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 cargo relative motion. This paper is also tioning in this study [42–44] has not fully addressed the aimed at addressing the physically grounded interrelations natural frequencies estimation for the system “crane boom- between the spherical pendulum with rotating pivot center payload” in the case of a fully rigid crane boom model. and Foucault pendulum. Mitrev and Grigorov [49, 50] have derived governing equations for load relative swaying taking into account 2. DAE System for Payload Swaying energy dissipation, centrifugal, Coriolis, inertia, and gravity forces [49, 50]. The Lagrange equations used here allow the It has been shown in Appendices A–O that the nonlinear simulation of a spherical pendulum with a movable pivot DAE system (formulae (C.1), (F.7)) for the relative coordi- center [49, 50]. Mitrev’s approach [49, 50]isbasedonthe nates 𝑥1(𝑡) (Figure 1(a)); 𝑦1(𝑡) (Figure 1(b)); 𝑧1(𝑡) of a swaying Shock and Vibration 3

0.03

0.02 0.02

0.01 0.01

x1 0 y1 5 10 15 0 5 10 15 t t −0.01 −0.01

−0.02 −0.02

(a) (b)

0.5

0.4 0.4 0.3

0.2 0.3 0.1 y2 x2 0 0.2 5 10 15 −0.1 t

−0.2 0.1

−0.3 0 −0.4 51510 t (c) (d)

0.03 y1

0.02 0.5 y1

0.01 0.4 0.3 x1 y2 −0.02 −0.01 0 0.01 0.02 0.2 x 1 0.1 −0.01

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 −0.02 x2

(0.492 + y1)∗cos(𝜑) + x1 ∗ sin(𝜑) (e) (f)

Figure 1: Computational relative (a-b, e) and computational absolute (c-d, f) coordinates of a swaying payload during uniform crane boom 2 slewing for the following numerical values of system parameters: 𝑚 = 0.1 kg; 𝑙 = 0.825 m; 𝑅 = 0.492 m; 𝑔 = 9.81 m/s ;and𝜔𝑒 = 0.209 rad/s. 4 Shock and Vibration payload during crane boom slewing (Figures 8–10)inthe The computational results in Figure 1, derived for DAE noninertial reference frame B is as follows: problem (1)-(2) solution, coincide with the linearized solu- tion of the payload swaying problem in Appendices A–I 𝑑2 (𝑥 (𝑡)) 𝑚( 1 ) (formula (H.1) in the noninertial reference frame B). 𝑑𝑡2 It is necessary to note that the posed DAE problem (1) describes payload motion in the vicinity of the lower position 𝑥 (𝑡) 𝑑(𝜑 (𝑡)) 2 =−𝑁(𝑡) ( 1 )+𝑚( 𝑒 ) 𝑥 (𝑡) of stable dynamical equilibrium assuming rope tension force 𝑙 𝑑𝑡 1 𝑁(𝑡) >0.Factually,theappliedropeisassumedasaunilateral geometric constraint (Appendix C) in the shape of a torsion 2 𝑑 (𝜑 (𝑡)) fiber. The upper position of the mechanical system is unstable +𝑚( 𝑒 )(𝑅+𝑦 (𝑡)) 𝑑𝑡2 1 (see [73–75, 78–80, 91, 93]) and corresponds to the case of 𝑁(𝑡) <0 within the parameters of the chosen mechanical 𝑑(𝜑 (𝑡)) 𝑑(𝑦 (𝑡)) system in Figures 8–10. Such an upper pendulum position +2𝑚( 𝑒 )( 1 ); 𝑑𝑡 𝑑𝑡 might conflict with the hypothesis about the unilateral nature of geometric constraint. All of the above mentioned upper 𝑑2 (𝑦 (𝑡)) pendulum position conditions are outside the objectives and 𝑚( 1 ) 𝑑𝑡2 scope of the present paper.

𝑦 (𝑡) 𝑑(𝜑 (𝑡)) 2 3. Experimental Procedure =−𝑁(𝑡) ( 1 )+𝑚( 𝑒 ) (𝑅 + 𝑦 (𝑡)) 𝑙 𝑑𝑡 1 The experimental procedure has been grounded on the usage 𝑑2 (𝜑 (𝑡)) of the assembly in Figures 2 and 3. −𝑚( 𝑒 )𝑥 (𝑡) The assembly in Figures 2 and 3 includes the following 𝑑𝑡2 1 machinecomponents:theverticalfixedshaft𝑂2𝐷 with height 𝐻=1 𝐵𝐷 𝑅 = 0.5 𝑑(𝜑 (𝑡)) 𝑑(𝑥 (𝑡)) m, the crane boom model with length m −2𝑚( 𝑒 )( 1 ); and diameter 6 mm, the cable 𝐵𝑀 with different fixed lengths 𝑑𝑡 𝑑𝑡 𝑙1 = 0.825 m(inFigure 2 and Figure 5(b)), 𝑙2 = 0.618 m(in Figure 5(a)), 𝑙3 = 0.412 m(inFigure 4(b)), and 𝑙4 = 0.206 m 𝑑2 (𝑧 (𝑡)) 𝑙−𝑧 (𝑡) 𝑚( 1 )=−𝑚𝑔+𝑁(𝑡) ( 1 ); (in Figure 4(a)). The crane boom model 𝐵𝐷 is attached to 2 𝑑𝑡 𝑙 the vertical fixed shaft 𝑂2𝐷 by bearing 𝐷.Thecable𝐵𝑀 𝐵𝐷 𝐵 2 2 2 2 is attached to the crane boom tip in the point .The 𝑙 =(𝑥1 (𝑡)) +(𝑦1 (𝑡)) +(𝑧1 (𝑡) −𝑙) ; free or running end 𝑀 of the cable 𝐵𝑀 is the payload 𝑀 𝑀 𝜑 (𝑡) =𝜔𝑡; 𝜑 (0) =0; attachment point. The load is a light emitting diode (LED) 𝑒 𝑒 𝑒 with diameter 2 mm and the battery with the battery voltage 3 V. The experimental swaying trajectory (—, 𝑦2 =𝑦2(𝑥2)) in 𝑑(𝜑𝑒 (0)) =𝜔;𝑁(0) =𝑚𝑔; Figures 2, 4,and5 is the experimental light emitting diode 𝑀 𝑑𝑡 𝑒 absolute trajectory in the inertial reference frame E. 𝑑(𝑥1 (0)) The laser pointer in Figure 2 is also attached to the crane 𝑥 (0) =0; =𝜔𝑅; 𝑦 (0) =0; 𝐵𝐷 𝐵 1 𝑑𝑡 𝑒 1 boom tip in the point .Thelaserpointeristhepartof the noninertial reference frames B and F and the pointer 𝑑(𝑦 (0)) 𝑑(𝑧 (0)) 1 =0; 𝑧 (0) =0; 1 =0. trajectoryismarkedas(---)inFigure 2.Theintroduction 𝑑𝑡 1 𝑑𝑡 of laser pointer 𝐵 allows estimation of dynamic deviation of (1) payload 𝑀. The horizontal regular fixed grid with canvas size 1200 mm × 700 mm is placed in the horizontal plane 𝑂2𝑥2𝑦2. The absolute coordinates 𝑥2(𝑡) (Figure 1(c)); 𝑦2(𝑡) The horizontal grid is formed by the square cells with size (Figure 1(d)); 𝑧2(𝑡) of payload 𝑀 in the inertial reference 20 mm × 20 mm. E frame may be computed as The experimental swaying trajectories (—, 𝑦2 =𝑦2(𝑥2)) in Figures 2, 4,and5 havebeenwrittenintheobscureroom 𝑥 (𝑡) =(𝑅+𝑦(𝑡)) (𝜑 (𝑡))+𝑥 (𝑡) (𝜑 (𝑡)); 2 1 cos 𝑒 1 sin 𝑒 with the introduction of an upper digital camera with a long exposure 60 s–90 s and the camera height above horizontal 𝑦 (𝑡) =(𝑅+𝑦(𝑡)) (𝜑 (𝑡))−𝑥 (𝑡) (𝜑 (𝑡)); (2) 2 1 sin 𝑒 1 cos 𝑒 grid level is 2.5 m.

𝑧2 (𝑡) =𝑧(𝑡) . 4. Comparison and Discussion of The absolute coordinates 𝑥2, 𝑦2 in the inertial reference DAE-Solution Based Theoretical and E 𝑥 𝑦 frame , which depend upon the relative coordinates 1, 1 in Experimental Results the noninertial reference frame B during the swaying of load 𝑀 (Figures 8–10), has been defined according to the above The comparison of DAE-solution based theoretical (- - -, 𝑦2 = mentioned Equations and has been shown in Figures 4 and 5 𝑦2(𝑥2)) and experimental (—, 𝑦2 =𝑦2(𝑥2))resultsisshown as (- - -, 𝑦2 =𝑦2(𝑥2)). in Figures 4 and 5. Shock and Vibration 5

Bearing Camera

R = 0.5 m

D Boom

Laser pointer m H=1

Slewing axis B 𝜔e = 0.209 rad/s m Lower plate

l = 0.825 O Regular grid 1200 mm ×700mm 2

with square cells 20 mm ×20mm m 2.5

Ast

O1 M

Light-emitting diode

Figure 2: The dimension experimental measurement system for payload 𝑀 swaying motion during crane boom uniform slewing.

2 In order to estimate the relative disagreement of the mechanical system parameters: 𝑅 = 0.492 m; 𝑔 = 9.81 m/s ; 0.5 derived DAE-solution based computational (- - -) and 𝑙 = 0.206 m; 𝑘=(𝑔/𝑙) ≈ 6.901 rad/s; 𝑇=14s; 𝜔𝑒 =2𝜋/𝑇≈ 𝑀 ∘ experimental (—) payload absolute trajectories in the 0.449 rad/s; 𝜑𝑒 = 180 ; 𝛼1 = 0.01014 rad; 𝑉𝐵 = 0.221 m/s; E dyn inertial reference frame we have computed the amplitude 𝑦 = 0.00209 m; ]1 =𝑘+𝜔𝑒 = 7.3496 rad/s; ]2 =𝑘− 𝛿 dyn discrepancy in the polar coordinate system by the following 𝜔𝑒 = 6.4520 rad/s; 𝐶1 = 0.01502 m; and 𝐶3 = 0.01711 m formula: (Figure 4(a)). 󵄨 󵄨 󵄨 󵄨 󵄨𝑟 −𝑟 󵄨 󵄨𝑟 −𝑟 󵄨 Numerical computations (- - -, 𝑦2 =𝑦2(𝑥2))in 1 󵄨 comp exper󵄨 󵄨 comp exper󵄨 𝛿=⟨ (( )+( ))⟩ , Figure 4(b) have been carried out for the following values of 2 𝑟 𝑟 2 comp exper mechanical system parameters: 𝑅 = 0.492 m; 𝑔 = 9.81 m/s ; 0.5 (3) 𝑙 = 0.412 m; 𝑘=(𝑔/𝑙) ≈ 4.879 rad/s; 𝑇=14s; 𝜔𝑒 =2𝜋/𝑇≈ 0.449 𝜑 = 180∘ 𝛼 = 0.01019 𝑉 = 0.221 where 𝑟 and 𝑟 arethemagnitudesoftheradius- rad/s; 𝑒 ; 1dyn rad; 𝐵 m/s; comp exper 𝑦 = 0.00419 ] =𝑘+𝜔= 5.3284 ] =𝑘− vectors, connecting point 𝑂2 and theoretical (- - -) and dyn m; 1 𝑒 rad/s; 2 𝜔 = 4.4308 𝐶 = 0.02072 𝐶 = 0.02492 experimental (—) curves, and computed for the same fixed 𝑒 rad/s; 1 m; and 3 m polar angle 𝜑𝑒. (Figure 4(b)). 𝑦 =𝑦(𝑥 ) The amplitude discrepancies 𝛿 have the following values: Numericalcomputations(---, 2 2 2 )inFigure 5(a) 𝛿 = 7.53 𝑙 = 0.206 𝛿 = 7.58 have been carried out for the following values of mechanical %for minFigure 4(a); %for 2 𝑙 = 0.412 minFigure 4(b); 𝛿 = 7.40%for𝑙 = 0.618 min system parameters: 𝑅 = 0.492 m; 𝑔 = 9.81 m/s ; 𝑙= 0.5 Figure 5(a);and𝛿 = 6.9%for𝑙 = 0.825 minFigure 5(b). 0.618 m; 𝑘=(𝑔/𝑙) ≈ 3.984 rad/s; 𝑇=20s; 𝜔𝑒 =2𝜋/𝑇≈ 𝑦 =𝑦(𝑥 ) 0.314 𝜑 = 180∘ 𝛼 = 0.00498 𝑉 = Numerical computations (- - -, 2 2 2 )in rad/s; 𝑒 ; 1dyn rad; 𝐵 0.1546 𝑦 = 0.00308 ] =𝑘+𝜔= 4.298 Figure 4(a) have been carried out for the following values of m/s; dyn m; 1 𝑒 rad/s; 6 Shock and Vibration

(a) (b)

(c) (d) Figure 3: Photos of dimension experimental measurement system (a)–(d) for payload 𝑀 swaying motion during crane boom uniform slewing.

