Transverse vibrations of arbitrary non-uniform beams

Free and steady state forced transverse vibrations of non-uniform beams are investigated with a proposed method, leading to a series solution. The obtained series is verified to be convergent and linearly independent in a convergence test and by the non-zero value of the corresponding Wronski determinant, respectively. The obtained solution is rigorous, which can be reduced to a classical solution for uniform beams. The proposed method can deal with arbitrary non-uniform Euler-Bernoulli beams in principle, but the methods in terms of special functions or elementary functions can only work in some special cases.     

Longitudinal waves in one dimensional non-uniform waveguides

Wave approach is used to analyze the longitudinal wave motion in one dimensional non-uniform waveguides. With assumptions of constant wave velocity and no wave conversion, there exist four types of non-uniform rods and corresponding traveling wave solutions are investigated. The obtained results indicate that the kinetic energy is preserved as a constant and the wave amplitude is inversely proportional to square root of the cross-sectional area of the rod. Under certain condition, there exists a cut-off frequency for the rod with variation in geometric or material properties, below which waves do not propagate along the non-uniform rod. For the rod with arbitrary variable cross-section, the conclusions are similar if the wave frequency is high enough. And a series solution of the wave motion is presented.     

Implementation of Convergence in Adaptive Global Digital Image Correlation

In FE based global digital image correlation (DIC) a finite element mesh is used to describe the deformation of the region of interest (ROI). However, the identification of an optimal mesh is a difficult problem and is often obtained by using “mechanical” pre-knowledge of the solution. In Finite Element Analysis (FEA) an optimal mesh can be found without any pre-knowledge of the solution by using mesh adaptivity, where an initial (non optimal) mesh is refined until the optimal solution is obtained. Refinement of the mesh can be based on error and/or convergence estimators. Despite the fundamental differences between FEA and DIC, in the present article the convergence procedure is successfully used in a recently published global FE based DIC method. In the used global DIC method elements can receive higher order shape functions, also known as p-elements. Using the aforementioned algorithm, also called p-DIC, refinement to a non-uniform higher order mesh is possible. Using the non-uniform mesh, an optimal mesh can be obtained for each section of the ROI. The presented study shows that a convergence scheme can be used to automatically control the mesh refinement in a global DIC approach. The convergence boundary, in percentage, is a more intuitive boundary than the absolute error boundary used in the original p-DIC approach. The procedure is validated using numerical examples and the robustness to experimental variables is investigated. Finally, the complete procedure is tested against a wide range of practical examples.     

由基模态构造任意支撑杆的多项式型轴向刚度

给出了当杆的横截面积均匀而材料线密度为已知多项式时，由基模态构造任意支撑方式下杆的多项式型的轴向刚度系致的方法，证明了所得轴向刚度的正值性．     

几何精确NURBS有限元中边界条件施加方式对精度影响的三维计算分析

非均匀有理B样条(NURBS)有限元法把计算机辅助几何设计(CAGD)中的NURBS几何构形方法与有限元方法有机结合起来,有效消除了有限元离散模型的几何误差,提高了计算精度。但是由于NURBS基函数不是插值函数,直接在控制节点上施加位移边界条件会引起较大误差。本文详细讨论了NURBS基函数的插值特性,在NURBS有限元分析中采用罚函数法施加位移边界条件,提高了收敛率和计算精度。结合典型三维弹性力学问题,对两种施加位移边界条件的方法进行了对比和分析。计算结果表明,直接施加位移边界条件会导致收敛率和精度的明显降低,而基于罚函数法的NURBS有限元分析则能达到最优收敛率,并具有更高的精度。     

Local stress singularities in mixed axisymmetric problems of the bending of circular cylinders

A method of analyzing the near-edge stress state in mixed problems of the deformation of an isotropic cylindrical body is proposed. The method is based on the expansion of the solution of three-dimensional problems of elasticity into a series of Lurie–Vorovich homogeneous basis functions. An asymptotic analysis is performed to find the principal part of the solution of the infinite systems of linear algebraic systems to which the problems are reduced. The type of the stress singularity at the edge of the cylinder is the same as in the mixed problems for a quarter plane. Kummer’s convergence acceleration method is used. The obtained results are validated by testing the boundary conditions and by comparing with results obtained by other authors     

