Solution For Free Vibration Analysis Research Articles

On the symplectic superposition method for new analytic bending, buckling, and free vibration solutions of rectangular nanoplates with all edges free

This paper extends an up-to-date symplectic superposition method for analytic bending, buckling, and free vibration solutions of rectangular nanoplates based on the nonlocal theory. By transferring the problems to the Hamiltonian system expression, the symplectic approach becomes applicable, which involves generation of symplectic eigenvalue problems and symplectic eigen expansion. An original problem is then converted into superposition of several elaborated subproblems that are solved by the symplectic approach. The analytic bending deflection, natural frequency, buckling load, and associated mode shape solutions are obtained by the equivalence between the original problem and the superposition. As a representative class of non-Levy-type nanoplates that were not solved by conventional analytic methods, the nanoplates with all edges free are focused on the rapid convergence of the present solutions confirming comprehensive results tabulated with the accuracy of five significant figures, which can serve as benchmarks for future comparison. Quantitative parameter analyses are performed with the analytic solutions to investigate the effects of the nonlocal parameter and plate dimension on the mechanical behaviors of free nanoplates. The present method has merits on tackling complex boundary-value problems of higher-order partial differential equations represented by those for nanoplates and may be further developed for more analytic solutions in future studies.

Analytical solution of free vibration of FG beam utilizing different types of beam theories: A comparative study

In engineering structures, to having the projected structure to serve all the engineering purposes, the theory to be used during the modeling stage is also of great importance. In the present work, an analytical solution of the free vibration of the beam composed of functionally graded materials (FGMs) is presented utilizing different beam theories. The comparison of supposed beam theory for free vibration of functionally graded (FG) beam is examined. For this aim, Euler-Bernoulli, Rayleigh, Shear, and Timoshenko beam theories are employed. The functionally graded material properties are assumed to vary continuously through the thickness direction of the beam with respect to the volume fraction of constituents. The governing equations of free vibration of FG beams are derived in the frameworks of four beam theories. Resulting equations are solved versus simply supported boundary conditions, analytically. To verify the results, comparisons are carried out with the available results. Parametrical studies are performed for discussing the effects of supposed beam theory, the variation of beam characteristics, and FGM properties on the free vibration of beams. In conclusion, it is found that the interaction between FGM properties and the supposed beam theory is of significance in terms of free vibration of the beams and that different beam theories need to be used depending on the characteristics of the beam in question.

On the symplectic superposition method for new analytic free vibration solutions of side-cracked rectangular thin plates

In this study, a first attempt is made to develop an up-to-date symplectic superposition method for some new analytic free vibration solutions of side-cracked rectangular thin plates that were not obtained by conventional semi-inverse methods. In contrast with the classical Lagrangian system and Euclidean space, the present method is implemented within the Hamiltonian-system framework and symplectic space. The solution procedure involves expressing the problems in the Hamiltonian system and dividing a side-cracked plate into several sub-plates that are analytically solved by the symplectic superposition method, where the imposed quantities are determined by the plate boundary conditions, free edge conditions along the crack, and interfacial continuity conditions between the sub-plates. In the analytic solution of a sub-plate, specifically, the symplectic eigenvalue problems are formulated, followed by the symplectic eigen expansion. The integration of the solutions of the sub-plates yields the final solution of a side-cracked plate. The rigorous mathematical techniques, without predetermination of solution forms, qualify the present method as an unusual approach for exploring more analytic solutions. Comprehensive natural frequency and mode shape solutions of the side-cracked plates under three representative boundary constraints are provided and well validated by other methods. The new analytic solutions obtained may serve as benchmarks for other potential solution methods.

