Exact Solution For Free Vibration Research Articles

Exact solution for free vibration of thick rectangular plates made of porous materials

Free vibration analysis for thick rectangular porous plate saturated by inviscid fluid is presented. Material properties of the plate are assumed to vary through the thickness according to a simple cosine law in term of a coefficient related to plate’s porosity. Based on Reddy’s third-order shear deformation plate theory and considering the effect of fluid in pore network of the porous medium, the equations of motion are obtained. The plate is assumed to be simply supported on two opposite edges and the rest of edges are free, clamped or simply supported. For the sake of analytical solution, a unique approach is employed to decouple the equations of system. Exact frequencies rising from the solution are obtained for a rectangular porous plate made of Berea sandstone. The roles played by mechanical constraints on edges, fluid, geometrical dimensions of plate as well as the effect of the coefficient of plate porosity are investigated. It is found that porosity function affects significantly on the fundamental frequency of the system. Also, the effect of fluid on the dynamic response of plate is studied in detail.

Exact solution for free vibration of elastically restrained cantilever non-uniform beams joined by a mass-spring system at the free end

This article focuses on the free vibration analysis of a non-uniform cantilever beam with an attached mass-spring system at the free end. One end of the beam is elastically restrained against rotation and translation. The height of the beam is assumed constant but the width of the beam exponentially varies. The governing differential equations of the beam, which is a partial differential equation with variable coefficients, and that of the mass-spring system, which is an ordinary differential equation, are found. The exact solution of the problem is then obtained using the pertinent boundary conditions. The eigenvalues and eigenfunctions of the problem which are frequencies and mode shapes of the system are derived for various properties of the system such as stiffness of springs and attached mass. Some limiting cases with available results in the literature are analyzed. The comparison of the proposed results and the reference results shows the accuracy of the solution.

Exact solution for free vibration of coupled double viscoelastic graphene sheets by viscoPasternak medium

Based on nonlocal theory, this article discusses vibration of CDVGS11Coupled double viscoelastic graphene sheets. systems. The properties of each single layer graphene sheet (SLGS) are assumed to be orthotropic and viscoelastic. The two SLGSs are simply supported and coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak layer. This model is aimed at representing dynamic interactions in nanocomposite materials with dissipation effect. By considering the Kirchhoff plate theory and Kelvin–Voigt model, the governing equation is derived using Hamilton's principle. The equation is solved analytically to obtain the complex natural frequency. The parametric study is thoroughly performed, concentrating on the series effects of viscoelastic damping structure, aspect ratio, visco-Pasternak medium, and mode number. In this system, in-phase (IPV) and out-of-phase (OPV) vibrations are investigated. The numerical results of this article show a perfect correspondence with those of the previous researches.

Free Vibrations of a Series of Beams Connected by Viscoelastic Layers

An exact solution for free vibrations of a series of uniform Euler-Bernoulli beams connected by Kelvin-Voigt is developed. The beams have the same length and end conditions but can have different material or geometric properties. An example of five concentric beams connected by viscoelastic layers is considered.

Dynamic stiffness matrix of a rectangular plate for the general case

The dynamic stiffness matrix of a rectangular plate for the most general case is developed by solving the bi-harmonic equation and finally casting the solution in terms of the force–displacement relationship of the freely vibrating plate. Essentially the frequency dependent dynamic stiffness matrix of the plate when all its sides are free is derived, making it possible to achieve exact solution for free vibration of plates or plate assemblies with any boundary conditions. Previous research on the dynamic stiffness formulation of a plate was restricted to the special case when the two opposite sides of the plate are simply supported. This restriction is quite severe and made the general purpose application of the dynamic stiffness method impossible. The theory developed in this paper overcomes this long-lasting restriction. The research carried out here is basically fundamental in that the bi-harmonic equation which governs the free vibratory motion of a plate in harmonic oscillation is solved in an exact sense, leading to the development of the dynamic stiffness method. It is significant that the ingeniously sought solution presented in this paper is completely general, covering all possible cases of elastic deformations of the plate. The Wittrick–Williams algorithm is applied to the ensuing dynamic stiffness matrix to provide solutions for some representative problems. A carefully selected sample of mode shapes is also presented.