]2 =𝑘−𝜔𝑒 = 3.670 rad/s; 𝐶1 = 0.01798 m; and 𝐶3 = payload trajectories (Figures 4 and 5)intheinertialreference 0.02106 m(Figure 5(a)). frame E, that have been computed with DAE-solution based For further estimation of the relative disagreement of theoreticalmodel(---)(1) and measured experimentally (—) derived DAE-solution based computational (- - -) and asshowninFigures4 and 5. experimental (—) payload 𝑀 absolute trajectories in the ItisimportanttonotethatthepayloadmotionDAE inertial reference frame E we have computed the confidence equations in the nonlinear problem (1) for the noninertial intervals in Figures 4 and 5 for the dimensionless parameters reference frame B may be derived with an introduction 𝑟 /𝑟 𝑡 comp exper and confidence probability 0.95. The Student’s - of Lagrange equations (Appendices A–C, F). However the test results yield the following confidence intervals: 0.9374 ≤ discussion of (1)ismoresuitableandinformativewithan (𝑟 /𝑟 ) ≤ 1.0315 𝑙 = 0.206 comp exper for minFigure 4(a); introduction of dynamic Coriolis theorem (Appendices A– 0.9014 ≤ (𝑟 /𝑟 ) ≤ 1.0272 𝑙 = 0.412 2 2 2 2 comp exper for min D, F). The first terms 𝑑 𝑥1/𝑑𝑡 ; 𝑑 𝑦1/𝑑𝑡 in (1) define the 0.9200 ≤ (𝑟 /𝑟 ) ≤ 1.0352 𝑙 = 0.618 Figure 4(b); comp exper for m vector a𝑀/B = ar (Appendix B,formula(B.4)) for the relative 0.9257 ≤ (𝑟 /𝑟 ) ≤ 1.0066 in Figure 5(a);and comp exper for acceleration of load 𝑀 in the noninertial reference frame 𝑙 = 0.825 minFigure 5(b). B. The straight-line terms (D.3)formula-basedtermsin(1) Both the relative discrepancy 𝛿 and the confidence inter- are linearly proportional to the relative payload coordinates vals show the satisfactory agreement between the absolute in the noninertial reference frame B.Thesestraight-line Shock and Vibration 7

y (m) y2 (m) 2 0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1 p. O2 (p. D) x (m) p. O2 (p. D) x2 (m) 2 0.0 0.0 2R = 1 (m) 2R = 1 (m)

−0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 l ≈ 0.412 (m); 𝜔 ≈ 0.449 (rad/s); k ≈ 4.88 (rad/s) l ≈ 0.206 (m); 𝜔e ≈ 0.449 (rad/s); k ≈ 6.9 (rad/s) e

Experimental y2 =y2(x2) Experimental y2 =y2(x2) Computational y2 =y2(x2) Computational y2 =y2(x2) (a) (b)

Figure 4: The comparison of the small swaying angle 𝛼1 theoretical (- - -, 𝑦2 =𝑦2(𝑥2)) and experimental (—, 𝑦2 = 𝑦2(𝑥2)) load absolute trajectories in the inertial reference frame E for fixed cable length 𝑙 = 0.206 m(a)andfixedcablelength𝑙 = 0.412 m(b).

y (m) 2 y2 (m) 0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1 p. O (p. D) p. O (p. D) 2 x (m) 2 x2 (m) 0.0 2 0.0 2R = 1 (m) 2R = 1 (m) −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5

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

Experimental y2 =y2(x2) Experimental y2 =y2(x2) Computational y2 =y2(x2) Computational y2 =y2(x2) (a) (b)

Figure 5: The comparison of the small swaying angle 𝛼1 theoretical (- - -, 𝑦2 =𝑦2(𝑥2)) and experimental (—, 𝑦2 =𝑦2(𝑥2)) load absolute trajectories in the inertial reference frame E for fixed cable length 𝑙 = 0.618 m(a)andfixedcablelength𝑙 = 0.825 m(b). terms in (1) have been determined by the contribution of Finite motions of a swaying load are shown in Figure 6 the normal or centripetal acceleration 𝜔B/E ×(𝜔B/E × in the case of 𝜔𝑒 =0rad/s. In Figure 6 the relative and n r𝐸/𝑀)=ae (B.6) of transportation for payload 𝑀 and by the absolute load trajectories are identically equal. The lingering appearance of corresponding D’Alembert centrifugal inertia remains of elliptical motion in Figure 6 and ellipses semiaxes n n force Φe =(−𝑚)ae (D.3)duetocraneboomtransport are defined by the initial conditions. rotation in the noninertial reference frame B. Additional −2𝜔 (𝑑𝑦 /𝑑𝑡) 2𝜔 (𝑑𝑥 /𝑑𝑡) terms 𝑒 1 and 𝑒 1 in (1) have been defined 6. Discussion of Analogy between Payload by the compound or Coriolis acceleration 2𝜔B/E × V𝑀/B = Swaying and Foucault’s Pendulum Motion acor of payload 𝑀 in the noninertial reference frame B.The rectangular Cartesian projections of Coriolis inertia force in The motion of Foucault’s pendulum is shown in Figure 7, the noninertial reference frame B is defined by formula (B.7). where 𝑥2𝑦2𝑧2 is the heliocentric inertial reference frame (not shown in Figure 7), 𝑥1𝑦1𝑧1 is the geocentric noninertial 5. Experimental Results for the Spherical reference frame, and 𝑥𝑦𝑧 is a noninertial reference frame, 𝑦 𝑧 𝐴 𝑧 Pendulum with the Fixed Pivot Center located at the geographic latitude 1 ; st is a local vertical; 𝐵𝐷 is the radius of the circle of the pinning point; 𝐵𝑀 is Experimental load swaying results for the fixed crane boom the cable length; and 𝜔𝑒 is the angular velocity of the Earth model (𝜔𝑒 =0rad/s) are shown in Figure 6. diurnal rotation. 8 Shock and Vibration

(a) (b)

Figure 6: Experimental trajectories of the spherical pendulum with fixed pivot center and fixed cable length 𝑙 = 0.825 m(a;b).

of relative amplitude extremes have angular velocity values z1 z equaltotheangularvelocityoftheEarthdiurnalrotation. D Moreover, for Foucault’s pendulum (Figure 7)thedirection B of rotation of major semiaxes of relative amplitude extremes 𝜔 is oppositely directed to the direction of the Earth diurnal e rotation. All the above mentioned means that the plane 𝐴 𝐵𝑀 ( st ) of Foucault’s pendula swaying remains fixed in the A g st heliocentric inertial reference frame 𝑥2𝑦2𝑧2. M The presented analogy between crane boom-driven pay- load swaying and Foucault’s pendulum motion essentially x y increases knowledge and awareness of the oscillation pro- cesses in such dynamic systems.

7. Discussion and Mechanical x y 1 1 Interpretation of Governing Equations for DAE-Based Nonlinear Figure 7: Foucault’s pendulum schematic view. and Linearized Models A governing nonlinear DAE-based system (1)forcraneboom uniform slewing has been proposed and numerically solved. It is important to note that the structure of linearized Aconservativesystem(1)and(H.4)-(H.5)withcomponents Φ Φ Φ equations (H.4)-(H.5) for the swaying payload during crane cor 𝑥 and cor 𝑦 has been derived. Both projections cor 𝑥 and Φ B F boom uniform rotation in the noninertial reference frame cor 𝑦 coincide in the noninertial reference frames and . F is analogous to the structure of the governing equations The occurrence of the first derivatives 𝑑𝑥/𝑑𝑡; 𝑑𝑦/𝑑𝑡 of for Foucault’s pendulum motion in the known published the load 𝑀 relative coordinates with respect to time in terms works by Zhuravlev and Petrov [100](formula(6),p.36of −2𝜔𝑒(𝑑𝑦/𝑑𝑡) and 2𝜔𝑒(𝑑𝑥/𝑑𝑡) determines that there is no [100]), Pardy [88] (formulae (12)–(14), pp. 850-851 of [88]), decay of the oscillations in (H.4)-(H.5), and (1)butonly and Condurache et al. [77] (formulae (1.1)–(1.3), p. 743-744 redirection of the relative velocity vector V𝑀/F = Vr of the of [77]). The above mentioned analogy between linearized load 𝑀 in the noninertial reference frames B and F. equations (H.4)-(H.5)andthegoverningequationsin[77, 88, The introduction of a linearized modelH.4 ( )-(H.5) 100] assumes the geometric analogy of relative swaying tra- allowed the determination of the natural frequencies (]1 ≠ jectories between Foucault’s pendulum and the boom-driven ]2)offreeoscillationsofpayload𝑀:]1 =𝑘+𝜔𝑒 and pendulum with rotating pivot centers (Figures 1(e), 12,and ]2 =𝑘−𝜔𝑒. 16). So there is the geometric analogy between computational It follows from the forms of the relative payload trajectory relative trajectories of load 𝑀 swaying during crane boom 𝑦=𝑦(𝑥)in the noninertial reference frame F and the uniform slewing (Figures 1(e), 12,and16) and relative trajec- absolute trajectory 𝑦2 =𝑦2(𝑥2) in the inertial reference frame 𝑀 𝐴 𝑥𝑦 𝐴 E 𝑀 tories of load swaying in the plane ( st )aroundp. st in that the resulting motion of the payload on the cable Figure 7,shownbyZhuravlevandPetrov[100](Figure4,p.39 𝑀𝐵, taking into account the Coriolis inertia force Φcor (D.5), of [100]), and Condurache and Martinusi [77](Figure14atp. will be the sum of two oscillations with natural frequencies ]1 754 and Figure 16 at p. 755 of [77]). The better understanding and ]2,andwithperiods𝑇1 =2𝜋/]1 and 𝑇2 =2𝜋/]2. ofFoucault’spenduladynamicsisprovidedwiththestudy It is worth noting that the frequencies ]1 and ]2 differ of computational relative trajectories in Figures 1(e), 12,and by 2𝜔𝑒. This means that for the small angle assumption 16. For Foucault’s pendulum (Figure 7)themajorsemiaxes (Appendices H-I)wehave]1 ≈ ]2 and the trajectory Shock and Vibration 9

z2 z z1 B R D

ℱ 𝛾

l N 𝛼1 ℬ 𝛼 𝜔 = 𝜔 e3 1dyn ℬ/ℰ ℱ/ℰ 𝜔 ẽ e Vr 𝛽 3

M ℱ d𝜑e e2 At dt y Adyn h O ℰ y1 O A ẽ2 1 st Δ ℬ ̂e3 ydyn mg ℰ γ O2

E ̂e2 sin(𝜑e) ℰ

̂e1 cos(𝜑e)

Figure 8: Swaying scheme of load 𝑀 on the cable 𝑀𝐵, which is fixedly attached at the point 𝐵 on the crane boom 𝐵𝑂2, in the vertical plane (𝑦𝑧) for the nonlinear model derivation in Cartesian coordinates.

of relative motion 𝑦=𝑦(𝑥)in the noninertial reference essentially differ from its harmonic oscillations neighbor for frame F on the expiration of the relative oscillations period quarter-period. 2𝜋/𝑘 𝛼 𝑦 time looks approximately like an ellipse (Figures 12(b) Due to the negligible quantity of dyn and dyn (G.3)– and 16(b))withsemiaxes,whicharegovernedbytheinitial (G.5), the average deviation of the load 𝑀 from the mechan- conditions. ical trajectory of the boom tip 𝐵 is negligible (Figures The major semiaxis of such an ellipse in Figures 12(b) 1(f), 2, 4 and 5). The basic dynamic load on the system and 16(b) is defined by the initial velocity of load 𝑀 in the “crane boom 𝐵𝑂2-load 𝑀” is created by the high-frequency noninertial reference frame F and the analytical value of the oscillations of load 𝑀, which are determined by the action major semiaxis is equal to 𝑉𝐵/𝑘= (𝜔 𝑒𝑅)/𝑘. The derived value of inertia force Φcor =−2𝑚(𝜔B/E × V𝑀/B),stipulatedby of 𝑉𝐵/𝑘 is approximately the sum of the absolute values of the Coriolis acceleration acor =2𝜔B/E × V𝑀/B of load 𝑀 the amplitudes in the analytical form (I.15)ofthelinearized in the noninertial reference frame F.Thebasicdynamic Cauchy problem (H.4)-(H.5). load in the system defines additional loads and vibrations of The minor semiaxis of the above mentioned ellipse crane boom 𝐵𝑂2 mechanical elements and support bearings, 𝑦 in Figures 12(b) and 16(b) is defined by the value dyn, complicates the automatic and manual control systems of the which is approximately the difference of amplitudes in the electromechanical crane boom 𝐵𝑂2, and also makes crane analytical form (I.15) of the linearized Cauchy problem (H.4)- operation much more difficult. (H.5). In this case the major semiaxis is approximately 5– It is also important to note that the stop of crane boom 20 times larger than the minor semiaxis because the angular 𝐵𝑂2 does not lead to instantaneous damping of the load velocity 𝜔𝑒 of the transport rotation of crane boom 𝐵𝑂2 is 𝑀 absolute oscillations in the inertial reference frame E much smaller than the natural frequency 𝑘 of the spherical (Figures 2, 4,and5). This phenomenon directly follows from oscillations of payload 𝑀 on the cable 𝑀𝐵 (Figures 12(b) and the experimental trajectory in Figures 2, 4,and5.Also, 16(b)). the natural spherical oscillations of load 𝑀 will occur with Itisalsonecessarytonotethatthemajorsemiaxisofthe the frequency 𝑘 andtheamplitude(thedifferencebetween ellipse in the relative motion rotates with an angular velocity the final position of load 𝑀 intherelativemotioninthe 𝜔𝑒 in the opposite direction of the transport rotation of crane noninertial reference frame F for half-period of vibration 𝐵𝑂 𝐴 𝑀 boom 2 in (Figures 12 and 16). and static equilibrium st of load on the cable). The It follows from Figures 1(f), 2, 4,and5 that the trajectory further oscillations for the stop of the crane boom 𝐵𝑂2 of absolute motion of load 𝑀 in the inertial reference frame slewing motion are the deviation of the real trajectory from E is almost a symmetric curve with 𝑦2 axial symmetry. the intended final position of load 𝑀. The results of physical The initial and the final motions of load 𝑀 for half-period simulation in Figures 2, 4,and5 show the necessity of add-on 10 Shock and Vibration

y, y1 Φ n e ℱ Ve

e2 V 𝜃 r M

O

e1 Adyn

ẽ2 x ydyn ℬ Φ τ N e ( ) ℱ B O1 𝛼2

Φ Ao cor Ast y2 x1 𝜃 ẽ1 ℬ ̂e2

ℰ 𝜔 𝜔 𝜔 e = ℬ/ℰ = ℱ/ℰ

𝜔 𝜑 = et E e

O2(D) ̂e1 x2 ℰ

Figure 9: Swaying scheme of load 𝑀 at the cable 𝑀𝐵, which is fixedly attached in the point 𝐵 of the crane boom 𝐵𝑂2, in the horizontal plane (𝑥1𝑦1) for the nonlinear model derivation in Cartesian coordinates.

devices development for the efficient suppression of load 𝑀 In Figure 6 of published work by Ju et al., 2006 [34], final oscillations. it was shown that the angle between the payload’s cable ∘ and the vertical line does not exceed 16 . Ju’s formula (15a) 8. Discussion and Comparison of Derived and in p. 382 of [34]assumesthatJu’sanglebetweenpayload’s Known Published Results cableandtheverticallinehastheharmoniclawofvariation with an introduction of a small perturbation term. Both Ju’s In Figure 1 and pp. 537-538 of published work by Sakawa et al., assumptions in [34] confirm the proposed linearized model 1981 [63], the small angle between payload’s cable and vertical (H.4)-(H.5). line has been introduced, that confirms proposed nonlinear Comparison of the derived linearized model (H.4)-(H.5) DAE-based system (1) and linearized model (H.4)-(H.5). with Figure 5 of published work by Mitrev and Grigorov In Figure 9 of p. 278 of published work by Maczynski 2008 in [49] shows that the Mitrev-derived ranges of payload swaying angles within Mitrev’s nonlinear model does not and Wojciech, 2003 [44], the computational finite element ∘ method(FEM)-basedresultswereshownfortheabso- exceed 4.8 . Such small values of swaying angles completely lute payload trajectories in the inertial reference frame E. confirmtheapplicabilityandcorrectnessofthesmallangle Maczynski’s Figure 9 in [44] outlines that the angle between assumption in (H.4)-(H.5). thepayload’scableandtheverticallinedoesnotexceed0.1 radandthatagreeswiththeproposedlinearizedmodel(H.4)- 9. Final Conclusions (H.5) for a small angle assumption. Computational trajectory for Maczynski-derived small load swaying [44]aftercrane A governing nonlinear DAE-based system for crane boom boom stop qualitatively coincides with the experimentally uniform slewing has been proposed, numerically solved, and observable load 𝑀 motion in Figures 2, 4,and5. experimentally verified. Shock and Vibration 11

