Effects Of Elastic Boundary Conditions Research Articles

Accurate and efficient analytical simulation of free vibration for embedded nonlocal CNTRC beams with general boundary conditions

This study aims to evaluate the free vibrational response of embedded restrained nanobeams enriched by nanocomposites based on an exact Fourier series approach. In order to capture the small-scale effects on the dynamical response, Eringen's differential form of nonlocal elasticity is used which employs a scale (nonlocal) parameter. Within the framework of Rayleigh and Bernoulli-Euler beam theories, including the effect of nonlocality and employing the Fourier sine series together with Stokes' transformation, systems of linear equations are obtained and solved using the coefficient matrices. The combined effects of elastic boundary conditions, elastic foundation, dispersion patterns and volume fractions of carbon nanotubes, and nonlocal parameter are examined by solving eigenvalue problems constructed with Fourier infinite series. Free vibration frequencies are calculated for carbon nanotube-reinforced nanobeams under different rigid or restrained boundary conditions, including Winkler-Pasternak type elastic foundation. A comprehensive parametric study is performed, focusing on various effects for the free vibrational response of the composite nanobeam reinforced with carbon nanotubes. It is concluded that adding a small amount of carbon nanotube material can reinforce the stiffness of the composite nanobeam, and its free vibration performance is significantly affected by the distribution patterns, elastic medium, and boundary conditions.

WITHDRAWN: Accurate and efficient analytical simulation of free vibration for embedded nonlocal CNTRC beams with general boundary conditions

This study aims to evaluate the free vibrational response of restrained nanobeam enriched by nanocomposites based on an exact Fourier series approach. In order to capture the small size effects on the dynamical response, Eringen’s differential form of nonlocal elasticity is used which employs the one size (nonlocal) parameter. Within the framework of Rayleigh and Bernoulli-Euler beam theories to include the effect of nonlocality and by employing Fourier sine series together with Stokes’ transformation, a system of linear equations is obtained then solved by using the coefficient matrix. The combined effects of elastic boundary conditions, elastic medium, dispersion patterns and nonlocal effects are examined by solving this eigen-value problem constructed with Fourier infinite series. Free vibration frequencies are calculated for carbon nanotube reinforced nanobeam under different rigid or restrained boundary conditions, including an elastic medium Winkler-Pasternak type. A comprehensive parametric study is performed, with the particular focus on the influences of distribution pattern, elastic boundary and medium parameter on the free vibrational response of the composite nanobeam with different graded distributions. It is concluded that the addition of a small amount of carbon nanotube material can reinforce the stiffness of the nanobeam, and its free vibration performance is significantly affected by the distribution patterns, elastic medium and boundary conditions.

Dynamic Analysis and Design of Metamaterial Plates With Crossed Acoustic Black Holes for Vibration Control

Abstract Acoustic black holes (ABHs) have shown great potential in vibration and noise control. Merging the ABH effect and the metamaterial can be a more efficient approach for vibration control. The aim of this article is to study the dynamics of a metamaterial plate with crossed acoustic black holes. The band gap properties of the infinite structure and the influence of the design variables are investigated by using the finite element method and the Floquet–Bloch theorem. The vibration transmission and frequency response functions of the finite structure are presented to reveal the vibration attenuation mechanism. The effect of elastic boundary conditions on the vibration properties of the metamaterial plate is also studied. Numerical results demonstrate that the vibration is remarkably weakened due to the band gap and local modes induced by the ABH effect. Then, experimental validation is given by using 3D printing techniques. Finally, we study the multi-objective optimal design problem of the ABH plate to reduce the vibration amplitude and the structural mass simultaneously. Optimization results provide more options for the trade-off design of metamaterial plates between the lightweight design and vibration suppression capability.

Effect of elastic support on the linear buckling response of quasi-isotropic cylindrical shells under axial compression

Cylindrical shells under compressive loading are highly sensitive to boundary conditions. Considering that these structures are connected by surrounding structural components with finite stiffness, an accurate evaluation of the effects of their boundary stiffness is crucial in their design. As such, this work investigates the effect of elastic boundary conditions on the linear buckling behaviour of cylindrical shells under compressive loading. To achieve this goal, a virtual testing investigation on the effect of translational and rotational constraints to the linear buckling response of a quasi-isotropic cylinder subjected to axial compression is performed. Subsequently, the effect of many kinds of constraints on linear buckling behaviour is discussed and interesting insights regarding a significant coupling effect between the radial and tangential translational constraints are given. Results obtained from virtual testing show that seven recurrent buckling mode shapes occur with seven corresponding similar linear buckling loads. Therefore, based on these similarities, seven groups of classical boundary conditions are introduced to classify all possible linear buckling behaviours exhibited by the cylinder under consideration. Finally, these findings can support the development of theoretical models for cascade, or flange, designs of multiple connecting cylinders.

