Free Vibration Modes Research Articles

Simple Formula to Study the Large Amplitude Free Vibrations of Beams and Plates

A simple formula to study the large amplitude free vibration behavior of structural members, such as beams and plates, is developed. The nonlinearity considered is of von Karman type, and after eliminating the space variable(s), the corresponding temporal equation is a homogeneous Duffing equation. The simple formula uses the tension(s) developed in the structural members due to large deflections along with the corresponding buckling load obtained when the structural members are subjected to the end axial or edge compressive load(s) and are equal in magnitude of the tension(s). The ratios of the nonlinear to the linear radian frequencies for beams and the nonlinear to linear time periods for plates are obtained as a function of the maximum amplitude ratio. The numerical results, for the first mode of free vibration obtained from the present simple formula compare very well to those available in the literature obtained by applying the standard analytical or numerical methods with relatively complex formulations.

Vibration analysis of solid ellipsoids and hollow ellipsoidal shells of revolution with variable thickness from a three-dimensional theory

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of complete ellipsoidal shells of revolution with variable thickness and solid ellipsoids. Unlike conventional shell theories, which are mathematically two-dimensional (2D), the present method is based upon the 3D dynamic equations of elasticity. Displacement components u r , u θ, and u z in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the ellipsoidal shells of revolution and solid ellipsoids are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to three or four-digit exactness is demonstrated for the first five frequencies of the ellipsoidal shells of revolution. Numerical results are presented for a variety of ellipsoidal shells with variable thickness. Frequencies for five solid ellipsoids of different axis ratios are also given. Spherical shells and solid spheres are special cases which are included. Comparisons are also made between the frequencies from the present 3D Ritz method, a 2D Ritz method, and a 3D finite element method. The multiple degeneracies (two or more modes having the same frequency) of spherical bodies are examined by analyzing some almost-spherical ellipsoids.

ACCURATE FREQUENCIES AND MODE SHAPES FOR MODERATELY THICK, CANTILEVERED, SKEW PLATES

Accurate free vibration frequencies and mode shapes are presented for complete sets of moderately thick, cantilevered skew plates of triangular, trapezoidal and parallelogram shape. These accurate results are obtained by using the Ritz method applied to the Mindlin plate theory. Two sets of functions are employed simultaneously for each of the three dependent variables: transverse displacement (w) and bending rotations (ϕxand ϕy). One set is the widely used algebraic polynomials. The other is the set of corner functions which provide the proper stress singularities in the reentrant clamped-free corner, and accelerates the convergence of the solutions. The extensive frequencies presented are exact to the four digits shown. Corresponding mode shapes are also shown, by means of nodal patterns, most of which are novel in the published literature.

Calculation of vibration transmission over bedrock using a waveguide finite element model

AbstractA finite element method is developed for the study of elastic wave propagation in layered ground environments. The formulation is based on a spectral finite‐element approach using a mixture of high‐order element shape functions and wave solutions. The numerical method provides solutions to vibration transmission on and within layered elastic waveguides. Examples of its use include the theoretical analysis of transmission of vibrations in the vicinity of the surface of the ground. The mathematical model is two dimensional, and the interior of the ground is modelled as an elastic layer overlying a rigid foundation. An analysis of the natural modes of free vibration in a single layer and two layers is presented and compared with known results. In addition the forced response of the layers, for which the surface is assumed to be subjected to a harmonic point force load is shown. These results also include an illustration of the attenuation of surface vibration due to ‘wave impedance blocks’ in the ‘near field’ of the source up to a frequency of 200 Hz for two soil types. Copyright © 2007 John Wiley & Sons, Ltd.

Damping by Bulk and Shear Viscosity of Confined Acoustic Phonons for Nanostructures in Aqueous Solution

A nanoparticle in aqueous solution is modeled as a homogeneous elastic isotropic continuum sphere in contact with an infinite viscous compressible Newtonian fluid. The frequencies and damping of the confined vibrational modes of the sphere are calculated for the material parameters of a CdSe nanoparticle in water and a poly(methyl methacrylate) nanosphere in water. Although the effects of viscosity are found to be negligible for macroscopic objects, for nanoscale objects, both the frequency and damping of the vibrational modes are significantly affected by the viscosity of the liquid. Furthermore, both shear viscosity and bulk viscosity play an important role. A model of the spherical satellite tobacco mosaic virus consisting of outer solid layers with a water core is also investigated, and the viscosity of the water core is found to significantly damp the free vibrational modes. The same approach can be applied for nonspherical geometries and also to viscoelastic nanoparticles.

センターポンツーン付シングルデッキ型浮屋根を有する円筒液体貯槽の地震時スロッシング応答

An analytical solution is presented to predict the sloshing response of a cylindrical liquid storage tank with a single-deck type floating roof under seismic excitation. The floating roof is considered to be composed of an inner deck stiffened with outer and center pontoons. The deck is idealized as an isotropic elastic plate with uniform stiffness and mass, while the outer and center pontoons are modeled as an elastic curved beam and a rigid circular plate, respectively. The contained liquid is assumed to be inviscid, incompressible and irrotational. The dynamic interaction between the floating roof and the liquid is taken into account exactly within the framework of linear potential theory. By expanding the response of the coupled deck-pontoon system into free vibration modes in air and employing the Fourier-Bessel expansion technique in cylindrical coordinates, the solution is obtained in an explicit form which will be useful for parametric understanding of the sloshing behavior and preliminary study in the early design stage. Numerical results are provided to investigate the effect of the presence of the center pontoon on the sloshing response of the floating roof.

