Research ArticleSpherical Pendulum Small Oscillations forSlewing Crane Motion

Alexander V Perig1 Alexander N Stadnik2 and Alexander I Deriglazov2

1 Manufacturing Processes and Automation Engineering Department Engineering Automation FacultyDonbass State Engineering Academy Shkadinova 72 Donetsk Region Kramatorsk 84313 Ukraine

2Department of Technical Mechanics Engineering Automation Faculty Donbass State Engineering Academy Shkadinova 72Donetsk Region Kramatorsk 84313 Ukraine

Correspondence should be addressed to Alexander V Perig olexanderperiggmailcom

Received 29 August 2013 Accepted 2 October 2013 Published 9 January 2014

Academic Editors N-I Kim M Seddeek and M Zappalorto

Copyright copy 2014 Alexander V Perig et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The present paper focuses on the Lagrange mechanics-based description of small oscillations of a spherical pendulum with auniformly rotating suspension center The analytical solution of the natural frequenciesrsquo problem has been derived for the caseof uniform rotation of a crane boom The payload paths have been found in the inertial reference frame fixed on earth and in thenoninertial reference frame which is connected with the rotating crane boomThe numerical amplitude-frequency characteristicsof the relative payloadmotion have been foundThemechanical interpretation of the terms in Lagrange equations has been outlinedThe analytical expression and numerical estimation for cable tension force have been proposed The numerical computationalresults which correlate very accurately with the experimental observations have been shown

1 Introduction

The solution of nonlinear differential equations for sphericalpendulum swaying requires the introduction of modernnumerical computational analysis techniques Such computertechniques allow us to produce approximate solutions of theposed problemsHowever the linearizedmodels for sphericalpendulum small swaying motion provide a clear mechanicalexplanation and the first numerical approximation of thecomplex spherical motion problems This fact confirms thesignificance of the small oscillation problems for the modernfield of lifting-and-transport machines A series of modernresearch efforts [1ndash22] have been focused on the solution of aspherical pendulum with moving pivot center

Abdel-Rahman and Nayfeh [1] have studied the dynamicproblem for a ldquocrane boom tip-payloadrdquo mechanical systemfocused on the reduction of payload lateral vibrations byreeling and unreeling the hoist cable In this work theauthors have derived two- and three-dimensional models fora spherical pendulum taking into account the linearity oftransport motion

Adamiec-Wojcik et al [2] have provided a numericalfinite-element simulation of the supporting frame of anoffshore crane portal construction Within the proposedmodel supporting frame elastic oscillations define both thepayload motion and the displacements of the suspensioncenter with a payload on the cable The derived model alsoincorporates ship vibration motion

Aston has obtained two independent motion equationsfor a spherical pendulum taking into account viscous frictioneffects This model neglects the components of the Coriolisinertia force [3]

Betsch et al [4] have described rotary crane dynam-ics in terms of redundant coordinates with an introduc-tion of differential-algebraic equations (DAEs) The authorsrsquoapproach focused on the simultaneous incorporation of alarge total number of redundant coordinates and the augmen-tation constraints do not fully describe the particular cases ofrotary crane dynamics

Blajer and Kołodziejczyk [5] have derived governingDAE equations for the open-loop control formulation thatforces the underactuated crane system to complete the partly

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 451804 10 pageshttpdxdoiorg1011552014451804

2 The Scientific World Journal

specified motion It is necessary to note that this paper dealsprimarily with controlled in-plane motions of the crane sys-tem with linear transport motion Blajer and Kołodziejczyk[6] have investigated the control of underactuated cranemechanical systems with servo constraints This work takesinto account the restoring load gravity force in the horizontalplane However the derived mathematical model assumesprimarily linear transportmotionwithout theCoriolis inertiaforce Blajer and Kołodziejczyk [7] have presented a concep-tual research article focused on the building of n-index DAEsystems for a relevant dynamic description of partly specifiedcrane system motion Blajer and Kołodziejczyk [8] haveintroduced a carefully argued computational technique fornumerical simulation of cranes based on the use of dependentcoordinates The advantages of dependent coordinatesrsquo usagehave been outlined It is necessary to note that with each newresearch effort the authors apply much more sophisticatedcomputational methods and derive more exact governingdynamic equations However this paper is also based onthe same simple pendulum model as the first approacheswherein the Coriolis inertia force and the restoring loadgravity force are ignored Blajer et al [9] have simulated atwo-tier framed human arm model with segments of equallength The application of this model seems to be veryuseful for the dynamic analysis of a swaying load in thehorizontal plane during rotation of a crane boom Blajer andKołodziejczyk [10] have developed improved DAE equationsfor more precise rotary crane dynamics simulation Butadditional dimensional analysis for the explicit forms of thegoverning equations would greatly enhance the work

Cha et al [11] have applied a topological modelingapproach to a dynamic simulation of multiple floating cranesin shipyards taking into account crane boom oscillations

Ellermann et al [12] have introduced the vector ofgeneralized gyroscopic andCoriolis forces for amathematicaldescription of nonlinear dynamics of floating cranes Thiswork used a comprehensivemodel with a set of 20 differentialequations but does not allow proper analysis of particularcases of floating jib crane-load system motions

Erneux and Kalmar-Nagy [13] have proposed a planependulum container crane model with linear transport trol-ley motion This model illustrates linear displacement of theplane pendulum suspension center

Glossiotis and Antoniadis [14] have derived a four-degrees-of-freedom (4DOF) model for the rotary cranesystem incorporating both hoisting and slewing payloadmotions and thus is able to handle cases where hoisting canbe simultaneously applied to the rotary motion

Ibrahim [15] has studied the issues of multiplicative noiseand noise-enhanced stability of spherical pendulums

Kortelainen andMikkola [16] have introduced data man-agement techniqueswithin an approachusing a semantic datamodel for payload oscillation problems

Mitrev [17] has applied his generalized approach with theintroduction of an inertia force vector to the case of lineartransport motion of the pendulum pivot center Grigorovand Mitrev [18] have investigated a spherical pendulumdynamic model for a numerical solution of a freely sus-pended swinging load problem They have derived absolute

trajectories of both the pendulum suspension center and theswinging load

Osinski andWojciech [19] have analyzed the hoistmotionof a mechanical system consisting of a load hoist rope andelastic crane jib This study mainly deals with rectilinear loadoscillations and lateral crane jib vibrations

Ren et al [20] have used a mixed approach combin-ing both analytical and finite-element techniques for theLagrangian function of a mechanical system They havedetermined the relations between load swaying crane boomflexibility and wave motions Schaub [21] has implemented arate-based ship-mounted crane payload pendulation controlsystem This research is focused on a kinematic approach topayload stabilizationwith an emphasis on the usage of inertialmeasurement unit (IMU) information

Uchiyama [22] has applied control theory techniques for arobust crane controller design that provides robustness withrespect to changes in cable length and transported payloadmass without iterative computations

At the same time these known articles [1ndash22] do not dojustice to load swaying descriptions using Coriolis effects inrelative motion The present paper focuses on these issues

The present paper focuses on the Lagrange mechanics-based description of small oscillations for a spherical pendu-lum with a uniformly rotating suspension center

The prime novelty statement of the present paper isbased on the rational introduction of a Cartesian coordinatesystem for study of small relative swaying of a payload andsubstantiation of uniformity of crane boom rotation

2 Kinematic Analysis

In order to solve the problem we introduce the Lagrangeequations for the motion of the mechanical system ldquocraneboom 119861119874

2-load119872rdquo which is shown in Figure 1

The computational scheme in Figure 1 may be describedwith an introduction of three degrees of freedom For gen-eralized coordinates we assume the rectangular coordinates119909 119910 and 119911 of the load and the angle 120593

119890= 120596119890119905 of crane

boom 1198611198742uniform rotation in the horizontal plane (119909119910)

with the constant angular velocity 120596119890around the vertical

axis 11987421199112 We denote the fixed inertial coordinate system

as 119909211991021199112and the moving noninertial frame of reference as

119909111991011199111 which is rigidly bounded with the crane boom 119861119874

2

Rotation of themoving noninertial frame of reference 119909111991011199111

around the fixed inertial coordinate system 119909211991021199112defines

the transportation motion The motion of load119872 relative tothe moving noninertial frame of reference 119909

111991011199111defines the

relative motionThe scalar of the load119872 transportation velocity is defined

as 119881119890

= 120596119890sdot 1198742119872 and transportation velocity vector V

119890

is perpendicular to O2M that is the scalar product(V119890O2M) = 0 (Figure 1) The relative velocity vector V

119903of

the load 119872 is defined as V119903= (119889119909

1119889119905 119889119910

1119889119905 119889119911

1119889119905) in

Figure 1 In the initial time 119905 = 0 load119872 has the vertical 119861119860stin-line position that is the load119872 initial position coincideswith the static equilibrium position119860 st for the load119872 on thecable119872119861

The Scientific World Journal 3

z2

z1

y2

y1

O2

O1

x2

x1

120593e

1205722

1205721

Adyn

Ast

= d120593edt

H

D

l

R

B

Δ

x

y

M

O

z

120579

rarrVr

rarrVe

rarr120596e120596e

rarrN

mrarrg

Figure 1 Computational spatial scheme of spherical pendulum119872 swaying on the cable119872119861 during crane boom 1198611198742slewing motion for the

linearized model derivation in Cartesian coordinates

At time 119905 = 0 the load 119872 has zero absolute velocity andthe initial relative velocity is 119881

119903119909(0) = +120596

119890sdot 119861119863 (Figure 1)

In Figure 1 we denote the angle 120579 which is the currentangle 119860 st1198742119872 that is the angle between the axis 119910

1and

the radius-vector O2M where sin(120579) = 11990911198742119872 cos(120579) =

(119877+1199101)1198742119872The vertical coordinate 119911

1in Figure 1 is defined

as 1199111= 119897 minus 119897 sdot cos(120572

1) where 120572

1is the angle between the cable

119872119861 and the verticalFor small angles 120572

1 the projections of the absolute

velocity vector may be written as

1198811199091

= (1198891199091

119889119905) minus 119881119890sdot cos (120579)

1198811199101

= (1198891199101

119889119905) + 119881119890sdot sin (120579)

1198811199111

= (1198891199111

119889119905) = (

1198891205721

119889119905) sdot 119897 sdot sin (120572

1)

(1)

Taking into account Figure 1 the system of (1) for theprojections of the absolute velocity vector takes the followingform

1198811199091

= (1198891199091

119889119905) minus (

119889120593119890

119889119905) (119877 + 119910

1)

1198811199101

= (1198891199101

119889119905) + (

119889120593119890

119889119905) 1199091

1198811199111

= (1198891199111

119889119905)

(2)

The square of absolute payload119872 velocitymay be writtenas 1198812abs = 119881

2

1199091

+ 1198812

1199101

+ 1198812

1199111

1198812

abs = (1198891199091

119889119905)

2

+ (1198891199101

119889119905)

2

+ (1198891199111

119889119905)

2

4 The Scientific World Journal

+ (119889120593119890

119889119905)

2

(1199092

1+ (1199101+ 119877)2

)

+ 2(119889120593119890

119889119905)(1199091(1198891199101

119889119905) minus (119910

1+ 119877) (

1198891199091

119889119905))

= ((1198891199091

119889119905) minus (

119889120593119890

119889119905) (119877 + 119910

1))

2

+ ((1198891199101

119889119905) + (

119889120593119890

119889119905) 1199091)

2

+ (1198891199111

119889119905)

2

(3)

3 Dynamic Analysis

The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figure 1 is governed by the vector equation

for the rate of change of moment of momentum H11987423

for thesystem ldquocrane boom 119861119874

2-load119872rdquo with respect to point119874

2in

the inertial reference frame 119909211991021199112

(119909211991021199112)119889H11987423

119889119905= sumM1198742 (4)

The vector equation (4) contains the following compo-nents

H11987423

= 1198681198742

33120596119890 (5)

1198681198742

33= (1198681198742

33)1198611198742

+ 119898(1199092

1+ (119877 + 119910

1)2

) (6)

sumM1198742 = M1198742119863119879

minusM1198742119865119879

= (1198721198742

119863119879minus1198721198742

119865119879) k (7)

where (119868119874233)1198611198742

is the element of mass moment of inertia forthe crane boom119861119874

2in inertial fixed on earth reference frame

119909211991021199112with respect to unit vector k and119898(119909

2

1+ (119877 + 119910

1)2) is

the element of mass moment of inertia for the payload119872 ininertial fixed on earth reference frame 119909

211991021199112with respect to

unit vector kThe external moment of gravitational forceM1198742(mg) = 0

in (4) and (7) becausemg uarrdarr kFor the system ldquocrane boom-payloadrdquo the cable reaction

force N is the internal force So in (4) and (7) we haveM1198742(N) = 0

Substitution of (5) (6) and (7) into (4) yields thefollowing scalar equation for the rate of change of momentof momentum 119867

1198742

3for the system ldquocrane boom 119861119874

2-load

119872rdquo with respect to point 1198742in the inertial reference frame

119909211991021199112

119889

119889119905(((1198681198742

33)1198611198742

+ 119898(1199092

1+ (119877 + 119910

1)2

)) (119889120593119890

119889119905))

= 1198721198742

119863119879minus1198721198742

119865119879

(8)

where driving1198721198742119863119879

and frictional1198721198742119865119879

torques are the tech-nically defined functions for specific electric drive systems

4 Nonlinear Mathematical Model withLagrange Equationsrsquo Introduction

Taking into account (3) for the square of absolute payload119872velocity 1198812abs and by adding the kinetic energy for a slewingcrane boom according to (6) we will have the followingexpression for ldquocrane boom-payloadrdquo kinetic energy

119879 =119898

2(((

1198891199091

119889119905) minus (

119889120593119890

119889119905) (119877 + 119910

1))

2

+((1198891199101

119889119905) + (

119889120593119890

119889119905) 1199091)

2

+ (1198891199111

119889119905)

2

)

+1198681198742

33

2(119889120593119890

119889119905)

2

(9)

For further analysis we will assume that the crane boomslewing angular velocity is the constant 119889120593

119890119889119905 = 120596

119890 We will

use the derived expression for kinetic energy for the left-handsides of Lagrange equations in the noninertial reference frame119909111991011199111

119889

119889119905(120597119879

1205971

) minus120597119879

1205971199091

= 1198761199091

119889

119889119905(120597119879

120597 1199101

) minus120597119879

1205971199101

= 1198761199101

119889

119889119905(120597119879

1205971

) minus120597119879

1205971199111

= 1198761199111

(10)

Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N we have the followingformulae for the generalized forces in the noninertial refer-ence frame 119909

111991011199111

1198761199091

= minus 119873(1199091

119897)

1198761199101

= minus 119873(1199101

119897)

1198761199111

= + 119873(119897 minus 1199111

119897) minus 119898119892

(11)

Taking into account (9) (11) the equations (10) in thenoninertial reference frame 119909

111991011199111will finally take the

following form

119898(11988921199091

1198891199052) minus 119898(

119889120593119890

119889119905)

2

1199091minus 119898(

1198892120593119890

1198891199052) (119877 + 119910

1)

minus 2119898(119889120593119890

119889119905)(

1198891199101

119889119905) = minus119873(

1199091

119897)

119898(11988921199101

1198891199052) minus 119898(

119889120593119890

119889119905)

2

(119877 + 1199101) + 119898(

1198892120593119890

1198891199052)1199091

+ 2119898(119889120593119890

119889119905)(

1198891199091

119889119905) = minus119873(

1199101

119897)

The Scientific World Journal 5

119898(11988921199111

1198891199052) = minus119898119892 + 119873(

119897 minus 1199111

119897)

1198972= 1199092

1+ 1199102

1+ (1199111minus 119897)2

(12)

After two times differentiation of forth equations insystem (12) we will have the following formula for the cablereaction force119873 (Figure 2) from third equation of (12)

119873 = ((119898119897 [((1199091(11988921199091

1198891199052) + (

1198891199091

119889119905)

2

+ 1199101(11988921199101

1198891199052) + (

1198891199101

119889119905)

2

)

times (1198972minus 1199092

1minus 1199102

1)

+ (1199091(1198891199091

119889119905) + 1199101(1198891199101

119889119905))

2

)])

times (1198972minus 1199092

1minus 1199102

1)minus2

)

+119898119892119897

radic(1198972 minus 1199092

1minus 1199102

1)

(13)

So with an introduction of (13) to the first and secondequations of system (12) we will have the explicit nonlinearODE system for payload119872 swaying inCartesian coordinates

5 Linearized Mathematical Model

Further for the small angle 1205721 we may assume that 119911

1and

119889211991111198891199052 are the small parameters So the nonlinear equation

(13) yields

119873 asymp 119898119892 (14)

Then taking into account (14) the linearized system (12)for the two variables 119909

1and 119910

1takes the following form

11988921199091

1198891199052minus (

119889120593119890

119889119905)

2

1199091minus 2(

119889120593119890

119889119905)(

1198891199101

119889119905) = minus119892(

1199091

119897)

11988921199101

1198891199052minus (

119889120593119890

119889119905)

2

(119877 + 1199101) + 2 (

119889120593119890

119889119905)(

1198891199091

119889119905) = minus119892(

1199101

119897)

(15)

So we derive the system (15) of differential equationsfor relative motion of load 119872 on cable 119861119872 with a movablesuspension center 119861 which is fixed at the crane boom1198611198742 We assume that the crane boom 119861119874

2rotates with a

constant angular velocity 120596119890around vertical axis 119863119874

2 We

then transfer the origin of coordinate system 119874119909119910 (Figure 1)to the point 119860dyn of dynamic equilibrium for load 119872 andassume that 119909

1= 119909 119910

1= 119910 + 119910dyn (Figure 3) Then

the second equation of system (15) defines the amount ofdynamic deflection 119910dyn = 119860 st119860dyn = (120596

2

119890119877119897)(119892 minus 120596

2

119890119897)

