Spherical Pendulum Small Oscillations for Slewing Crane Motion

Hindawi Publishing Corporation e Scientiﬁc World Journal Volume 2014, Article ID 451804, 10 pages http://dx.doi.org/10.1155/2014/451804

Research Article Spherical Pendulum Small Oscillations for Slewing Crane Motion

Alexander V. Perig,1 Alexander N. Stadnik,2 and Alexander I. Deriglazov2

1 Manufacturing Processes and Automation Engineering Department, Engineering Automation Faculty, Donbass State Engineering Academy, Shkadinova 72, Donetsk Region, Kramatorsk 84313, Ukraine 2 Department of Technical Mechanics, Engineering Automation Faculty, Donbass State Engineering Academy, Shkadinova 72, Donetsk Region, Kramatorsk 84313, Ukraine

Correspondence should be addressed to Alexander V. Perig; [email protected]

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

Academic Editors: N.-I. Kim, M. Seddeek, and M. Zappalorto

Copyright © 2014 Alexander V. Perig et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The present paper focuses on the Lagrange mechanics-based description of small oscillations of a spherical pendulum witha uniformly rotating suspension center. The analytical solution of the natural frequencies’ problem has been derived for the case of uniform rotation of a crane boom. The payload paths have been found in the inertial reference frame fixed on earth and inthe noninertial reference frame, which is connected with the rotating crane boom. The numerical amplitude-frequency characteristics of the relative payload motion have been found. The mechanical interpretation of the terms in Lagrange equations has been outlined. The analytical expression and numerical estimation for cable tension force have been proposed. The numerical computational results, which correlate very accurately with the experimental observations, have been shown.

