Frequency characteristics of a thin rotating cylindrical shell using the generalized differential quadrature method

In this paper, the generalized differential quadrature (GDQ) method is used for the first time to study the effects of boundary conditions on the frequency characteristics of a thin rotating cylindrical shell. The present analysis is based on Love-type shell theory and the governing equations of motion include the effects of initial hoop tension and the centrifugal and coriolis accelerations due to rotation. The displacement field is expressed as a product of unknown smooth continuous functions in the meridional direction and trigonometric functions along the circumferential direction so that the three-dimensional dynamic problem may be transformed mathematically into a one-dimensional problem. Based on this approach, the results are obtained for the effects of the boundary conditions on the frequency characteristics at different circumferential wave numbers and rotating speeds and various geometric properties; the effect of rotating speed on the relationship between frequency parameter and circumferential wave number is also discussed. To validate the accuracy and efficiency of the GDQ method, the results obtained are compared with those in the literature and very good agreement is achieved.     

Generalized differential quadrature for free vibration of rotating composite laminated conical shell with various boundary conditions

By using the generalized differential quadrature (GDQ) method, this paper presents the orthotropic influence of composite materials on frequency characteristics for a rotating thin truncated circular symmetrical cross-ply laminated composite conical shell with different boundary conditions. The present governing equations of free vibration include the effects of initial hoop tension and the centrifugal and Coriolis accelerations due to rotation. Frequency characteristics are obtained to study in detail the orthotropic influences. Effects of boundary condition, rotating speed, circumferential wave number and geometric property are also discussed. To ensure the accuracy of the present results by the GDQ method, comparisons are made with those available in open literature and very good agreements are achieved.     

考虑气体粘性的径流式叶轮流场计算

本文为通过考虑真实粘性气体特性改进径流式叶轮设计计算方法的一次尝试。第一部分给出了全张量形式的叶轮机械粘性气体流动方程组,从这一普通形式出发,还给出了粘性气体沿任意曲线方向的速度梯度方程;第二部分具体介绍纯径流式叶轮中的粘性流动方程组及其解法,其中简化模型的由来和求解时边界条件的处理是两个主要关键;第三部分介绍典型例题的计算结果与分析,初步结果是满意的。     

Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.     

Refined theory for vibrations and buckling of laminated isotropic annular plates

Hamilton's variational principle is used for the derivation of equations of transversally isotropic laminated annular plates motion. Nonlinear strain—displacements relations are considered. Linearized vibration and buckling equations are obtained for the annular plates uniformly compressed in the radial direction. The effects of transverse shear and rotational inertia are included. A closed form solution is given for the mode shapes in terms of Bessel, power and trigonometric functions. The eigenvalue equations are derived for natural frequencies and buckling loads of annular and circular plates elastically restrained against rotation along edges. Classical-type plate theory results are obtained then by letting the transverse shear stiffness go to infinity and rotational inertia go to zero. Numerical examples are presented by tables and figures for 2- and 3-layered plates with various geometrical and physical parameters. The transverse shear, rotational inertia and boundary conditions effects are discussed.     

Local adaptive differential quadrature for free vibration analysis of cylindrical shells with various boundary conditions

This paper presents the formulation and numerical analysis of circular cylindrical shells by the local adaptive differential quadrature method (LaDQM), which employs both localized interpolating basis functions and exterior grid points for boundary treatments. The governing equations of motion are formulated using the Goldenveizer–Novozhilov shell theory. Appropriate management of exterior grid points is presented to couple the discretized boundary conditions with the governing differential equations instead of using the interior points. The use of compactly supported interpolating basis functions leads to banded and well-conditioned matrices, and thus, enables large-scale computations. The treatment of boundary conditions with exterior grid points avoids spurious eigenvalues. Detailed formulations are presented for the treatment of various shell boundary conditions. Convergence and comparison studies against existing solutions in the literature are carried out to examine the efficiency and reliability of the present approach. It is found that accurate natural frequencies can be obtained by using a small number of grid points with exterior points to accommodate the boundary conditions.     

FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations

In this paper, the generalized differential quadrature (GDQ) method is applied to study the dynamic behavior of functionally graded materials (FGMs) and laminated doubly curved shells and panels of revolution with a free-form meridian. The First-order Shear Deformation Theory (FSDT) is used to analyze the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner-Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. Firstly, the differential quadrature (DQ) rule is introduced to determine the geometric parameters of the structures with a free-form meridian. Secondly, the discretization of the system by means of the GDQ technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner-Mindlin and the Toorani-Lakis theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed. Finally, different lamination schemes are considered to expand the combination of the two functionally graded four-parameter power-law distributions adopted. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the lamina thickness direction. A two-constituent functionally graded lamina consists of ceramic and metal those are graded through the lamina thickness. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of shell and panel structures considered.     

Vibrations characteristics of joined cylindrical-spherical shell with elastic-support boundary conditions

Based on the Reissner-Naghdi’s thin shell theory, this paper concentrates on the free vibration of a joined cylindrical-spherical shell with elastic-support boundary conditions by a domain decomposition method. The joined shell is first separated from the elastic-support boundary and then subdivided into some cylindrical and spherical shell segments along the axis of revolution. The elastic-support boundary is regarded as a combination of distributed linear springs and can be treated as a special interface as well as the interface between two adjacent shell segments. Through the variational operation of the whole energy functional, the discretized equations of motion are derived by the expansions of the displacement field for each shell segment with Fourier series and Chebyshev orthogonal polynomials in the circumferential and longitudinal direction, respectively. To analyze the numerical convergence and precision of the present method, a number of case studies have been conducted and the solutions are compared with those derived by ANSYS and those presented in the previous literature to confirm the reliability and accuracy.     

Dynamic relaxation method for nonlinear buckling analysis of moderately thick FG cylindrical panels with various boundary conditions

Using new approach proposed by Dynamic relaxation (DR) method, buckling analysis of moderately thick Functionally graded (FG) cylindrical panels subjected to axial compression is investigated for various boundary conditions. The mechanical properties of FG panel are assumed to vary continuously along the thickness direction by the simple rule of mixture and Mori-Tanaka model. The incremental form of nonlinear formulations are derived based on First-order shear deformation theory (FSDT) and large deflection von Karman equations. The DR method combined with the finite difference discretization technique is employed to solve the incremental form of equilibrium equations. The critical mechanical buckling load is determined based on compressive load-displacement curve by adding the incremental displacements in each load step to the displacements obtained from the previous ones. A detailed parametric study is carried out to investigate the influences of the boundary conditions, rule of mixture, grading index, radius-to-thickness ratio, length-to-radius ratio and panel angle on the mechanical buckling load. The results reveal that with increase of grading index the effect of radius-to-thickness ratio on the buckling load decreases. It is also observed that effect of distribution rules on the buckling load is dependent to the type of boundary conditions.

     

Sensitivity analysis of crack detection in beams by wavelet technique

This paper examines the sensitivity of wavelet technique in the detection of cracks in beam structures. Specifically, the effects of different crack characteristics, boundary conditions, and wavelet functions employed are investigated. Crack characteristics studied include the length, orientation and width of slit. The two different boundary conditions considered are simply supported and fixed end support, and the two types of wavelets compared in this study are the Haar and Gabor wavelets. The results show that the wavelet transform is a useful tool in detection of cracks in beam structures. The dimension of the crack projected along the longitudinal direction can be deduced from the analysis. The method is sensitive to the curvature of the deflection profile and is a function of the support condition. For detection of discrete cracks, Haar wavelets exhibit superior performance.     