Exact solution of free vibration of a uniform tensioned beam combined with both lateral and rotational linear sub-systems

A general exact solution of free vibration of a uniform tensioned beam with any number of both lateral and rotational sub-systems is addressed. This study is completed in complex domain, because of the presence of the non-proportional damping caused by the dampers utilized in sub-systems. The sub-systems constructed herein could degenerate to the conventional supports and sub-systems, and be arbitrarily placed. Accordingly, the solution is also suitable for multi-span beams. The argument principle method (APM), which does not need initial values and iteration, is employed to solve the characteristic equation in complex domain. Nevertheless, the presence of multi-valued square root functions due to the axial loading renders the function non-analytic in complex domain. A variable substitution method, in which the main branch of logarithmic function is chosen, is proposed to solve this problem. The results obtained are compared with those in literature. Furthermore, several typical case studies were completed, and the root loci of the frequency parameters attained are compared with those obtained by the finite element method (FEM). A good agreement is observed, which confirms the validity of the presented methodologies in this paper.

A new generic exact solution for free vibration of functionally graded moderately thick doubly curved shallow shell panel

Plates, cylindrical and spherical shells, and in general, doubly curved shell panels are the main parts of each new structural design because their geometrical characteristics give them high strength to weight ratio. Moreover, requiring high mechanical and heat resistant materials leads engineers to use functionally graded material to achieve both of these characteristics. Hence, having a reliable high precision method for free vibration analysis of such shell panels is the first step of each dynamical analysis. This paper used a set of potential functions and auxiliary variables to present an exact Levy-type closed-form solution for free vibration of a doubly curved FG shell panel. The first-order shear deformation theory beside both the Donnell and Sanders strain-displacement relations are considered for mathematical modeling. The results of the present method are checked with the literature and a finite element analysis in which good agreement is observed. Also, a benchmark study is presented that will be useful for future researches.

Free Vibration of Shallow-Spherical Shell

Shallow spherical shell is one of the structure shapes and is adopted extensively in engineering. Based on the exact solution of free vibration of the shallow-spherical shell, the frequency equations of the shallow spherical shell with clamped edge are derived and calculated numerically. Then the first third-order frequency curve and modal curve and solved. Comparing these with the results of the exact-solution method, the range of of the shallow-spherical shell when calculated by use of practical-solution method can be derived.

Exact vibration solutions for cross-ply laminated plates with two opposite edges simply supported using refined theories of variable order

Abstract This paper presents exact solutions for free vibration of rectangular cross-ply laminated plates with at least one pair of opposite edges simply supported using refined kinematic theories of variable order. Exact natural frequencies are obtained using an efficient and unified formulation where the solving set of second-order differential equations of motion and related boundary conditions are expressed at layer level in terms of so-called fundamental nuclei having invariant properties with respect to the order of the plate theory. The nuclei are then appropriately expanded according to the number of layers and the order of the theory and the resulting equations are transformed into a first-order model whose solution is obtained by using the state space concept. In this way, the mathematical effort needed to derive analytical solutions is highly reduced. Both higher-order equivalent single-layer and layer-wise theories are considered in this study. Comparisons with other exact solutions are presented and useful benchmark frequency results for symmetric and un-symmetric cross-ply laminates are provided.

Exact solution for free vibration of curved beams with variable curvature and torsion

For the purpose of investigating the free vibration response of the spatial curved beams, the governing equations are derived in matrix formats, considering the variable curvature and torsion. The theory includes all the effects of rotary inertia, shear and axial deformations. Frobenius' scheme and the dynamic stiffness method are then applied to solve these equations. A computer program is coded in Mathematica according to the proposed method. As a special case, the dynamic stiffness and further the natural frequencies of a cylindrical helical spring under fixed-fixed boundary condition are carried out. Comparison of the present results with the FEM results using body elements in I-DEAS shows good accuracy in computation and validity of the model. Further, the present model is used for reciprocal spiral rods with different boundary conditions, and the comparison with FEM results shows that only a limited number of terms in the resultant provide a relatively accurate solution.

Exact solutions for free vibration of circular cylindrical shells with classical boundary conditions

The obtaining of exact solutions for free vibration of circular cylindrical shells with classical boundary conditions, except for all edges being shear diaphragm, was known to be difficult or impossible. This work presents those exact solutions based on the Donnell–Mushtari shell theory . They are shown to be in simple and compact form, and not as complex as they have generally been regarded. The computed natural frequencies are in excellent agreement with those in literature and those of the highly accurate semi-analytical differential quadrature finite element method developed in this study. The exact solutions can be used as benchmarks for researchers to check their numerical methods and for engineers to design such shell structures. Two of the four eigenvalues are found to be much larger than the other two for some boundary conditions. High accurate membrane approximate solutions are also presented for the circular cylindrical shells.

