of theand the orientation of the rotary body reference system. The elastic nodal coordinate qqe

ibe the rotary body deformation with respect to the body reference xx, yy and zz. Therefore theted generalised coordinate are given by

q RT h qqeT T 50

boom system is divided into eight Timoshenko boom elements [13]. The element local ref-e system xxj, yyj and zzj is used to dene the element cross-section j, see Fig. 6(b). The sus-on ties are described as one truss element. The ropes are represented as a spring element. Thenodal displacements of the boom system which are initial conditions for the simulation ofion are calculated under acting load and the boom gravity force.

Fig. 6. (a) Rotary mobile crane and (b) schematic model with coordinate systems.

Table 2

Technical data of the rotary mechanism and hydraulic close system

Name Symbol Value

Rotating speed of the pump (rpm) n1 1680.0Forder volume of the variable pump (cm3) V1max 71.0Volume of the hydraulic motor (cm3) V2 90.0Translate factor between hydraulic motor and rotary mechanism i 3500.0Capacity of the hydraulic system (cm3/bar) Ch 0.08125Factor of the safe valve (cm3/bar) Kr 185.0Maximal controlled displacement of the variable pump (mm) xmax 43.62The limiting pressure of the system (bar) jpsetj 300.0Flow coecient of the valve (l/(min bar1=2 mm)) Bv 3.641System pressure for the control valve (bar) ps 100.0Constant low pressure of the closed circuit (bar) p0 20.0Piston area of the regulator cylinder (cm2) Ak 1.3Volume in chamber A of the regulator cylinder (cm3) va 70.0Volume in chamber B of the regulator cylinder (cm3) vb 70.0Spring constant in the regulator cylinder r (N/mm) ck 19.0Damping constant in the regulator cylinder (N s/mm) dk 0.042272Initial spring force in the regulator cylinder (N) f0 66.424Eective uid balk module (bar) Eol 16000Lapping of the control valve (mm) x0 1.0Piston mass of the regulator cylinder (kg) mk 1.0Mass of the control valve piston (kg) mv 0.1Spring constant in control valve (N/mm) cv 300.0Damping constant in control valve (N s/mm) dv 0.029709

Table 1

Technical data of the crane structure

Name Value

Load radius (m) 12.0

Length of the boom (m) 96.0

Mass of the boom (kg/m) 550.0

Mass of the superstructure (t) 53.0

Mass of the load (t) 55.0

Mass of the load hook (t) 5.526

Mass of the counterweight (t) 150.0

Eective boom inertia moment about yyj-axis of the element reference (m4) 0.0838Eective boom inertia moment about zzj-axis of the element reference (m4) 0.0665Reduced boom shear area about yyj-axis of the element reference (m2) 0.0034Reduced boom shear area about zzj-axis of the element reference (m2) 0.0032Eective area of the boom (m2) 0.0443

Area of the suspension ties (m2) 0.0084

Area of the ropes (m2) 0.0004

Length of the ropes (m) 71.51

Youngs modulus (N/m2) 1.08 1011

1502 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508

The desired rotary process of the superstructure with speed of 0.5 rpm and braking at t 10:0 scan be described using following step function:

dspeed 0:5 06 t < 10:00:0 10:06 t < 1

51

The following rotary processes of the superstructure under dierent control parameters Kp, Tpand Tv are carried out:

rotary 1 : Kp 5; Tp 0:10; Tv 0:4rotary 2 : Kp 10; Tp 0:12; Tv 0:4rotary 3 : Kp 25; Tp 0:128; Tv 0:4

Fig. 7 shows the corresponding rotary speed responses. The dierent controlled speed curvesare achieved. All curves are focussed to the intended rotary process (51), but with dierent ac-celerations and decelerations. The curve with rotary 1 is focussed at 4.0 and 14.0 s, the curve with

G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508 1503rotary 2 and 3 are at 2.0 and 12.0 s, respectively. The dynamic displacements of the boom tip indirection yy of the body reference system xx, yy and zz for dierent controlled speed curves are given inFig. 8. For comparison with the method of kinematic forcing that was used by some literature[6,8], two types of assumed trapezoidal velocity proles with the same acceleration and deceler-ation times 2.0 and 4.0 s are considered, see Fig. 7. It should be noted that these assumed inputvelocity proles could not occur in the simulation of the real rotary control process. However, forcomparison with the current study, the corresponding dynamic responses of the boom tip withthese velocity proles have been calculated in this paper using the method of kinematic forcing[6]. The results are also shown in Fig. 8.The inuences of dierent rotary speeds by the same accelerations and decelerations under the

same control parameters Kp 5, Tp 0:10 and Tv 0:4 are also studied, which are given in Fig.10. It is observed that the rotary speed with the same acceleration and deceleration has alsoFig. 7. Dierent rotary speeds of the crane superstructure.

1504 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508signicant inuence on the dynamic response of crane. With higher rotary speeds the super-structure of crane can have more kinetic energy, which will be transformed to dynamic response inbraking process. The relationship between the rotary speed and the maximum displacement of theboom tip is approximately linear, which is given in Fig. 10.Based on these results the following observations can be made:

1. At the same rotary speed, the acceleration time and braking time, which can be controlled by Kpand Tp, have strong inuence on the dynamic response. Using the assumed trapezoidal velocity

Fig. 8. Dynamic response of the boom tip.

Fig. 9. Dierent accelerations and decelerations.