Three-dimensional analysis of thermal stresses in four-parameter continuous grading fiber reinforced cylindrical panels

In this paper, three-dimensional analysis of thermal stresses in four-parameter continuous grading fiber reinforced cylindrical panel subjected to thermal load is studied. The cylindrical panel is assumed to be made of an orthotropic material. The continuous grading fiber reinforced panel has a smooth variation in matrix volume fraction in the radial direction. A generalization of the power-law distribution presented in literature is proposed. Symmetric and asymmetric volume fraction profiles are presented in this paper. Suitable temperature and displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the thermoelastic equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by generalized differential quadrature method. The fast rate of convergence of the method is demonstrated and to validate the results, comparisons are made with the available solutions for orthotropic shells. The main contribution of this work is to illustrate the influence of the power-law exponent, of the power-law distribution choice and of the choice of the four parameters on the thermal behaviour of continuous grading fiber reinforced cylindrical panels.     

The unified equations to obtain the exact solutions of piezoelectric plane beam subjected to arbitrary loads

The unified equations to obtain the exact solutions for piezoelectric plane beam subjected to arbitrary mechanical and electrical loads with various ends supported conditions is founded by solving functional equations. Comparing this general method with traditional trial-and-error method, the most advantage is it can obtain the exact solutions directly and does not need to guess and modify the form of stress function or electric displacement function repeatedly. Firstly, the governing equation for piezoelectric plane beam is derived. The general solution for the governing equation is expressed by six unknown functions. Secondly, in terms of boundary conditions of the two longitudinal sides of the beam, six functional equations are yielded. These equations are simplified to derive the unified equations to solve the boundary value problems of piezoelectric plane beam. Finally, several examples show the correctness and generalization of this method.     

求解复杂轴扭振动力特性的一种近似分析方法

应用模态摄动法求解非均质变截面复杂轴的扭转振动的模态特性。在这一方法中,选择与复杂轴约束条件相同、等截面均匀轴的扭转振动解析模态函数作为近似分析的基函数,采用Ritz展开和摄动分析相结合的方法,把复杂轴扭振的变系数微分方程的求解,转化为一组代数方程组的求解。虽然这一方法应用了结构振动分析中常用的Ritz展开原理,但避免了关于模态广义坐标的特征值问题的求解,通过代数方程组的求解,可逐一求得复杂轴的扭转振动的模态特性,简化了求解的难度。通过2个算例的分析表明,该方法简单实用,且具有较高的精度。     

Vibration of composite plates with internal supports

The free vibration of rectangular laminated composite plates with arbitrary support conditions along the edges, internal line supports and discrete point supports are studied using the Rayleigh-Ritz method. Polynomial approximation functions are selected to satisfy all essential boundary conditions along the edges of the plate and to vanish along line supports parallel to the co-ordinate axes. Straight line supports at an angle from the co-ordinate axes and curved line supports are modeled by introducing several point supports along the line. Zero displacement constraints at the point support locations are enforced using the Lagrange multiplier technique. The plate constitutive equations are expressed in terms of stiffness invariants and the fundamental natural frequency is maximized by selecting the appropriate lay-up. Several examples are presented to illustrate the versatility of the approach and provide results not previously available. The influence of the number of plies in the laminate, lay-up, material properties and plate aspect ratios are investigated.     

Analysis of the vibration behavior of FGM cylindrical shells including internal pressure and ring support effects based on Love-Kirchhoff theory with various boundary conditions

This paper presents the study on vibration behavior of functionally graded material (FGM) cylindrical shell with the effects of internal pressure and ring support. The FGM properties are graded along the thickness direction of the shell. The FGM shell equations with internal pressure and ring support are established based on strain-displacement relationship using Love-Kirchhoff shell theory. The governing equations of motion were solved by using energy functional and by applying Ritz method. The boundary conditions represented by end conditions of the FGM cylindrical shell are simply supported-simply supported (SS-SS), clamped-clamped (C-C), free-free (F-F), clamped-free (C-F), clamped-simply supported (C-SS), free-simply supported (F-SS), free-sliding (F-SL) and clamped-sliding (C-SL). To check the validity and accuracy of the present method, the results obtained are compared with those available in the literature. The influence of internal pressure, ring support position and the effect of the different boundary conditions on natural frequencies characteristics are studied. These results presented can be used as important benchmark for researchers to validate their numerical methods when studying natural frequencies of shells with internal pressure and ring support.     