On the symplectic superposition method for analytic free vibration solutions of right triangular plates

The analytic free vibration solutions of triangular plates are important for both rapid analyses and preliminary designs of similar structures. Due to the difficulty in solving the complex boundary value problems of the governing high-order partial differential equations, the current knowledge about the analytic solutions is limited. This study presents a first attempt to explore an up-to-date symplectic superposition method for analytic free vibration solutions of right triangular plates. Specifically, an original problem is regarded as the superposition of three fundamental subproblems of the corresponding rectangular plates that are solved by the symplectic eigenexpansion within the Hamiltonian-system framework, involving the coordinate transformation. The analytic frequency and mode shape solutions are then obtained by the requirement of the equivalence between the original problem and the superposition. By comparison with the numerical results for the right triangular plates under six different combinations of clamped and simply supported boundary constraints, the fast convergence and high accuracy of the present approach are well confirmed. Within the current solution framework, the extension to the problems of more polygonal plates is possible.

On New Analytic Free Vibration Solutions of Doubly Curved Shallow Shells by the Symplectic Superposition Method Within the Hamiltonian-System Framework

Abstract This study presents a first attempt to explore new analytic free vibration solutions of doubly curved shallow shells by the symplectic superposition method, with focus on non-Lévy-type shells that are hard to tackle by classical analytic methods due to the intractable boundary-value problems of high-order partial differential equations. Compared with the conventional Lagrangian-system-based expression to be solved in the Euclidean space, the present description of the problems is within the Hamiltonian system, with the solution procedure implemented in the symplectic space, incorporating formulation of a symplectic eigenvalue problem and symplectic eigen expansion. Specifically, an original problem is first converted into two subproblems, which are solved by the above strategy to yield the symplectic solutions. The analytic frequency and mode shape solutions are then obtained by the requirement of the equivalence between the original problem and the superposition of subproblems. Comprehensive results for representative non-Lévy-type shells are tabulated or plotted, all of which are well validated by satisfactory agreement with the numerical finite element method. Due to the strictness of mathematical derivation and accuracy of solution, the developed method provides a solid approach for seeking more analytic solutions.

A three-dimensional solution for free vibration of FGP-GPLRC cylindrical shells resting on elastic foundations: a comparative and parametric study

This paper presents a unified solution for free vibration analysis of thick functionally graded porous graphene platelet reinforced composite (FGP-GPLRC) cylindrical shells embedded in elastic foundations. The three-dimensional (3-D) theory of shell theory is introduced for theoretical formulation. The Rayleigh-Ritz method in conjugation with artificial spring technique are employed, where the arbitrary boundary conditions can be conveniently obtained. A unified solution which comprises of six different displacement functions is developed. The calculation performances including convergence rate and calculating efficiency with respect to different displacement functions are compared extensively. Besides, three elastic foundations (Winkler/ Pasternak/ Kerr foundations), four types of porosity distributions and three categories of GPL patterns are considered. Some benchmark results are provided for free vibration of FGP-GPLRC cylindrical shells resting on elastic foundations. At last, the effects of different boundary conditions, elastic foundations with various parameters, porosity coefficient, GPL weight fraction and geometrical parameters on the vibration are elucidated.

Free vibration analysis of functionally graded hybrid matrix/fiber nanocomposite conical shells using multiscale method

This study is dedicated to propose a numerical solution for free vibration analysis of nanocomposite conical shells. The First-order Shear Deformation Theory (FSDT) is used to achieve the governing equations. Then, an efficient numerical method, namely the Generalized Differential Quadrature Method (GDQM), is employed to solve the Donnell-type governing differential equations. The effects of the nanofillers aggregation are investigated throughout this research. It is considered that a single conical shell is composed of a variety of the constituent material. This type of material includes three phases: polymer matrix, macro-scale fibers and nano-scale Carbon Nano Tubes (CNTs). Two well-known approaches, namely Han and Chamis, are employed for the homogenization procedure to obtain the equivalent material properties. It is worth mentioning that the power pattern is employed for the functionally distribution of CNTs within the polymer matrix. Furthermore, the effects of different boundary conditions, geometric and material properties on the dimensionless frequency parameter of the multiscale nanocomposite conical shells are considered. Afterward, a parametric study is implemented to determine the deep influences of the volume fraction of the nanofillers throughout the inclusion on the vibrational behaviors of conical shells. It is found out that increasing the agglomeration parameter increases the frequency parameter of the conical structures. Moreover, increasing the power exponent of the volume fraction of FG-CNT material deals decrease of the frequency parameter of conical shells. Comparing the obtained results with the outputs of some well-known benchmarks proves the correctness, accuracy and high capability of the proposed formulation.

