Nonlinear Mechanical Model for the Description of Propellant Sloshing

Mechanical model for longitudinally excited sloshing motion.

“…This free surface displacement is assumed to be of the form r = (4) where a mn are also functions of time to be determined from the free surface boundary conditions. With the necessary expansions and using an approximation up to third-order terms, the coefficients for the one-half subharmonic antisymmetric sloshing mode can be found.…”

“…-JV is the (£J)th element of the vehicle inertia tensor (assumed to be constant) and o>/ is the jth element of G> (i,j -1,2,3). The dynamic equation of the vehicle rotational motion is assumed to be G> = J-^G/G)) X 6) + T(o>)] (4) where J~1 denotes the inverse of / and T is the vector torque exerted on the body. Since feedback control laws of the form T = T(G>) are sought in the following, T in Eq.…”

Response regimes in equivalent mechanical model of moderately nonlinear liquid sloshing

“…In the former case, nonlinear dynamical features may take place, for example multiple periodic solutions ('jump' phenomenon) [1], weakly quasi-periodic [2], [3] and strongly modulated responses [4]. Based on experimental results, it was pointed that cubic spring seems to describe the dynamical regimes in the best way [5]. When the hydraulic jumps are involved, significant hydraulic impacts are applied to the vessel inner walls [6].…”

“…Mechanical models of the liquid sloshing often use an infinite series of pendula or massspring systems to represent the free liquid surface oscillation using infinite series of sloshing modes. These models, primarily developed using linear sloshing theory, are shown for various types of tank geometries and excitation types, by Graham [49], Graham and Rodriguez [38] and others [1], [5], [39], [50]. Equivalent moment of inertia of a liquid in cylindrical containers has been estimated numerically by Partom ([51] and [52]) and verified experimentally by Werner and Coldwell) [53].…”

“…Figure 15the comparison show qualitative similarity between the weaklynonlinear dynamical regimes revealed with the help of the relatively simple reduced-order model introduced in the current study and the documented results. Amplitude 'jump' phenomenon in partially-filled liquid tanks under horizontal periodic ground motion associated with hardening and softening nonlinearities was already observed experimentally by Abramson[1] and analytically by Bauer[5].Hill and Frandsen [14] solved analytically third-order amplitude evolution equation in terms of Jacobian elliptic functions to describe the evolution of weakly nonlinear sloshing wave, under assumptions of finite fluid depth and non-breaking waves. Classic hard-spring and softspring Duffing-type responses were revealed.…”