Solution For Free Vibration Analysis Research Articles

Free Vibration Analysis of a Conservative Two-Mass System with General Odd Type Nonlinear Connection

In this paper, free vibration of a two-mass system with nonlinear connection investigated. The connections considered as sums of linear and odd type nonlinear springs. The mathematical model of this system is derived by two differential equations with nonlinear coupling. By introducing the intermediate variables equations of motion transformed into a single one-variable differential equation which known as the generalized Duffing equation. The energy balance method based on Galerkin–Petrov approach is applied to solve the generalized Duffing equation and new amplitude-frequency relationships obtained. Comparisons between results obtained from exact approach and results obtained from energy balance method based on Galerkin–Petrov approach show obtained solutions reach better accuracy rather than other solutions that obtained for the generalized Duffing equation and obtained solutions can be served as a most accurate solution for this problem. Then using the intermediate variables obtained solution extended to analysis of a two-mass system with general odd type nonlinear connection. Three examples with different types of the nonlinear connection between masses are considered. Comparisons show very good agreement between obtained analytical solutions and numerical ones. Obtained solution is capable to derive an accurate analytical solution for free vibration of a two-mass with any odd type of nonlinear connection.

A simple shear deformation theory for nonlocal beams

In this paper, a simple beam theory accounting for shear deformation effects with one unknown is proposed for static bending and free vibration analysis of isotropic nanobeams. The size-dependent behaviour is captured by using the nonlocal differential constitutive relations of Eringen. The governing equation of the present beam theory is obtained by using equilibrium equations of elasticity theory. The present theory has strong similarities with nonlocal Euler–Bernoulli beam theory in terms of the governing equation and boundary conditions. Analytical solutions for static bending and free vibration are derived for nonlocal beams with various types of boundary conditions. Verification studies indicate that the present theory is not only more accurate than Euler–Bernoulli beam theory, but also comparable with Timoshenko beam theory.

A unified spectro-geometric-Ritz solution for free vibration analysis of conical–cylindrical–spherical shell combination with arbitrary boundary conditions

In the present article, a spectro-geometric-Ritz solution for free vibration analysis of conical–cylindrical–spherical shell combinations with arbitrary boundary conditions is presented. The classical theory of small displacements of thin shells is employed to formulate the theoretical model. The admissible functions of each shell components are described as a combination of a two-dimensional (2-D) Fourier cosine series and auxiliary functions. As an innovative point of this work, the auxiliary functions are introduced to accelerate the convergence of the series representations and eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries and the junction between the shell components. The artificial spring technique is adopted here to model the boundary condition and coupling condition, respectively. All the expansion coefficients are considered as the generalized coordinates and determined by Ritz procedure. Convergence and comparison studies for both open and closed conical–cylindrical–spherical shells with arbitrary boundary conditions are carried out to verify the reliability and accuracy of the present solutions. Some selected mode shapes are illustrated to enhance the understanding of the research topic. It is found the present method exhibits stable and rapid convergence characteristics, and the present results, including the natural frequencies and the mode shapes, agree closely with those solutions obtained from the finite element analyses and results in the literature. The effects of the geometrical dimensions of the shell combinations on the natural frequencies are also investigated.

Vibration analysis of micro composite thin beam based on modified couple stress

In this article, analytical solution for free vibration of micro composite laminated beam on elastic medium based on modified couple stress are presented. The surrounding elastic medium is modeled as the Winkler elastic foundation. The governing equations and boundary conditions are obtained by using the principle of minimum potential energy for Euler- Bernoulli beam. For investigating the effect of different parameters including material length scale, beam thickness, some numerical results on different cross ply laminated beams such as (90,0,90), (0,90,0), (90,90,90) and (0,0,0) are presented on elastic medium. Free vibration analysis of a simply supported beam is considered utilizing the Fourier series. Also, the fundamental frequency is obtained using the principle of Hamilton for four types of cross ply laminations with hinged-hinged boundary conditions and different beam theories. The fundamental frequency for different thin beam theories are investigated by increasing the slenderness ratio and various foundation coefficients. The results prove that the modified couple stress theory increases the natural frequency under the various foundation for free vibration of composite laminated micro beams.

