Shock and Vibration

Volume 2014 (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

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

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

Received 11 February 2014; Revised 7 May 2014; Accepted 21 May 2014; Published 4 August 2014

Academic Editor: Mickaël 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.

Abstract.

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.

1. 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.

Today payload swaying problems attract the great attention of such applied mathematicians and mechanical engineers as Abdel-Rahman et al. [1–3], Adamiec-Wójcik et al. [4], Al-mousa et al. [5], Allan and Townsend [6], Aston [7], 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], Erneux and Kalmár-Nagy [21, 22], Ghigliazza and Holmes [23], Glossiotis and Antoniadis [24], Grigorov and Mitrev [25], Gusev and Vinogradov [26], Hong and Ngo [27], Hoon et al. [28], Ibrahim [29], Jerman et al. [30–33], Ju et al. [34], Kłosiński [35], Krukowski et al. [36, 37], Lenci et al. [38], Leung and Kuang [39], Loveykin et al. [40], Maleki et al. [41], Maczynski et al. [42–44], Marinović et al. [45], Masoud et al. [46, 47], Mijailović [48], Mitrev and Grigorov [49, 50], Morales et al. [51], Nakazono et al. [52], Neitzel et al. [53], Neupert et al. [54], O’Connor and Habibi [55], Omar and Nayfeh [56], Osiński and Wojciech [57], F. Palis and S. Palis [58], Perig et al. [59], Posiadała et al. [60], Ren et al. [61], Safarzadeh et al. [62], Sakawa et al. [63], Sawodny et al. [64], Spathopoulos et al. [65], Schaub [66], Solarz and Tora [67], Uchiyama et al. [68], Urbaś [69], and others.

Applied engineering problems in the field of lifting-and-transport 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. [1–3] 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 [1–3]. However, this research [1–3] gives inadequate attention to the effects of Coriolis inertia forces on the payload relative motion during crane boom slew motion.

Blajer et al. [11–17] have proposed thirteen index-three differential-algebraic equations (DAE) in the rotary crane state variables and control variables [11–17]. They have proposed governing DAE equations and stable solution techniques allowing the rotary crane to execute the prescribed load trajectory and the control commands to implement feed-forward control. In this approach [11–17] the governing equations have been derived without consideration of the Coriolis inertia force and non-zero horizontal projections of the load gravity force.

Hairer et al. [70, 71] and Øksendal [72] have developed computational techniques for DAE systems numerical integration.

Jerman et al. [30–33] have applied an enhanced mathematical model of slewing the crane motion with load pendulation, taking into account system stiffness coefficients [30–33]. All mathematical techniques and governing equations in these works [30–33] have been presented in implicit form.

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 [42–44]. Optimization problem of load positioning in this study [42–44] has not fully addressed the natural frequencies estimation for the system “crane boom-payload” 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 [1–100] 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.

It should be noted that spherical pendulum related research has also been further developed for Foucault pendulums in the works of Condurache and Martinusi [77], Gusev and Vinogradov [26], de Icaza-Herrera and Castoño [83], Pardy [88], Zanzottera et al. [99], Zhuravlev and Petrov [100], and others.

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.

2. DAE System for Payload Swaying

It has been shown in Appendices A–O that the nonlinear DAE system (formulae , ) for the relative coordinates (Figure 1(a)); (Figure 1(b)); of a swaying payload during crane boom slewing (Figures 8–10) in the noninertial reference frame is as follows:

(a)

(b)

(c)

(d)

(e)

(f)

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:  kg;  m;  m;  m/s²; and  rad/s.

The absolute coordinates (Figure 1(c)); (Figure 1(d)); of payload in the inertial reference frame may be computed as

The absolute coordinates , in the inertial reference frame , which depend upon the relative coordinates , in the noninertial reference frame 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 (- - -, ).

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 in the noninertial reference frame ).

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 . 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 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.

3. Experimental Procedure

The experimental procedure has been grounded on the usage of the assembly in Figures 2 and 3.

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

(a)

(b)

(c)

(d)

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

(a)

(b)

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

(a)

(b)

Figure 5: The comparison of the small swaying angle theoretical (- - -, ) and experimental (—, ) load absolute trajectories in the inertial reference frame for fixed cable length  m (a) and fixed cable length  m (b).

