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Research Article 3 DOF Spherical Pendulum Oscillations with a Uniform Slewing Pivot Center and a Small Angle Assumption

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Research Article 3 DOF Spherical Pendulum Oscillations with a Uniform Slewing Pivot Center and a Small Angle Assumption Hindawi Publishing Corporation Shock and Vibration Volume 2014, Article ID 203709, 32 pages http://dx.doi.org/10.1155/2014/203709 Research Article 3 DOF Spherical Pendulum Oscillations with a Uniform Slewing Pivot Center and a Small Angle Assumption Alexander V. Perig,1 Alexander N. Stadnik,2 Alexander I. Deriglazov,2 and Sergey V. Podlesny2 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, Shkadinova72,DonetskRegion,Kramatorsk84313,Ukraine Correspondence should be addressed to Alexander V. Perig; [email protected] Received 11 February 2014; Revised 7 May 2014; Accepted 21 May 2014; Published 4 August 2014 Academic Editor: Mickael¨ Lallart 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 addresses the derivation of a 3 DOF mathematical model of a spherical pendulum attached to a crane boom tip for uniform slewing motion of the crane. The governing nonlinear DAE-based system for crane boom uniform slewing has been proposed, numerically solved, and experimentally verified. The proposed nonlinear and linearized models have been derived with an introduction of Cartesian coordinates. The linearizedodel m with small angle assumption has an analytical solution. The relative and absolute payload trajectories have been derived. The amplitudes of load oscillations, which depend on computed initial conditions, have been estimated. The dependence of natural frequencies on the transport inertia forces and gravity forces has been computed. The conservative system, which contains first time derivatives of coordinates without oscillation damping, has been derived. The dynamic analogy between crane boom-driven payload swaying motion and Foucault’s pendulum motion has been grounded and outlined. For a small swaying angle, good agreement between theoretical and averaged experimental results was obtained. 1. Introduction Betsch et al. [8], Blackburn et al. [9, 10], Blajer et al. [11– 17], Cha et al. [18], Chin et al. [19], Ellermann et al. [20], Payload swaying dynamics during crane boom slewing is Erneux and Kalmar-Nagy´ [21, 22], Ghigliazza and Holmes within the objectives and scope of many academic and [23], Glossiotis and Antoniadis [24], Grigorov and Mitrev industrial research programs in the fields of mechanical, elec- [25], Gusev and Vinogradov [26], Hong and Ngo [27], trical, and control engineering, and theoretical mechanics. Hoon et al. [28], Ibrahim [29], Jerman et al. [30–33], Ju et al. Mathematical descriptions of relative and absolute payload [34], Kłosinski´ [35], Krukowski et al. [36, 37], Lenci et al. swaying motion during crane boom rotation require the [38], Leung and Kuang [39], Loveykin et al. [40], Maleki introduction of design models for a spherical pendulum with et al. [41], Maczynski et al. [42–44], Marinovicetal.[´ 45], a suspension point following a horizontal circular trajectory. Masoud et al. [46, 47], Mijailovic[´ 48], Mitrev and Grigorov Spherical pendula with moving pivot centers are the so-called [49, 50], Morales et al. [51], Nakazono et al. [52], Neitzel et al. “eternal” problems that have been posed by ancient builders [53], Neupert et al. [54], O’Connor and Habibi [55], Omar and civil engineers. and Nayfeh [56], Osinski´ and Wojciech [57], F. Palis and S. Today payload swaying problems attract the great atten- Palis [58], Perig et al. [59], Posiadała et al. [60], Ren et al. [61], tion of such applied mathematicians and mechanical engi- Safarzadeh et al. [62], Sakawa et al. [63], Sawodny et al. [64], neers as Abdel-Rahman et al. [1–3], Adamiec-Wojcik´ et al. Spathopoulos et al. [65], Schaub [66], Solarz and Tora [67], [4], Al-mousa et al. [5], Allan and Townsend [6], Aston [7], Uchiyama et al. [68], Urba´s[69], and others. 