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.     

非线性方程组求解的混沌最小二乘法及平面四杆函数机构综合

机构学问题的数学模型常可化为多元非线性方程组，一般求解多元非线性方程组需要初始值，而初始值的选择是相当困难的；同伦方法不需初始值就能求出全部解，为求解这一问题提供了可行的方法，但需要编写专用的程序，且计算量比较大；本文结合MATLAB6．5．1高级程序设计语言采用简单的最小二乘法迭代，并将非线性方程视为非线性的动力学系统，利用使得系统产生混沌的Julia集的点求解方程的全实数解，而Julia集的点集用二周期逆像函数求得，再在其邻域内求解即可。运用该算法编写了MATLAB程序，对平面四杆机构综合问题进行了研究，从而找到了实现最大精确点时该问题的全部的解，为实际机构的设计提供了多种选择方案，为机构学设计提供了全新的方法。     

机构综合的实数同伦方法研究

机构学问题的数学模型常可化为多元非线性方程组，一般求解多元非线性方程组需要初始值，而初始值的选择是相当困难的，同伦方法不需初始值就能求出全部解，为求解这一问题提供了可行的方法，但计算工作量大，需要编写专用的程序。同伦方法构造初始方程，可方便地求出其初始解，运用初始解为初始值，不需要构造同伦函数就可以求出非线性方程组的全部解或大部分实数解，这一发现虽不能理论证明，但可方便应用于机构学问题的求解中。我们运用这一发现，结合MAPLE与MATLAB编制了计算程序，对平面四杆机构的函数发生器综合问题进行了研究，从而找到了实现最大精确点时该问题的全部的解，为实际机构的设计提供了多种选择方案，为同伦方法提供了简便的实现方法。     

纤维增强FGM梁的自由振动和临界屈曲载荷分析

基于经典梁理论（CBT）研究轴向力作用下纤维增强功能梯度材料（FGM）梁的横向自由振动和临界屈曲载荷问题。首先考虑由混合律模型来表征纤维增强FGM梁的材料属性,其次利用Hamilton原理推导轴向力作用下纤维增强FGM梁横向自由振动和临界屈曲载荷的控制微分方程,并应用微分变换法（DTM）对控制微分方程及边界条件进行变换,计算了纤维增强FGM梁在固定-固定（C-C）、固定-简支（C-S）和简支-简支（S-S）3种边界条件下横向自由振动的无量纲固有频率和无量纲临界屈曲载荷。退化为各向同性梁和FGM梁,并与已有文献结果进行对比,验证了本文方法的有效性。最后讨论在不同边界条件下纤维增强FGM梁的刚度比、纤维体积分数和无量纲压载荷对无量纲固有频率的影响以及各参数对无量纲临界屈曲载荷的影响。     

牛顿混沌迭代方法及四杆导引机构综合应用举例

机构学问题的数学模型常可化为多元非线性方程组，一般求解多元非线性方程组需要初始值，而初始值的选择是相当困难的。同伦方法不需初始值就能求出全部解，为求解这一问题提供了可行的方法，但需要培写专用的程序，且计算量比较大。该文结合MATLAB7．1高级程序设计语言采用简单的牛顿迭代法迭代，并将非线性方程视为非线性的动力学系统，利用使得系统产生混沌的Julia集的点求解方程的全实数解，而Julia集的点集用二周期逆像函数求得，再在其邻城内求解即可。运用该算法培写了MATLAB程序，并以平面四杆刚体导引机构综合问题为例进行了运算，找到了实现最大精确点时该问题的全部解，为实际机构的设计提供了多种选择方案，为机构学设计提供了一种迭代新算法。     

An approximate load estimation method for three dimensional metal forming problems

Slip line field solutions have been presented only for plane strain and axisymmetric problems in the literature. In fact due to the very nature of the differential equations for the slip line field theory, it has never been possible to apply the theory to three dimensional problems. In this paper a three dimensional solution is presented for metal forming processes using an approximate load estimation method based on the slip line field theory. This has been done by extending the axisymmetric solution and modifying it in order to apply it to the three dimensional case. Using the idea of stream lines and stream surfaces, the deforming region has been defined. A generic stream surface was developed which was a ruled surface with a three dimensional shape. The slip line field theory could only be applied to flat surfaces in the case of axisymmetric problem. Here using some assumptions an approximate load estimation method was developed so that it was applied to the three dimensional stream surfaces and hence to the deforming region under considerations. This method was applied to the forward extrusion of elliptical sections from round billets. Experiments were also carried out by authors to verify the theory. The results obtained from the new formulations were compared to the results obtained from other analytical, numerical and experimental methods. It was shown that there exist good agreements between these results.     