KdV-Burgers方程的级数解

给出ＫｄＶ－Ｂｕｒｇｅｒｓ方程的有界行波解的精确级数解,采用Ａｄｏｍｉａｎ算子分解法分别求行二个区域ζ〈０和ζ〉０的级数解,然后利用对接连续条件构成整体级数解。所得级数解能精确满足对接连续条件,并由此得到确定级数的系数递推公式,无需解非红性高阶代数方程组。与某些精确解及其它方法比较,计算简捷具在对接点处是收剑的。对某些非线性波动现象的研究,可作为计算和分析的数学依据。     

Three-dimensional magneto-thermo-elastic analysis of functionally graded cylindrical shell

The present paper presents the three-dimensional magneto-thermo-elastic analysis of the functionally graded cylindrical shell immersed in applied thermal and magnetic fields under non-uniform internal pressure. The inhomogeneity of the shell is assumed to vary along the radial direction according to a power law function, whereas Poisson's ratio is supposed to be constant through the thickness. The existing equations in terms of the displacement components, temperature, and magnetic parameters are derived, and then the effective differential quadrature method(DQM) is used to acquire the analytical solution. Based on the DQM, the governing heat differential equations and edge boundary conditions are transformed into algebraic equations, and discretized in the series form. The effects of the gradient index and rapid temperature on the displacement,stress components, temperature, and induced magnetic field are graphically illustrated.The fast convergence of the method is demonstrated and compared with the results obtained by the finite element method(FEM).     

基于无网格法的Timoshenko曲梁动力分析

曲梁具有外形美观、受力性能良好的优点,故在工程中得到广泛应用。本文基于移动最小二乘近似和一阶剪切变形理论,提出一种对Timoshenko曲梁自由振动和受迫振动进行分析的无网格方法。通过一系列离散点离散曲梁,建立曲梁无网格模型,然后推导曲梁势能和动能方程,通过哈密顿原理给出曲梁自由振动和受迫振动的控制方程,因为本文方法不能直接施加边界条件,所以使用完全转换法处理本质边界条件,最后求解方程得到频率和振动模态。文末通过算例验证了本文方法的有效性,且通过收敛性分析表明本文方法具有较好的收敛性,并进一步分析了不同边界条件、跨高比和变截面变曲率对曲梁自由振动和受迫振动的影响,将计算结果与文献解或ABAQUS解进行对比分析,表明本文方法具有较高的精度,且适用于实际工程情况。     

Free vibration of non-uniform axially functionally graded beams using the asymptotic development method

The asymptotic development method is applied to analyze the free vibration of non-uniform axially functionally graded(AFG) beams, of which the governing equations are differential equations with variable coefficients. By decomposing the variable flexural stiffness and mass per unit length into reference invariant and variant parts, the perturbation theory is introduced to obtain an approximate analytical formula of the natural frequencies of the non-uniform AFG beams with different boundary conditions.Furthermore, assuming polynomial distributions of Young's modulus and the mass density, the numerical results of the AFG beams with various taper ratios are obtained and compared with the published literature results. The discussion results illustrate that the proposed method yields an effective estimate of the first three order natural frequencies for the AFG tapered beams. However, the errors increase with the increase in the mode orders especially for the cases with variable heights. In brief, the asymptotic development method is verified to be simple and efficient to analytically study the free vibration of non-uniform AFG beams, and it could be used to analyze any tapered beams with an arbitrary varying cross width.     

Free vibration analysis of elastic rods using global collocation

This paper investigates the performance of a novel global collocation method for the eigenvalue analysis of freely vibrated elastic structures when either basis or shape functions are used to approximate the displacement field. Although the methodology is generally applicable, numerical results are presented only for rods in which one-dimensional basis functions in the form of a power series, as well as equivalent Lagrange, Bernstein or Chebyshev polynomials are used. The new feature of the proposed methodology is that it can deal with any type of boundary conditions; therefore, the cases of two Dirichlet as well as one Dirichlet and one Neumann condition were successfully treated. The basic finding of this work is that all these polynomials lead to results identical to those obtained by the power series expansion; therefore, the solution depends on the position of the collocation points only.     