2 Shock and Vibration Applied engineering problems in the field of lifting-and- introduction of angular generalized coordinates which result transport machines mainly deal with rectilinear or rotational in nonlinearity of the problems and require a fourth-order motion of the spherical pendulum suspension center in deter- Runge-Kutta fixed-step integration method. mined and stochastic cases. Further improvement of rotating However the previously known studies [1–100]havegiven crane performance and efficiency requires the development inadequate attention to the dynamic analysis of a load sway- of mathematical models for an adequate description of pay- ing in the horizontal plane of vibrations while accounting for load and crane boom tip positions. Modern computational the effect of the Coriolis force on the trajectory of the relative approaches to the solution of payload positioning problems load motion of the cable. The present research addresses this have been investigated in the following works. situation. Abdel-Rahman et al. [1–3] have provided a comprehen- It should be noted that spherical pendulum related sive review of different types of cranes, the essential and research has also been further developed for Foucault pendu- widely used mathematical techniques for models of crane lums in the works of Condurache and Martinusi [77], Gusev dynamics and classical control methods [1–3]. However, this and Vinogradov [26], de Icaza-Herrera and Castono˜ [83], research [1–3] gives inadequate attention to the effects of Pardy [88], Zanzottera et al. [99], Zhuravlev and Petrov [100], Coriolis inertia forces on the payload relative motion during and others. crane boom slew motion. At first sight, the spherical pendulum with rotating Blajer et al. [11–17] have proposed thirteen index-three pivot center and Foucault pendulum are two vastly differ- differential-algebraic equations (DAE) in the rotary crane state variables and control variables [11–17]. They have pro- ent dynamic systems. The key difference between the two posed governing DAE equations and stable solution tech- dynamic systems is that dynamic analysis of the Foucault niques allowing the rotary crane to execute the prescribed pendulumisfocusedonaloadswayinginthefieldofthe load trajectory and the control commands to implement central gravity force, while crane boom slewing problems are feed-forward control. In this approach [11–17]thegoverning posed for the vertical gravity force. A commonality of the equations have been derived without consideration of the two dynamic systems is, in both cases, the effect of influence Coriolis inertia force and non-zero horizontal projections of of normal centrifugal and Coriolis forces on the shape of theloadgravityforce. the relative and absolute trajectories. The normal centrifugal Hairer et al. [70, 71] and Øksendal [72]havedeveloped forces depend on the relative coordinates and Coriolis forces computational techniques for DAE systems numerical inte- depend on the relative velocity of the swaying load. Moreover, gration. the appearance of Coriolis forces, that are dependent on Jerman et al. [30–33] have applied an enhanced mathe- relative payload velocity, retains the dynamic system as a matical model of slewing the crane motion with load pendu- conservative one because Coriolis acceleration remains at all lation, taking into account system stiffness coefficients30 [ – times perpendicular to the relative velocity of the load. It 33]. All mathematical techniques and governing equations in may therefore be concluded that there is a close coupling these works [30–33] have been presented in implicit form. between the spherical pendulum with rotating pivot center Juetal.haveimplementedfiniteelementsimulationfor and Foucault pendulum. Moreover the close relationship a flexible crane structure with a spherical pendulum34 [ ]. In between the two dynamic systems has not been properly this work [34], the spherical pendulum excitation is induced addressed in all previous known research, which emphasizes by vibration modes of the flexible crane structure. theactualityandrelevanceofthepresentpaper. Loveykin et al. have derived the law of an optimal control The present paper is focused on the study of the oscilla- for lifting-and-shifting machines under the assumption of tion processes taking place in the vicinity of a steady equi- minimization of quadratic performance criteria in the case librium position of a payload during crane boom uniform for two-phase coordinates, control and control rate [40]. slewing. The computational approach is based on the solution This work [40] made wide use of variation optimization of the initial value problem of particle dynamics for the techniques for pendulum oscillations in the vertical plane, determination of the relative load trajectory in the horizontal which involve the trolleys of crane frames with rectilinear plane of the vibrations during crane boom uniform rotation. motion. The approach used here takes into account both the relative Maczynski et al. have applied a numerically based finite rotation of the load vibrations in the vertical plane and the element method (FEM) approach for simulation of a “crane boom-payload” system without an explicit introduction of influence of Coriolis acceleration on the form of the trajectory inertia forces [42–44]. Optimization problem of load posi- of the swaying
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