Fully nonlinear differential equations for 3 DOF spherical for the load 𝑀, when the crane boom 𝐵𝑂2 is fixed. The point 𝐴 𝑥 =0 𝑦 =𝑦 𝑧 =Δ pendulum oscillations with a uniform slewing crane boom dyn with the coordinates 1 ; 1 dyn;and 1 is the around a fixed vertical axis of rotation have been derived dynamic equilibrium position for the load 𝑀 for the rotating 𝐵𝑂 𝐴 in relative Cartesian and spherical coordinates. The identical crane boom 2. We will assume the point dyn as the origin linearized differential equations in relative Cartesian coordi- of the noninertial reference frame F(𝑥𝑦𝑧).Thedirectionsof nates have been derived with an introduction of the Coriolis axes 𝑥, 𝑦, 𝑧 of the noninertial reference frame F are parallel dynamic theorem and Lagrange equations for a uniform to the axes 𝑥1, 𝑦1, 𝑧1 of the noninertial reference frame B in crane boom rotation and small swaying angle 𝛼1 assumptions. Figures 8–10.Theload𝑀 relative motion takes place along Linearized and nonlinear theoretical problems for relative the sphere surface with the fixed radius, equal to the cable swaying of the payload have been formulated in the form of length 𝐵𝑀 =𝑙. theinitialvalue(Cauchy)problem. The computational scheme in Figures 8–10 for the nonlin- An analytical solution of the linearized system has been ear model derivation may be described with an introduction derived. of three degrees of freedom. For generalized coordinates we assume the rectangular coordinates 𝑥1, 𝑦1,and𝑧1 of the load The influences of inertia forces from the centripetal and 𝑀 in the moving noninertial frame of reference B and the compound accelerations have been estimated. angle 𝜑𝑒 of crane boom 𝐵𝑂2 slewing in the horizontal plane A linearized conservative system, which contains first (𝑥1𝑦1) with the angular velocity 𝜔𝑒 around the vertical axis time derivatives of coordinates and no damping of oscilla- 𝑂2𝑧2. tions, has been derived. The relative velocity vector Vr = V𝑀/B of the load 𝑀 is The amplitudes of load oscillations, which depend on defined as Vr = V𝑀/B =(𝑑𝑥1/𝑑𝑡,1 𝑑𝑦 /𝑑𝑡,1 𝑑𝑧 /𝑑𝑡). computed initial conditions, have been estimated within a In order to derive the absolute velocity V𝑀/E of payload small angle assumption. The dependence of natural frequen- 𝑀 we will apply the vector method for absolute motion cies on the transport inertia forces and gravity forces has been assignment. We will define r𝑂2/𝑀 as the position vector, con- computed for the linearized systems. necting initial point 𝑂2 and terminal point 𝑀 in noninertial The formulae for the association of relative payload reference frame B (𝑚); consider motion in the noninertial reference frame F and absolute r𝑂 /𝑀 = r𝑂 /𝐵 + r𝐵/𝑀, payload motion in the inertial reference frame E have been 2 2 (A.1) outlined. where vector components in (1) are defined in nomenclature. The results of the numerical DAE-based investigation and In accordance with Figures 8–10 we have the following the performed physical simulation show a satisfactory fit for unit vectors’ expansions for position vectors r𝑂 /𝐵 and r𝐵/𝑀 frequencies and amplitudes of load oscillations. 2 in (A.1): The dynamic analogy between crane boom-driven pay- =(𝑅 (𝜑 )) ̂ +(𝑅 (𝜑 )) ̂ +(𝑅 (𝛾)) ̂ ; load swaying motion and Foucault’s pendulum motion has r𝑂2/𝐵 cos 𝑒 e1 sin 𝑒 e2 tan e3 been grounded and outlined. (A.2) The results of the present work are the foundation for =𝑥̃ +𝑦̃ +(𝑧 −𝑙)̃ , further investigations of payload 𝑀 swaying dynamics during r𝐵/𝑀 1e1 1e2 1 e3 (A.3) telescopic crane boom nonuniform slewing motion with where 𝑅, 𝜑𝑒, 𝑥1,and𝑦1 notations have been defined in variable cable length and for the different motions of the nomenclature and in Figures 8–10. pendulum pivot center. For further definition of the position vector r𝑂2/𝑀 in iner- tial reference frame E, we write unit vectors of noninertial Appendices reference frame B in (A.3) through the unit vectors of inertial reference frame E: A. Velocity Kinematics Analysis in ̃ = ( (𝜑 )) ̂ − ( (𝜑 )) ̂ ; Cartesian Coordinates e1 sin 𝑒 e1 cos 𝑒 e2 (A.4) ̃ = ( (𝜑 )) ̂ − ( (𝜑 )) ̂ ; For the nonlinear problem definition we study the coopera- e2 cos 𝑒 e1 sin 𝑒 e2 (A.5) 𝐵𝑂 tive motion of the mechanical system “crane boom 2-load ̃ = ̂ . 𝑀”whichisshowninFigures8–10. e3 e3 (A.6) In the nomenclature chapter we denote the fixed inertial After substitution of formulaeA.2 ( )–(A.6)in(A.1)and E 𝑥 𝑦 𝑧 frame of reference as 2 2 2, and the moving noninertial some algebraic manipulations the vector expression (A.1)for B 𝑥 𝑦 𝑧 frame of reference as 1 1 1, which is rigidly bounded the position vector r𝑂 /𝑀 in the inertial reference frame E will 𝐵𝑂 2 with the crane boom 2. Rotation of the moving noninertial take the following form: frame of reference B(𝑥1𝑦1𝑧1) around the fixed inertial frame E(𝑥 𝑦 𝑧 ) =((𝑅+𝑦) (𝜑 )+𝑥 (𝜑 )) ̂ of reference 2 2 2 defines the transportation motion for r𝑂2/𝑀 1 cos 𝑒 1 sin 𝑒 e1 payload 𝑀.Themotionofload𝑀 relative to the moving + ((𝑅 + 𝑦 ) (𝜑 )−𝑥 (𝜑 )) ̂ noninertial frame of reference B(𝑥1𝑦1𝑧1) defines the relative 1 sin 𝑒 1 cos 𝑒 e2 (A.7) motion of payload 𝑀.Thepoint𝐴 with the coordinates st +(𝑅 (𝛾) + 𝑧 −𝑙)̂ . 𝑥1 =0; 𝑦1 =0;and𝑧1 =0is the steady equilibrium position tan 1 e3 12 Shock and Vibration

The absolute velocity of payload 𝑀 we define by time where payload relative velocity in the noninertial reference differentiation of (A.7) assuming constancy of unit vectors frame B may be written as ̂e1, ̂e2,and̂e3 of the inertial reference frame E and constancy 𝑅 𝛾 𝑑𝑥 𝑑𝑦 𝑑𝑧 and : = =( 1 ) ̃ +( 1 ) ̃ +( 1 ) ̃ . V𝑀/B Vr 𝑑𝑡 e1 𝑑𝑡 e2 𝑑𝑡 e3 𝑑(r𝑂 /𝑀) (A.13) = = 2 . (A.8) V𝑀/E Vabs 𝑑𝑡 The velocity V𝑂 /E of the point 𝑂2 =point𝐸 in the inertial E 2 After differentiation and algebraic transformations in reference frame has zero value: (A.7)and(A.8)wehavethefollowingvectorexpressionfor =0. absolute payload 𝑀 velocity in the inertial reference frame V𝑂2/E (A.14) E: The last term in (A.12)isthevectorproduct𝜔B/E × V𝑀/E 𝜔 = r𝑂2/𝑀, where crane boom angular velocity vector is B/E 𝑑𝑦 𝑑𝜑 (𝑑𝜑𝑒/𝑑𝑡)̃e3 (A.6)andthepositionvectorr𝑂 /𝑀 in the nonin- = ((( 1 )+𝑥 ( 𝑒 )) (𝜑 ) B 2 𝑑𝑡 1 𝑑𝑡 cos 𝑒 ertial reference frame is as follows:

𝑑𝑥1 𝑑𝜑𝑒 =𝑥̃ +(𝑦 +𝑅)̃ +𝑧̃ . +(( )−(𝑅+𝑦)( )) (𝜑 )) ̂ r𝑂2/𝑀 1e1 1 e2 1e3 (A.15) 𝑑𝑡 1 𝑑𝑡 sin 𝑒 e1 𝑑𝑦 𝑑𝜑 Taking into account (A.13)–(A.15) the vector equation + ((( 1 )+𝑥 ( 𝑒 )) (𝜑 ) E 𝑑𝑡 1 𝑑𝑡 sin 𝑒 (A.12) in the inertial reference frame written through the unit vectors ̃e1, ̃e2, ̃e3 of the noninertial reference frame B 𝑑𝜑 𝑑𝑥 +((𝑅+𝑦 )( 𝑒 )−( 1 )) (𝜑 )) will have the following form: 1 𝑑𝑡 𝑑𝑡 cos 𝑒 𝑑𝑧 𝑑𝑥1 𝑑𝜑𝑒 × ̂ +( 1 ) ̂ . V𝑀/E =(( )−( )(𝑅+𝑦1)) ̃e1 e2 𝑑𝑡 e3 𝑑𝑡 𝑑𝑡 (A.16) (A.9) 𝑑𝑦 𝑑𝜑 𝑑𝑧 +(( 1 )+( 𝑒 )𝑥 ) ̃ +( 1 ) ̃ . 𝑑𝑡 𝑑𝑡 1 e2 𝑑𝑡 e3 The square of absolute payload 𝑀 velocity may be written as So, on the basis of (A.16), we have the following square of 𝑑𝑥 2 𝑑𝑦 2 𝑑𝑧 2 𝑑𝜑 2 absolute payload 𝑀 velocity as 𝑉2 =( 1 ) +( 1 ) +( 1 ) +( 𝑒 ) abs 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡 2 (V𝑀/E, V𝑀/E)=𝑉 𝑑𝜑 abs ×(𝑥2 +(𝑦 +𝑅)2)+2( 𝑒 ) 1 1 (A.10) 𝑑𝑥 𝑑𝜑 2 𝑑𝑡 =(( 1 )−( 𝑒 )(𝑅+𝑦 )) 𝑑𝑡 𝑑𝑡 1 𝑑𝑦 𝑑𝑥 ×(𝑥 ( 1 )−(𝑦 +𝑅)( 1 )) . 1 𝑑𝑡 1 𝑑𝑡 𝑑𝑦 𝑑𝜑 2 𝑑𝑧 2 +(( 1 )+( 𝑒 )𝑥 ) +( 1 ) . 𝑑𝑡 𝑑𝑡 1 𝑑𝑡 After algebraic transformations we have the square of (A.17) absolute payload 𝑀 velocity in the form of

2 Independently derived expressions for the scalar prod- (V𝑀/E, V𝑀/E)=𝑉 abs uct (V𝑀/E, V𝑀/E)in(A.11)and(A.17) completely coincide, 2 which confirms the correctness and accuracy of the pen- 𝑑𝑥1 𝑑𝜑𝑒 =(( )−( )(𝑅+𝑦 )) dulum absolute velocity V𝑀/E derivation and shows that 𝑑𝑡 𝑑𝑡 1 the scalar product (V𝑀/E, V𝑀/E) is the invariant expression, 𝑑𝑦 𝑑𝜑 2 𝑑𝑧 2 independent of choice of reference frame. +(( 1 )+( 𝑒 )𝑥 ) +( 1 ) . 𝑑𝑡 𝑑𝑡 1 𝑑𝑡 The second and third terms of (A.12) in the noninertial reference frame B determine the vector of the load 𝑀 (A.11) transportation velocity as Algebraic expressions (A.10)and(A.11) could be derived Ve = V𝑂 /E + 𝜔B/E × r𝑂 /𝑀. (A.18) on the basis of a velocity addition theorem for payload 𝑀 2 2 E compound motion in the inertial reference frame through 𝑀 theunitvectorsofthenoninertialreferenceframeB (see Taking into account (A.14)thescalaroftheload [59, 76, 81, 82, 84, 85, 87, 90–92, 94–98]): transportation velocity is defined as 󵄩 󵄩 󵄩 󵄩 = + + 𝜔 × , 󵄩 󵄩 =𝜔 󵄩 󵄩 . V𝑀/E V𝑀/B V𝑂2/E B/E r𝑂2/𝑀 (A.12) 󵄩Ve󵄩 B/E 󵄩r𝑂2/𝑀󵄩 (A.19) Shock and Vibration 13

z2

R D z1

ℬ H z 𝜔 𝜔 B ℬ/ℰ = ℱ/ℰ ℱ 𝜔 e =d𝜑e/dt

ℰ ̂e3 ℰ ̂e2 y2 O2 𝛼1 l

̂e1 ẽ3 𝜑e e ℰ N 3 n Ast 𝐚e ẽ1 𝛼2 ẽ O1 2 Adyn O 𝐚cor e2 ℱ τ Φ e x e 𝐚𝜏 1 2 e y ℬ M Ve Δ Φ y1 cor ℬ V x r 1 mg Φ n e

x ℱ

Figure 10: Computational spatial scheme of spherical pendulum 𝑀,swayingonthecable𝑀𝐵 during crane boom 𝐵𝑂2 slewing motion for the nonlinear model derivation in Cartesian coordinates.

