Exact Solution For Free Vibration Research Articles

Benchmark exact free vibration solutions of two-dimensional decagonal piezoelectric quasicrystal cylindrical shells

Abstract Quasicrystalline materials with piezoelectric effects show significant potential for advancing actuators, sensors and energy harvesters. In this paper, the free vibration characteristics of two-dimensional decagonal piezoelectric quasicrystal cylindrical shells (PQCSs) are investigated in the framework of symplectic mechanics system. By introducing an original vector and its dual variable vector as the fundamental unknowns, the governing equations are reduced into a set of low-order ordinary differential equations system, thus the free vibration analysis is transformed into an eigenvalue problem within the symplectic space. By using the symplectic mathematics, the exact solutions for free vibration of PQCSs are finally obtained and expanded as a series of symplectic eigensolutions. Finally, accurate natural frequency and analytical vibration mode shapes for arbitrary classical boundary conditions are obtained simultaneously. The accuracy of the obtained solutions is verified by comparing with existing results in open literature. In addition, the effects of geometrical parameters, temperature rise, external voltage and coupling fields on the natural frequency and vibration mode shapes are investigated in numerical examples. Results indicate that the phason field exhibits significant influences on the natural frequencies and cannot be neglected in free vibration analysis of PQCSs. Furthermore, all the results can be served as benchmarks for the development of new analytical and numerical approaches.

Free vibration of arbitrarily shaped plates with complex cutouts

In this paper, an exact model for free vibration analysis of arbitrarily shaped plates with various cutouts under general elastic boundary conditions is established. It combines the artificial connecting spring technique with the domain segmentation integral method (DSIM) proposed by the author. The arbitrarily shaped plate with cutouts is decomposed into several arbitrarily shaped subdomains based on the cutouts. The continuity conditions at the interconnecting interfaces of the subdomains are realized by artificial coupling springs. The energy expressions of the free vibration of the subdomain with arbitrary shape is calculated analytically or semi-analytically by using the DSIM. That is, in order to simplify the energy expressions into definite integrals, the arbitrarily shaped integration domain of the energy integrals is divided into several trapezoid domains with curved sides, and then the penalty method and Gram–Schmidt orthogonalization process are combined to generate the admissible functions for calculation. When the contour curve equations of the plate enable the energy integrals to be analytically calculated, the model can obtain the exact solution of free vibration of an arbitrarily shaped plate with various cutouts under general elastic boundary conditions; otherwise, the semi-analytic results can be obtained by combining Gauss–Legendre method. The convergence and accuracy of the proposed model are verified by comparing the numerical examples with results available in the literatures. The vibration characteristics of several complex shaped plates with various cutouts are studied. Taking elliptical plates with a rhombic cutout as examples, the effects of cutout size and position on the vibration characteristics of the plate are discussed. These new results can serve as benchmarks for further research.

Benchmark solutions for stochastic dynamic responses of rectangular Mindlin plates

There exist some researches on the benchmark responses of moderately thick plates based on Mindlin theory under static loads. In practice, rectangular Mindlin plates are often subjected to various random excitations. However, the benchmark solutions of analytically stochastic dynamic responses for rectangular Mindlin plates under random excitations were not investigated. In this paper, analytical power spectral density functions and root mean squares for stationary and nonstationary stochastic responses of rectangular Mindlin plates are derived, which provide benchmark solutions for other numerical methods and experimental design. Firstly, based on the closed-form free vibration analyses of rectangular Mindlin plates with two opposite simply supported edges, exact natural frequencies and modal shapes are obtained. Then, the pseudo excitation method is introduced into the decoupled differential equations of motion. By combining the exact solutions of free vibration, the analytical method for random vibration analysis of elastic Mindlin plate is proposed, and the analytical stochastic responses are derived. Due to the low efficiency of the AM based on symbolic operation, the efficient discrete analytical method, which can ensure that the results are exact in the spatial domain, is further developed. Finally, the performance of the proposed approaches is verified by comparing with the commercial software, Monte Carlo simulation, and literature. The remarkable effects of spatial distributions of excitations on the stochastic responses of Mindlin plates are illustrated. The important influences of parameters including relative thickness, elastic modulus, and boundary constraint degree on responses are revealed.

