3D Nonlinear Simulation of Laundry Machine Auto-Balancing Mechanism: Dynamic Analysis of Imbalance Settlement

Abstract

This thesis presents a 3D mathematical model of an auto-balancing mechanism of a laundry unit. The auto-balancing mechanism consists of balancing balls freely moving in two guiding tracks, which compensate for the centrifugal forces produced by unbalanced laundry in a rotating drum. The main objective is to derive the 3D system, incorporating spatial degrees of freedom and its effects, and to compare its performance with a simplified 2D system, which has constrained these spatial degrees of freedom. The behaviour of the laundry unit is simulated utilising implicit and explicit methods, specifically the Newmark Newton-Raphson method (NNR) and the central difference method (CDM). Furthermore, the efficiency of obtaining tangential matrices is evaluated by comparing automatic differentiation (AD) with analytical differentiation. A parametric study reveals that while both 2D and 3D systems yield identical results for a centred unbalanced mass, the 3D system uniquely captures a critical state involving an axially offset unbalanced mass. Due to additional nonlinearities, the 3D system is computationally demanding. The comparison demonstrates that the CDM provides better time efficiency than the NNR without compromising the accuracy of displacement amplitudes. However, the Newmark method with a modified Newton-Raphson variant is more efficient when the critical time step in the CDM is evaluated at every step. Additionally, utilising analytical differentiation results in shorter execution times compared to AD. Finally, to eliminate physically impossible states and validate the proposed mathematical model, future research should include interactions between balls and experimental measurements on a physical prototype.