A unified semi-analytical method for vibration analysis of shells of revolution stiffened by rings with T cross-section

A unified semi-analytical method is presented to study free and forced vibrations of arbitrary shells of revolution stiffened by rings with T cross-section. Combined shells of revolution and rings with rectangular and L cross-sections are included in the titled structure. The ring-stiffened shell is firstly decomposed to several shell segments and rings with T cross-section according to intersections between the shell and rings, and shell segments are further divided into some narrower segments to be treated as conical shells. Rings with T cross-section are modeled as two cylindrical shells (left and right flanges) and one annular plate (web), rather than conventional curved beams. Conical segments, cylindrical segments and annular plates are uniformly analyzed by employing Flügge thin shell theory and expanding displacements as power series. To consider both classic and elastic boundary conditions, the artificial spring technology is employed to restrain displacements at edges. Then, boundary conditions at two edges and a set of continuity conditions of adjacent segments are assembled to the final governing equation. Through comparing free and forced vibration results of present method with the ones in the literature and calculated by finite element method, rapid convergence and high accuracy are demonstrated. Furthermore, effects of elastic boundary conditions and rings are discussed. Results reveal that mode shapes of shells are significantly affected by rings with T cross-section, and in-plane and out-plane deformations of the web of rings cannot be neglected, especially for rings with large height.

Wave based method for vibration analysis of double-walled cylindrical shells

Wave based method (WBM) is presented for vibration analysis of double-walled cylindrical shells coupled by annular plates. To establish the governing equation, the double-walled shell is divided into several cylindrical shell segments and annular plates according to junctions of discontinuities, and they can be easily synthesized through continuity conditions at the junctions. Flügge shell theory and thin plate theory are utilized to describe motions of cylindrical segments and annular plates, displacements of which are expanded as wave functions. In addition, artificial springs are employed to restrain displacements at boundaries to uniformly consider elastic and classic boundary conditions. To test the validity of WBM, free and forced vibration results of WBM are compared with those of literature and finite element method, which demonstrates high accuracy of present method. Some mode shapes are also given for visual understanding. Furthermore, effects of elastic boundary conditions and annular plates on free and forced vibrations are investigated. Results show that axial and circumferential displacements are of great effect on fundamental and beam mode frequencies, and effects of the number of annular plates are obviously greater than the thickness of annular plates. The results of present paper can be used as benchmarks for further researches.

Silicon as an anisotropic mechanical material: Deflection of thin crystalline plates

While silicon is an anisotropic material it is often in literature treated as an isotropic material when it comes to plate calculations. This leads to considerable errors in the calculated deflection. To overcome this problem, we present an in-depth analysis of the bending behavior of thin crystalline plates. An analysis of the compliance tensor for the 32 different crystal classes shows, that for thin plates, only 5 different types of plates exist. An anisotropic plate equation valid for crystalline thin plates is derived and solved for circular, elliptic, rectangular and square plates using both exact analytical expressions and approximate expressions calculated by the Galerkin method. The results are applied to plates made on silicon (001), (011) and (111) substrates, respectively, and analytical equations for the deflection, strain energy and resonance frequency of such plates are presented. These expressions are in excellent agreement with anisotropic finite element calculations. The calculated deflection differs less than 0.1%, for both circular and rectangular plates, compared to finite element calculations. The results are presented as ready-to-use facilitating accurate analytical models involving crystalline plates, such as those often found in the field of micro electro mechanical systems. The effect of elastic boundary conditions is taken into account by using an effective radius of the plate.

Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bernoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method (DTM) has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.

English

    Laminated composite plate with high mechanical strength forms one of the main parts of the plane. These laminated composite plates get vibrated free or forced during their use. In this study, three different kinds of composite plates are used to determine the effect of elastic boundary conditions on the free vibration of the plates. The classical finite element method is used for solution. Special computer program using Matlab are utilized to obtain the solutions. It is observed that the elastic boundary condition have significant effect on the vibration of the plate. Investigation of this free vibration analysis has proved that to obtain a good design, reasonable materials based on similar analyses are to be selected.   Key words: Composite plate, finite element method, vibration analysis, elastic boundary condition.  

An improved formulation for predicting low-frequency noise transmitted through double-pane windows.

With superior sound insulation properties over a single-panel configuration, a double-panel structure with the presence of a cavity has found a wide range of applications for sound insulation. A classical method for combining a finite element method (FEM) with a boundary element method (BEM) is used to examine the transmission of low-frequency noise through double-pane windows in the present study. The technique of component mode synthesis is applied to adjust the stiffness matrix in the FEM formulation in order to examine the effects of elastic boundary conditions on the sound transmission through these structures. However, the Green function for predicting sound propagation in a rectangular long enclosure is used in favor of the free-field Green function for predicting the pressure inside the cavity of the window panes. The predicted pressure in the cavity is then coupled with the FEM formulation for the window panes and the BEM formulations for the sound fields at the outer surfaces of window panes. A parametric study is conducted systematically to allow a detailed examination for the characteristics of sound insulation of a double-pane window at different frequency bands especially for the low-frequency components.