The assembly in Figures 2 and 3 includes the following machine components: the vertical fixed shaft with height  m, the crane boom model with length  m and diameter 6 mm, the cable with different fixed lengths  m (in Figure 2 and Figure 5(b)),  m (in Figure 5(a)),  m (in Figure 4(b)), and  m (in Figure 4(a)). The crane boom model is attached to the vertical fixed shaft 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 (—, ()) in Figures 2, 4, and 5 is the experimental light emitting diode absolute trajectory in the inertial reference frame .

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 and 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 . The horizontal grid is formed by the square cells with size 20 mm × 20 mm.

The experimental swaying trajectories (—, ) 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.

4. Comparison and Discussion of DAE-Solution Based Theoretical and Experimental Results

The comparison of DAE-solution based theoretical (- - -, ) and experimental (—, ) 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 we have computed the amplitude discrepancy in the polar coordinate system by the following formula:

where and are the magnitudes of the radius-vectors, connecting point and theoretical (- - -) and experimental (—) curves, and computed for the same fixed polar angle .

The amplitude discrepancies have the following values: for  m in Figure 4(a); for  m in Figure 4(b); for  m in Figure 5(a); and % for  m in Figure 5(b).

Numerical computations (- - -, ) in Figure 4(a) have been carried out for the following values of mechanical system parameters:  m;  m/s²;  m;  rad/s;  s;  rad/s; °;  rad;  m/s;  m;  rad/s;  rad/s;  m; and  m (Figure 4(a)).

Numerical computations (- - -, ) in Figure 4(b) have been carried out for the following values of mechanical system parameters:  m;  m/s²;  m;  rad/s;  s;  rad/s; °;  rad;  m/s;  m;  rad/s;  rad/s;  m; and  m (Figure 4(b)).

Numerical computations (- - -, ) in Figure 5(a) have been carried out for the following values of mechanical system parameters:  m;  m/s²;  m; ≈ 3.984 rad/s;  s;  rad/s; °;  rad;  m/s;  m;  rad/s;  rad/s;  m; and  m (Figure 5(a)).

For further estimation of the relative disagreement of derived DAE-solution based computational (- - -) and experimental (—) payload absolute trajectories in the inertial reference frame we have computed the confidence intervals in Figures 4 and 5 for the dimensionless parameters and confidence probability 0.95. The Student’s -test results yield the following confidence intervals: for  m in Figure 4(a); for  m in Figure 4(b); for  m in Figure 5(a); and for  m in Figure 5(b).

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 , 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 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 ; in (1) define the vector (Appendix B, formula ) for the relative acceleration of load in the noninertial reference frame . The straight-line terms formula-based terms in (1) are linearly proportional to the relative payload coordinates in the noninertial reference frame . These straight-line terms in (1) have been determined by the contribution of the normal or centripetal acceleration   of transportation for payload and by the appearance of corresponding D’Alembert centrifugal inertia force   due to crane boom transport rotation in the noninertial reference frame . Additional terms and in (1) have been defined by the compound or Coriolis acceleration of payload in the noninertial reference frame . The rectangular Cartesian projections of Coriolis inertia force in the noninertial reference frame is defined by formula .

5. Experimental Results for the Spherical Pendulum with the Fixed Pivot Center

Experimental load swaying results for the fixed crane boom model ( rad/s) are shown in Figure 6.

(a)

(b)

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

Finite motions of a swaying load are shown in Figure 6 in the case of  rad/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.

6. Discussion of Analogy between Payload Swaying and Foucault’s Pendulum Motion

The motion of Foucault’s pendulum is shown in Figure 7, where is the heliocentric inertial reference frame (not shown in Figure 7), is the geocentric noninertial reference frame, and is a noninertial reference frame, located at the geographic latitude ; 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.

Figure 7: Foucault’s pendulum schematic view.

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

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

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

It is important to note that the structure of linearized equations - for the swaying payload during crane boom uniform rotation in the noninertial reference frame 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 - 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 () around p. 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 () of Foucault’s pendula swaying remains fixed in the heliocentric inertial reference frame .

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.

7. 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 - with components and has been derived. Both projections and coincide in the noninertial reference frames and .

The occurrence of the first derivatives ; of the load relative coordinates with respect to time in terms and determines that there is no decay of the oscillations in -, and (1) but only redirection of the relative velocity vector of the load in the noninertial reference frames and .

The introduction of a linearized model - allowed the determination of the natural frequencies () of free oscillations of payload and .

