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Numerical JModelica.org-based approach to a simulation of Coriolis effects on guided boom-driven payload swaying during non-uniform rotary crane boom slewing

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Abstract

The present article is focused on the study of the swaying motion of a payload during non-uniform guided slewing of a crane boom with an emphasis on the Coriolis effects. A new mathematical model of the motion of the mechanical system “guided crane boom–payload” has been derived with an introduction of Blajer’s projection method. The governing dynamic system has been shown in the form of an implicit system of differential-algebraic equations. A time-optimal control problem has been formulated for the guided system with an open-loop control and with explicit limitations on payload swaying and on the control input. The numerical solution of the posed optimization problem has been found with an introduction of Optimica and JModelica.org freeware. Verification of numerically derived results has been realized through the comparison of computational and experimental absolute trajectories of a swaying payload. A satisfactory agreement between theoretical and experimental results was found. A dynamic analogy between the governing system of equations and Foucault pendulum-like systems was found and outlined. The proposed open code algorithms, derived numerical data and computational plots enhance and expand our knowledge about the dynamics of guided boom-driven payload swaying during rotary crane boom slewing.

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Abbreviations

DAE:

Differential algebraic equation

point A st :

Point of static equilibrium of payload M at the cable BM

point M :

Point of position of payload M in the current moment of time

point D :

Initial point of crane boom BD and center of circle for trajectory of transport and absolute motion for the point B

point B :

Terminal point of crane boom BD

m :

Mass of payload M (kg)

g :

Scalar value of gravitational acceleration (m/s2)

u(t):

Control input voltage at the windings of the rotary AC-servo motor (V)

u max :

Maximum value of control input voltage (V)

t :

Current swaying time for payload M (s)

T :

Period of slewing for crane boom BD (s)

t f :

The final time of rotation of crane boom BD with attached to p. B payload M at final angular displacement φ e (s)

T φ :

Time constant of rotary AC-servo motor, where the numerical value of time constant T φ = 0.015 (s) is provided in Table 2, page 1181 of Ref. [10] by Terashima et al. (2007) (s)

K φ :

Gain of rotary AC-servo motor, where the numerical value of gain K φ = 0.315 [1/(sV)] is provided in Table 2, page 1181 of Ref. [10] by Terashima et al. (2007) [1/(sV)]

J:

Minimized functional for final time of crane boom rotation (s)

J = t f :

Notation for time-optimal control problem (s)

(x 1(t), …, x 8(t)):

The phase variables of dynamic system, where dimensions of the phase variables are determined by formulae (14)–(21) and are as follows: (x 1(t), x 3(t) and x 5(t)) are in (m); x 7(t) is in (rad); (x 2(t), x 4(t) and x 6(t)) are in (m/s); x 8(t) is in (rad/s)

ε :

The maximum value of payload swaying during crane boom BD slewing at the certain final angular displacement (φ e)f, which is determined by the restrictive inequality (x 1(t))2 + (x 3(t))2 + (x 5(t))2 ≤ ε on the phase variables (x 1(t), x 3(t), x 5(t)) (m2)

\( \mathscr{E} \) :

Motionless fixed on earth inertial reference frame \( O_{2} x_{2}^{ * } y_{2}^{ * } z_{2}^{ * } \) \( \left( {O_{2} x_{{\text{abs}}} y_{{\text{abs}}} z_{{\text{abs}}} } \right) \) with origin in p. O 2 and the orthogonal unit vectors ê1; ê2; ê3 , where \( x_{2}^{ * } = x_{{\text{abs}}} \); \( y_{2}^{ * } = y_{{\text{abs}}} \) \( z_{2}^{ * } = z_{{\text{abs}}} \) are the absolute coordinates of payload M with respect to Earth (m)

\( \mathscr{B} \) :

Non-inertial reference frame (O 1 x 1 y 1 z 1), which is connected with rotating crane boom BD with origin in p. O 1 (p. A st) and the orthogonal unit vectors ẽ 1 ; ẽ 2 ; ẽ 3 , where x 1; y 1 ; z 1 are the relative coordinates of payload M with respect to crane boom BD (m)

R = BD:

Boom length in the horizontal plane (ê 1 , ê 2 ) (m)

l = l BM = ||r B/M||:

Length of the cable BM, i.e., the magnitude of the position vector r B/M (m)

r comp :

The magnitude of the radius-vector, connecting point O2 and theoretical (J = t f)-based computational absolute trajectory of payload M in Figs. 6 and 7

r exper = r emp :

