A trapezoidal Fourier p-element for membrane vibrations

A trapezoidal Fourier p-element for the analysis of membrane transverse vibrations is investigated. Trigonometric functions are used as shape functions instead of polynomials to avoid ill-conditioning problems. The element matrices are analytically integrated in closed form. With the enrichment degrees of freedom in Fourier series, the accuracy of natural frequencies obtained is increased in a stable manner. One element can predict many modes accurately. Since a triangle can be divided into three trapezoidal elements, the range of application is wider than the previously derived rectangular Fourier p-element. The natural modes of a square membrane consisting of two trapezoidal elements are computed as test cases and convergence is very fast with an increasing number of trigonometric terms. Comparison of natural modes calculated by the trapezoidal Fourier p-element and the conventional finite elements is carried out. The results show that the trapezoidal Fourier p-element produces higher accurate natural frequencies than the conventional finite elements with the same number of degrees of freedom.     

Two-dimensional viscoelastic vibration by analytic Fourier p-elements

Three new Fourier p-elements of rectangular, skew and trapezoidal shapes are given analytically for plane viscoelastic vibration problems. The natural frequencies of the plane viscoelastic structures with complex Young’s modulus are computed by a complex eigenvalue solver. With the additional Fourier degrees of freedom, the accuracy of the computed natural frequencies is greatly increased. Since trigonometric functions are used as enriching functions instead of polynomials in the proposal elements, the ill-conditioning problems associated with polynomials of higher degree in the traditional p-version finite element method are avoided. The two mapped plane coordinates in the Jacobian are uncoupled for trapezoidal elements whose element matrices can then be integrated analytically. A triangle can easily be divided into three trapezoids. Therefore, any plane viscoelastic problem with polygonal shape can be analyzed by a combination of rectangular and trapezoidal elements. Numerical examples show that the convergence of the present elements is very fast with respect to the number of trigonometric terms. The natural frequencies of several polygonal viscoelastic plates subject to in-plane vibration are presented.     

Free vibration analysis of circular thin plates with stepped thickness by the DSC element method

Novel formulation is presented by using the discrete singular convolution (DSC) for free vibration analysis of circular thin plates with uniform and stepped thickness. Different from the commonly used ones in literature, regularity conditions are not needed at the circular plate center point to avoid singularity. DSC circular and annular thin plate elements are established. For the DSC circular plate element with radius of R₁, the stiffness equation is first formulated in region [−R₁, R₁] with even number of nodes and then reduced to region [0, R₁] by using either symmetric or anti-symmetric conditions. The proposed DSC circular and annular plate elements are used for obtaining frequencies of uniform/stepped circular thin plates or annular thin plates with different boundary conditions. Comparison of the present DSC results to existing analytic and numerical solutions verifies the proposed formulations. The present research extends the DSC method to free vibration of circular thin plates with stepped thicknesses.     

Lepidium sativum natural seed plant extract in the structural and physical characteristics of polyvinyl alcohol

A virgin sample of pure polyvinyl alcohol (PVA) in combination with other polymeric samples containing pre-calculated amounts of lepidium sativum natural plant seed extract was successfully prepared. Structural modification was investigated through a combination of both Fourier transformation infrared (FT-IR) combined with X-ray diffraction (XRD). Both techniques indicate interaction and complexation of the polymeric matrix through changes in vibrational bands at 1331 and 849 cm^?1 owing to the CH₂ wagging vibration and carbon stretching vibrations respectively. EDAX analysis supported with atomic absorption experimental data were used to investigate the structure and type of elements present in the natural seeds. Obtained data explain the complexation process between the polymeric material and plant extract attributed to the presence of Fe⁺² ions even in ppm level.     

General analysis and application to redundant arches under static loading

The aim of this paper is to define an exact formulation of a curved beam finite element for static analysis. The basic equations are combined in the coupled fundamental system in terms of radial displacement v, tangential displacement u and rotation ϕ. An original procedure for solving the fundamental system of equations is used. A finite element formulation based on shape functions that satisfy the homogeneous form of the fundamental system of differential equations is developed. The effects of bending moment, axial extension and transverse shear are taken into account. The exact elastic solution renders the element obtained free of shear and membrane locking.An efficient numerical procedure is presented for determining the pressure curve in the case of circular arches under static loading and arbitrary bonding conditions. The solution obtained is applicable to the analysis of both thin and thick curved beams.Several examples of arches with various loading and boundary conditions are investigated to illustrate the validity and the accuracy of the method. Finally, the effect of the arch rise on the structural response is pointed out.     