Free vibration of thick rectangular orthotropic plates with clamped edges- using asymptotic analysis of infinite systems

Exact solutions for free vibration of thick rectangular orthotropic plates when their all edges are clamped are sought through asymptotic analysis of infinite systems without resorting to the usual truncation of series solution. The use of modified trigonometric functions made it possible to obtain a general solution for the problem which has the same form for all four cases of symmetry of the quarter plate. Thus, an infinite system of linear algebraic equations is derived for the unknown coefficients of the series representing the solution for each case. This is in sharp contrast to previous publications based on series-solution which does not allow the satisfaction of the quasi-regularity condition of the corresponding infinite system, and therefore, the method used earlier, was not amenable to asymptotic solution of the infinite system. In this investigation, the quasi-regularity of the infinite system is proved, but importantly, an algorithm for determining the natural frequencies of the plate based on the theorem of the existence of the solution for the quasi-regular system is presented. The asymptotic behaviour of the non-trivial solution of the homogeneous quasi-regular infinite system is ascertained by generalising the asymptotic law of Koialovich which essentially led to the development of the algorithm. Numerical examples are given with significant conclusions drawn.

An accurate model for free vibration of porous magneto-electro-thermo-elastic functionally graded cylindrical shells subjected to multi-field coupled loadings

An accurate model for vibration of a porous magneto-electro-thermo-elastic functionally graded (METE-FG) cylindrical shell made of barium titanate (BaTiO3) and cobalt diiron tetraoxide (CoFe2O4) with magneto-electro-thermal loadings is proposed within the framework of Hamiltonian system. Four types of porosity distribution profiles in the thickness direction are considered. By introducing a new total eigenvector, the higher-order governing differential equations are transformed into a set of lower-order equations. The exact solution for free vibration of METE-FG shells can be expanded in terms of specific symplectic eigenfunctions having seven possible explicit forms. Subsequently, analytical frequency equations and vibration mode shapes for METE-FG shells with various boundary conditions are derived simultaneously. A comparison study is presented to demonstrate the accuracy of the proposed model and very good agreement is observed. The effects of material properties and magneto-electro-thermal loadings on free vibration characteristics of METE-FG cylindrical shells are analyzed and discussed in detail.

An improved separation-of-variable method for the free vibration of orthotropic rectangular thin plates

A separation-of-variable method was proposed in 2009 by the present authors and co-workers (Xing YF, Liu B. New exact solutions for free vibrations of thin orthotropic rectangular plates. Compos Struct 2009;89:567–74), but this traditional separation-of-variable method is incapable for free boundary conditions. Hence, this work is to improve the traditional separation-of-variable method to make it capable of dealing with free boundary conditions. This new method is called the improved separation-of-variable method, in which the characteristic differential governing equation is solved directly, and the free boundary conditions also have separable forms, which are achieved with the Rayleigh’s variational principle. In addition, the relationships between spatial eigenvalues and temporal eigenvalue are quite concise. For plates without free edges, the proposed method reduces to the traditional separation-of-variable method. Numerical experiments validate the improved method. The present technique of handling free boundary conditions is a general methodology for eigenvalue problems such as free vibration and eigenbuckling.

Exact solution of free vibration of adjacent buildings interconnected by visco-elastic dampers

ABSTRACT With consideration of a high-rise coupled building system, a flexible beams-based analytical model is setup to characterize the dynamic behavior of the system. The general motion equation for the two beams interconnected by multiple viscous/visco-elastic dampers is rewritten into a non-dimensional form to identify the minimal set of parameters governing the dynamic characteristics. The corresponding exact solution suitable for arbitrary boundary conditions is presented. Furthermore, the methodology for computing the coefficients of the modal shape function is proposed. As an example, the explicit expression of the modal shape function is derived, provided only one damper is adopted to connect the adjacent buildings. Finally, to validate the proposed methodologies, three case studies are performed, in which the existence of the overdamping and the optimal damping coefficient are revealed. In the case of using one damper in connecting two similar buildings, the estimating equations for the first modal damping ratio are formulated.