Effects of imperfections of the buckling response of composite shells

The results of an experimental and analytical study of the effects of initial imperfections on the buckling response and failure of unstiffened thin-walled compression-loaded graphite-epoxy cylindrical shells are presented. The shells considered in the study have six different shell-wall laminates two different shell-radius-to-thickness ratios. The shell-wall laminates include four different orthotropic laminates and two different quasi-isotropic laminates. The shell-radius-to-thickness ratios includes shell-radius-to-thickness ratios equal to 100 and 200. The numerical results include the effects of traditional and nontraditional initial imperfections and selected shell parameter uncertainties. The traditional imperfections include the geometric shell-wall mid-surface imperfections that are commonly discussed in the literature on thin shell buckling. The nontraditional imperfections include shell-wall thickness variations, local shell-wall ply-gaps associated with the fabrication process, shell-end geometric imperfections, nonuniform applied end loads, and variations in the boundary conditions including the effects of elastic boundary conditions. The cylinder parameter uncertainties considered include uncertainties in geometric imperfection measurements, lamina fiber volume fraction, fiber and matrix properties, boundary conditions, and applied end load distribution. Results that include the effects of these traditional and nontraditional imperfections and uncertainties on the nonlinear response characteristics, buckling loads and failure of the shells are presented. The analysis procedure includes a nonlinear static analysis that predicts the stable response characteristics of the shells, and a nonlinear transient analysis that predicts the unstable response characteristics. In addition, a common failure analysis is used to predict material failures in the shells.

On the use of a component mode synthesis technique to investigate the effects of elastic boundary conditions on the transmission loss of baffled plates

In this paper, it is shown that the structural and acoustic response of an elastically-supported baffled plate excited by a plane wave or a diffuse field can be solved efficiently using a free-interface component mode synthesis technique, in which the elastic support is considered as a mass-less structural component characterized solely by a flexibility matrix. The presented method is applied to the calculation of the transmission loss of plates with different boundary stiffness and is shown to be well adapted to parameter studies and optimization. The convergence of the technique is investigated and an example of parameter study is presented.

Effects of imperfections on the buckling response of compression-loaded composite shells

The results of an experimental and analytical study of the effects of initial imperfections on the buckling and postbuckling response of three unstiffened thin-walled compression-loaded graphite-epoxy cylindrical shells with different orthotropic and quasi-isotropic shell-wall laminates are presented. The results identify the effects of traditional and non-traditional initial imperfections on the non-linear response and buckling loads of the shells. The traditional imperfections include the geometric shell-wall mid-surface imperfections that are commonly discussed in the literature on thin shell buckling. The non-traditional imperfections include shell-wall thickness variations, local shell-wall ply-gaps associated with the fabrication process, shell-end geometric imperfections, non-uniform applied end loads, and variations in the boundary conditions including the effects of elastic boundary conditions. A high-fidelity non-linear shell analysis procedure that accurately accounts for the effects of these traditional and non-traditional imperfections on the non-linear responses and buckling loads of the shells is described. The analysis procedure includes a non-linear static analysis that predicts stable response characteristics of the shells and a non-linear transient analysis that predicts unstable response characteristics.

Thermodynamic Theory of Single‐Crystal Lead Titanate with Consideration of Elastic Boundary Conditions

A phenomenological free‐energy function including the effects of elastic boundary conditions was presented and used to investigate the single‐domain, single‐crystal properties of the ferroelectric perovskite, PbTiO3. In particular, the effects of tensile and compressive hydrostatic stress on the spontaneous polarization, Curie point, dielectric susceptibility, and piezoelectric property coefficients were examined. The calculated shift of the Curie point with hydrostatic stress, along with the entropy and enthalpy of the ferroelectric‐paraelectric phase transition, was found to be in good agreement with experimental measurements. The isothermal variation of the relative dielectric susceptibility and piezoelectric coefficients with hydrostatic stress exhibited the expected behavior near the ferroelectric‐paraelectric phase transition.

Effect of elastic boundary conditions on morphotropic Pb(Zr,Ti)O3 piezoelectrics.

Most of the technologically important transducers are made of poled-polycrystalline lead zirconate--lead titanate Pb(Zr,Ti)${\mathrm{O}}_{3}$ compositions near the morphotropic phase boundary where the [Zr]:[Ti] ratio is approximately 1:1. The effect of elastic boundary conditions on single-domain stability and properties of compositions near the morphotropic phase boundary in the ${\mathrm{PbZrO}}_{3}$-${\mathrm{PbTiO}}_{3}$ phase diagram is of considerable interest, since in a polycrystalline ensemble, the elastic and electric boundary conditions are uncertain. In this work, a phenomenological free-energy function which we developed earlier is used to determine the influence of elastic boundary conditions upon the relative stability points and the rhombohedral (R3m)--tetragonal (P4mm) degeneracy required for the morphotropy at [Zr]:[Ti] ratio near 1:1.