Analytical elastostatic stiffness modeling of parallel manipulators considering the compliance of the link and joint

This paper discusses the analytical elastostatic stiffness modeling of parallel manipulators (PMs) considering the compliance of the link and joint. The proposed modeling is implemented in three steps: (1) the limb constraint wrenches are formulated based on screw theory; (2) the strain energy of the link and the joint is formulated using material mechanics and a mapping matrix, respectively, and the concentrated limb stiffness matrix corresponding to the constraint wrenches is obtained by summing the strain energy of the links and joints in the limb; and (3) the overall stiffness matrix is assembled based on the deformation compatibility equations. The strain energy factor index (SEFI) is adopted to describe the influence of the elastic components on the stiffness performance of the mechanism. Matrix structural analysis (MSA) using Timoshenko beam elements is applied to obtain analytical expressions for the compliance matrices of different joints through a three-step process: (1) formulate the element stiffness equation for each element; (2) extend the element stiffness equation to obtain the element contribution matrix, allowing the extended overall stiffness matrix to be obtained by summing the element contribution matrices; and (3) determine the stiffness matrices of joints by extracting the node stiffness matrix from the extended overall stiffness matrix and then releasing the degrees of freedom of twist. A comparison with MSA using Euler–Bernoulli beam elements demonstrates the superiority of using Timoshenko beam elements. The 2PRU-UPR PM is presented to illustrate the effectiveness of the proposed approach. Finally, the global SEFI and scatter matrix are used to identify the elastic component with the weakest stiffness performance, providing a new approach for effectively improving the stiffness performance of the mechanism.     

Semi‐analytical integration of the 8‐node plane element stiffness matrix using symbolic computation

The semi‐analytical integration of an 8‐node plane strain finite element stiffness matrix is presented in this work. The element is assumed to be super‐parametric, having straight sides. Before carrying out the integration, the integral expressions are classified into several groups, thus avoiding duplication of calculations. Symbolic manipulation and integration is used to obtain the basic formulae to evaluate the stiffness matrix. Then, the resulting expressions are postprocessed, optimized, and simplified in order to reduce the computation time. Maple symbolic‐manipulation software was used to generate the closed expressions and to develop the corresponding Fortran code. Comparisons between semi‐analytical integration and numerical integration were made. It was demonstrated that semi‐analytical integration required less CPU time than conventional numerical integration (using Gaussian‐Legendre quadrature) to obtain the stiffness matrix. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Methods of analysing ropes. The extension–torsion method

Two new approaches are used for calculating the stress–strain state of a rope and its stiffnesses. The first approach relies on the theory of fibrous composites and Saint-Venant's solution for a cylinder with helical anisotropy. The second approach is based on the solution by the finite element method of the three-dimensional problem of elasticity theory for a solid inhomogeneous cylinder formed by a finite number of elastic fibres arranged in helical lines and connected by a weak filler (in the sense that its Young's modulus is several orders of magnitude less than the Young's modulus of the fibre). The behaviour of the stiffness when the modulus of elasticity of the filler tends to zero is analysed, and the results of the limiting transition are discussed. The numerical results obtained are compared with calculations by other well-known applied theories.     

Semi‐analytical integration of the elastic stiffness matrix of an axisymmetric eight‐noded finite element

The finite element method (FEM) is a numerical method for approximate solution of partial differential equations with appropriate boundary conditions. This work describes a methodology for generating the elastic stiffness matrix of an axisymmetric eight‐noded finite element with the help of Computer Algebra Systems. The approach is described as “semi analytical” because the formulation mimics the steps taken using Gaussian numerical integration techniques. The semianalytical subroutines developed herein run 50[percnt] faster than the conventional Gaussian integration approach. The routines, which are made publically available for download,¹ should help FEM researchers and engineers by providing significant reductions of CPU times when dealing with large finite element models. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Exact integration formulas for the finite volume element method on simplicial meshes

This article considers the technological aspects of the finite volume element method for the numerical solution of partial differential equations on simplicial grids in two and three dimensions. We derive new classes of integration formulas for the exact integration of generic monomials of barycentric coordinates over different types of fundamental shapes corresponding to a barycentric dual mesh. These integration formulas constitute an essential component for the development of high‐order accurate finite volume element schemes. Numerical examples are presented that illustrate the validity of the technology. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