Vibration of rectangular plates: fundamental mode and integer multiple of the fundamental period of vibration

This paper is concerned with a vibration analysis of the transverse vibrations of rectangular plates with freely-supported boundaries and corners fixed, in an imagistic way, to describe the deformation state for the specific natural frequency and mode shape for each degree-of-freedom. Two materials, i.e. aluminum and steel, showing the homogeneous properties in the direction of plate thickness were chosen. The exact solutions for free vibrations of the homogeneous rectangular plate were obtained and the fundamental mode and integer multiple of the fundamental period of vibration were provided. The comparisons show that the analytical model predicts natural frequencies reasonably well for a rectangular plate.

Exact Solution for Free Vibration and Buckling of Sandwich S-FGM Plates on Pasternak Elastic Foundation with Various Boundary Conditions

In the present study, free vibration and buckling characteristics of a sandwich functionally graded material (FGM) plate resting on the Pasternak elastic foundation have been investigated. The formulation is based on non-polynomial higher-order shear deformation theory with inverse hyperbolic shape function. A new modified sigmoid law is presented to compute the effective material properties of sandwich FGM plate. The governing equilibrium equations have been derived using Hamilton’s principle. Non-dimensional frequencies and critical buckling loads are evaluated by considering different boundary conditions based on admissible functions satisfying the desired primary and secondary variables. Comprehensive parametric studies have been performed to analyze the influence of geometric configuration, volume fraction exponent, elastic medium parameter, and non-dimensional load parameter on the non-dimensional frequency and critical buckling load. These parametric studies have been done for various boundary conditions and different configurations of the sandwich plate. The computed results can be used as a benchmark for future comparison of sandwich S-FGM plates.

An exact solution for free vibration of cross-ply laminated composite cylindrical shells with elastic restraint ends

In this paper, a new exact solution based on the classical shell theory (CST) for free vibration of cross-ply laminated composite cylindrical shells with elastic restraint ends is proposed. The present exact solution can be summed up in the following steps: Firstly, the displacement functions are constructed by the governing differential equations with the exact closed form solutions; Then, the artificial spring technology is introduced to simulate the general boundary conditions of the two end edges of shell; Thirdly, the equation for natural frequencies is obtained by means of the method of reverberation-ray matrix (MRRM); Lastly, the vibration results are presented by the modified golden section search (MGSS) algorithm. By comparing the present method with published papers, the accuracy of present method is verified. On the basis of that, some new exact nature frequencies and mode shapes of the cross-ply laminated composite cylindrical shells with various classical boundary conditions and elastic restraints are performed and they can be served as the benchmark data for the future.

Exact Solutions for Free Vibration of Cylindrical Shells by a Symplectic Approach

In engineering applications, the vibration analysis of cylindrical shells is highly significant for the design of engineering structures subjected to external dynamic excitations. In practice, the explicit expression would make a better and more quick design comparing to the numerical results. However, the studies on analytical method were concentrated on the semi-inverse method. To overcome the limitation, we introduce a new analytical treatment for the free vibration of cylindrical shells. Under the framework of the Hamiltonian system, an analytical symplectic approach is introduced into the free vibration of cylindrical shells. In the symplectic space, exact solutions are obtained in a rational and systematic way without any trial functions. Analytical frequency equation for arbitrary boundary conditions is derived. Highly accurate natural frequencies and analytical vibration mode functions are obtained simultaneously. In this paper, the governing equations for Donnell’s and Reissner’s shell theories can be successfully transformed into the Hamiltonian form by introducing a full state vector. The free vibration of cylindrical shells is reduced into a symplectic eigenproblem. Numerical results are compared with those previously published in literature and good agreement is observed. In addition, the proposed method can be extended to micro/nanoscale cylindrical shells.

