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

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 linearized model 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.


Introduction
Payload swaying dynamics during crane boom slewing is within the objectives and scope of many academic and industrial research programs in the fields of mechanical, electrical, and control engineering, and theoretical mechanics.Mathematical descriptions of relative and absolute payload swaying motion during crane boom rotation require the introduction of design models for a spherical pendulum with a suspension point following a horizontal circular trajectory.Spherical pendula with moving pivot centers are the so-called "eternal" problems that have been posed by ancient builders and civil engineers.
Applied engineering problems in the field of lifting-andtransport machines mainly deal with rectilinear or rotational motion of the spherical pendulum suspension center in determined and stochastic cases.Further improvement of rotating crane performance and efficiency requires the development of mathematical models for an adequate description of payload and crane boom tip positions.Modern computational approaches to the solution of payload positioning problems have been investigated in the following works.
Abdel-Rahman et al. [1][2][3] have provided a comprehensive review of different types of cranes, the essential and widely used mathematical techniques for models of crane dynamics and classical control methods [1][2][3].However, this research [1][2][3] gives inadequate attention to the effects of Coriolis inertia forces on the payload relative motion during crane boom slew motion.
Ju et al. have implemented finite element simulation for a flexible crane structure with a spherical pendulum [34].In this work [34], the spherical pendulum excitation is induced by vibration modes of the flexible crane structure.
Loveykin et al. have derived the law of an optimal control for lifting-and-shifting machines under the assumption of minimization of quadratic performance criteria in the case for two-phase coordinates, control and control rate [40].This work [40] made wide use of variation optimization techniques for pendulum oscillations in the vertical plane, which involve the trolleys of crane frames with rectilinear motion.
Maczynski et al. have applied a numerically based finite element method (FEM) approach for simulation of a "crane boom-payload" system without an explicit introduction of inertia forces [42][43][44].Optimization problem of load positioning in this study [42][43][44] has not fully addressed the natural frequencies estimation for the system "crane boompayload" in the case of a fully rigid crane boom model.
Mitrev and Grigorov [49,50] have derived governing equations for load relative swaying taking into account energy dissipation, centrifugal, Coriolis, inertia, and gravity forces [49,50].The Lagrange equations used here allow the simulation of a spherical pendulum with a movable pivot center [49,50].Mitrev's approach [49,50] is based on the introduction of angular generalized coordinates which result in nonlinearity of the problems and require a fourth-order Runge-Kutta fixed-step integration method.
However the previously known studies  have given inadequate attention to the dynamic analysis of a load swaying in the horizontal plane of vibrations while accounting for the effect of the Coriolis force on the trajectory of the relative load motion of the cable.The present research addresses this situation.
At first sight, the spherical pendulum with rotating pivot center and Foucault pendulum are two vastly different dynamic systems.The key difference between the two dynamic systems is that dynamic analysis of the Foucault pendulum is focused on a load swaying in the field of the central gravity force, while crane boom slewing problems are posed for the vertical gravity force.A commonality of the two dynamic systems is, in both cases, the effect of influence of normal centrifugal and Coriolis forces on the shape of the relative and absolute trajectories.The normal centrifugal forces depend on the relative coordinates and Coriolis forces depend on the relative velocity of the swaying load.Moreover, the appearance of Coriolis forces, that are dependent on relative payload velocity, retains the dynamic system as a conservative one because Coriolis acceleration remains at all times perpendicular to the relative velocity of the load.It may therefore be concluded that there is a close coupling between the spherical pendulum with rotating pivot center and Foucault pendulum.Moreover the close relationship between the two dynamic systems has not been properly addressed in all previous known research, which emphasizes the actuality and relevance of the present paper.
The present paper is focused on the study of the oscillation processes taking place in the vicinity of a steady equilibrium position of a payload during crane boom uniform slewing.The computational approach is based on the solution of the initial value problem of particle dynamics for the determination of the relative load trajectory in the horizontal plane of the vibrations during crane boom uniform rotation.The approach used here takes into account both the relative rotation of the load vibrations in the vertical plane and the influence of Coriolis acceleration on the form of the trajectory of the swaying cargo relative motion.This paper is also aimed at addressing the physically grounded interrelations between the spherical pendulum with rotating pivot center and Foucault pendulum.

DAE System for Payload Swaying
It has been shown in Appendices A-O that the nonlinear DAE system (formulae (C.1), (F.7)) for the relative coordinates  1 () (Figure 1 The absolute coordinates  2 () The absolute coordinates  2 ,  2 in the inertial reference frame E, which depend upon the relative coordinates  1 ,  1 in the noninertial reference frame B during the swaying of load  (Figures 8-10), has been defined according to the above mentioned Equations and has been shown in Figures 4 and 5 as (---,  2 =  2 ( 2 )).
The computational results in Figure 1, derived for DAE problem (1)-( 2) solution, coincide with the linearized solution of the payload swaying problem in Appendices A-I (formula (H.1) in the noninertial reference frame B).
It is necessary to note that the posed DAE problem (1) describes payload motion in the vicinity of the lower position of stable dynamical equilibrium assuming rope tension force () > 0. Factually, the applied rope is assumed as a unilateral geometric constraint (Appendix C) in the shape of a torsion fiber.The upper position of the mechanical system is unstable (see [73-75, 78-80, 91, 93]) and corresponds to the case of () < 0 within the parameters of the chosen mechanical system in Figures 8-10.Such an upper pendulum position might conflict with the hypothesis about the unilateral nature of geometric constraint.All of the above mentioned upper pendulum position conditions are outside the objectives and scope of the present paper.