G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508 1505proles with the same acceleration and deceleration times, the results are lower as the real ve-locity proles. The reason is that during start-up of the rotary process and braking the shockacceleration and deceleration are very higher than the assumed trapezoidal velocity proles andinduce strong dynamic responses (see Fig. 9).It is clear that the assumed trapezoidal velocity proles with the same acceleration and de-celeration times for sudden start-up and braking, in comparison to the current study, could notmore accurately describe the acceleration and deceleration histories during start-up andbraking of the rotary mechanism.

2. With the same rotating acceleration and deceleration, the rotary speed has also signicant in-

Fig. 10. Maximum boom tip displacements with dierent rotary speeds.uence on the dynamic response of crane.3. The maximum responses occur during the braking processes.

In addition to the dynamic responses of the crane structure, the state parameters and theoutputs of the hydraulic drive system can be determined easily. Fig. 11 shows the motor outputmoments during various rotary processes.As expected, during start-up and braking the motor outputs the maximum shock moments. The

safety relief valves ensure that these shock moments remain lower than the allowable values. Thepressure histories in the chambers of the piston are given in Fig. 12. It is observed that the pressurecurves of chamber a and b are symmetric to the value of 50 bar.

7. Conclusion

A simulation method for determining the dynamic response of rotary cranes with considerationof the hydraulic drive system is developed. With the steel structure, hydraulic drive and controlsystem, a comprehensive crane model is established. Based on the theory of exible multibody

1506 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508system, the nite element method, the hydraulic and control theory, the system equations forcomplete dynamic calculations are established.With this method, simulations of a hydraulic mobile crane in rotating processes are carried out.

The inputs for the simulations are the desired rotary speeds and control parameters. In com-parison with the assumed velocity proles in the method of kinematic forcing, the currentmethod is very convenient for crane simulations. The results show that the dynamic responsesduring start-up and braking of the crane rotation with the assumed trapezoidal velocity prolesfor sudden acceleration and deceleration could not more accurately described. The results support

Fig. 11. Motor output moments.

Fig. 12. Pressures pa and pb in chamber a and b of the piston.

[4] H. Peeken, K. Menninger, Dynamische Kraafte beim Drehen eines Mobilkranes, Foordern und Heben 24 (12) (1974)11431147.

169178.

[9] W.A. Guunthner, M. Kleeberger, Zum Stand der Berechnung von Gittermast-Fahrzeugkranen, dhf 43 (3) (1997)

G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508 15075661.

[10] M. Kleeberger, Nichtlineare dynamische Berechnung von Gittermast-Fahrzeugkranen, Muunchen, TechnischeUniversitaat, Dissertation, 1996.

[11] J. Maier, Untersuchung zur nichtlinearen Berechnung dynamischer Belas-tungsvorga nge an Turmdrehktanen,

Muunchen, Technische Universitaat, Dissertation, 1999.[12] G. Sun, Berechnung von Gittermast-Fahrzeugkranen unter Beruucksichtigung der Antriebs- und Regelungssysteme,

Muunchen, Technische Universitaat, Dissertation, 2001.[13] M. Guunthner, Statische Berechnung vom Gittermast-Auslegerkranen mit Hilfe niter Turmelemente unter

Beruucksigtigung der Elastizitaat des Kranwagens und von Messungen, Muunchen, Technische Universitaat,Dissertation, 1985.[5] K. Maczynski, S. Wojciech, Bin diskretes Modell fuur Teleskopdrehkrane zur Anylyse der Bewegung der Last beimDrehen des Kranes, Hebezeuge und Foordermittel 21 (1981) 333337.

[6] B. Posiadala, B. Skalmierski, L. Tomski, Motion of the lifted load brought by a kinematic forcing of the crane

telescopic boom, Mech. Mach. Theory 25 (1990) 547555.

[7] B. Posiadala, Inuence of crane support system on motion of the lifted load, Mech. Mach. Theory 32 (1) (1997) 9

20.

[8] W.S.M. Lau, K.H. Low, Motion analysis of a suspended mass attached to a crane, Comput. Struct. 52 (1) (1994)the proposition that a more accurate and convenient calculating instead of the kinematic forc-ing for the dynamic response of crane system is expected. The simulation results also show that,in addition to the accelerations and decelerations of crane movements, the rotary speed has alsosignicant inuence on crane dynamic responses.With this method, not only the dynamic responses of the structure, but also the state param-

eters of the drive system, such as motor output moment history, dynamic response of valves, oilpressure, oil ow and control stability, which are signicant for design of crane drive system, canbe studied simultaneously.

Acknowledgements

The rst author appreciates the nancial support provided by The Institute of MaterialHandling, Material Flow and Logistics, Technical University of Munich, Germany, for thisproject. The authors are very grateful to Prof. Dr. -Ing, Willibald A. Guunthner for supporting thiswork.

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1508 G. Sun, M. Kleeberger / Mechanism and Machine Theory 38 (2003) 14891508

Dynamic responses of hydraulic mobile crane with consideration of the drive systemIntroductionThe principle of complete dynamic calculation for mechanismModelling of the driven mechanism systemModelling of the drive system with control

Flexible multibody formulationMathematical models for hydraulic drive system with controlModelling the hydrostatic closed-loop systemModelling the regulator unit systemModelling the controller

Total system equationsNumerical simulation of a mobile hydraulic rotary craneConclusionAcknowledgementsReferences