Non-linear free vibration of elliptic sector plates by a curved triangular p-element

A curved triangular p-element is developed and applied to geometrically non-linear free vibration of isotropic elliptic sector plates. The formulation takes into account shear deformation, rotary inertia, and geometric non-linearity. The element can describe the geometry of the elliptic sector plate exactly and is therefore suitable for this type of plate. The shifted Legendre orthogonal polynomials are used as enriching hierarchic shape functions. The element stiffness and mass matrices are integrated numerically using the Gauss–Legendre quadrature. The non-linear equations of free motion are obtained using the harmonic balance method and solved iteratively by the linearized updated mode method. Numerical results for the linear and non-linear frequencies of clamped elliptic sector plates are obtained. The efficiency of the proposed p-element is demonstrated through convergence studies and comparisons with published results. The effects of the ellipse eccentricity, thickness ratio, and sector angle on the backbone curves are examined. It is shown that these parameters influence the hardening behavior.     

Finite strip method for the free vibration and buckling analysis of plates with abrupt changes in thickness and complex support conditions

The free vibration problem of a stepped plate supported on non-homogeneous Winkler elastic foundation with elastically mounted masses is formulated based on Hamilton's principle. The stepped plate is modelled by finite strip method. To overcome the problem of excessive continuity of common beam vibration functions at the location of abrupt change of plate thickness, a set of C¹ continuous functions have been chosen as the longitudinal interpolation functions in the finite strip analysis. The C¹ continuous functions are obtained by augmenting the relevant beam vibration modes with piecewise cubic polynomials. As these displacement functions are built up from beam vibration modes with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as the appropriate order of continuity. The method is further extended to the buckling analysis of rectangular stepped plates. Numerical results also show that the method is versatile, efficient and accurate.     

Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open cross section

A finite element formulation is present for the nonlinear free vibration of thin-walled curved beams with non-symmetric open across section. The kinetic and potential energies are derived by the virtual principle. The energy functional includes the effect of flexural–torsional coupling, the torsion warping and the shear center location. For finite element analysis, cubic polynomials are utilized as the shape functions of the two nodal thin-walled curved elements. Each node possesses seven degrees freedom including the warping degree of freedom. The nonlinear eigenvalue problem has been solved by the direct iteration technique. The results are compared with those for straight beams as available in the literature. The results for nonlinear free vibration analysis of curved beams for various radii and subtended angle are presented.     

Free vibration of non-uniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods

In this research, the free vibration analysis of cylindrical shells with circumferential stiffeners, i.e. rings with non-uniform stiffeners eccentricity and unequal stiffeners spacing is investigated using analytical, experimental and finite elements (FE) methods. Ritz method is applied in analytical solution while stiffeners treated as discrete elements. The polynomial functions are used for Ritz functions and natural frequency results for simply supported stiffened cylindrical shell with equal rings spacing and constant eccentricity is compared with other's analytical and experimental results, which showed good agreement. Also, a stiffened shell with unequal rings spacing and non-uniform eccentricity with free–free boundary condition is considered using analytical, experimental and FE methods. In experimental method, modal testing is performed to obtain modal parameters, including natural frequencies, mode shapes and damping in each mode. In FE method, two types of modeling, including shell and beam elements and solid element are used, applying ANSYS software. The analytical and the FE results are compared with the experimental one, showing good agreements. Because of insufficient experimental modal data for non-uniformly stiffeners distribution, the results of modal testing obtained in this study could be as useful reference for validating the accuracy of other analytical and numerical methods for free vibration analysis.     

Free vibration analysis of arbitrarily shaped membranes using the trigonometric p-version of the finite-element method

The trigonometric p-version of the finite-element method is formulated and applied to free-vibration analysis of arbitrarily shaped membranes. This is accomplished using a trigonometrically enriched curved quadrilateral element. New shape functions are expressed in terms of trigonometric sine functions. These functions are numerically more stable than orthogonal polynomials as the order is increased. In this method, large elements are used and curved segments are represented accurately by using the blending function method. The accuracy of the solution is sought by increasing the trigonometric order while fixing the mesh. Frequency values are first found for a closed sectorial membrane and comparisons are made with exact values. Frequency values are also found for an open sectorial membrane and comparisons are made with exact values and with values from linear and isoparametric quadratic quadrilateral finite-element solutions. Furthermore, highly accurate frequency values are found for a square membrane with a central circular opening. It is first shown that the method converges rapidly downwards as the trigonometric order is increased. It is also shown that, for equivalent numbers of d.o.f.s, the method gives a much higher accuracy than linear and isoparametric quadrilateral finite-element solutions.     