Experimental Procedure
The experimental procedure has been grounded on the usage of the assembly in Figures 2 and 3.
The assembly in Figures 2 and 3 includes the following machine components: the vertical fixed shaft  2  with height  = 1 m, the crane boom model  with length  = 0.5 m and diameter 6 mm, the cable  with different fixed lengths  1 = 0.825 m (in Figure 2  The laser pointer in Figure 2 is also attached to the crane boom  tip in the point .The laser pointer is the part of the noninertial reference frames B and F and the pointer trajectory is marked as (---) in Figure 2. The introduction of laser pointer  allows estimation of dynamic deviation of payload .The horizontal regular fixed grid with canvas size 1200 mm × 700 mm is placed in the horizontal plane  2  2  2 .The horizontal grid is formed by the square cells with size 20 mm × 20 mm.
The experimental swaying trajectories (-,  2 =  2 ( 2 )) in Figures 2, 4, and 5 have been written in the obscure room with the introduction of an upper digital camera with a long exposure 60 s-90 s and the camera height above horizontal grid level is 2.5 m.

Comparison and Discussion of DAE-Solution Based Theoretical and Experimental Results
The comparison of DAE-solution based theoretical (---,  2 =  2 ( 2 )) and experimental (-,  2 =  2 ( 2 )) results is shown in Figures 4 and 5.In order to estimate the relative disagreement of the derived DAE-solution based computational (---) and experimental (-) payload  absolute trajectories in the inertial reference frame E we have computed the amplitude discrepancy  in the polar coordinate system by the following formula: where  comp and  exper are the magnitudes of the radiusvectors, connecting point  2 and theoretical (---) and experimental (-) curves, and computed for the same fixed polar angle   .
The amplitude discrepancies  have the following values:  = 7.53% for  = 0.206 m in Figure 4 Both the relative discrepancy  and the confidence intervals show the satisfactory agreement between the absolute payload trajectories (Figures 4 and 5) in the inertial reference frame E, that have been computed with DAE-solution based theoretical model (---) (1) and measured experimentally (-) as shown in Figures 4 and 5.
It is important to note that the payload motion DAE equations in the nonlinear problem (1) for the noninertial reference frame B may be derived with an introduction of Lagrange equations (Appendices A-C, F).However the discussion of (1) is more suitable and informative with an introduction of dynamic Coriolis theorem (Appendices A-D, F).The first terms  2  1 / 2 ;  2  1 / 2 in (1) define the vector a /B = a r (Appendix B, formula (B.

Experimental Results for the Spherical Pendulum with the Fixed Pivot Center
Experimental load swaying results for the fixed crane boom model (  = 0 rad/s) are shown in Figure 6.
Finite motions of a swaying load are shown in Figure 6 in the case of   = 0rad/s.In Figure 6 the relative and absolute load trajectories are identically equal.The lingering remains of elliptical motion in Figure 6 and ellipses semiaxes are defined by the initial conditions.

Discussion of Analogy between Payload Swaying and Foucault's Pendulum Motion
The motion of Foucault's pendulum is shown in Figure 7, where  2  2  2 is the heliocentric inertial reference frame (not shown in Figure 7),   It is important to note that the structure of linearized equations (H.4)-(H.5)for the swaying payload during crane boom uniform rotation in the noninertial reference frame F is analogous to the structure of the governing equations for Foucault's pendulum motion in the known published works by Zhuravlev and Petrov [100] (formula (6), p. 36 of [100]), Pardy [88] (formulae ( 12)-( 14), pp.850-851 of [88]), and Condurache et al. [77] (formulae (1.1)-(1.3),p. 743-744 of [77]).The above mentioned analogy between linearized equations (H.4)-(H.5)and the governing equations in [77,88,100] assumes the geometric analogy of relative swaying trajectories between Foucault's pendulum and the boom-driven pendulum with rotating pivot centers (Figures 1(e), 12, and 16).So there is the geometric analogy between computational relative trajectories of load  swaying during crane boom uniform slewing (Figures 1(e), 12, and 16) and relative trajectories of load  swaying in the plane ( st ) around p.  st in Figure 7, shown by Zhuravlev and Petrov [100] (Figure 4, p. 39 of [100]), and Condurache and Martinusi [77] (Figure 14 at p. 754 and Figure 16 at p. 755 of [77]).The better understanding of Foucault's pendula dynamics is provided with the study of computational relative trajectories in Figures 1(e), 12, and 16.For Foucault's pendulum (Figure 7) the major semiaxes of relative amplitude extremes have angular velocity values equal to the angular velocity of the Earth diurnal rotation.Moreover, for Foucault's pendulum (Figure 7) the direction of rotation of major semiaxes of relative amplitude extremes is oppositely directed to the direction of the Earth diurnal rotation.All the above mentioned means that the plane ( st ) of Foucault's pendula swaying remains fixed in the heliocentric inertial reference frame  2  2  2 .
The presented analogy between crane boom-driven payload swaying and Foucault's pendulum motion essentially increases knowledge and awareness of the oscillation processes in such dynamic systems.