The transportation velocity vector Ve is perpendicular to Assuming (A.21)equation(A.23)willtakethefollowing B r𝑂2/𝑀 and r𝐷/𝑀 (Figures 8–10); that is, form in the noninertial reference frame : ( , )=( + 𝜔 × , )=0; Ve r𝑂2/𝑀 V𝑂2/E B/E r𝑂2/𝑀 r𝑂2/𝑀

(Ve, r𝐷/𝑀)=(V𝑂 /E + 𝜔B/E × r𝑂 /𝑀, r𝐷/𝑀)=0. 𝑑𝜑𝑒 𝑑𝜑𝑒 2 2 Ve =(−( )(𝑅+𝑦1)) ̃e1 +(( )𝑥1) ̃e2. (A.24) (A.20) 𝑑𝑡 𝑑𝑡 In nomenclature and in Figure 9 we denote the current angle 𝜃=∠(̃e2,(r𝑂 /𝑀) ),where 2 𝑥1𝑦1 Taking into account (A.13)and(A.24), the formula (A.12) 𝑥1 (𝑅 +1 𝑦 ) sin (𝜃) = 󵄩 󵄩; cos (𝜃) = 󵄩 󵄩 . (A.21) yields again (A.16)and(A.17). 󵄩 󵄩 󵄩 󵄩 𝑀 󵄩r𝑂2/𝑀󵄩 󵄩r𝑂2/𝑀󵄩 Sothesquareofabsolutepayload velocity (A.10), In the noninertial reference frame B the vector of the (A.11), and (A.17) has been derived with three independent methods, which confirms the accuracy and correctness of load 𝑀 transportation velocity Ve is defined as 󵄩 󵄩 󵄩 󵄩 expressions (A.10), (A.11), and (A.17). Ve = (− 󵄩Ve󵄩 cos (𝜃)) ̃e1 + (󵄩Ve󵄩 sin (𝜃)) ̃e2. (A.22) Taking into account (A.19), (A.22)takesthefollowing form: B. Acceleration Kinematics Analysis 󵄩 󵄩 Ve =(−𝜔B/E 󵄩r𝑂 /𝑀󵄩 cos (𝜃)) ̃e1 󵄩 2 󵄩 Further dynamic analysis, with the introduction of Newton’s (A.23) 󵄩 󵄩 second law, requires us to study the accelerations of payload +(𝜔B/E 󵄩r𝑂 /𝑀󵄩 sin (𝜃)) ̃e2. 󵄩 2 󵄩 𝑀,showninFigure 10. 14 Shock and Vibration

The standard vector equation for the acceleration addi- C. The Geometric Constraints tion for payload 𝑀 in the inertial reference frame E has the Imposed on the Payload form (see [59, 76, 81, 82, 84, 85, 87, 90–92, 94–98]): The geometric constraint, imposed on the payload 𝑀 is = + + 𝛼 × a𝑀/E a𝑀/B a𝐸/E B/E r𝐸/𝑀 showninFigures8–10 in the form of the cable 𝐵𝑀. The length 𝑙=𝑙 =‖ ‖ 𝐵𝑀 + 𝜔 ×(𝜔 × ) 𝐵𝑀 r𝐵/𝑀 of the cable determines the geometrical B/E B/E r𝐸/𝑀 (B.1) constraint in this problem: +2𝜔 × . B/E V𝑀/B 2 2 2 2 𝑙 =𝑥1 +𝑦1 +(𝑧1 −𝑙) . (C.1) In our case we assume point 𝐸 as the pole for payload transportation motion, located at the vertical axis 𝑂2𝑧2.So On the basis of Figure 10 we may derive that the second term in (B.1) for the inertial reference frame E and in the noninertial reference frame B takes the form 2 2 𝑙 =(𝑙sin (𝛼1) cos (𝛼2)) (C.2) a𝐸/E = a𝐸/B =0. (B.2) 2 2 +(𝑙sin (𝛼1) sin (𝛼2)) +(𝑙cos (𝛼1)) , So taking into account the nomenclature and (B.2), (B.1) 𝛼 𝛼 in the inertial reference frame E takes the following form: where angles 1 and 2 in Figures 8–10 are the spherical coordinates of spherical pendulum 𝑀. 𝜏 n aabs = ar + ae + ae + acor. (B.3) The comparison of formulae (C.1)and(C.2)allowsus to determine the Cartesian coordinates 𝑥1; 𝑦1;and𝑧1 in the Equations (B.1)and(B.2) contain the following accelera- noninertial reference frame B as tions. The vector of payload 𝑀 relative acceleration is defined 𝑥1 =𝑙sin (𝛼1) cos (𝛼2); in the noninertial reference frame B as 𝑦1 =𝑙sin (𝛼1) sin (𝛼2); (C.3) ar = a𝑀/B 𝑧1 =𝑙−𝑙cos (𝛼1). 𝑑2𝑥 𝑑2𝑦 𝑑2𝑧 (B.4) =( 1 ) ̃ +( 1 ) ̃ +( 1 ) ̃ . 2 e1 2 e2 2 e3 𝑑𝑡 𝑑𝑡 𝑑𝑡 The geometric constraint of(C.1)–(C.3) can be derived on 𝜏 the basis of (C.3)as The vector of tangential acceleration ae for transporta- tion of payload 𝑀 has the same direction as the vector of the 𝑙2 2 (𝛼 )=𝑥2 +𝑦2; 𝑀 𝜏 ↑↑ sin 1 1 1 load transportation velocity Ve,thatis,ae Ve and is (C.4) B defined in the noninertial reference frame as 𝑧1 =𝑙−𝑙cos (𝛼1). 𝜏 ae = 𝛼B/E × r𝐸/𝑀 Equations (C.4) yield the following partial derivatives of 2 2 𝑑 𝜑 𝑑 𝜑 𝛼1 with respect to Cartesian coordinates 𝑥1; 𝑦1;and𝑧1 in the =(−( 𝑒 )(𝑅+𝑦 )) ̃ +(( 𝑒 )𝑥 ) ̃ . B 𝑑𝑡2 1 e1 𝑑𝑡2 1 e2 noninertial reference frame as (B.5) 𝜕𝛼 𝑥 1 = 1 ; 2 n 𝜕𝑥1 𝑙 sin (𝛼1) cos (𝛼1) The vector of the normal or centripetal acceleration ae of transportation for payload 𝑀 is directed towards the axis 𝜕𝛼 𝑦 𝜏 n 1 = 1 ; 𝑂2𝑧2 and at the same time ae and ae are the coplanar 2 (C.5) 𝜕𝑦1 𝑙 sin (𝛼1) cos (𝛼1) vectors, located in the horizontal plane (𝑥2𝑦2), where n 𝜕𝛼1 1 ae = 𝜔B/E ×(𝜔B/E × r𝐸/𝑀) = . 𝜕𝑧1 𝑙 sin (𝛼1) 2 2 (B.6) 𝑑𝜑𝑒 𝑑𝜑𝑒 =(−( ) 𝑥 ) ̃ −(( ) (𝑅 + 𝑦 )) ̃ . 𝑥 𝑦 𝑧 𝑑𝑡 1 e1 𝑑𝑡 1 e2 Absolute coordinates 2, 2,and 2 in inertial reference frame E which depend upon the relative coordinates 𝑥1, 𝑦1, 𝑧 B The vector of the Coriolis (compound) acceleration of and 1 in noninertial reference frame during the swaying 𝑀 payload 𝑀 is directed in accordance with the vector product of load are defined according to the following equations law (Figures 8–10): =2𝜔 × acor B/E V𝑀/B 𝑥2 =(𝑅+𝑦1) cos (𝜑𝑒)+𝑥1 sin (𝜑𝑒);

𝑑𝜑𝑒 𝑑𝑦1 𝑑𝜑𝑒 𝑑𝑥1 =(−2( )( )) ̃ +(2( )( )) ̃ . 𝑦2 =(𝑅+𝑦1) sin (𝜑𝑒)−𝑥1 cos (𝜑𝑒); (C.6) 𝑑𝑡 𝑑𝑡 e1 𝑑𝑡 𝑑𝑡 e2 (B.7) 𝑧2 =𝑧1 =𝑙−𝑙cos (𝛼1). Shock and Vibration 15

𝑂 (𝐼 2 ) D. Forces Imposed on the Payload where 33 𝐵𝑂2 is the element of mass moment of inertia for the crane boom 𝐵𝑂2 in inertial fixed on earth reference frame Among the forces imposed on the payload 𝑀 (Figure 3(a)) 2 2 E with respect to axis ̂e3; 𝑚(𝑥 +(𝑅+𝑦1) ) is the element of we have an active force of gravity mg,thecablereactionforce 1 𝜏 mass moment of inertia for the payload 𝑀 in inertial fixed on N, the tangential inertial force Φe , the normal or centrifugal n earth reference frame E with respect to axis ̂e3. inertial force Φe , and the Coriolis inertial force Φcor.Taking 𝑂 The external moment of gravitational force M 2 (mg)=0 into account formulae (B.1)–(B.7) we will express below all ↑↓ ̂ imposed forces in the noninertial reference frame B in (E.4)becausemg e3. For the system “crane boom-payload” the cable reaction =(−𝑚𝑔)̃ ; force N istheinternalforce.Soin(E.1)and(E.4)wehave mg e3 (D.1) 𝑂 M 2 (N)=0. (𝑧 −𝑙) 𝑥1 𝑦1 1 Substitution of (E.2), (E.3), and (E.4)into(E.1)yieldsthe N =−𝑁( ) ̃e1 −𝑁( ) ̃e2 −𝑁( ) ̃e3; (D.2) 𝑙 𝑙 𝑙 following scalar equation for the rate of change of moment 𝑂2 of momentum H for the system “crane boom 𝐵𝑂2-load 𝑀” 𝑑𝜑 2 3 n n 𝑒 with respect to point 𝑂2 in the inertial reference frame E: Φe = (−𝑚) ae =(𝑚( ) 𝑥1) ̃e1 𝑑𝑡 𝑑 𝑑𝜑 𝑂2 2 2 𝑒 (D.3) (((𝐼33 ) +𝑚(𝑥1 +(𝑅+𝑦1) )) ( )) 2 𝑑𝑡 𝐵𝑂2 𝑑𝑡 𝑑𝜑𝑒 (E.5) +(𝑚( ) (𝑅 +1 𝑦 )) ̃e2; 𝑑𝑡 𝑂2 𝑂2 =𝑀𝐷𝑇 −𝑀𝐹𝑇, 2 𝑂 𝑂 𝑑 𝜑 2 2 𝜏 𝜏 𝑒 where driving 𝑀𝐷𝑇 and frictional 𝑀𝐹𝑇 torques (see nomen- Φe = (−𝑚) ae =(𝑚( )(𝑅+𝑦1)) ̃e1 𝑑𝑡2 clature section) are the technically defined functions for (D.4) specific electric drive systems (see1 [ –3, 42–44, 59, 63, 67, 76, 𝑑2𝜑 −(𝑚( 𝑒 )𝑥 ) ̃ ; 81, 82, 84, 85, 87, 90–92, 94–98]). 𝑑𝑡2 1 e2 𝑑𝜑 𝑑𝑦 F. Derivation of the Fully Nonlinear Φ = (−𝑚) =(2𝑚( 𝑒 )( 1 )) ̃ cor acor 𝑑𝑡 𝑑𝑡 e1 Equations in Relative Cartesian Coordinates (D.5) of the Noninertial Reference frame B 𝑑𝜑 𝑑𝑥 −(2𝑚( 𝑒 )( 1 )) ̃ , 𝑑𝑡 𝑑𝑡 e2 The vector differential equation for relative motion of payload 𝑀 in the noninertial reference frame B is as follows: −(𝑥 /𝑙) −(𝑦 /𝑙) −((𝑧 −𝑙)/𝑙) n 𝜏 where 1 ; 1 and 1 are the direction 𝑚a𝑀/B = mg + N + Φe + Φe + Φcor; (F.1) cosines of the cable reaction force N in the noninertial reference frame B.TheforceN is directed from point 𝑀 to 𝑚a𝑀/B = mg + N + (−𝑚) (𝜔B/E ×(𝜔B/E × r𝐸/𝑀)) point B; that is, the force N ↑↓ r𝐵/𝑀 is oppositely directed to + (−𝑚) (𝛼 × )+(−𝑚) (2𝜔 × ). the r𝐵/𝑀 (A.3). B/E r𝐸/𝑀 B/E V𝑀/B (F.2) E. Forces Imposed on the Crane The vector differential equation (F.1)-(F.2) yields three Boom-Payload System scalar ordinary differential equations (ODEs) for payload 𝑀 swaying motion. The slewing motion of the mechanical system “crane boom We will project (F.1)and(F.2)totheaxes𝑥1, 𝑦1,and𝑧1 in 𝐵𝑂2-load 𝑀”inFigures8–10 is governed by the vector the noninertial reference frame B equation for the rate of change of moment of momentum 2 2 𝑂 𝑑 𝑥1 𝑥1 𝑑𝜑𝑒 2 𝐵𝑂 𝑀 𝑚( )=−𝑁( )+𝑚( ) 𝑥 H3 for the system “crane boom 2-load ”withrespect 𝑑𝑡2 𝑙 𝑑𝑡 1 to point 𝑂2 in the inertial reference frame E: 2 𝑑 𝜑𝑒 𝑑𝜑𝑒 𝑑𝑦1 E 𝑂2 +𝑚( )(𝑅+𝑦 )+2𝑚( )( ); 𝑑H 𝑂 2 1 3 = ∑ 2 . (E.1) 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑡 M 𝑑2𝑦 𝑦 𝑑𝜑 2 𝑚( 1 )=−𝑁( 1 )+𝑚( 𝑒 ) (𝑅 + 𝑦 ) The vector equation (E.1) contains the following compo- 𝑑𝑡2 𝑙 𝑑𝑡 1 nents: 2 𝑂 𝑂 𝑑 𝜑𝑒 𝑑𝜑𝑒 𝑑𝑥1 2 =𝐼 2 𝜔 ; −𝑚( )𝑥 −2𝑚( )( ); H3 33 B/E (E.2) 𝑑𝑡2 1 𝑑𝑡 𝑑𝑡 𝑂 𝑂 2 2 𝐼 2 = (𝐼 2 ) +𝑚(𝑥 + (𝑅+𝑦) ) ; 2 33 33 𝐵𝑂 1 1 (E.3) 𝑑 𝑧 𝑙−𝑧 2 𝑚( 1 )=−𝑚𝑔+𝑁( 1 ). 𝑑𝑡2 𝑙 𝑂2 𝑂2 𝑂2 𝑂2 𝑂2 ∑ M = M − M =(𝑀 −𝑀 ) ̂e3, (E.4) 𝐷𝑇 𝐹𝑇 𝐷𝑇 𝐹𝑇 (F.3) 16 Shock and Vibration

y2 t =0s 0 𝜔 0 r 0 ℬ/ℰ( )× O2/M( ) 𝜑e(0) = 0 rad

̂e2 ℰ 𝜔ℬ/ℰ(0) A e B, A M dyn 2 y y1 x2 O (E D st ẽ2 2 , ) ̂e1 ℬ e ℱ ẽ1 1 ⏐⏐r ⏐⏐ ⏐⏐ O2/M(0)⏐⏐ = R ⏐⏐ ⏐⏐ ydyn

VM/ℬ(0) x1 x

Figure 11: The computational scheme for initial conditions assignment.

