scholarly journals Bulging Modes of Circular Bottom Plates in Rigid Cylindrical Containers Filled with a Liquid

1997 ◽  
Vol 4 (1) ◽  
pp. 51-68 ◽  
Author(s):  
Marco Amabili
Keyword(s):  
Free Surface ◽  
Liquid Surface ◽  
Dynamic Pressure ◽  
Ritz Method ◽  
Free Vibrations ◽  
Free Liquid Surface ◽  
Mode Shapes ◽  
Bottom Plate ◽  
Free Surface Waves ◽  
Theoretical Approaches

In this article the free vibrations of the bottom plate of an otherwise rigid circular cylindrical tank filled with liquid are studied, considering only the bulging modes (when the amplitude of the plate displacement is predominant with respect to that of the free surface). The tank axis is vertical, thus the free liquid surface is orthogonal to the tank axis. The liquid is assumed to be inviscid, and the contribution of the free surface waves to the dynamic pressure on the free liquid surface is neglected. Wet and dry mode shapes of the plate are assumed to be the same, so that the natural frequencies are obtained by using the nondimensionalized added virtual mass incremental (NA VMI) factors and the modal properties of dry plates. This simplifies computations compared to other existing theoretical approaches. NAVMI factors express the nondimensionalized ratio between the reference kinetic energy of the liquid and that of the plate and have the advantage that, due to their nondimensional form, they can be computed once and for all. Numerical results for simply supported and clamped bottom plates, as well as for supported plates with an elastic moment edge constraint are given. For more accurate results, and to exceed the limits of the assumed modes approach, the Rayleigh-Ritz method is applied and results are compared to those obtained by using the NAVMI factors and other existing methods in the literature.

Free Vibrations of Thick, Complete Conical Shells of Revolution From a Three-Dimensional Theory

2005 ◽  
Vol 72 (5) ◽  
pp. 797-800 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Ritz Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Shells Of Revolution ◽  
Dimensional Theory ◽  
Conical Shells ◽  
Cylindrical Coordinate

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution in which the bottom edges are normal to the midsurface of the shells based upon the circular cylindrical coordinate system using the Ritz method. Comparisons are made between the frequencies and the corresponding mode shapes of the conical shells from the authors' former analysis with bottom edges parallel to the axial direction and the present analysis with the edges normal to shell midsurfaces.

Analysis of In-Plane Modal Characteristics of an Annular Disk With Multiple Point Supports

2009 ◽  
Author(s):  
S. Bashmal ◽  
R. Bhat ◽  
S. Rakheja
Keyword(s):  
Finite Element ◽  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Element Model ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Multiple Point ◽  
Annular Disk ◽  
Free Boundary Conditions ◽  
Boundary Characteristic

In-plane free vibrations of an isotropic, elastic annular disk constrained at some points on the inner and outer boundaries are investigated. The presented study is relevant to various practical problems including disks clamped by bolts along the inner and outer edges or the railway wheel vibrations. The boundary characteristic orthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. The boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to realize clamped conditions at discrete points. The frequency parameters for different point constraint conditions are evaluated and compared with those computed from a finite element model to demonstrate the validity of the proposed method. The computed mode shapes are presented for a disk with different point constraints at the inner and outer boundaries to demonstrate the free in-plane vibration behavior of the disk. Results show that addition of point supports causes some of the modes to split into two different frequencies with different mode shapes. The effects of different orientations of multiple point supports on the frequency parameters and mode shapes are also discussed.

scholarly journals In-Plane free Vibration Analysis of an Annular Disk with Point Elastic Support

2011 ◽  
Vol 18 (4) ◽  
pp. 627-640 ◽  
Author(s):  
S. Bashmal ◽  
R. Bhat ◽  
S. Rakheja
Keyword(s):  
Boundary Conditions ◽  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Outer Edge ◽  
Mode Shapes ◽  
Annular Disk ◽  
Different Boundary Conditions ◽  
Boundary Characteristic

In-plane free vibrations of an elastic and isotropic annular disk with elastic constraints at the inner and outer boundaries, which are applied either along the entire periphery of the disk or at a point are investigated. The boundary characteristic orthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. Boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to account for different boundary conditions. The frequency parameters for different boundary conditions of the outer edge are evaluated and compared with those available in the published studies and computed from a finite element model. The computed mode shapes are presented for a disk clamped at the inner edge and point supported at the outer edge to illustrate the free in-plane vibration behavior of the disk. Results show that addition of point clamped support causes some of the higher modes to split into two different frequencies with different mode shapes.