Discussion and Mechanical Interpretation of Governing Equations for DAE-Based Nonlinear and Linearized Models
A governing nonlinear DAE-based system (1) for crane boom uniform slewing has been proposed and numerically solved.
A conservative system (1) and (H.4)-(H.5)with components Φ cor  and Φ cor  has been derived.Both projections Φ cor  and Φ cor  coincide in the noninertial reference frames B and F. The occurrence of the first derivatives /; / of the load  relative coordinates with respect to time in terms −2  (/) and 2  (/) determines that there is no decay of the oscillations in (H.4)-(H.5),and (1) but only redirection of the relative velocity vector V /F = V r of the load  in the noninertial reference frames B and F.
The introduction of a linearized model (H.4)-(H.5)allowed the determination of the natural frequencies (] It follows from the forms of the relative payload trajectory  = () in the noninertial reference frame F and the absolute trajectory  2 =  2 ( 2 ) in the inertial reference frame E that the resulting motion of the payload  on the cable , taking into account the Coriolis inertia force Φ cor (D.5), will be the sum of two oscillations with natural frequencies ] 1 and ] 2 , and with periods It is worth noting that the frequencies ] 1 and ] 2 differ by 2  .This means that for the small angle assumption (Appendices H-I) we have ] 1 ≈ ] 2 and the trajectory The minor semiaxis of the above mentioned ellipse in Figures 12(b) and 16(b) is defined by the value  dyn , which is approximately the difference of amplitudes in the analytical form (I.15) of the linearized Cauchy problem (H.4)-(H.5).In this case the major semiaxis is approximately 5-20 times larger than the minor semiaxis because the angular velocity   of the transport rotation of crane boom  2 is much smaller than the natural frequency  of the spherical oscillations of payload  on the cable  (Figures 12(b It is also necessary to note that the major semiaxis of the ellipse in the relative motion rotates with an angular velocity   in the opposite direction of the transport rotation of crane boom  2 in (Figures 12 and 16).
It follows from Figures 1(f), 2, 4, and 5 that the trajectory of absolute motion of load  in the inertial reference frame E is almost a symmetric curve with  2 axial symmetry.The initial and the final motions of load  for half-period essentially differ from its harmonic oscillations neighbor for quarter-period.
Due to the negligible quantity of  dyn and  dyn (G.3)-(G.5), the average deviation of the load  from the mechanical trajectory of the boom tip  is negligible (Figures 1(f), 2, 4 and 5).The basic dynamic load on the system "crane boom  2 -load " is created by the high-frequency oscillations of load , which are determined by the action of inertia force Φ cor = −2 ( B/E × V /B ), stipulated by the Coriolis acceleration a cor = 2 B/E × V /B of load  in the noninertial reference frame F. The basic dynamic load in the system defines additional loads and vibrations of crane boom  2 mechanical elements and support bearings, complicates the automatic and manual control systems of the electromechanical crane boom  2 , and also makes crane operation much more difficult.
It is also important to note that the stop of crane boom  2 does not lead to instantaneous damping of the load  absolute oscillations in the inertial reference frame E (Figures 2, 4, and 5).This phenomenon directly follows from the experimental trajectory in Figures 2, 4, and 5. Also, the natural spherical oscillations of load  will occur with the frequency  and the amplitude (the difference between the final position of load  in the relative motion in the noninertial reference frame F for half-period of vibration and static equilibrium  st of load  on the cable).The further oscillations for the stop of the crane boom  2 slewing motion are the deviation of the real trajectory from the intended final position of load .The results of physical simulation in Figures 2, 4, and 5 show the necessity of add-on devices development for the efficient suppression of load  final oscillations.

Discussion and Comparison of Derived and Known Published Results
In Figure 1 and pp.537-538 of published work by Sakawa et al., 1981 [63], the small angle between payload's cable and vertical line has been introduced, that confirms proposed nonlinear DAE-based system (1) and linearized model (H.4)-(H.5).
In Figure 9 of p. 278 of published work by Maczynski and Wojciech, 2003 [44], the computational finite element method (FEM)-based results were shown for the absolute payload trajectories in the inertial reference frame E. Maczynski's Figure 9 in [44] outlines that the angle between the payload's cable and the vertical line does not exceed 0.1 rad and that agrees with the proposed linearized model (H.4)-(H.5)for a small angle assumption.Computational trajectory for Maczynski-derived small load swaying [44] after crane boom stop qualitatively coincides with the experimentally observable load  motion in Figures 2, 4, and 5.
In Figure 6 of published work by Ju et al., 2006 [34], it was shown that the angle between the payload's cable and the vertical line does not exceed 16 ∘ .Ju's formula (15a) in p. 382 of [34] assumes that Ju's angle between payload's cable and the vertical line has the harmonic law of variation with an introduction of a small perturbation term.Both Ju's assumptions in [34] confirm the proposed linearized model (H.4)-(H.5).
Comparison of the derived linearized model (H.4)-(H.5)with Figure 5 of published work by Mitrev and Grigorov 2008 in [49] shows that the Mitrev-derived ranges of payload swaying angles within Mitrev's nonlinear model does not exceed 4.8 ∘ .Such small values of swaying angles completely confirm the applicability and correctness of the small angle assumption in (H.4)-(H.5).

Final Conclusions
A governing nonlinear DAE-based system for crane boom uniform slewing has been proposed, numerically solved, and experimentally verified.
Fully nonlinear differential equations for 3 DOF spherical pendulum oscillations with a uniform slewing crane boom around a fixed vertical axis of rotation have been derived in relative Cartesian and spherical coordinates.The identical linearized differential equations in relative Cartesian coordinates have been derived with an introduction of the Coriolis dynamic theorem and Lagrange equations for a uniform crane boom rotation and small swaying angle  1 assumptions.
Linearized and nonlinear theoretical problems for relative swaying of the payload have been formulated in the form of the initial value (Cauchy) problem.
An analytical solution of the linearized system has been derived.
The influences of inertia forces from the centripetal and compound accelerations have been estimated.A linearized conservative system, which contains first time derivatives of coordinates and no damping of oscillations, has been derived.
The amplitudes of load oscillations, which depend on computed initial conditions, have been estimated within a small angle assumption.The dependence of natural frequencies on the transport inertia forces and gravity forces has been computed for the linearized systems.
The formulae for the association of relative payload motion in the noninertial reference frame F and absolute payload motion in the inertial reference frame E have been outlined.
The results of the numerical DAE-based investigation and the performed physical simulation show a satisfactory fit for frequencies and amplitudes of load oscillations.
The dynamic analogy between crane boom-driven payload swaying motion and Foucault's pendulum motion has been grounded and outlined.
The results of the present work are the foundation for further investigations of payload  swaying dynamics during telescopic crane boom nonuniform slewing motion with variable cable length and for the different motions of the pendulum pivot center.