It follows from the forms of the relative payload trajectory in the noninertial reference frame and the absolute trajectory in the inertial reference frame that the resulting motion of the payload on the cable , taking into account the Coriolis inertia force  , will be the sum of two oscillations with natural frequencies and , and with periods and .

It is worth noting that the frequencies and differ by . This means that for the small angle assumption (Appendices H-I) we have and the trajectory of relative motion in the noninertial reference frame on the expiration of the relative oscillations period time looks approximately like an ellipse (Figures 12(b) and 16(b)) with semiaxes, which are governed by the initial conditions.

The major semiaxis of such an ellipse in Figures 12(b) and 16(b) is defined by the initial velocity of load in the noninertial reference frame and the analytical value of the major semiaxis is equal to . The derived value of is approximately the sum of the absolute values of the amplitudes in the analytical form of the linearized Cauchy problem -.

The minor semiaxis of the above mentioned ellipse in Figures 12(b) and 16(b) is defined by the value , which is approximately the difference of amplitudes in the analytical form of the linearized Cauchy problem -. 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 is much smaller than the natural frequency of the spherical oscillations of payload on the cable (Figures 12(b) and 16(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 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 is almost a symmetric curve with 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 and  –, 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 -load ” is created by the high-frequency oscillations of load , which are determined by the action of inertia force , stipulated by the Coriolis acceleration of load in the noninertial reference frame . The basic dynamic load in the system defines additional loads and vibrations of crane boom mechanical elements and support bearings, complicates the automatic and manual control systems of the electromechanical crane boom , and also makes crane operation much more difficult.

It is also important to note that the stop of crane boom does not lead to instantaneous damping of the load absolute oscillations in the inertial reference frame (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 for half-period of vibration and static equilibrium of load on the cable). The further oscillations for the stop of the crane boom 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.

8. 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 -.

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 . 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 - 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 -.

Comparison of the derived linearized model - 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 -.

9. 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 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 and absolute payload motion in the inertial reference frame 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.

Appendices

A. Velocity Kinematics Analysis in Cartesian Coordinates

For the nonlinear problem definition we study the cooperative motion of the mechanical system “crane boom -load ” which is shown in Figures 8–10.

In the nomenclature chapter we denote the fixed inertial frame of reference as , and the moving noninertial frame of reference as , which is rigidly bounded with the crane boom . Rotation of the moving noninertial frame of reference around the fixed inertial frame of reference defines the transportation motion for payload . The motion of load relative to the moving noninertial frame of reference defines the relative motion of payload . The point with the coordinates ; ; and is the steady equilibrium position for the load , when the crane boom is fixed. The point with the coordinates ; ; and is the dynamic equilibrium position for the load for the rotating crane boom . We will assume the point as the origin of the noninertial reference frame . The directions of axes , , of the noninertial reference frame are parallel to the axes , , of the noninertial reference frame 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 , , and of the load in the moving noninertial frame of reference and the angle of crane boom slewing in the horizontal plane () with the angular velocity around the vertical axis .

The relative velocity vector of the load is defined as .

In order to derive the absolute velocity of payload we will apply the vector method for absolute motion assignment. We will define as the position vector, connecting initial point and terminal point in noninertial reference frame (); 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 and in (A.1):

where , , , and notations have been defined in nomenclature and in Figures 8–10.

For further definition of the position vector in inertial reference frame , we write unit vectors of noninertial reference frame in (A.3) through the unit vectors of inertial reference frame :

After substitution of formulae (A.2)–(A.6) in (A.1) and some algebraic manipulations the vector expression (A.1) for the position vector in the inertial reference frame will take the following form:

The absolute velocity of payload we define by time differentiation of (A.7) assuming constancy of unit vectors , , and of the inertial reference frame 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 :

The square of absolute payload velocity may be written as

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 through the unit vectors of the noninertial reference frame (see [59, 76, 81, 82, 84, 85, 87, 90–92, 94–98]):

where payload relative velocity in the noninertial reference frame may be written as

The velocity of the point = point in the inertial reference frame has zero value:

The last term in (A.12) is the vector product , where crane boom angular velocity vector is (A.6) and the position vector in the noninertial reference frame is as follows:

Taking into account (A.13)–(A.15) the vector equation (A.12) in the inertial reference frame written through the unit vectors , , of the noninertial reference frame will have the following form:

So, on the basis of (A.16), we have the following square of absolute payload velocity as

Independently derived expressions for the scalar product () in (A.11) and (A.17) completely coincide, which confirms the correctness and accuracy of the pendulum absolute velocity derivation and shows that the scalar product () is the invariant expression, independent of choice of reference frame.

