Efficient Evaluation of the Elastic Forces and the Jacobian in the Absolute Nodal Coordinate Formulation

This paper develops a new procedure for evaluating the elastic forces, the elastic energy and the jacobian of the elastic forces in the absolute nodal coordinate formulation. For this procedure, it is fundamental to use some invariant sparse matrices that are integrated in advance and have the property of transforming the evaluation of the elastic forces in a matrix multiplication process. The use of the invariant matrices avoids the integration over the volume of the element for every evaluation of the elastic forces. Great advantages can be achieved from these invariant matrices when evaluating the elastic energy and calculating the jacobian of the elastic forces as well. The exact expression of the jacobian of the differential system of equations of motion is obtained, and some advantages of using the absolute nodal coordinate formulation are pointed out. Numerical results show that there is important time saving as a result of the use of the invariant matrices.     

On the analysis of simple shear problem using the micro-polar hypoelasticity Cosserat theory

In this paper, an analysis of kinematics of the isotropic elastic Cosserat continuum is presented in infinitesimal and finite deformations. Emphasis is given on the applicability of corotational stress rates for hypoelasticity in micro-polar continua. A non-linear finite element analysis is performed with an explicit formulation of tangent stiffness matrices in the case of Truesdell stress and couple stress rates. A comprehensive path-dependent procedure is employed based on the arc-length method to calculate the stability points and handle the snap-back problem. Finally, the accuracy and efficiency of method are illustrated by numerical examples.     

On the Principal Boundary Value Problem in the Two-Dimensional Anisotropic Theory of Elasticity

By definition, the principal problem of the two-dimensional theory of elasticity consists in solving the equation for the Airy’s stress function in a region with its first order derivatives assigned at a boundary. In this paper, an indirect formulation of this problem based on integral equations with weakly singular kernels is proposed. In a bounded region with a Lyapunov boundary it is reduced to the solution of weakly singular integral equations. Differential properties of its solution are investigated.     

An Internal Damping Model for the Absolute Nodal Coordinate Formulation

Introducing internal damping in multibody system simulations is important as real-life systems usually exhibit this type of energy dissipation mechanism. When using an inertial coordinate method such as the absolute nodal coordinate formulation, damping forces must be carefully formulated in order not to damp rigid body motion, as both this and deformation are described by the same set of absolute nodal coordinates. This paper presents an internal damping model based on linear viscoelasticity for the absolute nodal coordinate formulation. A practical procedure for estimating the parameters that govern the dissipation of energy is proposed. The absence of energy dissipation under rigid body motion is demonstrated both analytically and numerically. Geometric nonlinearity is accounted for as deformations and deformation rates are evaluated by using the Green–Lagrange strain–displacement relationship. In addition, the resulting damping forces are functions of some constant matrices that can be calculated in advance, thereby avoiding the integration over the element volume each time the damping force vector is evaluated.     

Modeling Method and Application of Rational Finite Element Based on Absolute Nodal Coordinate Formulation

In multibody system dynamics, the absolute nodal coordinate formulation(ANCF)uses power functions as interpolating polynomials to describe the displacement field. It can get accurate results for flexible bodies that undergo large deformation and large rotation. However, the power functions are irrational representation which cannot describe the complex shapes precisely, especially for circular and conic sections. Different from the ANCF representation,the rational absolute nodal coordinate formulation(RANCF) utilizes rational basis functions to describe geometric shapes, which allows the accurate representation of complicated displacement and deformation in dynamics modeling. In this paper, the relationships between the rational surface and volume and the RANCF finite element are provided, and the generalized transformation matrices are established correspondingly. Using these transformation matrices, a new four-node three-dimensional RANCF plate element and a new eight-node three-dimensional RANCF solid element are proposed based on the RANCF. Numerical examples are given to demonstrate the applicability of the proposed elements. It is shown that the proposed elements can depict the geometric characteristics and structure configurations precisely, and lead to better convergence in comparison with the ANCF finite elements for the dynamic analysis of flexible bodies.     

两类基于局部标架梁单元的闭锁缓解方法

对于大转动、大变形柔性体的刚柔耦合动力学问题,基于李群SE(3)局部标架(local frame formulation,LFF)的建模方法能够规避刚体运动带来的几何非线性问题,离散数值模型中广义质量矩阵与切线刚度矩阵满足刚体变换的不变性,可明显地提高柔性多体系统动力学问题的计算效率.有限元方法中,闭锁问题是导致单元收...     