Free Vibration and Tension Buckling of Circular Plates With Diametral Point Forces

1990 ◽  
Vol 57 (4) ◽  
pp. 995-999 ◽  
Author(s):  
E. F. Ayoub ◽  
A. W. Leissa
Keyword(s):  
Circular Plate ◽  
Tensile Force ◽  
Ritz Method ◽  
Compressive Force ◽  
Free Vibrations ◽  
Plane Elasticity ◽  
Mode Shapes ◽  
Simply Supported ◽  
Symmetry Classes ◽  
Tensile Forces

This paper presents the first known results for the free vibrations of a circular plate subjected to a pair of static, concentrated forces acting on the boundary at opposite ends of a diameter. The closed-form exact solution of the plane elasticity problem is used to provide the in-plane stress distribution for the vibration problem. A proper procedure using the Ritz method is developed for solving the latter problem for clamped, simply supported, or free boundary conditions. Numerical results are given for the vibration frequencies of a simply supported circular plate, which separate into four symmetry classes of mode shapes. Compressive buckling loads for each symmetry class are determined as a special case as the frequencies decrease to zero with increasing compressive force. Tracking the frequency versus loading data with increasing tensile forces shows that buckling due to tensile force can also occur, and the critical value of the force is found.

scholarly journals Free Vibrations of a Trapezoidal Plate with an Internal Line Hinge

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
María Virginia Quintana ◽  
Ricardo Oscar Grossi
Keyword(s):  
Plate Theory ◽  
Ritz Method ◽  
Free Vibrations ◽  
Rectangular Domain ◽  
Internal Line ◽  
Thin Plates ◽  
Mode Shapes ◽  
Approximate Analytical Solutions ◽  
Elastically Restrained Edges ◽  
Elastically Restrained

This paper deals with a general variational formulation for the determination of natural frequencies and mode shapes of free vibrations of laminated thin plates of trapezoidal shape with an internal line hinge restrained against rotation. The analysis was carried out by using the kinematics corresponding to the classical laminated plate theory (CLPT). The eigenvalue problem is obtained by employing a combination of the Ritz method and the Lagrange multipliers method. The domain of the plate is transformed into a rectangular domain in the computational space by using nonorthogonal triangular coordinates and the transverse displacements are approximated with a set of simple polynomials automatically generated and expressed in the triangular coordinates. The developed algorithm allows obtaining approximate analytical solutions for mentioned plate with different geometries, aspect ratio, position of the line hinge, and boundary conditions including translational and rotational elastically restrained edges. It allows studying the influence of the mentioned line on the vibration frequencies and respective mode shapes. The algorithm can easily be programmed and it is numerically stable. Additionally, as a particular case, the results of triangular plates can be easily generated.

Dynamic Modeling of Air Cushion Vehicles

2007 ◽  
Author(s):  
M. Pollack ◽  
B. Connell ◽  
J. Wilson ◽  
W. Milewski
Keyword(s):  
Free Surface ◽  
Pressure Distribution ◽  
Dynamic Pressure ◽  
Strong Dependence ◽  
Low Frequency ◽  
Coupled System ◽  
Lumped Parameter Model ◽  
Mode Shapes ◽  
Resonant Frequencies ◽  
Air Cushion

The motion of an Air Cushion Vehicle (ACV) is a complex process involving the nonlinear dynamics of the ship, free surface waves, air cushion cavity, skirt, and air flow hydraulics (e.g. orifice behavior of the bag feed holes). The overall system is tightly coupled, with the loading of the ship dependent on the pressure field within the cavity, and the dynamics of the cavity dependent on the motion of the ship, free surface, and skirt. Principle excitation to the system is through the free surface motion and the fan flow. The large dimensions of the system introduce low frequency acoustic and mechanical resonances, which lead to complex and coupled system dynamics. The focus of this paper is on analytical modeling of an ACV and its physics to enable verification of a numerical model which is under development. The initial focus is on the dynamics of the air cushion cavity, with emphasis on its resonant frequencies and mode shapes. The mode shapes are important because they define the nature of the dynamic pressure distribution acting on the ship, and associated heave, pitch, and roll excitation. The strong dependence of the cavity resonant characteristics on the impedance of the skirt, which bounds the cavity, is first demonstrated by assessing limiting cases of a high impedance skirt (e.g. massive or rigid) and of a low impedance skirt (e.g. light or soft). The changes in resonant frequency and dynamic pressure distribution associated with the changes in skirt impedance are illustrated. Because the actual skirt impedance will lie between these two idealized cases, we also develop a lumped parameter model of the skirt dynamics. Initial parametric studies with this model, which investigate variations in the skirt mass, further demonstrate the strong dependence of the resonant frequencies and pressure distributions on the skirt impedance.

scholarly journals Construction of basic functions for problems of fluid oscillations in a tank

2019 ◽  
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Liquid Surface ◽  
Singular Integral Equations ◽  
Spherical Shape ◽  
Free Liquid Surface ◽  
Surface Shape ◽  
Slight Rise ◽  
Oscillation Frequencies ◽  
The Impact