The second and third terms of (A.12) in the noninertial reference frame 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 is perpendicular to and (Figures 8–10); that is,

In nomenclature and in Figure 9 we denote the current angle , where

In the noninertial reference frame the vector of the load transportation velocity is defined as

Taking into account (A.19), (A.22) takes the following form:

Assuming (A.21) equation (A.23) will take the following form in the noninertial reference frame :

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.

The standard vector equation for the acceleration addition for payload in the inertial reference frame has the form (see [59, 76, 81, 82, 84, 85, 87, 90–92, 94–98]):

In our case we assume point as the pole for payload transportation motion, located at the vertical axis . So the second term in (B.1) for the inertial reference frame and in the noninertial reference frame takes the form

So taking into account the nomenclature and (B.2), (B.1) in the inertial reference frame takes the following form:

Equations (B.1) and (B.2) contain the following accelerations.

The vector of payload relative acceleration is defined in the noninertial reference frame as

The vector of tangential acceleration for transportation of payload has the same direction as the vector of the load transportation velocity , that is, and is defined in the noninertial reference frame as

The vector of the normal or centripetal acceleration of transportation for payload is directed towards the axis and at the same time and are the coplanar vectors, located in the horizontal plane (), 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 of the cable determines the geometrical constraint in this problem:

On the basis of Figure 10 we may derive that

where angles and 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 ; ; and in the noninertial reference frame as

The geometric constraint of (C.1)–(C.3) can be derived on the basis of (C.3) as

Equations (C.4) yield the following partial derivatives of with respect to Cartesian coordinates ; ; and in the noninertial reference frame as

Absolute coordinates , , and in inertial reference frame which depend upon the relative coordinates , , and in noninertial reference frame during the swaying of load are defined according to the following equations (Figures 8–10):

D. Forces Imposed on the Payload

Among the forces imposed on the payload (Figure 3(a)) we have an active force of gravity , the cable reaction force , the tangential inertial force , the normal or centrifugal inertial force , and the Coriolis inertial force . Taking into account formulae (B.1)–(B.7) we will express below all imposed forces in the noninertial reference frame

where ; and are the direction cosines of the cable reaction force N in the noninertial reference frame . The force N is directed from point to point ; that is, the force N ↑↓ is oppositely directed to the (A.3).

E. Forces Imposed on the Crane Boom-Payload System

The slewing motion of the mechanical system “crane boom -load ” in Figures 8–10 is governed by the vector equation for the rate of change of moment of momentum for the system “crane boom -load ” with respect to point in the inertial reference frame :

The vector equation (E.1) contains the following components:

where is the element of mass moment of inertia for the crane boom in inertial fixed on earth reference frame with respect to axis ; is the element of mass moment of inertia for the payload in inertial fixed on earth reference frame with respect to axis .

The external moment of gravitational force in (E.4) because mg ↑↓ .

For the system “crane boom-payload” the cable reaction force N is the internal force. So in (E.1) and (E.4) we have (N) = 0.

Substitution of (E.2), (E.3), and (E.4) into (E.1) yields the following scalar equation for the rate of change of moment of momentum for the system “crane boom -load ” with respect to point in the inertial reference frame :

where driving and frictional torques (see nomenclature section) are the technically defined functions for specific electric drive systems (see [1–3, 42–44, 59, 63, 67, 76, 81, 82, 84, 85, 87, 90–92, 94–98]).

F. Derivation of the Fully Nonlinear Equations in Relative Cartesian Coordinates of the Noninertial Reference frame

The vector differential equation for relative motion of payload in the noninertial reference frame is as follows:

The vector differential equation (F.1)-(F.2) yields three scalar ordinary differential equations (ODEs) for payload swaying motion.

We will project (F.1) and (F.2) to the axes , , and in the noninertial reference frame

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 .

