Hamilton's law of variable mass system and time finite element formulations for time-varying structures based on the law

Traditional principles of mechanics are primarily conceived for constant mass systems, which are only valid if mass is gained or lost at null velocity with respect to an inertial reference frame for variable mass systems, thus the numerical algorithms for time-varying structures based on these principles are only suitable for this special case. In this paper, Hamilton's law of variable mass system is derived based on Meshchersky's fundamental equation, and two classes of novel time finite element formulations for linear systems with arbitrary continuous time-varying parameters are developed based on the previous law. The formulations are verified extensively through numerical examples in which the convergence and effectiveness of algorithms are evaluated. Numerical examples demonstrate that compared with the algorithms for time-varying structures that developed based on traditional principles of mechanics, the proposed algorithms provide extended capabilities in both time-varying mass problems that mass is gained or lost at any velocity (such as rocket problem) and moving-mass problems (such as vehicle-bridge interaction problem) besides the time-varying stiffness and damping problems, the proposed algorithms have a wider range of application. In particular, Hamilton's law of variable mass system provides a solid theoretical foundation for further research on the algorithm design for time-varying structures.