A generic mathematical model of single point cutting tools in terms of grinding parameters

The paper presents the analytical geometric details of the mathematical modeling of a single point cutting tool with a generic profile. The grinding angles and the ground depths on the tool are allowed to vary along the tool flanks and face, altering the cutting angles from point to point. The surface modeling begins with the creation of a tool blank model. Then unbounded surfaces are considered and transformed to get the cutting tool surfaces. The intersection of these surfaces gives the complete model of the tool. Starting from the basic model where the tool face and flank are planar, the generalization of the geometric design has been done in two steps to give free-form shapes to the tool surfaces, termed as the two generations of the generic profile. Then a forward and inverse mapping has been presented for the basic model and the two generations of the generic tool to relate the grinding angles with the prevalent nomenclatures (ASA, ORS and NRS). The model has been validated and the variation of tool angles with the grinding parameters has been illustrated with an example.     

A new design concept for milling tools of spherical surfaces obtained by kinematic generation

In this paper, the authors show an original methodology for optimization of tool geometry used in the milling of spherical surfaces. This methodology is based on thorough theoretical research, which highlights the kinematic characteristics of spherical surface generation by milling. These characteristics lead to the conclusion that the constructive angles of tools used in spherical surface milling must have the same values on both cutting edges.     

Mathematical modeling of interrelationships among cutting angles,setting angles and working angles of single-point cutting tools

The angles of a cutting tool play a key role in determining its cutting efficiency. However, once the tool has been mounted on a machine tool, setting errors inevitably cause the working angles of the tool to deviate slightly from the designed cutting angles. Accordingly, this study develops mathematical models to analyze the interrelationships among the tool angles, the setting angles and the working angles. In the proposed approach, a process of homogeneous coordinate transformation is employed to develop a kinematic model of the cutting tool such that the working angles can be derived given the tool angles and setting angles. Mathematical formulae are then developed to inversely derive the tool angles required to generate the specified working angles given prior knowledge of the setting angles. An illustrative numerical example is presented to demonstrate the validity of the proposed approach. Overall, the numerical results confirm that the methodology presented in this study provides a comprehensive, simple and versatile means of modeling a variety of conventional single-point cutting tools.     

一种新的大位移井钻柱几何非线性分析方法

提出了基于实测的井深及相应的井斜角和方位角来获得确保井内钻柱参考构形长度不变的井眼轴线插值方法．当以空间大位移井的井眼轴线为钻柱的参考构形时,钻柱内的初始内力可以由井眼轴线的曲率和挠率确定．利用基于在空间自然坐标系下的包含所有单元刚体位移和常应变模式的位移函数,严格地按虚功原理推出了具有初始曲率和挠率的钻柱单元内由初始内力所引起的等效节点力计算公式,为大位移井钻柱的几何非线性处理提供了理论依据．澄清了钻柱有限元分析中的若干基本概念．为随后进行的以井眼轴线为参考构形的小变形分析,计算钻柱的自重和基于自然坐标系下的线性刚度矩阵及一致载荷列阵提供了保证．     

Symplectic invariants near hyperbolic-hyperbolic points

We construct symplectic invariants for Hamiltonian integrable systems of 2 degrees of freedom possessing a fixed point of hyperbolic-hyperbolic type. These invariants consist in some signs which determine the topology of the critical Lagrangian fibre, together with several Taylor series which can be computed from the dynamics of the system.We show how these series are related to the singular asymptotics of the action integrals at the critical value of the energy-momentum map. This gives general conditions under which the non-degeneracy conditions arising in the KAM theorem (Kolmogorov condition, twist condition) are satisfied. Using this approach, we obtain new asymptotic formulae for the action integrals of the C. Neumann system. As a corollary, we show that the Arnold twist condition holds for generic frequencies of this system.      

Nonlinear process machine interaction in tool grinding

The manufacturing of high performance cutting tools, like drills and chamfers, is done by a deep grinding process. This tool grinding process shows nonlinear relations between predefined process parameters, as cutting depth, feed speed and cutting speed, and the process characteristics, as process forces, surface properties and stability. Within an interdisciplinary research project a process model is built up to represent the process machine interaction and to predict process forces, deformation of the workpiece and geometry errors caused by this deformation. In this paper, the process model and the coupling of the included submodules will be described. The focus is thereby on the explanation of the contact module and the implementation in the dynamical model. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A planar cubic Bézier spiral

A planar cubic Bézier curve segment that is a spiral, i.e., its curvature varies monotonically with arc-length, is discussed. Since this curve segment does not have cusps, loops, and inflection points (except for a single inflection point at its beginning), it is suitable for applications such as highway design, in which the clothoid has been traditionally used. Since it is polynomial, it can be conveniently incorporated in CAD systems that are based on B-splines, Bézier curves, or NURBS (nonuniform rational B-splines) and is thus suitable for general curve design applications in which fair curves are important.     

Automated generation of LFT-based parametric uncertainty descriptions from generic aircraft models

A computer assisted modelling methodology is developed for the generation of linearized models with parametric uncertainties described by Linear Fractional Transformations (LFTs). The starting point of the uncertainty modelling is a class of generic nonlinear aircraft models with explicit parametric dependence used for simulation purposes. The proposed methodology integrates specialized software tools for object-oriented modelling, for simulation, and for numerical as well as symbolic computations. The methodology has many generic features being applicable to similar nonlinear model classes.     

