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 2-load ”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 , ,and of the load and the angle = of crane pendulum container crane model with linear transport trol- boom 2 uniform rotation in the horizontal plane () ley motion. This model illustrates linear displacement of the with the constant angular velocity around the vertical planependulumsuspensioncenter. axis 22. We denote the fixed inertial coordinate system Glossiotis and Antoniadis [14]havederivedafour- as 222 and the moving noninertial frame of reference as degrees-of-freedom (4DOF) model for the rotary crane 111, which is rigidly bounded with the crane boom 2. system, incorporating both hoisting and slewing payload Rotation of the moving noninertial frame of reference 111 motions and thus is able to handle cases where hoisting can around the fixed inertial coordinate system 222 defines be simultaneously applied to the rotary motion. the transportation motion. The motion of load relative to Ibrahim [15] has studied the issues of multiplicative noise the moving noninertial frame of reference 111 defines the and noise-enhanced stability of spherical pendulums. relative motion. Kortelainen and Mikkola [16]haveintroduceddataman- The scalar of the load transportation velocity is defined agement techniques within