A mathematical and numerical model for variable cross-section hull girder with time-varying mass systems applied in marine vessels

• Modified Euler-Bernoulli beam with time-varying mass functions. • Improved Kane's dynamic method with time-varying mass system applied in variable cross-section hull girder with support from the semi-analytical approach for the vibration characteristic analysis of variable cross-section Euler-Bernoulli beam. • Time-varying wet surface and trim characteristics of the hull girder considered in timevarying hydrodynamic calculations. • Complex loads including external and internal excitation loads coupled with dynamic model. For some types of special-purpose marine vessels, such as dredgers, the variation of mass onboard derived from their working conditions may induce unforeseen loads within a short time, which will lead to unexpected structural responses and fatigue damage. For dredgers, additional loads induced by the mass-variation in their specific working conditions are of significant concern and have not been effectively considered during the design stage. This paper proposes a mathematical and numerical model for structural dynamic analysis of variable cross-section hull girder with time-varying mass characteristics. It leverages the modified Euler-Bernoulli beam theory to accommodate variable mass functions and employs a semi-analytical approach for the analysis of vibration characteristics in the variable cross-section beam. The excitation loads acting on the hull girder are composed of engine loads, propeller loads, and hydrodynamic loads respectively defined in the dynamic model. Furthermore, an improved Kane's dynamic equation is integrated into the mathematical and numerical model, tailored for time-varying mass systems, serving as the primary dynamic solver module. A customized program, written in FORTRAN language, is developed based on the proposed model. In addition, a user-defined case study is given in this paper. The varying wet surface and trim characteristics of the ship hull within a short period are also taken into consideration via dividing wet surface into ten shifting waterlines and loading conditions in variable mass properties. Hydrodynamic analysis results pre-calculated by SESAM are transferred into the program. Finally, dynamic response results including displacement and angular responses of each pre-defined rigid cross-section in the hull girder are generated by the self-developed program, which can be used for further FEA analysis to achieve detailed stress and deformation results. The proposed mathematical and numerical model can be used in the design stage for ships and offshore vessels that have time-varying mass features to evaluate their special structural responses during variable mass operations.