A note on vibration of a cantilever plate immersed in water

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.

Hydroelastic vibration of rectangular plates

Abstract: This paper is concerned with the virtual mass effect on the natural frequencies and mode shapes of rectangular plates due to the presence of the water on one side of the plate. The approximate formula, which mainly depends on the so-called nondimensionalized added virtual mass incremental factor, can be used to estimate natural frequencies in water from natural frequencies in vacuo. However, the approximate formula is valid only when the wet mode shapes are almost the same as the one in vacuo. Moreover, the nondimensionalized added virtual mass incremental factor is in general a function of geometry, material properties of the plate and mostly boundary conditions of the plate and water domain. In this paper, the added virtual mass incremental factors for rectangular plates are obtained using the Rayleigh-Ritz method combined with the Green function method. Two cases of interfacing boundary conditions, which are free-surface and rigid-wall conditions, and two cases of plate boundary conditions, simply supported and clamped cases, are considered in this paper. It is found that the theoretical results match the experimental results. To investigate the validity of the approximate formula, the exact natural frequencies and mode shapes in water are calculated by means of the virtual added mass matrix. It is found that the approximate formula predicts lower natural frequencies in water with a very good accuracy.

The free vibration analysis of submerged cantilever plates

Abstract: Based on empirical added mass formulation, this work presents a simple procedure to determine the vibration frequencies and mode shapes of submerged cantilever plates. Once the added mass formulation is derived, the procedure can be used to analyze free vibration response easily. An analytical and numerical study is also performed for the vibrations of cantilever plates in air and in water, with these results compared with experimental and numerical data from pertinent literature. Besides, the frequency parameters of the submerged plate for various aspect ratios and thickness ratios are given in design data sheet form and are appropriate for engineering design applications.

Numerical simulation of fluid added mass effect on a francis turbine runner

Abstract: In this paper, a numerical simulation to analyze the influence of the surrounding water in a turbine runner has been carried out using finite element method (FEM). First, the sensitivity of the FEM model on the element shape and mesh density has been analysed. Secondly, with the optimized FEM model, the modal behaviour with the runner vibrating in air and in water has been calculated. The added mass effect by comparing the natural frequencies and mode shapes in both cases has been determined. The numerical results obtained have been compared with experimental results available. The comparison shows a good agreement in the natural frequency values and in the mode shapes. The added mass effect due to the fluid structure interaction has been discussed in detail. Finally, the added mass effect on the submerged runner is quantified using a non-dimensional parameter so that the results can be

Dynamic characteristics and forced response of an electrostatically-actuated microbeam subjected to fluid loading

Abstract: In this paper flexural vibrations of an electrostatically actuated cantilever microbeam in an incompressible inviscid stationary fluid have been studied. By applying “Three dimensional aerodynamic theory” pressure jump across the microbeam has been investigated and the inertial effects of fluid on microbeam dynamics have been modeled as a mass added to microbeam mass. Magnitude of the added mass has been calculated for various aspect ratios of cantilever microbeams and compared with those of clamped-clamped microbeams. To investigate the dynamic characteristics, it has been considered that the microbeam has been deflected by a DC voltage, V DC and then the dynamic characteristics and forced response of the system have been considered about these conditions. Galerkin-based step by step linearization method (SSLM) and Galerkin-based reduced order model have been applied to solve the nonlinear static and dynamic governing equations, respectively. Water by neglecting viscidity effects, as an instant has been considered as a surrounding fluid and the frequency response of the microbeam has been compared with that of vacuum conditions. It has been shown that because of the added mass effects in watery environment, the natural frequencies of the microbeam decrease. Because of the higher dielectric coefficient and increasing electrical stiffness and decreasing total stiffness consequently, maximum amplitude of the microbeam vibrations increases in watery environment, compared with vacuum. Moreover, it has been shown that increasing the DC voltage, increases the electrical stiffness and maximum amplitude of the microbeam vibrations, consequently, It has been shown that in higher voltages (near pull-in voltage), the rate of variation of resonance frequency and maximum amplitude is stronger than lower voltages.

The finite element method in engineering science

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Vibration of plates

Abstract: : Contents: Fundamental Equations of Classical Plate Theory; Circular Plates; Elliptical Plates; Rectangular Plates; Parallelogram Plates; Other Quadrilateral Plates; Triangular Plates; Plates of Other Shapes; Anisotropic Plates; Plates With Inplane Forces; Plates With Variable Thickness; and Other Considerations.

Application of the simultaneous iteration method to undamped vibration problems

Abstract: The simultaneous iteration method of obtaining eigenvalues and eigenvectors is employed for the solution of undamped vibration problems. This method is of significance when a few of the dominant eigenvalues and eigenvectors are required from a large matrix, and hence is particularly suitable for vibration problems involving a large number of degrees of freedom. It is shown that advantage may be taken of both the symmetry and the band form of the mass and stiffness matrices, thus making it feasible to process on a computer larger order vibration problems than can be processed using transformation methods. A method of allowing for body freedom is given and some numerical tests are discussed.

Fluid finite elements for added mass calculations

Abstract: The equations for the free undamped vibration of a structure in an ideal incompressible fluid medium and their finite element formulation are briefly reviewed. The relevant matrices (stiffness and loading) for two prismatic fluid elements are given explicitly and some numerical results are presented.