Stress singularities in an anisotropic body of revolution

To fill the gap in the literature on the application of three-dimensional elasticity theory to geometrically induced stress singularities, this work develops asymptotic solutions for Williams-type stress singularities in bodies of revolution that are made of rectilinearly anisotropic materials. The Cartesian coordinate system used to describe the material properties differs from the coordinate system used to describe the geometry of a body of revolution, so the problems under consideration are very complicated. The eigenfunction expansion approach is combined with a power series solution technique to find the asymptotic solutions by directly solving the three-dimensional equilibrium equations in terms of the displacement components. The correctness of the proposed solution is verified by convergence studies and by comparisons with results obtained using closed-form characteristic equations for an isotropic body of revolution and using the commercial finite element program ABAQUS for orthotropic bodies of revolution. Thereafter, the solution is employed to comprehensively examine the singularities of bodies of revolution with different geometries, made of a single material or bi-materials, under different boundary conditions.     

Bremmer series for the multimodal sound propagation in inhomogeneous waveguides

The method of Bremmer series is presented and implemented for the multimodal sound propagation in a waveguide with varying cross-section. The solution is constructed iteratively by summing terms of increasing order, with each order representing the number of scattering events. This formulation generalizes to higher dimensions the series decomposition proposed by Bremmer in one-dimensional inhomogeneous media, and it also gives an insight on the complex wave scattering of the coupled guided modes. The accuracy and convergence of the solution are inspected and a comparison is made with the one-way coupled mode equation derived assuming no reflected waves. It is notably shown that the first-order Bremmer series, simply obtained by quadrature, is a relevant alternative to classical WKB or one-way approximations.     

Stability analysis of functionally graded rectangular plates under nonlinearly varying in-plane loading resting on elastic foundation

In this research work, an exact analytical solution for buckling of functionally graded rectangular plates subjected to non-uniformly distributed in-plane loading acting on two opposite simply supported edges is developed. It is assumed that the plate rests on two-parameter elastic foundation and its material properties vary through the thickness of the plate as a power function. The neutral surface position for such plate is determined, and the classical plate theory based on exact neutral surface position is employed to derive the governing stability equations. Considering Levy-type solution, the buckling equation reduces to an ordinary differential equation with variable coefficients. An exact analytical solution is obtained for this equation in the form of power series using the method of Frobenius. By considering sufficient terms in power series, the critical buckling load of functionally graded plate with different boundary conditions is determined. The accuracy of presented results is verified by appropriate convergence study, and the results are checked with those available in related literature. Furthermore, the effects of power of functionally graded material, aspect ratio, foundation stiffness coefficients and in-plane loading configuration together with different combinations of boundary conditions on the critical buckling load of functionally graded rectangular thin plate are studied.     

A new efficient method to evaluate exact stiffness and mass matrices of non-uniform beams resting on an elastic foundation

In this paper, a new efficient method to evaluate the exact stiffness and mass matrices of a non-uniform Bernoulli–Euler beam resting on an elastic Winkler foundation is presented. The non-uniformity may result from variable cross-section and/or from inhomogeneous linearly elastic material. It is assumed that there is no abrupt variation in the cross-section of the beam so that the Euler–Bernoulli theory is valid. The method is based on the integration of the exact shape functions which are derived from the solution of the axial deformation problem of a non-uniform bar and the bending problem of a non-uniform beam which are both formulated in terms of the two displacement components. The governing differential equations are uncoupled with variable coefficients and are solved within the framework of the analog equation concept. According to this, the two differential equations with variable coefficients are replaced by two linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under ideal load distributions. The key point of the method is the evaluation of the two ideal loads which in this work is achieved by approximating them by two polynomials. More specifically, the axial ideal load is approximated by a linear polynomial while the transverse one by a cubic polynomial. The numerical implementation of the method is simple, and the results are compared favorably to those obtained by exact solutions available in literature.     

Uniform Convergence Series to Solve Nonlinear Partial Differential Equations: Application to Beam Dynamics

Extended trigonometric series of uniform convergence are proposed as a method to solve the nonlinear dynamic problemsgoverned by partial differential equations. In particular, the method isapplied to the solution of a uniform beam supported at its ends withnonlinear rotational springs and subjected to dynamic loads. The beam isassumed to be both material and geometrically linear and the end springs are of the Duffing type. The action may be a continuous load q = q(x, t) within a certain range and/or concentrated dynamic moments at the boundaries. The adopted solution satisfies the differential equation, the initial conditions, andthe nonlinear boundary conditions. It has been previously demonstrated that, due to the uniform convergence of the series, the method yieldsarbitrary precision results. An illustration example shows theefficiency of the method.     