浮屋根式円筒液体貯槽の地震時スロッシング応答 : FEMと解析解の結合解法

A computational method based on coupled FE (finite element) and analytical solutions is proposed for predicting the sloshing response of a floating roof in a cylindrical liquid storage tank subjected to seismic excitation. The liquid is assumed to be inviscid, incompressible and irrotational, while the floating roof is considered as an orthotropic elastic plate of arbitrary axisymmetric form. The method of solution is based on applying the FEM to evaluate the free vibration modes of the floating roof in air into which the response of the floating roof is expanded. The liquid-floating roof coupling analysis is then performed semi-analytically based on linear potential theory. The method is easier to apply and computationally more efficient than the conventional 3-D analysis based on FE modeling for both the floating roof and the liquid. Numerical results are presented to illustrate the applicability of the proposed method and to investigate the effect of rib-stiffening on the sloshing response of rib-stiffened floating roofs.

타원형 또는 원형 단면을 가진 원형 링의 3차원적 진동해석

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (circumferentially closed), circular rings with an elliptical or circular cross-section. Displacement components <TEX>$u_r,\;u_\theta\;and\;u_z$</TEX> in the radial, circumferential, and axial directions, respectively, are taken to be periodic in <TEX>${\theta}$</TEX> and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the circular rings are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. Novel numerical results are presented for the circular rings having an elliptical cross-section based upon 3-D theory. Comparisons are also made between the frequencies from the present 3-D Ritz method and ones obtained from thin and thick ring theories, experiments, and another 3-D method.

Sloshing in a Cylindrical Liquid Storage Tank With a Floating Roof Under Seismic Excitation

An analytical solution is presented to predict the sloshing response of a cylindrical liquid storage tank with a floating roof under seismic excitation. The contained liquid is assumed to be inviscid, incompressible, and irrotational, while the floating roof is idealized as an isotropic elastic plate with uniform stiffness and mass. The dynamic interaction between the floating roof and the liquid is taken into account exactly within the framework of linear potential theory. By expanding the response of the floating roof into free-vibration modes in air and employing the Fourier–Bessel expansion method in cylindrical coordinates, the solution is obtained in an explicit form, which will be useful for parametric understanding of the sloshing behavior and preliminary study in the early design stage. Numerical results are also provided to investigate the effect of the stiffness and mass of the floating roof on the sloshing response.

Flow-induced internal resonances and mode exchange in horizontal cantilevered pipe conveying fluid (II)

The Newtonian method is employed to obtain nonlinear mathematical model of motion of a horizontally cantilevered and inflexible pipe conveying fluid. The order magnitudes of relevant physical parameters are analyzed qualitatively to establish a foundation on the further study of the model. The method of multiple scales is used to obtain eigenfunctions of the linear free-vibration modes of the pipe. The boundary conditions yield the characteristic equations from which eigenvalues can be derived. It is found that flow velocity in the pipe may induced the 3:1, 2:1 and 1:1 internal resonances between the first and second modes such that the mechanism of flow-induced internal resonances in the pipe under consideration is explained theoretically. The 3:1 internal resonance first occurs in the system and is, thus, the most important since it corresponds to the minimum critical velocity.

Adaptive Control of Nonlinear Free Vibrations of Composite Plates Using Piezoelectric Actuators

A coupled structural-electrical finite element modal formulation is employed for the control of nonlinear free vibrations of beams and composite plates. Multiple modes of the nonlinear free vibration are considered in closed-loop simulations. Two different controllers are designed and investigated for the suppression of nonlinear free vibrations. The first is the position output feedback controller comprising a linear quadratic regulator (LQR) and an extended Kalman filter (EKF). The EKF is used to consider the nonlinear state-space matrix and has a gain sequence evaluated online. The second controller is the output feedback adaptive LQR/EKF controller. This adaptive controller includes an adaptive modal frequency identification and state estimation algorithm. Numerical simulations show that the adaptive LQR/EKF controller with system identification is effective in suppressing nonlinear free vibrations of a beam and a composite plate with unpredictable sudden frequency changes. The placement of piezoelectric self-sensing actuators is based on three approaches: one is the norm of optimal feedback control gain matrix method, the second is the H 2 norm, and the last one is the norm of Kalman filter estimation gain method.

Natural frequencies of thick, complete, circular rings with an elliptical or circular cross-section from a three-dimensional theory

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (circumferentially closed), circular rings with an elliptical or circular cross-section. Displacement components ur, uθ, and uz in the radial, circumferential, and axial directions, respectively, are taken to be periodic in θ and in time, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the circular rings are formulated, and upper-bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the rings. Novel numerical results are presented for the circular rings having an elliptical cross-section based upon 3D theory. Comparisons are also made between the frequencies from the present 3D Ritz method and ones obtained from thin and thick ring theories, experiments, and other 3D methods.