1. Introduction Adamiec-Wojcik´ et al. [2]haveprovidedanumerical finite-element simulation of the supporting frame of an The solution of nonlinear differential equations for spherical offshore crane portal construction. Within the proposed pendulum swaying requires the introduction of modern model, supporting frame elastic oscillations define both the numerical computational analysis techniques. Such computer payload motion and the displacements of the suspension techniquesallowustoproduceapproximatesolutionsofthe center with a payload on the cable. The derived model also posedproblems.However,thelinearizedmodelsforspherical incorporates ship vibration motion. pendulum small swaying motion provide a clear mechanical Aston has obtained two independent motion equations explanation and the first numerical approximation of the forasphericalpendulumtakingintoaccountviscousfriction complex spherical motion problems. This fact confirms the effects. This model neglects the components of the Coriolis significance of the small oscillation problems for the modern inertia force [3]. field of lifting-and-transport machines. A series of modern Betsch et al. [4] have described rotary crane dynam- research efforts [1–22] have been focused on the solution of a ics in terms of redundant coordinates with an introduc- spherical pendulum with moving pivot center. tion of differential-algebraic equations (DAEs). The authors’ Abdel-Rahman and Nayfeh [1]havestudiedthedynamic approachfocusedonthesimultaneousincorporationofa problem for a “crane boom tip-payload” mechanical system large total number of redundant coordinates and the augmen- focused on the reduction of payload lateral vibrations by tation constraints do not fully describe the particular cases of reeling and unreeling the hoist cable. In this work, the rotary crane dynamics. authors have derived two- and three-dimensional models for Blajer and Kołodziejczyk [5] have derived governing a spherical pendulum taking into account the linearity of DAE equations for the open-loop control formulation that transport motion. forces the underactuated crane system to complete the partly 2 The Scientific World Journal specified motion. It is necessary to note that this paper deals trajectories of both the pendulum suspension center and the primarily with controlled in-plane motions of the crane sys- swinging load. tem with linear transport motion. Blajer and Kołodziejczyk Osinski´ and Wojciech [19]haveanalyzedthehoistmotion [6] have investigated the control of underactuated crane of a mechanical system consisting of a load, hoist rope, and mechanical systems with servo constraints. This work takes elastic crane jib. This study mainly deals with rectilinear load into account the restoring load gravity force in the horizontal oscillations and lateral crane jib vibrations. plane. However, the derived mathematical model assumes Ren et al. [20]haveusedamixedapproachcombin- primarily linear transport motion without the Coriolis inertia ing both analytical and finite-element techniques for the force. Blajer and Kołodziejczyk [7]havepresentedaconcep- Lagrangian function of a mechanical system. They have tual research article focused on the building of n-index DAE determined the relations between load swaying, crane boom systems for a relevant dynamic description of partly specified flexibility, and wave motions. Schaub [21] has implemented a crane system motion. Blajer and Kołodziejczyk [8]have rate-based ship-mounted crane payload pendulation control introduced a carefully argued computational technique for system. This research is focused on a kinematic approach to numerical simulation of cranes based on the use of dependent payload stabilization with an emphasis on the usage of inertial coordinates. The advantages of dependent coordinates’ usage measurement unit (IMU) information. have been outlined. It is necessary to note that with each new Uchiyama [22]hasappliedcontroltheorytechniquesfora research effort, the authors apply much more sophisticated robust crane controller design that provides robustness with computational methods and derive more exact governing respect to changes in cable length and transported payload dynamic equations. However, this paper is also based on mass without iterative computations. thesamesimplependulummodelasthefirstapproaches, At the same time, these known articles [1–22]donotdo wherein the Coriolis inertia force and the restoring load justice to load swaying descriptions using Coriolis effects in gravity force are ignored. Blajer et al. [9]havesimulateda relative motion. The present paper focuses on these issues. two-tier framed human arm model with segments of equal The present paper focuses on the Lagrange mechanics- length. The application of this model seems to be very based description of small oscillations for a spherical pendu- useful for the dynamic analysis of a swaying load in the lum with a uniformly rotating suspension center. horizontal plane during rotation of a crane boom. Blajer and The prime novelty statement of the present paper is Kołodziejczyk [10] have developed improved DAE equations based on the rational introduction of a Cartesian coordinate for more precise rotary crane dynamics simulation. But, system for study of small relative swaying of a payload and additional dimensional analysis for the explicit forms of the substantiation of uniformity of crane boom rotation. governing equations would greatly enhance the work. Cha et al. [11] have applied a topological modeling approach to a dynamic simulation of multiple floating cranes 2. Kinematic Analysis in shipyards taking into account crane boom oscillations. Ellermann et al. [12] have introduced the vector of In order to solve the problem, we introduce the Lagrange generalized gyroscopic and Coriolis forces for a mathematical equations for the motion of the mechanical system “crane description of nonlinear dynamics of floating cranes. This boom 𝐵𝑂2-load 𝑀”whichisshowninFigure1. work used a comprehensive model with a set of 20 differential The computational scheme in Figure 1 may be described equations but does not allow proper analysis of particular with an introduction of three degrees of freedom. For gen- cases of floating jib crane-load system motions. eralized coordinates, we assume the rectangular coordinates Erneux and Kalmar-Nagy´ [13]haveproposedaplane 𝑥, 𝑦,and𝑧 of the load and the angle 𝜑𝑒 =𝜔𝑒𝑡 of crane pendulum container crane model with linear transport trol- boom 𝐵𝑂2 uniform rotation in the horizontal plane (𝑥𝑦) ley motion. This model illustrates linear displacement of the with the constant angular velocity 𝜔𝑒 around the vertical planependulumsuspensioncenter. axis 𝑂2𝑧2. We denote the fixed inertial coordinate system Glossiotis and Antoniadis [14]havederivedafour- as 𝑥2𝑦2𝑧2 and the moving noninertial frame of reference as degrees-of-freedom (4DOF) model for the rotary crane 𝑥1𝑦1𝑧1, which is rigidly bounded with the crane boom 𝐵𝑂2. system, incorporating both hoisting and slewing payload Rotation of the moving noninertial frame of reference 𝑥1𝑦1𝑧1 motions and thus is able to handle cases where hoisting can around the fixed inertial coordinate system 𝑥2𝑦2𝑧2 defines be simultaneously applied to the rotary motion. the transportation motion. The motion of load 𝑀 relative to Ibrahim [15] has studied the issues of multiplicative noise the moving noninertial frame of reference 𝑥1𝑦1𝑧1 defines the and noise-enhanced stability of spherical pendulums. relative motion. Kortelainen and Mikkola [16]haveintroduceddataman- The scalar of the load 𝑀 transportation velocity is defined agement techniques within an approach using a semantic data as 𝑉𝑒 =𝜔𝑒 ⋅𝑂2𝑀, and transportation velocity vector V𝑒 model for payload oscillation problems. is perpendicular to O2M,thatis,thescalarproduct Mitrev [17] has applied his generalized approach with the (V𝑒, O2M)=0(Figure 1). The relative velocity vector V𝑟 of introduction of an inertia force vector to the case of linear the load 𝑀 is defined as V𝑟 =(𝑑𝑥1/𝑑𝑡,1 𝑑𝑦 /𝑑𝑡,1 𝑑𝑧 /𝑑𝑡) in 𝑡=0 𝑀 𝐵𝐴 transportmotionofthependulumpivotcenter.Grigorov Figure 1. In the initial time ,load has the vertical st and Mitrev [18] have investigated a spherical pendulum in-line position; that is, the load 𝑀 initial position coincides 𝐴 𝑀 dynamicmodelforanumericalsolutionofafreelysus- with the static equilibrium position st for the load on the pended swinging load problem. They have derived absolute cable 𝑀𝐵. The Scientific World Journal 3

R D z1

H z

→ B 𝜔e 𝜔e =d𝜑e/dt

y2 O2

𝛼1

l 𝜃

→ 𝜑 N e Ast 𝛼2 O1 Adyn O

y x2 → Ve M Δ y1

→ x1 Vr → mg x

Figure 1: Computational spatial scheme of spherical pendulum 𝑀,swayingonthecable𝑀𝐵 during crane boom 𝐵𝑂2 slewing motion for the linearized model derivation in Cartesian coordinates.

