Denis Blackmore

The Sweep-Envelope Differential Equation Algorithm and Its Application to NC Machining Verification

Abstract A new method, called the sweep-envelope differential equation, for characterizing swept volume boundaries is introduced. This method is used as the theoretical foundation of an algorithm for computing swept volumes using the trajectories of the sweep-envelope differential equation which start at the initial grazing points of the moving object. The major advantages of this algorithm are: (1) the grazing point set need essentially only be computed at the initial position of the object—the remaining grazing points are generated by the flow of the sweep-envelope equation—so the computation complexity is drastically reduced; and (2) it provides automatic connectivity for computed boundary points that facilitates integration with standard algorithms and cad software for visual realization and Boolean operations. Examples are presented that illustrate successful integration of a prototype program (based on the algorithm) with commercial NC verification software.

Analysis of swept volume via Lie groups and differential equations

The development of useful mathematical techniques for an alyzing swept volumes, together with efficient means of im plementing these methods to produce serviceable models, has important applications to numerically controlled (NC) machin ing, robotics, and motion planning, as well as other areas of automation. In this article a novel approach to swept volumes is delineated—one that fully exploits the intrinsic geometric and group theoretical structure of Euclidean motions in or der to formulate the problem in the context of Lie groups and differential equations. Precise definitions of sweep and swept volume are given that lead naturally to an associated ordinary differential equation. This sweep differential equation is then shown to be related to the Lie group structure of Euclidean motions and to generate trajectories that completely determine the geometry of swept volumes. It is demonstrated that the notion of a sweep differential equation leads to criteria that provide useful insights concern ing the geometric and topologic features of swept volumes. Several new results characterizing swept volumes are obtained. For example, a number of simple properties that guarantee that the swept volume is a Cartesian product of elementary mani folds are identified. The criteria obtained may be readily tested with the aid of a computer.

A perspective on vibration-induced size segregation of granular materials

Abstract Segregation of particulate mixtures is a problem of great consequence in industries involved with the handling and processing of granular materials in which homogeneity is generally required. While there are several factors that may be responsible for segregation in bulk solids, it is well accepted that nonuniformity in particle size is a fundamental contributor. When the granular material is exposed to vibrations, the question of whether or not convection is an essential ingredient for size segregation is addressed by distinguishing between the situation where vibrations are not sufficiently energetic to promote a mean flow of the bulk solid, and those cases where a convective flow does occur. Based on experimental and simulation results in the literature, as well as dynamical systems analysis of a recent model of a binary granular mixture, it is proposed that “void-filling” beneath large particles is a universal mechanism promoting segregation, while convection essentially provides a means of mixing enhancement.

Fractal geometry model for wear prediction

Abstract Fractal characterization of surface topography is applied to the study of contact mechanics and wear processes. The structure function method is used to find the fractal dimension D and the topothesy L . We develop a fractal geometry model, which predicts the wear rate in terms of these two fractal parameters for wear prediction. Using this model we show that the wear rate V r and the true contact area A r have the relationship V r αA r m ( D ) , where m ( D ) is a function of D and has a value between 0.5 and 1. We next study the optimum ( i.e. the lowest wear rate) fractal dimension in a wear process. It is found that the optimum fractal dimension is affected by the contact area, material properties and scale amplitude. Experimental results of wear testing show good agreement with the predictions based on the model.

Analysis and modelling of deformed swept volumes

Abstract The sweep differential equation approach and the boundary-flow method developed for the analysis and representation of swept volumes are extended to include objects experiencing deformation. It is found that the theoretical framework can be generalized quite naturally to deformed swept volumes by the enlargement of the Lie group structure of the sweeps. All the usual results, including the boundary-flow formula, are shown to have extensions for swept volumes with deformation. Several special classes of deformation are identified, and their particular properties are studied insofar as they pertain to swept volumes. A program for obtaining deformed swept volumes of planar polygons is described, and is then applied to several examples to demonstrate its effectiveness.

