Research Article | Open Access

Alexander V. Perig, Alexander N. Stadnik, Alexander I. Deriglazov, Sergey V. Podlesny, "3 DOF Spherical Pendulum Oscillations with a Uniform Slewing Pivot Center and a Small Angle Assumption", *Shock and Vibration*, vol. 2014, Article ID 203709, 32 pages, 2014. https://doi.org/10.1155/2014/203709

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

**Academic Editor:**Mickaël Lallart

#### 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)**

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.

**(a)**

**(b)**

**(c)**

**(d)**

**(a)**

**(b)**

**(a)**

**(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/s^{2}; 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/s^{2}; 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/s^{2}; 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)**

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.

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