The derived system (F.3)and(E.5) is the nonlinear ODE the following Lagrange equations (see [59, 76, 81, 82, 84, 85, system. The nonlinearity of (F.3) is determined by the pres- 87, 90–92, 94–98]) in the noninertial reference frame B: 𝑁=𝑁(𝑡) ence of the unknown function ,variableboom 𝑑 𝜕𝑇 𝜕𝑇 𝜑 ( )− =𝑄 ; slewing angle 𝑒, variable boom slewing angular velocity 𝑥1 2 2 𝑑𝑡 𝜕𝑥1̇ 𝜕𝑥1 𝑑𝜑𝑒/𝑑𝑡, and variable angular acceleration 𝑑 𝜑𝑒/𝑑𝑡 . 𝑑 𝜕𝑇 𝜕𝑇 In order to verify the correctness of the derived system ( )− =𝑄 ; 𝑦1 (F.6) (F.3) we will utilize second-kind Lagrange equations. Taking 𝑑𝑡 𝜕𝑦1̇ 𝜕𝑦1 into account equations (A.10), (A.11), and (A.17)forthe 𝑀 𝑑 𝜕𝑇 𝜕𝑇 square of absolute payload velocity (V𝑀/E, V𝑀/E), and by ( )− =𝑄 . 𝑧1 adding the kinetic energy (see [59, 76, 81, 82, 84, 85, 87, 90– 𝑑𝑡 𝜕𝑧̇1 𝜕𝑧1 92, 94–98]) for a slewing crane boom according to (E.2)- Taking into account (F.4)-(F.5), (F.6) in the noninertial (E.3), we will have the following expression for “crane boom- B payload” kinetic energy: reference frame will finally take the following form: 𝑑2𝑥 𝑑𝜑 2 𝑑2𝜑 𝑚( 1 )−𝑚( 𝑒 ) 𝑥 −𝑚( 𝑒 )(𝑅+𝑦 ) 𝑚 𝑑𝑥 𝑑𝜑 2 2 1 2 1 𝑇= ((( 1 )−( 𝑒 )(𝑅+𝑦 )) 𝑑𝑡 𝑑𝑡 𝑑𝑡 2 𝑑𝑡 𝑑𝑡 1 𝑑𝜑 𝑑𝑦 𝑥 −2𝑚( 𝑒 )( 1 )=−𝑁( 1 ); 𝑑𝑦 𝑑𝜑 2 𝑑𝑧 2 𝑑𝑡 𝑑𝑡 𝑙 +(( 1 )+( 𝑒 )𝑥 ) +( 1 ) ) 𝑑𝑡 𝑑𝑡 1 𝑑𝑡 (F.4) 𝑑2𝑦 𝑑𝜑 2 𝑑2𝜑 𝑚( 1 )−𝑚( 𝑒 ) (𝑅 + 𝑦 )+𝑚( 𝑒 )𝑥 𝑂 2 1 2 1 (F.7) 𝐼 2 𝑑𝜑 2 𝑑𝑡 𝑑𝑡 𝑑𝑡 + 33 ( 𝑒 ) . 2 𝑑𝑡 𝑑𝜑 𝑑𝑥 𝑦 +2𝑚( 𝑒 )( 1 )=−𝑁( 1 ); 𝑑𝑡 𝑑𝑡 𝑙 Taking into account the nonlinearity and nonconser- 𝑑2𝑧 𝑙−𝑧 vatism of the cable reaction force N,andtheequationsfor 𝑚( 1 )=−𝑚𝑔+𝑁( 1 ). geometric constraints (C.1)–(C.5), we have the following for- 𝑑𝑡2 𝑙 mulae for the generalized forces in the noninertial reference frame B: The derived ODE system (F.7) coincides with the system (F.3), which confirms the accuracy and correctness of the fully 𝑥1 nonlinear equations (F.3) in relative Cartesian coordinates for 𝑄𝑥 =−𝑁( ); 1 𝑙 payload 𝑀 swaying motion in the noninertial reference frame B. 𝑦1 𝑄𝑦 =−𝑁( ); 1 𝑙 (F.5) G. Uniform Crane Boom Rotation 𝑙−𝑧1 𝑄𝑧 =+𝑁( )−𝑚𝑔. Assumption: The Introduction of 1 𝑙 the Noninertial Reference Frame F We will derive the same nonlinear differential equations We now address the case of uniform crane boom slewing. We (F.3) for relative system motion with an introduction of assume that the crane boom 𝐵𝑂2 rotates with the constant Shock and Vibration 17

angular velocity 𝜔𝑒 around vertical axis 𝐷𝑂2; that is, that the 𝑥1, 𝑦1,and𝑧1 in noninertial reference frame B according to noninertial reference frame B uniformly rotates around the the following equations (Figures 8–10): unit vector ̂e3 of the inertial reference frame E.Suchcase takes place when the right-hand side of (E.5)iszero,thatis, 𝑥1 =𝑥; for the steady state of crane boom rotation with 𝑦 =𝑦 +𝑦; 1 dyn (G.5) 𝑑2𝜑 𝑑𝜑 𝑒 =0; 𝑒 =𝜔 = . (G.1) 𝑧 =Δ+𝑧. 𝑑𝑡2 𝑑𝑡 𝑒 const 1

So, taking into account (G.1), the third terms, containing With (G.3)–(G.5) the nonlinear system (G.2)inthe 2 2 F 𝑑 𝜑𝑒/𝑑𝑡 ,vanishinthe1stand2ndequationsofsystem(F.7). noninertial reference frame takes the following form The second terms of (F.7) are linearly coordinate-dependent (Figures 8–10): on 𝑥1, 𝑦1, and the forth terms of (F.7) are linearly velocity- 𝑑𝑥 /𝑑𝑡 𝑑𝑦 /𝑑𝑡 𝑑2𝑥 𝑑𝜑 2 dependent on 1 and 1 . So in the noninertial 𝑚( )−𝑚( 𝑒 ) 𝑥 reference frame B we have 𝑑𝑡2 𝑑𝑡 𝑑2𝑥 𝑑𝜑 2 𝑑𝜑 𝑑𝑦 𝑥 𝑚( 1 )−𝑚( 𝑒 ) 𝑥 −2𝑚( 𝑒 )( )=−𝑁( ); 𝑑𝑡2 𝑑𝑡 1 𝑑𝑡 𝑑𝑡 𝑙

2 2 𝑑𝜑 𝑑𝑦 𝑥 𝑑 𝑦 𝑑𝜑𝑒 −2𝑚( 𝑒 )( 1 )=−𝑁( 1 ); 𝑚( )−𝑚( ) (𝑅 + 𝑦 +𝑦) (G.6) 𝑑𝑡 𝑑𝑡 𝑙 𝑑𝑡2 𝑑𝑡 dyn 𝑑2𝑦 𝑑𝜑 2 𝑑𝜑 𝑑𝑥 𝑦 +𝑦 𝑚( 1 )−𝑚( 𝑒 ) (𝑅 + 𝑦 ) +2𝑚( 𝑒 )( )=−𝑁( dyn ); 𝑑𝑡2 𝑑𝑡 1 (G.2) 𝑑𝑡 𝑑𝑡 𝑙 𝑑𝜑 𝑑𝑥 𝑦 𝑑2𝑧 𝑙−Δ−𝑧 +2𝑚( 𝑒 )( 1 )=−𝑁( 1 ); 𝑚( )=−𝑚𝑔+𝑁( ). 𝑑𝑡 𝑑𝑡 𝑙 𝑑𝑡2 𝑙 𝑑2𝑧 𝑙−𝑧 𝑚( 1 )=−𝑚𝑔+𝑁( 1 ). 𝛼 𝑑𝑡2 𝑙 H. The Small Swaying Angle 1 Assumption

We now address the case of the small swaying angle 𝛼1.In The nonlinear system (G.2) has been presented in the the case of small swaying angle 𝛼1 the system (C.3) defines 𝑥1 B noninertial reference frame for the case of uniform crane and 𝑦1 as the small variables, and 𝑧1 =0.Having𝑧1 =0we boom slewing. conclude that Δ=0and all time derivatives are zero, that is, For further physical analysis of the swaying problem and the vertical load velocity 𝑑𝑧1/𝑑𝑡 =0 and the vertical payload 2 2 for ease of building the analytical solution we introduce the acceleration 𝑑 𝑧1/𝑑𝑡 =0. F noninertial reference frame . The origin of the noninertial Thus, the 3rd equation of system (F.7) yields that the cable F reference frame we connect with the so-called point reaction force N approximately coincides the gravitational 𝐴 𝑀 𝑦 dyn of dynamic equilibrium for load ,wherethe - force mg;thatis,N ≈ mg.Asaresultthesystem(G.2), B distance between the noninertial references frames and containing three ODEs, transforms into a linearized system F we define through the numerical solution of the following with two independent equations for the relative Cartesian relative equilibrium transcendental equation: coordinates 𝑥1 and 𝑦1 in the noninertial reference frame B. We then cancel the mass 𝑚 of load 𝑀 from the system of 𝑚𝑔𝑙 ⋅ (𝛼 )=𝑚𝜔2 (𝑅+𝑙⋅ (𝛼 ))⋅𝑙⋅ (𝛼 ). sin 1dyn 𝑒 sin 1dyn cos 1dyn (G.6). Consider (G.3) 𝑑2𝑥 𝑑𝜑 2 𝑥 𝑑𝜑 𝑑𝑦 1 −( 𝑒 ) 𝑥 +𝑔( 1 )−2( 𝑒 )( 1 )=0; This equationG.3 ( ) has been derived as the momentum 𝑑𝑡2 𝑑𝑡 1 𝑙 𝑑𝑡 𝑑𝑡 sum for the gravitational force mg and the normal inertial Φ n 𝐵 𝑀 ( )=𝑀 (Φ n) 𝑑2𝑦 𝑑𝜑 2 𝑦 𝑑𝜑 𝑑𝑥 force e about point ,thatis, 𝐵 mg 𝐵 e . 1 −( 𝑒 ) (𝑅 + 𝑦 )+𝑔( 1 )+2( 𝑒 )( 1 )=0. 𝛼 2 1 Taking into account the numerically derived angle 1dyn 𝑑𝑡 𝑑𝑡 𝑙 𝑑𝑡 𝑑𝑡 𝑦 from (G.3) we find the value of the horizontal distance dyn (H.1) 𝐴 𝐴 between points st and dyn (Figures 8–10) according to the following equation: So we have derived the system (H.1) of differential equations for relative motion of load 𝑀 on cable 𝐵𝑀 with a 𝑦 =𝐴 𝐴 =𝑙 (𝛼 ). dyn st dyn sin 1dyn (G.4) movablesuspensioncenter𝐵, which is attached to the crane boom 𝐵𝑂2 in the noninertial reference frame B We then Relative coordinates 𝑥, 𝑦,and𝑧 in noninertial reference transfertheoriginofcoordinatesystem𝑂𝑥𝑦𝑧 (Figures 8–10) F 𝐴 𝑀 frame have been connected with the relative coordinates to the point dyn of dynamic equilibrium for load ;thatis, 18 Shock and Vibration

we make the transition from the noninertial reference frame a𝑀/F = ar (B.4) for the relative acceleration of load 𝑀 in B to the noninertial reference frame F and simplify (G.5)to the noninertial reference frame F. The straight-line terms 2 2 (𝜔𝑒 − (𝑔/𝑙))𝑥 and (𝜔𝑒 − (𝑔/𝑙))𝑦 in (H.4)-(H.5)arelinear 𝑥 =𝑥; 1 proportional to the relative payload coordinates in the nonin- ertial reference frame F.Thesestraight-linetermshavebeen 𝑦1 =𝑦 +𝑦; (H.2) dyn determined by the contribution of the normal or centripetal 𝜔 ×(𝜔 × )= n 𝑧1 =𝑧. acceleration B/E B/E r𝐸/𝑀 ae of transportation for payload 𝑀 andbyappearanceofcorrespondingD’Alembert n n The second equation of system (H.1) defines the amount centrifugal inertia force Φe =(−𝑚)ae duetocraneboom of dynamic deflection, that is, the 𝑦-distance between the transport rotation in the noninertial reference frame B. noninertial reference frames B and F as The third terms −2𝜔𝑒(𝑑𝑦/𝑑𝑡) and 2𝜔𝑒(𝑑𝑥/𝑑𝑡) in (H.4)-(H.5) 2 2 have been defined by the compound or Coriolis acceleration 𝜔 𝑅𝑙 𝜔 𝑅 2𝜔 × = 𝑀 𝑦 =𝐴 𝐴 = 𝑒 =− 𝑒 , B/E V𝑀/B acor of payload in the noninertial dyn st dyn 2 2 (H.3) 𝑔−𝜔𝑒 𝑙 (𝜔𝑒 −(𝑔/𝑙)) reference frame F. The rectangular Cartesian projections of Coriolis inertia force in the noninertial reference frame F are 𝜔 =𝑑𝜑/𝑑𝑡 where 𝑒 𝑒 . defined by formula (D.5). Then with the introduction ofH.2 ( )and(H.3)into(H.1) we have the normal system of two linear homogeneous differential equations of second order for relative motion of I. Analytical Solution of the Linearized System 𝑀 F the load in the noninertial reference frame : After generation of the determinant of natural frequencies 𝑑2𝑥 𝑑𝜑 2 𝑔 𝑑𝜑 𝑑𝑦 matrix for the system (H.4)-(H.5)wewillwritethefollowing −(( 𝑒 ) −( )) 𝑥 − 2 ( 𝑒 )( )=0; 𝑑𝑡2 𝑑𝑡 𝑙 𝑑𝑡 𝑑𝑡 characteristic biquadratic equation of fourth order for system (H.4)-(H.5)intheform: (H.4) 2 2 𝑑 𝑦 𝑑𝜑 𝑔 𝑑𝜑 𝑑𝑥 2 −(( 𝑒 ) −( )) 𝑦 + 2 ( 𝑒 )( )=0. 4 𝑔 2 2 𝑔 2 2 𝜆 +2(( ) +𝜔 ) 𝜆 + (( ) −𝜔 ) =0. (I.1) 𝑑𝑡 𝑑𝑡 𝑙 𝑑𝑡 𝑑𝑡 𝑙 𝑒 𝑙 𝑒

The same system in the noninertial reference frame F can Using (I.1)weadjusttheroots𝜆1 and 𝜆2 of the secular be derived from (G.6) for the case of the small swaying angle equation: 𝛼1. Taking into account that 𝜔𝑒 =𝑑𝜑𝑒/𝑑𝑡 = const we will write (H.4)as 𝜆1 =±]1𝑖; 𝜆2 =±]2𝑖, 2 𝑑 𝑥 𝑔 𝑑𝑦 ]1 =𝑘+𝜔𝑒; ]2 =𝑘−𝜔𝑒; −(𝜔2 −( ))𝑥−2𝜔 ( )=0; (I.2) 2 𝑒 𝑒 𝑑𝑡 𝑙 𝑑𝑡 𝑔 (H.5) 𝑘=√ ;𝜔𝑒 =𝑘.̸ 𝑑2𝑦 𝑔 𝑑𝑥 𝑙 −(𝜔2 −( )) 𝑦 + 2𝜔 ( )=0. 𝑑𝑡2 𝑒 𝑙 𝑒 𝑑𝑡 Taking into account (I.2), the law of relative load 𝑀 motion in the noninertial reference frame F takes the Taking into account (C.6),theabsolutecoordinates𝑥2, 𝑦2, following form: and 𝑧2 in inertial reference frame E, which depend upon the relative coordinates 𝑥, 𝑦,and𝑧 in the noninertial reference 𝑥 (𝑡) =𝐶1 sin (]1𝑡) +2 𝐶 cos (]1𝑡) frame F during the swaying of load 𝑀, will be defined according to the following equations (Figures 8–10): +𝐶3 sin (]2𝑡) +4 𝐶 cos (]2𝑡) ; (I.3) 𝑥 =(𝑅+𝑦 +𝑦) (𝜑 )+𝑥 (𝜑 ); 2 dyn cos 𝑒 sin 𝑒 𝑦 (𝑡) =𝐶1 cos (]1𝑡) −2 𝐶 sin (]1𝑡) 𝑦 =(𝑅+𝑦 +𝑦) (𝜑 )−𝑥 (𝜑 ); −𝐶 (] 𝑡) + 𝐶 (] 𝑡) . 2 dyn sin 𝑒 cos 𝑒 (H.6) 3 cos 2 4 sin 2