Component-wise analysis of laminated anisotropic composites

This paper proposes a one-dimensional (1D) refined formulation for the analysis of laminated composites which can model single fibers and related matrices, layers and multilayers. Models built by means of an arbitrary combination of these four components lead to a component-wise analysis. Different scales can be used in different portions of the structure and this leads to a global–local approach. In this work, computational models were developed in the framework of finite element approximations. The 1D FE formulation used has hierarchical features, that is, 3D stress/strain fields can be detected by increasing the order of the 1D model used. The Carrera Unified Formulation (CUF) was exploited to obtain advanced displacement-based theories where the order of the unknown variables over the cross-section is a free parameter of the formulation. Taylor- and Lagrange-type polynomials were used to interpolate the displacement field over the element cross-section. Lagrange polynomials permitted the use of only pure displacements as unknown variables. The related finite element led straightforwardly to the assembly of the stiffness matrices at the structural element interfaces (matrix-to-fiber, matrix-to-layer, layer-to-layer etc). Preliminary assessments with solid model results are proposed in this paper; various numerical examples were carried out on cross-ply symmetrical fiber-reinforced laminates [0/90/0] and a more complex composite C-shaped model. The examples show that the proposed models can analyze laminated structures by combining fibers, matrices, layers and multilayers and by referring to a unique structural finite element formulation.     

Critical points in the formulation of pharmaceutical swellable controlled release dosage forms-Influence of particle size

Swellable matrix represents one of the most employed controlled release systems. These dosage forms provide slow release of drugs to reduce the fluctuation of drug concentration in plasma in order to improve the efficiency of treatment and/or to reduce adverse effects. The application of the concepts of statistical physics has allowed discovering the existence of critical points in the formulation of swellable matrices. These points, representing the volume fractions of the tablet components where the properties of the matrix diverge or change suddenly, provide important knowledge of how to rationalize the design of swellable matrices. The critical points are generally related to the percolation threshold of one of the components of the formulation, which corresponds to a geometrical phase transition of this component, passing from isolation to spanning the whole system. The last section of the paper is devoted to more recent findings concerning the influence of particle size of the components on the percolation threshold of the matrix forming polymer, and therefore on the release behaviour of the matrix. Knowledge of the excipient percolation threshold allows a more rational design of swellable matrices, according to the guidelines of the regulatory authorities concerning science-based formulation and quality by design.     

Applications of the dual integral formulation in conjunction with fast multipole method to the oblique incident wave problem

In this paper, the dual integral formulation is derived for the modified Helmholtz equation in the propagation of oblique incident wave passing a thin barrier (zero thickness) by employing the concept of fast multipole method (FMM) to accelerate the construction of an influence matrix. By adopting the addition theorem, the four kernels in the dual formulation are expanded into degenerate kernels that separate the field point and the source point. The source point matrices decomposed in the four influence matrices are similar to each other or only to some combinations. There are many zeros or the same influence coefficients in the field point matrices decomposed in the four influence matrices, which can avoid calculating the same terms repeatedly. The separable technique reduces the number of floating‐point operations from O((N)²) to O(N log^a(N)), where N is the number of elements and a is a small constant independent of N. Finally, the FMM is shown to reduce the CPU time and memory requirement, thus enabling us to apply boundary element method (BEM) to solve water scattering problems efficiently. Two‐moment FMM formulation was found to be sufficient for convergence in the singular equation. The results are compared well with those of conventional BEM and analytical solutions and show the accuracy and efficiency of the FMM. Copyright © 2008 John Wiley & Sons, Ltd.     

A frequency-domain FEM approach based on implicit Green’s functions for non-linear dynamic analysis

The present paper describes an efficient algorithm to integrate the equations of motion implicitly in the frequency domain. The standard FEM displacement model (Galerkin formulation) is employed to perform space discretization, and the time-marching process is carried out through an algorithm based on the Green’s function of the mechanical system in nodal coordinates. In the present formulation, mechanical system Green’s functions are implicitly calculated in the frequency domain. Once the Green’s functions related matrices are computed, a time integration procedure, which demands low computational effort when applied to non-linear mechanical systems, becomes available. At the end of the paper numerical examples are presented in order to illustrate the accuracy of the present approach.     

Nonlinear behaviour of piezoceramics under weak electric fields: Part-I: 3-D finite element formulation

Piezoceramic materials exhibit different types of nonlinearities under different combinations of electric and mechanical fields. When excited near resonance in the presence of weak electric fields, they exhibit typical nonlinearities similar to a Duffing oscillator such as jump phenomena and presence of superharmonics in the response spectra. In order to model such nonlinearities, a nonlinear electric enthalpy density function (using quadratic and cubic terms) valid for a general 3-D piezoelectric continuum has been proposed in this work. Linear (i.e. proportional) and nonlinear damping models have also been proposed. The coupled nonlinear finite element equations have been derived using variational formulation. The classical linearization technique has been used to derive the linearized stiffness and damping matrices which helps in assembling the nonlinear matrices and solution of resulting nonlinear equation. The general 3-D finite element formulation is discussed in this paper. In a companion paper by Samal et al., numerical results on various typical examples are shown to match very well with the experimental observations.     