In order to verify the correctness of the derived system (F.3) we will utilize second-kind Lagrange equations. Taking into account equations (A.10), (A.11), and (A.17) for the square of absolute payload velocity (, ), and by adding the kinetic energy (see [59, 76, 81, 82, 84, 85, 87, 90–92, 94–98]) for a slewing crane boom according to (E.2)-(E.3), we will have the following expression for “crane boom-payload” kinetic energy:

Taking into account the nonlinearity and nonconservatism of the cable reaction force N, and the equations for geometric constraints (C.1)–(C.5), we have the following formulae for the generalized forces in the noninertial reference frame :

We will derive the same nonlinear differential equations (F.3) for relative system motion with an introduction of the following Lagrange equations (see [59, 76, 81, 82, 84, 85, 87, 90–92, 94–98]) in the noninertial reference frame :

Taking into account (F.4)-(F.5), (F.6) in the noninertial reference frame will finally take the following form:

The derived ODE system (F.7) coincides with the system (F.3), which confirms the accuracy and correctness of the fully nonlinear equations (F.3) in relative Cartesian coordinates for payload swaying motion in the noninertial reference frame .

G. Uniform Crane Boom Rotation Assumption: The Introduction of the Noninertial Reference Frame

We now address the case of uniform crane boom slewing. We assume that the crane boom rotates with the constant angular velocity around vertical axis ; that is, that the noninertial reference frame uniformly rotates around the unit vector of the inertial reference frame . 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 , vanish in the 1st and 2nd equations of system (F.7). The second terms of (F.7) are linearly coordinate-dependent on , , and the forth terms of (F.7) are linearly velocity-dependent on and . So in the noninertial reference frame we have

The nonlinear system (G.2) has been presented in the noninertial reference frame 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 . The origin of the noninertial reference frame we connect with the so-called point of dynamic equilibrium for load , where the -distance between the noninertial references frames and 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 about point , that is, .

Taking into account the numerically derived angle from (G.3) we find the value of the horizontal distance between points and (Figures 8–10) according to the following equation:

Relative coordinates , , and in noninertial reference frame have been connected with the relative coordinates , , and in noninertial reference frame according to the following equations (Figures 8–10):

With (G.3)–(G.5) the nonlinear system (G.2) in the noninertial reference frame takes the following form (Figures 8–10):

H. The Small Swaying Angle Assumption

We now address the case of the small swaying angle . In the case of small swaying angle the system (C.3) defines and as the small variables, and . Having we conclude that and all time derivatives are zero, that is, the vertical load velocity and the vertical payload acceleration .

Thus, the 3rd equation of system (F.7) yields that the cable reaction force N approximately coincides the gravitational force mg; that is, . As a result the system (G.2), containing three ODEs, transforms into a linearized system with two independent equations for the relative Cartesian coordinates and in the noninertial reference frame . 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 in the noninertial reference frame We then transfer the origin of coordinate system (Figures 8–10) to the point of dynamic equilibrium for load ; that is, we make the transition from the noninertial reference frame to the noninertial reference frame 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 and 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 :

The same system in the noninertial reference frame can be derived from (G.6) for the case of the small swaying angle . Taking into account that we will write (H.4) as

Taking into account (C.6), the absolute coordinates , , and in inertial reference frame , which depend upon the relative coordinates , , and in the noninertial reference frame 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 (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 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 ; in (H.4)-(H.5) define the vector (B.4) for the relative acceleration of load in the noninertial reference frame . The straight-line terms ( and in (H.4)-(H.5) are linear proportional to the relative payload coordinates in the noninertial reference frame . These straight-line terms have been determined by the contribution of the normal or centripetal acceleration of transportation for payload and by appearance of corresponding D’Alembert centrifugal inertia force due to crane boom transport rotation in the noninertial reference frame . The third terms and in (H.4)-(H.5) have been defined by the compound or Coriolis acceleration of payload in the noninertial reference frame . The rectangular Cartesian projections of Coriolis inertia force in the noninertial reference frame are defined by formula (D.5).

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 and of the secular equation:

Taking into account (I.2), the law of relative load motion in the noninertial reference frame takes the following form:

Time derivatives of (I.3) yield the following projections of relative payload velocity in the noninertial reference frame :

In order to determine the arbitrary constants of integration in (I.3)-(I.4) we define the initial conditions of the problem according to Figures 8–10 and Figure 11.

Figure 11: The computational scheme for initial conditions assignment.

(a)

(b)

(c)

(d)

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

In the initial time load has the vertical in-line position (Figures 8–10); that is, the load initial position coincides with the static equilibrium position for the load on the cable . This means that the initial coordinates of payload are as follows:

At time the load has zero absolute velocity in the inertial reference frame ; that is, .