Buckling of symmetrically laminated rectangular plates under parabolic edge compressions

Buckling analysis of symmetrically laminated rectangular plates with parabolic distributed in-plane compressive loadings along two opposite edges is performed using the Rayleigh-Ritz method. Classical laminated plate theory is adopted. Stress functions satisfying all stress boundary conditions are constructed based on the Chebyshev polynomials. Displacement functions for buckling analysis are constructed by Chebyshev polynomials multiplying with functions that satisfy either simply supported or clamped boundary condition along four edges. Methodology and procedures are worked out in detail. Buckling loads for symmetrically laminated plates with four combinations of boundary conditions are obtained. The proposed method is verified by comparing results to data obtained by the differential quadrature method (DQM) and the finite element method (FEM). Numerical example also shows that the double sine series displacement for simply supported symmetrically laminated plates having bending-twisting coupling may overestimate the stiffness, thus providing higher buckling loads.     

一种分析薄壁箱梁剪滞效应的高精度法

考虑了三个不同的剪滞纵向位移差函数以反映薄壁箱梁不同宽度翼板的剪滞变化幅度 ,提出一种分析薄壁箱梁剪滞效应的高精度法。应用能量变分原理 ,导出了箱梁受横向荷载作用下的剪滞控制微分方程和边界条件 ,获得闭合解 ,通过高阶有限条法计算验证了本文方法的正确性。并探讨了不同纵向位移差函数对剪力滞的影响。     

凸轮机构通用样条函数运动规律的研究

样条函数运动规律具有良好的通用性和适应性 ,可以不受数量限制地增加运动控制条件 ,能够满足大多数凸轮机构运动设计的需要 ,并非常适用于凸轮机构的动力学综合。本文对样条函数在凸轮机构运动规律方面的应用作了深入的研究 ,给出了样条函数运动规律的通用表达式、实现方法及各种情况下的边界条件 ,开发了适用于各种情况的通用样条函数运动规律设计软件 ,并给出了应用实例     

Stress singularities at angular corners in first-order shear deformation plate theory

The first-known Williams-type singularities caused by homogeneous boundary conditions in the first-order shear deformation plate theory (FSDPT) are thoroughly examined. An eigenfunction expansion method is used to solve the three equilibrium equations in terms of displacement components. Asymptotic solutions for both moment singularity and shear-force singularity are developed. The characteristic equations for moment singularity and shear-force singularity and the corresponding corner functions due to ten different combinations of boundary conditions are explicated in this study. The validity of the present solution is confirmed by comparing with the singularities in the exact solution for free vibrations of Mindlin sector plates with simply supported radial edges, and with the singularities in the three-dimensional elasticity solution for a completely free wedge. The singularity orders of moments and shear forces caused by various boundary conditions are also thoroughly discussed. The singularity orders of moments and shear forces are compared according to FSDPT and classic plate theory.     

Modelling and simulation of the superelastic behaviour of shape memory alloys using the element-free Galerkin method

This paper is concerned with the application of the element-free Galerkin method to simulate the superelastic behaviour of shape memory alloys (SMA). The meshfree shape functions are derived from a moving least-squares interpolation scheme. A thermomechanical SMA constitutive law is used to describe the superelastic effect. The incremental displacement-based formulation for large deformation is developed by employing the meshfree shape functions and the continuum tangent stiffness tensor in the weak form of the equilibrium equations. By eliminating the unknown constrained nodal variables from the discrete equations, an effective approach is developed for the imposition of the essential boundary conditions. The numerical tests show that the proposed meshfree scheme can successfully reproduce the superelastic behaviour of shape memory alloys.