Free vibration analysis of spatial Bernoulli–Euler and Rayleigh curved beams using isogeometric approach

This paper deals with the linear free vibration analysis of Bernoulli–Euler and Rayleigh curved beams using isogeometric approach. The geometry of the beam as well as the displacement field are defined using the NURBS basis functions which present the basic concept of the isogeometric analysis. A novel approach based on the fundamental relations of the differential geometry and Cauchy continuum beam model is presented and applied to derive the stiffness and consistent mass matrices of the corresponding spatial curved beam element. In the Bernoulli–Euler beam element only translational and torsional inertia are taken into account, while the Rayleigh beam element takes all inertial terms into consideration. Due to their formulation, isogeometric beam elements can be used for the dynamic analysis of spatial curved beams. Several illustrative examples have been chosen in order to check the convergence and accuracy of the proposed method. The results have been compared with the available data from the literature as well as with the finite element solutions.     

Modeling and homogenization of textile reinforced composites based on the isogeometric analysis

In this contribution, the isogeometric analysis is used to compute the effective material properties of textile reinforced composites. The isogeometric analysis based on non-uniform rational B-splines (NURBS) provides an efficient approach for numerical modeling because there is no need for a mesh generation. There are further advantages such as the availability of a geometry representation based on NURBS in computer-aided design software and the possibility to apply different refinement methods which do not change the geometry of the numerical model. These properties motivate the combination of the isogeometric analysis with the homogenization method. Therefor, the unit cell model representing the inner architecture of a textile reinforced composite is defined using NURBS. In order to compute the effective mechanical properties of the heterogeneous material, the homogenization method with periodic boundary conditions is applied. Finally, two examples demonstrate the advantages of this approach. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Study of the parameters in electrical discharge machining through response surface methodology approach

Whereas the efficiency of traditional cutting processes is limited by the mechanical properties of the processed material and the complexity of the workpiece geometry, electrical discharge machining (EDM) being a thermal erosion process, is subject to no such constraints. The lack of correlations between the cutting rate, the surface finish and the physical material parameters of this process made it difficult to use. This paper highlights the development of a comprehensive mathematical model for correlating the interactive and higher order influences of various electrical discharge machining parameters through response surface methodology (RSM), utilizing relevant experimental data as obtained through experimentation. The adequacy of the above the proposed models have been tested through the analysis of variance (ANOVA). Optimal combination of these parameters was obtained for achieving controlled EDM of the workpieces.     

Homoclinic twist bifurcations with ℤ2 symmetry

Summary We analyze bifurcations occurring in the vicinity of a homoclinic twist point for a generic two-parameter family of ℤ₂ equivariant ODEs in four dimensions. The results are compared with numerical results for a system of two coupled Josephson junctions with pure capacitive load.     

Bi-Lipschitz geometry of complex surface singularities

We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface with particular emphasis on when it is metrically conical.      

The opaque nature of generic examples: The structure of student teachers’ arguments in multiplicative reasoning

The study aims to explore the structural aspects of generic examples, to get better insight into what makes them potentially opaque for learners. We have analyzed 27 written arguments, for which student teachers (grades 1–10) were asked to use a generic example to prove a given statement in multiplication. Using Toulmin’s framework, we developed five categories of arguments based on their structure: examples, empirical arguments, leap arguments, embedded arguments, and other arguments. Also, we conclude that none of the student teachers provided arguments that we recognize as complete generic examples. The results bring us to a discussion about features of generic examples making them difficult to come to grips with, having implications for how teacher educators can support student teachers’ learning to prove. From this, we propose a definition of generic examples that attends to the criteria suggested in previous research, yet, emphasizing their structural nature.     

Transformations of Border Strips and Schur Function Determinants

We introduce the notion of the cutting strip of an outside decomposition of a skew shape, and show that cutting strips are in one-to-one correspondence with outside decompositions for a given skew shape. Outside decompositions are introduced by Hamel and Goulden and are used to give an identity for the skew Schur function that unifies the determinantal expressions for the skew Schur functions including the Jacobi-Trudi determinant, its dual, the Giambelli determinant and the rim ribbon determinant due to Lascoux and Pragacz. Using cutting strips, one obtains a formula for the number of outside decompositions of a given skew shape. Moreover, one can define the basic transformations which we call the twist transformation among cutting strips, and derive a transformation theorem for the determinantal formula of Hamel and Goulden. The special case of the transformation theorem for the Giambelli identity and the rim ribbon identity was obtained by Lascoux and Pragacz. Our transformation theorem also applies to the supersymmetric skew Schur function.     

Numerical manifold method for vibration analysis of Kirchhoff's plates of arbitrary geometry

Many rectangular plate elements developed in the history of finite element method (FEM) have displayed excellent numerical properties, yet their applications have been limited due to inability to conform to the arbitrary geometry of plates and shells. Numerical manifold method (NMM), considered to be a generalization of FEM, can easily solve this issue by viewing a mesh made up of rectangular elements as mathematical cover. In this study, ACM element (Adini and Clough element from A. Adini, R.W. Clough, Analysis of plate bending by the finite element method, University of California, 1960), a typical rectangular plate element is first integrated in the framework of NMM. Then, vibration analysis of arbitrary shaped thin plates is conducted employing the tailored NMM. Using the definition of integral of scalar functions on manifolds, we developed a mathematically rigorous mass lumping scheme for creating a symmetric and positive definite lumped mass matrix that is easy to inverse. A series of numerical experiments have been studied and analyzed, including free and forced vibration of thin plates with various shapes, validating the proposed mass lumping scheme can supersede the consistent mass formulation in those cases.