An improved Fourier series solution for free vibration analysis of the moderately thick laminated composite rectangular plate with non-uniform boundary conditions

In this investigation, an improved Fourier series method is presented for the free vibration analysis of the moderately thick laminated composite rectangular plate with non-uniform boundary conditions, a class of problems which are rarely attempted in the literatures. Under the current framework, the displacement and rotation functions are generally sought, regardless of boundary conditions, in spectral form, as a double Fourier cosine series and three supplementary functions. These supplementary functions are introduced to remove the potential discontinuities associated with the original displacement functions along the edges when they are viewed as periodic functions defined within the entire coordinates of laminate plate. The boundary conditions can be readily realized by setting the stiffness of the five types restraining springs. All the series expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh-Ritz technique. Unlike most of the existing studies, the presented method can be readily and universally applied to a wide spectrum of plate vibration problems involving different boundary conditions, varying material, and geometric properties with no need of modifying the basic functions or adapting solution procedures. The excellent accuracy of the current result is validated by comparing the present results with those Finite Element Method (FEM) data. Numerous new results for free vibration of moderately thick rectangular plates with various multi-points supported and non-uniform partially supported boundary conditions are presented, which may serve as benchmark solution for future researches in the field.

Free vibration of three-dimensional anisotropic layered composite nanoplates based on modified couple-stress theory

Based on the modified couple-stress theory, three-dimensional analytical solutions of free vibration of a simply supported, multilayered and anisotropic composite nanoplate are derived by solving an eigenvalue system and using the propagator matrix method. By expanding the solutions of the extended displacements in terms of two-dimensional Fourier series, the final governing equations of motion with modified couple-stress effect are reduced to an eigenvalue system of ordinary differential equations. Analytical expressions for the natural frequencies and mode shapes of multilayered anisotropic composite plates with modified couple-stress effect are then derived via the propagator matrix method. Numerical examples are carried out for homogeneous thick-plates and sandwich composite plates to show the effect of the non-local parameter in different layers and stacking sequence on the mode shapes. The present solutions can serve as benchmarks to various thick-plate theories and numerical methods, and could be further useful for designing layered composite structures involving small scale.

Analytical solution for free vibrations of rotating cylindrical shells having free boundary conditions

In this paper free vibrations of rotating cylindrical shells with both ends free are studied. The model used also allows for considering a flexible foundation supporting the shell in the sense of a radial and circumferential distributed stiffness. Furthermore, a circumferential tension (hoop stress) which may be due to pressurisation or centrifugal forces is taken into account. Natural frequencies and mode shapes are determined exactly for both stationary shells and for shells rotating with a constant angular speed around the cylinder axis. Trigonometric functions are assumed for the circumferential mode shape profiles, and a sum of eight weighted exponential functions is assumed for the axial mode shape profiles. The functional form of the axial profiles is shown to greatly vary with the roots of a characteristic bi-quartic polynomial that occurs in the process of satisfying the equations of motion. In the previously published work it has been very often assumed that the roots are two real, two imaginary, and two pairs of complex conjugates. In the present study, a total of eight types of roots are shown to determine the whole set of mode shapes, either for stationary or for rotating shells. The results using the developed analytical model are compared with results of experimental studies and very good agreement is obtained. Also, a parametric study is carried out where effects of the elastic foundation stiffnesses and the rotation speed are examined.

An analytical three-dimensional solution for free vibration of a magneto-electro-elastic plate considering the nonlocal effect

An exact closed-form solution for the three-dimensional free vibrational response of a simply-supported and multilayered magneto-electro-elastic plate considering the nonlocal effect is presented. The solution is derived using the pseudo-Stroh formulation and propagator matrix method. Various numerical examples are presented for a homogeneous elastic plate, piezoelectric plate, magnetostrictive plate, and sandwich plates made of piezoelectric and magnetostrictive materials. The natural frequencies and the corresponding mode shapes of the multilayered plates show the influence of stacking sequence and the important role that the nonlocal effect plays in dynamic analysis of nanostructures. This exact solution is useful for it provides benchmark results to assess the accuracy of nonlocal thin plate models and finite element formulations.

Size-dependent static bending and free vibration of 0–3 polarized PLZT microcantilevers

In this paper, analytical solutions for size-dependent static bending and free vibration of a pure 0–3 polarized PbLaZrTi (PLZT) cantilever are developed. This paper also makes the first attempt to investigate the static bending of a cantilever metal beam bonded with discretized 0–3 polarized PLZT actuator based on the modified couple stress theory and composite laminated beam theory. These models involve an internal material length scale parameter used to capture the size effect. In the limit when the internal material length scale parameter goes to zero, this model reduces to classical (local) solutions available in the literature. Exact solutions for the normalized static deflection are obtained as a function of the actuator thickness and the internal material length scale parameter. The simulations show that the size-dependent results developed by the present models have a remarkable difference with those got by the classical solutions when the ratio of the actuator thickness to the internal material length scale parameter is small. It is also observed that an increase in the stiffness parameter of the substrate beam gives rise to an increase in the effect of the material length scale parameter on tip deflections of the cantilever metal beam.

