Solution For Free Vibration Analysis Research Articles

New Analytic Free Vibration Solutions of L-Shaped Moderately Thick Plates by Symplectic Superposition

L-shaped moderately thick plates have widespread applications in diverse engineering structures. Exploring the benchmark analytic solutions for the free vibration of L-shaped moderately thick plates is important in order to accurately analyze and efficiently design structures. Nevertheless, analytical solutions, which serve as the benchmarks, have been rarely documented in previous literature due to the challenge of finding suitable solutions that satisfy both the governing higher-order partial differential equations (PDEs) and the boundary conditions of the plates. The symplectic superposition method was employed in our recently published study to present the benchmark vibration solutions of clamped rectangular moderately thick plates. In this study, we expand upon this method to solve the free vibration problem of L-shaped moderately thick plates. By employing domain decomposition, we construct an irregular domain by combining multiple rectangular domains. First, the construction of the superposition system is carried out, followed by the import of the sub-problems into the Hamiltonian system, utilizing the fundamental governing equations of the plate. Then, the sub-problems are resolved through the application of the symplectic geometry methodology in an analytical manner. Ultimately, the analytical solutions for frequencies and mode shapes are derived through ensuring the equivalence between the initial problem and the combination of sub-problems. The finite element method is used to validate and present a comprehensive analysis of the natural frequencies and mode shapes obtained from this method. This method possesses the benefits of rapid convergence and accurate precision, rendering it well suited for the analytic modeling of a broader range of plate-related problems.

Numerical and Analytical Solution for Nonlinear Free Vibration of Tapered Beams

Numerical and Analytical Solution for Nonlinear Free Vibration of Tapered Beams

Analytical and isogeometric solutions of flexoelectric microbeams based on a layerwise beam theory

The theory of layerwise beam, which considers both microstructure and flexoelectric effects, is used in the curvature-based flexoelectricity theory. The displacement field function of the layerwise beam model is given. The governing equations and boundary conditions are obtained using a variational formulation based on Hamilton's principle. Analytical solutions for static bending and free vibration of a simply supported layerwise beam are derived. Isogeometric analysis (IGA) with higher-order continuity considering the flexoelectric effect is presented to numerically solve the static bending and free vibration problems. The results obtained from the analytical and IGA approaches are compared through several examples in which the microstructure and flexoelectric effects on the mechanical responses of layerwise nanobeams are analyzed.

Levy-type analytical series solution for the three-dimensional free vibrations of functionally graded material rectangular plates with piezoelectric layers

Analytical solutions founded on three-dimensional theories play a crucial role in evaluating the credibility and precision of different plate theories and numerical methodologies. While Levy-type analytical solutions are widely recognized, they have been primarily confined to purely elastic plates. This study introduces a Levy-type analytical series solution for three-dimensional vibrations in a sandwich rectangular plate featuring a functionally graded material (FGM) core, along with piezoelectric material (PM) layers on the top and bottom surfaces. The behaviors of the FGM and PM layers were described using three-dimensional elasticity and piezoelasticity theories, respectively. In this study, the displacement functions and electric potential of each layer were expanded by Fourier series and polynomial auxiliary functions. An analytical series solution was then established by satisfying the governing equations of each layer, the mechanical and electric boundary conditions on the six faces of the plate, and the continuity conditions on the interfaces between the PM and FGM layers. To validate the proposed solutions, in-depth convergence studies were conducted for the vibration frequencies of the first six modes of sandwich square plates with various boundary conditions on the other pair of side faces. The well-converged results were then compared with published data based on various plate theories to verify the accuracy of these published data. Finally, accurate nondimensional frequencies were tabulated for the first six modes of sandwich rectangular plates with various aspect ratios, thickness-to-width ratios, PM-to-FGM layer thickness ratios, power law indices for the FGM layer, and six combinations of boundary conditions. These new numerical results when piezoelectric coupling is considered should be very useful to future analytical and numerical studies.