The magnitude of the radius-vector, connecting point O 2 and experimentally (or empirically) derived absolute trajectory of payload M in Figs. 6 and 7

r circle :

The radius of standard circular trajectory of p. B (p. Ast) in Figs. 6 and 7

\( \varphi_{\text{e}} = \angle \left( {{\hat{\mathbf{e}}}_{{\mathbf{1}}} ,{\tilde{\mathbf{e}}}_{{\mathbf{2}}} } \right) \) :

Angle of rotation of non-inertial reference frame \({\mathscr{B}}\) around axis \( O_{2} z_{2}^{ * } \,\;\left( {O_{2} z_{{\text{abs}}} } \right) \) of inertial reference frame \({\mathscr{E}}\), i.e., the angle of transport rotation for crane boom BD around the vertical axis \( O_{2} z_{2}^{ * } \,\left( {O_{2} z_{{\text{abs}}} } \right) \), i.e., crane boom slewing angle (rad)

α1 = \( \angle \)(ẽ 3 , r B/M):

Swaying angle between the vertical axis O 1 z 1 and the position vector r B/M, i.e., the first spherical coordinate of spherical pendulum M (rad)

α2 :

Swaying angle between the planes (x 1 z 1) and (BMO 1), i.e., the second spherical coordinate of spherical pendulum M (rad)

(α1, α2, φ e):

The angular coordinates of spherical pendulum M with rotating pivot center in p. B connect to crane boom tip (rad)

\( {\boldsymbol{\upomega}}_{{{\mathscr{B}}/{\mathscr{E}} }} \) :

Angular velocity vector of reference frame \({\mathscr{B}}\) with respect to frame \({\mathscr{E}}\), i.e., the vector of slewing velocity of crane boom BD (rad/s)

\( \omega_{{{\mathscr{B}}/{\mathscr{E}} }} \) = ωe = dφ e/dt :

Scalar of transport angular velocity of reference frame \({\mathscr{B}}\) with respect to frame \({\mathscr{E}}\), i.e., the value of slewing velocity of crane boom BD (rad/s)

ω max :

Maximum value of angular velocity of reference frame \({\mathscr{B}}\) with respect to frame \({\mathscr{E}}\), i.e., the maximum value of slewing velocity of crane boom BD (rad/s)

\( {\boldsymbol{\upomega}}_{{{\mathscr{B}}/{\mathscr{E}} }} \) = ω e  = (dφ e/dt)ê 3 :

Transport slewing angular velocity vector of reference frame \({\mathscr{B}}\) with respect to frame \({\mathscr{E}}\), i.e., the vector of slewing velocity of crane boom BD (rad/s)

\({\boldsymbol{\varepsilon}}_{{{\mathscr{B}}/{\mathscr{E}} }} = {\boldsymbol{\upalpha}}_{{{\mathscr{B}}/{\mathscr{E}} }}\) :

Angular acceleration vector of reference frame \({\mathscr{B}}\) with respect to frame \({\mathscr{E}}\), i.e., the vector of slewing acceleration of crane boom BD (rad/s2)

\( \alpha_{{{\mathscr{B}}/{\mathscr{E}} }} \) = εe = d2 φ e/dt 2 :

Scalar of transport angular acceleration of reference frame \({\mathscr{B}}\) with respect to frame \({\mathscr{E}}\), i.e., the value of slewing acceleration of crane boom BD (rad/s2)

\({\boldsymbol{\varepsilon}}_{{{\mathscr{B}}/{\mathscr{E}} }} \) = ε e  = (d2 φ e /dt 2)e 3 :

Transport angular acceleration vector of reference frame \({\mathscr{B}}\) with respect to frame \({\mathscr{E}}\), i.e., the vector of slewing acceleration of crane boom BD (rad/s2)

\( {\textbf{V}}_{M/{\mathscr{E}}}\) = V abs :

Velocity of point M in inertial fixed on earth reference frame \({\mathscr{E}}\), i.e., absolute velocity of payload M (m/s)

\( {\textbf{V}}_{{M/{\mathscr{B}}}} \) = V r :

Velocity of point M in non-inertial reference frame \({\mathscr{B}}\), i.e., relative velocity of payload M (m/s)

\( {\mathbf{V}}_{{\mathbf{e}}} = {\mathbf{V}}_{{O_{2} /{\mathscr{E}}}} + \,\omega_{{{\mathscr{B}}/{\mathscr{E}} }} \times {\textbf{r}}_{{O_{2} /M}} \) :

