Internal Resonances and Dynamic Responses in Equivalent Mechanical Model of Partially Liquid-Filled Vessel

Abstract

The paper treats oscillations of a liquid in partially filled vessel under horizontal harmonic ground excitation. Such excitation may lead to hydraulic impacts. The liquid sloshing mass is modeled by equivalent pendulum, which can impact the vessel walls. We use parameters of the equivalent pendulum for well-explored case of cylindrical vessels. The hydraulic impacts are modeled by high-power potential function. Conditions for internal resonances are presented. A non-resonant behavior and dynamic response related to 3:1 internal resonance are explored. When the excitation amplitude exceeds a critical value, the system exhibits multiple steady state solutions. Quasi-periodic solutions appear in relatively narrow range of parameters. Numerical continuation links between resonant regimes found asymptotically for small excitation amplitude, and high-amplitude responses with intensive impacts. NonTrivial J J J J J J        Jwhere coefficients ij J are expressed as follows:

Response regimes in equivalent mechanical model of moderately nonlinear liquid sloshing

“…Hydraulic jumps and wave collisions with the vessel shells express the strong non-linearities in the system [7]- [10].…”

“…Equivalent mechanical models for linear small-amplitude sloshing were based on pendulums and mass-spring systems were studied for different excitations and tank shapes [1], [38], [39]. However, since nonlinear sloshing regimes cannot be described by the linear models, major attempts were made to formulate nonlinear reduced order equivalent mechanical models [7], [8], [10], [26], [40], [41] that may predict and describe nonlinear responses observed experimentally. For instance, nonlinear rotary sloshing in a scale model of Centaur propellant tank at low fill level was modeled by combination of ordinary linear pendulum and a spherical pendulum (i.e.…”

“…In previous works, we treated hydraulic impacts regime with the help of reduced-order mechanical models. Both pendulum [10] and mass-spring systems [43], [44] were used to describe the vibro-impact regimes induced under a horizontal ground excitation. Tank elasticity and strongly nonlinear sloshing were considered, and different dynamical regimes were described with asymptotic tools that were validated numerically.…”

Response regimes in equivalent mechanical model of strongly nonlinear liquid sloshing

“…Besides, we take into account only the first sloshing mode, which has the largest modal mass (see [15,18] and Section 4 below). Then, the liquid sloshing in the tank with total mass M is approximately described by oscillations of the mass-spring-dashpot system with mass m, stiffness k, linear viscosity c and displacement y with respect to the vessel centerline.…”

“…The vessel mass includes also the non-sloshing portion of the liquid. Dynamical responses in the weakly nonlinear case are considered in previous work [18] and not addressed here. Our main goal is to get insight into the strongly nonlinear (impact) responses of the considered system in the framework of the lumped-mass model.…”