An analytical closed-form solution for free vibration and stability analysis of curved sandwich panels made of porous metal-foam core and nanocomposite reinforced face-sheets

This paper aims to obtain an analytical solution for free vibration and stability analysis of shallow curved sandwich panels made of a porous metal-foam core and nanocomposites reinforced sheets resting on a Winkler-Pasternak elastic foundation. In this regard, two nanocomposites, carbon nanotubes (CNTs) and graphene nanoplatelets (GNPs) are considered for reinforcing the face-sheets. The governing equations are developed by considering the transversal shear strains based on the first-order shear deformation theory (FSDT). The obtained system of differential equations representing the dynamic equilibrium and compatibility of the proposed curved sandwich panels are solved analytically, using the principal of Galerkin method. In order to examine the results of developed formulation, some benchmark examples are adopted from the existing literature and the obtained results corresponding to the considered examples are compared with those are presented in the references. Using the achieved numerical results in the framework of a comprehensive parametric study, the effects of various material and geometrical factors and also various boundary conditions on the natural frequencies and buckling loads of the proposed sandwich panels are assessed. By performing the extensive parametric study it was observed that several results corresponding to the free vibration and stability of the proposed sandwich panels can be achieved by the present method without spending much time for computations. On the contrary, in order to achieve the similar results using the numerical methods, a time-consuming process is required which can justify the novelty of the present work.

Analytical solution for free vibration of symmetric Terfenol-D layered functionally graded beam with different boundary conditions

Analytical solution for free vibration of symmetric Terfenol-D layered functionally graded beam with different boundary conditions

New analytic free vibration solutions of non-Lévy-type porous FGM rectangular plates within the symplectic framework

This study reports new analytic free vibration solutions of non-Lévy-type porous functionally graded material (FGM) rectangular plates, i.e., those without two opposite edges simply supported, by the symplectic superposition method (SSM). Such analytic solutions are not accessible through conventional analytic methods as seeking analytic solutions that meet both the governing partial differential equations and various non-Lévy-type boundary conditions is an acknowledged challenge. The primary advantage of the SSM is that it does not require any assumption of solution forms, but is conducted via rigorous mathematical treatments, such as the symplectic space-based separation of variables and the symplectic eigen expansion. Therefore, the intractable problems can be analytically solved. The rapid convergence and high accuracy of the analytic solutions obtained from the SSM have been demonstrated. Comprehensive natural frequency results of porous FGM plates with all possible combinations of clamped and free edges are presented and validated by the finite element method. The effects of various parameters, e.g., porosity distributions, are also investigated by the analytic solutions obtained. All tabulated results can serve as benchmarks for the future studies focusing on porous FGM plates with non-Lévy-type boundary conditions.

Symplectic superposition solutions for free in-plane vibration of orthotropic rectangular plates with general boundary conditions

This work reports new analytic free in-plane vibration solutions for orthotropic non-Lévy-type rectangular plates, i.e., those without two opposite edges simply supported, by the symplectic superposition method (SSM), which has never been applied to in-plane elasticity problems in any existing works. Such analytic solutions are not accessible through conventional analytic methods as seeking analytic solutions that meet both the governing partial differential equations and various non-Lévy-type boundary conditions is an acknowledged challenge in mechanical analysis of plates. The clamped and free plates are considered as two most representative cases of non-Lévy-type plates. The SSM is implemented in the Hamiltonian system-based symplectic space, where the separation of variables and the symplectic eigen expansion prove to be well-grounded. These two mathematical treatments are adopted to first gain the analytic solutions of two elementary problems. The final analytic free in-plane vibration solutions are obtained by superposition of the two elementary problems. Comprehensive new natural frequencies and vibration modes are studied and validated by reference solutions from the finite element method or other approaches. The rigorous solution procedure, fast convergence, and highly accurate results render the present framework capable of serving as benchmarks for future comparison and applicable to analytic investigation of more plate problems.