Exact Solutions for Free Vibrations and Buckling of Double Tapered Columns With Elastic Foundation and Tip Mass

The present study aims at the free vibration analysis of double tapered columns. Foundation is assumed to be elastic and the effects of self-weight and tip mass with significant moment of inertia are considered. The governing equation of motion is obtained using the Hamilton principle, based on both the Euler–Bernoulli and Timoshenko beam models. Applying the power series method of Frobenius, the base solutions of the governing equations are obtained in the form of a power series via general recursive relations. Applying the boundary conditions, the natural frequencies of the beam/column are obtained using both models. The obtained results are compared with literature and a very good agreement is achieved. Subsequently, comprehensive studies are performed to provide an insight into the variation of the natural frequencies and instability conditions of the beam with respect to the tip mass, self-weight, taper ratio, slenderness, and foundation stiffness and eventually some general conclusions are drawn.

Solution methods of exact solutions for free vibration of rectangular orthotropic thin plates with classical boundary conditions

The exact solutions for free vibrations of orthotropic rectangular thin plates are presented in an elegant way using separation of variables method. And the exact solutions of three configurations (G-G-C-C, SS-G-C-C and C-C-C-G) are solved for the first time. In separation of variables method, the general formulation of the natural mode function and the relations among two spatial eigenvalues and a temporal eigenvalue are directly obtained from the characteristic differential equation, and the coefficients of exact mode functions and two exact eigenvalue equations are determined by two pairs of opposite edge conditions. It was regarded before 2009 that there were not exact solutions for rectangular thin plates with at least two adjacent clamped edges, and the others to be arbitrary combinations of clamped (C), simply supported (SS) and guided (G) conditions. For these configurations, the exact eigensolutions are solved successfully and the first ten exact frequencies are tabulated for a few of aspect ratios.

Exact solutions of free vibration of simply supported functionally graded piezoelectric sandwich cylinders using a modified Pagano method

The three-dimensional free vibration analysis of simply supported, functionally graded piezoelectric material sandwich circular hollow cylinders with different surface conditions is presented. The material properties of each functionally graded piezoelectric material layer are regarded as heterogeneous through the thickness coordinate, and obey an exponent-law dependent on this. Pagano’s method is modified to be feasible for this study of functionally graded piezoelectric material sandwich cylinders, in which a displacement-based formulation is replaced by a mixed one; a set of the complex-valued solutions of the system equations is transferred to the corresponding set of real-valued solutions using Euler’s formula; a successive approximation method is adopted to approximately transform each functionally graded piezoelectric material layer into homogeneous piezoelectric layers with an equal and small thickness for each layer in comparison with the mid-surface radius, and with the homogeneous material properties determined in an average thickness sense; and a transfer matrix method is developed so that the general solutions of system equations can be obtained layer-by-layer, which is significantly less time-consuming than usual. A parametric study of the influence of the mid-surface radius-to-thickness ratio, open- and closed-circuit surface conditions, the thickness ratio of each layer, and the material-property gradient index on the natural frequencies of functionally graded piezoelectric material sandwich cylinders is carried out.

Exact solutions of free vibration of rotating multilayered FGM cylinders

A modified Pagano method is developed for the three-dimensional (3D) free vibration analysis of simply-supported, multilayered functionally graded material (FGM) circular hollow cylinders with a constant rotational speed with respect to the meridional direction of the cylinders. The material properties of each FGM layer constituting the cylinders are regarded as heterogeneous through the thickness coordinate, and then specified to obey a power-law distribution of the volume fractions of the constituents, and the effects of centrifugal and Coriolis accelerations, as well as the initial hoop stress due to rotation, are considered. The Pagano method, which was developed for the static and dynamic analyses of multilayered composite plates, is modified in that a displacement-based formulation is replaced by a mixed formulation, the complex-valued solutions of the system equations are transferred to the real-valued solutions, a successive approximation method is adopted to extend its application to FGM cylinders, and a propagator matrix method is developed to reduce the time needed for its implementation. These modifications make the Pagano method feasible for multilayered FGM cylinders, and the computation in the implementation is independent of the total number of the layers, thus becoming less time-consuming than usual.