At time 𝑡=0,theload𝑀 has zero absolute velocity and Taking into account Figure 1,thesystemof(1)forthe the initial relative velocity is 𝑉𝑟𝑥(0) = +𝜔𝑒 ⋅𝐵𝐷(Figure 1). projections of the absolute velocity vector takes the following In Figure 1, we denote the angle 𝜃,whichisthecurrent form: angle 𝐴 𝑂2𝑀,thatis,theanglebetweentheaxis𝑦1 and st 𝑑𝑥 𝑑𝜑 the radius-vector O2M,wheresin(𝜃) =1 𝑥 /𝑂2𝑀;cos(𝜃) = 1 𝑒 𝑉𝑥 =( )−( )(𝑅+𝑦1), (𝑅+𝑦1)/𝑂2𝑀.Theverticalcoordinate𝑧1 in Figure 1 is defined 1 𝑑𝑡 𝑑𝑡 as 𝑧1 =𝑙−𝑙⋅cos(𝛼1),where𝛼1 is the angle between the cable 𝑀𝐵 𝑑𝑦1 𝑑𝜑𝑒 and the vertical. 𝑉𝑦 =( )+( )𝑥1, (2) 1 𝑑𝑡 𝑑𝑡 For small angles 𝛼1, the projections of the absolute velocity vector may be written as 𝑑𝑧1 𝑉𝑧 =( ). 1 𝑑𝑡 𝑑𝑥1 𝑉𝑥 =( )−𝑉𝑒 ⋅ cos (𝜃) , 1 𝑑𝑡 The square of absolute payload 𝑀 velocity may be written 𝑑𝑦 2 2 2 2 1 as 𝑉 =𝑉𝑥 +𝑉𝑦 +𝑉𝑧 : 𝑉𝑦 =( )+𝑉𝑒 ⋅ sin (𝜃) , (1) abs 1 1 1 1 𝑑𝑡 2 2 2 𝑑𝑧1 𝑑𝛼1 2 𝑑𝑥1 𝑑𝑦1 𝑑𝑧1 𝑉𝑧 =( )=( )⋅𝑙⋅sin (𝛼1). 𝑉 =( ) +( ) +( ) 1 𝑑𝑡 𝑑𝑡 abs 𝑑𝑡 𝑑𝑡 𝑑𝑡 4 The Scientific World Journal

𝑑𝜑 2 +( 𝑒 ) (𝑥2 +(𝑦 +𝑅)2) 4. Nonlinear Mathematical Model with 𝑑𝑡 1 1 Lagrange Equations’ Introduction 𝑑𝜑 𝑑𝑦 𝑑𝑥 +2( 𝑒 )(𝑥 ( 1 )−(𝑦 +𝑅)( 1 )) Taking into account (3)forthesquareofabsolutepayload𝑀 𝑑𝑡 1 𝑑𝑡 1 𝑑𝑡 𝑉2 velocity abs, and by adding the kinetic energy for a slewing 𝑑𝑥 𝑑𝜑 2 crane boom according to (6), we will have the following =(( 1 )−( 𝑒 )(𝑅+𝑦 )) expression for “crane boom-payload” kinetic energy: 𝑑𝑡 𝑑𝑡 1 𝑚 𝑑𝑥 𝑑𝜑 2 𝑑𝑦 𝑑𝜑 2 𝑑𝑧 2 𝑇= ((( 1 )−( 𝑒 )(𝑅+𝑦 )) +(( 1 )+( 𝑒 )𝑥 ) +( 1 ) . 2 𝑑𝑡 𝑑𝑡 1 𝑑𝑡 𝑑𝑡 1 𝑑𝑡 𝑑𝑦 𝑑𝜑 2 𝑑𝑧 2 (3) +(( 1 )+( 𝑒 )𝑥 ) +( 1 ) ) 𝑑𝑡 𝑑𝑡 1 𝑑𝑡 (9)

3. Dynamic Analysis 𝑂 𝐼 2 𝑑𝜑 2 + 33 ( 𝑒 ) . The slewing motion of the mechanical system “crane boom 2 𝑑𝑡 𝐵𝑂2-load 𝑀”inFigure1 is governed by the vector equation 𝑂 for the rate of change of moment of momentum H 2 for the For further analysis, we will assume that the crane boom 3 𝑑𝜑 /𝑑𝑡 =𝜔 system “crane boom 𝐵𝑂2-load 𝑀” with respect to point 𝑂2 in slewing angular velocity is the constant 𝑒 𝑒.Wewill the inertial reference frame 𝑥2𝑦2𝑧2: use the derived expression for kinetic energy for the left-hand sides of Lagrange equations in the noninertial reference frame (𝑥 𝑦 𝑧 ) 𝑥 𝑦 𝑧 2 2 2 𝑂2 1 1 1: 𝑑H3 𝑂 2 (4) = ∑M . 𝑑 𝜕𝑇 𝜕𝑇 𝑑𝑡 ( )− =𝑄 , 𝑥1 𝑑𝑡 𝜕𝑥1̇ 𝜕𝑥1 The vector equation (4) contains the following compo- 𝑑 𝜕𝑇 𝜕𝑇 nents: ( )− =𝑄 , 𝑦1 (10) 𝑑𝑡 𝜕𝑦1̇ 𝜕𝑦1 𝑂 𝑂 2 =𝐼 2 𝜔 , H3 33 𝑒 (5) 𝑑 𝜕𝑇 𝜕𝑇 ( )− =𝑄 . 2 𝑧 𝑂2 𝑂2 2 𝑑𝑡 𝜕𝑧̇ 𝜕𝑧 1 𝐼33 = (𝐼33 ) +𝑚(𝑥1 + (𝑅+𝑦1) ) , (6) 1 1 𝐵𝑂2 𝑂 𝑂 𝑂 𝑂 𝑂 Taking into account the nonlinearity and nonconser- ∑ 2 = 2 − 2 =(𝑀 2 −𝑀 2 ) , M M𝐷𝑇 M𝐹𝑇 𝐷𝑇 𝐹𝑇 k (7) vatism of the cable reaction force N,wehavethefollowing