Relative swaying paths of payload 119872 in the plane ofthe introduced relative coordinates 119909 and 119910 are shown inFigure 3 for successive increasing swaying times 119905

1= 125 s

(Figure 3(a)) 1199052= 3 s (Figure 3(b)) 119905

3= 10 s (Figure 3(c))

and 1199054= 25 s (Figure 3(d))

Then we have the normal system of two linear homo-geneous differential equations of second order for relativemotion of the load119872

(

1198632minus (1205962

119890minus (

119892

119897)) minus2120596

119890119863

2120596119890119863 119863

2minus (1205962

119890minus (

119892

119897))

)

times (119909 (119905)

119910 (119905)) = 0 119863 =

119889

119889119905

(16)

The determinant of natural frequencies matrix for thesystem (16) has the following form

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205822+ (

119892

119897) minus 1205962

119890minus2 sdot 120596

119890sdot 120582

2 sdot 120596119890sdot 120582 120582

2+ (

119892

119897) minus 1205962

119890

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (17)

Using (17) we adjust the roots 1205821and 120582

2of secular

equation

1205821= plusmn]1sdot 119894 120582

2= plusmn]2sdot 119894 ]

1= 119896 + 120596

119890

]2= 119896 minus 120596

119890 119896 = radic

119892

119897 120596119890

= 119896

(18)

The initial conditions for the problem are as follows

119909 (0) = 0 119863119909 (0) = +119881119861= +120596119890119877

119910 (0) = minus119910dyn 119863119910 (0) = 0

(19)

Taking into account (18) and (19) the law of relativemotion for payload takes the following form

119909 (119905) = 1198601

119894

2(119890minus]1119894119905minus 119890

]1119894119905) + 119860

2

119894

2(119890minus]2119894119905minus 119890

]2119894119905)

119910 (119905) = 1198601

1

2(119890minus]1119894119905+ 119890

]1119894119905) minus 119860

2

1

2(119890minus]2119894119905+ 119890

]2119894119905)

(20)

After substitution of values of1198601and119860

2in (20) we have

the following law of payload119872 relative swaying motion withthe uniformly rotating pivot center

119909 (119905) = (

119881119861minus 119910dyn (119896 minus 120596

119890)

2119896)

119894

2(119890minus]1119894119905minus 119890

]1119894119905)

+ (

119881119861+ 119910dyn (119896 + 120596

119890)

2119896)

119894

2(119890minus]2119894119905minus 119890

]2119894119905)

119910 (119905) = (

119881119861minus 119910dyn (119896 minus 120596

119890)

2119896)1

2(119890minus]1119894119905+ 119890

]1119894119905)

minus (

119881119861+ 119910dyn (119896 + 120596

119890)

2119896)1

2(119890minus]2119894119905+ 119890

]2119894119905)

(21)

6 The Scientific World Journal

102

101

1

099

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)

(a)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)

1012

1010

1008

1006

1004

1002

1

0998

0996

0994

(b)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)

1004

1002

1003

1001

1

0999

l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)

(c)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)

10012

10010

10008

10006

10004

10002

1

09996

09994

09998

l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448 (rads)

(d)

Figure 2 Time dependencies of dimensionless cable reactions 119873119898119892 in the first approximation for fixed cable lengths 119897 = 0206m (a)119897 = 0412m (b) 119897 = 0618m (c) and 119897 = 0825m (d)

The relative trajectories for load119872on the cable in Figure 3have been derived for the following numerical values 119877 =

0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)05

asymp

3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120572

1dyn =

000221 rad 119881119861

= 0103ms 119910dyn = 000182m ]1

=

119896 + 120596119890

= 3658 rads ]2

= 119896 minus 120596119890

= 3239 rads(Figure 3)

Numerical plots in Figure 3 show the payload119872 swayingin the noninertial reference frame (119874119909119910119911) In order tovisualize payload119872 absolute swaying path (Figures 4 and 5)

The Scientific World Journal 7

t1 = 125 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(a)

t2 = 3s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(b)

t3 = 10 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(c)

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

t4 = 25s

Relative trajectory of load M swayingfor linearized model

(d)

Figure 3The relative trajectories for load119872 on the cable during uniform rotational motion of the crane boom 1198611198742for the moments of time

1199051= 125 s (a) 119905

2= 3 s (b) 119905

3= 10 s (c) and 119905

4= 25 s (d)

in the fixed inertial reference frame (1198742119909211991021199112) we will

address the following transition formula

(1199092(119905)

1199102(119905)

) = (sin (120596

119890119905) cos (120596

119890119905)

minus cos (120596119890119905) sin (120596

119890119905)) (

119909 (119905)

119910 (119905) + 119877 + 119910dyn)

(22)

Derived computational plot for absolute payload swayingpath in the absolute coordinatesrsquo plane (119909

21199102) is shown in

Figure 5(a) for load119872 swaying time 1199051= 15 s and crane boom

transport slewing angle120593119890= 180

∘ In Figure 5(b) we show theexperimental plot for absolute payload swaying path in theabsolute coordinatesrsquo plane (119909

21199102) for load 119872 swaying time

1199052= 15 s and crane boom transport slewing angle 120593

119890= 180

∘which was built with an introduction of experimental setupin Figure 4

6 Comparison of Derived andKnown Published Results

The computational dimensionless cable force119873119898119892 derivedin the first approximation and shown in Figure 2 outlinesthe minor change of the cable tension force in the vicinityof the ordinate (119873119898119892) = 1 (119873119898119892) isin [09875 10233]

for 119897 = 0206m in Figure 2(a) (119873119898119892) isin [0994 10122]

for 119897 = 0412m in Figure 2(b) (119873119898119892) isin [0998 1004] for119897 = 0618m in Figure 2(c) and (119873119898119892) isin [099933 10013]

for 119897 = 0825m in Figure 2(d) So the assumption (14)really takes place as shown in Figure 2 The same effect for(119873119898119892)Mitrev isin [094 104] has been derived by Mitrev andGrigorov [23 Mitrevrsquos Figure 7 in page 89] that confirmsassumption (14) An analogous effect for dimensionless cable

8 The Scientific World Journal

Camera

Bearing

Boom

Laser pointer

Slewing axis

CableLower plate

M

Light-emitting diodeRegular grid 1200mm times 700mmwith square cells 200mm times 20mm

B

D

O2

Figure 4 The experimental measurement system for payload 119872

swaying during crane boom slewing

force (119873119898119892)Maczynski isin [0996 1005] has been derived byMaczynski and Wojciech [24 Maczynskirsquos Figure 10 in page278] which also reaffirms the assumption (14)

In published work by Sakawa et al ([25] Figure 3 at page552 Figure 4 at page 552 Figure 5 at page 554 and Figure 6at page 554) the small Sakawa values 119889119911119889119905 le 01 (Figures 3and 4) and 119889119911119889119905 le 005 (Figures 5 and 6) of 119911-projectionsfor payload velocities have been shown which confirms thesmall angle assumption119873 asymp 119898119892 (14)

7 Discussion and Analysis of Basic Results forMathematical and Physical Simulation

The relative trajectories for load119872 swaying in Figure 3 showequality of maximum values for relative 119909 and 119910 payloadcoordinates

119909max = 119910max = 1198601+ 1198602=

120596119890(119877 + 119910dyn)

119896 (23)

The formula (23) outlines that values 119909max and 119910max inFigure 3 increase with growth of boom slewing velocity 120596

119890

and with increased cable length 119897 The 119910dyn influence on thevalues of 119909max and 119910max is negligible

Equation (18) shows that the free oscillation frequencies]1and ]

2depend on the boom slewing velocity 120596

119890and

natural frequency 119896 of the payload with a fixed pivot point119861 The frequencies ]

1and ]

2essentially depend on the

pendulum length 119897 because 119896 ≫ 120596119890 Kinematically saying

the miscoordination of frequencies ]1and ]

2is ]1minus ]2=

2120596119890and is determined by the crane boom slewing velocity

120596119890 Dynamically saying the value of miscoordination is

governed by the Coriolis acceleration-induced DrsquoAlembertrsquosinertia force

Computations shown in Figure 3 for relative payload119872 swaying show that the clockwise direction of rotationfor relative oscillations is opposite in direction from thecounterclockwise direction of rotation for the crane boomThis effect follows from the conservation of position of theplane 119910

21199112of oscillations of load 119872 in the fixed inertial

coordinate system 119909211991021199112

The comparative analysis of computational (1) and exper-imental (2) absolute payload trajectories is shown in Fig-ure 5(c)The quantity of the local maximums in Figure 5(a) is8 for theoretical curve 1 So the theoretical period for relativeload swaying is 119879theor = 187 s The quantity of the localmaximums in Figure 5(b) is 75 for experimental curve 2Theperiod for experimental curve 2 in Figure 5(b) is 119879experim =

2 s The discrepancy between the values of theoretical andexperimental periods does not exceed 7

The experimental absolute trajectory (2) in Figures 5(b)and 5(c) outlines the additional relative swaying of payload119872 after crane boom stop which has the elliptical form inthe vicinity of the point (0 minus05) in the plane 119909

21199102for 120593119890=

180∘ The remaining payload motions after crane boom stop

are the small oscillations of the payload with the fixed pivotcenter

8 Final Conclusions

The closed system of nonlinear ODEs for payload relativeswaying has been derived in relative Cartesian coordinateswith an introduction of Lagrange equations and a geometricconstraint equation

The explicit form of the dependence of the cable reactionforce as a function of the relative payload coordinates 119909

1 1199101

1199111and their first and second time derivatives has been found

Derived complex dependency of the cable reaction force onthe abovementioned variables determines the nonlinearity ofthe present problem

The deviation of cable reaction force frompayload gravityforce is small This fact allowed the derivation of a linearizedODE system for payload swaying in relative Cartesian coor-dinates and verified the small relative Cartesian coordinatesrsquoassumption The analytical expressions for relative payloadpath have been derived

Derived results have good agreement with experimentsand known published data

The Scientific World Journal 9

05

04

03

02

01

0

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

t = 15 s

1

Point O2

2R = 1 (m)

1

PointnP i OO2OO

y2 (m)

x2 (m)(Point D)

(Point M)

(a)

05

04

03

02

01

02R = 1 (m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

2

4

3

Rotational directionPoint O2

t = 18 s

(Point D)

(Point M)

(Point B)

x2 (m)

y2 (m)

(b)

1 2

4

Point O2

2R = 1 (m)

y2(m)

x2(m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

(Point D)

(Point M) (Point M)

05

04

03

02

01

0

(c)

Figure 5 Calculated 1 (a c) and experimental 2 (b c) absolute trajectories for load119872 where 3 is the trajectory of boom point 119861 (b) 4 is thecenter 119874

2(D) of boom rotation (b c)

Disclosure

The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors

Conflict of Interests

The authors Alexander V Perig Alexander N Stadnik andAlexander I Deriglazov declare that there is no conflict ofinterests regarding the publication of this paper

References

[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002

[2] W Adamiec-Wojcik I P Fałat A Maczynski and S WojciechldquoLoad stabilisation in an A-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009

[3] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquoMethods in AppliedMechanics and Engineering vol170 no 3-4 pp 343ndash3353 1999

[4] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009

[5] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004

[6] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007

[7] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008

[8] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing aload prescribed motionrdquo The Archive of Mechanical Engineer-ing vol 3 pp 209ndash222 2009 httpamemeilpweduplindexphpengcontentdownload3731748fileart 2009 3 02pdf

[9] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010

[10] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011

10 The Scientific World Journal

[11] J H Cha S H Ham K Y Lee and M I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution ofMechanical Engineers K vol 224no 4 pp 365ndash373 2010

[12] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002

[13] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007

[14] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K vol 217no 2 pp 99ndash109 2003

[15] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006

[16] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers K vol 224 no 4 pp 341ndash352 2010

[17] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity Science A vol 8 no 9 pp 1395ndash1400 2007

[18] B Grigorov and R Mitrev ldquoA NewtonmdashEuler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009

[19] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998

[20] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008

[21] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008

[22] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers I vol 223 no 4 pp 575ndash580 2009

[23] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008

[24] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003

[25] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981

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RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

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DistributedSensor Networks

International Journal of

2 The Scientific World Journal

specified motion It is necessary to note that this paper dealsprimarily with controlled in-plane motions of the crane sys-tem with linear transport motion Blajer and Kołodziejczyk[6] have investigated the control of underactuated cranemechanical systems with servo constraints This work takesinto account the restoring load gravity force in the horizontalplane However the derived mathematical model assumesprimarily linear transportmotionwithout theCoriolis inertiaforce Blajer and Kołodziejczyk [7] have presented a concep-tual research article focused on the building of n-index DAEsystems for a relevant dynamic description of partly specifiedcrane system motion Blajer and Kołodziejczyk [8] haveintroduced a carefully argued computational technique fornumerical simulation of cranes based on the use of dependentcoordinates The advantages of dependent coordinatesrsquo usagehave been outlined It is necessary to note that with each newresearch effort the authors apply much more sophisticatedcomputational methods and derive more exact governingdynamic equations However this paper is also based onthe same simple pendulum model as the first approacheswherein the Coriolis inertia force and the restoring loadgravity force are ignored Blajer et al [9] have simulated atwo-tier framed human arm model with segments of equallength The application of this model seems to be veryuseful for the dynamic analysis of a swaying load in thehorizontal plane during rotation of a crane boom Blajer andKołodziejczyk [10] have developed improved DAE equationsfor more precise rotary crane dynamics simulation Butadditional dimensional analysis for the explicit forms of thegoverning equations would greatly enhance the work

Cha et al [11] have applied a topological modelingapproach to a dynamic simulation of multiple floating cranesin shipyards taking into account crane boom oscillations

Ellermann et al [12] have introduced the vector ofgeneralized gyroscopic andCoriolis forces for amathematicaldescription of nonlinear dynamics of floating cranes Thiswork used a comprehensivemodel with a set of 20 differentialequations but does not allow proper analysis of particularcases of floating jib crane-load system motions

Erneux and Kalmar-Nagy [13] have proposed a planependulum container crane model with linear transport trol-ley motion This model illustrates linear displacement of theplane pendulum suspension center

Glossiotis and Antoniadis [14] have derived a four-degrees-of-freedom (4DOF) model for the rotary cranesystem incorporating both hoisting and slewing payloadmotions and thus is able to handle cases where hoisting canbe simultaneously applied to the rotary motion

Ibrahim [15] has studied the issues of multiplicative noiseand noise-enhanced stability of spherical pendulums

Kortelainen andMikkola [16] have introduced data man-agement techniqueswithin an approachusing a semantic datamodel for payload oscillation problems

Mitrev [17] has applied his generalized approach with theintroduction of an inertia force vector to the case of lineartransport motion of the pendulum pivot center Grigorovand Mitrev [18] have investigated a spherical pendulumdynamic model for a numerical solution of a freely sus-pended swinging load problem They have derived absolute

trajectories of both the pendulum suspension center and theswinging load

Osinski andWojciech [19] have analyzed the hoistmotionof a mechanical system consisting of a load hoist rope andelastic crane jib This study mainly deals with rectilinear loadoscillations and lateral crane jib vibrations

Ren et al [20] have used a mixed approach combin-ing both analytical and finite-element techniques for theLagrangian function of a mechanical system They havedetermined the relations between load swaying crane boomflexibility and wave motions Schaub [21] has implemented arate-based ship-mounted crane payload pendulation controlsystem This research is focused on a kinematic approach topayload stabilizationwith an emphasis on the usage of inertialmeasurement unit (IMU) information

Uchiyama [22] has applied control theory techniques for arobust crane controller design that provides robustness withrespect to changes in cable length and transported payloadmass without iterative computations

At the same time these known articles [1ndash22] do not dojustice to load swaying descriptions using Coriolis effects inrelative motion The present paper focuses on these issues

The present paper focuses on the Lagrange mechanics-based description of small oscillations for a spherical pendu-lum with a uniformly rotating suspension center

The prime novelty statement of the present paper isbased on the rational introduction of a Cartesian coordinatesystem for study of small relative swaying of a payload andsubstantiation of uniformity of crane boom rotation

2 Kinematic Analysis

In order to solve the problem we introduce the Lagrangeequations for the motion of the mechanical system ldquocraneboom 119861119874

2-load119872rdquo which is shown in Figure 1

The computational scheme in Figure 1 may be describedwith an introduction of three degrees of freedom For gen-eralized coordinates we assume the rectangular coordinates119909 119910 and 119911 of the load and the angle 120593

119890= 120596119890119905 of crane

boom 1198611198742uniform rotation in the horizontal plane (119909119910)

with the constant angular velocity 120596119890around the vertical

axis 11987421199112 We denote the fixed inertial coordinate system

as 119909211991021199112and the moving noninertial frame of reference as

119909111991011199111 which is rigidly bounded with the crane boom 119861119874

2

Rotation of themoving noninertial frame of reference 119909111991011199111

around the fixed inertial coordinate system 119909211991021199112defines

the transportation motion The motion of load119872 relative tothe moving noninertial frame of reference 119909

111991011199111defines the

relative motionThe scalar of the load119872 transportation velocity is defined

as 119881119890

= 120596119890sdot 1198742119872 and transportation velocity vector V

119890

is perpendicular to O2M that is the scalar product(V119890O2M) = 0 (Figure 1) The relative velocity vector V

119903of

the load 119872 is defined as V119903= (119889119909

1119889119905 119889119910

1119889119905 119889119911

1119889119905) in

Figure 1 In the initial time 119905 = 0 load119872 has the vertical 119861119860stin-line position that is the load119872 initial position coincideswith the static equilibrium position119860 st for the load119872 on thecable119872119861

The Scientific World Journal 3

z2

z1

y2

y1

O2

O1

x2

x1

120593e

1205722

1205721

Adyn

Ast

= d120593edt

H

D

l

R

B

Δ

x

y

M

O

z

120579

rarrVr

rarrVe

rarr120596e120596e

rarrN

mrarrg

Figure 1 Computational spatial scheme of spherical pendulum119872 swaying on the cable119872119861 during crane boom 1198611198742slewing motion for the

linearized model derivation in Cartesian coordinates

At time 119905 = 0 the load 119872 has zero absolute velocity andthe initial relative velocity is 119881

