Finite amplitude vibrations of a sharp-edged beam immersed in a viscous fluid near a solid surface

Abstract

In this paper, we study finite amplitude bending vibrations of a slender thin beam immersed in a quiescent viscous liquid and oscillating near a solid surface. We focus on the regime of low Knudsen and squeeze numbers and moderately large Keulegan-Carpenter number, for which neither squeeze film models nor unsteady Stokes hydrodynamics are suitable to describe the flow physics. In this case, the distributed hydrodynamic loading experienced by the oscillating beam is represented by a complex-valued hydrodynamic function, which explicitly depends on the Keulegan-Carpenter number to account for convection-driven nonlinearities in the fluid-structure interaction. We conduct a parametric study on the two-dimensional computational fluid dynamics of a rigid lamina oscillating in the vicinity of a solid surface to establish a handleable semianalytical formula for the hydrodynamic function in terms of the key nondimensional parameters. We validate the proposed modeling approach through experiments on centimeter-size compliant cantilevers vibrating underwater under base excitation at varying distances from a rigid wall.

Influence of squeeze-film damping on higher-mode microcantilever vibrations in liquid

“…[13]. If the vibration amplitudes are orders of magnitude smaller than b, i.e., KeuleganCarpenter numbers « 1 [15], the effect becomes independent of the amplitude [16]. Furthermore, the continuum hypothesis is valid because the mean-free-path of the molecules in liquid is very small compared to the dominant length b and the gap size g, i.e., Knudsen numbers « 1 [15].…”

“…If the vibration amplitudes are orders of magnitude smaller than b, i.e., KeuleganCarpenter numbers « 1 [15], the effect becomes independent of the amplitude [16]. Furthermore, the continuum hypothesis is valid because the mean-free-path of the molecules in liquid is very small compared to the dominant length b and the gap size g, i.e., Knudsen numbers « 1 [15]. Experimental investigations of cantilevers with dimensions ranging from centimeters to micrometers, immersed in water, buffer, organic solvents and oils are reported in the literature [3,[15][16][17][18][19].…”

Hydrodynamic coupling of two sharp-edged beams vibrating in a viscous fluid

“…This decomposition allows for the independent analysis of structural vibration problems, without the need of further flow physics simulations, through the mere implementation of the hydrodynamic functions that are developed as part of this study. Based on the analysis in [42], we expect equation (2.3) to be accurate for oscillations as large as the width in the case of β of the order of one. As β increases, the advection term in the Navier-Stokes equation increases and the range of validity of the approximation is expected to drastically decrease, whereby only oscillations of the order of 10 −2 b should be resolved for β approaching the limit value of 10 4 considered in this study.…”

“…We solve the system of two coupled equations in equation (2.5) through the Galerkin method, whereby we first diagonalize equation (2.5) to obtain two decoupled boundary value problems, and then we project the corresponding displacement fields following [29,30,42,61]. For convenience, we rewrite equation (2.5) in the following compact form …”

“…We retain the computational model, domain and solution strategy used in [29,30,42,61], by simply replacing the single lamina studied therein with two laminae. CFD analysis provides pressure and velocity fields in the computational domain and the time histories of the forces exerted by the fluid on the two laminae.…”