Considerable number of studies and publications is devoted to issues of dynamic behavior of liquids, the impact on the surface tension of a liquid in partially filled tanks in particular. The study of liquid vibrations in partially fluid-filled cylindrical containers with the presence of a free surface is an important technical task. The influence of the free surface curvature of the tank filler on the oscillation frequency is taken into account. It is assumed that the liquid is incompressible and inviscid, and its motion is irrotational. The method to solve a boundary value problem for determining fluid oscillations in a reservoir has been developed, and an integral presentation of an unknown velocity potential is proposed. The geometrical characteristics of the free liquid surface have been determined. It is taken into account that the free liquid surface deviates from the equilibrium position and assumes a spherical shape. A system of singular integral equations has been obtained for unknown values of the potential and flow. The method of boundary elements with constant approximation of an unknown density on the elements has been used to solve the system numerically. The oscillation frequencies for the zero harmonic are determined in accordance with the level of the free-surface elevation. It has been determined that the deviation of the free surface shape from the flat and even a slight rise in the free surface level leads to noticeable changes in the vibration frequencies. The vibrational modes obtained in the study mostly coincide with the modes for a flat free surface and can serve as the basic system of functions in the studies of free and forced fluid vibrations in tanks, as well as, in the study of the intrinsic and forced sloshing in the reservoirs provided surface tension is taken into account.

scholarly journals Analysis of turbulence on cylinder with additional fairing with free surface

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Siqin Chen ◽  
Xiaomin Li
Keyword(s):  
Free Surface ◽  
Drag Coefficient ◽  
Liquid Surface ◽  
Lift Coefficient ◽  
Flow Characteristics ◽  
Reynolds Numbers ◽  
Free Liquid Surface ◽  
Uniform Grid ◽  
Different Depths ◽  
High Reynolds

In this study, two dimensional unsteady flow of cylinder and cylinder with additional fairing close to a free surface was numerically investigated. The governing momentum equations were solved by using the Semi Implicit Method for Pressure Linked Equations(SIMPLE). The Volume of Fluid(VOF) method applied to simulate a free surface. Non- uniform grid structures were used in the simulation with denser grids near the cylinder. Under the conditions of Reynolds number 150624, 210874, 210874 and 331373, the cylinders were simulated with different depths of invasion. It was shown that the flow characteristics were influenced by submergence depth and Reynolds numbers. When the cylinder close to the free surface, the drag coefficient, lift coefficient and Strouhal numbers will increase due to the effect of free liquid surface on vortex shedding. With additional fairing, can effectively reduce the influence of the free surface on the drag coefficient. Fairing will reduce lift coefficient at high Reynolds numbers, but increase lift coefficient when Reynolds numbers is small. Fairing can effectively reduce Strouhal numbers, thus can well suppress the vortex induced vibration.

scholarly journals Analysis of Free Vibration Characteristics of Cylindrical Shells with Finite Submerged Depth Based on Energy Variational Principle

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2162
Author(s):  
Rui Nie ◽  
Tianyun Li ◽  
Xiang Zhu ◽  
Cheng Zhang
Keyword(s):  
Free Surface ◽  
Free Vibration ◽  
Natural Frequency ◽  
Liquid Surface ◽  
Energy Functional ◽  
Calculation Model ◽  
Vibration Characteristics ◽  
Free Liquid Surface ◽  
Immersion Depth ◽  
Infinite Domain

Based on the principle of energy variation, a calculation model for the free vibration characteristics of a cylindrical shell with a finite submerged depth considering the influence of the free liquid surface is established in this paper. First, the Euler beam function is used instead of the shell axial displacement function to obtain the shell kinetic energy and potential energy. Then, by using the mirror image method, the analytical expression of the fluid velocity potential considering the free surface is obtained, and the flow field is added to the system energy functional in the form of fluid work. Then the energy functional is changed to obtain the shell–liquid coupled vibration equation. Solving the equation can obtain the natural frequencies and modes of the structure. The comparison with the finite element calculation results verifies the accuracy of the calculation model in this paper. The research on the influence of the free liquid surface shows that compared to the infinite domain, the free liquid surface destroys the symmetry of the entire system, resulting in a difference in the natural frequency of the positive and negative modes of the shell, and the circumferential mode shapes are no longer mutually uncoupled trigonometric functions. The existence of free liquid surface will also increase the natural frequency of the same order mode, and the closer to the free surface, the natural frequency is greater. As the immersion depth increases, the free vibration characteristics will quickly tend to the result of infinite domain. Additionally, when the immersion depth is equal to or greater than four times the radius of the shell structure, it can be considered that the free liquid surface has no effect. These law and phenomena have also been explained from the mechanism. The method in this paper provides a new analytical solution pattern for solving this type of problem.

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