119903119909(0) = +120596

119890sdot 119861119863 (Figure 1)

In Figure 1 we denote the angle 120579 which is the currentangle 119860 st1198742119872 that is the angle between the axis 119910

1and

the radius-vector O2M where sin(120579) = 11990911198742119872 cos(120579) =

(119877+1199101)1198742119872The vertical coordinate 119911

1in Figure 1 is defined

as 1199111= 119897 minus 119897 sdot cos(120572

1) where 120572

1is the angle between the cable

119872119861 and the verticalFor small angles 120572

1 the projections of the absolute

velocity vector may be written as

1198811199091

= (1198891199091

119889119905) minus 119881119890sdot cos (120579)

1198811199101

= (1198891199101

119889119905) + 119881119890sdot sin (120579)

1198811199111

= (1198891199111

119889119905) = (

1198891205721

119889119905) sdot 119897 sdot sin (120572

1)

(1)

Taking into account Figure 1 the system of (1) for theprojections of the absolute velocity vector takes the followingform

1198811199091

= (1198891199091

119889119905) minus (

119889120593119890

119889119905) (119877 + 119910

1)

1198811199101

= (1198891199101

119889119905) + (

119889120593119890

119889119905) 1199091

1198811199111

= (1198891199111

119889119905)

(2)

The square of absolute payload119872 velocitymay be writtenas 1198812abs = 119881

2

1199091

+ 1198812

1199101

+ 1198812

1199111

1198812

abs = (1198891199091

119889119905)

2

+ (1198891199101

119889119905)

2

+ (1198891199111

119889119905)

2

4 The Scientific World Journal

+ (119889120593119890

119889119905)

2

(1199092

1+ (1199101+ 119877)2

)

+ 2(119889120593119890

119889119905)(1199091(1198891199101

119889119905) minus (119910

1+ 119877) (

1198891199091

119889119905))

= ((1198891199091

119889119905) minus (

119889120593119890

119889119905) (119877 + 119910

1))

2

+ ((1198891199101

119889119905) + (

119889120593119890

119889119905) 1199091)

2

+ (1198891199111

119889119905)

2

(3)

3 Dynamic Analysis

The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figure 1 is governed by the vector equation

for the rate of change of moment of momentum H11987423

for thesystem ldquocrane boom 119861119874

2-load119872rdquo with respect to point119874

2in

the inertial reference frame 119909211991021199112

(119909211991021199112)119889H11987423

119889119905= sumM1198742 (4)

The vector equation (4) contains the following compo-nents

H11987423

= 1198681198742

33120596119890 (5)

1198681198742

33= (1198681198742

33)1198611198742

+ 119898(1199092

1+ (119877 + 119910

1)2

) (6)

sumM1198742 = M1198742119863119879

minusM1198742119865119879

= (1198721198742

119863119879minus1198721198742

119865119879) k (7)

where (119868119874233)1198611198742

is the element of mass moment of inertia forthe crane boom119861119874

2in inertial fixed on earth reference frame

119909211991021199112with respect to unit vector k and119898(119909

2

1+ (119877 + 119910

1)2) is

the element of mass moment of inertia for the payload119872 ininertial fixed on earth reference frame 119909

211991021199112with respect to

unit vector kThe external moment of gravitational forceM1198742(mg) = 0

in (4) and (7) becausemg uarrdarr kFor the system ldquocrane boom-payloadrdquo the cable reaction

force N is the internal force So in (4) and (7) we haveM1198742(N) = 0

Substitution of (5) (6) and (7) into (4) yields thefollowing scalar equation for the rate of change of momentof momentum 119867

1198742

3for the system ldquocrane boom 119861119874

2-load

119872rdquo with respect to point 1198742in the inertial reference frame

119909211991021199112

119889

119889119905(((1198681198742

33)1198611198742

+ 119898(1199092

1+ (119877 + 119910

1)2

)) (119889120593119890

119889119905))

= 1198721198742

119863119879minus1198721198742

119865119879

(8)

where driving1198721198742119863119879

and frictional1198721198742119865119879

torques are the tech-nically defined functions for specific electric drive systems

4 Nonlinear Mathematical Model withLagrange Equationsrsquo Introduction

Taking into account (3) for the square of absolute payload119872velocity 1198812abs and by adding the kinetic energy for a slewingcrane boom according to (6) we will have the followingexpression for ldquocrane boom-payloadrdquo kinetic energy

119879 =119898

2(((

1198891199091

119889119905) minus (

119889120593119890

119889119905) (119877 + 119910

1))

2

+((1198891199101

119889119905) + (

119889120593119890

119889119905) 1199091)

2

+ (1198891199111

119889119905)

2

)

+1198681198742

33

2(119889120593119890

119889119905)

2

(9)

For further analysis we will assume that the crane boomslewing angular velocity is the constant 119889120593

119890119889119905 = 120596

119890 We will

use the derived expression for kinetic energy for the left-handsides of Lagrange equations in the noninertial reference frame119909111991011199111

119889

119889119905(120597119879

1205971

) minus120597119879

1205971199091

= 1198761199091

119889

119889119905(120597119879

120597 1199101

) minus120597119879

1205971199101

= 1198761199101

119889

119889119905(120597119879

1205971

) minus120597119879

1205971199111

= 1198761199111

(10)

Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N we have the followingformulae for the generalized forces in the noninertial refer-ence frame 119909

111991011199111

1198761199091

= minus 119873(1199091

119897)

1198761199101

= minus 119873(1199101

119897)

1198761199111

= + 119873(119897 minus 1199111

119897) minus 119898119892

(11)

Taking into account (9) (11) the equations (10) in thenoninertial reference frame 119909

111991011199111will finally take the

following form

119898(11988921199091

1198891199052) minus 119898(

119889120593119890

119889119905)

2

1199091minus 119898(

1198892120593119890

1198891199052) (119877 + 119910

1)

minus 2119898(119889120593119890

119889119905)(

1198891199101

119889119905) = minus119873(

1199091

119897)

119898(11988921199101

1198891199052) minus 119898(

119889120593119890

119889119905)

2

(119877 + 1199101) + 119898(

1198892120593119890

1198891199052)1199091

+ 2119898(119889120593119890

119889119905)(

1198891199091

119889119905) = minus119873(

1199101

119897)

The Scientific World Journal 5

119898(11988921199111

1198891199052) = minus119898119892 + 119873(

119897 minus 1199111

119897)

1198972= 1199092

1+ 1199102

1+ (1199111minus 119897)2

(12)

After two times differentiation of forth equations insystem (12) we will have the following formula for the cablereaction force119873 (Figure 2) from third equation of (12)

119873 = ((119898119897 [((1199091(11988921199091

1198891199052) + (

1198891199091

119889119905)

2

+ 1199101(11988921199101

1198891199052) + (

1198891199101

119889119905)

2

)

times (1198972minus 1199092

1minus 1199102

1)

+ (1199091(1198891199091

119889119905) + 1199101(1198891199101

119889119905))

2

)])

times (1198972minus 1199092

1minus 1199102

1)minus2

)

+119898119892119897

radic(1198972 minus 1199092

1minus 1199102

1)

(13)

So with an introduction of (13) to the first and secondequations of system (12) we will have the explicit nonlinearODE system for payload119872 swaying inCartesian coordinates

5 Linearized Mathematical Model

Further for the small angle 1205721 we may assume that 119911

1and

119889211991111198891199052 are the small parameters So the nonlinear equation

(13) yields

119873 asymp 119898119892 (14)

Then taking into account (14) the linearized system (12)for the two variables 119909

1and 119910

1takes the following form

11988921199091

1198891199052minus (

119889120593119890

119889119905)

2

1199091minus 2(

119889120593119890

119889119905)(

1198891199101

119889119905) = minus119892(

1199091

119897)

11988921199101

1198891199052minus (

119889120593119890

119889119905)

2

(119877 + 1199101) + 2 (

119889120593119890

119889119905)(

1198891199091

119889119905) = minus119892(

1199101

119897)

(15)

So we derive the system (15) of differential equationsfor relative motion of load 119872 on cable 119861119872 with a movablesuspension center 119861 which is fixed at the crane boom1198611198742 We assume that the crane boom 119861119874

2rotates with a

constant angular velocity 120596119890around vertical axis 119863119874

2 We

then transfer the origin of coordinate system 119874119909119910 (Figure 1)to the point 119860dyn of dynamic equilibrium for load 119872 andassume that 119909

1= 119909 119910

1= 119910 + 119910dyn (Figure 3) Then

the second equation of system (15) defines the amount ofdynamic deflection 119910dyn = 119860 st119860dyn = (120596

2

119890119877119897)(119892 minus 120596

2

119890119897)

Relative swaying paths of payload 119872 in the plane ofthe introduced relative coordinates 119909 and 119910 are shown inFigure 3 for successive increasing swaying times 119905

1= 125 s

(Figure 3(a)) 1199052= 3 s (Figure 3(b)) 119905

3= 10 s (Figure 3(c))

and 1199054= 25 s (Figure 3(d))

Then we have the normal system of two linear homo-geneous differential equations of second order for relativemotion of the load119872

(

1198632minus (1205962

119890minus (

119892

119897)) minus2120596

119890119863

2120596119890119863 119863

2minus (1205962

119890minus (

119892

119897))

)

times (119909 (119905)

119910 (119905)) = 0 119863 =

119889

119889119905

(16)

The determinant of natural frequencies matrix for thesystem (16) has the following form

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205822+ (

119892

119897) minus 1205962

119890minus2 sdot 120596

119890sdot 120582

2 sdot 120596119890sdot 120582 120582

2+ (

119892

119897) minus 1205962

119890

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (17)

Using (17) we adjust the roots 1205821and 120582

2of secular

equation

1205821= plusmn]1sdot 119894 120582

2= plusmn]2sdot 119894 ]

1= 119896 + 120596

119890

]2= 119896 minus 120596

119890 119896 = radic

119892

119897 120596119890

= 119896

(18)

The initial conditions for the problem are as follows

119909 (0) = 0 119863119909 (0) = +119881119861= +120596119890119877

119910 (0) = minus119910dyn 119863119910 (0) = 0

(19)

Taking into account (18) and (19) the law of relativemotion for payload takes the following form

119909 (119905) = 1198601

119894

2(119890minus]1119894119905minus 119890

]1119894119905) + 119860

2

119894

2(119890minus]2119894119905minus 119890

]2119894119905)

119910 (119905) = 1198601

1

2(119890minus]1119894119905+ 119890

]1119894119905) minus 119860

2

1

2(119890minus]2119894119905+ 119890

]2119894119905)

(20)

After substitution of values of1198601and119860

2in (20) we have

the following law of payload119872 relative swaying motion withthe uniformly rotating pivot center

119909 (119905) = (

119881119861minus 119910dyn (119896 minus 120596

119890)

2119896)

119894

2(119890minus]1119894119905minus 119890

]1119894119905)

+ (

119881119861+ 119910dyn (119896 + 120596

119890)

2119896)

119894

2(119890minus]2119894119905minus 119890

]2119894119905)

119910 (119905) = (

119881119861minus 119910dyn (119896 minus 120596

119890)

2119896)1

2(119890minus]1119894119905+ 119890

]1119894119905)

minus (

119881119861+ 119910dyn (119896 + 120596

119890)

2119896)1

2(119890minus]2119894119905+ 119890

]2119894119905)

(21)

6 The Scientific World Journal

102

101

1

099

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)

(a)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)

1012

1010

1008

1006

1004

1002

1

0998

0996

0994

(b)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)

1004

1002

1003

1001

1

0999

l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)

(c)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)

10012

10010

10008

10006

10004

10002

1

09996

09994

09998

l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448 (rads)

(d)

Figure 2 Time dependencies of dimensionless cable reactions 119873119898119892 in the first approximation for fixed cable lengths 119897 = 0206m (a)119897 = 0412m (b) 119897 = 0618m (c) and 119897 = 0825m (d)

The relative trajectories for load119872on the cable in Figure 3have been derived for the following numerical values 119877 =

0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)05

asymp

3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120572

1dyn =

000221 rad 119881119861

= 0103ms 119910dyn = 000182m ]1

=

119896 + 120596119890

= 3658 rads ]2

= 119896 minus 120596119890

= 3239 rads(Figure 3)

Numerical plots in Figure 3 show the payload119872 swayingin the noninertial reference frame (119874119909119910119911) In order tovisualize payload119872 absolute swaying path (Figures 4 and 5)

The Scientific World Journal 7

t1 = 125 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(a)

t2 = 3s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(b)

t3 = 10 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(c)

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

t4 = 25s

Relative trajectory of load M swayingfor linearized model

(d)

Figure 3The relative trajectories for load119872 on the cable during uniform rotational motion of the crane boom 1198611198742for the moments of time

1199051= 125 s (a) 119905

2= 3 s (b) 119905

3= 10 s (c) and 119905

4= 25 s (d)

in the fixed inertial reference frame (1198742119909211991021199112) we will

address the following transition formula

(1199092(119905)

1199102(119905)

) = (sin (120596

119890119905) cos (120596

119890119905)

minus cos (120596119890119905) sin (120596

119890119905)) (

119909 (119905)

119910 (119905) + 119877 + 119910dyn)

(22)

Derived computational plot for absolute payload swayingpath in the absolute coordinatesrsquo plane (119909

21199102) is shown in

Figure 5(a) for load119872 swaying time 1199051= 15 s and crane boom

transport slewing angle120593119890= 180

∘ In Figure 5(b) we show theexperimental plot for absolute payload swaying path in theabsolute coordinatesrsquo plane (119909

21199102) for load 119872 swaying time

1199052= 15 s and crane boom transport slewing angle 120593

119890= 180

∘which was built with an introduction of experimental setupin Figure 4

6 Comparison of Derived andKnown Published Results

The computational dimensionless cable force119873119898119892 derivedin the first approximation and shown in Figure 2 outlinesthe minor change of the cable tension force in the vicinityof the ordinate (119873119898119892) = 1 (119873119898119892) isin [09875 10233]

for 119897 = 0206m in Figure 2(a) (119873119898119892) isin [0994 10122]

for 119897 = 0412m in Figure 2(b) (119873119898119892) isin [0998 1004] for119897 = 0618m in Figure 2(c) and (119873119898119892) isin [099933 10013]

for 119897 = 0825m in Figure 2(d) So the assumption (14)really takes place as shown in Figure 2 The same effect for(119873119898119892)Mitrev isin [094 104] has been derived by Mitrev andGrigorov [23 Mitrevrsquos Figure 7 in page 89] that confirmsassumption (14) An analogous effect for dimensionless cable

8 The Scientific World Journal

Camera

Bearing

Boom

Laser pointer

Slewing axis

CableLower plate

M

Light-emitting diodeRegular grid 1200mm times 700mmwith square cells 200mm times 20mm

B

D

O2

Figure 4 The experimental measurement system for payload 119872

swaying during crane boom slewing

force (119873119898119892)Maczynski isin [0996 1005] has been derived byMaczynski and Wojciech [24 Maczynskirsquos Figure 10 in page278] which also reaffirms the assumption (14)

In published work by Sakawa et al ([25] Figure 3 at page552 Figure 4 at page 552 Figure 5 at page 554 and Figure 6at page 554) the small Sakawa values 119889119911119889119905 le 01 (Figures 3and 4) and 119889119911119889119905 le 005 (Figures 5 and 6) of 119911-projectionsfor payload velocities have been shown which confirms thesmall angle assumption119873 asymp 119898119892 (14)

7 Discussion and Analysis of Basic Results forMathematical and Physical Simulation

The relative trajectories for load119872 swaying in Figure 3 showequality of maximum values for relative 119909 and 119910 payloadcoordinates

119909max = 119910max = 1198601+ 1198602=

120596119890(119877 + 119910dyn)

119896 (23)

The formula (23) outlines that values 119909max and 119910max inFigure 3 increase with growth of boom slewing velocity 120596

119890

and with increased cable length 119897 The 119910dyn influence on thevalues of 119909max and 119910max is negligible

Equation (18) shows that the free oscillation frequencies]1and ]

2depend on the boom slewing velocity 120596

119890and

natural frequency 119896 of the payload with a fixed pivot point119861 The frequencies ]

1and ]

2essentially depend on the

pendulum length 119897 because 119896 ≫ 120596119890 Kinematically saying

the miscoordination of frequencies ]1and ]

2is ]1minus ]2=

2120596119890and is determined by the crane boom slewing velocity

120596119890 Dynamically saying the value of miscoordination is

governed by the Coriolis acceleration-induced DrsquoAlembertrsquosinertia force

Computations shown in Figure 3 for relative payload119872 swaying show that the clockwise direction of rotationfor relative oscillations is opposite in direction from thecounterclockwise direction of rotation for the crane boomThis effect follows from the conservation of position of theplane 119910

21199112of oscillations of load 119872 in the fixed inertial

coordinate system 119909211991021199112

The comparative analysis of computational (1) and exper-imental (2) absolute payload trajectories is shown in Fig-ure 5(c)The quantity of the local maximums in Figure 5(a) is8 for theoretical curve 1 So the theoretical period for relativeload swaying is 119879theor = 187 s The quantity of the localmaximums in Figure 5(b) is 75 for experimental curve 2Theperiod for experimental curve 2 in Figure 5(b) is 119879experim =

2 s The discrepancy between the values of theoretical andexperimental periods does not exceed 7

The experimental absolute trajectory (2) in Figures 5(b)and 5(c) outlines the additional relative swaying of payload119872 after crane boom stop which has the elliptical form inthe vicinity of the point (0 minus05) in the plane 119909