悬索大挠度风振响应的有限体积法分析

基于Lagrange原理,建立了一套新的悬索大挠度动力特性和动力响应分析的有限体积法列式,推导了结点力向量、质量矩阵和单元刚度矩阵的显式表达式。该列式的一个显著特点是直接利用工程应变定义结构变形,其物理意义明确,列式简单,适用于各种垂度和荷载情况的悬索大挠度动力分析。实例动力特性和随机风振响应分析表明,该有限体积列式不仅计算效率高,而且具有良好的计算精度。     

Free vibration of axially loaded,rotating Timoshenko shaft systems by the wave-train closure principle

A systematic approach for the free vibration analysis of a rotating Timoshenko shaft system subjected to axial forces is presented in this paper. The system has multiple point discontinuities such as elastic supports, rotor masses, and cross-sectional changes. Wave reflection and transmission matrices are employed to characterize the wave motions between the sub-spans of the shaft system. These matrices are combined with the field transfer matrices expressed in wave forms to obtain the characteristic equation in a straightforward manner. The solutions are exact since effects of attenuating wave components are included in the formulation. The wave propagation-based matrix algebra leads to recursive algorithms which are suitable for computer coding. Three examples are presented to illustrate the numerical procedure.     

哈密顿系统正则变换在时变最优控制中的应用

利用哈密顿系统正则变换和生成函数理论求解线性时变最优控制问题,构造了新的最优控制律形式并提出了控制增益计算的保结构算法. 利用生成函数求解最优控制导出的哈密顿系统两端边值问题,并构造线性时变系统的最优控制律,由第2类生成函数所构造的最优控制律避免了末端时刻出现无穷大反馈增益. 控制系统设计中需求解生成函数满足的时变矩阵微分方程组. 根据生成函数与哈密顿系统状态转移矩阵之间的关系,从正则变换的辛矩阵描述出发,导出了求解这组微分方程组的保结构递推算法.为了保持递推计算中的辛矩阵结构,哈密顿系统状态转移矩阵的计算中利用了Magnus级数.      

Stochastic response determination of structural systems modeled via dependent coordinates: a frequency domain treatment based on generalized modal analysis

Generalized independent coordinates are typically utilized within an analytical dynamics framework to model the motion of structural and mechanical engineering systems. Nevertheless, for complex systems, such as multi-body structures, an explicit formulation of the equations of motion by utilizing generalized, independent, coordinates can be a daunting task. In this regard, employing a set of redundant coordinates can facilitate the formulation of the governing dynamics equations. In this setting, however, standard response analysis techniques cannot be applied in a straightforward manner. For instance, defining and determining a transfer function within a frequency domain response analysis framework is challenging due to the presence of singular matrices, and thus, the machinery of generalized matrix inverses needs to be employed. An efficient frequency domain response analysis methodology for structural dynamical systems modeled via dependent coordinates is developed herein. This is done by resorting to the Moore–Penrose generalized matrix inverse in conjunction with a recently proposed extended modal analysis treatment. It is shown that not only the formulation is efficient in drastically reducing the computational cost when compared to a straightforward numerical evaluation of the involved generalized inverses, but also facilitates the derivation and implementation of the celebrated random vibration input–output frequency domain relationship between the excitation and the response power spectrum matrices. The validity of the methodology is demonstrated by considering a multi-degree-of-freedom shear type structure and a multi-body structural system as numerical examples.

     

Kinematostatic modeling of compliant parallel mechanisms

The benefits of compliant mechanisms in terms of precision are not easy to exploit because of the limitations of the existing kinematic models used to analyze them. In practice, compliant mechanisms are more sensitive to external wrenches than conventional mechanisms. In this paper, based on the kinematic constraints and the static equilibrium between the joint coordinates and the external wrenches, a general kinematostatic model of compliant parallel mechanisms is presented. Then, this model is differentiated to provide a quasi-static model that makes it possible to calculate the variation of the pose as a linear function of the motion of the actuators and the variation of the external loads through two new matrices: the quasi-static Jacobian matrix and the Cartesian compliance matrix that give a simple and meaningful formulation of the model of the mechanism.     

Derivation of geometric stiffness and mass matrices for finite element hybrid models

An unifying approach in deriving the geometric stiffness and mass matrices for finite element hybrid models is presented. The variational formulation is based on a modified Reissner Principle. Numerical verification is illustrated through a simple beam example. The element interchangeability in the finite element method and an alternative way of deriving the loading vector for the hybrid stress model are also discussed.