𝑧2 =𝑧. Time derivatives of (I.3) yield the following projections of relative payload 𝑀 velocity in the noninertial reference frame So the linearized ODE system (H.4)-(H.5)hasbeen F: derived for the case of uniform crane boom slewing (G.1)and 𝑑 (𝑥 (𝑡)) the small swaying angle 𝛼1 (H.2). =+𝐶] (] 𝑡) − 𝐶 ] (] 𝑡) 𝑑𝑡 1 1 cos 1 2 1 sin 1 It is important to note that the left-hand sides of payload motion equations in linearized problem (H.4)-(H.5)forthe +𝐶3]2 cos (]2𝑡) −4 𝐶 ]2 sin (]2𝑡) ; F noninertial reference frame may be derived with an intro- (I.4) 𝑑(𝑦(𝑡)) duction of Lagrange equations (F.6). However the discussion =−𝐶] (] 𝑡) − 𝐶 ] (] 𝑡) of (H.4)-(H.5)ismoresuitableandinformativewithan 𝑑𝑡 1 1 sin 1 2 1 cos 1 introduction of dynamic Coriolis theorem (F.1)-(F.2). The 2 2 2 2 +𝐶] (] 𝑡) + 𝐶 ] (] 𝑡) . first terms 𝑑 𝑥/𝑑𝑡 ; 𝑑 𝑦/𝑑𝑡 in (H.4)-(H.5) define the vector 3 2 sin 2 4 2 cos 2 Shock and Vibration 19

In order to determine the arbitrary constants of integra- The substitution ofI.12 ( )into(I.3)and(I.4)yieldsthe tion 𝐶1,...,𝐶4 in (I.3)-(I.4) we define the initial conditions following system for derivation the analytical expressions for of the problem according to Figures 8–10 and Figure 11. the arbitrary constants of integration 𝐶1,...,𝐶4: 𝑡=0 𝑀 𝐵𝐴 In the initial time load has the vertical st in- 0=𝐶 +𝐶 ; line position (Figures 8–10); that is, the load 𝑀 initial position 2 4 𝐴 coincides with the static equilibrium position st for the load 𝑉𝐵 =𝐶1]1 +𝐶2]2; 𝑀 on the cable 𝑀𝐵. This means that the initial coordinates of (I.13) −𝑦 =𝐶 −𝐶 ; payload 𝑀 are as follows: dyn 1 3

𝐴 (𝑅, 0,) 0 E; 0=−𝐶2]1 +𝐶4]2. point st in the reference frame (I.5) The algebraic expressions for the arbitrary constants 𝐴 (0, 0, 0) B; point st in the reference frame (I.6) of integration 𝐶1,...,𝐶4 can be derived as the following solution of the system (I.13): point 𝐴 (0, −𝑦 ,0) in the reference frame F. (I.7) st dyn 𝑉 −𝑦 ] 𝐵 dyn 2 𝐶1 = ;𝐶2 =0; At time 𝑡=0the load 𝑀 haszeroabsolutevelocityinthe ]1 + ]2 inertial reference frame E;thatis,V𝑀/E(0) = 0. 𝑉𝐵 +𝑦 ]1 In order to determine the load 𝑀 relative velocity in the 𝐶 = dyn ;𝐶=0; 3 ] + ] 4 noninertial reference frame F, which is equal to the load 1 2 𝑀 B (I.14) relative velocity in the noninertial reference frame ,we 𝑉 −𝑦 (𝑘− 𝜔 ) 𝐵 dyn 𝑒 address the velocity addition theorem (A.12)andformula or 𝐶1 = ;𝐶2 =0; (A.14): 2𝑘 𝑉 +𝑦 (𝑘+ 𝜔 ) 𝐵 dyn 𝑒 0 = V𝑀/B (0) + 𝜔B/E (0) × r𝑂 /𝑀 (0) . 𝐶3 = ;𝐶4 =0. 2 (I.8) 2𝑘 𝑀 The vector equation (I.8) determines the initial relative So, the relative motion equations for pendulum sway- ing take the final form as stated below: velocity V𝑀/B(0) of the load 𝑀 in the inertial reference frame E as 𝑉 −𝑦 (𝑘− 𝜔 ) 𝑥 (𝑡) =( 𝐵 dyn 𝑒 ) (] 𝑡) 2𝑘 sin 1 (0) = (0) =−𝜔 (0) × (0) V𝑀/B V𝑀/F B/E r𝑂2/𝑀 󵄩 󵄩 (I.9) 𝑉 +𝑦 (𝑘+ 𝜔 ) 󵄩 󵄩 𝐵 dyn 𝑒 =−𝜔B/E (0) × 󵄩r𝑂 /𝑀 (0)󵄩 ̂e2. +( ) sin (]2𝑡) ; 󵄩 2 󵄩 2𝑘 (I.15) The direction of the vector for the initial relative velocity 𝑉𝐵 −𝑦 (𝑘− 𝜔 𝑒) (0) ̂ 𝑦 (𝑡) =( dyn ) (] 𝑡) V𝑀/B is directed opposite to the unit vector e2 in the 2𝑘 cos 1 inertial reference frame E;thatis,V𝑀/B(0) ↑↓ ̂e2.Theinitial direction of the vector V𝑀/B(0) is invariant in all reference 𝑉 +𝑦 (𝑘+ 𝜔 ) E B F −( 𝐵 dyn 𝑒 ) (] 𝑡) . systems , ,and . So in the noninertial reference frame 2𝑘 cos 2 B we have V𝑀/𝐵(0) ↑↑ ̃e1: The obtained analytical equationsI.15 ( ), derived for the 󵄩 󵄩 (0) = (0) =+𝜔 (0) × 󵄩 (0)󵄩 ̃ . linearized model (H.4)-(H.5), determine the shape of the rel- V𝑀/B V𝑀/F B/E 󵄩r𝑂2/𝑀 󵄩 e1 (I.10) ative payload 𝑀 swaying trajectory as is shown in Figure 12. Numerical computations in Figure 12 have been carried out Taking into account that ‖𝜔B/E(0)‖ = 𝜔𝑒 and for the following values of mechanical system parameters: ‖r𝑂 /𝑀(0)‖ = 𝑅 in (I.10), we have the following scalar 2 0.5 2 𝑅 = 0.492 m; 𝑔 = 9.81 m/s ; 𝑙=0.825m; 𝑘=(𝑔/𝑙) ≈ equation in the noninertial reference frames B and F: ∘ 3.448 rad/s; 𝑇=30s; 𝜔𝑒 = 2𝜋/𝑇 ≈ 0.209 rad/s; 𝜑𝑒 = 180 ; 𝛼 = 0.00221 𝑉 = 0.103 𝑦 = 0.00182 𝑑𝑥 1dyn rad; 𝐵 m/s; dyn m; (0) = (0) = (0) =𝜔𝑅. (I.11) ]1 =𝑘+𝜔𝑒 = 3.6578 rad/s; ]2 =𝑘−𝜔𝑒 = 3.2388 rad/s; V𝑀/B V𝑀/F 𝑑𝑡 𝑒 𝐶1 = 0.01408 m; and 𝐶3 = 0.01591 m(Figure 12). The above mentioned numerical values of analytical relations Thus, with an introduction ofI.5 ( )–(I.11), the initial (I.15)subjecttotheinitialconditions(I.5)–(I.12) define a conditions for ODE system (H.4)-(H.5) in the noninertial 𝑀 F theoretically computed law of relative motion for load . reference frame are as follows: After the elimination of variable 𝑡 from the system of (I.15) 𝑀 𝑑𝑥 wewillhavetherelativetrajectoryforload (Figure 12). The 𝑥 (0) =0; (0) =𝜔𝑒𝑅; system (H.6) and the computational relative trajectories for 𝑑𝑡 𝑀 (I.12) load swaying in Figure 12 show that there is the contra- 𝑑𝑦 rotation of the noninertial reference frames B and F to crane 𝑦 (0) =−𝑦 ; (0) =0. dyn 𝑑𝑡 boom 𝐵𝑂2 slewing. 20 Shock and Vibration

Relative trajectory of load M swaying for linearized model

t2 =3s Relative trajectory of load M swaying 0.010 y for linearized model 0.005 0.007 t1 = 1.25 s y 0.002 −0.02 −0.01 0 0.01 0.02 −0.005 −0.02 −0.01−0.002 0.01 0.02 x x −0.010

y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 0.209 rad/s y(x): l = 0.825 m;k≈3.448rad/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 t =10s 3 t4 =25s

0.02 0.02

y y

0.01 0.01

−0.02 −0.01 0 0.01 0.02 −0.02 −0.01 00.01 0.02 x x −0.01 −0.01

−0.02 −0.02

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

Figure 12: The relative trajectories for load 𝑀 swaying on the cable during uniform rotational motion of the crane boom 𝐵𝑂2 for the moments of times 𝑡1 = 1.25 s(a),𝑡2 =3s(b),𝑡3 =10s, (c) and 𝑡4 =25s (d), derived with linearized model introduction.

J. Velocity Kinematics Analysis in payload 𝑀 (Figure 13). The advantage of the introduction of Relative Spherical Coordinates of the natural noninertial comoving reference frames C, D,and G the Noninertial Reference Frame B: is grounded on the orthogonality of the reaction force N to the unit vectors e𝜏𝛼 and e𝜏𝛼 , that is, on the fact that N ⊥ e𝜏𝛼 The Introduction of the Noninertial 1 2 1 and N ⊥ e𝜏𝛼 . C D G 2 Reference Frames , ,and The vector equation (A.12) for velocity addition will take the form The fully nonlinear equations (F.7) in relative Cartesian coordinates of the noninertial reference frame B have the unknown function 𝑁=𝑁(𝑡), that is, undetermined 𝑀/E = 𝑀/C + 𝑀/D + 𝜔B/E × 𝑂 /𝑀, Lagrangian coefficient. The reaction force N of the cable 𝐵𝑀 V V V r 2 (J.1) in Figures 8–10 and Figure 13 is the nonconservative force. So for derivation of solvable nonlinear equations we 𝑀 = = should remove the cable tension force N from the payload where V𝑀/C V𝛼1 and V𝑀/D V𝛼2 are the relative payload motion equations with the introduction of Newton’s second velocities in the natural noninertial comoving reference law and the natural noninertial comoving reference frames frames C and D. C, D,andG, which correspond to the spherical angular In Figure 13 and by using (J.1)wehavethatthenatural coordinates (𝛼1, 𝛼2,and𝜑𝑒) and moving together with components of payload 𝑀 velocity in the natural noninertial Shock and Vibration 21

e 𝜏𝛼2 y1

𝒟 V e𝜏𝜑 𝛼2

𝒞 e cos 𝛼1 𝒢 V𝜑 𝜏𝛼1 𝛼2 V cos 𝛼 𝛼1 1 𝜃 𝛼2 M sin 𝛼1 en𝛼 ẽ 1 2 e ℬ n𝛼2 𝒞 𝒞 𝒟 𝒢 ( ) e B O1 n𝜑 eb𝛼 𝛼2 1

e e b𝛼2 ‖ b𝜑 x1 y2 ẽ1

̂e ℬ 2 𝜃

𝛼 ℰ 1 𝜑e

O (D E 2 , )

̂e1 x2 ℰ

Figure 13: Swaying scheme for velocity kinematics analysis in spherical coordinates of load 𝑀 at the cable 𝑀𝐵, which is fixedly attached in the point 𝐵 to the crane boom 𝐵𝑂2, in the horizontal plane (𝑥1𝑦1) for nonlinear model derivation in relative spherical coordinates. comoving reference frames C, D,andG have the following So formula (K.1)willtakethefollowingform: values:

𝑑𝛼1 = + + ( )+ ( ) a𝑀/E a𝑀/C a𝑀/D acor V𝛼1 acor V𝛼2 𝑉𝛼 =𝑙 ; 1 𝑑𝑡 (K.2) + 𝜔 ×(𝜔 × )+𝛼 × . D/E D/E r𝑂2/𝑀 D/E r𝑂2/𝑀 𝑑𝛼2 𝑉𝛼 =𝑙sin (𝛼1) ; (J.2) 2 𝑑𝑡 The directions of vectors have been shown in Figures 13, 𝑑𝜑𝑒 𝑉 =𝑂𝑀 . 14,and15. The scalar values of vectors 𝜔C/E and 𝜔D/E are as 𝜑 2 𝑑𝑡 follows:

K. Acceleration Kinematics 𝑑𝜑 𝑑𝛼 𝜔 = 𝑒 + 2 ; Analysis in Relative Spherical C/E 𝑑𝑡 𝑑𝑡 (K.3) Coordinates in the Noninertial Reference 𝑑𝜑 Frames C, D,andG 𝜔 = 𝑒 . D/E 𝑑𝑡 The vector equation (B.1) for acceleration addition will take the following form (Figure 14): Taking into account (K.3), the magnitudes of Coriolis acceleration vectors in the natural noninertial comoving = + +2𝜔 × a𝑀/E a𝑀/C a𝑀/D C/E V𝑀/C reference frames C, D,andG in Figure 14 have the following values: +2𝜔D/E × V𝑀/D (K.1) 𝑑𝛼 𝑑𝜑 𝑑𝛼 + 𝜔D/E ×(𝜔D/E × r𝑂 /𝑀)+𝛼D/E × r𝑂 /𝑀, 1 𝑒 2 2 2 a (𝑉𝛼 )=2𝑙( )( + ) cos (𝛼1); (K.4) cor 1 𝑑𝑡 𝑑𝑡 𝑑𝑡 ( )= ( )=2𝜔 × =2𝜔 × where acor V𝑀/C acor V𝛼1 C/E V𝑀/C C/E V𝛼1 ( )= ( )=2𝜔 × =2𝜔 × 𝑑𝛼2 𝑑𝜑𝑒 and acor V𝑀/D acor V𝛼2 D/E V𝑀/D D/E V𝛼2 (𝑉 )=2𝑙( )( ) (𝛼 ). acor 𝛼2 sin 1 (K.5) in Figure 14. 𝑑𝑡 𝑑𝑡 22 Shock and Vibration

(V ) 𝐚cor 𝛼1

y1 𝐚𝜏𝛼2

𝐚𝜏𝜑 𝛼2 cos 𝛼1 𝐚𝜏𝛼1 𝜃 𝛼2 sin 𝛼 M ẽ2 1 𝐚n𝛼1 𝜃 ℬ (V ) 𝐚cor 𝛼2

𝐚n𝛼2 ( ) 𝐚n𝜑 B O1 𝛼2

y2 x1 ẽ1 ℬ ̂e2 𝜃

𝛼1 ℰ 𝛼1 𝜑e O (D E 2 , )

̂e 1 x2 ℰ

Figure 14: Swaying scheme for acceleration kinematics analysis in spherical coordinates of load 𝑀 at the cable 𝑀𝐵, which is fixedly attached in the point 𝐵 to the crane boom 𝐵𝑂2, in the horizontal plane (𝑥1𝑦1) for nonlinear model derivation in relative spherical coordinates.

y1

Φ n𝜑 Φ (V ) cor 𝛼2

𝜃 Φ sin 𝛼1 n𝛼 Φ 1 𝜃 n𝛼2 𝛼2 M Φ cos 𝛼 𝜏𝛼1 1 𝜃 ẽ 𝜃 ℬ 2 𝛼2 Φ 𝜏𝜑 N ( ) B O1 𝛼2 Φ 𝜏𝛼2 Φ (V ) cor 𝛼1

y2 x ẽ1 1 𝜃 ℬ ̂e2

ℰ 𝜑e

O2(D, E) x ̂e1 2 ℰ

Figure 15: Swaying scheme for dynamic analysis in spherical coordinates of load 𝑀 at the cable 𝑀𝐵, which is fixedly attached in the point 𝐵 to the crane boom 𝐵𝑂2, in the horizontal plane (𝑥1𝑦1) for nonlinear model derivation in relative spherical coordinates. Shock and Vibration 23