21199102for 120593119890=

180∘ The remaining payload motions after crane boom stop

are the small oscillations of the payload with the fixed pivotcenter

8 Final Conclusions

The closed system of nonlinear ODEs for payload relativeswaying has been derived in relative Cartesian coordinateswith an introduction of Lagrange equations and a geometricconstraint equation

The explicit form of the dependence of the cable reactionforce as a function of the relative payload coordinates 119909

1 1199101

1199111and their first and second time derivatives has been found

Derived complex dependency of the cable reaction force onthe abovementioned variables determines the nonlinearity ofthe present problem

The deviation of cable reaction force frompayload gravityforce is small This fact allowed the derivation of a linearizedODE system for payload swaying in relative Cartesian coor-dinates and verified the small relative Cartesian coordinatesrsquoassumption The analytical expressions for relative payloadpath have been derived

Derived results have good agreement with experimentsand known published data

The Scientific World Journal 9

05

04

03

02

01

0

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

t = 15 s

1

Point O2

2R = 1 (m)

1

PointnP i OO2OO

y2 (m)

x2 (m)(Point D)

(Point M)

(a)

05

04

03

02

01

02R = 1 (m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

2

4

3

Rotational directionPoint O2

t = 18 s

(Point D)

(Point M)

(Point B)

x2 (m)

y2 (m)

(b)

1 2

4

Point O2

2R = 1 (m)

y2(m)

x2(m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

(Point D)

(Point M) (Point M)

05

04

03

02

01

0

(c)

Figure 5 Calculated 1 (a c) and experimental 2 (b c) absolute trajectories for load119872 where 3 is the trajectory of boom point 119861 (b) 4 is thecenter 119874

2(D) of boom rotation (b c)

Disclosure

The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors

Conflict of Interests

The authors Alexander V Perig Alexander N Stadnik andAlexander I Deriglazov declare that there is no conflict ofinterests regarding the publication of this paper

References

[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002

[2] W Adamiec-Wojcik I P Fałat A Maczynski and S WojciechldquoLoad stabilisation in an A-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009

[3] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquoMethods in AppliedMechanics and Engineering vol170 no 3-4 pp 343ndash3353 1999

[4] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009

[5] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004

[6] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007

[7] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008

[8] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing aload prescribed motionrdquo The Archive of Mechanical Engineer-ing vol 3 pp 209ndash222 2009 httpamemeilpweduplindexphpengcontentdownload3731748fileart 2009 3 02pdf

[9] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010

[10] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011

10 The Scientific World Journal

[11] J H Cha S H Ham K Y Lee and M I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution ofMechanical Engineers K vol 224no 4 pp 365ndash373 2010

[12] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002

[13] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007

[14] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K vol 217no 2 pp 99ndash109 2003

[15] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006

[16] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers K vol 224 no 4 pp 341ndash352 2010

[17] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity Science A vol 8 no 9 pp 1395ndash1400 2007

[18] B Grigorov and R Mitrev ldquoA NewtonmdashEuler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009

[19] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998

[20] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008

[21] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008

[22] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers I vol 223 no 4 pp 575ndash580 2009

[23] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008

[24] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003

[25] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

The Scientific World Journal 3

z2

z1

y2

y1

O2

O1

x2

x1

120593e

1205722

1205721

Adyn

Ast

= d120593edt

H

D

l

R

B

Δ

x

y

M

O

z

120579

rarrVr

rarrVe

rarr120596e120596e

rarrN

mrarrg

Figure 1 Computational spatial scheme of spherical pendulum119872 swaying on the cable119872119861 during crane boom 1198611198742slewing motion for the

linearized model derivation in Cartesian coordinates

At time 119905 = 0 the load 119872 has zero absolute velocity andthe initial relative velocity is 119881

119903119909(0) = +120596

119890sdot 119861119863 (Figure 1)

In Figure 1 we denote the angle 120579 which is the currentangle 119860 st1198742119872 that is the angle between the axis 119910

1and

the radius-vector O2M where sin(120579) = 11990911198742119872 cos(120579) =

(119877+1199101)1198742119872The vertical coordinate 119911

1in Figure 1 is defined

as 1199111= 119897 minus 119897 sdot cos(120572

1) where 120572

1is the angle between the cable

119872119861 and the verticalFor small angles 120572

1 the projections of the absolute

velocity vector may be written as

1198811199091

= (1198891199091

119889119905) minus 119881119890sdot cos (120579)

1198811199101

= (1198891199101

119889119905) + 119881119890sdot sin (120579)

1198811199111

= (1198891199111

119889119905) = (

1198891205721

119889119905) sdot 119897 sdot sin (120572

1)

(1)

Taking into account Figure 1 the system of (1) for theprojections of the absolute velocity vector takes the followingform

1198811199091

= (1198891199091

119889119905) minus (

119889120593119890

119889119905) (119877 + 119910

1)

1198811199101

= (1198891199101

119889119905) + (

119889120593119890

119889119905) 1199091

1198811199111

= (1198891199111

119889119905)

(2)

The square of absolute payload119872 velocitymay be writtenas 1198812abs = 119881

2

1199091

+ 1198812

1199101

+ 1198812

1199111

1198812

abs = (1198891199091

119889119905)

2

+ (1198891199101

119889119905)

2

+ (1198891199111

119889119905)

2

4 The Scientific World Journal

+ (119889120593119890

119889119905)

2

(1199092

1+ (1199101+ 119877)2

)

+ 2(119889120593119890

119889119905)(1199091(1198891199101

119889119905) minus (119910

1+ 119877) (

1198891199091

119889119905))

= ((1198891199091

119889119905) minus (

119889120593119890

119889119905) (119877 + 119910

1))

2

+ ((1198891199101

119889119905) + (

119889120593119890

119889119905) 1199091)

2

+ (1198891199111

119889119905)

2

(3)

3 Dynamic Analysis

The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figure 1 is governed by the vector equation

for the rate of change of moment of momentum H11987423

for thesystem ldquocrane boom 119861119874

2-load119872rdquo with respect to point119874

2in

the inertial reference frame 119909211991021199112

(119909211991021199112)119889H11987423

119889119905= sumM1198742 (4)

The vector equation (4) contains the following compo-nents

H11987423

= 1198681198742

33120596119890 (5)

1198681198742

33= (1198681198742

33)1198611198742

+ 119898(1199092

1+ (119877 + 119910

1)2

) (6)

sumM1198742 = M1198742119863119879

minusM1198742119865119879

= (1198721198742

119863119879minus1198721198742

119865119879) k (7)

where (119868119874233)1198611198742

is the element of mass moment of inertia forthe crane boom119861119874

2in inertial fixed on earth reference frame

119909211991021199112with respect to unit vector k and119898(119909

2

1+ (119877 + 119910

1)2) is

the element of mass moment of inertia for the payload119872 ininertial fixed on earth reference frame 119909

211991021199112with respect to

unit vector kThe external moment of gravitational forceM1198742(mg) = 0

in (4) and (7) becausemg uarrdarr kFor the system ldquocrane boom-payloadrdquo the cable reaction

force N is the internal force So in (4) and (7) we haveM1198742(N) = 0

Substitution of (5) (6) and (7) into (4) yields thefollowing scalar equation for the rate of change of momentof momentum 119867

1198742

3for the system ldquocrane boom 119861119874

2-load

119872rdquo with respect to point 1198742in the inertial reference frame

119909211991021199112

119889

119889119905(((1198681198742

33)1198611198742

+ 119898(1199092

1+ (119877 + 119910

1)2

)) (119889120593119890

119889119905))

= 1198721198742

119863119879minus1198721198742

119865119879

(8)

where driving1198721198742119863119879

and frictional1198721198742119865119879

torques are the tech-nically defined functions for specific electric drive systems

4 Nonlinear Mathematical Model withLagrange Equationsrsquo Introduction

Taking into account (3) for the square of absolute payload119872velocity 1198812abs and by adding the kinetic energy for a slewingcrane boom according to (6) we will have the followingexpression for ldquocrane boom-payloadrdquo kinetic energy

119879 =119898

2(((

1198891199091

119889119905) minus (

119889120593119890

119889119905) (119877 + 119910

1))

2

+((1198891199101

119889119905) + (

119889120593119890

119889119905) 1199091)

2

+ (1198891199111

119889119905)

2

)

+1198681198742

33

2(119889120593119890

119889119905)

2

(9)

For further analysis we will assume that the crane boomslewing angular velocity is the constant 119889120593

119890119889119905 = 120596

119890 We will

use the derived expression for kinetic energy for the left-handsides of Lagrange equations in the noninertial reference frame119909111991011199111

119889

119889119905(120597119879

1205971

) minus120597119879

1205971199091

= 1198761199091

119889

119889119905(120597119879

120597 1199101

) minus120597119879

1205971199101

= 1198761199101

119889

119889119905(120597119879

1205971

) minus120597119879

1205971199111

= 1198761199111

(10)

Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N we have the followingformulae for the generalized forces in the noninertial refer-ence frame 119909

111991011199111

1198761199091

= minus 119873(1199091

119897)

1198761199101

= minus 119873(1199101

119897)

1198761199111

= + 119873(119897 minus 1199111

119897) minus 119898119892

(11)

Taking into account (9) (11) the equations (10) in thenoninertial reference frame 119909

111991011199111will finally take the

following form

119898(11988921199091

1198891199052) minus 119898(

119889120593119890

119889119905)

2

1199091minus 119898(

1198892120593119890

1198891199052) (119877 + 119910

1)

minus 2119898(119889120593119890

119889119905)(

1198891199101

119889119905) = minus119873(

1199091

119897)

119898(11988921199101

1198891199052) minus 119898(

119889120593119890

119889119905)

2

(119877 + 1199101) + 119898(

1198892120593119890

1198891199052)1199091

+ 2119898(119889120593119890

119889119905)(

1198891199091

119889119905) = minus119873(

1199101

119897)

The Scientific World Journal 5

119898(11988921199111

1198891199052) = minus119898119892 + 119873(

119897 minus 1199111

119897)

1198972= 1199092

1+ 1199102

1+ (1199111minus 119897)2

(12)

After two times differentiation of forth equations insystem (12) we will have the following formula for the cablereaction force119873 (Figure 2) from third equation of (12)

119873 = ((119898119897 [((1199091(11988921199091

1198891199052) + (

1198891199091

119889119905)

2

+ 1199101(11988921199101

1198891199052) + (

1198891199101

119889119905)

2

)

times (1198972minus 1199092

1minus 1199102

1)

+ (1199091(1198891199091

119889119905) + 1199101(1198891199101

119889119905))

2

)])

times (1198972minus 1199092

1minus 1199102

1)minus2

)

+119898119892119897

radic(1198972 minus 1199092

1minus 1199102

1)

(13)

So with an introduction of (13) to the first and secondequations of system (12) we will have the explicit nonlinearODE system for payload119872 swaying inCartesian coordinates

5 Linearized Mathematical Model

Further for the small angle 1205721 we may assume that 119911

1and

119889211991111198891199052 are the small parameters So the nonlinear equation

(13) yields

119873 asymp 119898119892 (14)

Then taking into account (14) the linearized system (12)for the two variables 119909

1and 119910

1takes the following form

11988921199091

1198891199052minus (

119889120593119890

119889119905)

2

1199091minus 2(

119889120593119890

119889119905)(

1198891199101

119889119905) = minus119892(

1199091

119897)

11988921199101

1198891199052minus (

119889120593119890

119889119905)

2

(119877 + 1199101) + 2 (

119889120593119890

119889119905)(

1198891199091

119889119905) = minus119892(

1199101

119897)

(15)

So we derive the system (15) of differential equationsfor relative motion of load 119872 on cable 119861119872 with a movablesuspension center 119861 which is fixed at the crane boom1198611198742 We assume that the crane boom 119861119874

2rotates with a

constant angular velocity 120596119890around vertical axis 119863119874

2 We

then transfer the origin of coordinate system 119874119909119910 (Figure 1)to the point 119860dyn of dynamic equilibrium for load 119872 andassume that 119909

1= 119909 119910

1= 119910 + 119910dyn (Figure 3) Then

the second equation of system (15) defines the amount ofdynamic deflection 119910dyn = 119860 st119860dyn = (120596

2

119890119877119897)(119892 minus 120596

2

119890119897)

Relative swaying paths of payload 119872 in the plane ofthe introduced relative coordinates 119909 and 119910 are shown inFigure 3 for successive increasing swaying times 119905

1= 125 s

(Figure 3(a)) 1199052= 3 s (Figure 3(b)) 119905

3= 10 s (Figure 3(c))

and 1199054= 25 s (Figure 3(d))

Then we have the normal system of two linear homo-geneous differential equations of second order for relativemotion of the load119872

(

1198632minus (1205962

119890minus (

119892

119897)) minus2120596

119890119863

2120596119890119863 119863

2minus (1205962

119890minus (

119892

119897))

)

times (119909 (119905)

119910 (119905)) = 0 119863 =

119889

119889119905

(16)

The determinant of natural frequencies matrix for thesystem (16) has the following form

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205822+ (

119892

119897) minus 1205962

119890minus2 sdot 120596

119890sdot 120582

2 sdot 120596119890sdot 120582 120582

2+ (

119892

119897) minus 1205962

119890

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (17)

Using (17) we adjust the roots 1205821and 120582

2of secular

equation

1205821= plusmn]1sdot 119894 120582

2= plusmn]2sdot 119894 ]

1= 119896 + 120596

119890

]2= 119896 minus 120596

119890 119896 = radic

119892

119897 120596119890

= 119896

(18)

The initial conditions for the problem are as follows

119909 (0) = 0 119863119909 (0) = +119881119861= +120596119890119877

119910 (0) = minus119910dyn 119863119910 (0) = 0

(19)

Taking into account (18) and (19) the law of relativemotion for payload takes the following form

119909 (119905) = 1198601

119894

2(119890minus]1119894119905minus 119890

]1119894119905) + 119860

2

119894

2(119890minus]2119894119905minus 119890

]2119894119905)

119910 (119905) = 1198601

1

2(119890minus]1119894119905+ 119890

]1119894119905) minus 119860

2

1

2(119890minus]2119894119905+ 119890

]2119894119905)

(20)

After substitution of values of1198601and119860

2in (20) we have

the following law of payload119872 relative swaying motion withthe uniformly rotating pivot center

119909 (119905) = (

119881119861minus 119910dyn (119896 minus 120596

119890)

2119896)

119894

2(119890minus]1119894119905minus 119890

]1119894119905)

+ (

119881119861+ 119910dyn (119896 + 120596

119890)

2119896)

119894

2(119890minus]2119894119905minus 119890

]2119894119905)

119910 (119905) = (

119881119861minus 119910dyn (119896 minus 120596

119890)

2119896)1

2(119890minus]1119894119905+ 119890

]1119894119905)

minus (

119881119861+ 119910dyn (119896 + 120596

119890)

2119896)1

2(119890minus]2119894119905+ 119890

]2119894119905)

(21)

6 The Scientific World Journal

102

101

1

099

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)

(a)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)

1012

1010

1008

1006

1004

1002

1

0998

0996

0994

(b)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)

1004

1002

1003

1001

1

0999

l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)

(c)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)

10012

10010

10008

10006

10004

10002

1

09996

09994

09998

l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448 (rads)

(d)

Figure 2 Time dependencies of dimensionless cable reactions 119873119898119892 in the first approximation for fixed cable lengths 119897 = 0206m (a)119897 = 0412m (b) 119897 = 0618m (c) and 119897 = 0825m (d)

The relative trajectories for load119872on the cable in Figure 3have been derived for the following numerical values 119877 =

0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)05

asymp

3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120572

1dyn =

000221 rad 119881119861

= 0103ms 119910dyn = 000182m ]1

=

119896 + 120596119890

= 3658 rads ]2

= 119896 minus 120596119890

= 3239 rads(Figure 3)

Numerical plots in Figure 3 show the payload119872 swayingin the noninertial reference frame (119874119909119910119911) In order tovisualize payload119872 absolute swaying path (Figures 4 and 5)

The Scientific World Journal 7

t1 = 125 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(a)

t2 = 3s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(b)

t3 = 10 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(c)

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

t4 = 25s

Relative trajectory of load M swayingfor linearized model

(d)

Figure 3The relative trajectories for load119872 on the cable during uniform rotational motion of the crane boom 1198611198742for the moments of time

1199051= 125 s (a) 119905

2= 3 s (b) 119905

3= 10 s (c) and 119905

4= 25 s (d)

in the fixed inertial reference frame (1198742119909211991021199112) we will

address the following transition formula

(1199092(119905)

1199102(119905)

) = (sin (120596

119890119905) cos (120596

119890119905)

minus cos (120596119890119905) sin (120596

119890119905)) (

119909 (119905)

119910 (119905) + 119877 + 119910dyn)

(22)

Derived computational plot for absolute payload swayingpath in the absolute coordinatesrsquo plane (119909

21199102) is shown in

Figure 5(a) for load119872 swaying time 1199051= 15 s and crane boom

transport slewing angle120593119890= 180

∘ In Figure 5(b) we show theexperimental plot for absolute payload swaying path in theabsolute coordinatesrsquo plane (119909

21199102) for load 119872 swaying time

1199052= 15 s and crane boom transport slewing angle 120593

119890= 180

∘which was built with an introduction of experimental setupin Figure 4

6 Comparison of Derived andKnown Published Results

The computational dimensionless cable force119873119898119892 derivedin the first approximation and shown in Figure 2 outlinesthe minor change of the cable tension force in the vicinityof the ordinate (119873119898119892) = 1 (119873119898119892) isin [09875 10233]

for 119897 = 0206m in Figure 2(a) (119873119898119892) isin [0994 10122]

for 119897 = 0412m in Figure 2(b) (119873119898119892) isin [0998 1004] for119897 = 0618m in Figure 2(c) and (119873119898119892) isin [099933 10013]

for 119897 = 0825m in Figure 2(d) So the assumption (14)really takes place as shown in Figure 2 The same effect for(119873119898119892)Mitrev isin [094 104] has been derived by Mitrev andGrigorov [23 Mitrevrsquos Figure 7 in page 89] that confirmsassumption (14) An analogous effect for dimensionless cable