𝑂 formulae for the generalized forces in the noninertial refer- (𝐼 2 ) 𝑥 𝑦 𝑧 where 33 𝐵𝑂2 is the element of mass moment of inertia for ence frame 1 1 1: the crane boom 𝐵𝑂2 in inertial fixed on earth reference frame 2 2 𝑥1 𝑥2𝑦2𝑧2 with respect to unit vector k and 𝑚(𝑥1 +(𝑅+𝑦1) ) is 𝑄𝑥 =−𝑁( ), 1 𝑙 the element of mass moment of inertia for the payload 𝑀 in inertial fixed on earth reference frame 𝑥2𝑦2𝑧2 with respect to 𝑦1 𝑄𝑦 =−𝑁( ), unit vector k. 1 𝑙 (11) 𝑂 The external moment of gravitational force M 2 (mg)=0 ↑↓ 𝑙−𝑧1 in (4)and(7)becausemg k. 𝑄𝑧 =+𝑁( )−𝑚𝑔. For the system “crane boom-payload,” the cable reaction 1 𝑙 force N istheinternalforce.Soin(4)and(7), we have 𝑂2 Taking into account (9), (11), the equations (10)inthe M (N)=0. noninertial reference frame 𝑥1𝑦1𝑧1 will finally take the Substitution of (5), (6), and (7)into(4)yieldsthe following form: following scalar equation for the rate of change of moment 𝑂 2 2 2 2 of momentum 𝐻3 for the system “crane boom 𝐵𝑂2-load 𝑑 𝑥 𝑑𝜑 𝑑 𝜑 𝑀 𝑂 𝑚( 1 )−𝑚( 𝑒 ) 𝑥 −𝑚( 𝑒 )(𝑅+𝑦 ) ” with respect to point 2 in the inertial reference frame 𝑑𝑡2 𝑑𝑡 1 𝑑𝑡2 1 𝑥2𝑦2𝑧2: 𝑑𝜑 𝑑𝑦 𝑥 𝑑 𝑑𝜑 −2𝑚( 𝑒 )( 1 )=−𝑁( 1 ), 𝑂2 2 2 𝑒 (((𝐼33 ) +𝑚(𝑥1 +(𝑅+𝑦1) )) ( )) 𝑑𝑡 𝑑𝑡 𝑙 𝑑𝑡 𝐵𝑂2 𝑑𝑡 (8) 2 2 2 𝑂 𝑂 𝑑 𝑦1 𝑑𝜑𝑒 𝑑 𝜑𝑒 =𝑀 2 −𝑀 2 , 𝑚( )−𝑚( ) (𝑅 + 𝑦 )+𝑚( )𝑥 𝐷𝑇 𝐹𝑇 𝑑𝑡2 𝑑𝑡 1 𝑑𝑡2 1

𝑂 𝑂 𝑀 2 𝑀 2 𝑑𝜑 𝑑𝑥 𝑦 where driving 𝐷𝑇 and frictional 𝐹𝑇 torques are the tech- +2𝑚( 𝑒 )( 1 )=−𝑁( 1 ), nically defined functions for specific electric drive systems. 𝑑𝑡 𝑑𝑡 𝑙 The Scientific World Journal 5

𝑑2𝑧 𝑙−𝑧 𝑀 𝑚( 1 )=−𝑚𝑔+𝑁( 1 ), Relative swaying paths of payload in the plane of 𝑑𝑡2 𝑙 the introduced relative coordinates 𝑥 and 𝑦 are shown in Figure 3 for successive increasing swaying times 𝑡1 = 1.25 s 2 2 2 2 𝑙 =𝑥1 +𝑦1 +(𝑧1 −𝑙) . (Figure 3(a)), 𝑡2 =3s(Figure3(b)), 𝑡3 =10s(Figure3(c)), 𝑡 =25 (12) and 4 s(Figure3(d)). Then, we have the normal system of two linear homo- After two times differentiation of forth equations in geneous differential equations of second order for relative system (12), we will have the following formula for the cable motion of the load 𝑀: 𝑁 reaction force (Figure 2) from third equation of (12): 2 2 𝑔 𝐷 −(𝜔 −( )) −2𝜔𝑒𝐷 2 2 𝑒 𝑙 𝑑 𝑥 𝑑𝑥 ( 𝑔 ) 𝑁 = ((𝑚𝑙 [((𝑥 ( 1 )+( 1 ) 2 2 1 2 2𝜔 𝐷𝐷−(𝜔 −( )) 𝑑𝑡 𝑑𝑡 𝑒 𝑒 𝑙 (16) 𝑑2𝑦 𝑑𝑦 2 𝑥 (𝑡) 𝑑 +𝑦 ( 1 )+( 1 ) ) ×( )=0, 𝐷= . 1 𝑑𝑡2 𝑑𝑡 𝑦 (𝑡) 𝑑𝑡

×(𝑙2 −𝑥2 −𝑦2) The determinant of natural frequencies matrix for the 1 1 system (16)hasthefollowingform: 𝑑𝑥 𝑑𝑦 2 (13) 1 1 󵄨 2 𝑔 2 󵄨 +(𝑥1 ( )+𝑦1 ( )) )]) 󵄨𝜆 +( )−𝜔 −2 ⋅ 𝜔 ⋅𝜆 󵄨 𝑑𝑡 𝑑𝑡 󵄨 𝑙 𝑒 𝑒 󵄨 󵄨 󵄨 =0. (17) 󵄨 2 𝑔 2󵄨 󵄨 2⋅𝜔𝑒 ⋅𝜆 𝜆 +( )−𝜔𝑒 󵄨 2 2 2 −2 󵄨 𝑙 󵄨 ×(𝑙 −𝑥1 −𝑦1 ) ) Using (17), we adjust the roots 𝜆1 and 𝜆2 of secular 𝑚𝑔𝑙 equation: + . √ 2 2 2 𝜆 =±] ⋅𝑖, 𝜆 =±] ⋅𝑖, ] =𝑘+𝜔, (𝑙 −𝑥1 −𝑦1 ) 1 1 2 2 1 𝑒 (18) So with an introduction of (13)tothefirstandsecond 𝑔 ]2 =𝑘−𝜔𝑒,𝑘=√ ,𝜔𝑒 =𝑘.̸ equations of system (12), we will have the explicit nonlinear 𝑙 ODE system for payload 𝑀 swaying in Cartesian coordinates. The initial conditions for the problem are as follows:

5. Linearized Mathematical Model 𝑥 (0) =0, 𝐷𝑥(0) =+𝑉𝐵 =+𝜔𝑒𝑅, (19) 𝛼 𝑧 𝑦 (0) =−𝑦 ,𝐷𝑦(0) =0. Further for the small angle 1, we may assume that 1 and dyn 2 2 𝑑 𝑧1/𝑑𝑡 are the small parameters. So the nonlinear equation (13)yields Taking into account (18)and(19), the law of relative motion for payload takes the following form: 𝑁≈𝑚𝑔. (14) 𝑖 −] 𝑖𝑡 ] 𝑖𝑡 𝑖 −] 𝑖𝑡 ] 𝑖𝑡 𝑥 (𝑡) =𝐴 (𝑒 1 −𝑒 1 )+𝐴 (𝑒 2 −𝑒 2 ), Then, taking into account14 ( ), the linearized system (12) 1 2 2 2 for the two variables 𝑥1 and 𝑦1 takes the following form: (20) 1 −] 𝑖𝑡 ] 𝑖𝑡 1 −] 𝑖𝑡 ] 𝑖𝑡 𝑦 (𝑡) =𝐴 (𝑒 1 +𝑒 1 )−𝐴 (𝑒 2 +𝑒 2 ). 𝑑2𝑥 𝑑𝜑 2 𝑑𝜑 𝑑𝑦 𝑥 1 2 2 2 1 −( 𝑒 ) 𝑥 −2( 𝑒 )( 1 )=−𝑔( 1 ), 𝑑𝑡2 𝑑𝑡 1 𝑑𝑡 𝑑𝑡 𝑙 After substitution of values of 𝐴1 and 𝐴2 in (20), we have 𝑀 𝑑2𝑦 𝑑𝜑 2 𝑑𝜑 𝑑𝑥 𝑦 the following law of payload relative swaying motion with 1 −( 𝑒 ) (𝑅 + 𝑦 )+2( 𝑒 )( 1 )=−𝑔( 1 ). the uniformly rotating pivot center: 𝑑𝑡2 𝑑𝑡 1 𝑑𝑡 𝑑𝑡 𝑙 𝑉 −𝑦 (𝑘 − 𝜔 ) (15) 𝐵 𝑒 𝑖 −] 𝑖𝑡 ] 𝑖𝑡 𝑥 (𝑡) =( dyn ) (𝑒 1 −𝑒 1 ) 2𝑘 2 So we derive the system (15) of differential equations 𝑀 𝐵𝑀 forrelativemotionofload on cable with a movable 𝑉 +𝑦 (𝑘 + 𝜔 ) 𝐵 𝑒 𝑖 −] 𝑖𝑡 ] 𝑖𝑡 suspension center 𝐵,whichisfixedatthecraneboom +( dyn ) (𝑒 2 −𝑒 2 ), 2𝑘 2 𝐵𝑂2. We assume that the crane boom 𝐵𝑂2 rotates with a constant angular velocity 𝜔𝑒 around vertical axis 𝐷𝑂2.We (21) 𝑉 −𝑦 (𝑘 − 𝜔 ) 𝑂𝑥𝑦 𝐵 𝑒 1 −] 𝑖𝑡 ] 𝑖𝑡 then transfer the origin of coordinate system (Figure 1) 𝑦 (𝑡) =( dyn ) (𝑒 1 +𝑒 1 ) 𝐴 𝑀 2𝑘 2 to the point dyn of dynamic equilibrium for load and assume that 𝑥1 =𝑥; 𝑦1 =𝑦+𝑦 (Figure 3). Then, dyn 𝑉 +𝑦 (𝑘 + 𝜔 ) 𝐵 𝑒 1 −] 𝑖𝑡 ] 𝑖𝑡 the second equation of system (15) defines the amount of −( dyn ) (𝑒 2 +𝑒 2 ). 𝑦 =𝐴 𝐴 =(𝜔2𝑅𝑙)/(𝑔2 −𝜔 𝑙) dynamic deflection dyn st dyn 𝑒 𝑒 . 2𝑘 2 6 The Scientific World Journal

1.012

1.02 1.010

) 1.008 ) N/mg

N/mg 1.006 1.01 1.004

1.002

1 1

0.998 Dimensionless cable reaction ( reaction cable Dimensionless Dimensionless cable reaction ( reaction cable Dimensionless 0.996 0.99 0.994 0246810 0246810 t (s) t (s) (m); (rad/s); (rad/s) (m); (rad/s); (rad/s) l ≈ 0.206 𝜔e ≈ 0.449 k≈6.9 l ≈ 0.412 𝜔e ≈ 0.449 k ≈ 4.88 (a) (b)