Analytical solution for free vibration of flexible 2D rectangular tanks

This paper provides a brief literature review on dynamic behavior of liquid storage containers. Moreover, the free vibration of two-dimensional deformable rectangular tanks fully-filled with a compressible fluid is analytically investigated, and the exact solutions are derived. In this study, the fluid-structure interaction is rigorously considered, and the fluid is assumed to be irrotational, inviscid and compressible. Furthermore, a closed-form expression is introduced for evaluating the fundamental frequency of the flexible 2D rectangular tanks filled with compressible liquid. In addition, an approximate formula is suggested to determine the corresponding pressure distribution on tank walls. Finally, numerical examples are performed to verify the presented closed-form relations. To study the effect of various parameters on natural frequencies and mode shapes of the fluid-structure system, a sensitivity analysis is also conducted.

Analytical Solutions for Free Vibration and Buckling of Composite Beams Using a Higher Order Beam Theory

To satisfy the interfacial shear force continuity conditions, a new model is proposed for the two-layer composite beam with partial interaction by modifying Reddy's higher order beam theory. The governing differential equations for free vibration and buckling are formulated using the Hamilton's principle, the natural frequencies and axial forces are thus analytically obtained by Laplace transform technique. The analytical results are verified through the comparison with those of several other models common in use; and the presented model is found to be a finer one than the Reddy's. A parametric study is also performed to investigate the effects of geometry and material parameters.

Analytical Solution for Free Vibration of a Variable Cross-section Nonlocal Nanobeam

In this article, small scale effects on free vibration analysis of non-uniform nanobeams is discussed. Small scale effects are modelled after Eringen’s nonlocal elasticity theory while the non-uniformity is presented by exponentially varying width among the beams length with constant thickness. Analytical solution is achieved for free vibration with different boundary conditions. It is shown that section variation accompanying small scale effects has a noticeable effect on natural frequencies of non-uniform beams at nano scale. First, five natural frequencies of single-layered graphene nanoribbons (GNRs) with various boundary conditions are obtained for different nonlocal and nonuniform parameters which shows a great sensitivity to non-uniformity in different shape modes.

New analytic free vibration solutions of rectangular thin plates resting on multiple point supports

Free vibration solution of a free rectangular thin plate resting on multiple point supports has been a topic of fundamental importance in mechanical engineering. It is well known that various approximate/numerical methods have been developed to solve the problems, but exact analytic solutions, as the benchmarks, are rarely reported in the literature. This is attributed to the difficulty in seeking the solutions that satisfy the governing fourth-order partial differential equation (PDE) with the completely free boundary conditions as well as the support conditions. In this paper, we present a successful endeavor to address the issue with our recently developed symplectic superposition method for free vibration problems. A general set of equations are obtained for determining the natural frequencies and mode shapes of the plates with any point supports. Seventeen typical combinations of support locations are investigated for the plates with three or four point supports. The obtained solutions are all shown by the numerical results listed for comparison with those by the well-accepted finite element method (FEM), and very good agreement is observed. This study provides some useful benchmark solutions of the point-supported free rectangular plates, and demonstrates an effective analytic approach to solving similar boundary value problems which have not been well settled by other analytic methods.

Using an equivalent continuum model for 3D dynamic analysis of nanocomposite plates

Most of the early studies on plates vibration are focused on two-dimensional theories, these theories reduce the dimensions of problems from three to two by introducing some assumptions in mathematical modeling leading to simpler expressions and derivation of solutions. However, these simplifications inherently bring errors and therefore may lead to unreliable results for relatively thick plates. The main objective of this research paper is to present 3-D elasticity solution for free vibration analysis of continuously graded carbon nanotube-reinforced (CGCNTR) rectangular plates resting on two-parameter elastic foundations. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. In this study, an equivalent continuum model based on the Eshelby-Mori-Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight carbon nanotubes (CNTs). The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The formulations are based on the three-dimensional elasticity theory. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the present approach. The novelty of the present work is to exploit Eshelby-Mori-Tanaka approach in order to reveal the impacts of the volume fractions of oriented CNTs, different CNTs distributions, various coefficients of foundation and different combinations of free, simply supported and clamped boundary conditions on the vibrational characteristics of CGCNTR rectangular plates. The new results can be used as benchmark solutions for future researches.