Effect of adsorbate viscoelasticity on dynamical responses of laminated microcantilever resonators

Understanding the mechanical properties of bio-adsorbates at the pathological state is important in mechanobiology and medical diagnosis. With the development of biodetection technology, the mechanical properties of bio-adsorbates can be detected at microscale. The paper presents an integro-partial-differential theoretical model to elucidate the effect of adsorbate viscoelasticity on vibrational properties of elastic–viscoelastic microcantilever. A differential method in the temporal domain and a variable separation method or a differential quadrature method (DQM) in the spatial domain are combined to obtain the relevant analytical solutions for free vibration with a complete adsorption and numerical solutions for forced vibration with a local adsorption. The present results agree well with the previous studies in the case of complete elastic adsorption. The study shows that the amplitude of a weakly damped free vibration of the microcantilever is positively correlated with the reciprocal of relaxation time of adsorbate film; the dissipative mechanical property of adsorption film with a standard linear solid constitutive relation endows the forced vibration of the microcantilever subjected to a transverse harmonic load with a double-peak resonance; and the sensitivity of microcantilever-based dynamic detection can be improved remarkably if the excitation frequency, film-to-substrate thickness/modulus ratio, adsorption position/area are carefully controlled.

Exact Solution for Free Vibration Analysis of FGM Beams

Exact Solution for Free Vibration Analysis of FGM Beams

Analytical solutions of bending and free vibration of moderately thick micro-plate via two-variable strain gradient theory

In this paper, the strain gradient theory (SGT) and two-variable refined plate theory are implemented simultaneously to analyze bending and free vibration of micro-plate. The proposed theory is called two-variable strain gradient theory (TV-SGT). The TV-SGT, which can be used for both thin and moderately thick micro-plates, predicts parabolic variation of transverse shear stresses across the plate thickness and does not need the shear correction factor. The governing equations are obtained using a variational approach and solved analytically to find the deflection of simply supported micro-plate under uniformly distributed loading and natural frequency of this plate. Thereby, the effects of length scale parameters and dimensions on the bending deflection and natural frequency of the micro-plate are investigated. Results show that, strengthen micro-plate has higher shear rigidity. Moreover, it is inferred that TV-SGT is well defined to analyze the micro-plate.

Analytical and finite element solutions of free and forced vibration of unrestrained and braced thin-walled beams

In this article, vibration of thin-walled beams with arbitrary open cross-section shape is investigated. Based on the beam element model accounting for warping and flexural–torsional coupling, analytical solutions for different boundary conditions are derived for higher free vibration modes in bending, torsion and flexural–torsional coupled modes. In the model, the effects of rotational inertial kinematic terms are considered. The finite element approach of the model is also investigated. Three-dimensional beams with seven degrees of freedom per node are adopted in the mesh process. Free vibration and forced vibration analyses are possible. In forced vibration, the behaviour of the beams is studied in the frequency domain using the steady-state method (modal analysis). Damping is considered using the Rayleigh model. The model is validated by comparing the results to benchmark solutions found in the literature and to other recent numerical and experimental results. Additional finite element simulations are performed by means of commercial softwares (Abaqus and Adina). In slender unrestrained beams, the vibration behaviour is predominated by torsion and lateral bending modes. In design, recourse to braces is a good compromise. This solution is discussed, and improvement of the vibration behaviour in the presence of intermediate braces is confirmed. Application of higher vibration modes in building and bridge design is outlined. The effects of the number and distribution of the intermediate braces to improve structural stability against vibration behaviour is outlined.