1.004 1.0012 ) 1.003 1.0010 N/mg ) 1.0008

1.002 N/mg 1.0006

1.0004 1.001 1.0002

1 1 Dimensionless cable reaction ( reaction cable Dimensionless 0.9998

0.999 ( reaction cable Dimensionless 0.9996

0.9994

0246810 0246810 t (s) t (s) (m); (rad/s); (rad/s) (m); (rad/s); (rad/s) l ≈ 0.618 𝜔e ≈ 0.314 k≈3.984 l ≈ 0.825 𝜔e ≈ 0.209 k≈3.448 (c) (d)

Figure 2: Time dependencies of dimensionless cable reactions 𝑁/𝑚𝑔 in the first approximation for fixed cable lengths 𝑙 = 0.206 m(a); 𝑙 = 0.412 m(b);𝑙 = 0.618 m(c);and𝑙 = 0.825 m (d).

The relative trajectories for load 𝑀 on the cable in Figure 3 𝑘+𝜔𝑒 = 3.658 rad/s; ]2 =𝑘−𝜔𝑒 = 3.239 rad/s have been derived for the following numerical values: 𝑅= (Figure 3). 2 0,5 0.492 m; 𝑔 = 9.81 m/s ; 𝑙 = 0.825 m; 𝑘 = (𝑔/𝑙) ≈ Numerical plots in Figure 3 show the payload 𝑀 swaying 3.448 𝑇=30 𝜔 = 2𝜋/𝑇 ≈ 0.209 𝛼 = (𝑂𝑥𝑦𝑧) rad/s; s; 𝑒 rad/s; 1dyn in the noninertial reference frame . In order to 0.00221 𝑉 = 0.103 𝑦 = 0.00182 ] = 𝑀 rad; 𝐵 m/s; dyn m; 1 visualize payload absolute swaying path (Figures 4 and 5) The Scientific World Journal 7

Relative trajectory of load M swaying Relative trajectory of load M swaying for linearized model for linearized model

t2 = 3s t1 = 1.25s

y(x): l = 0.825 (m); k≈3.448(rad/s); 𝜔e ≈ 0.209 (rad/s) y(x): l = 0.825 (m); k≈3.448(rad/s); 𝜔e ≈ 0.209 (rad/s) (a) (b)

Relative trajectory of load M swaying Relative trajectory of load M swaying for linearized model for linearized model t4 = 25s t3 = 10 s

y(x): l = 0.825 (m); k≈3.448(rad/s); 𝜔e ≈ 0.209 (rad/s) y(x): l = 0.825 (m); k≈3.448(rad/s); 𝜔e ≈ 0.209 (rad/s) (c) (d)

Figure 3: The relative trajectories for load 𝑀 on the cable during uniform rotational motion of the crane boom 𝐵𝑂2 for the moments of time 𝑡1 = 1.25 s(a),𝑡2 =3s(b),𝑡3 =10s(c),and𝑡4 =25s (d).

in the fixed inertial reference frame (𝑂2𝑥2𝑦2𝑧2),wewill 6. Comparison of Derived and address the following transition formula: Known Published Results

𝑥2 (𝑡) sin (𝜔𝑒𝑡) cos (𝜔𝑒𝑡) 𝑥 (𝑡) ( )=( )( ). The computational dimensionless cable force 𝑁/𝑚𝑔, derived 𝑦2 (𝑡) − cos (𝜔𝑒𝑡) sin (𝜔𝑒𝑡) 𝑦 (𝑡) +𝑅+𝑦 dyn in the first approximation and shown in Figure 2, outlines (22) the minor change of the cable tension force in the vicinity Derived computational plot for absolute payload swaying of the ordinate (𝑁/𝑚𝑔): =1 (𝑁/𝑚𝑔) ∈ [0.9875; 1.0233] path in the absolute coordinates’ plane (𝑥2𝑦2) is shown in for 𝑙 = 0.206 minFigure2(a); (𝑁/𝑚𝑔) ∈ [0.994; 1.0122] Figure 5(a) for load 𝑀 swaying time 𝑡1 =15sandcraneboom for 𝑙 = 0.412 minFigure2(b); (𝑁/𝑚𝑔) ∈ [0.998; 1.004] for ∘ transport slewing angle 𝜑𝑒 = 180 .InFigure5(b),weshowthe 𝑙 = 0.618 minFigure2(c);and(𝑁/𝑚𝑔) ∈ [0.99933; 1.0013] experimental plot for absolute payload swaying path in the for 𝑙 = 0.825 minFigure2(d). So the assumption (14) absolute coordinates’ plane (𝑥2𝑦2) for load 𝑀 swaying time really takes place as shown in Figure 2.Thesameeffectfor 𝑡 =15 𝜑 = 180∘ (𝑁/𝑚𝑔) ∈ [0.94; 1.04] 2 s and crane boom transport slewing angle 𝑒 , Mitrev has been derived by Mitrev and which was built with an introduction of experimental setup Grigorov, [23, Mitrev’s Figure 7 in page 89], that confirms in Figure 4. assumption (14). An analogous effect for dimensionless cable 8 The Scientific World Journal

