Model Based Analysis of Overhead Crane and Inverted Pendulum

Abstract

Because miscellaneous expenses such as an entrance fee, the facilities fee for use, the quay fee for use, the cargo-handling machine fee for use incur for coming alongside the pier in the quay, a large ship has to raise cargo work efficiency to the maximum. In this context, the container is required to stop its shaking, when the cart stops. When a cart of overhead crane suddenly stops, the cargo would swing for a while. On the other hand, by operating a cart in a procedure of the Full-Accelerate => Half-Decelerate => Half-Accelerate => Full-Decelerate, the cargo would not swing when the cart stops. This situation can be found by solving the following state-space representation, which is comprised of a cart position, cart speed, a pendulum angle of inclination, cart driving force. X ?=AX+BU u- External Force (N) y ?=v ?- Mass Ratio?m_2?((m_1+m_2 ) ) (m_1=10(kg)&&m_2=1(kg) ) v ?=??+u y- Cart Position (m) ? ?=q v- Cart Velocity (m/s) q ?=-?-u ?- Pendulum Angle (rad) q- Pendulum Angular Velocity (rad/sec) Meanwhile, GEKKO Optimization Suite is a system seeking the solutions of the algebraic equation with non-linear Solver represented by (IPOPT, APOPT, BPOPT, SNOPT, MINOS) developed in Brigham Young University, U.S.A. After the calculation of GEKKO Optimization Suite, the optimal results are obtained, in which the pendulum stands still when the cart stops. Next, the external force history, which is obtained from GEKKO Optimization Suite is tried to replicate by OpenModelica modeling. As a result, the multiplication of cart mass and the cart acceleration obtained from OpenModelica accorded for a driving force history obtained from GEKKO Optimization Suite well. Finally, the same trend is obtained in the case of the inverted pendulum either, with the following state-space representation, of which configuration is almost same with that of overhead crane. X ?=AX+BU u- External Force (N) y ?=v ?- Mass Ratio?m_2?((m_1+m_2 ) ) (m_1=10(kg)&&m_2=1(kg) ) v ?=-??+u y- Cart Position (m) ? ?=q v- Cart Velocity (m/s) q ?=?-u ?- Pendulum Angle (rad) q- Pendulum Angular Velocity (rad/sec)