Free vibration of a functionally graded annular sector plate integrated with piezoelectric layers

Based on the first order shear deformation theory, free vibration behavior of functionally graded (FG) annular sector plates integrated with piezoelectric layers is investigated. The distribution of electric potential along the thickness direction of piezoelectric layers which is assumed to be a combination of linear and sinusoidal functions, satisfies both open and closed circuit electrical boundary conditions. Through a reformulation of governing equations and harmonic motion assumption, a novel decoupling method is suggested to transform the six second order coupled partial differential equations of motion into two eighth order and fourth order equations. A Fourier series method is then employed to present analytical solutions for free vibration of smart FG annular sector plates with simply supported radial edges and arbitrarily supported circular edges. The results, which can be used as a benchmark and suitable for design purposes, are verified with those reported in the literature. Finally, by presenting extensive ranges of frequencies, the effects of geometric parameters, power law index, FG and piezoelectric materials, electrical and mechanical boundary conditions as well as the piezoelectric layer thickness on vibration response of smart annular sector plates are discussed in detail.

An exact analytical solution for free in-plane vibration of sector plates with simply supported radial edges

In this paper, an exact analytical solution is proposed to investigate the free in-plane vibration of sector plates. The highly complex and coupled differential equations of in-plane vibration motion are decoupled by invoking Helmholtz decomposition. The method of separation of variables is used to solve the resulting uncoupled Helmholtz equations. Applying the boundary conditions at both radial edges, the compatible eigenvalue problems for sector plates are obtained. Then, applying the boundary conditions at inner and outer circumferential edges, the corresponding characteristic equations can be solved to obtain the natural frequencies and the associated mode shapes. The results from the proposed method agree very well with the results from the available literature and the FEM. Significant influence of the boundary conditions and the geometric parameters on the free in-plane vibration characteristics of sector plates is investigated and addressed. The superficial frequency veering “aberration” between neighboring modes is discussed and found in fact to be an eigenvalue crossing phenomenon. It is proved that for a sector plate, only with simply supported radial edges, there exists an exact analytical solution. The proposed exact analytical solution can provide not only the detailed characteristics of the free in-plane vibration of sector plates but also the benchmark for further study.

A semi analytical solution for free vibration analysis of combined spherical and cylindrical shells with non-uniform thickness based on Ritz method

Abstract The paper investigated the free vibration of combined spherical-cylindrical-spherical (CSCS) shell with non-uniform thickness based on Ritz method. The energy method and first-order shear deformation theory are adopted to derive the formulas. The displacement functions are improved on base of domain decomposition method, in which the unified Jacobi polynomials are introduced to represent the displacement functions component along axial direction. The displacement functions component along circumferential direction is still Fourier series. In addition, the spring stiffness method is formed to be a unified format to deal with general boundary conditions. Then the final solutions are obtained by using Ritz method. To prove the validity of this method, the results of the same condition are compared with those obtained by FEM, published literatures and experiment. In addition, some meaningful examples are provided to reveal the free vibration characteristics of stepped spherical-cylindrical-spherical shell. The results of this paper can be used as the reference data for future research in related field.

Analytical solution for free vibration of stiffened functionally graded cylindrical shell structure resting on elastic foundation

In this work, the analytical solution for free vibration of functionally graded cylindrical shell with stiffeners resting on Winkler–Pasternak foundation with several boundary conditions is reported. Material properties of the cylindrical shell are assumed to be varying smoothly and continuously across the thickness according to the power law distribution of the volume fraction of constituents. The governing equations of the stiffened functionally graded cylindrical shell resting on elastic foundation are obtained by using Hamilton’s principle, the first-order shear deformation theory and the Lekhnitsky smeared stiffeners technique. The purpose of the present study is to illustrate a simple approach to get the natural frequency of the stiffened functionally graded cylindrical shell with several boundary conditions by using the Galerkin method with beam functions of the axial displacement fields. A good agreement is obtained as the present results in comparison with those of publications in the existing literature. In numerical investigations, some influences of volume fraction index, cylindrical shell’s geometric ratios, boundary conditions and elastic foundation parameters on the fundamental frequency of the stiffened cylindrical shell resting on elastic foundation are given.