Efficiency analysis of straight fin with variable heat transfer coefficient and thermal conductivity

In this study, one type of applicable analytical method, differential transformation method (DTM), is used to evaluate the efficiency and behavior of a straight fin with variable thermal conductivity and heat transfer coefficient. Fins are widely used to enhance heat transfer between primary surface and the environment in many industrial applications. The performance of such a surface is significantly affected by variable thermal conductivity and heat transfer coefficient, particularly for large temperature differences. General heat transfer equation related to the fin is derived and dimensionalized. The concept of differential transformation is briefly introduced, and then this method is employed to derive solutions of nonlinear equations. Results are evaluated for several cases such as: laminar film boiling or condensation, forced convection, laminar natural convection, turbulent natural convection, nucleate boiling, and radiation. The obtained results from DTM are compared with the numerical solution to verify the accuracy of the proposed method. The effects of design parameters on temperature and efficiency are evaluated by some figures. The major aim of the present study, which is exclusive for this article, is to find the effect of the modes of heat transfer on fin efficiency. It has been shown that for radiation heat transfer, thermal efficiency reaches its maximum value.     

Transverse vibration of a uniform Euler-Bernoulli beam under varying axial force using differential transformation method

This paper presents the application of techniques of differential transformation method (DTM) to analyze the transverse vibration of a uniform Euler-Bernoulli beam under varying axial force. The governing differential equation of the transverse vibration of a uniform Euler-Bernoulli beam under varying axial force is derived and verified. The varying axial force was extended to the more general case which was high polynomial consisted of many terms. The concepts of DTM were briefly introduced. Numerical calculations are carried out and compared with previous published results. The accuracy and the convergence in solving the problem by DTM are discussed.     

Large deflection of constant curvature cantilever beam under follower load

This paper describes a method to analyze for the large deflections of curved prismatic cantilever beams with uniform curvature subjected to a follower load at the tip. The large deflection, the deflection dependent follower load and the initial curved geometry are the important features of the beam considered in this work. Shear force formulation proposed by Lee [Large deflections of cantilever beams of non-linear elastic material under a combined loading. Int J Non-Linear Mech 2002;37(3)] is used for deriving the governing equations. Using this approach, the resulting two point boundary value problem (TPBVP) can be reduced to an initial value problem (IVP). Fourth order Runge-Kutta method along with one parameter reverse shooting method is applied to the numerical solution to the problem. A novel approach presented in this paper of integrating from the free end to the fixed end of the cantilever beam simply replaces the two parameter shooting with a single parameter shooting yielding several advantages. This solution technique is demonstrated for various types of follower tip loads on curved and straight cantilever beams and is validated with existing solutions in the literature.     

Buckling analysis of laminated plates using the extended Kantorovich method and a system of first-order differential equations

The extended Kantorovich method using multi-term displacement functions is applied to the buckling problem of laminated plates with various boundary conditions. The out-of-plane displacement of the buckled plate is written as a series of products of functions of parameter x and functions of parameter y. With known functions in parameter x or parameter y, a set of governing equations and a set of boundary conditions are obtained after applying the variational principle to the total potential energy of the system. The higher order differential equations are then transformed into a set of first-order differential equations and solved for the buckling load and mode. Since the governing equations are first-order differential equations, solutions can be obtained analytically with the out-of-plane displacement written in the form of an exponential function. The solutions from the proposed technique are verified with solutions from the literature and FEM solutions. The bucking loads correspond very well to other available solutions in most of the comparisons. The buckling modes also compare very well with the finite element solutions. The proposed solution technique transforms higher-order differential equations to first-order differential equations, and they are analytically solved for out-of-plane displacement in the form of an exponential function. Therefore, the proposed solution technique yields a solution which can be considered as an analytical solution.     