In order to determine the load relative velocity in the noninertial reference frame , which is equal to the load relative velocity in the noninertial reference frame , we address the velocity addition theorem (A.12) and formula (A.14):

The vector equation (I.8) determines the initial relative velocity of the load in the inertial reference frame as

The direction of the vector for the initial relative velocity is directed opposite to the unit vector in the inertial reference frame ; that is, . The initial direction of the vector is invariant in all reference systems , , and . So in the noninertial reference frame we have :

Taking into account that and in (I.10), we have the following scalar equation in the noninertial reference frames and :

Thus, with an introduction of (I.5)–(I.11), the initial conditions for ODE system (H.4)-(H.5) in the noninertial reference frame are as follows:

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 :

The algebraic expressions for the arbitrary constants of integration 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:

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:  m;  m/s²;  m;  rad/s;  s;  rad/s; °;  rad;  m/s;  m;  rad/s;  rad/s;  m; and  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 contra-rotation of the noninertial reference frames and to crane boom slewing.

J. Velocity Kinematics Analysis in Relative Spherical Coordinates of the Noninertial Reference Frame : The Introduction of the Noninertial Reference Frames , , and

The fully nonlinear equations (F.7) in relative Cartesian coordinates of the noninertial reference frame have the unknown function , that is, undetermined Lagrangian coefficient. The reaction force N of the cable in Figures 8–10 and Figure 13 is the nonconservative force.

Figure 13: 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 , in the horizontal plane () for nonlinear model derivation in relative spherical coordinates.

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 , , and , which correspond to the spherical angular coordinates (, , and ) and moving together with payload (Figure 13). The advantage of the introduction of the natural noninertial comoving reference frames , , and is grounded on the orthogonality of the reaction force N to the unit vectors and , that is, on the fact that and .

The vector equation (A.12) for velocity addition will take the form

where and are the relative payload velocities in the natural noninertial comoving reference frames and .

In Figure 13 and by using (J.1) we have that the natural components of payload velocity in the natural noninertial comoving reference frames , , and have the following values:

K. Acceleration Kinematics Analysis in Relative Spherical Coordinates in the Noninertial Reference Frames , , and

The vector equation (B.1) for acceleration addition will take the following form (Figure 14):

where and 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 , in the horizontal plane () for nonlinear model derivation in relative spherical coordinates.

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 and are as follows:

Figure 15: Swaying scheme for dynamic analysis in spherical coordinates of load at the cable , which is fixedly attached in the point to the crane boom , in the horizontal plane () for nonlinear model derivation in relative spherical coordinates.

(a)

(b)

(c)

(d)

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

Taking into account (K.3), the magnitudes of Coriolis acceleration vectors in the natural noninertial comoving reference frames , , and in Figure 14 have the following values:

The tangential accelerations in the natural noninertial comoving reference frames , , have the values as stated below:

The normal accelerations in the natural noninertial comoving reference frames , , have the values as stated below:

L. Forces Imposed on the Payload in Relative Spherical Coordinates in the Noninertial Reference Frames , , and

Among the forces, imposed on the payload in Figure 15, we have an active force of gravity , the cable reaction force , the tangential inertial forces ; ; , the normal or centrifugal inertial forces ; ; , and the Coriolis inertial forces and .

Taking into account formulae (K.3)–(K.7) we will express below all imposed forces in the noninertial reference frames , , :

M. Derivation of the Fully Nonlinear Equations in Relative Spherical Coordinates in the Noninertial Reference Frames , , and

Taking into account derived expressions (L.1), the introduction of D’Alembert’s principle in the noninertial reference frames , , yields the following scalar equations in the projections to the axes and :

where

N. Uniform Crane Boom Rotation Assumption with Nonlinear Equations in Relative Spherical Coordinates in the Noninertial Reference Frames

Assuming formulae (G.1), we will have the following nonlinear ODE system in the noninertial reference frame , 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 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’ and . 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(b), 17(d), 17(f), 18(b), 18(d), and 18(f). Numerical plots in Figure 16 have been carried out for the following values of mechanical system parameters:  m;  m/s²;  m;  rad/s;  s;  rad/s; °;  rad;  rad;  m/s;  m;  rad/s; and  rad/s (Figure 16).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 17: The comparison of relative trajectories of load swaying for linearized (a), (c), (e), and nonlinear (b), (d), and (f) models in the cases of (a-b), (c-d), and (e-f).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 18: The comparison of relative trajectories of load swaying for linearized (a), (c), (e), and nonlinear (b), (d), and (f) models in the cases of (a-b), (c-d), and (e-f).

