Linear vibration analysis of cantilever plates partially submerged in fluid

Abstract

Dynamic characteristics, such as natural frequencies and mode shapes, of cantilever plates, partially in contact with a fluid, are investigated. In the analysis of the linear fluid–structure system, it is assumed that the fluid is ideal, and fluid forces are associated with inertial effects of the surrounding fluid. This implies that the fluid pressure on the wetted surface of the structure is in phase with the structural acceleration. Furthermore, the infinite frequency limit is assumed on the free surface. The in vacuo dynamic properties of the plates are obtained by use of a standard finite-element software. In the wet part of the analysis, it is assumed that the plate structure preserves its in vacuo mode shapes when in contact with the surrounding fluid and that each mode shape gives rise to a corresponding surface pressure distribution of the cantilever plate. The fluid–structure interaction effects are calculated in terms of the generalized added-mass values independent of frequency (i.e., infinite frequency generalized added-masses), by use of a boundary-integral equation method together with the method of images in order to impose the Φ = 0 boundary condition on the free surface. To assess the influence of the surrounding fluid on the dynamic characteristics, the wet natural frequencies and associated mode shapes were calculated, and they compared very well with the available experimental data and numerical predictions.

Introduction

An accurate understanding of the dynamic interaction between an elastic structure and fluid is necessary in various engineering problems. Examples include vibration of floating structures (ships, offshore platforms, etc.) excited by wave impact, water-retaining structures (dams, storage vessels, etc.) under earthquake loading, etc. The local resonant vibration behavior of individual plates has also been a great concern to the shipbuilder and operator for many years. Therefore, considerable effort has been made in the study of the vibrational response of plates, partially or totally immersed in the fluid.

Lindholm et al. (1965) experimentally investigated the resonance frequencies of cantilever plates in air, totally or partially immersed in water, and compared the measurements with the theoretical predictions. They evaluated the fluid actions using a strip-theory approach. Meyerhoff (1970) calculated the added mass of thin rectangular plates in infinite fluid, and described the potential flow around a flat rectangular plate by use of dipole singularities. The finite-element method has also been applied to solve the fluid–structure interaction problems for completely submerged elastic plates (see, for example, Muthuveerappan et al., 1979; Rao et al., 1993). On the other hand, Fu and Price (1987) studied the dynamic behavior of a vertical or horizontal cantilever plate totally or partially immersed in fluid. In their analysis, they calculated the generalized fluid loading to assess the influences of free surface and submerged plate length on the dynamic characteristics. Recently, Liang et al. (2001) adopted an empirical added-mass formulation to determine the frequencies and mode shapes of submerged cantilever plates, and compared their results with the available experimental data and numerical predictions.

In this work, the dynamic characteristics (i.e., wet natural frequencies and mode shapes) of vibrating cantilever plates, partially or totally immersed within the fluid, as illustrated in Fig. 1, are studied, assuming a linear fluid–structure system. In this investigation, it is assumed that the fluid is ideal, i.e., inviscid and incompressible, and its motion is irrotational. Furthermore, the fluid forces are associated with the inertial effect of the fluid. This means that the fluid pressure on the wetted surface of the structure is in phase with the acceleration. In the analysis, it is assumed that the dry plate vibrates in its in vacuo eigenmodes when it is in contact with fluid, and that each mode gives rise to a corresponding surface pressure distribution on the wetted part of the structure. The in vacuo dynamic analysis entails the vibration of the cantilever plate in the absence of any external force and structural damping, and the corresponding dynamic characteristics (e.g., natural frequencies and principal mode shapes) of the plate structure were obtained using ANSYS, a standard finite-element software.

At the fluid–structure interface, the body-boundary condition requires that the normal velocity of the fluid be equal to that of the structure. The normal velocities on the wetted surface are expressed in terms of the modal structural displacements, obtained from the in vacuo dynamic analysis. By use of a boundary-integral equation method in conjunction with the method of images (i.e., imposing infinite frequency limit condition on the free surface), the fluid pressure is eliminated from the problem and the fluid–structure interaction forces are calculated solely in terms of the generalized added-mass coefficients that are independent of the vibrational frequency. As a result of the boundary condition imposed on the free surface, the hydrodynamic damping becomes zero.

In this analysis, the wet surface is idealized by use of appropriate boundary elements, referred to as hydrodynamic panels. The generalized structural mass matrix is merged with the generalized added-mass matrix and then the total generalized mass matrix is used in solving the eigenvalue problem for the partially or totally immersed cantilever plate. To assess the influence of the surrounding fluid on the dynamic behavior of the plate structure, wet natural frequencies and associated mode shapes are calculated.

Comparison of the predicted dynamic characteristics with available experimental measurements (Lindholm et al., 1965) and numerical calculations (Fu and Price, 1987) shows very good agreement. Furthermore, the influences of various effects such as the submergence depth, plate aspect ratio (a/b) and thickness ratio (t/b) on the dynamic behavior are also investigated.