ℋ︁‐matrices for the convection‐diffusion equation

Hierarchical matrices (ℋ︁‐matrices) provide a technique for the sparse approximation of large, fully populated matrices. This technique has been shown to be applicable to stiffness matrices arising in boundary element method applications where the kernel function displays certain smoothness properties. The error estimates for an approximation of the kernel function by a separable function can be carried over directly to error estimates for an approximation of the stiffness matrix by an ℋ︁‐matrix, using a certain standard partitioning and admissibility condition for matrix blocks. Similarly, ℋ︁‐matrix techniques can be applied in the finite element context where it is the inverse of the stiffness matrix that is fully populated. Here one needs a separable approximation of Green's function of the underlying boundary value problem in order to prove approximability by matrix blocks of low rank. Unfortunately, Green's function for the convection‐diffusion equation does not satisfy the required smoothness properties, hence prohibiting a straightforward generalization of the separable approximation through Taylor polynomials. We will use Green's function to motivate a modification in the (hierarchical) partitioning of the index set and as a consequence the resulting hierarchy of block partitionings as well as the admissibility condition. We will illustrate the effect of the proposed modifications by numerical results.     

Effective moduli of elasticity of a composite composed of anisotropic layers

Relations are obtained for the effective moduli of elasticity and Poisson's ratios of a laminated fiber-reinforced composite, each layer of which has at least orthorhombic symmetry. The elastic properties of the composite in terms of the elastic constants of the layer are expressed exactly, and the elastic constants of the individual layer in terms of the values for the fiber and the matrix are expressed approximately. Two approximations are considered: one corresponds to the Hashin-Shtrikman variational approach, while in the second the comparison material is assigned elastic properties equal to the Voigt or Reuss means of the values for each layer. A numerical example is worked for the combination boron fibers-epoxy resin. The results of the calculation are compared with the exact solution of the problem for a composite composed of alternating layers of boron and epoxy resin.     

Exact solutions for coupled analysis of thin-walled functionally graded beams with non-symmetric single- and double-cells

In the present work, the exact solutions for coupled analysis for bending and torsional case thin-walled functionally graded (FG) beams with non-symmetric single- and double-cells are presented for the first time. For this purpose, an accurate and efficient method is proposed to obtain the FG member stiffness matrix based on the series expansions of displacement components. Three types of material distributions are considered and the beam mechanical properties are graded along the wall thickness according to a power law of the volume fraction. The present beam model is on the basis of the Euler-Bernoulli beam theory and the Vlasov one for bending and torsional problems, respectively. The explicit expressions for displacement parameters are derived using the power series approach from the four coupled equilibrium equations. Finally, the FG member stiffness matrix is determined from the seven force-displacement relations. In order to show the accuracy and super convergence of the thin-walled FG beam element developed by this study, the numerical solutions are presented and compared with results obtained from the finite beam element based on the approximate interpolation polynomials and other available results. Especially, the effects of various structural parameters such as material distribution type, volume fraction index, boundary condition, and material ratio on the spatially coupled responses of FG box beams with non-symmetric single- and double-cells are parametrically investigated.     

Sensitivity analysis of weakened beams on elastic foundation with Green's functions

The approach of Sensitivity Analysis with Green's Functions (SAGF) [1,2] was developed to predict changes in deformations, stresses or eigenfrequencies of structures resulting from stiffness modifications or cracking by considering only the weakened or damaged parts of the structures. This approach results in a local analysis instead of a global analysis by recalculating the whole structure. Consequently, it is computationally less time-consuming than the conventional methods based on a global analysis. The key idea of the SAGF approach is based on the comparison between the elastic strain energies of the original and the weakened structures and the substitution of the virtual displacements by the corresponding Green's functions [1, 2]. Furthermore, an approximate approach for the sensitivity analysis was suggested which is described in [1, 3] in details. This approach enables us to predict the changes in the structural responses due to the stiffness weakening in the beam or in the elastic Winkler foundation by considering only the internal forces or the deflections of the original unweakened system. In addition, an iterative method was developed to enhance the accuracy of the SAGF approach. In this paper, the local SAGF method for sensitivity analysis of elastic beams on Winker foundation with stiffness weakening is presented. The accuracy and the efficiency of the proposed method are verified by using a numerical example. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