8 The Scientific World Journal

Camera

Bearing

Boom

Laser pointer

Slewing axis

CableLower plate

M

Light-emitting diodeRegular grid 1200mm times 700mmwith square cells 200mm times 20mm

B

D

O2

Figure 4 The experimental measurement system for payload 119872

swaying during crane boom slewing

force (119873119898119892)Maczynski isin [0996 1005] has been derived byMaczynski and Wojciech [24 Maczynskirsquos Figure 10 in page278] which also reaffirms the assumption (14)

In published work by Sakawa et al ([25] Figure 3 at page552 Figure 4 at page 552 Figure 5 at page 554 and Figure 6at page 554) the small Sakawa values 119889119911119889119905 le 01 (Figures 3and 4) and 119889119911119889119905 le 005 (Figures 5 and 6) of 119911-projectionsfor payload velocities have been shown which confirms thesmall angle assumption119873 asymp 119898119892 (14)

7 Discussion and Analysis of Basic Results forMathematical and Physical Simulation

The relative trajectories for load119872 swaying in Figure 3 showequality of maximum values for relative 119909 and 119910 payloadcoordinates

119909max = 119910max = 1198601+ 1198602=

120596119890(119877 + 119910dyn)

119896 (23)

The formula (23) outlines that values 119909max and 119910max inFigure 3 increase with growth of boom slewing velocity 120596

119890

and with increased cable length 119897 The 119910dyn influence on thevalues of 119909max and 119910max is negligible

Equation (18) shows that the free oscillation frequencies]1and ]

2depend on the boom slewing velocity 120596

119890and

natural frequency 119896 of the payload with a fixed pivot point119861 The frequencies ]

1and ]

2essentially depend on the

pendulum length 119897 because 119896 ≫ 120596119890 Kinematically saying

the miscoordination of frequencies ]1and ]

2is ]1minus ]2=

2120596119890and is determined by the crane boom slewing velocity

120596119890 Dynamically saying the value of miscoordination is

governed by the Coriolis acceleration-induced DrsquoAlembertrsquosinertia force

Computations shown in Figure 3 for relative payload119872 swaying show that the clockwise direction of rotationfor relative oscillations is opposite in direction from thecounterclockwise direction of rotation for the crane boomThis effect follows from the conservation of position of theplane 119910

21199112of oscillations of load 119872 in the fixed inertial

coordinate system 119909211991021199112

The comparative analysis of computational (1) and exper-imental (2) absolute payload trajectories is shown in Fig-ure 5(c)The quantity of the local maximums in Figure 5(a) is8 for theoretical curve 1 So the theoretical period for relativeload swaying is 119879theor = 187 s The quantity of the localmaximums in Figure 5(b) is 75 for experimental curve 2Theperiod for experimental curve 2 in Figure 5(b) is 119879experim =

2 s The discrepancy between the values of theoretical andexperimental periods does not exceed 7

The experimental absolute trajectory (2) in Figures 5(b)and 5(c) outlines the additional relative swaying of payload119872 after crane boom stop which has the elliptical form inthe vicinity of the point (0 minus05) in the plane 119909

21199102for 120593119890=

180∘ The remaining payload motions after crane boom stop

are the small oscillations of the payload with the fixed pivotcenter

8 Final Conclusions

The closed system of nonlinear ODEs for payload relativeswaying has been derived in relative Cartesian coordinateswith an introduction of Lagrange equations and a geometricconstraint equation

The explicit form of the dependence of the cable reactionforce as a function of the relative payload coordinates 119909

1 1199101

1199111and their first and second time derivatives has been found

Derived complex dependency of the cable reaction force onthe abovementioned variables determines the nonlinearity ofthe present problem

The deviation of cable reaction force frompayload gravityforce is small This fact allowed the derivation of a linearizedODE system for payload swaying in relative Cartesian coor-dinates and verified the small relative Cartesian coordinatesrsquoassumption The analytical expressions for relative payloadpath have been derived

Derived results have good agreement with experimentsand known published data

The Scientific World Journal 9

05

04

03

02

01

0

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

t = 15 s

1

Point O2

2R = 1 (m)

1

PointnP i OO2OO

y2 (m)

x2 (m)(Point D)

(Point M)

(a)

05

04

03

02

01

02R = 1 (m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

2

4

3

Rotational directionPoint O2

t = 18 s

(Point D)

(Point M)

(Point B)

x2 (m)

y2 (m)

(b)

1 2

4

Point O2

2R = 1 (m)

y2(m)

x2(m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

(Point D)

(Point M) (Point M)

05

04

03

02

01

0

(c)

Figure 5 Calculated 1 (a c) and experimental 2 (b c) absolute trajectories for load119872 where 3 is the trajectory of boom point 119861 (b) 4 is thecenter 119874

2(D) of boom rotation (b c)

Disclosure

The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors

Conflict of Interests

The authors Alexander V Perig Alexander N Stadnik andAlexander I Deriglazov declare that there is no conflict ofinterests regarding the publication of this paper

References

[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002

[2] W Adamiec-Wojcik I P Fałat A Maczynski and S WojciechldquoLoad stabilisation in an A-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009

[3] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquoMethods in AppliedMechanics and Engineering vol170 no 3-4 pp 343ndash3353 1999

[4] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009

[5] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004

[6] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007

[7] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008

[8] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing aload prescribed motionrdquo The Archive of Mechanical Engineer-ing vol 3 pp 209ndash222 2009 httpamemeilpweduplindexphpengcontentdownload3731748fileart 2009 3 02pdf

[9] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010

[10] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011

10 The Scientific World Journal

[11] J H Cha S H Ham K Y Lee and M I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution ofMechanical Engineers K vol 224no 4 pp 365ndash373 2010

[12] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002

[13] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007

[14] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K vol 217no 2 pp 99ndash109 2003

[15] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006

[16] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers K vol 224 no 4 pp 341ndash352 2010

[17] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity Science A vol 8 no 9 pp 1395ndash1400 2007

[18] B Grigorov and R Mitrev ldquoA NewtonmdashEuler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009

[19] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998

[20] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008

[21] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008

[22] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers I vol 223 no 4 pp 575ndash580 2009

[23] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008

[24] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003

[25] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981

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4 The Scientific World Journal

+ (119889120593119890

119889119905)

2

(1199092

1+ (1199101+ 119877)2

)

+ 2(119889120593119890

119889119905)(1199091(1198891199101

119889119905) minus (119910

1+ 119877) (

1198891199091

119889119905))

= ((1198891199091

119889119905) minus (

119889120593119890

119889119905) (119877 + 119910

1))

2

+ ((1198891199101

119889119905) + (

119889120593119890

119889119905) 1199091)

2

+ (1198891199111

119889119905)

2

(3)

3 Dynamic Analysis

The slewing motion of the mechanical system ldquocrane boom1198611198742-load 119872rdquo in Figure 1 is governed by the vector equation

for the rate of change of moment of momentum H11987423

for thesystem ldquocrane boom 119861119874

2-load119872rdquo with respect to point119874

2in

the inertial reference frame 119909211991021199112

(119909211991021199112)119889H11987423

119889119905= sumM1198742 (4)

The vector equation (4) contains the following compo-nents

H11987423

= 1198681198742

33120596119890 (5)

1198681198742

33= (1198681198742

33)1198611198742

+ 119898(1199092

1+ (119877 + 119910

1)2

) (6)

sumM1198742 = M1198742119863119879

minusM1198742119865119879

= (1198721198742

119863119879minus1198721198742

119865119879) k (7)

where (119868119874233)1198611198742

is the element of mass moment of inertia forthe crane boom119861119874

2in inertial fixed on earth reference frame

119909211991021199112with respect to unit vector k and119898(119909

2

1+ (119877 + 119910

1)2) is

the element of mass moment of inertia for the payload119872 ininertial fixed on earth reference frame 119909

211991021199112with respect to

unit vector kThe external moment of gravitational forceM1198742(mg) = 0

in (4) and (7) becausemg uarrdarr kFor the system ldquocrane boom-payloadrdquo the cable reaction

force N is the internal force So in (4) and (7) we haveM1198742(N) = 0

Substitution of (5) (6) and (7) into (4) yields thefollowing scalar equation for the rate of change of momentof momentum 119867

1198742

3for the system ldquocrane boom 119861119874

2-load

119872rdquo with respect to point 1198742in the inertial reference frame

119909211991021199112

119889

119889119905(((1198681198742

33)1198611198742

+ 119898(1199092

1+ (119877 + 119910

1)2

)) (119889120593119890

119889119905))

= 1198721198742

119863119879minus1198721198742

119865119879

(8)

where driving1198721198742119863119879

and frictional1198721198742119865119879

torques are the tech-nically defined functions for specific electric drive systems

4 Nonlinear Mathematical Model withLagrange Equationsrsquo Introduction

Taking into account (3) for the square of absolute payload119872velocity 1198812abs and by adding the kinetic energy for a slewingcrane boom according to (6) we will have the followingexpression for ldquocrane boom-payloadrdquo kinetic energy

119879 =119898

2(((

1198891199091

119889119905) minus (

119889120593119890

119889119905) (119877 + 119910

1))

2

+((1198891199101

119889119905) + (

119889120593119890

119889119905) 1199091)

2

+ (1198891199111

119889119905)

2

)

+1198681198742

33

2(119889120593119890

119889119905)

2

(9)

For further analysis we will assume that the crane boomslewing angular velocity is the constant 119889120593

119890119889119905 = 120596

119890 We will

use the derived expression for kinetic energy for the left-handsides of Lagrange equations in the noninertial reference frame119909111991011199111

119889

119889119905(120597119879

1205971

) minus120597119879

1205971199091

= 1198761199091

119889

119889119905(120597119879

120597 1199101

) minus120597119879

1205971199101

= 1198761199101

119889

119889119905(120597119879

1205971

) minus120597119879

1205971199111

= 1198761199111

(10)

Taking into account the nonlinearity and nonconser-vatism of the cable reaction force N we have the followingformulae for the generalized forces in the noninertial refer-ence frame 119909

111991011199111

1198761199091

= minus 119873(1199091

119897)

1198761199101

= minus 119873(1199101

119897)

1198761199111

= + 119873(119897 minus 1199111

119897) minus 119898119892

(11)

Taking into account (9) (11) the equations (10) in thenoninertial reference frame 119909

111991011199111will finally take the

following form

119898(11988921199091

1198891199052) minus 119898(

119889120593119890

119889119905)

2

1199091minus 119898(

1198892120593119890

1198891199052) (119877 + 119910

1)

minus 2119898(119889120593119890

119889119905)(

1198891199101

119889119905) = minus119873(

1199091

119897)

119898(11988921199101

1198891199052) minus 119898(

119889120593119890

119889119905)

2

(119877 + 1199101) + 119898(

1198892120593119890

1198891199052)1199091

+ 2119898(119889120593119890

119889119905)(

1198891199091

119889119905) = minus119873(

1199101

119897)

The Scientific World Journal 5

119898(11988921199111

1198891199052) = minus119898119892 + 119873(

119897 minus 1199111

119897)

1198972= 1199092

1+ 1199102

1+ (1199111minus 119897)2

(12)

After two times differentiation of forth equations insystem (12) we will have the following formula for the cablereaction force119873 (Figure 2) from third equation of (12)

119873 = ((119898119897 [((1199091(11988921199091

1198891199052) + (

1198891199091

119889119905)

2

+ 1199101(11988921199101

1198891199052) + (

1198891199101

119889119905)

2

)

times (1198972minus 1199092

1minus 1199102

1)

+ (1199091(1198891199091

119889119905) + 1199101(1198891199101

119889119905))

2

)])

times (1198972minus 1199092

1minus 1199102

1)minus2

)

+119898119892119897

radic(1198972 minus 1199092

1minus 1199102

1)

(13)

So with an introduction of (13) to the first and secondequations of system (12) we will have the explicit nonlinearODE system for payload119872 swaying inCartesian coordinates

5 Linearized Mathematical Model

Further for the small angle 1205721 we may assume that 119911

1and

119889211991111198891199052 are the small parameters So the nonlinear equation

(13) yields

119873 asymp 119898119892 (14)

Then taking into account (14) the linearized system (12)for the two variables 119909

1and 119910

1takes the following form

11988921199091

1198891199052minus (

119889120593119890

119889119905)

2

1199091minus 2(

119889120593119890

119889119905)(

1198891199101

119889119905) = minus119892(

1199091

119897)

11988921199101

1198891199052minus (

119889120593119890

119889119905)

2

(119877 + 1199101) + 2 (

119889120593119890

119889119905)(

1198891199091

119889119905) = minus119892(

1199101

119897)

(15)

So we derive the system (15) of differential equationsfor relative motion of load 119872 on cable 119861119872 with a movablesuspension center 119861 which is fixed at the crane boom1198611198742 We assume that the crane boom 119861119874

2rotates with a

constant angular velocity 120596119890around vertical axis 119863119874

2 We

then transfer the origin of coordinate system 119874119909119910 (Figure 1)to the point 119860dyn of dynamic equilibrium for load 119872 andassume that 119909

1= 119909 119910

1= 119910 + 119910dyn (Figure 3) Then

the second equation of system (15) defines the amount ofdynamic deflection 119910dyn = 119860 st119860dyn = (120596

2

119890119877119897)(119892 minus 120596

2

119890119897)

Relative swaying paths of payload 119872 in the plane ofthe introduced relative coordinates 119909 and 119910 are shown inFigure 3 for successive increasing swaying times 119905

1= 125 s

(Figure 3(a)) 1199052= 3 s (Figure 3(b)) 119905

3= 10 s (Figure 3(c))

and 1199054= 25 s (Figure 3(d))

Then we have the normal system of two linear homo-geneous differential equations of second order for relativemotion of the load119872

(

1198632minus (1205962

119890minus (

119892

119897)) minus2120596

119890119863

2120596119890119863 119863

2minus (1205962

119890minus (

119892

119897))

)

times (119909 (119905)

119910 (119905)) = 0 119863 =

119889

119889119905

(16)

The determinant of natural frequencies matrix for thesystem (16) has the following form

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205822+ (

119892

119897) minus 1205962

119890minus2 sdot 120596

119890sdot 120582

2 sdot 120596119890sdot 120582 120582

2+ (

119892

119897) minus 1205962

119890

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (17)

Using (17) we adjust the roots 1205821and 120582

2of secular

equation

1205821= plusmn]1sdot 119894 120582

2= plusmn]2sdot 119894 ]

1= 119896 + 120596

119890

]2= 119896 minus 120596

119890 119896 = radic

119892

119897 120596119890

= 119896

(18)

The initial conditions for the problem are as follows

119909 (0) = 0 119863119909 (0) = +119881119861= +120596119890119877

119910 (0) = minus119910dyn 119863119910 (0) = 0

(19)

Taking into account (18) and (19) the law of relativemotion for payload takes the following form

119909 (119905) = 1198601

119894

2(119890minus]1119894119905minus 119890

]1119894119905) + 119860

2

119894

2(119890minus]2119894119905minus 119890

]2119894119905)

119910 (119905) = 1198601

1

2(119890minus]1119894119905+ 119890

]1119894119905) minus 119860

2

1

2(119890minus]2119894119905+ 119890

]2119894119905)

(20)

After substitution of values of1198601and119860

2in (20) we have

the following law of payload119872 relative swaying motion withthe uniformly rotating pivot center

119909 (119905) = (

119881119861minus 119910dyn (119896 minus 120596

119890)

2119896)

119894

2(119890minus]1119894119905minus 119890

]1119894119905)

+ (

119881119861+ 119910dyn (119896 + 120596

119890)

2119896)

119894

2(119890minus]2119894119905minus 119890

]2119894119905)

119910 (119905) = (

119881119861minus 119910dyn (119896 minus 120596

119890)

2119896)1

2(119890minus]1119894119905+ 119890

]1119894119905)

minus (

119881119861+ 119910dyn (119896 + 120596

119890)

2119896)1

2(119890minus]2119894119905+ 119890

]2119894119905)

(21)

6 The Scientific World Journal

102

101

1

099

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)

(a)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)

1012

1010

1008

1006

1004

1002

1

0998

0996

0994

(b)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)

1004

1002

1003

1001

1

0999

l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)

(c)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)

10012

10010

10008

10006

10004

10002

1

09996

09994

09998

l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448 (rads)

(d)

Figure 2 Time dependencies of dimensionless cable reactions 119873119898119892 in the first approximation for fixed cable lengths 119897 = 0206m (a)119897 = 0412m (b) 119897 = 0618m (c) and 119897 = 0825m (d)

The relative trajectories for load119872on the cable in Figure 3have been derived for the following numerical values 119877 =

0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)05

asymp

3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120572

1dyn =

000221 rad 119881119861

= 0103ms 119910dyn = 000182m ]1

=

119896 + 120596119890

= 3658 rads ]2

= 119896 minus 120596119890

= 3239 rads(Figure 3)

Numerical plots in Figure 3 show the payload119872 swayingin the noninertial reference frame (119874119909119910119911) In order tovisualize payload119872 absolute swaying path (Figures 4 and 5)

The Scientific World Journal 7

t1 = 125 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(a)

t2 = 3s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(b)

t3 = 10 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(c)

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

t4 = 25s

Relative trajectory of load M swayingfor linearized model

(d)

Figure 3The relative trajectories for load119872 on the cable during uniform rotational motion of the crane boom 1198611198742for the moments of time

1199051= 125 s (a) 119905

2= 3 s (b) 119905

3= 10 s (c) and 119905

4= 25 s (d)

in the fixed inertial reference frame (1198742119909211991021199112) we will

address the following transition formula

(1199092(119905)

1199102(119905)

) = (sin (120596

119890119905) cos (120596

119890119905)

minus cos (120596119890119905) sin (120596

119890119905)) (

119909 (119905)

