Modeling and Stability Analysis of Sloshing on Liquid-Propelled Reusable Launch Vehicles

Abstract

The modeling of the liquid dynamics within the tanks of liquid-propelled vehicles is presented in this publication. The proposed model is valid for high-g environments, and planetary atmospheric flight, i.e., applicable to expendable \gls{LV:launch}s, \gls{RLV}s and planetary landers. The oscillatory dynamics occurring at the top of the liquid's mass is approximated as the sum of $n$-swinging pendula representing the harmonic response of the $n$ modes. The interaction between the rigid body and pendula is tackled using a classic methodology from multibody systems, which is valid for a generic spacecraft with $m$ tanks and $n$ associated modes, i.e., the iterative Newton-Euler. The rationale behind the computation of the symbolic 3-DoF model, and subsequent simplifications to a 2-DoF planar model, is thoroughly discussed. The former model is suitable for simulation, while the latter finds its applications for control design and stability analysis. The symbolic computations were used to validate the Symbolic Multibody Dynamics toolbox belonging to the \gls{GNC} systems department of the DLR. This library can perform analytical computations of EoMs for different configurations of multibody space systems, and at the same time, it is able to provide models that can be easily implemented in simulation environments. The parameters associated to the liquids within the tanks of a space mission are calculated. They were derived using a set of analytical formulas corresponding to the geometry of the propellant and oxidizer tanks of a RLV. The stability analysis of the pair of complex poles and zeros associated to each mode is also studied, furthermore the position of the sloshing masses with regards to the danger zone is computed and analyzed. Finally, a CL analysis of the modes for a controller synthesized using H-inf designed is presented in terms of the stability margins.