Transport velocity of point M in inertial reference frame \({\mathscr{E}}\), i.e., the linear velocity of the “frozen” point M, which being “frozen” to crane boom BD “lost its relative velocity” and moves together with slewing non-inertial reference frame \({\mathscr{B}}\) with respect to inertial reference frame \({\mathscr{E}}\) (m/s)

\( V_{{x_{1} }} \); \( V_{{y_{1} }} \); \( V_{{z_{1} }} \) :

X 1-, y 1-, z 1- projections of payload M velocity in non-inertial reference frame \({\mathscr{B}}\) (m/s)

\( {\mathbf{a}}_{{M/{\mathscr{E}}}}\) = a abs :

Acceleration of point M in inertial fixed on earth reference frame \({\mathscr{E}}\), i.e., absolute acceleration of payload M (m/s2)

\( {\mathbf{a}}_{{M/{\mathscr{B}}}} \) = a r :

Acceleration of point M in non-inertial reference frame \({\mathscr{B}}\), i.e., relative acceleration of payload M (m/s2)

\( {\mathbf{a}}_{{\mathbf{e}}}^{{\varvec{\uptau}}} = {\boldsymbol{\upalpha}}_{{{\mathscr{B}}/{\mathscr{E}} }} \times {\textbf{r}}_{{o_{2} /M}} \) :

Tangential acceleration of transportation for payload M (m/s2)

\( {\mathbf{a}}_{{\mathbf{e}}}^{\text{n}} = {\boldsymbol{\upomega}}_{{{\mathscr{B}}/{\mathscr{E}} }} \times \left( {{\boldsymbol{\upomega}}_{{{\mathscr{B}}/{\mathscr{E}} }} \times {\textbf{r}}_{{o_{2} /M}} } \right) \) :

Normal or centripetal acceleration of transportation for payload M (m/s2)

\( {\mathbf{a}}_{{\text{cor}}} = 2{\boldsymbol{\upomega}}_{{{\mathscr{B}}/{\mathscr{E}} }} \times {\textbf{V}}_{{M/{\mathscr{B}}}} \) :

Coriolis (compound) acceleration of payload M (m/s2)

mg :

Gravitational force (N)

N :

Reaction force of the cable BM (N)

Φ τ e  = (–m)a τ e :

Tangential inertial force (tangential force of moving space) for payload M (N)

Φ n e  = (–m)a n e :

Normal or centrifugal inertial force (normal force of moving space) for payload M [N]

Φ cor  = (–m)a cor :

Coriolis inertial force (compound centrifugal force) for payload M (N)

δ :

Relative dimensionless amplitude discrepancy between computational and experimental absolute trajectories of point M in the inertial reference frame \({\mathscr{E}}\)

δ max :

The maximum value of δ

JModelica (JModelica.org):

Computational freeware code for time-optimal control problem solution, available at http://www.jmodelica.org/

Optimica:

The integrated part of JModelica (JModelica.org) freeware

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Acknowledgments

Authors thank “anonymous” referees for their valuable notes and suggestions.

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Authors and Affiliations

  1. Informatics and Engineering Graphics Department, Donbass State Engineering Academy, Shkadinova 72, Kramatorsk, Donetsk Region, 84313, Ukraine

    Alexander A. Kostikov

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

    Alexander V. Perig & Ruslan R. Lozun

  3. Computer and Information Technology Department, Donbass State Engineering Academy, Shkadinova 72, Kramatorsk, Donetsk Region, 84313, Ukraine

    Denys Yu. Mikhieienko & Ruslan R. Lozun

Authors
  1. Alexander A. Kostikov
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  2. Alexander V. Perig
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  3. Denys Yu. Mikhieienko
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  4. Ruslan R. Lozun
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Correspondence to Alexander V. Perig.

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The authors Alexander A. Kostikov, Alexander V. Perig, Denys Yu. Mikhieienko, and Ruslan R. Lozun declare that there is no conflict of interests regarding the publication of this paper.

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Technical Editor: Kátia Lucchesi Cavalca Dedini.

Appendices

Appendix 1: Listing of the *.mop Optimica’s file

Appendix 2: Listing of the *.py JModelica’s file

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Kostikov, A.A., Perig, A.V., Mikhieienko, D.Y. et al. Numerical JModelica.org-based approach to a simulation of Coriolis effects on guided boom-driven payload swaying during non-uniform rotary crane boom slewing. J Braz. Soc. Mech. Sci. Eng. 39, 737–756 (2017). https://doi.org/10.1007/s40430-016-0554-2

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