A New Analytical Approach for Free Vibration, Buckling and Forced Vibration of Rectangular Nanoplates Based on Nonlocal Elasticity Theory

In this paper, an analytical Hamiltonian-based model for the dynamic analysis of rectangular nanoplates is proposed using the Kirchhoff plate theory and Eringen’s nonlocal theory. In a symplectic space, the dynamic problem is reduced to solving a unified Hamiltonian dual equation formed by a total unknown vector consisting of displacements, rotation angles, bending moments and generalized shear forces. The exact solutions for free vibration, buckling and steady state forced vibration are established by the eigenvalue analysis and expansion of eigenfunction without any trial functions. In addition, the explicit expressions of the characteristic equations, mode functions and steady state response of the nanoplate with two opposite edges that are simply supported or guided supported are obtained. To verify the accuracy and reliability of the present method, numerical results are compared with published solutions and excellent agreement is obtained. Comprehensive benchmark results that consider the nonlocal effect on the dynamic behaviors of rectangular nanoplates are also presented in dimensionless tabular and graphical forms.

Exact Solutions of Fully Nonstationary Random Vibration for Rectangular Kirchhoff Plates Using Discrete Analytical Method

This paper proposes the discrete analytical method (DAM) to determine exactly and efficiently the fully nonstationary random responses of rectangular Kirchhoff plates under temporally and spectrally nonstationary acceleration excitation of earthquake ground motions. First, the fully nonstationary power spectral density (PSD) model is suggested by replacing the filtered frequency and damping of Gaussian filtered white-noise model with the time-variant ones. The exact solutions of free vibration of thin plates with two opposite edges simply supported boundary conditions are introduced. Then, the full analytical procedure for random vibration analysis of the plate is established by using a pseudo excitation method (PEM) that can consider all modal auto-correlation and cross-correlation terms. Owing to involving a series of Duhamel time integrals of single degree of freedom systems, it is difficult to fully analytically evaluate the PSD of time-variant responses such as the transverse deflection, velocity, acceleration and stress components. Thus, DAM that combines the PEM with precise integration technique is developed to enhance the computational efficiency. Finally, comparison of the results by the DAM with Monte Carlo simulations and the analytical stationary random vibration analysis demonstrates the high efficiency and accuracy of DAM. Moreover, the fully nonstationary excitation imposes a remarkable effect on the response PSD of rectangular Kirchhoff plates.

A new Hamiltonian-based approach for free vibration of a functionally graded orthotropic circular cylindrical shell embedded in an elastic medium

Exact solutions for free vibration of functionally graded (FG) orthotropic circular cylindrical shells embedded in an elastic medium with arbitrary classical boundary conditions are obtained by a Hamiltonian-based method. Based on the Reissner shell theory and symplectic mathematics, all six possible general solutions instead of trial functions in classical inverse or semi-inverse methods are determined analytically. The determination of natural frequency and vibration mode shape is reduced to an eigen-problem of the Hamiltonian matrix in symplectic space. Numerical examples are provided to verify the validity of the present method. Some new results are also given.

Benchmark solutions of stationary random vibration for rectangular thin plate based on discrete analytical method

This paper aims to accurately and efficiently achieve the benchmark solutions of stationary stochastic responses for rectangular thin plate. Firstly, the exact solutions of free vibration for thin plate with SSSS, SSSC, SCSC, SFSF, SSSF and SCSF boundary conditions are introduced to random vibration analysis. Based on pseudo excitation method (PEM), the analytical power spectral density (PSD) functions of the transverse deflection, velocity, acceleration and stress responses for thin plate under random base acceleration excitation are derived. Subsequently, to enhance computational efficiency, the discrete analytical method (DAM) that realizes the discretization for the modal coordinates and frequency domain is proposed. Finally, the efficiency of DAM and the accuracy of benchmark solutions are scrutinized by comparison with the analytical solutions and finite element solutions.