119910 (119905) + 119877 + 119910dyn)

(22)

Derived computational plot for absolute payload swayingpath in the absolute coordinatesrsquo plane (119909

21199102) is shown in

Figure 5(a) for load119872 swaying time 1199051= 15 s and crane boom

transport slewing angle120593119890= 180

∘ In Figure 5(b) we show theexperimental plot for absolute payload swaying path in theabsolute coordinatesrsquo plane (119909

21199102) for load 119872 swaying time

1199052= 15 s and crane boom transport slewing angle 120593

119890= 180

∘which was built with an introduction of experimental setupin Figure 4

6 Comparison of Derived andKnown Published Results

The computational dimensionless cable force119873119898119892 derivedin the first approximation and shown in Figure 2 outlinesthe minor change of the cable tension force in the vicinityof the ordinate (119873119898119892) = 1 (119873119898119892) isin [09875 10233]

for 119897 = 0206m in Figure 2(a) (119873119898119892) isin [0994 10122]

for 119897 = 0412m in Figure 2(b) (119873119898119892) isin [0998 1004] for119897 = 0618m in Figure 2(c) and (119873119898119892) isin [099933 10013]

for 119897 = 0825m in Figure 2(d) So the assumption (14)really takes place as shown in Figure 2 The same effect for(119873119898119892)Mitrev isin [094 104] has been derived by Mitrev andGrigorov [23 Mitrevrsquos Figure 7 in page 89] that confirmsassumption (14) An analogous effect for dimensionless cable

8 The Scientific World Journal

Camera

Bearing

Boom

Laser pointer

Slewing axis

CableLower plate

M

Light-emitting diodeRegular grid 1200mm times 700mmwith square cells 200mm times 20mm

B

D

O2

Figure 4 The experimental measurement system for payload 119872

swaying during crane boom slewing

force (119873119898119892)Maczynski isin [0996 1005] has been derived byMaczynski and Wojciech [24 Maczynskirsquos Figure 10 in page278] which also reaffirms the assumption (14)

In published work by Sakawa et al ([25] Figure 3 at page552 Figure 4 at page 552 Figure 5 at page 554 and Figure 6at page 554) the small Sakawa values 119889119911119889119905 le 01 (Figures 3and 4) and 119889119911119889119905 le 005 (Figures 5 and 6) of 119911-projectionsfor payload velocities have been shown which confirms thesmall angle assumption119873 asymp 119898119892 (14)

7 Discussion and Analysis of Basic Results forMathematical and Physical Simulation

The relative trajectories for load119872 swaying in Figure 3 showequality of maximum values for relative 119909 and 119910 payloadcoordinates

119909max = 119910max = 1198601+ 1198602=

120596119890(119877 + 119910dyn)

119896 (23)

The formula (23) outlines that values 119909max and 119910max inFigure 3 increase with growth of boom slewing velocity 120596

119890

and with increased cable length 119897 The 119910dyn influence on thevalues of 119909max and 119910max is negligible

Equation (18) shows that the free oscillation frequencies]1and ]

2depend on the boom slewing velocity 120596

119890and

natural frequency 119896 of the payload with a fixed pivot point119861 The frequencies ]

1and ]

2essentially depend on the

pendulum length 119897 because 119896 ≫ 120596119890 Kinematically saying

the miscoordination of frequencies ]1and ]

2is ]1minus ]2=

2120596119890and is determined by the crane boom slewing velocity

120596119890 Dynamically saying the value of miscoordination is

governed by the Coriolis acceleration-induced DrsquoAlembertrsquosinertia force

Computations shown in Figure 3 for relative payload119872 swaying show that the clockwise direction of rotationfor relative oscillations is opposite in direction from thecounterclockwise direction of rotation for the crane boomThis effect follows from the conservation of position of theplane 119910

21199112of oscillations of load 119872 in the fixed inertial

coordinate system 119909211991021199112

The comparative analysis of computational (1) and exper-imental (2) absolute payload trajectories is shown in Fig-ure 5(c)The quantity of the local maximums in Figure 5(a) is8 for theoretical curve 1 So the theoretical period for relativeload swaying is 119879theor = 187 s The quantity of the localmaximums in Figure 5(b) is 75 for experimental curve 2Theperiod for experimental curve 2 in Figure 5(b) is 119879experim =

2 s The discrepancy between the values of theoretical andexperimental periods does not exceed 7

The experimental absolute trajectory (2) in Figures 5(b)and 5(c) outlines the additional relative swaying of payload119872 after crane boom stop which has the elliptical form inthe vicinity of the point (0 minus05) in the plane 119909

21199102for 120593119890=

180∘ The remaining payload motions after crane boom stop

are the small oscillations of the payload with the fixed pivotcenter

8 Final Conclusions

The closed system of nonlinear ODEs for payload relativeswaying has been derived in relative Cartesian coordinateswith an introduction of Lagrange equations and a geometricconstraint equation

The explicit form of the dependence of the cable reactionforce as a function of the relative payload coordinates 119909

1 1199101

1199111and their first and second time derivatives has been found

Derived complex dependency of the cable reaction force onthe abovementioned variables determines the nonlinearity ofthe present problem

The deviation of cable reaction force frompayload gravityforce is small This fact allowed the derivation of a linearizedODE system for payload swaying in relative Cartesian coor-dinates and verified the small relative Cartesian coordinatesrsquoassumption The analytical expressions for relative payloadpath have been derived

Derived results have good agreement with experimentsand known published data

The Scientific World Journal 9

05

04

03

02

01

0

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

t = 15 s

1

Point O2

2R = 1 (m)

1

PointnP i OO2OO

y2 (m)

x2 (m)(Point D)

(Point M)

(a)

05

04

03

02

01

02R = 1 (m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

2

4

3

Rotational directionPoint O2

t = 18 s

(Point D)

(Point M)

(Point B)

x2 (m)

y2 (m)

(b)

1 2

4

Point O2

2R = 1 (m)

y2(m)

x2(m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

(Point D)

(Point M) (Point M)

05

04

03

02

01

0

(c)

Figure 5 Calculated 1 (a c) and experimental 2 (b c) absolute trajectories for load119872 where 3 is the trajectory of boom point 119861 (b) 4 is thecenter 119874

2(D) of boom rotation (b c)

Disclosure

The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors

Conflict of Interests

The authors Alexander V Perig Alexander N Stadnik andAlexander I Deriglazov declare that there is no conflict ofinterests regarding the publication of this paper

References

[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002

[2] W Adamiec-Wojcik I P Fałat A Maczynski and S WojciechldquoLoad stabilisation in an A-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009

[3] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquoMethods in AppliedMechanics and Engineering vol170 no 3-4 pp 343ndash3353 1999

[4] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009

[5] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004

[6] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007

[7] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008

[8] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing aload prescribed motionrdquo The Archive of Mechanical Engineer-ing vol 3 pp 209ndash222 2009 httpamemeilpweduplindexphpengcontentdownload3731748fileart 2009 3 02pdf

[9] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010

[10] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011

10 The Scientific World Journal

[11] J H Cha S H Ham K Y Lee and M I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution ofMechanical Engineers K vol 224no 4 pp 365ndash373 2010

[12] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002

[13] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007

[14] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K vol 217no 2 pp 99ndash109 2003

[15] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006

[16] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers K vol 224 no 4 pp 341ndash352 2010

[17] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity Science A vol 8 no 9 pp 1395ndash1400 2007

[18] B Grigorov and R Mitrev ldquoA NewtonmdashEuler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009

[19] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998

[20] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008

[21] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008

[22] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers I vol 223 no 4 pp 575ndash580 2009

[23] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008

[24] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003

[25] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 5

119898(11988921199111

1198891199052) = minus119898119892 + 119873(

119897 minus 1199111

119897)

1198972= 1199092

1+ 1199102

1+ (1199111minus 119897)2

(12)

After two times differentiation of forth equations insystem (12) we will have the following formula for the cablereaction force119873 (Figure 2) from third equation of (12)

119873 = ((119898119897 [((1199091(11988921199091

1198891199052) + (

1198891199091

119889119905)

2

+ 1199101(11988921199101

1198891199052) + (

1198891199101

119889119905)

2

)

times (1198972minus 1199092

1minus 1199102

1)

+ (1199091(1198891199091

119889119905) + 1199101(1198891199101

119889119905))

2

)])

times (1198972minus 1199092

1minus 1199102

1)minus2

)

+119898119892119897

radic(1198972 minus 1199092

1minus 1199102

1)

(13)

So with an introduction of (13) to the first and secondequations of system (12) we will have the explicit nonlinearODE system for payload119872 swaying inCartesian coordinates

5 Linearized Mathematical Model

Further for the small angle 1205721 we may assume that 119911

1and

119889211991111198891199052 are the small parameters So the nonlinear equation

(13) yields

119873 asymp 119898119892 (14)

Then taking into account (14) the linearized system (12)for the two variables 119909

1and 119910

1takes the following form

11988921199091

1198891199052minus (

119889120593119890

119889119905)

2

1199091minus 2(

119889120593119890

119889119905)(

1198891199101

119889119905) = minus119892(

1199091

119897)

11988921199101

1198891199052minus (

119889120593119890

119889119905)

2

(119877 + 1199101) + 2 (

119889120593119890

119889119905)(

1198891199091

119889119905) = minus119892(

1199101

119897)

(15)

So we derive the system (15) of differential equationsfor relative motion of load 119872 on cable 119861119872 with a movablesuspension center 119861 which is fixed at the crane boom1198611198742 We assume that the crane boom 119861119874

2rotates with a

constant angular velocity 120596119890around vertical axis 119863119874

2 We

then transfer the origin of coordinate system 119874119909119910 (Figure 1)to the point 119860dyn of dynamic equilibrium for load 119872 andassume that 119909

1= 119909 119910

1= 119910 + 119910dyn (Figure 3) Then

the second equation of system (15) defines the amount ofdynamic deflection 119910dyn = 119860 st119860dyn = (120596

2

119890119877119897)(119892 minus 120596

2

119890119897)

Relative swaying paths of payload 119872 in the plane ofthe introduced relative coordinates 119909 and 119910 are shown inFigure 3 for successive increasing swaying times 119905

1= 125 s

(Figure 3(a)) 1199052= 3 s (Figure 3(b)) 119905

3= 10 s (Figure 3(c))

and 1199054= 25 s (Figure 3(d))

Then we have the normal system of two linear homo-geneous differential equations of second order for relativemotion of the load119872

(

1198632minus (1205962

119890minus (

119892

119897)) minus2120596

119890119863

2120596119890119863 119863

2minus (1205962

119890minus (

119892

119897))

)

times (119909 (119905)

119910 (119905)) = 0 119863 =

119889

119889119905

(16)

The determinant of natural frequencies matrix for thesystem (16) has the following form

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205822+ (

119892

119897) minus 1205962

119890minus2 sdot 120596

119890sdot 120582

2 sdot 120596119890sdot 120582 120582

2+ (

119892

119897) minus 1205962

119890

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (17)

Using (17) we adjust the roots 1205821and 120582

2of secular

equation

1205821= plusmn]1sdot 119894 120582

2= plusmn]2sdot 119894 ]

1= 119896 + 120596

119890

]2= 119896 minus 120596

119890 119896 = radic

119892

119897 120596119890

= 119896

(18)

The initial conditions for the problem are as follows

119909 (0) = 0 119863119909 (0) = +119881119861= +120596119890119877

119910 (0) = minus119910dyn 119863119910 (0) = 0

(19)

Taking into account (18) and (19) the law of relativemotion for payload takes the following form

119909 (119905) = 1198601

119894

2(119890minus]1119894119905minus 119890

]1119894119905) + 119860

2

119894

2(119890minus]2119894119905minus 119890

]2119894119905)

119910 (119905) = 1198601

1

2(119890minus]1119894119905+ 119890

]1119894119905) minus 119860

2

1

2(119890minus]2119894119905+ 119890

]2119894119905)

(20)

After substitution of values of1198601and119860

2in (20) we have

the following law of payload119872 relative swaying motion withthe uniformly rotating pivot center

119909 (119905) = (

119881119861minus 119910dyn (119896 minus 120596

119890)

2119896)

119894

2(119890minus]1119894119905minus 119890

]1119894119905)

+ (

119881119861+ 119910dyn (119896 + 120596

119890)

2119896)

119894

2(119890minus]2119894119905minus 119890

]2119894119905)

119910 (119905) = (

119881119861minus 119910dyn (119896 minus 120596

119890)

2119896)1

2(119890minus]1119894119905+ 119890

]1119894119905)

minus (

119881119861+ 119910dyn (119896 + 120596

119890)

2119896)1

2(119890minus]2119894119905+ 119890

]2119894119905)

(21)

6 The Scientific World Journal

102

101

1

099

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)

(a)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)

1012

1010

1008

1006

1004

1002

1

0998

0996

0994

(b)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)

1004

1002

1003

1001

1

0999

l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)

(c)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)

10012

10010

10008

10006

10004

10002

1

09996

09994

09998

l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448 (rads)

(d)

Figure 2 Time dependencies of dimensionless cable reactions 119873119898119892 in the first approximation for fixed cable lengths 119897 = 0206m (a)119897 = 0412m (b) 119897 = 0618m (c) and 119897 = 0825m (d)

The relative trajectories for load119872on the cable in Figure 3have been derived for the following numerical values 119877 =

0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)05

asymp

3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120572

1dyn =

000221 rad 119881119861

= 0103ms 119910dyn = 000182m ]1

=

119896 + 120596119890

= 3658 rads ]2

= 119896 minus 120596119890

= 3239 rads(Figure 3)

Numerical plots in Figure 3 show the payload119872 swayingin the noninertial reference frame (119874119909119910119911) In order tovisualize payload119872 absolute swaying path (Figures 4 and 5)

The Scientific World Journal 7

t1 = 125 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(a)

t2 = 3s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(b)

t3 = 10 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(c)

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

t4 = 25s

Relative trajectory of load M swayingfor linearized model

(d)

Figure 3The relative trajectories for load119872 on the cable during uniform rotational motion of the crane boom 1198611198742for the moments of time

1199051= 125 s (a) 119905

2= 3 s (b) 119905

3= 10 s (c) and 119905

4= 25 s (d)

in the fixed inertial reference frame (1198742119909211991021199112) we will

address the following transition formula

(1199092(119905)

1199102(119905)

) = (sin (120596

119890119905) cos (120596

119890119905)

minus cos (120596119890119905) sin (120596

119890119905)) (

119909 (119905)

119910 (119905) + 119877 + 119910dyn)

(22)

Derived computational plot for absolute payload swayingpath in the absolute coordinatesrsquo plane (119909

21199102) is shown in

Figure 5(a) for load119872 swaying time 1199051= 15 s and crane boom

transport slewing angle120593119890= 180

∘ In Figure 5(b) we show theexperimental plot for absolute payload swaying path in theabsolute coordinatesrsquo plane (119909

21199102) for load 119872 swaying time

1199052= 15 s and crane boom transport slewing angle 120593

119890= 180

∘which was built with an introduction of experimental setupin Figure 4

6 Comparison of Derived andKnown Published Results

The computational dimensionless cable force119873119898119892 derivedin the first approximation and shown in Figure 2 outlinesthe minor change of the cable tension force in the vicinityof the ordinate (119873119898119892) = 1 (119873119898119892) isin [09875 10233]

for 119897 = 0206m in Figure 2(a) (119873119898119892) isin [0994 10122]

for 119897 = 0412m in Figure 2(b) (119873119898119892) isin [0998 1004] for119897 = 0618m in Figure 2(c) and (119873119898119892) isin [099933 10013]

for 119897 = 0825m in Figure 2(d) So the assumption (14)really takes place as shown in Figure 2 The same effect for(119873119898119892)Mitrev isin [094 104] has been derived by Mitrev andGrigorov [23 Mitrevrsquos Figure 7 in page 89] that confirmsassumption (14) An analogous effect for dimensionless cable

8 The Scientific World Journal

Camera

Bearing

Boom

Laser pointer

Slewing axis

CableLower plate

M

Light-emitting diodeRegular grid 1200mm times 700mmwith square cells 200mm times 20mm

B

D

O2

Figure 4 The experimental measurement system for payload 119872

swaying during crane boom slewing

force (119873119898119892)Maczynski isin [0996 1005] has been derived byMaczynski and Wojciech [24 Maczynskirsquos Figure 10 in page278] which also reaffirms the assumption (14)

In published work by Sakawa et al ([25] Figure 3 at page552 Figure 4 at page 552 Figure 5 at page 554 and Figure 6at page 554) the small Sakawa values 119889119911119889119905 le 01 (Figures 3and 4) and 119889119911119889119905 le 005 (Figures 5 and 6) of 119911-projectionsfor payload velocities have been shown which confirms thesmall angle assumption119873 asymp 119898119892 (14)

7 Discussion and Analysis of Basic Results forMathematical and Physical Simulation

The relative trajectories for load119872 swaying in Figure 3 showequality of maximum values for relative 119909 and 119910 payloadcoordinates

119909max = 119910max = 1198601+ 1198602=

120596119890(119877 + 119910dyn)

119896 (23)

The formula (23) outlines that values 119909max and 119910max inFigure 3 increase with growth of boom slewing velocity 120596

119890

and with increased cable length 119897 The 119910dyn influence on thevalues of 119909max and 119910max is negligible

Equation (18) shows that the free oscillation frequencies]1and ]

2depend on the boom slewing velocity 120596

119890and

natural frequency 119896 of the payload with a fixed pivot point119861 The frequencies ]

1and ]

2essentially depend on the

pendulum length 119897 because 119896 ≫ 120596119890 Kinematically saying

the miscoordination of frequencies ]1and ]

2is ]1minus ]2=

2120596119890and is determined by the crane boom slewing velocity

120596119890 Dynamically saying the value of miscoordination is

governed by the Coriolis acceleration-induced DrsquoAlembertrsquosinertia force

Computations shown in Figure 3 for relative payload119872 swaying show that the clockwise direction of rotationfor relative oscillations is opposite in direction from thecounterclockwise direction of rotation for the crane boomThis effect follows from the conservation of position of theplane 119910