𝑙 𝑦 and with increased cable length .The dyn influence on the values of 𝑥max and 𝑦max is negligible. Equation (18) shows that the free oscillation frequencies ]1 and ]2 depend on the boom slewing velocity 𝜔𝑒 and Camera natural frequency 𝑘 of the payload with a fixed pivot point 𝐵. The frequencies ]1 and ]2 essentially depend on the pendulum length 𝑙 because 𝑘≫𝜔𝑒. Kinematically saying, the miscoordination of frequencies ]1 and ]2 is ]1 − ]2 = 2𝜔𝑒 and is determined by the crane boom slewing velocity 𝜔𝑒. Dynamically saying, the value of miscoordination is governed by the Coriolis acceleration-induced D’Alembert’s Laser pointer Bearing inertia force. ComputationsshowninFigure3 for relative payload Boom 𝑀 swaying show that the clockwise direction of rotation B for relative oscillations is opposite in direction from the D counterclockwise direction of rotation for the crane boom. This effect follows from the conservation of position of the O plane 𝑦2𝑧2 of oscillations of load 𝑀 in the fixed inertial 2 Slewing axis coordinate system 𝑥2𝑦2𝑧2. (1) Lower plate The comparative analysis of computational and exper- Cable imental (2) absolute payload trajectories is shown in Fig- ure 5(c). The quantity of the local maximums in Figure 5(a) is M 8 for theoretical curve 1. So the theoretical period for relative 𝑇 = 1.87 load swaying is theor s. The quantity of the local maximums in Figure 5(b) is 7.5 for experimental curve 2. The Light-emitting diode 𝑇 = period for experimental curve 2 in Figure 5(b) is experim Regular grid 1200 mm ×700mm 2 with square cells 200 mm ×20mm s. The discrepancy between the values of theoretical and experimental periods does not exceed 7%. The experimental absolute trajectory (2) in Figures 5(b) and 5(c) outlines the additional relative swaying of payload 𝑀 Figure 4: The experimental measurement system for payload 𝑀 after crane boom stop, which has the elliptical form in the vicinity of the point (0, −0.5) in the plane 𝑥2𝑦2 for 𝜑𝑒 = swaying during crane boom slewing. ∘ 180 . The remaining payload motions after crane boom stop are the small oscillations of the payload with the fixed pivot center. (𝑁/𝑚𝑔) ∈ [0.996; 1.005] force Maczynski has been derived by Maczynski and Wojciech, [24, Maczynski’s Figure 10 in page 8. Final Conclusions 278], which also reaffirms the assumption14 ( ). In published work by Sakawa et al. ([25], Figure 3 at page The closed system of nonlinear ODEs for payload relative 552, Figure 4 at page 552, Figure 5 at page 554, and Figure 6 swaying has been derived in relative Cartesian coordinates at page 554), the small Sakawa values 𝑑𝑧/𝑑𝑡 ≤0.1 (Figures 3 with an introduction of Lagrange equations and a geometric and 4) and 𝑑𝑧/𝑑𝑡 ≤ 0.05 (Figures 5 and 6) of 𝑧-projections constraint equation. for payload velocities have been shown, which confirms the small angle assumption 𝑁≈𝑚𝑔(14). The explicit form of the dependence of the cable reaction forceasafunctionoftherelativepayloadcoordinates𝑥1, 𝑦1, 𝑧1 and their first and second time derivatives has been found. 7. Discussion and Analysis of Basic Results for Derived complex dependency of the cable reaction force on Mathematical and Physical Simulation the above mentioned variables determines the nonlinearity of the present problem. The relative trajectories for load 𝑀 swaying in Figure 3 show equality of maximum values for relative 𝑥 and 𝑦 payload The deviation of cable reaction force from payload gravity coordinates: force is small. This fact allowed the derivation of a linearized ODE system for payload swaying in relative Cartesian coor- 𝜔 (𝑅 + 𝑦 ) dinates and verified the small relative Cartesian coordinates’ 𝑥 =𝑦 =𝐴 +𝐴 = 𝑒 dyn . (23) max max 1 2 𝑘 assumption. The analytical expressions for relative payload path have been derived. The formula (23)outlinesthatvalues𝑥max and 𝑦max in Derived results have good agreement with experiments Figure 3 increase with growth of boom slewing velocity 𝜔𝑒 and known published data. The Scientific World Journal 9

y (m) t=15s y2 (m) t=18s 2 0.5 0.5 3 (Point B) 0.4 0.4

0.3 0.3 1 2 0.2 (Point M) 0.2 4 (Point M)

0.1 PointPin O 0.1 Point O2 Rotational direction 2 x (m) (Point D) 2 (Point D) x2 (m) 0 0 2R = 1 (m) 2R = 1 (m)

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 (a) (b)

y2(m) 0.5

0.4

0.3 1 2 0.2 (Point M) (Point M) 4 0.1 Point O2 (Point D) x2(m) 0 2R = 1 (m)

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 (c)

Figure 5: Calculated 1 (a, c) and experimental 2 (b, c) absolute trajectories for load 𝑀, where 3 is the trajectory of boom point 𝐵 (b); 4 is the center 𝑂2 (D) of boom rotation (b, c).