2 The tangential accelerations in the natural noninertial 𝑑𝛼2 Φ𝑛𝛼 =𝑚𝑎𝑛𝛼 =𝑚( ) 𝑙 sin (𝛼1); comoving reference frames C, D, G havethevaluesasstated 2 2 𝑑𝑡 below: 𝑑𝜑 2 󵄩 󵄩 Φ =𝑚𝑎 =𝑚( 𝑒 ) 󵄩𝑂 𝑀󵄩 ; 𝑛𝜑 𝑛𝜑 𝑑𝑡 󵄩 2 󵄩 𝑑2𝛼 1 Φ (𝑉𝛼 )=𝑚𝑎 (𝑉𝛼 ) 𝑎𝜏𝛼 =( )𝑙; cor 1 cor 1 1 𝑑𝑡2 𝑑𝛼1 𝑑𝜑𝑒 𝑑𝛼2 2 =2𝑚𝐿( )( + ) cos (𝛼1); 𝑑 𝛼2 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑎𝜏𝛼 =( )𝑙sin (𝛼1); (K.6) 2 𝑑𝑡2 Φ (𝑉 )=𝑚𝑎 (𝑉 ) cor 𝛼2 cor 𝛼2 𝑑2𝜑 𝑎 =( 𝑒 )(𝑂 𝑀) . 𝑑𝛼 𝑑𝜑 𝜏𝜑 𝑑𝑡2 2 =2𝑚𝐿( 2 )( 𝑒 ) (𝛼 ). 𝑑𝑡 𝑑𝑡 sin 1 (L.1)

The normal accelerations in the natural noninertial comoving reference frames C, D, G havethevaluesasstated M. Derivation of the Fully Nonlinear below: Equations in Relative Spherical Coordinates in the Noninertial Reference Frames C, D,andG 2 𝑑𝛼1 𝑎𝑛𝛼 =( ) 𝑙; 1 𝑑𝑡 Taking into account derived expressions (L.1), the introduc- tion of D’Alembert’s principle in the noninertial reference 𝑑𝛼 2 frames C, D, G yields the following scalar equations in the 𝑎 =( 2 ) 𝑙 (𝛼 ); 𝑛𝛼 sin 1 (K.7) 𝜏𝛼 𝜏𝛼 2 𝑑𝑡 projections to the axes e 1 and e 2 : 𝑑𝜑 2 𝑎 =( 𝑒 ) (𝑂 𝑀) . 𝑛𝜑 𝑑𝑡 2 :0=−Φ +Φ (𝛼 )+Φ (𝑉 ) (𝛼 ) e𝜏𝛼1 𝜏𝛼1 𝑛𝛼2 cos 1 cor 𝛼2 cos 1

+Φ𝑛𝜑 sin (𝛼2 +𝜃)cos (𝛼1) L. Forces Imposed on the Payload in Relative +Φ𝜏𝜑 cos (𝛼2 +𝜃)cos (𝛼1)−mg sin (𝛼1)=0; Spherical Coordinates in the Noninertial C D G :0=−Φ −Φ (𝑉 ) Reference Frames , ,and e𝜏𝛼2 𝜏𝛼2 cor 𝛼1 𝑀 Among the forces, imposed on the payload in Figure 15, +Φ𝑛𝜑 cos (𝛼2 +𝜃)−Φ𝜏𝜑 sin (𝛼2 +𝜃), we have an active force of gravity mg,thecablereactionforce (M.1) N, the tangential inertial forces Φ𝜏𝛼 ; Φ𝜏𝛼 ; Φ𝜏𝜑,thenormal Φ Φ 1 Φ 2 or centrifugal inertial forces 𝑛𝛼1 ; 𝑛𝛼2 ; 𝑛𝜑, and the Coriolis inertial forces Φ (𝑉𝛼 ) and Φ (𝑉𝛼 ). cor 1 cor 2 where Taking into account formulae (K.3)–(K.7) we will express below all imposed forces in the noninertial reference frames C, D, G: 𝑙 sin (𝛼1) cos (𝛼2) sin (𝜃) = 󵄩 󵄩 ; 󵄩𝑂2𝑀󵄩 (M.2) 𝑑2𝛼 (𝑅 + 𝑙 sin (𝛼1) sin (𝛼2)) 1 (𝜃) = 󵄩 󵄩 . Φ𝜏𝛼 =𝑚𝑎𝜏𝛼 =𝑚( )𝑙; cos 󵄩 󵄩 1 1 𝑑𝑡2 󵄩𝑂2𝑀󵄩

2 𝑑 𝛼2 Φ𝜏𝛼 =𝑚𝑎𝜏𝛼 =𝑚( )𝑙sin (𝛼1); 2 2 𝑑𝑡2 N. Uniform Crane Boom Rotation Assumption with Nonlinear Equations in Relative 𝑑2𝜑 𝑒 󵄩 󵄩 Spherical Coordinates in the Noninertial Φ𝜏𝜑 =𝑚𝑎𝜏𝜑 =𝑚( ) 󵄩𝑂2𝑀󵄩 ; 𝑑𝑡2 Reference Frames B 2 𝑑𝛼1 Assuming formulae (G.1), we will have the following non- Φ𝑛𝛼 =𝑚𝑎𝑛𝛼 =𝑚( ) 𝑙; 1 1 𝑑𝑡 linear ODE system in the noninertial reference frame B, 24 Shock and Vibration

Relative trajectory of load M swaying for nonlinear model 0.015 t2 =3s Relative trajectory of load M swaying for nonlinear model 0.010

0.008 0.005 t1 = 1.25 s 0.005 0 0.002 −0.02 −0.01 0.01 x 0.02 −0.005 0.825 ∗ sin(𝛼 )∗cos(𝛼 ) −0.02 −0.01 0 0.01 0.02 1 2 0.825 ∗ sin(𝛼1)∗cos(𝛼2)

y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 0.209 rad/s y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 0.209 rad/s (a) (b) Relative trajectory of load M swaying Relative trajectory of load M swaying for nonlinear model for nonlinear model

0.03 0.03 t4 =25s t3 =10s

0.02 0.02

y 0.01 0.01

−0.02 −0.01 0 0.01 0.02 −0.02 −0.01 0 0.01 0.02 x 0.825 ∗ sin(𝛼1)∗cos(𝛼2) 0.825 ∗ sin(𝛼1)∗cos(𝛼2) −0.01 −0.01

−0.02 −0.02

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

Figure 16: The relative trajectories for load 𝑀 swaying on the cable during uniform rotational motion of the crane boom 𝐵𝑂2 for the moments of times 𝑡1 = 1.25 s(a),𝑡2 =3s(b),𝑡3 =10s(c),and𝑡4 =25s (d), derived with nonlinear model introduction.

describing payload 𝑀 relative swaying in the case of uniform the noninertial reference frame F and with introduction of crane boom slewing: constraint equations (C.3):

𝑙 sin (𝛼1 (0)) cos (𝛼2 (0))=0; 𝑑2𝛼 1 𝑑𝛼 2 1 ( 1 )− ( 2 ) (2𝛼 )− 𝜔2 (2𝛼 ) 2 sin 1 𝑒 sin 1 𝑙 (𝛼 (0)) (𝛼 (0))=−𝑦 ; 𝑑𝑡 2 𝑑𝑡 2 sin 1 sin 2 dyn 𝑅 𝑔 𝑑𝛼 (0) −( )𝜔2 (𝛼 ) (𝛼 )=−( ) (𝛼 ); 𝑙( 1 ) (𝛼 (0)) (𝛼 (0)) 𝑙 𝑒 cos 1 sin 2 𝑙 sin 1 𝑑𝑡 cos 1 cos 2 𝑑2𝛼 𝑑𝛼 𝑑𝛼 𝑑𝛼 (0) ( 2 ) (𝛼 )+2( 1 )(𝜔 +( 2 )) (𝛼 ) −𝑙( 2 ) (𝛼 (0)) (𝛼 (0))=𝜔𝑅; 𝑑𝑡2 sin 1 𝑑𝑡 𝑒 𝑑𝑡 cos 1 𝑑𝑡 sin 1 sin 2 𝑒 𝑅 𝑑𝛼 (0) −( )𝜔2 (𝛼 )=0. 𝑙( 1 ) (𝛼 (0)) (𝛼 (0)) 𝑙 𝑒 cos 2 𝑑𝑡 cos 1 sin 2 (N.1) 𝑑𝛼 (0) +𝑙( 2 ) (𝛼 (0)) (𝛼 (0))=0. 𝑑𝑡 sin 1 cos 2 The initial conditions for the nonlinear system (N.1) may be formulated on the basis of the initial conditions in (N.2) Shock and Vibration 25

Relative trajectory of load M swaying Relative trajectory of load M swaying for linearized model for nonlinear model t =25s 0.08 4 t4 =25s 0.08 0.06 y 0.06 0.04 0.04 0.02

0.02 −0.08 −0.06−0.04−0.02 0 0.02 0.04 0.06 0.08 −0.02 x −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 sin cos −0.04 −0.02 0.825 ∗ (𝛼1)∗ (𝛼2)

−0.06 −0.04

−0.08 −0.06

y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 0.6 rad/s y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 0.6 rad/s (a) (b) Relative trajectory of load M swaying Relative trajectory of load M swaying for nonlinear model for linearized model 0.2 t4 =25s t4 =25s 0.15

y 0.10 0.1 0.05

−0.15 −0.10 −0.05 0 0.05 0.10 0.15 −0.05 x −0.15 −0.10 −0.05 0 0.05 0.10 0.15 −0.10 0.825 ∗ sin(𝛼1)∗cos(𝛼2)

−0.15 −0.1

y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 1.2 rad/s y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 1.2 rad/s (c) (d) Relative trajectory of load M swaying Relative trajectory of load M swaying for linearized model for nonlinear model t4 =25s t4 =25s 0.3 0.3 0.2 y 0.2 0.1

0.1 −0.3 −0.2 −0.1 0 0.10.2 0.3 −0.1 x −0.2 −0.1 0 0.1 0.2 −0.2 0.825 ∗ sin(𝛼1)∗cos(𝛼2) −0.1 −0.3

y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 1.8 rad/s y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 1.8 rad/s (e) (f)

Figure 17: The comparison of relative trajectories of load 𝑀 swaying for linearized (a), (c), (e), and nonlinear (b), (d), and (f) models in the cases of 𝜔𝑒/𝜔𝑒0 ≈ 2.871 (a-b), 𝜔𝑒/𝜔𝑒0 ≈ 5.742 (c-d), and 𝜔𝑒/𝜔𝑒0 ≈ 8.612 (e-f). 26 Shock and Vibration

Relative trajectory of load M swaying Relative trajectory of load M swaying for linearized model for nonlinear model t =25s 4 0.6 t4 =25s 0.4 0.5 y 0.4 0.2 0.3 0.2 0 −0.6 −0.4 −0.2 0.2 0.4 0.1 x −0.2 −0.4 −0.2 0 0.2 0.4 −0.1 0.825 ∗ sin(𝛼1)∗cos(𝛼2) −0.4 −0.2 −0.3

y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 2.4 rad/s y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 2.4 rad/s (a) (b) Relative trajectory of load M swaying Relative trajectory of load M swaying for linearized model for nonlinear model t =25s t4 =25s 0.8 4 0.8 0.6 0.6 y 0.4 0.4

0.2 0.2

0 0.2 0.4 0.6 0.8 −0.8 −0.6 −0.4 −0.2 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −0.2 0.825 ∗ sin(𝛼 )∗cos(𝛼 ) x −0.2 1 2 −0.4 −0.4 −0.6 −0.6 −0.8

y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 3.0 rad/s y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 3.0 rad/s (c) (d) Relative trajectory of load M swaying Relative trajectory of load M swaying for nonlinear model for linearized model t4 =25s t4 =25s 0.8 1 0.6

y 0.5 0.4

0.2

0

0.2 0.4 0.6 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −1.2 −1.0 −0.8 −0.6 −0.4 −0.2 sin cos x −0.2 0.825 ∗ (𝛼1)∗ (𝛼2) −0.5 −0.4

−1 −0.6

y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 3.6 rad/s y(x): l = 0.825 m;k≈3.448rad/s;𝜔e ≈ 3.6 rad/s (e) (f)

Figure 18: The comparison of relative trajectories of load 𝑀 swaying for linearized (a), (c), (e), and nonlinear (b), (d), and (f) models in the cases of 𝜔𝑒/𝜔𝑒0 ≈ 11.483 (a-b), 𝜔𝑒/𝜔𝑒0 ≈ 14.354 (c-d), and 𝜔𝑒/𝜔𝑒0 ≈ 17.225 (e-f). Shock and Vibration 27

The solution of the system (N.2) defines the initial conditions for the nonlinear ODE system (N.1)forpayload 50 spherical coordinates’ 𝛼1 and 𝛼2. The solution set member of the system (N.2)hasthefollowingform: 𝑦 dyn 40 𝛼1 (0) = arcsin ( ); (%) 𝑙 1 𝛿 l = 0.825 m;k≈3.448rad/s 𝜋 𝛼 (0) =− ; 2 2 30 𝑑𝛼 (0) (N.3) 1 =0; 𝑑𝑡 20 𝑑𝛼2 (0) 𝜔𝑒𝑅 = . discrepancy Amplitude 𝑑𝑡 𝑦 dyn 10 The numerical solution of the Cauchy problem (N.1)– (N.3) for the nonlinear problem in spherical coordinates determines the shape of the relative payload 𝑀 swaying trajectory as is shown in Figures 16, 17(b), 17(d), 17(f), 18(b), 246810 12 14 16 18(d),and18(f).NumericalplotsinFigure 16 have been Dimensionless crane boom slewing velocity 𝜔e/𝜔e0 carried out for the following values of mechanical system 2 Figure 19: The computational points (IIII)andregression parameters: 𝑅 = 0.492 m; 𝑔 = 9.81 m/s ; 𝑙 = 0.825 m; 0.5 diagram (—) for the relative dimensionless amplitude discrepancy 𝑘=(𝑔/𝑙) ≈ 3.448 rad/s; 𝑇=30s; 𝜔𝑒 = 2𝜋/𝑇 ≈ 0.209 rad/s; 𝛿 ∘ 1 between averaged radii of computational relative trajectories 𝜑𝑒 = 180 ; 𝛼1(0) ≈ 0.00221 rad; 𝛼2(0) ≈ −1.57079 rad; of payload 𝑀 in the noninertial reference frame F,derivedfor 𝑉 = 0.103 𝑦 = 0.00182 𝑑(𝛼 (0))/𝑑𝑡 =0 𝑙= 𝐵 m/s; dyn m; 1 rad/s; linearized and nonlinear models in the case of cable length 0.825 and 𝑑(𝛼2(0))/𝑑𝑡 = 56.565608 rad/s (Figure 16). m.

