Spatial resonance of a liquid-filled vibrating beaker

Abstract

Mahony & Smith (1972) put forward a model to explain the phenomenon of energy transfer between nearly resonant oscillations at greatly differing frequencies. However, their model of ‘spatial resonance’ is restricted to situations where the geometry of the system is very simple. The present paper shows how to derive Mahony & Smith's equations in a general manner, and compares the theoretical predictions for a situation with circular symmetry with existing experimental results (Huntley 1972). In addition, it suggests a simple method for evaluating the resonance frequencies when a liquid-filled beaker is vibrated in one of its bell modes.

Nonlinear low-frequency gravity waves in a water-filled cylindrical vessel subjected to high-frequency excitations

“…Comparing with previous studies concentrating only on the water waves in the fluid domain excited by prescribed vessel motions, the theoretical model presented in this paper considers nonlinear FSI; therefore, new findings amend the previous researches: -Mahony & Smith [5] developed an acoustic air-water wave interaction model which, theoretically, could not be directly applied to the glass shell-water system by Huntley [6]. Our study considers the elastic shell-water system, based on which the nonlinear responses of the shell-modulated vibration mode are theoretically predicted other than a single harmonic linear response was assumed by Huntley [6]; -by deductions from the basic Hamiltonian of the coupled system, the motion of shell and fluid is explicitly described as function of external excitation, other than the conjectural components assumed by Mahony & Smith [5].…”

“…Our study considers the elastic shell-water system, based on which the nonlinear responses of the shell-modulated vibration mode are theoretically predicted other than a single harmonic linear response was assumed by Huntley [6]; -by deductions from the basic Hamiltonian of the coupled system, the motion of shell and fluid is explicitly described as function of external excitation, other than the conjectural components assumed by Mahony & Smith [5]. The developed model therefore gives quantitative prediction of the critical force-frequency curve other than simple qualitative predictions; -this study demonstrated that parametric nonlinearity exists in the system, which extends the instability region to excitation frequencies equal or below the natural frequency with sufficient large excitation amplitudes, as verified by experimental observations.…”

“…Their investigation developed an equation governing the critical amplitude of solid boundary vibration with respect to excitation frequency, which was validated by the experimental results. Huntley [6] extended Mahony & Smith's model to real three-dimensional cases. As reported by Wei et al [7], Wang observed and systematically studied the same phenomenon while doing some experimental research on the dragon washbasin.…”

Nonlinear low frequency water waves in a cylindrical shell subjected to high frequency excitations – Part II: Theoretical analysis

“…They studied a simple 2-dimensional (2-D) rectangular enclosure half-filled with water and then driven as an acoustic resonance tube. Considering nonlinear aerial and water wave interactions, they developed an equation governing the critic acoustic pressure amplitude with respect to excitation frequency when this phenomenon occurs, which was validated by experimental results of Huntley [1] and Franklin et al [5]. Huntley [6] extends Mahony and Smith's model to real three-dimensional (3-D) cases.…”

Chemical Sensor Based on Nonlinearity: Principle and Application

“…The nature of ripples under a periodic perturbation has been investigated. [47][48][49] However, the spatial information based on "ripple" has not yet been reported as an analytical tool.…”