21199112of oscillations of load 119872 in the fixed inertial

coordinate system 119909211991021199112

The comparative analysis of computational (1) and exper-imental (2) absolute payload trajectories is shown in Fig-ure 5(c)The quantity of the local maximums in Figure 5(a) is8 for theoretical curve 1 So the theoretical period for relativeload swaying is 119879theor = 187 s The quantity of the localmaximums in Figure 5(b) is 75 for experimental curve 2Theperiod for experimental curve 2 in Figure 5(b) is 119879experim =

2 s The discrepancy between the values of theoretical andexperimental periods does not exceed 7

The experimental absolute trajectory (2) in Figures 5(b)and 5(c) outlines the additional relative swaying of payload119872 after crane boom stop which has the elliptical form inthe vicinity of the point (0 minus05) in the plane 119909

21199102for 120593119890=

180∘ The remaining payload motions after crane boom stop

are the small oscillations of the payload with the fixed pivotcenter

8 Final Conclusions

The closed system of nonlinear ODEs for payload relativeswaying has been derived in relative Cartesian coordinateswith an introduction of Lagrange equations and a geometricconstraint equation

The explicit form of the dependence of the cable reactionforce as a function of the relative payload coordinates 119909

1 1199101

1199111and their first and second time derivatives has been found

Derived complex dependency of the cable reaction force onthe abovementioned variables determines the nonlinearity ofthe present problem

The deviation of cable reaction force frompayload gravityforce is small This fact allowed the derivation of a linearizedODE system for payload swaying in relative Cartesian coor-dinates and verified the small relative Cartesian coordinatesrsquoassumption The analytical expressions for relative payloadpath have been derived

Derived results have good agreement with experimentsand known published data

The Scientific World Journal 9

05

04

03

02

01

0

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

t = 15 s

1

Point O2

2R = 1 (m)

1

PointnP i OO2OO

y2 (m)

x2 (m)(Point D)

(Point M)

(a)

05

04

03

02

01

02R = 1 (m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

2

4

3

Rotational directionPoint O2

t = 18 s

(Point D)

(Point M)

(Point B)

x2 (m)

y2 (m)

(b)

1 2

4

Point O2

2R = 1 (m)

y2(m)

x2(m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

(Point D)

(Point M) (Point M)

05

04

03

02

01

0

(c)

Figure 5 Calculated 1 (a c) and experimental 2 (b c) absolute trajectories for load119872 where 3 is the trajectory of boom point 119861 (b) 4 is thecenter 119874

2(D) of boom rotation (b c)

Disclosure

The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors

Conflict of Interests

The authors Alexander V Perig Alexander N Stadnik andAlexander I Deriglazov declare that there is no conflict ofinterests regarding the publication of this paper

References

[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002

[2] W Adamiec-Wojcik I P Fałat A Maczynski and S WojciechldquoLoad stabilisation in an A-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009

[3] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquoMethods in AppliedMechanics and Engineering vol170 no 3-4 pp 343ndash3353 1999

[4] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009

[5] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004

[6] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007

[7] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008

[8] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing aload prescribed motionrdquo The Archive of Mechanical Engineer-ing vol 3 pp 209ndash222 2009 httpamemeilpweduplindexphpengcontentdownload3731748fileart 2009 3 02pdf

[9] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010

[10] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011

10 The Scientific World Journal

[11] J H Cha S H Ham K Y Lee and M I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution ofMechanical Engineers K vol 224no 4 pp 365ndash373 2010

[12] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002

[13] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007

[14] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K vol 217no 2 pp 99ndash109 2003

[15] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006

[16] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers K vol 224 no 4 pp 341ndash352 2010

[17] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity Science A vol 8 no 9 pp 1395ndash1400 2007

[18] B Grigorov and R Mitrev ldquoA NewtonmdashEuler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009

[19] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998

[20] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008

[21] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008

[22] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers I vol 223 no 4 pp 575ndash580 2009

[23] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008

[24] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003

[25] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 The Scientific World Journal

102

101

1

099

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)l asymp 0206 (m) 120596e asymp 0449 (rads) k asymp 69 (rads)

(a)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)l asymp 0412 (m) 120596e asymp 0449 (rads) k asymp 488 (rads)

1012

1010

1008

1006

1004

1002

1

0998

0996

0994

(b)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)

1004

1002

1003

1001

1

0999

l asymp 0618 (m) 120596e asymp 0314 (rads) k asymp 3984 (rads)

(c)

0 2 4 6 8 10

Dim

ensio

nles

s cab

le re

actio

n (N

mg)

t (s)

10012

10010

10008

10006

10004

10002

1

09996

09994

09998

l asymp 0825 (m) 120596e asymp 0209 (rads) k asymp 3448 (rads)

(d)

Figure 2 Time dependencies of dimensionless cable reactions 119873119898119892 in the first approximation for fixed cable lengths 119897 = 0206m (a)119897 = 0412m (b) 119897 = 0618m (c) and 119897 = 0825m (d)

The relative trajectories for load119872on the cable in Figure 3have been derived for the following numerical values 119877 =

0492m 119892 = 981ms2 119897 = 0825m 119896 = (119892119897)05

asymp

3448 rads 119879 = 30 s 120596119890= 2120587119879 asymp 0209 rads 120572

1dyn =

000221 rad 119881119861

= 0103ms 119910dyn = 000182m ]1

=

119896 + 120596119890

= 3658 rads ]2

= 119896 minus 120596119890

= 3239 rads(Figure 3)

Numerical plots in Figure 3 show the payload119872 swayingin the noninertial reference frame (119874119909119910119911) In order tovisualize payload119872 absolute swaying path (Figures 4 and 5)

The Scientific World Journal 7

t1 = 125 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(a)

t2 = 3s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(b)

t3 = 10 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(c)

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

t4 = 25s

Relative trajectory of load M swayingfor linearized model

(d)

Figure 3The relative trajectories for load119872 on the cable during uniform rotational motion of the crane boom 1198611198742for the moments of time

1199051= 125 s (a) 119905

2= 3 s (b) 119905

3= 10 s (c) and 119905

4= 25 s (d)

in the fixed inertial reference frame (1198742119909211991021199112) we will

address the following transition formula

(1199092(119905)

1199102(119905)

) = (sin (120596

119890119905) cos (120596

119890119905)

minus cos (120596119890119905) sin (120596

119890119905)) (

119909 (119905)

119910 (119905) + 119877 + 119910dyn)

(22)

Derived computational plot for absolute payload swayingpath in the absolute coordinatesrsquo plane (119909

21199102) is shown in

Figure 5(a) for load119872 swaying time 1199051= 15 s and crane boom

transport slewing angle120593119890= 180

∘ In Figure 5(b) we show theexperimental plot for absolute payload swaying path in theabsolute coordinatesrsquo plane (119909

21199102) for load 119872 swaying time

1199052= 15 s and crane boom transport slewing angle 120593

119890= 180

∘which was built with an introduction of experimental setupin Figure 4

6 Comparison of Derived andKnown Published Results

The computational dimensionless cable force119873119898119892 derivedin the first approximation and shown in Figure 2 outlinesthe minor change of the cable tension force in the vicinityof the ordinate (119873119898119892) = 1 (119873119898119892) isin [09875 10233]

for 119897 = 0206m in Figure 2(a) (119873119898119892) isin [0994 10122]

for 119897 = 0412m in Figure 2(b) (119873119898119892) isin [0998 1004] for119897 = 0618m in Figure 2(c) and (119873119898119892) isin [099933 10013]

for 119897 = 0825m in Figure 2(d) So the assumption (14)really takes place as shown in Figure 2 The same effect for(119873119898119892)Mitrev isin [094 104] has been derived by Mitrev andGrigorov [23 Mitrevrsquos Figure 7 in page 89] that confirmsassumption (14) An analogous effect for dimensionless cable

8 The Scientific World Journal

Camera

Bearing

Boom

Laser pointer

Slewing axis

CableLower plate

M

Light-emitting diodeRegular grid 1200mm times 700mmwith square cells 200mm times 20mm

B

D

O2

Figure 4 The experimental measurement system for payload 119872

swaying during crane boom slewing

force (119873119898119892)Maczynski isin [0996 1005] has been derived byMaczynski and Wojciech [24 Maczynskirsquos Figure 10 in page278] which also reaffirms the assumption (14)

In published work by Sakawa et al ([25] Figure 3 at page552 Figure 4 at page 552 Figure 5 at page 554 and Figure 6at page 554) the small Sakawa values 119889119911119889119905 le 01 (Figures 3and 4) and 119889119911119889119905 le 005 (Figures 5 and 6) of 119911-projectionsfor payload velocities have been shown which confirms thesmall angle assumption119873 asymp 119898119892 (14)

7 Discussion and Analysis of Basic Results forMathematical and Physical Simulation

The relative trajectories for load119872 swaying in Figure 3 showequality of maximum values for relative 119909 and 119910 payloadcoordinates

119909max = 119910max = 1198601+ 1198602=

120596119890(119877 + 119910dyn)

119896 (23)

The formula (23) outlines that values 119909max and 119910max inFigure 3 increase with growth of boom slewing velocity 120596

119890

and with increased cable length 119897 The 119910dyn influence on thevalues of 119909max and 119910max is negligible

Equation (18) shows that the free oscillation frequencies]1and ]

2depend on the boom slewing velocity 120596

119890and

natural frequency 119896 of the payload with a fixed pivot point119861 The frequencies ]

1and ]

2essentially depend on the

pendulum length 119897 because 119896 ≫ 120596119890 Kinematically saying

the miscoordination of frequencies ]1and ]

2is ]1minus ]2=

2120596119890and is determined by the crane boom slewing velocity

120596119890 Dynamically saying the value of miscoordination is

governed by the Coriolis acceleration-induced DrsquoAlembertrsquosinertia force

Computations shown in Figure 3 for relative payload119872 swaying show that the clockwise direction of rotationfor relative oscillations is opposite in direction from thecounterclockwise direction of rotation for the crane boomThis effect follows from the conservation of position of theplane 119910

21199112of oscillations of load 119872 in the fixed inertial

coordinate system 119909211991021199112

The comparative analysis of computational (1) and exper-imental (2) absolute payload trajectories is shown in Fig-ure 5(c)The quantity of the local maximums in Figure 5(a) is8 for theoretical curve 1 So the theoretical period for relativeload swaying is 119879theor = 187 s The quantity of the localmaximums in Figure 5(b) is 75 for experimental curve 2Theperiod for experimental curve 2 in Figure 5(b) is 119879experim =

2 s The discrepancy between the values of theoretical andexperimental periods does not exceed 7

The experimental absolute trajectory (2) in Figures 5(b)and 5(c) outlines the additional relative swaying of payload119872 after crane boom stop which has the elliptical form inthe vicinity of the point (0 minus05) in the plane 119909

21199102for 120593119890=

180∘ The remaining payload motions after crane boom stop

are the small oscillations of the payload with the fixed pivotcenter

8 Final Conclusions

The closed system of nonlinear ODEs for payload relativeswaying has been derived in relative Cartesian coordinateswith an introduction of Lagrange equations and a geometricconstraint equation

The explicit form of the dependence of the cable reactionforce as a function of the relative payload coordinates 119909

1 1199101

1199111and their first and second time derivatives has been found

Derived complex dependency of the cable reaction force onthe abovementioned variables determines the nonlinearity ofthe present problem

The deviation of cable reaction force frompayload gravityforce is small This fact allowed the derivation of a linearizedODE system for payload swaying in relative Cartesian coor-dinates and verified the small relative Cartesian coordinatesrsquoassumption The analytical expressions for relative payloadpath have been derived

Derived results have good agreement with experimentsand known published data

The Scientific World Journal 9

05

04

03

02

01

0

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

t = 15 s

1

Point O2

2R = 1 (m)

1

PointnP i OO2OO

y2 (m)

x2 (m)(Point D)

(Point M)

(a)

05

04

03

02

01

02R = 1 (m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

2

4

3

Rotational directionPoint O2

t = 18 s

(Point D)

(Point M)

(Point B)

x2 (m)

y2 (m)

(b)

1 2

4

Point O2

2R = 1 (m)

y2(m)

x2(m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

(Point D)

(Point M) (Point M)

05

04

03

02

01

0

(c)

Figure 5 Calculated 1 (a c) and experimental 2 (b c) absolute trajectories for load119872 where 3 is the trajectory of boom point 119861 (b) 4 is thecenter 119874

2(D) of boom rotation (b c)

Disclosure

The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors

Conflict of Interests

The authors Alexander V Perig Alexander N Stadnik andAlexander I Deriglazov declare that there is no conflict ofinterests regarding the publication of this paper

References

[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002

[2] W Adamiec-Wojcik I P Fałat A Maczynski and S WojciechldquoLoad stabilisation in an A-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009

[3] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquoMethods in AppliedMechanics and Engineering vol170 no 3-4 pp 343ndash3353 1999

[4] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009

[5] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004

[6] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007

[7] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008

[8] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing aload prescribed motionrdquo The Archive of Mechanical Engineer-ing vol 3 pp 209ndash222 2009 httpamemeilpweduplindexphpengcontentdownload3731748fileart 2009 3 02pdf

[9] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010

[10] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011

10 The Scientific World Journal

[11] J H Cha S H Ham K Y Lee and M I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution ofMechanical Engineers K vol 224no 4 pp 365ndash373 2010

[12] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002

[13] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007

[14] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K vol 217no 2 pp 99ndash109 2003

[15] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006

[16] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers K vol 224 no 4 pp 341ndash352 2010

[17] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity Science A vol 8 no 9 pp 1395ndash1400 2007

[18] B Grigorov and R Mitrev ldquoA NewtonmdashEuler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009

[19] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998

[20] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008

[21] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008

[22] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers I vol 223 no 4 pp 575ndash580 2009

[23] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008

[24] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003

[25] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 7

t1 = 125 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(a)

t2 = 3s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(b)

t3 = 10 s

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

Relative trajectory of load M swayingfor linearized model

(c)

y(x) l = 0825 (m) k asymp 3448 (rads) 120596e asymp 0209 (rads)

t4 = 25s

Relative trajectory of load M swayingfor linearized model

(d)

Figure 3The relative trajectories for load119872 on the cable during uniform rotational motion of the crane boom 1198611198742for the moments of time

1199051= 125 s (a) 119905

2= 3 s (b) 119905

3= 10 s (c) and 119905

4= 25 s (d)

in the fixed inertial reference frame (1198742119909211991021199112) we will

address the following transition formula

(1199092(119905)

1199102(119905)

) = (sin (120596

119890119905) cos (120596

119890119905)

minus cos (120596119890119905) sin (120596

119890119905)) (

119909 (119905)

119910 (119905) + 119877 + 119910dyn)

(22)

Derived computational plot for absolute payload swayingpath in the absolute coordinatesrsquo plane (119909

21199102) is shown in

Figure 5(a) for load119872 swaying time 1199051= 15 s and crane boom

transport slewing angle120593119890= 180

∘ In Figure 5(b) we show theexperimental plot for absolute payload swaying path in theabsolute coordinatesrsquo plane (119909

21199102) for load 119872 swaying time

1199052= 15 s and crane boom transport slewing angle 120593

119890= 180

∘which was built with an introduction of experimental setupin Figure 4

6 Comparison of Derived andKnown Published Results

The computational dimensionless cable force119873119898119892 derivedin the first approximation and shown in Figure 2 outlinesthe minor change of the cable tension force in the vicinityof the ordinate (119873119898119892) = 1 (119873119898119892) isin [09875 10233]

for 119897 = 0206m in Figure 2(a) (119873119898119892) isin [0994 10122]

for 119897 = 0412m in Figure 2(b) (119873119898119892) isin [0998 1004] for119897 = 0618m in Figure 2(c) and (119873119898119892) isin [099933 10013]

for 119897 = 0825m in Figure 2(d) So the assumption (14)really takes place as shown in Figure 2 The same effect for(119873119898119892)Mitrev isin [094 104] has been derived by Mitrev andGrigorov [23 Mitrevrsquos Figure 7 in page 89] that confirmsassumption (14) An analogous effect for dimensionless cable

8 The Scientific World Journal

Camera

Bearing

Boom

Laser pointer

Slewing axis

CableLower plate

M

Light-emitting diodeRegular grid 1200mm times 700mmwith square cells 200mm times 20mm

B

D

O2

Figure 4 The experimental measurement system for payload 119872

swaying during crane boom slewing

force (119873119898119892)Maczynski isin [0996 1005] has been derived byMaczynski and Wojciech [24 Maczynskirsquos Figure 10 in page278] which also reaffirms the assumption (14)

In published work by Sakawa et al ([25] Figure 3 at page552 Figure 4 at page 552 Figure 5 at page 554 and Figure 6at page 554) the small Sakawa values 119889119911119889119905 le 01 (Figures 3and 4) and 119889119911119889119905 le 005 (Figures 5 and 6) of 119911-projectionsfor payload velocities have been shown which confirms thesmall angle assumption119873 asymp 119898119892 (14)

7 Discussion and Analysis of Basic Results forMathematical and Physical Simulation

The relative trajectories for load119872 swaying in Figure 3 showequality of maximum values for relative 119909 and 119910 payloadcoordinates

119909max = 119910max = 1198601+ 1198602=

120596119890(119877 + 119910dyn)

119896 (23)

The formula (23) outlines that values 119909max and 119910max inFigure 3 increase with growth of boom slewing velocity 120596

119890

and with increased cable length 119897 The 119910dyn influence on thevalues of 119909max and 119910max is negligible

Equation (18) shows that the free oscillation frequencies]1and ]

2depend on the boom slewing velocity 120596

119890and

natural frequency 119896 of the payload with a fixed pivot point119861 The frequencies ]