A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution

In this paper, a new method based on an accurate analytical series solution for free vibration of triangular isotropic and orthotropic plates is presented. The proposed solution is expressed in terms of undetermined arbitrary coefficients, which are exactly satisfied by the governing differential equation in free vibration. The approach used is based on an innovative extension of the superposition method through the application of a modified system of trigonometric functions. The boundary conditions for bending displacements and bending rotations on the sides of the triangular plate led to an infinite system of linear algebraic equations in terms of the undetermined coefficients. Following this development, the paper then presents an algorithm to solve the boundary value problem for isotropic and orthotropic triangular plates for any kinematic boundary conditions. Of course, the boundary conditions with zero displacements and zero rotations on all sides correspond to the case when the plate is fully clamped all around. The convergence of the proposed method is examined by numerical simulation applying stringent accuracy requirements to fulfill the prescribed boundary conditions. Some of the computed numerical results are compared with published results and finally, the paper draws significant conclusions.

Analytical solutions of free vibration for rectangular thin plate and right-angle triangle plate on the Winkler elastic foundation based on the symplectic superposition method

Abstract Based on the symplectic superposition method, the free vibration models of rectangular and right-angle triangle plates on the Winkler elastic foundation are established in the present paper, and the modes and frequencies are studied. In addition, the theoretical calculation model and finite element analysis model of rectangular thin plate and right-angle triangle plate on elastic foundation are established by using Mathematica software and ABAQUS software. It proves that the symplectic superposition method converges very fast and has a good consistency with the finite element simulation results. Analytical results show that foundation stiffness, aspect ratio, and boundary condition have great influences on vibration frequency and mode shape for structures. This paper solved the free vibration problem of rectangular plate and right-angle triangle plate on elastic foundation by using symplectic superposition method. Compared with the inverse or semi-inverse method, this method avoids the process of assuming the form about the solution, hence the result of this method is completely rational.

Free vibration of non-Lévy-type functionally graded doubly curved shallow shells: New analytic solutions

• Symplectic superposition framework for free vibration of FG shells is presented. • New analytic solutions of non-Lévy-type FG doubly curved shallow shells are obtained. • Frequencies of FG shells with different boundaries are provided as benchmarks. • Rational and rigorous procedure, fast convergence, and high accuracy are validated. Analytic solutions are of great significance to serve as benchmarks for validating numerical methods and to guide efficient preliminary structural designs. In this paper, we concentrate on seeking new analytic free vibration solutions of non-Lévy-type functionally graded (FG) doubly curved shallow shells, which were not available in the literature to the best of our knowledge. The main challenge in gaining such analytic solutions comes from both the governing system of partial differential equations under complicated boundary conditions. Here, we extend an analytic symplectic superposition method (SSM), which we proposed originally for plate problems, to solve the current tough issue. The solution methodology is implemented via the following three steps: converting an original free vibration problem into two elementary problems and formulating the equations of each elementary problem in the Hamiltonian system; utilizing specific mathematical techniques in the symplectic space for the analytic solutions of the elementary problems; superposing the solutions of the elementary problems to yield the eventual analytic free vibration solutions. The primary advantage of the SSM is that it does not require any assumption of solution forms, but is conducted via rigorous derivation, which is realized by the exceptional mathematical treatments, such as the symplectic space-based separation of variables and the symplectic eigen expansion. Comprehensive new free vibration solutions are presented for non-Lévy-type FG doubly curved shallow shells, through which the solutions of FG cylindrical shell panels are also reported as extreme cases. All tabulated results are well validated by the finite element method via ABAQUS software, guaranteeing the accuracy of the present solutions for benchmark use. Quantitative analyses on the natural frequencies are also conducted to reveal the effects of boundary conditions, FG models, and aspect ratios.