Exact solutions for the free in-plane vibration of rectangular plates with two opposite edges simply supported

Two distinct types of simple support boundary conditions are formulated for rectangular plates undergoing in-plane free vibration. Each type of simple support is shown to be analogous to the well known simple support edge conditions encountered in the free transverse vibration analysis of rectangular plates. It is shown that exact solutions may be obtained for plate free vibration eigenvalues and mode shapes when two opposite plate edges are given either type of simple support, the other two edges being given any combination of classical edge conditions. A complete analysis is provided for plate in-plane vibration with pairs of clamped or free edge conditions imposed on the non-simply supported boundaries and exact eigenvalues are tabulated for a large range of plate aspect ratios. This appears to be the first thorough study of these in-plane vibration problems with exact solutions.

シングルデッキ型浮屋根を有する円筒液体貯槽の地震時スロッシング応答

An explicit analytical solution is derived to predict the sloshing response of a cylindrical liquid storage tank with a single-deck type floating roof under seismic excitation. The floating roof is composed of an inner deck which may be idealized as an isotropic elastic plate with uniform thickness and connected to an outer pontoon which can be considered as an elastic curved beam. The contained liquid is assumed to be inviscid, incompressible and irrotational. By expanding the response of the floating roof into free vibration modes in air and applying the Fourier-Bessel expansion technique in cylindrical coordinates, the solution is obtained in an explicit form which is exact within the framework of linear potential theory. Numerical results are presented to investigate the effect of the type (single-deck or double-deck) and stiffness of the floating roof on the sloshing response.Copyright © 2007 by ASME

Free vibration analysis of complete paraboloidal shells of revolution with variable thickness and solid paraboloids from a three-dimensional theory

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid paraboloids and complete (that is, without a top opening) paraboloidal shells of revolution with variable wall thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell may be free or may be subjected to any degree of constraint. Displacement components u r , u θ and u z in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the paraboloidal shells of revolution are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the complete, shallow and deep paraboloidal shells of revolution with variable thickness. Numerical results are presented for a variety of paraboloidal shells having uniform or variable thickness, and being either shallow or deep. Frequencies for five solid paraboloids of different depth are also given. Comparisons are made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory.

Accurate analytical type solutions for free vibration frequencies and mode shapes of multi-span bridge decks: the span-by-span approach

Utilizing the superposition method an accurate analytical type solution is obtained for the free vibration frequencies and mode shapes of multi-span bridge decks. In this approach to the problem, a separate solution is obtained for the lateral displacement of each individual span. Simple support is provided at the outer extremities of the deck. Conditions of plate continuity and zero lateral displacement are enforced at inter-span support locations. Edges running parallel to the long centre line of the bridge are free. Eigenvalues and mode shapes are presented for a typical bridge deck of three spans, though it is shown that decks of any number of spans may be analysed. Advantages of this approach over an earlier one in which the entire deck was considered as a single plate with intermediate line supports, are discussed. The mathematical procedure developed constitutes one part of an overall study of bridge dynamics. Extension of the analysis to handle orthotropic bridge decks is easily accomplished.

Free Vibration of Thermally Buckled Composite Sandwich Plates

Abstract A high precision triangular plate element is developed for the free vibration analysis of thermally buckled composite sandwich plates. Due to an uneven thermal expansion in the two principal material directions, the buckling mode of the plate may change from one pattern to another in the postbuckling region for certain fiber orientation and aspect ratio of the plate. Because of this buckle pattern change, the sequence of natural frequencies of the plate is also suddenly altered. By examining the buckling and free vibration modes of the plate, a clear picture of buckle pattern change and vibration mode shifting is presented. Numerical results show that if the shape of a free vibration mode is similar to the plate buckling mode then the natural frequency of that mode will drop to zero when the temperature reaches the buckling temperature.

두꺼운 완전 원추형 회전셸의 3차원적 진동해석

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution, Unlike conventional shell theories, which are mathematically two-dimensional (2-D). the present method is based upon the 3-D dynamic equations of elasticity. Displacement components <TEX>$u_{r},\;u_{z},\;and\;u_{\theta}$</TEX> in the radial, axial, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in , and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the conical shells are formulated, the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of theconical shells. Novel numerical results are presented for thick, complete conical shells of revolution based upon the 3-D theory. Comparisons are also made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory.

Free in-plane vibration analysis of rectangular plates with elastic support normal to the boundaries

The superposition method is employed to analyze the effects of elastic edge support on the in-plane free vibration frequencies and mode shapes of rectangular plates. The elastic edge support is considered to be uniformly distributed along the plate edges and symmetrically distributed with respect to the plate central axes. In addition, exact solutions are also obtained for a family of simply supported plates whose eigenvalues constitute upper limits for those of the elastically supported plates. An array of highly accurate eigenvalues for the completely free plate is also computed in order to provide lower limit eigenvalues. Eigenvalues are computed and tabulated for elastically supported plates of various aspect ratios and dimensionless elastic support coefficients.

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

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.