Research into 2D Dynamics and Control of Small Oscillations of a Cross-Beam during Transportation by Two Overhead Cranes

A new mathematical model of a 3DOF 2D mechanical system “transported cross-beam, two moving bridge cranes” has been proposed. Small system oscillations have been derived through the introduction of Lagrange equations. The numerical estimation of 3DOF system motion has been carried out with equation-based Modelica language. The present article uses the Lagrange method and numerical and optimization methods, realized with JModelica.org and Optimica freeware. The absolute swaying of the cross-beam with respect to the displacement of the two moving bridge cranes was estimated. The phase portraits of the 3DOF system for linear and angular coordinates were presented. An open loop optimal control problem was posed for the motion of the bridge cranes. A “bang-bang” control strategy was implemented for the derivation of an optimal control solution, which enables the travel of two bridge cranes at a prescribed distance for minimum time and minimum swaying of a heavy cross-beam. The derived results of the numerical simulation can be easily practically realized by crane operators with good agreement with simple engineering estimations. The proposed control strategy enables synchronous motion of two bridge cranes with a cross-beam that practically solves the posed problem of unwanted excessive oscillations of a heavy cross-beam during transportation

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

The present paper addresses the derivation of a 3 DOF mathematical model of a spherical pendulum attached to a crane boom tip for uniform slewing motion of the crane. The governing nonlinear DAE-based system for crane boom uniform slewing has been proposed, numerically solved, and experimentally verified. The proposed nonlinear and linearized models have been derived with an introduction of Cartesian coordinates. The linearized model with small angle assumption has an analytical solution. The relative and absolute payload trajectories have been derived. The amplitudes of load oscillations, which depend on computed initial conditions, have been estimated. The dependence of natural frequencies on the transport inertia forces and gravity forces has been computed. The conservative system, which contains first time derivatives of coordinates without oscillation damping, has been derived. The dynamic analogy between crane boom-driven payload swaying motion and Foucault’s pendulum motion has been grounded and outlined. For a small swaying angle, good agreement between theoretical and averaged experimental results was obtained