非对称开口截面薄壁圆形曲梁的非线性自由振动分析

为具有非对称开口截面的薄壁曲梁的非线性自由振动分析构建了一个有限元公式,由虚功原理推导出动能和势能,并考虑了弯曲-扭转耦合、扭曲和剪心位置等影响。有限元分析中,2结点薄壁曲线构件的形状函数采用三次多项式计算。每个结点具有7个自由度,其中包括扭曲自由度。采用直接迭代法计算非线性特征值。将计算结果与直线梁的计算结果相对比。同时归纳了具有多种半径和对角的曲梁的非线性自由振动分析结果。     

四边支承矩形薄板自振频率计算

从弹性薄板的基本振动微分方程出发,利用可以满足各边边界条件的振型函数,根据能量法推导出四边支承六种不同边界条件的矩形薄板的最低自振频率计算公式,为楼板结构的自振频率计算和舒适性设计提供了指导。     

Static and dynamic analysis of thin plates in bending— A novel finite element model

The static and dynamic behavior of thin flat plates in bending have been studied by means of a recently developed¹ doubly curved triangular shell element. The element's formulation is based on a modified mixed variational principle, wherein the primal variable σ (vector of shell stress resultants) and (boundary displacement vector) are required to satisfy a priori: (1) the complete shallow shell equilibrium equations, and (2) interelement C¹ displacement continuity. Several well-selected plate structures have been analyzed and the numerical results obtained indicate that the new element scheme competes most favorably with recently developed as well as with well-established elements included in commercial general-purpose finite element codes.     

Robust control of plate vibration via active constrained layer damping

In this paper, the theoretical modeling of a plate partially treated with active constrained layer damping (ACLD) treatments and its vibration control in an H_∞ approach is discussed. Vibration of the flat plate is controlled with patches of ACLD treatments, each consisting of a viscoelastic damping layer which is sandwiched between the piezo-electric constrained layer and the host plate. The piezo-electric constrained layer acts as an actuator to actively control the shear deformation of the viscoelastic damping layer according to the vibration response of the plate excited by external disturbances. In the first part of this paper, the Mindlin–Reissner plate theory is adopted to express the shear deformation characteristics of the viscoelastic damping layer, meanwhile GHM (Golla–Hughes–McTavish) model of viscoelastic damping material and FEM (finite element model) are incorporated to describe the dynamics of the plate partially treated with ACLD treatment. In the second part, particular emphasis is placed on the vibration control of the first four modes of the treated plate using H_∞ robust control method. For this purpose, an H_∞ robust controller is designed to accommodate uncertainties of the ACLD parameters, particularly those of the viscoelastic damping core which arise from the variation of the operation temperature and frequency. Disturbances and measurement noise are rejected in the closed loop by H_∞ robust controller. In the experimental validation, external disturbances of different types are employed to excite the treated plate. The results of the experimental clearly demonstrate that the proposed modeling method is correct and the ACLD treatments are very effective in fast damping out the structural vibration as compared to the conventional passive constrained layer damping (PCLD).     

采用高斯径向基函数和一阶剪切变形理论分析普通复合材料叠合板的固有频率

采用高斯径向基函数和一阶剪切变形理论对普通复合材料叠合板的固有频率进行了计算。采用几个数值计算案例来说明此法的收敛性和准确性。结果发现:当采用形状参数c=1·7×NS(NS为每一侧的节点数)计算固有频率时,收敛速度最快,而且与现有文献中的结果一致。这同时也证明了采用高斯径向基函数对普通复合材料层合板进行振动分析的高度准确性。     

斗栱

中国古建筑以木结构闻名于世，而木结构的精髓在斗栱。斗栱是一系列木构件的组合。斗和栱是这一系列构件中最基本的两个。介绍了斗栱的组成、型制以及型制所涉及的几个重要概念。阐述了斗栱的作用，不仅能提高梁、枋的承载力，还能与梁柱构成一个延性很好的节点，减震消能，而且装饰效果很强。     