Using eshelby–mori–tanaka scheme for 3D free vibration analysis of sandwich curved panels with functionally graded nanocomposite face sheets and finite length

This article deals with free vibration analysis of thick nanocomposite laminated curved panels with finite length resting on two‐parameter elastic foundations, based on the three‐dimensional elasticity theory. The main objective of this research paper is to present a 3D elasticity solution for free vibration analysis of thick laminated curved panels with continuously graded carbon nanotube‐reinforced sheets. The structure is supported by an elastic foundation with Winkler's (normal) and Pasternak's (shear) coefficients. The volume fractions of oriented, straight single‐walled carbon nanotubes are assumed to be a three‐parameter power‐law distribution which is graded in the radial direction of the panels. An equivalent continuum model based on the Eshelby‐Mori‐Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight carbon nanotubes. Because of using two‐dimensional generalized differential quadrature method, the present approach makes possible vibration analysis of cylindrical panels with two opposite axial edges simply supported and arbitrary boundary conditions including Free, Simply supported and Clamped at the curved edges. The convergence of the method is demonstrated and comparisons are made between the present results and results reported by well‐known references and have confirmed accuracy and efficiency of the present approach. This study serves as a benchmark for assessing the validity of numerical methods or two‐dimensional theories used to analysis of laminated curved panels. POLYM. COMPOS., 38:E563–E576, 2017. © 2016 Society of Plastics Engineers

A new method for analytical solution of inplane free vibration of rectangular orthotropic plates based on the analysis of infinite systems

In this paper, a new method for in-plane free vibration analysis of rectangular orthotropic plate is presented. The cases for completely clamped and completely free plates are considered in detail. The boundary value problem is essentially reduced to infinite systems of linear algebraic equations. New mathematical theory and associated results are presented for investigating the asymptotic behavior of non-trivial solution of quasi-regular infinite system. By enhancing the theory, an effective and accurate algorithm for determining the natural frequencies and modes shapes vibrations is developed. The accuracy and computational efficiency of the method are demonstrated by examples.

Static and dynamic analyses of nanoscale rectangular plates incorporating surface energy

In this paper, the Gurtin–Murdoch continuum theory is applied to develop a new continuum mechanics model for static and dynamic analyses of nanoscale rectangular plates. The relevant governing equations are established from basic principles. Analytical solutions for static and free vibration of nanoscale rectangular plates are presented for selected boundary conditions. A finite element method for the analysis of rectangular nanoplates is also developed to solve general cases that cannot be solved analytically. Expressions for stiffness and mass matrices and the load vector are derived by using a weighted residual formulation. A selected set of numerical results is presented to investigate the size-dependent static and free vibration response of plates and the influence of surface material properties and boundary conditions.

Natural Frequency of Functionally Graded Plates Resting on Elastic Foundation Using Finite Element Method

This paper deals with the numerical solution for free vibration analysis of simply supported functionally graded material (FGMs) plates on elastic foundation. The displacement filed based on the higher order shear deformation theory (HSDT) with 13 degree of freedom is used. The foundation is described by the Pasternak (two-parameter) model. A power law distribution is used to explain the variation of mechanical and physical properties through the thickness of the FGMs plate. The effect of volume fraction index, geometric configuration and foundation parameter on the natural frequency of FGMs plate is investigated. The results reveal that the Pasternak (shear) elastic foundation significantly affects the natural frequency.

Vibration characteristics of generally laminated composite curved beams with single through-the-width delamination

In this paper, we present analytical and finite element solutions for free vibration analysis of delaminated composite curved beams by taking into account the effects of shear deformation, rotary inertia, deepness term and material coupling. The piecewise-linear spring model has been used to model the behavior of the delaminated regions. The governing equations along with the boundary, continuity and equilibrium conditions at the delamination boundaries called as constraints have been obtained using Hamilton’s principle. Beam displacement fields have been expanded using simple Legendre polynomials to obtain the analytical solution regardless of boundary conditions. The solution for the natural frequencies and mode shapes are presented by incorporating the constraints using Lagrange multipliers. The convergence and accuracy of the present formulations are validated against several numerical examples in the literature and very good agreements have been observed. Moreover, the effect of delamination size and locations, layups configurations, boundary conditions and material anisotropy on the dynamic characteristics of the delaminated composite curved beams have been investigated which may serve as a benchmark for the researchers in this field.

Hamiltonian system-based analytic modeling of the free rectangular thin plates’ free vibration

Analytic free vibration solutions of free rectangular thin plates with or without an elastic foundation are obtained by using an up-to-date Hamiltonian system-based symplectic superposition method. Such boundary value problems are known to be very difficult and they were generally solved by the approximate/numerical methods. The present analytic solutions are expected to be the benchmarks for future verification of other methods. The advantage of the method is that the solution procedure is conducted in the symplectic space, where the symplectic eigen expansion is valid and the predetermination of the solution form is avoided. This significantly extends the approach to the analytic solutions of similar problems.