O. Comparative Estimation and Discussion of the Amplitude Responses, Computed from the Linear and Nonlinear Models

For comparison of the linear (H.4)-(H.5) and nonlinear (N.1) models, we introduce the relative dimensionless amplitude discrepancy between averaged radii (Figures 17(b), 17(d), 17(f), 18(b), 18(d), and 18(f)) and (Figures 17(a), 17(c), 17(e), 18(a), 18(c), and 18(e)) of computational relative trajectories of payload in the noninertial reference frame (Figures 17-18):

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.

Figure 19: The computational points (○○○○) and regression diagram (—) for the relative dimensionless amplitude discrepancy between averaged radii of computational relative trajectories of payload in the noninertial reference frame , derived for linearized and nonlinear models in the case of cable length  m.

The computational relative trajectories of load in the noninertial reference frame , shown in Figures 12, 16, 17 and 18 outline that both averaged radii (Figures 17(b), 17(d), 17(f), 18(b), 18(d) and 18(f)) and (Figures 17(a), 17(c), 17(e), 18(a), 18(c), and 18(e)) and amplitudes of relative trajectories increase with increasing dimensionless crane boom slewing velocity , where  rad/s is the experimentally chosen technological crane boom slewing velocity.

It is shown in Figures 12, 16, 17, and 18 that there is a persistent trend .

The computational diagram in Figure 19 shows that the relative dimensionless amplitude discrepancy between the linear and nonlinear models increases with increasing dimensionless crane boom slewing velocity . It is shown in Figure 19 that % for the  rad/s, that is, for . It is shown in Figure 19 that % for the  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 .