Free vibration analysis of arbitrary shaped thick plates by differential cubature method

In this paper, a new numerical solution technique, the differential cubature method, is applied to solve the free vibration problems of arbitrary shaped thick plates. The basic idea of the differential cubature method is to express a linear differential operation such as a continuous function or any order of partial derivative of a multivariable function, as a weighted linear sum of discrete function values chosen within the overall domain of a problem. By using the differential cubature procedure, the governing differential equations and boundary conditions are transformed into sets of linear homogeneous algebraic equations. This is an eigenvalue problem, of which the eigenvalues can be calculated numerically. The subspace iterative method is employed in search of the free vibration frequency parameters. Detailed formulations are presented, and the method is examined here for its suitability for solving the vibration problems of moderately thick plates governed by Mindlin shear deformation theory. The applicability, efficiency and simplicity of the method are demonstrated through solving some example plate vibration problems of different shapes. The numerical accuracy of the method is ascertained by comparing the vibration frequency solutions with those of existing literatures.     

Non-linear bending of thin, ideally elastic rods

A number of methods are available for the solution of elastica problems including the analytical elliptic-integral approach, various predictor-corrector methods and discrete analyses based on non-linear finite-element theory. In this paper an alternative discrete approach is proposed based on obtaining, by Dynamic Relaxation, finite difference solutions to the governing differential equations.Results from the method are presented for the large deflection behaviour of a cantilever beam and a circular ring and satisfactory correlation is demonstrated with the results of previously published exact analyses.     

Numerical experiments on determination of spatially concentrated time-varying loads on a beam: an iterative regularization method

This paper treats a solution for the ill-posed (inverse) load determination problem for a time-varying load on a beam. The ill-posed nature of the problem causes numerical instability. Conventional numerical approach for solutions results in arbitrarily large errors in solution. The Tikhonov regularization method, which is a non-iterative stabilization technique, has been widely adopted for overcoming the ill-posed nature (or numerical instability). However, in this paper, we introduce an “iterative” regularization method, specifically, the iterated Tikhonov regularization method. The iterated method is applied to the present load determination problem. The result of the iterative method is compared with that of the (non-iterative) Tikhonov regularization. The rate of convergence for the introduced iterative method turned out to be very fast. The accuracy and applicability of the introduced method are examined through a numerical experiment.     

混沌最小二乘法及其在机构近似综合中的应用

结合MATLAB6.5.1高级程序设计语言采用简单的最小二乘法迭代,并将非线性方程视为非线性的动力学系统,利用使系统产生混沌的Julia集的点求解方程的全实数解,而Julia集的点集用二周期逆像函数求得,再在其邻域内求解即可.运用该算法编写了MATLAB程序,对平面四杆机构近似综合问题进行了研究,从而找到了实现最大精确点时该问题的全部的解,为实际机构的设计提供了多种选择方案,为机构学设计提供了全新的方法.     

A heuristic method for the combined location routing and inventory problem

The combined location routing and inventory problem (CLRIP) is used to allocate depots from several potential locations, to schedule vehicles’ routes to meet customers’ demands, and to determine the inventory policy based on the information of customers’ demands, in order to minimize the total system cost. Since finding the optimal solution(s) for this problem is a nonpolynomial (NP) problem, several heuristics for searching local optima have been proposed. However, the solutions for these heuristics are trapped in local optima. Global search heuristic methods, such as tabu search, simulated annealing method, etc., have been known for overcoming the combinatorial problems such as CLRIP, etc. In this paper, the CLRIP is decomposed into two subproblems: depot location-allocation problem, and routing and inventory problem. A heuristic method is proposed to find solutions for CLRIP. First of all, an initial solution for CLRIP is determined. Then a hybrid heuristic combining tabu search with simulated annealing sharing the same tabu list is used to improve the initial solution for each subproblem separately and alternatively. The proposed heuristic method is tested and evaluated via simulation. The results show the proposed heuristic method is better than the existing methods and global search heuristic methods in terms of average system cost.     

