Linear Dynamics of a Container-Liquid Coupled System Moving in a Plane Based on Equivalent Mechanical Model Using Multibody System Transfer Matrix Method

A liquid-propelled rocket undergoes vibrations, from ignition to lift-off from the launch pad, due to the combined effect of thrust forces, ground winds, and propellant sloshing within tanks. To accurately obtain the system’s vibration response, the precise computation of its vibration characteristics is an essential prerequisite. The multibody system transfer matrix method (MSTMM) can elegantly model a linear rigid-flexible coupled system for its eigenvalue problem without discretization, which leads to even exact solution for systems composed of beam-like elements. However, the existing MSTMM mainly focuses on structural dynamics or rotor dynamics and has not considered linear vibrations of a rigid container coupled to internal fluids. In this paper, a rigid moving container and internal liquid are treated as a subsystem undergoing planar motion, in which an equivalent mechanical model is adopted to handle the sloshing liquid. Then, the transfer equation and transfer matrix of the superelement are deduced in the context of the linear MSTMM. The Lagrange’s equations of the second kind and the Newton-Euler method are used to establish the linearized equation of motion (EoM) of the superelement, respectively. Both methods lead to the same result, which validates the derivation. Then, a linear relationship between state vectors of the input and output end of the subsystem can be obtained by rewriting the established EoM, resulting in the subsystem transfer equation. This paper extends the library of the element transfer matrices in the context of MSTMM. The derived transfer matrix of the subsystem comprising the rigid container and the internal sloshing liquid can then be assembled according to the automatic deduction theorem of the system overall transfer equation, enabling the solution of an eigenvalue problem of a rigid-flexible-fluid coupled launch vehicle system.