1and ]

2essentially depend on the

pendulum length 119897 because 119896 ≫ 120596119890 Kinematically saying

the miscoordination of frequencies ]1and ]

2is ]1minus ]2=

2120596119890and is determined by the crane boom slewing velocity

120596119890 Dynamically saying the value of miscoordination is

governed by the Coriolis acceleration-induced DrsquoAlembertrsquosinertia force

Computations shown in Figure 3 for relative payload119872 swaying show that the clockwise direction of rotationfor relative oscillations is opposite in direction from thecounterclockwise direction of rotation for the crane boomThis effect follows from the conservation of position of theplane 119910

21199112of oscillations of load 119872 in the fixed inertial

coordinate system 119909211991021199112

The comparative analysis of computational (1) and exper-imental (2) absolute payload trajectories is shown in Fig-ure 5(c)The quantity of the local maximums in Figure 5(a) is8 for theoretical curve 1 So the theoretical period for relativeload swaying is 119879theor = 187 s The quantity of the localmaximums in Figure 5(b) is 75 for experimental curve 2Theperiod for experimental curve 2 in Figure 5(b) is 119879experim =

2 s The discrepancy between the values of theoretical andexperimental periods does not exceed 7

The experimental absolute trajectory (2) in Figures 5(b)and 5(c) outlines the additional relative swaying of payload119872 after crane boom stop which has the elliptical form inthe vicinity of the point (0 minus05) in the plane 119909

21199102for 120593119890=

180∘ The remaining payload motions after crane boom stop

are the small oscillations of the payload with the fixed pivotcenter

8 Final Conclusions

The closed system of nonlinear ODEs for payload relativeswaying has been derived in relative Cartesian coordinateswith an introduction of Lagrange equations and a geometricconstraint equation

The explicit form of the dependence of the cable reactionforce as a function of the relative payload coordinates 119909

1 1199101

1199111and their first and second time derivatives has been found

Derived complex dependency of the cable reaction force onthe abovementioned variables determines the nonlinearity ofthe present problem

The deviation of cable reaction force frompayload gravityforce is small This fact allowed the derivation of a linearizedODE system for payload swaying in relative Cartesian coor-dinates and verified the small relative Cartesian coordinatesrsquoassumption The analytical expressions for relative payloadpath have been derived

Derived results have good agreement with experimentsand known published data

The Scientific World Journal 9

05

04

03

02

01

0

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

t = 15 s

1

Point O2

2R = 1 (m)

1

PointnP i OO2OO

y2 (m)

x2 (m)(Point D)

(Point M)

(a)

05

04

03

02

01

02R = 1 (m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

2

4

3

Rotational directionPoint O2

t = 18 s

(Point D)

(Point M)

(Point B)

x2 (m)

y2 (m)

(b)

1 2

4

Point O2

2R = 1 (m)

y2(m)

x2(m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

(Point D)

(Point M) (Point M)

05

04

03

02

01

0

(c)

Figure 5 Calculated 1 (a c) and experimental 2 (b c) absolute trajectories for load119872 where 3 is the trajectory of boom point 119861 (b) 4 is thecenter 119874

2(D) of boom rotation (b c)

Disclosure

The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors

Conflict of Interests

The authors Alexander V Perig Alexander N Stadnik andAlexander I Deriglazov declare that there is no conflict ofinterests regarding the publication of this paper

References

[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002

[2] W Adamiec-Wojcik I P Fałat A Maczynski and S WojciechldquoLoad stabilisation in an A-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009

[3] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquoMethods in AppliedMechanics and Engineering vol170 no 3-4 pp 343ndash3353 1999

[4] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009

[5] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004

[6] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007

[7] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008

[8] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing aload prescribed motionrdquo The Archive of Mechanical Engineer-ing vol 3 pp 209ndash222 2009 httpamemeilpweduplindexphpengcontentdownload3731748fileart 2009 3 02pdf

[9] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010

[10] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011

10 The Scientific World Journal

[11] J H Cha S H Ham K Y Lee and M I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution ofMechanical Engineers K vol 224no 4 pp 365ndash373 2010

[12] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002

[13] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007

[14] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K vol 217no 2 pp 99ndash109 2003

[15] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006

[16] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers K vol 224 no 4 pp 341ndash352 2010

[17] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity Science A vol 8 no 9 pp 1395ndash1400 2007

[18] B Grigorov and R Mitrev ldquoA NewtonmdashEuler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009

[19] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998

[20] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008

[21] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008

[22] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers I vol 223 no 4 pp 575ndash580 2009

[23] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008

[24] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003

[25] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 The Scientific World Journal

Camera

Bearing

Boom

Laser pointer

Slewing axis

CableLower plate

M

Light-emitting diodeRegular grid 1200mm times 700mmwith square cells 200mm times 20mm

B

D

O2

Figure 4 The experimental measurement system for payload 119872

swaying during crane boom slewing

force (119873119898119892)Maczynski isin [0996 1005] has been derived byMaczynski and Wojciech [24 Maczynskirsquos Figure 10 in page278] which also reaffirms the assumption (14)

In published work by Sakawa et al ([25] Figure 3 at page552 Figure 4 at page 552 Figure 5 at page 554 and Figure 6at page 554) the small Sakawa values 119889119911119889119905 le 01 (Figures 3and 4) and 119889119911119889119905 le 005 (Figures 5 and 6) of 119911-projectionsfor payload velocities have been shown which confirms thesmall angle assumption119873 asymp 119898119892 (14)

7 Discussion and Analysis of Basic Results forMathematical and Physical Simulation

The relative trajectories for load119872 swaying in Figure 3 showequality of maximum values for relative 119909 and 119910 payloadcoordinates

119909max = 119910max = 1198601+ 1198602=

120596119890(119877 + 119910dyn)

119896 (23)

The formula (23) outlines that values 119909max and 119910max inFigure 3 increase with growth of boom slewing velocity 120596

119890

and with increased cable length 119897 The 119910dyn influence on thevalues of 119909max and 119910max is negligible

Equation (18) shows that the free oscillation frequencies]1and ]

2depend on the boom slewing velocity 120596

119890and

natural frequency 119896 of the payload with a fixed pivot point119861 The frequencies ]

1and ]

2essentially depend on the

pendulum length 119897 because 119896 ≫ 120596119890 Kinematically saying

the miscoordination of frequencies ]1and ]

2is ]1minus ]2=

2120596119890and is determined by the crane boom slewing velocity

120596119890 Dynamically saying the value of miscoordination is

governed by the Coriolis acceleration-induced DrsquoAlembertrsquosinertia force

Computations shown in Figure 3 for relative payload119872 swaying show that the clockwise direction of rotationfor relative oscillations is opposite in direction from thecounterclockwise direction of rotation for the crane boomThis effect follows from the conservation of position of theplane 119910

21199112of oscillations of load 119872 in the fixed inertial

coordinate system 119909211991021199112

The comparative analysis of computational (1) and exper-imental (2) absolute payload trajectories is shown in Fig-ure 5(c)The quantity of the local maximums in Figure 5(a) is8 for theoretical curve 1 So the theoretical period for relativeload swaying is 119879theor = 187 s The quantity of the localmaximums in Figure 5(b) is 75 for experimental curve 2Theperiod for experimental curve 2 in Figure 5(b) is 119879experim =

2 s The discrepancy between the values of theoretical andexperimental periods does not exceed 7

The experimental absolute trajectory (2) in Figures 5(b)and 5(c) outlines the additional relative swaying of payload119872 after crane boom stop which has the elliptical form inthe vicinity of the point (0 minus05) in the plane 119909

21199102for 120593119890=

180∘ The remaining payload motions after crane boom stop

are the small oscillations of the payload with the fixed pivotcenter

8 Final Conclusions

The closed system of nonlinear ODEs for payload relativeswaying has been derived in relative Cartesian coordinateswith an introduction of Lagrange equations and a geometricconstraint equation

The explicit form of the dependence of the cable reactionforce as a function of the relative payload coordinates 119909

1 1199101

1199111and their first and second time derivatives has been found

Derived complex dependency of the cable reaction force onthe abovementioned variables determines the nonlinearity ofthe present problem

The deviation of cable reaction force frompayload gravityforce is small This fact allowed the derivation of a linearizedODE system for payload swaying in relative Cartesian coor-dinates and verified the small relative Cartesian coordinatesrsquoassumption The analytical expressions for relative payloadpath have been derived

Derived results have good agreement with experimentsand known published data

The Scientific World Journal 9

05

04

03

02

01

0

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

t = 15 s

1

Point O2

2R = 1 (m)

1

PointnP i OO2OO

y2 (m)

x2 (m)(Point D)

(Point M)

(a)

05

04

03

02

01

02R = 1 (m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

2

4

3

Rotational directionPoint O2

t = 18 s

(Point D)

(Point M)

(Point B)

x2 (m)

y2 (m)

(b)

1 2

4

Point O2

2R = 1 (m)

y2(m)

x2(m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

(Point D)

(Point M) (Point M)

05

04

03

02

01

0

(c)

Figure 5 Calculated 1 (a c) and experimental 2 (b c) absolute trajectories for load119872 where 3 is the trajectory of boom point 119861 (b) 4 is thecenter 119874

2(D) of boom rotation (b c)

Disclosure

The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors

Conflict of Interests

The authors Alexander V Perig Alexander N Stadnik andAlexander I Deriglazov declare that there is no conflict ofinterests regarding the publication of this paper

References

[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002

[2] W Adamiec-Wojcik I P Fałat A Maczynski and S WojciechldquoLoad stabilisation in an A-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009

[3] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquoMethods in AppliedMechanics and Engineering vol170 no 3-4 pp 343ndash3353 1999

[4] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009

[5] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004

[6] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007

[7] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008

[8] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing aload prescribed motionrdquo The Archive of Mechanical Engineer-ing vol 3 pp 209ndash222 2009 httpamemeilpweduplindexphpengcontentdownload3731748fileart 2009 3 02pdf

[9] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010

[10] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011

10 The Scientific World Journal

[11] J H Cha S H Ham K Y Lee and M I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution ofMechanical Engineers K vol 224no 4 pp 365ndash373 2010

[12] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002

[13] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007

[14] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K vol 217no 2 pp 99ndash109 2003

[15] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006

[16] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers K vol 224 no 4 pp 341ndash352 2010

[17] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity Science A vol 8 no 9 pp 1395ndash1400 2007

[18] B Grigorov and R Mitrev ldquoA NewtonmdashEuler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009

[19] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998

[20] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008

[21] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008

[22] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers I vol 223 no 4 pp 575ndash580 2009

[23] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008

[24] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003

[25] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

The Scientific World Journal 9

05

04

03

02

01

0

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

t = 15 s

1

Point O2

2R = 1 (m)

1

PointnP i OO2OO

y2 (m)

x2 (m)(Point D)

(Point M)

(a)

05

04

03

02

01

02R = 1 (m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

2

4

3

Rotational directionPoint O2

t = 18 s

(Point D)

(Point M)

(Point B)

x2 (m)

y2 (m)

(b)

1 2

4

Point O2

2R = 1 (m)

y2(m)

x2(m)

minus05 minus04 minus03 minus02 minus01 0 01 02 03 04 05

(Point D)

(Point M) (Point M)

05

04

03

02

01

0

(c)

Figure 5 Calculated 1 (a c) and experimental 2 (b c) absolute trajectories for load119872 where 3 is the trajectory of boom point 119861 (b) 4 is thecenter 119874

2(D) of boom rotation (b c)

Disclosure

The submission of the authorsrsquo paper implies that it has notbeen previously published that it is not under considerationfor publication elsewhere and that it will not be publishedelsewhere in the same form without the written permissionof the editors

Conflict of Interests

The authors Alexander V Perig Alexander N Stadnik andAlexander I Deriglazov declare that there is no conflict ofinterests regarding the publication of this paper

References

[1] EMAbdel-Rahman andAHNayfeh ldquoPendulation reductionin boom cranes using cable length manipulationrdquo NonlinearDynamics vol 27 no 3 pp 255ndash269 2002

[2] W Adamiec-Wojcik I P Fałat A Maczynski and S WojciechldquoLoad stabilisation in an A-framemdasha type of an offshore cranerdquoThe Archive of Mechanical Engineering vol 56 no 1 pp 37ndash592009

[3] P J Aston ldquoBifurcations of the horizontally forced sphericalpendulumrdquoMethods in AppliedMechanics and Engineering vol170 no 3-4 pp 343ndash3353 1999

[4] P Betsch M Quasem and S Uhlar ldquoNumerical integration ofdiscretemechanical systems withmixed holonomic and controlconstraintsrdquo Journal of Mechanical Science and Technology vol23 no 4 pp 1012ndash1018 2009

[5] W Blajer and K Kołodziejczyk ldquoA geometric approach tosolving problems of control constraints theory and a DAEframeworkrdquoMultibody System Dynamics vol 11 no 4 pp 343ndash364 2004

[6] W Blajer and K Kołodziejczyk ldquoControl of underactuatedmechanical systems with servo-constraintsrdquoNonlinear Dynam-ics vol 50 no 4 pp 781ndash791 2007

[7] W Blajer and K Kołodziejczyk ldquoModeling of underactuatedmechanical systems in partly specified motionrdquo Journal ofTheoretical and Applied Mechanics vol 46 no 2 pp 383ndash3942008

[8] W Blajer and K Kołodziejczyk ldquoThe use of dependent coor-dinates in modeling and simulation of cranes executing aload prescribed motionrdquo The Archive of Mechanical Engineer-ing vol 3 pp 209ndash222 2009 httpamemeilpweduplindexphpengcontentdownload3731748fileart 2009 3 02pdf

[9] W Blajer A Czaplicki K Dziewiecki and Z Mazur ldquoInfluenceof selected modeling and computational issues on muscle forceestimatesrdquoMultibody System Dynamics vol 24 no 4 pp 473ndash492 2010

[10] W Blajer and K Kołodziejczyk ldquoImproved DAE formulationfor inverse dynamics simulation of cranesrdquo Multibody SystemDynamics vol 25 no 2 pp 131ndash143 2011

10 The Scientific World Journal

[11] J H Cha S H Ham K Y Lee and M I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution ofMechanical Engineers K vol 224no 4 pp 365ndash373 2010

[12] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002

[13] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007

[14] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K vol 217no 2 pp 99ndash109 2003

[15] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006

[16] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers K vol 224 no 4 pp 341ndash352 2010

[17] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity Science A vol 8 no 9 pp 1395ndash1400 2007

[18] B Grigorov and R Mitrev ldquoA NewtonmdashEuler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009

[19] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998

[20] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008

[21] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008

[22] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers I vol 223 no 4 pp 575ndash580 2009

[23] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008

[24] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003

[25] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 The Scientific World Journal

[11] J H Cha S H Ham K Y Lee and M I Roh ldquoApplicationof a topological modelling approach of multi-body systemdynamics to simulation of multi-floating cranes in shipyardsrdquoProceedings of the Institution ofMechanical Engineers K vol 224no 4 pp 365ndash373 2010

[12] K Ellermann E Kreuzer and M Markiewicz ldquoNonlineardynamics of floating cranesrdquo Nonlinear Dynamics vol 27 no2 pp 107ndash183 2002

[13] T Erneux and T Kalmar-Nagy ldquoNonlinear stability of a delayedfeedback controlled container cranerdquo Journal of Vibration andControl vol 13 no 5 pp 603ndash616 2007

[14] G Glossiotis and I Antoniadis ldquoPayload sway suppression inrotary cranes by digital filtering of the commanded inputsrdquoProceedings of the Institution of Mechanical Engineers K vol 217no 2 pp 99ndash109 2003

[15] R A Ibrahim ldquoExcitation-induced stability and phase transi-tion a reviewrdquo Journal of Vibration and Control vol 12 no 10pp 1093ndash1170 2006

[16] J Kortelainen and A Mikkola ldquoSemantic data model inmultibody system simulationrdquo Proceedings of the Institution ofMechanical Engineers K vol 224 no 4 pp 341ndash352 2010

[17] R Mitrev ldquoMathematical modelling of translational motion ofrail-guided cart with suspended payloadrdquo Journal of ZhejiangUniversity Science A vol 8 no 9 pp 1395ndash1400 2007

[18] B Grigorov and R Mitrev ldquoA NewtonmdashEuler approach to thefreely suspended load swinging problemrdquo Recent Journal vol10 no 26 pp 109ndash112 2009

[19] MOsinski and SWojciech ldquoApplication of nonlinear optimisa-tion methods to input shaping of the hoist drive of an off-shorecranerdquo Nonlinear Dynamics vol 17 no 4 pp 369ndash386 1998

[20] H L Ren X LWang Y J Hu and C G Li ldquoDynamic responseanalysis of a moored crane-ship with a flexible boomrdquo Journalof Zhejiang University Science A vol 9 no 1 pp 26ndash31 2008

[21] H Schaub ldquoRate-based ship-mounted crane payload pendula-tion control systemrdquo Control Engineering Practice vol 16 no 1pp 132ndash145 2008

[22] N Uchiyama ldquoRobust control for overhead cranes by partialstate feedbackrdquo Proceedings of the Institution of MechanicalEngineers I vol 223 no 4 pp 575ndash580 2009

[23] R Mitrev and B Grigorov ldquoDynamic behaviour of a spheri-cal pendulum with spatially moving pivotrdquo Zeszyty NaukowePolitechniki Poznanskiej Budowa Maszyn i Zarządzanie Pro-dukcją vol 9 pp 81ndash91 2008

[24] A Maczynski and S Wojciech ldquoDynamics of a mobile craneand optimisation of the slewing motion of its upper structurerdquoNonlinear Dynamics vol 32 no 3 pp 259ndash290 2003

[25] Y Sakawa Y Shindo and Y Hashimoto ldquoOptimal control of arotary cranerdquo Journal of Optimization Theory and Applicationsvol 35 no 4 pp 535ndash557 1981

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of