Comparison between three optimization methods for the minimization of maximum bending load and springback in wiping die bending obtained by an experimental approach

The sheet metal bending process is widely used in the automotive industries, and it is actually one of the most important manufacturing processes. The robustness and the reliability of the bending operation, like many other forming operations, depend of several parameters (geometry, material, and process). In this paper, the die radius and the clearance between the punch and the sheet are optimised in order to reduce the maximum bending load and the springback. Two optimization problems are formulated, and three optimization procedures based on the response surface method are proposed and used to find the optimum solutions. Global and local approximations are used to replace the initial optimization problem, which is implicit by an explicit problem, and the optimum is localised using two algorithms: a sequential quadratic programming and an evolution strategies. The objective functions are evaluated experimentally into a limited points number, which are defined using a design of experiments technique. Good results are obtained from the three optimization procedures. The ability of each technique to find the optimal solution is evaluated, and the results show a good agreement between those three methods.     

Vibration analysis of the plates subject to distributed dynamic loads by using spectral element method

A spectral element method (SEM) is introduced for the vibration analysis of rectangular plates under distributed dynamic loads. In this paper, the spectral plate element matrix (often called the dynamic stiffness matrix) is formulated from the relation between the forces and displacement along the opposite two parallel edges. The distributed dynamic load is discretized into a sequence of equivalent line loads. The plate is then considered as a connection of two spectral plate element with the joint node line along which the equivalent line load acts. The spatial coordinate dependence of each equivalent line load is then removed through the spatial Fourier transformation so that the plate (2-D) problem becomes a simplified equivalent beam like (1-D) problem. The remaining solution procedures is therefore the same as that used for beam problems. Numerical tests show that the present SEM provides very accurate solutions when compared to finite element solutions.     

A simple sixth order numerical method for solving the Reynolds equation

A numerical iterative method is given for the solution of the Reynolds equation subject to separable boundary conditions. The iteration error for a given eigenfunction is sixth order in the step size. The method has been tested in the case of the finite exponential bearing pad for which exact analytical solutions are available. The dimensionless load capacity per unit width, W, can be calculated to 0.1% accuracy if ten terms are used in the sum for W. Only the first five terms need be calculated using the iteration, thereafter, analytical approximations given in the article may be used. Not more than twenty steps are needed in any given iteration to obtain the accuracy stated above.Calculations of the dimensionless load capacities of plane bearings show that the method is at least one order of magnitude faster than the method based on a two-dimensional rectangular array of points. The superiority is even more marked when calculations are made for a family of plane bearings having the same inclination but different breadth/length ratios.     

Analysis of annular Reissner/Mindlin plates using differential quadrature method

The axisymmetric flexure responses of moderately thick annular plates under static loading are investigated. The shear deformation is considered using the first-order Reissner/Mindlin plate theory and the solutions are obtained using the differential quadrature (DQ) method. In the solution process, the governing differential equations and boundary conditions for the problem are initially discretized by the DQ algorithm into a set of linear algebraic equations. The solutions of the problem are then determined by solving the set of algebraic equations. This study considers the plate subjected to various combinations of clamped, simply-supported, free and guided boundary conditions and different loading manners. The accuracy of the method is demonstrated through direct comparison of the present results with the corresponding exact solutions available in the literature.     

Non-linear analysis of skew thin plate by finite difference method

This paper deals with a discrete analysis capability for predicting the geometrically nonlinear behavior of skew thin plate subjected to uniform pressure. The differential equations are discretized by means of the finite difference method which are used to determine the deflections and the in-plane stress functions of plates and reduced to several sets of linear algebraic simultaneous equations. For the geometrically non-linear, large deflection behavior of the plate, the non-linear plate theory is used for the analysis. An iterative scheme is employed to solve these quasi-linear algebraic equations. Several problems are solved which illustrate the potential of the method for predicting the finite deflection and stress. For increasing lateral pressures, the maximum principal tensile stress occurs at the center of the plate and migrates toward the corners as the load increases. It was deemed important to describe the locations of the maximum principal tensile stress as it occurs. The load-deflection relations and the maximum bending and membrane stresses for each case are presented and discussed.