O. Comparative Estimation and Discussion of the Amplitude Responses, the experimentally chosen technological crane boom slewing Computed from the Linear and velocity. Nonlinear Models It is shown in Figures 12, 16, 17,and18 that there is a persistent trend 𝑟 >𝑟 . Forcomparisonofthelinear(H.4)-(H.5) and nonlinear (N.1) linearized nonlinear The computational diagram in Figure 19 shows that the models, we introduce the relative dimensionless amplitude 𝛿 𝛿 𝑟 relative dimensionless amplitude discrepancy 1 between discrepancy 1 between averaged radii nonlinear (Figures 𝑟 the linear and nonlinear models increases with increasing 17(b), 17(d), 17(f), 18(b), 18(d),and18(f))and linearized dimensionless crane boom slewing velocity 𝜔𝑒/𝜔𝑒0.Itis (Figures 17(a), 17(c), 17(e), 18(a), 18(c),and18(e))ofcompu- shown in Figure 19 that 𝛿1 <11%forthe𝜔𝑒 <0.8rad/s, that tational relative trajectories of payload 𝑀 in the noninertial is, for 𝜔𝑒 < 3.828 𝜔𝑒0.ItisshowninFigure 19 that 𝛿1 <11% reference frame F (Figures 17-18): for the 𝛼1 < 0.145 rad. 󵄨 󵄨 1 󵄨𝑟 −𝑟 󵄨 So we draw a conclusion that the small angle assumption 𝛿 =⟨ ((󵄨 nonlinear linearized󵄨) 1 2 𝑟 for the linearized model (H.4)-(H.5) adequately describes nonlinear nonlinear load swaying problem (N.1)duringuniformcrane 󵄨 󵄨 (O.1) 󵄨𝑟 −𝑟 󵄨 boom slewing for the above mentioned limiting values of 𝜔𝑒 +(󵄨 nonlinear linearized󵄨))⟩ . 𝛼 𝑟 and 1. linearized

The results of numerical amplitude comparison, com- Nomenclature puted from the linear and nonlinear models in accordance with the formula (O.1) and Figures 12, 16, 17,and18,areshown DOF: Degree of freedom in Figure 19. ODE: Ordinary differential equation The computational relative trajectories of load 𝑀 in the DAE: Differential algebraic equation F 𝐴 noninertial reference frame , shown in Figures 12, 16, 17 point st: Point of static equilibrium of payload 𝑟 𝑀 𝐵𝑀 and 18 outline that both averaged radii nonlinear (Figures 17(b), at the cable 𝑟 𝐴 17(d), 17(f), 18(b), 18(d) and 18(f))and linearized (Figures 17(a), point dyn: Point of dynamic equilibrium of 17(c), 17(e), 18(a), 18(c),and18(e)) and amplitudes of relative payload 𝑀 at the cable 𝐵𝑀 trajectories increase with increasing dimensionless crane point 𝑀: Point of position of payload 𝑀 in the boom slewing velocity 𝜔𝑒/𝜔𝑒0,where𝜔𝑒0 ≈ 0.209 rad/s is current moment of time 28 Shock and Vibration

𝐸 𝐵𝐸 point : Initial point of crane boom r𝑂 /𝐵: Position vector, connecting initial 𝐵 𝐵𝐸 2 point : Terminal point of crane boom point 𝑂2 and terminal point 𝐵 in point 𝐷: Center of circle for trajectory of noninertial reference frame B (m) transport and absolute motion for the 𝑅=𝐵𝐷: Boom projection to the horizontal 𝐵 point plane (̂e1, ̂e2)(m) 𝑚 𝑀 :Massofpayload(kg) 𝑙=𝑙𝐵𝑀 =‖r𝐵/𝑀‖: Length of the cable, that is, the 𝑔:Scalarvalueofgravitational 2 magnitude of the position vector acceleration (m/s ) r𝐵/𝑀 (m) 𝜋 0.5 : Ratio of perimeter of circle to 𝑘=(𝑔/𝑙) : Natural frequency of undamped diameter oscillations of payload 𝑀 at the cable 𝐶 𝐶 𝐶 𝐶 1; 2; 3 and 4: Constants of integration (m) 𝐵𝑀 (1/s) 𝜆 𝜆 1; 2: Roots of secular equation 𝜏=2𝜋/𝑘: Periodofundampedoscillationsof 𝑡 𝑀 :Swayingtimeforpayload(s) payload 𝑀 at the cable 𝐵𝑀 (s) E 𝑂2𝑥2𝑦2𝑧2 : Inertial reference frame ( ) 𝜑𝑒 =∠(̂e1, ̃e2): Angle of rotation of noninertial fixed on earth reference frame B around axis 𝑂2𝑧2 ̂ ̂ ̂ e1; e2; e3: Orthogonal unit vectors defining the of inertial reference frame E,thatis, E inertial reference frame the angle of transport rotation for 𝑥2 𝑦2 𝑧2 ; ; : Theabsolutecoordinatesofpayload crane boom 𝐵𝑂2 around the vertical 𝑀 with respect to Earth (m) axis 𝑂2𝑧2, that is, crane boom slewing B : Noninertial reference frame angle (rad) 𝑂 𝑥 𝑦 𝑧 ( 1 1 1 1)whichisconnectedwith 𝛾: Angle between the position vector rotating crane boom 𝑂2𝐵 and the ̂ ̂ r𝑂2/𝐵 and horizontal plane (e1, e2)or origin of coordinates point 𝑂1 (̃e1, ̃e2), that is, inclination angle of coincides with point 𝐴 of static st crane boom 𝑂2𝐵 with horizontal equilibrium for load 𝑀 at the cable ̂ ̂ ̃ ̃ 𝐵𝑀 plane (e1, e2)or(e1, e2)(rad) 𝛼1 =∠(̃e3, r𝐵/𝑀): Swaying angle between the vertical 𝑥1; 𝑦1; 𝑧1: The relative coordinates of payload 𝑀 axis 𝑂1𝑧1 and the position vector with respect to crane boom 𝑂2𝐵 (m) ̃ ̃ ̃ r𝐵/𝑀,thatis,thefirstspherical e1; e2; e3: The orthogonal unit vectors defining 𝑀 B coordinate of spherical pendulum the noninertial reference frame (rad) F: Noninertial reference frame (𝑂𝑥𝑦𝑧) 𝛼2: Swaying angle between the planes with coordinate axes parallel to 𝑥 𝑧 𝐵𝑀𝑂 𝑂 𝑥 𝑦 𝑧 ( 1 1)and( 1), that is, the ( 1 1 1 1)andtheoriginof second spherical coordinate of coordinates point 𝑂 coincides with 𝑀 𝐴 spherical pendulum (rad) point dyn of dynamic equilibrium 𝛽=∠(𝐴𝐵𝐴 ) 𝑀 𝑀 𝐵𝑀 𝑡 dyn : Angle of ascent of payload over for load at the cable 𝐴 dynamical equilibrium level dyn e1; e2; e3: Orthogonal unit vectors defining the (rad) noninertial reference frame F 𝜃: Angle between the vertical plane, C: Natural noninertial comoving connected with crane boom 𝑂2𝐵,and reference frame which corresponds the vertical plane, connected with to the swaying angle 𝛼1 and moving position vector r𝑂 /𝑀 of payload 𝑀 together with payload 𝑀 2 in inertial reference frame E (rad) e𝜏𝛼1 ; e𝑛𝛼1 ; e𝑏𝛼1 : Orthogonal unit vectors defining the noninertial reference frame C; ℎ=𝑙(1−cos(𝛼1)) = Vertical coordinate of payload in D: Natural noninertial comoving 𝑧1: noninertial reference frame B (m) reference frame which corresponds 𝛼 𝛼 1dyn: Angular distance between the axis to the swaying angle 2 and moving 𝐴 𝑧 𝐵𝐴 together with payload 𝑀 st 1 and the line dyn (rad) 𝑦 =𝑙 (𝛼 ) e𝜏𝛼 ; e𝑛𝛼 ; e𝑏𝛼 : Orthogonal unit vectors defining the dyn sin 1dyn : Length of perpendicular from point 2 2 2 𝐴 𝐴 𝑧 noninertial reference frame D dyn to the axis st 1 (m) G: Natural noninertial comoving Δ= Vertical distance between the point 𝑙(1 − (𝛼 )) 𝐴 𝐴 reference frame which corresponds to cos 1dyn : st and the point dyn,thatis,the theangleofcraneboomrotation𝜑𝑒 vertical distance between the ̃ ̃ and moving together with payload 𝑀 horizontal planes (e1,e2)and(e1,e2) B e𝜏𝜑; e𝑛𝜑; e𝑏𝜑: Orthogonal unit vectors defining the of the noninertial reference frames noninertial reference frame G and F correspondingly (m) E B r𝐵/𝑀: Position vector, connecting initial 𝜔 = 𝜔B/E: Angular velocity vector of reference point 𝐵 and terminal point 𝑀 in frame B with respect to frame E noninertial reference frame B (m) (rad/s) Shock and Vibration 29

F E𝜔 = 𝜔 𝑄 𝑄 𝑄 F/E: Angular velocity vector of reference 𝑥1 ; 𝑦1 ; 𝑧1 : Generalized forces (N) frame F with respect to frame E ]1; ]2: First and second natural frequencies (rad/s) of payload relative oscillations (1/s) 𝜔B/E =𝜔F/E = Scalar of transport angular velocity of 𝜏1; 𝜏2: First and second periods of payload 𝜔𝑒 =𝑑𝜑𝑒/𝑑𝑡: reference frames B and F with respect to frame E (rad/s) relative oscillations (s) 𝜔B/E = 𝜔F/E = Transport angular velocity vector of mg:Gravitationalforce(N) 𝜔𝑒 =(𝑑𝜑𝑒/𝑑𝑡)̂e3: reference frames B and F with N: Reaction force of the cable 𝐵𝑀 (N) respect to frame E (rad/s) 𝜏 𝜏 Φe =(−𝑚)ae E B : Tangential inertial force (tangential 𝜀 = 𝜀B/E = Angular acceleration vector of force of moving space) for payload 𝑀 𝛼B/E: reference frame B with respect to 2 (N) frame E (rad/s ) n 𝜏 Φe =(−𝑚)ae : Normal or centrifugal inertial force E F 𝜀 = 𝜀F/E = Angular acceleration vector of (normal force of moving space) for 𝛼F/E: reference frame F with respect to payload 𝑀 (N) 2 frame E (rad/s ) Φ =(−𝑚) cor acor: Coriolis inertial force (compound 𝛼B/E =𝛼F/E = Scalar of transport angular 𝑀 2 2 centrifugal force) for payload (N) 𝜀 =𝑑𝜑 /𝑑𝑡 acceleration of reference frames B 𝑂 𝑒 𝑒 : 𝐼 2 : Element of mass moment of inertia and F with respect to frame E 33 2 for the system “crane boom 𝐵𝑂2-load (rad/s ) 𝑀”ininertialfixedonearth 𝜀B/E = 𝜀F/E = 𝜀e = Transport angular acceleration vector reference frame E with respect to 2 2 B F 2 (𝑑 𝜑𝑒/𝑑𝑡 )e3: of reference frames and with ̂ E 2 axis e3 (kg-m ) respect to frame (rad/s ) 𝑂2 𝑂2 E 𝑀 H3 =𝐼33 𝜔B/E: Moment of momentum for the V = V𝑀/E = Velocity of point 𝑀 in inertial fixed E system “crane boom 𝐵𝑂2-load 𝑀”, Vabs: on earth reference frame ,thatis, absolute velocity of payload 𝑀 (m/s) that is, vector component of the B 𝑀 = = 𝑀 angular momentum for “crane boom V V𝑀/B Velocity of point in noninertial 𝐵𝑂 𝑀 F 𝑀 B F 2-load ” with respect to point V = V𝑀/F = reference frames and ,thatis, 𝑂2 in inertial fixed on earth reference relative velocity of payload 𝑀 (m/s) 2 Vr: frame E (kg-m /s) = + 𝑀 Ve V𝑂 /E Transportvelocityofpoint in 𝑂2 𝑂2 2 ∑ M =𝑀 ̂e3: Vector of the resultant external 𝜔B/E × r𝑂 /𝑀: inertial reference frame E (m/s) 2 moment codirectional to axis ̂e3 with 𝑉𝑥 ; 𝑉𝑦 ; 𝑉𝑧 : 𝑥1-, 𝑦1-, 𝑧1-projections of payload 𝑀 1 1 1 respect to point 𝑂2,wherepoint𝑂2 is velocity in noninertial reference fixedintheinertialfixedonearth B frame (m/s) reference frame E (N-m) 𝑉𝑥; 𝑉𝑦; 𝑉𝑧: 𝑥-, 𝑦-, 𝑧-projections of payload 𝑀 𝑂2 𝑀 :Drivingtorqueforthesystem“crane velocity in noninertial reference 𝐷𝑇 boom 𝐵𝑂2-load 𝑀”withrespectto frame F (m/s) 𝑀 point 𝑂2,wherepoint𝑂2 is fixed in E = = 𝑀 a a𝑀/E aabs: Acceleration of point in inertial theinertialfixedonearthreference E fixed on earth reference frame ,that frame E (N-m) 𝑀 is, absolute acceleration of payload 𝑂2 𝑂2 2 M =𝑀 ̂e3: Vector of the driving torque for the (m/s ) 𝐷𝑇 𝐷𝑇 𝑀 system “crane boom 𝐵𝑂2-load 𝑀” B = 𝑀 a a𝑀/B = Acceleration of point in codirectional to axis ̂e3 with respect F 𝑀 noninertial reference frames B and 𝑂 𝑂 a = a𝑀/F = ar: to point 2,wherepoint 2 is fixed in F, that is, relative acceleration of theinertialfixedonearthreference 𝑀 2 payload (m/s ) frame E with respect to axis ̂e3 (N-m) E 𝐸 𝑂 = = 𝐸 𝐷 2 a a𝐸/E Acceleration of points and in 𝑀𝐹𝑇: Frictional torque for the system E 𝐷 = = inertial fixed on earth reference “crane boom 𝐵𝑂2-load 𝑀”with a a𝐷/E 0: E frame ,thatis,theabsolute respect to point 𝑂2,wherepoint𝑂2 is acceleration of points 𝐸 and 𝐷, 2 fixedintheinertialfixedonearth located at rotation axis 𝐸𝐷 (m/s ) reference frame E (N-m) 𝜏 𝑂 𝑂 ae = 𝛼B/E × r𝐸/𝑀: Tangential acceleration of 2 2 M =𝑀 ̂e3: Vector of the frictional torque for the 𝑀 2 𝐹𝑇 𝐹𝑇 transportation for payload (m/s ) system “crane boom 𝐵𝑂2-load 𝑀” 𝑇 : Kinetic energy of the system “crane codirectional to axis ̂e3 with respect 𝐵𝑂 𝑀 boom 2-load ”(J=N-m) to point 𝑂2,wherepoint𝑂2 is fixed in Π: Potential energy of the system “crane theinertialfixedonearthreference boom 𝐵𝑂2-load 𝑀”(J=N-m) frame E with respect to axis ̂e3 (N-m) 30 Shock and Vibration

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