Timoshenko悬臂梁自由振动特性的近似分析方法

本文讨论了用模态摄动法分析Timoshenko悬臂梁的自由振动特性。Timoshenko梁是目前较为流行的梁模型之一,Timoshenko梁方程一般较复杂,很难给出解的解析形式。利用模态摄动法,将Timoshenko梁方程转化为一组非线性代数方程组的求解,不仅可以简化计算过程,而且计算结果具有较高的精度,对任意边界条件下的Timoshenko梁都适用。通过算例,讨论了剪切变形和转动惯量对悬臂梁的固有频率的影响。     

The effects of phenol and some alkyl phenolics on batch anaerobic methanogenesis

Phenol and seven alkylphenols (o-, m- and p-cresol, 2.5-, 2.6-, 3.4- and 3,5-dimethylphenol) were added at various concentrations to aliquots of domestic anaerobic sludge in Hungate serum bottles and these were incubated at 37°C. The concentration of methane in the headspace gas was monitored to determine if the phenolics were fermented to methane or if they inhibited the anaerobic process. Only phenol and p-cresol were fermented to methane. At 500 mg l⁻¹ (but not at 300 mg l⁻¹) 2,5-, 3,4- and 3,5-dimethylphenol reduced the rate and the amount of methane produced. The cresols were inhibitory at 1000 mg l⁻¹ but not at 400 mg l⁻¹.In cultures supplemented with acetate and propionate (VOA), and in unsupplemented cultures, phenol at concentrations up to 500 mg l⁻¹ was fermented to methane. Between 800 and 1200 mg l⁻¹ phenol, methane production was neither enhanced nor inhibited relative to control cultures containing no phenol. Inhibition of methane production was evident when phenol was present at 2000 mg l⁻¹. Thus the methanogens are less susceptible to phenol inhibition than are the phenol-degrading acid formers. In similar experiments with p-cresol: enhanced methane production was observed at concentrations of 400 mg l⁻¹; no enhancement or inhibition was observed at 600 mg l⁻¹; and inhibition was noted when p-cresol was present at 1000 mg l⁻¹.     

Dynamic stiffness analysis of toroidal shells

The free vibration of a toroidal shell is studied using the dynamic stiffness method. The dynamic stiffness method eliminates both spatial discretization error and mesh generation. Moreover, with a finite number of degrees of freedom, the dynamic stiffness method can predict an infinite number of natural frequencies. The dynamic behavior of the toroidal shell is modeled by DMV (Donnell-Mushtari-Vlasov) linear thin shell theory in the present paper. However, the procedure can be adapted to be used with any other linear thin shell theory without difficulty. Since a close form solution of toroidal shell using DMV theory is not (yet) possible, in order to obtain the desired dynamic stiffness matrix, a finite number of Fourier's series terms are taken in the circumferential direction and the unknown longitudinal displacements are then solved from the reduced governing equations exactly. The solution obtained from the dynamic stiffness method can be regarded as semi-analytical due to the Fourier approximation. With the dynamic stiffness matrices in hands, a toroidal shell with different boundary conditions and connections (to other toroidal shells) can be analyzed. This paper presents the procedure and assumption made in order to obtain the dynamic stiffness matrix of a toroidal shell in harmonic oscillation. Also some numerical examples will be given and discussed.     

Free vibration analysis of cracked thin plates by quasi-convex coupled isogeometric-meshfree method

The free vibration analysis of cracked thin plates via a quasi-convex coupled isogeometric-meshfree method is presented. This formulation employs the consistently coupled isogeometric-meshfree strategy where a mixed basis vector of the convex B-splines is used to impose the consistency conditions throughout the whole problem domain. Meanwhile, the rigid body modes related to the mixed basis vector and reproducing conditions are also discussed. The mixed basis vector simultaneously offers the consistent isogeometric-meshfree coupling in the coupled region and the quasi-convex property for the meshfree shape functions in the meshfree region, which is particularly attractive for the vibration analysis. The quasi-convex meshfree shape functions mimic the isogeometric basis function as well as offer the meshfree nodal arrangement flexibility. Subsequently, this approach is exploited to study the free vibration analysis of cracked plates, in which the plate geometry is exactly represented by the isogeometric basis functions, while the cracks are discretized by meshfree nodes and highly smoothing approximation is invoked in the rest of the problem domain. The efficacy of the present method is illustrated through several numerical examples.