Analytical solution for free vibration analysis of GPL-RP beam integrated with piezoelectric layers

This report presents an analytical approach to the natural frequency analysis of a porous beam consisting of a host porous layer reinforced with graphene platelets (GPLs), namely GPL-reinforced porous core, and two piezoelectric outer layers. In the modelling, symmetric distributions of both porosity and GPLs in the core are supposed. The effective mechanical properties of the GPL-reinforced porous core are estimated by Halpin–Tsai model and the rule of mixture. The electric potential in each piezoelectric layer is assumed to vary linearly across its thickness. Two types of electrical boundary conditions, which are open- and closed-circuits, are considered for the free surfaces of the piezoelectric layers. Parabolic shear deformation beam theory associated with Hamilton’s principle is adopted to derive the governing equations of the free vibration. Afterwards these equations are solved analytically by Navier’s solution. Comparative and comprehensive studies are carried out to examine the accuracy and effects of parameters and conditions, such as GPL weight fraction, porosity coefficient, and electrical boundary conditions on the natural frequencies of the beam.

Finite Element Free Vibration Analysis of Clamped Functionally Graded Plate

The main objective of this research is to present analytical solutions for free vibration analysis of moderately thick plates composed of functionally graded materials. Finite element analysis using first-order shear deformation plate theory (FSDT) has been used in the research. Nine noded-isoparametric lagrangian plate bending elements are used in the analysis. Young’s modulus and density per unit volume are assumed to vary continuously through the plate thickness according to power-law distribution. The results have been validated with the existing literature available from other analytical and numerical techniques. The effect of different plate parameters such as aspect ratios, thickness to length ratios, and gradient indices on the natural frequencies of FG rectangular plates is presented. Only clamped boundary conditions have been investigated. It is found that the frequency parameter depends considerably on the said parameters.

Dynamic Analysis of Simply Supported Functionally Graded Plates

Functionally graded materials (FGMs) are considered as a special composite material that is synthesized for a specific purpose by gradual mixing of two or more different materials thus with varying material properties through their thickness. Due to smooth and continuous varying material properties from one face to another, FGMs are usually superior to the conventional composite materials in mechanical behavior. Hence the dynamic analysis of FGM plates is essential. In this paper, numerical solutions for free vibration analysis of moderately thick plates composed of functionally graded materials with simply supported boundary conditions using finite element analysis have been studied. First-order shear deformation plate theory (FSDT) has been applied in the research. Eight noded isoparametric serendipity plate bending elements have been used. Young’s modulus and density per unit volume are assumed to vary continuously through the plate thickness according to a power-law distribution. The results have been validated with the existing literature available from other analytical and numerical techniques. The effect of different plate parameters such as aspect ratios, edge constraints, thickness to length ratios, and gradient indices on the natural frequencies of FG rectangular plates are presented. It is found that the natural frequency depends considerably on the said parameters.

Analytical solution of free vibration of viscoelastic perforated nanobeam

Analytical solution of free vibration of viscoelastic perforated nanobeam

On new benchmark free vibration solutions of rectangular sandwich panels within the symplectic solution framework

In this paper, the first attempt is made to obtain some new analytic free vibration solutions of rectangular sandwich panels within the symplectic solution framework, which are difficult to tackle within the conventional analytic solution framework. The sandwich panels with honeycomb and truss cores are first treated as equivalent thick plates. The governing dual equation is then established within the Hamiltonian system. Subsequently, the original problem is converted into two subproblems whose analytic solutions are acquired by applying the variable separation and symplectic eigen expansion. The superposition yields the final analytic free vibration solution, with the emerging coefficients determined according to the equivalence between the original problem and the superposition. The natural frequency and mode shape solutions by the present symplectic superposition method are quantitatively shown via numerical and graphical results, respectively, and are all well validated by consistency with classical solutions, experimental results, or the numerical solutions by the refined finite element modeling. Besides providing the new results that can serve as benchmarks, the effects of the size parameters on the natural frequencies of the sandwich panels are also analyzed. Since the developed method gives up the assumption of any trial solutions and follows a rigorous derivation to yield new analytic solutions, it provides opportunities for solving more intricate problems of sandwich panels and shells.