Analytical Solution for Free Vibration of Annular Mindlin Plate with a Circumferential Open Crack

Mindlin plate theory is employed to obtain the free vibration response of an annular moderately thick plate with a circumferential open crack with fixed-free boundary conditions. To model the crack, a set of continuously distributed rotational springs are employed at the crack location. The corresponding spring stiffness value is a function of the crack depth and is given as a closed-form function. To obtain the vibration behaviour, the eigenvalue problem is solved to obtain the natural frequencies and mode shapes. The current method is verified by comparing the results with those obtained from finite element analysis. Through a parametric study, the effects of the crack depth and its radial location on the natural frequencies and mode shapes are investigated. The results show that for a constant crack depth, the reduction in natural frequency is a strong function of the radial location of the crack.

Hamiltonian system-based new analytic free vibration solutions of cylindrical shell panels

This paper deals with the classical challenging free vibration problems of non-Lévy-type cylindrical shell panels, i.e., those without two opposite edges simply supported, by a Hamiltonian system-based symplectic superposition method. The governing equations of a vibrating cylindrical panel are formulated within the Hamiltonian system framework such that the symplectic eigen problems are constructed, which yield analytic solutions of two types of fundamental problems. By the equivalence between the superposition of the fundamental problems and the original problem, new analytic frequency and mode shape solutions of the panels with four different combinations of boundary conditions are derived. Comprehensive benchmark results are tabulated and plotted, which are useful for validation of other numerical/approximate methods. The primary advantage of the developed approach that no pre-determination of solution forms is needed enables one to pursue more analytic solutions of intractable shell problems.

A semi-analytical solution for free vibration analysis of rotating carbon nanotube with various boundary conditions based on nonlocal theory

A nonlocal theory of elasticity is developed to study free vibration of rotating single walled carbon nanotubes (SWCNTs) with different boundary conditions. The equations of motion are obtained applying Hamilton’s concept, which contain the nonlocal Eringen’s theory of elasticity, Love’s shell assumptions, the influence of the centrifugal and the Coriolis accelerations, and the preliminary hoop stress. The equations of motion as well as the boundary condition equations are transformed into a set of algebraic equation applying Fourier decomposition and the DQ method in the longitudinal direction. Numerical results, presented for the frequency parameters and the associated vibration wavenumbers of a series of SWCNT and with different nonlocal parameters, confirm the validity of the solution presented. Furthermore, the model is used to clarify the effects of rotating speed, boundary conditions, length-to-radius ratio and nonlocal parameter on the natural frequency.

New analytic free vibration solutions of orthotropic rectangular plates by a novel symplectic approach

This paper aims at analytically solving the free vibration problems of orthotropic rectangular plates, which have found broad applications as a class of crucial aerostructures. A novel symplectic superposition method is presented to give analytic solutions of plates without two opposite edges simply supported (i.e., non-Levy-type plates). Since these solutions are very difficult to obtain within the classical Lagrangian system framework, the new approach proceeds in the Hamiltonian system, in physics, and in the symplectic space, in mathematics. The validity of separation of variables and symplectic eigenexpansion guarantees the realization of new analytic solutions in a rigorous step-by-step way, which cannot be achieved by some classical analytic methods where the predetermination of solution forms is usually inevitable. Representative challenging problems, including those incorporating two adjacent free edges (e.g., cantilever and free plates), are successfully solved and validated by the refined finite element modeling. Comprehensive natural frequency and mode shape results show the accuracy of the present solutions. The generality of the symplectic superposition method enables one to pursue more analytic solutions of similar problems that are governed by higher-order partial differential equations.

Vibration approximate analytical solutions of circular plate consideration of complex pre-stress distribution

The influence of the complex pre-stress on the circular plate is investigated herein, with which to solve the non-uniform pre-stress distribution problem. According to the strain–stress equation, the motion differential equations of the circular thin plate with complex pre-stress distribution are derived. Based on the Rayleigh–Ritz theory of energy method, the complex pre-stress distribution function and vibration-displacement function are expanded into the cosine trigonometric series, and the approximate analytical solutions of structural free vibration for circular plate are proposed. A circular plate with simply supported boundary condition, for example, the effectiveness of the proposed method is confirmed through numerical calculations and the finite element method verification. The influence of different type’s distribution of welding residual stress on the natural frequency and mode shape for circular plate structure are compared. The proposed approach in present article can be used in arbitrary pre-stress distribution problem.