A General Formulation of Stress Distribution in Cylinders Subjected to Non-Uniform External Pressure

This work presents a two-dimensional stress analysis for elastic solid cylinders subjected to combined loading. The loading is generally formed with a number of concentrated and partially distributed forces all applied radially on the outer surface. The distributed forces cause pressures with non-uniform intensity along the circumferential direction. The cylinder is assumed to be long so that a state of plane-strain is valid. To obtain the stress distribution for the problem of partially distributed forces a new approach is followed first introduced in this paper. It is based on the expressions formed after using the theory of simple radial stress distribution when point-forces are applied on the cylinder and leads to the solution after direct integration. The total stresses due to both concentrated and distributed forces are obtained using the method of superposition. Apart from its simplified formulation, this general solution is always preferable since it proved to have a great advantage. As a result of not containing Fourier series, it eliminates some problems of convergence of the series at the boundaries that appear due to the Gibbs phenomena when the boundary conditions are a discontinuous function. Numerical results are presented for some interesting cases of loading conditions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Study of pure transverse motion in free cylinders and plates in flexural vibration by Ritz’s method

In this work the flexural vibration of a free cylinder of any aspect ratio is analysed. A general solution by powers series of the coordinates is proposed here to represent the displacements, with restrictions on the powers of the radial coordinates which prevent potential energy and stress singularities at the axis of the cylinder. By means of an analytic method, it is concluded that certain points of the cylinder have no axial motion. As a result of the pure transverse movement and of the fact that the cylinder bends, it is inferred that the axis is extended. Furthermore, in the symmetric modes, the points situated at the centres of the bases are displaced in the same direction and sense, and hence the distance between them does not vary in time. Flexural natural frequencies are numerically calculated by Ritz’s method with the general solution series proposed. Since the series used are more adequate, convergence is better than with classic series. The results are verified by FEM. Some consequences are extended to a rectangular plate, whose points of the middle surface vibrate transversally in the double-symmetric mode. In order to verify the theoretical results, a set of experiments with a laser interferometer is carried out. The experimental frequencies agree with the theoretical values.     

Constrained parameter-splitting perturbation method for the improved solutions to the nonlinear vibrations of Euler–Bernoulli cantilevers

In this paper, a new method named constrained parameter-splitting perturbation method for improving the solutions obtained from the parameter-splitting perturbation method is proposed for solving the problems in some extremal cases, such as the strongly nonlinear vibration of an Euler–Bernoulli cantilever. The proposed method takes the advantages of both the perturbation method and the harmonic balance method. The idea is that the solution obtained by the parameter-splitting perturbation method is substituted into the equation of motion and then the accumulative error of the equation is minimized for determining the unknown splitting parameters under the constraints constructed under the frame of harmonic balance method. The forced vibration of an oscillator with cubic geometric nonlinearity and inertia nonlinearity and the forced vibration of a planar microcantilever beam with a lumped tip mass are studied as examples to reveal the efficacy of the proposed method. The inspection of the steady-state response including its stability is conducted by means of comparing the frequency-response curves obtained by the proposed method with those obtained by the numerical continuation method and harmonic balance method, respectively, to show the efficacy and the advantages of the proposed method. Meanwhile, the nonlinear ordering effect on the solutions of the proposed method is also studied by comparing the results obtained by using different nonlinear orderings in the systems. In the last, we found through convergence examinations that it is necessary to have corrections to the erroneous solution which are obtained by harmonic balance method and Floquet theory in stability analysis.

     

Explicit solution to the exact Riemann problem and application in nonlinear shallow‐water equations

The Riemann solver is the fundamental building block in the Godunov‐type formulation of many nonlinear fluid‐flow problems involving discontinuities. While existing solvers are obtained either iteratively or through approximations of the Riemann problem, this paper reports an explicit analytical solution to the exact Riemann problem. The present approach uses the homotopy analysis method to solve the nonlinear algebraic equations resulting from the Riemann problem. A deformation equation defines a continuous variation from an initial approximation to the exact solution through an embedding parameter. A Taylor series expansion of the exact solution about the embedding parameter provides a series solution in recursive form with the initial approximation as the zeroth‐order term. For the nonlinear shallow‐water equations, a sensitivity analysis shows fast convergence of the series solution and the first three terms provide highly accurate results. The proposed Riemann solver is implemented in an existing finite‐volume model with a Godunov‐type scheme. The model correctly describes the formation of shocks and rarefaction fans for both one and two‐dimensional dam‐break problems, thereby verifying the proposed Riemann solver for general implementation. Copyright © 2007 John Wiley & Sons, Ltd.