轴对称弹性体的有限元分析

轴对称弹性力学问题的有限元分析长期以来都是采用三角圆环有限元和线性形状函数.由于积分困难,常用近似积分求得刚度矩阵,这种近似积分对于靠近旋转对称轴的元素,误差很大,所以,长期以来,被认为不满意的办法.也有用精确积分计算刚度矩阵的.但本文指出,这种积分只适用于有中孔的轴对称体.对于实心的轴对称体而言,这种刚度矩阵都不收敛,计算是无效的.本文提出了一种新的形状函数,当径向座标r接近于零时,这种形状函数的径向位移u自然地接近于零.如果用这种新的形状函数,则由此计算求得的刚度矩阵,不论三角圆环有限元的位置是否靠近轴线.都是存在的.这种有限元,就能用于计算实心的轴对称体的问题.     

非线性分析中的全量刚度矩阵与增量刚度矩阵

本文详细导出了非线性分析中的全量(割线)刚度矩阵和增量(切线)刚度矩阵的一般表达式,并由此进一步讨论了它们二者之间的数学关系.本文的结果对于非线性方程的求解,及非线性、线性稳定性分析都具有重要帮助作用.     

The modified model of Koiter’s type for nonlinearly elastic shell

In this paper, we proposed a modified model of Koiter’s type for nonlinearly elastic shell. The change of metric tensor and the change of curvature tensor play an important role in constructing the linearly and nonlinearly elastic shell model of Koiter’s type. The approximate expressions of them once were proposed by Ciarlet. In this paper, the exact full expressions of the change of metric tensor and the change of curvature tensor are provided by tensor analysis. The former coincides with Ciarlet’s expression. And the latter is more exact than Ciarlet’s expression. Thus the modified model is better than Ciarlet’s model. At the same time, a numerical experiment of special hemispherical shell is provided to validate the modified model of Koiter’s type.     

Exact relations for effective tensors of composites: Necessary conditions and sufficient conditions

Typically the elastic and electrical properties of composite materials are strongly microstructure dependent. So it comes as a nice surprise to come across exact formulae for effective moduli that are universally valid no matter what the microstructure. Such exact formulae provide useful benchmarks for testing numerical and actual experimental data and for evaluating the merit of various approximation schemes. They can also be regarded as fundamental invariances existing in a given physical context. Classic examples include Hill's formulae for the effective bulk modulus of a two‐phase mixture when the phases have equal shear moduli, Levin's formulae linking the effective thermal expansion coefficient and effective bulk modulus of two‐phase mixtures, and Dykhne's result for the effective conductivity of an isotropic two‐dimensional polycrystalline material. Here we present a systematic theory of exact relations embracing the known exact relations and establishing new ones. The search for exact relations is reduced to a search for matrix subspaces having a structure of special Jordan algebras. One of many new exact relations is for the effective shear modulus of a class of three‐dimensional polycrystalline materials. We present complete lists of exact relations for three‐dimensional thermoelectricity and for three‐dimensional thermopiezoelectric composites that include all exact relations for elasticity, thermoelasticity, and piezoelectricity as particular cases. © 2000 John Wiley & Sons, Inc.     

Finite expressions for the characteristic matrices of weakly curved elastic layers

The propagation of two-dimensional waves in cylindrical and spherical, homogeneous, elastic layers is investigated. For these layers finite formulas are found for the characteristic matrices. Comparison of these matrices and use of asymptotic representations for the Hankel functions make it possible to derive expressions in the case of weakly curved elastic layers. The expressions obtained correspond to analogous formulas in the form of matrix series.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 156–169, 1981.     