Comparative dynamic and static studies for cross-ply shells based on a deep thick shell theory

Exact solutions for free vibration and static deflection of cross-ply laminated cylindrical panels and circular cylindrical shells are obtained using deep first order shell theory. The state space approach has been used to generate the exact solutions for different sets of boundary conditions. Comparison between the results has been made using the deep first order shell theory and the less accurate one, where the 1 + z/ R terms have been approximated by unity. The effects of various parameters including shear deformation, shallowness of the shell, orthotropicity ratio, lamination scheme and boundary condition on shell frequencies and centre deflections are studied. The fundamental frequencies of the deep shell theory are compared with 3D theory of elasticity available in the literature.

Three-dimensional exact solutions for free vibrations of simply supported magneto-electro-elastic cylindrical panels

This work investigates the free vibrations of magneto-electro-elastic cylindrical panels based on three-dimensional theory. Firstly, the general solutions for transversely isotropic magneto-electro-elastic materials are introduced and the displacement functions in the general solutions are expanded in trigonometric functions along the circumferential and axial directions. Then an ordinary differential equation of the displacement functions in radial direction is derived and solved. As a result, the frequency equations are obtained through the traction-free conditions on the cylindrical surfaces of the panel as well as the electric and magnetic conditions. For the torsion and thickness-shear modes, the frequency equations in simpler forms are presented. It is found that the magneto-electro-elastic coupling effects disappeared in torsion vibration. Meanwhile, the frequencies of pure elastic materials and magneto-electro-elastic materials have an explicit relation for the thickness-shear modes. The aforementioned solutions satisfy all the governing equations and boundary conditions point by point and they are three-dimensionally exact. Finally the numerical example demonstrates the present method and is compared with those from finite element method. Parametric investigation is also conducted to show the behavior of free vibrations of cylindrical panels.

An exact solution for free vibration of thin functionally graded rectangular plates

The aim of this article is to find an exact analytical solution for free vibration characteristics of thin functionally graded rectangular plates with different boundary conditions. The governing equations of motion are obtained based on the classical plate theory. Using an analytical method, three partial differential equations of motion are reformulated into two new decoupled equations. Based on the Navier solution, a closed-form solution is presented for natural frequencies of functionally graded simply supported rectangular plates. Then, considering Levy-type solution, natural frequencies of functionally graded plates are presented for various boundary conditions. Three mode shapes of a functionally graded rectangular plate are also presented for different boundary conditions. In addition, the effects of aspect ratio, thickness—length ratio, power law index, and boundary conditions on the vibration characteristics of functionally graded rectangular plates are discussed in details. Finally, it has been shown that the effects of in-plane displacements on natural frequencies of functionally graded plates under different boundary conditions have been studied.

An exact solution for free vibrations of a non-uniform beam carrying multiple elastic-supported rigid bars

The purpose of this paper is to utilize the numerical assembly method (NAM) to determine the exact natural frequencies and mode shapes of a multi-step beam carrying multiple rigid bars, with each of the rigid bars possessing its own mass and rotary inertia, to the beam at one and supported by a translational spring and/or a rotational spring at another point. Where the of each rigid bar with the beam does not coincide with the center of gravity the rigid bar or the supporting of the springs. The effects of the distance between the fixed point of each rigid bar and its center of gravity (i.e., eccentricity), and the distance between the fixed point and each linear spring (i.e., offset) are studied. For a beam carrying multiple various concentrated elements, the magnitude of each lumped mass and stiffness of each linear spring are the well-known key parameters affecting the free vibration characteristics of the (loaded) beam in the existing literature, however, the numerical results of this paper reveal that the eccentricity of each rigid bar and the offset of each linear spring are also the predominant parameters.

Exact Solutions for Free Vibrations of Functionally Graded Thick Plates on Elastic Foundations

Free vibration analysis of functionally graded thick plates on elastic foundation is carried out based on three-dimensional elasticity. An isotropic plate is assumed with Young's modulus and mass density varying exponentially through the thickness, and Poisson's ratio remaining constant. The state space method is used to derive an exact solution for a simply supported plate by expanding the state variables in trigonometric dual series. The effects of interaction between the plate surface and a Pasternak foundation are treated as the traction boundary conditions of the plate. The solution procedure is validated by comparing with established results for a homogeneous plate. Finally, the influences of foundation stiffness, foundation support, and gradient index on the natural frequencies are investigated.