Nomenclature

DOF: | Degree of freedom
ODE: | Ordinary differential equation
DAE: | Differential algebraic equation
point : | Point of static equilibrium of payload at the cable
point : | Point of dynamic equilibrium of payload at the cable
point : | Point of position of payload in the current moment of time
point : | Initial point of crane boom
point : | Terminal point of crane boom
point : | Center of circle for trajectory of transport and absolute motion for the point
: | Mass of payload (kg)
: | Scalar value of gravitational acceleration (m/s²)
: | Ratio of perimeter of circle to diameter
; ; and : | Constants of integration (m)
; : | Roots of secular equation
: | Swaying time for payload (s)
: | Inertial reference frame () fixed on earth
; ; : | Orthogonal unit vectors defining the inertial reference frame
; ; : | The absolute coordinates of payload with respect to Earth (m)
: | Noninertial reference frame () which is connected with rotating crane boom and the origin of coordinates point coincides with point of static equilibrium for load at the cable
; ; : | The relative coordinates of payload with respect to crane boom (m)
; ; : | The orthogonal unit vectors defining the noninertial reference frame
: | Noninertial reference frame () with coordinate axes parallel to () and the origin of coordinates point coincides with point of dynamic equilibrium for load at the cable
; ; : | Orthogonal unit vectors defining the noninertial reference frame
: | Natural noninertial comoving reference frame which corresponds to the swaying angle and moving together with payload
; ; : | Orthogonal unit vectors defining the noninertial reference frame ;
: | Natural noninertial comoving reference frame which corresponds to the swaying angle and moving together with payload
; ; : | Orthogonal unit vectors defining the noninertial reference frame
: | Natural noninertial comoving reference frame which corresponds to the angle of crane boom rotation and moving together with payload
; ; : | Orthogonal unit vectors defining the noninertial reference frame
: | Position vector, connecting initial point and terminal point in noninertial reference frame (m)
: | Position vector, connecting initial point and terminal point in noninertial reference frame (m)
: | Boom projection to the horizontal plane (, ) (m)
: | Length of the cable, that is, the magnitude of the position vector (m)
: | Natural frequency of undamped oscillations of payload at the cable (1/s)
: | Period of undamped oscillations of payload at the cable (s)
: | Angle of rotation of noninertial reference frame around axis of inertial reference frame , that is, the angle of transport rotation for crane boom around the vertical axis , that is, crane boom slewing angle (rad)
: | Angle between the position vector and horizontal plane (, ) or (, ), that is, inclination angle of crane boom with horizontal plane (, ) or (, ) (rad)
: | Swaying angle between the vertical axis and the position vector , that is, the first spherical coordinate of spherical pendulum (rad)
: | Swaying angle between the planes () and (), that is, the second spherical coordinate of spherical pendulum (rad)
: | Angle of ascent of payload over dynamical equilibrium level (rad)
: | Angle between the vertical plane, connected with crane boom , and the vertical plane, connected with position vector of payload in inertial reference frame (rad)
: | Vertical coordinate of payload in noninertial reference frame (m)
: | Angular distance between the axis and the line (rad)
: | Length of perpendicular from point to the axis (m)
: | Vertical distance between the point and the point , that is, the vertical distance between the horizontal planes (,) and (,) of the noninertial reference frames and correspondingly (m)
: | Angular velocity vector of reference frame with respect to frame (rad/s)
: | Angular velocity vector of reference frame with respect to frame (rad/s)
: | Scalar of transport angular velocity of reference frames and with respect to frame (rad/s)
: | Transport angular velocity vector of reference frames and with respect to frame (rad/s)
: | Angular acceleration vector of reference frame with respect to frame (rad/s²)
: | Angular acceleration vector of reference frame with respect to frame (rad/s²)
: | Scalar of transport angular acceleration of reference frames and with respect to frame (rad/s²)
: | Transport angular acceleration vector of reference frames and with respect to frame (rad/s²)
: | Velocity of point in inertial fixed on earth reference frame , that is, absolute velocity of payload (m/s)
: | Velocity of point in noninertial reference frames and , that is, relative velocity of payload (m/s)
: | Transport velocity of point in inertial reference frame (m/s)
; ; : | -, -, -projections of payload velocity in noninertial reference frame (m/s)
; ; : | -, -, -projections of payload velocity in noninertial reference frame (m/s)
: | Acceleration of point in inertial fixed on earth reference frame , that is, absolute acceleration of payload (m/s²)
= : | Acceleration of point in noninertial reference frames and , that is, relative acceleration of payload (m/s²)
: | Acceleration of points and in inertial fixed on earth reference frame , that is, the absolute acceleration of points and , located at rotation axis (m/s²)
: | Tangential acceleration of transportation for payload (m/s²)
: | Normal or centripetal acceleration of transportation for payload (m/s²)
: | Coriolis (compound) acceleration of payload (m/s²)
: | Kinetic energy of the system “crane boom -load ” (J = N-m)
Π: | Potential energy of the system “crane boom -load ” (J = N-m)
; ; : | Generalized forces (N)
; : | First and second natural frequencies of payload relative oscillations (1/s)
; : | First and second periods of payload relative oscillations (s)
mg: | Gravitational force (N)
N: | Reaction force of the cable (N)
: | Tangential inertial force (tangential force of moving space) for payload (N)
: | Normal or centrifugal inertial force (normal force of moving space) for payload (N)
: | Coriolis inertial force (compound centrifugal force) for payload (N)
: | Element of mass moment of inertia for the system “crane boom -load ” in inertial fixed on earth reference frame with respect to axis (kg-m²)
: | Moment of momentum for the system “crane boom -load ”, that is, vector component of the angular momentum for “crane boom -load ” with respect to point in inertial fixed on earth reference frame (kg-m²/s)
: | Vector of the resultant external moment codirectional to axis with respect to point , where point is fixed in the inertial fixed on earth reference frame (N-m)
: | Driving torque for the system “crane boom -load ” with respect to point , where point is fixed in the inertial fixed on earth reference frame (N-m)
: | Vector of the driving torque for the system “crane boom -load ” codirectional to axis with respect to point , where point is fixed in the inertial fixed on earth reference frame with respect to axis (N-m)
: | Frictional torque for the system “crane boom -load ” with respect to point , where point is fixed in the inertial fixed on earth reference frame (N-m)
: | Vector of the frictional torque for the system “crane boom -load ” codirectional to axis with respect to point , where point is fixed in the inertial fixed on earth reference frame with respect to axis (N-m)
: | Relative dimensionless amplitude discrepancy between computational and experimental absolute trajectories of payload in the inertial reference frame
: | Relative dimensionless amplitude discrepancy between averaged radii of computational relative trajectories of payload in the noninertial reference frame , derived for linearized and nonlinear models.

Disclosure

The submission of the authors’ paper implies that it has not been previously published, that it is not under consideration for publication elsewhere, and that it will not be published elsewhere in the same form without the written permission of the editors.

Conflict of Interests

The authors Alexander V. Perig, Alexander N. Stadnik, Alexander I. Deriglazov, and Sergey V. Podlesny declare that there is no conflict of interests regarding the publication of this paper.

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