A. Velocity Kinematics Analysis in Cartesian Coordinates
For the nonlinear problem definition we study the cooperative motion of the mechanical system "crane boom  2 -load " which is shown in Figures 8-10.
In the nomenclature chapter we denote the fixed inertial frame of reference E as  2  2  2 , and the moving noninertial frame of reference B as  1  1  1 , which is rigidly bounded with the crane boom  2 .Rotation of the moving noninertial frame of reference B( 1  1  1 ) around the fixed inertial frame of reference E( 2  2  2 ) defines the transportation motion for payload .The motion of load  relative to the moving noninertial frame of reference B( 1  1  1 ) defines the relative motion of payload .The point  st with the coordinates  1 = 0;  1 = 0; and  1 = 0 is the steady equilibrium position for the load , when the crane boom  2 is fixed.The point  dyn with the coordinates  1 = 0;  1 =  dyn ; and  1 = Δ is the dynamic equilibrium position for the load  for the rotating crane boom  2 .We will assume the point  dyn as the origin of the noninertial reference frame F().The directions of axes , ,  of the noninertial reference frame F are parallel to the axes  1 ,  1 ,  1 of the noninertial reference frame B in Figures 8-10.The load  relative motion takes place along the sphere surface with the fixed radius, equal to the cable length  = .
The computational scheme in Figures 8-10 for the nonlinear model derivation may be described with an introduction of three degrees of freedom.For generalized coordinates we assume the rectangular coordinates  1 ,  1 , and  1 of the load  in the moving noninertial frame of reference B and the angle   of crane boom  2 slewing in the horizontal plane ( 1  1 ) with the angular velocity   around the vertical axis  2  2 .
The relative velocity vector V r = V /B of the load  is defined as In order to derive the absolute velocity V /E of payload  we will apply the vector method for absolute motion assignment.We will define r  2 / as the position vector, connecting initial point  2 and terminal point  in noninertial reference frame B (); consider where vector components in (1) are defined in nomenclature.
In accordance with Figures 8-10 we have the following unit vectors' expansions for position vectors r  2 / and r / in (A.1): where ,   ,  1 , and  1 notations have been defined in nomenclature and in Figures 8-10.
For further definition of the position vector r  2 / in inertial reference frame E, we write unit vectors of noninertial reference frame B in (A.3) through the unit vectors of inertial reference frame E: After substitution of formulae (A.2)-(A.6) in (A.1) and some algebraic manipulations the vector expression (A.1) for the position vector r  2 / in the inertial reference frame E will take the following form: The absolute velocity of payload  we define by time differentiation of (A.7) assuming constancy of unit vectors ê1 , ê2 , and ê3 of the inertial reference frame E and constancy  and : After differentiation and algebraic transformations in (A.7) and (A.8) we have the following vector expression for absolute payload  velocity in the inertial reference frame E: (A.9) The square of absolute payload  velocity may be written as (A.10) After algebraic transformations we have the square of absolute payload  velocity in the form of Algebraic expressions (A.10) and (A.11) could be derived on the basis of a velocity addition theorem for payload  compound motion in the inertial reference frame E through the unit vectors of the noninertial reference frame B (see [59,76,81,82,84,85,87,[90][91][92][94][95][96][97][98]): where payload relative velocity in the noninertial reference frame B may be written as The velocity V  2 /E of the point  2 = point  in the inertial reference frame E has zero value: The last term in (A.12) is the vector product  B/E × r  2 / , where crane boom angular velocity vector is  B/E = (  /)ẽ 3 (A.6) and the position vector r  2 / in the noninertial reference frame B is as follows: Taking into account (A.13)-(A.15) the vector equation (A.12) in the inertial reference frame E written through the unit vectors ẽ1 , ẽ2 , ẽ3 of the noninertial reference frame B will have the following form: So, on the basis of (A.16), we have the following square of absolute payload  velocity as (A.17) Independently derived expressions for the scalar product (V /E , V /E ) in (A.11) and (A.17) completely coincide, which confirms the correctness and accuracy of the pendulum absolute velocity V /E derivation and shows that the scalar product (V /E , V /E ) is the invariant expression, independent of choice of reference frame.
The second and third terms of (A.12) in the noninertial reference frame B determine the vector of the load  transportation velocity as Taking into account (A.14) the scalar of the load  transportation velocity is defined as The transportation velocity vector V e is perpendicular to r  2 / and r / (Figures 8 -10); that is, In nomenclature and in Figure 9 we denote the current angle  = ∠(ẽ Taking into account (A.13) and (A.24), the formula (A.12) yields again (A.16) and (A.17).
So the square of absolute payload  velocity (A.10), (A.11), and (A.17) has been derived with three independent methods, which confirms the accuracy and correctness of expressions (A.10), (A.11), and (A.17).

B. Acceleration Kinematics Analysis
Further dynamic analysis, with the introduction of Newton's second law, requires us to study the accelerations of payload , shown in Figure 10.

Shock and Vibration
The standard vector equation for the acceleration addition for payload  in the inertial reference frame E has the form (see [59,76,81,82,84,85,87,[90][91][92][94][95][96][97][98]): In our case we assume point  as the pole for payload transportation motion, located at the vertical axis  The vector of payload  relative acceleration is defined in the noninertial reference frame B as The vector of tangential acceleration a e  for transportation of payload  has the same direction as the vector of the load  transportation velocity V e , that is, a e  ↑↑ V e and is defined in the noninertial reference frame B as The vector of the normal or centripetal acceleration a e n of transportation for payload  is directed towards the axis  2  2 and at the same time a e  and a e n are the coplanar vectors, located in the horizontal plane ( 2  2 ), where The vector of the Coriolis (compound) acceleration of payload  is directed in accordance with the vector product law

C. The Geometric Constraints Imposed on the Payload
The geometric constraint, imposed on the payload  is shown in Figures 8-10 in the form of the cable .The length  =   = ‖r / ‖ of the cable  determines the geometrical constraint in this problem: On the basis of Figure 10 we may derive that The geometric constraint of (C.1)-(C.3)can be derived on the basis of (C.3) as 1 =  −  cos ( 1 ) .
(C.4) Equations (C.4) yield the following partial derivatives of  1 with respect to Cartesian coordinates  1 ;  1 ; and  1 in the noninertial reference frame B as ; .
Absolute coordinates  2 ,  2 , and  2 in inertial reference frame E which depend upon the relative coordinates  1 ,  1 , and  1 in noninertial reference frame B during the swaying of load  are defined according to the following equations (Figures 8-10 where −( 1 /); −( 1 /) and −(( 1 − )/) are the direction cosines of the cable reaction force N in the noninertial reference frame B. The force N is directed from point  to point B; that is, the force N ↑↓ r / is oppositely directed to the r / (A.3).

