Time-varying mass-spring model of sloshing in emptying containers under excitation

Motivated by propellant-tank dynamics during rocket launch, this paper develops a mass-spring model for liquid sloshing in cylindrical containers that are being drained while subjected to external excitation. The liquid is partitioned into impulsive and convective zones and represented by an equivalent mechanical system whose masses and stiffnesses vary with the decreasing fill height h ( t ). The governing equations contain an additional damping-like term associated with the transfer of fluid from the convective (sloshing) domain to the impulsive domain during emptying. After nondimensionalization and reduction to a single dominant sloshing mode, the dynamics is described by a forced second-order ODE with a time-dependent natural frequency Ω( h ( t )). The frequency-liquid height (Ω − h ) parameter plane is introduced and its analysis shows that the instantaneous sloshing frequency possesses a well-defined maximum at an intermediate fill level and approaches an asymptote for tall tanks, implying multiple resonance passages during drain-down depending on the excitation frequency and the emptying history. Numerical simulations under harmonic forcing illustrate the distinct responses for static, non-resonant, and resonance-passage excitations. The equation is transformed via a Liouville substitution and approximated using Bellman’s asymptotic form, producing a closed-form expression for the time-varying amplitude and phase in both free and forced cases. Comparison with numerical solutions demonstrates good accuracy of the approximation for moderate-to-high fill levels.