Transient and steady state of a rising bubble in a viscoelastic fluid

A finite element code based on the level-set method is used to perform direct numerical simulations (DNS) of the transient and steady-state motion of bubbles rising in a viscoelastic liquid modelled by the Oldroyd-B constitutive equation. The role of the governing dimensionless parameters, the capillary number (Ca), the Deborah number (De) and the polymer concentration parameter c, in both the rising speed and the deformation of the bubbles is studied. Simulations show that there exists a critical bubble volume at which there is a sharp increase in the terminal velocity with increasing bubble volume, similar to the behaviour observed in experiments, and that the shape of both the bubble and its wake structure changes fundamentally at that critical volume value. The bubbles with volumes smaller than the critical volume are prolate shaped while those with volumes larger than the critical volume have cusp-like trailing ends. In the latter situation, we show that there is a net force in the upward direction because the surface tension no longer integrates to zero. In addition, the structure of the wake of a bubble with a volume smaller than the critical volume is similar to that of a bubble rising in a Newtonian fluid, whereas the wake structure of a bubble with a volume larger than the critical value is strikingly different. Specifically, in addition to the vortex ring located at the equator of the bubble similar to the one present for a Newtonian fluid, a vortex ring is also present in the wake of a larger bubble, with a circulation of opposite sign, thus corresponding to the formation of a negative wake. This not only coincides with the appearance of a cusp-like trailing end of the rising bubble but also propels the bubble, the direction of the fluid velocity behind the bubble being in the opposite direction to that of the bubble. These DNS results are in agreement with experiments.

Trimming swept volumes

Abstract The trimming problem for swept volumes — concerning the excision of points ostensibly on the boundary that actually lie in the swept volume interior — is investigated in detail. Building upon several techniques that have appeared in the literature, efficient methods for both local and global trimming of swept volume are developed. These methods are shown to be computationally cost effective when combined with the sweep-envelope differential equation algorithm for the approximate calculation and graphical rendering of swept volumes for quite general objects and sweeps. Examples are presented to demonstrate the efficacy of the trimming strategies.

A new fractal model for anisotropic surfaces

Abstract A new fractal-based functional model for anisotropic rough surfaces is used to devise and test two methods for the approximate computation of the fractal dimension of surfaces, and as an instrument for simulating the topography of engineering surfaces. A certain type of statistical self-affinity is proved for the model, and this property serves as the basis for one of the methods of approximating fractal dimension. The other technique for calculating fractal dimension is derived from a Holder type condition satisfied by the model. Algorithms for implementing both of these new schemes for computing a proximate values of fractal dimension are developed and compared with standard procedures. Both the functional model and its corresponding modified Gaussian height distribution are used for simulating fractal surfaces and several examples are adduced that strongly resemble some common anisotropic engineering surfaces.

Application of Flows and Envelopes to NC Machining

Abstract The characterization and representation of swept volume has important applications in NC machining theory and practice. Modern YC programs are quite versatile: they provide programmers with the capacity of performing a variety of possible types of motion for the cutting tool such as linear. circular, helical. parabolic and cubic interpolation. Using the method of sweep differential equations, techniques which incorporate the method of envelopes are developed for constructing the swept volume of simple tools which are generated by typical motions in YC programs. The technique for linear. circular and helical motions is shown to effectively reduce the dimension of the problem by two. For completely general motions it is shown that the procedure can be best effectuated by first imbedding the configuration in a space of one more dimension. Several examples are included to illustrate the implementation of the methods which are introduced.

Fractal geometry modeling with applications in surface characterisation and wear prediction

Manufactured surfaces such as those produced by electrical discharge machining, waterjet cutting and ion-nitriding coating can be characterized by fractal geometry. A modified Gaussian random fractal model coupled with structure functions is used to relate surface topography with fractal geometry via fractal geometry via fractal dimension (D) and topothesy (L). This fractal characterization of surface topography complements and improves conventional statistical and random process methods of surface characterization, Our fractal model for surface topography is shown to predict a primary relationship between D and the bearing area curve, while L affects this curve to a smaller degree. A fractal geometry model for wear prediction is proposed, which predicts the wear rate in terms of these two fractal parameters. Using this model we show that the wear rate Vr and the true contact area Ar have the relationship Vrα (Ar)m(D), where m(D) is a function of D and has a value between 0.5 and 1. We next study the optimum (ie the lowest wear rate) fractal diemsnsion in a wear process. It is found that the optimum fractal dimension is affected by the contact area, material properties, and scale amplitude. Experimental results of bearing area curves and wear testing show good agreement with the two models.