E. Forces Imposed on the Crane Boom-Payload System
The slewing motion of the mechanical system "crane boom  2 -load " in Figures 8-10 is governed by the vector equation for the rate of change of moment of momentum H  2 3 for the system "crane boom  2 -load " with respect to point  2 in the inertial reference frame E: The vector equation (E.1) contains the following components: where ( is the element of mass moment of inertia for the crane boom  2 in inertial fixed on earth reference frame E with respect to axis ê3 ; ( 2  1 + ( +  1 ) 2 ) is the element of mass moment of inertia for the payload  in inertial fixed on earth reference frame E with respect to axis ê3 .
For the system "crane boom-payload" the cable reaction force N is the internal force.So in (E.1) and (E.4) we have M  2 (N) = 0.
Substitution of (E.

F. Derivation of the Fully Nonlinear Equations in Relative Cartesian Coordinates of the Noninertial Reference frame B
The vector differential equation for relative motion of payload  in the noninertial reference frame B is as follows: a /B = mg + N + Φ e n + Φ e  + Φ cor ; (F.1) The vector differential equation (F.1)-(F.2) yields three scalar ordinary differential equations (ODEs) for payload  swaying motion.

Shock and Vibration
Figure 11: The computational scheme for initial conditions assignment.
The derived system (F.3) and (E.5) is the nonlinear ODE system.The nonlinearity of (F.3) is determined by the presence of the unknown function  = (), variable boom slewing angle   , variable boom slewing angular velocity   /, and variable angular acceleration  2   / 2 .

G. Uniform Crane Boom Rotation Assumption: The Introduction of the Noninertial Reference Frame F
We now address the case of uniform crane boom slewing.We assume that the crane boom  2 rotates with the constant angular velocity   around vertical axis  2 ; that is, that the noninertial reference frame B uniformly rotates around the unit vector ê3 of the inertial reference frame E. Such case takes place when the right-hand side of (E.5) is zero, that is, for the steady state of crane boom rotation with So, taking into account (G.1), the third terms, containing  2   / 2 , vanish in the 1st and 2nd equations of system (F.7).The second terms of (F.7) are linearly coordinate-dependent on  1 ,  1 , and the forth terms of (F.7) are linearly velocitydependent on  1 / and  1 /.So in the noninertial reference frame B we have (G. 2) The nonlinear system (G.2) has been presented in the noninertial reference frame B for the case of uniform crane boom slewing.
For further physical analysis of the swaying problem and for ease of building the analytical solution we introduce the noninertial reference frame F. The origin of the noninertial reference frame F we connect with the so-called point  dyn of dynamic equilibrium for load , where the distance between the noninertial references frames B and F we define through the numerical solution of the following relative equilibrium transcendental equation: This equation (G.3) has been derived as the momentum sum for the gravitational force mg and the normal inertial force Φ e n about point , that is,   (mg) =   (Φ e n ).Taking into account the numerically derived angle  1dyn from (G.3) we find the value of the horizontal distance  dyn between points  st and  dyn (Figures 8-10) according to the following equation: Relative coordinates , , and  in noninertial reference frame F have been connected with the relative coordinates (G.6)

H. The Small Swaying Angle 𝛼 1 Assumption
We now address the case of the small swaying angle  1 .In the case of small swaying angle  1 the system (C.3)defines  1 and  1 as the small variables, and  1 = 0. Having  1 = 0 we conclude that Δ = 0 and all time derivatives are zero, that is, the vertical load velocity  1 / = 0 and the vertical payload acceleration  2  1 / 2 = 0. Thus, the 3rd equation of system (F.7)yields that the cable reaction force N approximately coincides the gravitational force mg; that is, N ≈ mg.As a result the system (G.2),containing three ODEs, transforms into a linearized system with two independent equations for the relative Cartesian coordinates  1 and  1 in the noninertial reference frame B. We then cancel the mass  of load  from the system of (G.6).Consider So we have derived the system (H.1) of differential equations for relative motion of load  on cable  with a movable suspension center , which is attached to the crane boom  2 in the noninertial reference frame B We then transfer the origin of coordinate system  (Figures 8-10) to the point  dyn of dynamic equilibrium for load ; that is, Shock and Vibration we make the transition from the noninertial reference frame B to the noninertial reference frame F and simplify (G.5) to The second equation of system (H.1)defines the amount of dynamic deflection, that is, the -distance between the noninertial reference frames B and F as where   =   /.
Then with the introduction of (H.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 the load  in the noninertial reference frame F: (H.4) The same system in the noninertial reference frame F can be derived from (G.6) for the case of the small swaying angle  1 .Taking into account that   =   / = const we will write (H.4) as Taking into account (C.6), the absolute coordinates  2 ,  2 , and  2 in inertial reference frame E, which depend upon the relative coordinates , , and  in the noninertial reference frame F during the swaying of load , will be defined according to the following equations (Figures 8-10): So the linearized ODE system (H.4)-(H.5)has been derived for the case of uniform crane boom slewing (G.1) and the small swaying angle  1 (H.2).
It is important to note that the left-hand sides of payload motion equations in linearized problem (H.4)-(H.5)for the noninertial reference frame F may be derived with an introduction of Lagrange equations (F.6).However the discussion of (H.4)-(H.5) is more suitable and informative with an introduction of dynamic Coriolis theorem (F.1)-(F.2).The first terms  2 / 2 ;  2 / 2 in (H.4)-(H.5)define the vector