A semi-analytical method for free vibration of semi-closed shells of revolution with variable curvature

This study is concerned with the free vibration analysis of semi-closed shells of revolution with variable curvature, with a focus on the analytical solution to the natural frequency of such shells. A semi-closed shell of revolution with variable curvature is firstly divided into some narrow small shell segments, then the first small shell segment with a closed shell apex is approximated with a semi-closed conical shell, while the remaining small shell segments are approximated with a series of progressively larger truncated conical shells. Since each small conical shell is part of the large semi-closed shell of revolution with variable curvature and has to vibrate in sync with the large shell, the analytical solution to the axisymmetric free vibration of the large shell can be transformed into the analytical solutions to the synchronous free vibration of these small conical shells. The general solutions for the natural frequencies of all small conical shells are analytically derived under the condition that conical shells have zero meridional curvature. The undetermined constants in the general solutions are determined by letting all natural frequencies of these small conical shells be equal. Finally, the analytical solution for the axisymmetric free vibration of semi-closed shells of revolution with variable curvature is presented and is proved to be basically reliable by conducting a confirmatory experiment based on polymer 3D-printing technique.

An extended separation-of-variable method for free vibration of rectangular Reddy plates

The Reddy plate theory is widely used in the vibration analysis of thick plates, but there are few closed-form analytical solutions for free vibration of rectangular Reddy plates with arbitrary homogenous boundary conditions. This work is to develop the extended separation-of-variable (SOV) method for the free vibration analysis of rectangular Reddy plates. In the present method, the mode functions are in an explicit and closed form, and the natural frequencies corresponding to two-direction eigenfunctions are mathematically assumed to be independent of each other. According to two pairs of boundary conditions, the closed-form mode functions and frequency equations are achieved. The present method is numerically compared with other approximate and analytical methods, and some valuable observations are presented. The present solutions are exact for the plates with at least two opposite sides simply supported, and agree well with the results in literatures for other boundary conditions.

A Unified Solution for Free Vibration Analysis of Beam-Plate-Shell Combined Structures with General Boundary Conditions

A semi-analytical method is presented to analyze free vibration response of beam-plate-shell combined structures with general boundary conditions. Based on the beam-plate-shell energy theory, the coupled annular plate-conical-cylindrical-spherical shell with stiffened rings and bulkheads regarded as the theoretical model is constructed. The unified displacement admissible functions of each substructure are expanded as modified Fourier series and auxiliary convergence functions along generatrix direction and Fourier series along circumferential direction. Virtual spring technology is adopted to express the energy stored at the junction of adjacent substructures and both boundaries. The energy variational procedure and Ritz method are used to obtain the vibrational governing equation of the combined structure. The present method provides an analytical way for the vibrational response of complicated combined structures. The convergence, accuracy and reliability are validated by comparing the free vibrational response with those of the references and finite element method. Some numerical examples show effects of different boundary conditions on the free vibration. And the influence of stiffened rings and bulkheads treated as Euler-beams and annular plates is also discussed from quantity, size and spatial distribution, offering a feasible way to design the reinforced structures and optimize the bulkheads in engineering problems.

Analytical Solution for Bending and Free Vibrations of an Orthotropic Nanoplate based on the New Modified Couple Stress Theory and the Third-order Plate Theory

In the present work, the equations of motion of a thin orthotropic nanoplate were obtained based on the new modified couple stress theory and the third-order shear deformation plate theory. The nanoplate was considered as a size-dependent orthotropic plate. The governing equations were derived using the dynamic version of Hamilton’s principle and natural boundary conditions were formulated. An analytical solution in the form of a double Fourier series was obtained for a simply supported rectangular nanoplate. The eigenvalue problem was set and solved. It was analytically shown that the displacements of the median surface points in the plane of the plate do not depend on the material length scale parameters in the same directions; these in-plane directional displacements depend on the material length scale parameter in the out-of-plane direction only. On the other hand, the out-of-plane directional displacement depends on the length scale parameter in the plane directions only. The cross-section rotation angles depend on all length scale parameters. It was shown that the size-dependent parameters only have a noticeable effect on the deformed state of the plate if their order is not less than the order (plate height)-1.