Static and dynamic analysis of cracked or weakened structures

Stiffness modifications in engineering structures, for example due to damage and cracking, will inevitably also lead to changes in deformations, internal forces, natural frequencies and mode shapes of the structures. In this paper, an efficient and simple method for sensitivity analysis of cracked or weakened structures under time-harmonic loading is presented. The method is based on a comparison between the strain energy and the kinetic energy of an uncracked structure and that of a cracked structure in conjunction with the application of exact or approximate Green's functions as described in [3] for the static case. The present analysis enables the prediction of any changes in the displacements and stresses and has a lower computational effort as compared to available classical methods, because only the damaged region has to be re-considered in the method. Green's functions are taken as a basis of the approach, which have the ability to weight the influence of the stiffness modifications in a region of a structure and show how sensitive other regions respond to the stiffness modifications. Based on linear elastic fracture mechanics, cracked or damaged regions are approximated by spring models in the analytical solution of some simple beam problems, while cracked finite elements are used for complicated cases where analytical solutions cannot be obtained. Sensitivity analysis with Green's functions (SAGF) approach is applied to static and dynamic analysis of cracked and weakened structures, which consist of homogeneous materials or fiber reinforced composites like reinforced concretes. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The variability of dynamic responses of beams resting on elastic foundation subjected to vehicle with random system parameters

This paper investigates the variability of dynamic responses of a beam resting on an elastic foundation, which is subjected to a vehicle with uncertain parameters, such as random mass, stiffness, damping of the vehicle and random fields of mass density, and the elastic modulus of the beam and stiffness of elastic foundation. The vehicle is modeled as a two-degree-of-freedom spring-damper-mass system. The equations of motion of the beam was constructed using a finite element method. The mass and elastic properties of the beam, and the stiffness of foundation are assumed to be Gaussian random fields and were simulated by the spectral represent method. Masses, stiffness of the spring, and the damping coefficient of the vehicle are assumed as Gaussian random variables. The numerical analyses were performed using the finite element method (FEM) in conjunction with the Monte Carlo simulation (MCS). The variability of dynamic responses of the beam were investigated with various cases of random parameters. For each sample, the equations of motions were solved with the Wilson-q integral method to find dynamic responses. The influence of random system parameters and their correlation on the response variability is discussed in detail.     

Efficient numerical methods for initial-value solid-state laser problems

The difficulties of solving initial-value solid-state laser problems numerically arise from both stiffness of the problems and near-to-zero nonnegative exact solutions. Stability and non-negativity must be maintained simultaneously in the numerical solutions. Backward differentiation formulas (BDFs) is capable of dealing with stiff problems ,but is of small oscillation when time-step is large. Therefore unfortunately BDFs suffers from severe time-step restriction . In this paper,we present an optimized numerical approach, with which 3-dimensional laser problems can be solved faster and much more efficiently. These techniques can not only be used for solid-state laser systems, but can also be applied to solve other stiff problems which have near-to-zero nonnegative exact solutions. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact frequency response of two-node coupled bending-torsional beam element with attachments

This paper addresses the frequency response of coupled bending-torsional beams carrying an arbitrary number of in-span viscoelastic dampers and attached masses. Using the elementary coupled bending-torsion theory, along with appropriate generalized functions to treat the discontinuities of the response variables at the application points of dampers/masses, exact analytical expressions are derived for the frequency response of the beam under harmonically-varying, arbitrarily-placed point/polynomial loads. On this basis, the exact 6 × 6 dynamic stiffness matrix and 6 × 1 load vector of a two-node coupled bending-torsional beam finite element, with any number of in-span dampers/masses and harmonic loads, are obtained in a closed analytical form. Finally, the modal frequency response functions of the beam are built by a complex modal analysis approach, upon deriving pertinent orthogonality conditions for the modes. In this context, the modal impulse response functions are also obtained for time-domain analysis under arbitrary loads.     

Quadrature for meshless Nitsche's method

In this article, we study effect of numerical integration on Galerkin meshless method (GMM), applied to approximate solutions of elliptic partial differential equations with essential boundary conditions (EBC). It is well‐known that it is difficult to impose the EBC on the standard approximation space used in GMM. We have used the Nitsche's approach, which was introduced in context of finite element method, to impose the EBC. We refer to this approach as the meshless Nitsche's method (MNM). We require that the numerical integration rule satisfies (a) a “discrete Green's identity” on polynomial spaces, and (b) a “conforming condition” involving the additional integration terms introduced by the Nitsche's approach. Based on such numerical integration rules, we have obtained a convergence result for MNM with numerical integration, where the shape functions reproduce polynomials of degree k ≥ 1. Though we have presented the analysis for the nonsymmetric MNM, the analysis could be extended to the symmetric MNM similarly. Numerical results have been presented to illuminate the theoretical results and to demonstrate the efficiency of the algorithms.Copyright © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 265–288, 2014