I. Analytical Solution of the Linearized System
After generation of the determinant of natural frequencies matrix for the system (H.4)-(H.5)we will write the following characteristic biquadratic equation of fourth order for system (H.4)-(H.5) in the form: Using (I.1) we adjust the roots  1 and  2 of the secular equation: Taking into account (I.2), the law of relative load  motion in the noninertial reference frame F takes the following form: Time derivatives of (I.3) yield the following projections of relative payload  velocity in the noninertial reference frame F: (I.4)

Shock and Vibration 19
In order to determine the arbitrary constants of integration  1 , . . .,  4 in (I.3)-(I.4)we define the initial conditions of the problem according to Figures 8-10 and Figure 11.
In the initial time  = 0 load  has the vertical  st inline position ; that is, the load  initial position coincides with the static equilibrium position  st for the load  on the cable .This means that the initial coordinates of payload  are as follows: point  st (, 0, 0) in the reference frame E; (I.5) point  st (0, 0, 0) in the reference frame B; point  st (0, − dyn , 0) in the reference frame F. (I.7) At time  = 0 the load  has zero absolute velocity in the inertial reference frame E; that is, V /E (0) = 0.
In order to determine the load  relative velocity in the noninertial reference frame F, which is equal to the load  relative velocity in the noninertial reference frame B, we address the velocity addition theorem (A.12) and formula (A.14): The vector equation (I.8) determines the initial relative velocity V /B (0) of the load  in the inertial reference frame E as The direction of the vector for the initial relative velocity V /B (0) is directed opposite to the unit vector ê2 in the inertial reference frame E; that is, V /B (0) ↑↓ ê2 .The initial direction of the vector V /B (0) is invariant in all reference systems E, B, and F. So in the noninertial reference frame B we have V / (0) ↑↑ ẽ1 : Taking into account that ‖ B/E (0)‖ =   and ‖r  2 / (0)‖ =  in (I.10), we have the following scalar equation in the noninertial reference frames B and F: Thus, with an introduction of (I.5)-(I.11), the initial conditions for ODE system (H.4)-(H.5) in the noninertial reference frame F are as follows: (I.12) The substitution of (I.12) into (I.3) and (I.4) yields the following system for derivation the analytical expressions for the arbitrary constants of integration  1 , . . .,  4 : The algebraic expressions for the arbitrary constants of integration  1 , . . .,  4 can be derived as the following solution of the system (I.13): So, the relative motion equations for pendulum  swaying take the final form as stated below: (I.15) The obtained analytical equations (I.15), derived for the linearized model (H.4)-(H.5),determine the shape of the relative payload  swaying trajectory as is shown in Figure 12.Numerical computations in Figure 12 have been carried out for the following values of mechanical system parameters:  = 0.492 m;  = 9.81 m/s 2 ;  = 0.825 m;  = (/) 0.5 ≈ 3.448 rad/s;  = 30 s;   = 2/ ≈ 0.209 rad/s;   = 180 ∘ ;  1dyn = 0.00221 rad;   = 0.103 m/s;  dyn = 0.00182 m; ] 1 =  +   = 3.6578 rad/s; ] 2 =  −   = 3.2388 rad/s;  1 = 0.01408 m; and  3 = 0.01591 m (Figure 12).The above mentioned numerical values of analytical relations (I.15) subject to the initial conditions (I.5)-(I.12)define a theoretically computed law of relative motion for load .After the elimination of variable  from the system of (I.15) we will have the relative trajectory for load  (Figure 12).The system (H.6) and the computational relative trajectories for load  swaying in Figure 12 show that there is the contrarotation of the noninertial reference frames B and F to crane boom  2 slewing.So for derivation of solvable nonlinear equations we should remove the cable tension force N from the payload  motion equations with the introduction of Newton's second law and the natural noninertial comoving reference frames C, D, and G, which correspond to the spherical angular coordinates ( 1 ,  2 , and   ) and moving together with payload  (Figure 13).The advantage of the introduction of the natural noninertial comoving reference frames C, D, and G is grounded on the orthogonality of the reaction force N to the unit vectors e  1 and e  2 , that is, on the fact that N ⊥ e  1 and N ⊥ e  2 .
The vector equation (A.12) for velocity addition will take the form where V /C = V  1 and V /D = V  2 are the relative payload velocities in the natural noninertial comoving reference frames C and D.
In Figure 13 and by using (J.

