Levenberg-Marquardt’s and Gauss-Newton algorithms for parameter optimisation of the motion of a point mass rolling on a turntable

Abstract

The problem of a point mass in a rotating system subjected to inertial force is theoretically and analytically solved, to get a more complete understanding of the different scenarios of motion in rotating system with friction being neglected. Subsequently, the solution is quantitatively verified by experimental data, solving non-linear least squares problems, based on Levenberg–Marquardt’s and Gauss–Newton methods by minimising the sum of squares of errors between the data and model prediction. The process of model fitting is closely related to parameter identification. The optimisation parameters of the model (initial velocity of a point mass and angular velocity) are estimated. Experimental observation of the trajectories of a point mass rolling on a turntable is analysed from a video capture webcam in a mechanics laboratory. A good fit of the theoretical study with experimental data using optimisation methods output shows that there are instruments to directly verify rather abstract mechanical formulation in a non-inertial frame. It also demonstrates that a relatively simple theoretical background can be used for describing real different types of motion in mechanics and for explaining experimental results. Moreover, combining the theoretical description of the problem with experimental data and computational optimisation procedures gives a very easy understanding of the different scenarios of motion in a rotating system and parameters identification, which can be obtained in classical mechanics. On the other hand, this study represents an important and instructive topic in classical mechanics that cannot be omitted in physical modelling on the undergraduate university level.