Disclosure [4] P. Betsch, M. Quasem, and S. Uhlar, “Numerical integration of discrete mechanical systems with mixed holonomic and control The submission of the authors’ paper implies that it has not constraints,” Journal of Mechanical Science and Technology,vol. been previously published, that it is not under consideration 23,no.4,pp.1012–1018,2009. for publication elsewhere, and that it will not be published [5] W. Blajer and K. Kołodziejczyk, “A geometric approach to elsewhere in the same form without the written permission solving problems of control constraints: theory and a DAE of the editors. framework,” Multibody System Dynamics,vol.11,no.4,pp.343– 364, 2004. Conflict of Interests [6] W. Blajer and K. Kołodziejczyk, “Control of underactuated mechanical systems with servo-constraints,” Nonlinear Dynam- The authors Alexander V. Perig, Alexander N. Stadnik, and ics,vol.50,no.4,pp.781–791,2007. Alexander I. Deriglazov declare that there is no conflict of [7] W. Blajer and K. Kołodziejczyk, “Modeling of underactuated interests regarding the publication of this paper. mechanical systems in partly specified motion,” Journal of Theoretical and Applied Mechanics,vol.46,no.2,pp.383–394, 2008. References [8] W. Blajer and K. Kołodziejczyk, “The use of dependent coordinates in modeling and simulation of cranes executing a [1] E. M. Abdel-Rahman and A. H. Nayfeh, “Pendulation reduction load prescribed motion,” The Archive of Mechanical Engineer- in boom cranes using cable length manipulation,” Nonlinear ing, vol. 3, pp. 209–222, 2009, http://ame.meil.pw.edu.pl/index Dynamics,vol.27,no.3,pp.255–269,2002. .php?/eng/content/download/373/1748/file/art 2009 3 02.pdf. [2] W. Adamiec-Wojcik´ I, P. Fałat, A. Maczynski,´ and S. Wojciech, [9] W.Blajer, A. Czaplicki, K. Dziewiecki, and Z. Mazur, “Influence “Load stabilisation in an A-frame—a type of an offshore crane,” of selected modeling and computational issues on muscle force The Archive of Mechanical Engineering,vol.56,no.1,pp.37–59, estimates,” Multibody System Dynamics,vol.24,no.4,pp.473– 2009. 492, 2010. [3] P. J. Aston, “Bifurcations of the horizontally forced spherical [10] W. Blajer and K. Kołodziejczyk, “Improved DAE formulation pendulum,” Methods in Applied Mechanics and Engineering,vol. for inverse dynamics simulation of cranes,” Multibody System 170, no. 3-4, pp. 343–3353, 1999. Dynamics,vol.25,no.2,pp.131–143,2011. 10 The Scientific World Journal

[11] J. H. Cha, S. H. Ham, K. Y. Lee, and M. I. Roh, “Application of a topological modelling approach of multi-body system dynamics to simulation of multi-floating cranes in shipyards,” Proceedings of the Institution of Mechanical Engineers K,vol.224, no. 4, pp. 365–373, 2010. [12] K. Ellermann, E. Kreuzer, and M. Markiewicz, “Nonlinear dynamics of floating cranes,” Nonlinear Dynamics,vol.27,no. 2, pp. 107–183, 2002. [13] T. Erneux and T. Kalmar-Nagy,´ “Nonlinear stability of a delayed feedback controlled container crane,” Journal of Vibration and Control,vol.13,no.5,pp.603–616,2007. [14] G. Glossiotis and I. Antoniadis, “Payload sway suppression in rotary cranes by digital filtering of the commanded inputs,” Proceedings of the Institution of Mechanical Engineers K,vol.217, no.2,pp.99–109,2003. [15] R. A. Ibrahim, “Excitation-induced stability and phase transition: a review,” Journal of Vibration and Control,vol.12,no.10, pp. 1093–1170, 2006. [16] J. Kortelainen and A. Mikkola, “Semantic data model in multibody system simulation,” Proceedings of the Institution of Mechanical Engineers K,vol.224,no.4,pp.341–352,2010. [17] R. Mitrev, “Mathematical modelling of translational motion of rail-guided cart with suspended payload,” Journal of Zhejiang University Science A,vol.8,no.9,pp.1395–1400,2007. [18] B. Grigorov and R. Mitrev, “A Newton—Euler approach to the freely suspended load swinging problem,” Recent Journal,vol. 10, no. 26, pp. 109–112, 2009. [19] M. Osinski´ and S. Wojciech, “Application of nonlinear optimisation methods to input shaping of the hoist drive of an off-shore crane,” Nonlinear Dynamics, vol. 17, no. 4, pp. 369–386, 1998. [20] H. L. Ren, X. L. Wang, Y.J. Hu, and C. G. Li, “Dynamic response analysis of a moored crane-ship with a flexible boom,” Journal of Zhejiang University Science A,vol.9,no.1,pp.26–31,2008. [21] H. Schaub, “Rate-based ship-mounted crane payload pendulation control system,” Control Engineering Practice,vol.16,no.1, pp.132–145,2008. [22] N. Uchiyama, “Robust control for overhead cranes by partial state feedback,” Proceedings of the Institution of Mechanical Engineers I,vol.223,no.4,pp.575–580,2009. [23] R. Mitrev and B. Grigorov, “Dynamic behaviour of a spherical pendulum with spatially moving pivot,” Zeszyty Naukowe Politechniki Poznanskiej,´ Budowa Maszyn i Zarządzanie Pro- dukcją,vol.9,pp.81–91,2008. [24] A. Maczynski and S. Wojciech, “Dynamics of a mobile crane and optimisation of the slewing motion of its upper structure,” Nonlinear Dynamics,vol.32,no.3,pp.259–290,2003. [25] Y. Sakawa, Y. Shindo, and Y. Hashimoto, “Optimal control of a rotary crane,” Journal of Optimization Theory and Applications, vol. 35, no. 4, pp. 535–557, 1981. International Journal of

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