K. Acceleration Kinematics Analysis in Relative Spherical
Coordinates in the Noninertial Reference Frames C, D, and G The vector equation (B.1) for acceleration addition will take the following form (Figure 14): where 14.
So formula (K.1) will take the following form: The directions of vectors have been shown in Figures 13,14,and 15.The scalar values of vectors  C/E and  D/E are as follows: Taking into account (K.3), the magnitudes of Coriolis acceleration vectors in the natural noninertial comoving reference frames C, D, and G in Figure 14 Figure 14: Swaying scheme for acceleration kinematics analysis in spherical coordinates of load  at the cable , which is fixedly attached in the point  to the crane boom  2 , in the horizontal plane ( 1  1 ) for nonlinear model derivation in relative spherical coordinates.The tangential accelerations in the natural noninertial comoving reference frames C, D, G have the values as stated below: The normal accelerations in the natural noninertial comoving reference frames C, D, G have the values as stated below: Among the forces, imposed on the payload  in Figure 15, we have an active force of gravity mg, the cable reaction force N, the tangential inertial forces Φ  1 ; Φ  2 ; Φ  , the normal or centrifugal inertial forces Φ  1 ; Φ  2 ; Φ  , and the Coriolis inertial forces Φ cor (  1 ) and Φ cor (  2 ).
Taking into account formulae (K.3)-(K.7)we will express below all imposed forces in the noninertial reference frames C, D, G:  describing payload  relative swaying in the case of uniform crane boom slewing: The initial conditions for the nonlinear system (N.1) may be formulated on the basis of the initial conditions in the noninertial reference frame F and with introduction of constraint equations (C.3):The solution of the system (N.2) defines the initial conditions for the nonlinear ODE system (N.1) for payload spherical coordinates'  1 and  2 .The solution set member of the system (N.2) has the following form: 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  swaying trajectory as is shown in Figures 16,17  (O.1) The results of numerical amplitude comparison, computed from the linear and nonlinear models in accordance with the formula (O.1) and Figures 12,16,17,and 18, are shown in Figure 19.
The computational relative trajectories of load  in the noninertial reference frame F, shown in Figures 12,16 The computational diagram in Figure 19 shows that the relative dimensionless amplitude discrepancy  1 between the linear and nonlinear models increases with increasing dimensionless crane boom slewing velocity   / 0 .It is shown in Figure 19 that  1 < 11% for the   < 0.8 rad/s, that is, for   < 3.828  0 .It is shown in Figure 19 that  1 < 11% for the  1 < 0.145 rad.
So we draw a conclusion that the small angle assumption for the linearized model (H.4)-(H.5)adequately describes nonlinear load swaying problem (N.1) during uniform crane boom slewing for the above mentioned limiting values of   and  1 .

Figure 1 :
Figure 1: Computational relative (a-b, e) and computational absolute (c-d, f) coordinates of a swaying payload during uniform crane boom slewing for the following numerical values of system parameters:  = 0.1 kg;  = 0.825 m;  = 0.492 m;  = 9.81 m/s 2 ; and   = 0.209 rad/s.
and Figure 5(b)),  2 = 0.618 m (in Figure 5(a)),  3 = 0.412 m (in Figure 4(b)), and  4 = 0.206 m (in Figure 4(a)).The crane boom model  is attached to the vertical fixed shaft  2  by bearing .The cable  is attached to the crane boom  tip in the point .The free or running end  of the cable  is the payload  attachment point.The load  is a light emitting diode (LED) with diameter 2 mm and the battery with the battery voltage 3 V.The experimental swaying trajectory (-,  2 =  2 ( 2 )) in Figures 2, 4, and 5 is the experimental light emitting diode  absolute trajectory in the inertial reference frame E.

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

Figure 6 :Figure 7 :
Figure 6: Experimental trajectories of the spherical pendulum with fixed pivot center and fixed cable length  = 0.825 m (a; b).

Figure 8 :
Figure 8: Swaying scheme of load  on the cable , which is fixedly attached at the point  on the crane boom  2 , in the vertical plane () for the nonlinear model derivation in Cartesian coordinates.

Figure 9 :
Figure 9: Swaying scheme of load  at the cable , which is fixedly attached in the point  of the crane boom  2 , in the horizontal plane ( 1  1 ) for the nonlinear model derivation in Cartesian coordinates.

Figure 10 :
Figure 10: Computational spatial scheme of spherical pendulum , swaying on the cable  during crane boom  2 slewing motion for the nonlinear model derivation in Cartesian coordinates.

2
2 .So the second term in (B.1) for the inertial reference frame E and in the noninertial reference frame B takes the form a /E = a /B = 0. (B.2) So taking into account the nomenclature and (B.2), (B.1) in the inertial reference frame E takes the following form: a abs = a r + a e  + a e n + a cor .(B.3) Equations (B.1) and (B.2) contain the following accelerations.

a
/F = a r (B.4) for the relative acceleration of load  in the noninertial reference frame F. The straight-line terms (  2 − (/)) and (  2 − (/)) in (H.4)-(H.5)are linear proportional to the relative payload coordinates in the noninertial reference frame F. These straight-line terms have been determined by the contribution of the normal or centripetal acceleration  B/E ×( B/E ×r / ) = a e n of transportation for payload  and by appearance of corresponding D' Alembert centrifugal inertia force Φ e n = (−)a e n due to crane boom transport rotation in the noninertial reference frame B. The third terms −2  (/) and 2  (/) in (H.4)-(H.5)have been defined by the compound or Coriolis acceleration 2 B/E × V /B = a cor of payload  in the noninertial reference frame F. The rectangular Cartesian projections of Coriolis inertia force in the noninertial reference frame F are defined by formula (D.5).

Figure 12 :
Figure 12: The relative trajectories for load  swaying on the cable during uniform rotational motion of the crane boom  2 for the moments of times  1 = 1.25 s (a),  2 = 3 s (b),  3 = 10 s, (c) and  4 = 25 s (d), derived with linearized model introduction.

Figure 15 :
Figure15: Swaying scheme for dynamic analysis in spherical coordinates of load  at the cable , which is fixedly attached in the point  to the crane boom  2 , in the horizontal plane ( 1  1 ) for nonlinear model derivation in relative spherical coordinates.
Imposed on the Payload in Relative Spherical Coordinates in the Noninertial Reference Frames C, D, and G

Figure 16 :
Figure 16: The relative trajectories for load  swaying on the cable during uniform rotational motion of the crane boom  2 for the moments of times  1 = 1.25 s (a),  2 = 3 s (b),  3 = 10 s (c), and  4 = 25 s (d), derived with nonlinear model introduction.

