A Study of Roll Induced by Crane Motion on Ships: A Case Study of the Use of the Moving Frame Method

Abstract

This paper presents a new method in dynamics — The Moving Frame Method (MFM) — and applies it to analyze the roll, yaw and pitch of a ship at sea, as induced by an onboard moving crane. The MFM, founded on Lie Group Theory, Cartan’s Moving Frames and a compact notation from geometrical physics, enables this expedited extraction of the equations of motion. Next, the method deploys the power of the special Euclidean Group SE(3) and a restricted variation to be used in Hamilton’s Principle, to extract the equations of motion. The mathematical model is then simplified to get a clearer picture of the parameters that impact the motion of the crane. The equations of interest are numerically solved by using fourth order Runge-Kutta method to obtain the specific data for the motion induced by the crane. Then, The Cayley-Hamilton theorem is used to reconstruct the rotation matrix. To supplement the paper, a webpage is coded with a model of the crane and ship, to graphically visualize the motion in 3D. It is imperative to note that while there are many approaches to dynamics, the MFM presents a consistent method, from 2D to 3D, and across sub-disciplines. The simplification is what has enabled undergraduate students to undertake this project.