An analytical symplectic approach to the vibration analysis of orthotropic graphene sheets

A nonlocal continuum orthotropic plate model is proposed to study the vibration behavior of single-layer graphene sheets (SLGSs) using an analytical symplectic approach. A Hamiltonian system is established by introducing a total unknown vector consisting of the displacement amplitude, rotation angle, shear force, and bending moment. The high-order governing differential equation of the vibration of SLGSs is transformed into a set of ordinary differential equations in symplectic space. Exact solutions for free vibration are obtianed by the method of separation of variables without any trial shape functions and can be expanded in series of symplectic eigenfunctions. Analytical frequency equations are derived for all six possible boundary conditions. Vibration modes are expressed in terms of the symplectic eigenfunctions. In the numerical examples, comparison is presented to verify the accuracy of the proposed method. Comprehensive numerical examples for graphene sheets with Levy-type boundary conditions are given. A parametric study of the natural frequency is also included.

Exact solution for free vibrations of spinning nanotube based on nonlocal first order shear deformation shell theory

Abstractt Spinning nanotubes are the main part of power transmission units in nano-machines. So, having proper information about the dynamic behavior of them are very important. So, as the basis of each dynamics problem free vibration of spinning nanotube is analyzed in this paper. For this purpose, Sanders first order shear deformation shell theory is combined with nonlocal elasticity and equations of motion are found. For the first time, equations are exactly solved for all classical combinations of boundary conditions. Validity and accuracy of present method are checked with literature in different tables and figures. As benchmark analysis, effects of changing geometrical parameters, spinning speed, nonlocal parameter, and boundary conditions on natural frequencies and critical speeds of nanotube is studied which will be helpful for future investigations.

Exact solutions for free vibrations of axially inhomogeneous Timoshenko beams with variable cross section

A novel method is proposed to simplify the governing equations for the free vibration of Timoshenko beams with both geometrical nonuniformity and material inhomogeneity along the beam axis. For a wide class of Timoshenko beams, this method enables us to reduce the coupled governing differential equations with variable coefficients to a pair of uncoupled second-order differential equations of Sturm–Liouville type with respect to the rotation angle due to bending. The reduced equations contain two important parameters, one describing the variations of translational inertia and bending rigidity along the beam axis, and the other reflecting the comprehensive effect of rotatory inertia and shear deformation. A series of exact analytical solutions are derived from the reduced equations for the first time, and several examples are also provided as benchmarks.

A beam with arbitrarily placed lateral dampers: Evolution of complex modes with damping

Free vibration of a beam with multiple arbitrarily placed lateral viscous dampers is investigated to gain insight into the intrinsic dynamic features of non-proportional damped systems. In terms of virtual boundary condition method, complex modes of a damper-beam system are achieved, and the solution is also suitable for the beams that have different boundary conditions. The features of the wave numbers satisfying the frequency equation were discussed in theory. The orthogonality analysis conducted in this paper provides two orthogonality conditions for complex modes. Pseudoundamped natural frequencies, damping ratios and complex modes are surveyed via numerical study. The analysis on the evolution of complex modes shows that the increasing damping would lead to over damped modes, and the mode shape that corresponds to the small one of a pair of real-valued natural frequencies is close to the static deformation shape of a beam subjected to static forces located at the positions of the dampers. For the rest modes that would never be over damped with increasing damping, the mode shapes and corresponding psuedoundamped natural frequencies will converge to that of a beam with rolling supports located at where dampers are placed. The exact solution of free vibration of a multiple-span beam is presented in addition.

Benchmark solution for free vibration of thick open cylindrical shells on Pasternak foundation with general boundary conditions

In the present article, a new three-dimensional exact solution for free vibration of thick open cylindrical shells on Pasternak foundation with general boundary conditions is presented. The three-dimensional elasticity theory is employed to formulate the theoretical model. The admissible functions of the thick shells are described as a combination of a three-dimensional (3-D) Fourier cosine series and auxiliary functions. Compared with the traditional Fourier series, the improved Fourier series can eliminate all the relevant discontinuities of the displacements and their derivatives at the edges regardless of boundary conditions. The excellent accuracy and reliability of the current solutions are demonstrated by numerical examples and comparison of the present results with those available in the literature and obtained by using ABAQUS which is based on the finite element method. Numerous new results for thick open cylindrical shells on Pasternak foundation with elastic boundary conditions are presented. In addition, comprehensive studies on the effects of the elastic restraint parameters, geometric parameters and elastic foundation coefficients are also reported.