50Figure 19 :
Figure19: The computational points (IIII) and regression diagram (-) for the relative dimensionless amplitude discrepancy  1 between averaged radii of computational relative trajectories of payload  in the noninertial reference frame F, derived for linearized and nonlinear models in the case of cable length  = 0.825 m.
point  st : Point of static equilibrium of payload  at the cable  point  dyn : Point of dynamic equilibrium of payload  at the cable  point : Point of position of payload  in the current moment of time E  F =  F/E : Angular velocity vector of reference frame F with respect to frame E (rad/s)  B/E =  F/E =   =   /: Scalar of transport angular velocity of reference frames B and F with respect to frame E (rad/s)  B/E =  F/E =   = (  /)ê 3 : Transport angular velocity vector of reference frames B and F with respect to frame E (rad/s) E  B =  B/E =  B/E : Angular acceleration vector of reference frame B with respect to frame E (rad/s 2 ) E  F =  F/E =  F/E : Angular acceleration vector of reference frame F with respect to frame E (rad/s 2 )  B/E =  F/E =   =  2   / 2 : Scalar of transport angular acceleration of reference frames B and F with respect to frame E (rad/s 2 )  B/E =  F/E =  e = ( 2   / 2 )e 3 : Transport angular acceleration vector of reference frames B and F with respect to frame E (rad/s 2 ) EV  = V /E = V abs :Velocity of point  in inertial fixed on earth reference frame E, that is, absolute velocity of payload  (m/s)B V  = V /B = F V  = V /F = V r :Velocity of point  in noninertial reference frames B and F, that is, relative velocity of payload  (m/s) V e = V  2 /E +  B/E × r  2 / : Transport velocity of point  in inertial reference frame E (m/s) 1  1  1 is the geocentric noninertial reference frame, and  is a noninertial reference frame, located at the geographic latitude  1 ;  st  is a local vertical;  is the radius of the circle of the pinning point;  is the cable length; and   is the angular velocity of the Earth diurnal rotation.
2 , (r  2 / )  1  1 e = (− B/E      r  2 /      cos ()) ẽ1 + ( B/E      r  2 /      sin ()) ẽ2 .(A.23) Assuming (A.21) equation (A.23) will take the following form in the noninertial reference frame B: 1 and  2 in Figures 8-10 are the spherical coordinates of spherical pendulum .The comparison of formulae (C.1) and (C.2) allows us to determine the Cartesian coordinates  1 ;  1 ; and  1 in the noninertial reference frame B as 2 3 for the system "crane boom  2 -load " with respect to point  2 in the inertial reference frame E: 1)we have that the natural components of payload  velocity in the natural noninertial Figure13: Swaying scheme for velocity kinematics analysis in spherical coordinates of load  at the cable , which is fixedly attached in the point  to the crane boom  2 , in the horizontal plane ( 1  1 ) for nonlinear model derivation in relative spherical coordinates.

Uniform Crane Boom Rotation Assumption with Nonlinear Equations in Relative Spherical Coordinates in the Noninertial Reference Frames B
Taking into account derived expressions (L.1), the introduction of D' Alembert's principle in the noninertial reference frames C, D, G yields the following scalar equations in the projections to the axes e  1 and e  2 : e  1 : 0 = −Φ  1 + Φ  2 cos ( 1 ) + Φ cor (  2 ) cos ( 1 ) + Φ  sin ( 2 + ) cos ( 1 ) + Φ  cos ( 2 + ) cos ( 1 ) − mg sin ( 1 ) = 0;e  2 : 0 = −Φ  2 − Φ cor (  1 ) + Φ  cos ( 2 + ) − Φ  sin ( 2 + ) ,N.Assuming formulae (G.1), we will have the following nonlinear ODE system in the noninertial reference frame B, 1 ;   1 ;   1 :  1 -,  1 -,  1 -projections of payload  velocity in noninertial reference frame B (m/s)   ;   ;   : -, -, -projections of payload  velocity in noninertial reference frame F (m/s) E a  = a /E = a abs : Acceleration of point  in inertial fixed on earth reference frame E, that is, absolute acceleration of payload  (m/s 2 ) Acceleration of point  in noninertial reference frames B and F, that is, relative acceleration of payload  (m/s 2 ) Acceleration of points  and  in inertial fixed on earth reference frame E, that is, the absolute acceleration of points  and , located at rotation axis  (m/s 2 ) a e   1 ;   1 ;   1 : Generalized forces (N) ] 1 ; ] 2 : First and second natural frequencies of payload relative oscillations (1/s)  1 ;  2 : Element of mass moment of inertia for the system "crane boom  2 -load " in inertial fixed on earth reference frame E with respect to axis ê3 (kg-m 2 ) H 33  B/E : Moment of momentum for the system "crane boom  2 -load ", that is, vector component of the angular momentum for "crane boom  2 -load " with respect to point  2 in inertial fixed on earth reference frame E (kg-m 2 /s) ∑ M  2 =   2 ê3 : Vector of the resultant external moment codirectional to axis ê3 with respect to point  2 , where point  2 is fixed in the inertial fixed on earth reference frame E (N-m)  e boom  2 -load " with respect to point  2 , where point  2 is fixed in the inertial fixed on earth reference frame E (N-m) M  ê3 : Vector of the driving torque for the system "crane boom  2 -load " codirectional to axis ê3 with respect to point  2 , where point  2 is fixed in the inertial fixed on earth reference frame E with respect to axis ê3 (N-m)   2 -load " with respect to point  2 , where point  2 is fixed in the inertial fixed on earth reference frame E (N-m) M  ê3 : Vector of the frictional torque for the system "crane boom  2 -load " codirectional to axis ê3 with respect to point  2 , where point  2 is fixed in the inertial fixed on earth reference frame E with respect to axis ê3 (N-m) B a  = a /B = F a  = a /F = a r : E a  = a /E = E a  = a /E = 0:  =  B/E × r / : Tangential acceleration of transportation for payload  (m/s 2 ) :Kinetic energy of the system "crane boom  2 -load " (J = N-m) Π:Potential energy of the system "crane boom  2 -load " (J = N-m)  : D r i v i n g t o r q u e f o r t h e s y s t e m " c r a n