Optimized normal and distance matching for heterogeneous object modeling

This paper presents a new optimization methodology of material blending for heterogeneous object modeling by matching the material governing features for designing a heterogeneous object. The proposed method establishes point-to-point correspondence represented by a set of connecting lines between two material directrices. To blend the material features between the directrices, a heuristic optimization method developed with the objective is to maximize the sum of the inner products of the unit normals at the end points of the connecting lines and minimize the sum of the lengths of connecting lines. The geometric features with material information are matched to generate non-self-intersecting and non-twisted connecting surfaces. By subdividing the connecting lines into equal number of segments, a series of intermediate piecewise curves are generated to represent the material metamorphosis between the governing material features. Alternatively, a dynamic programming approach developed in our earlier work is presented for comparison purposes. Result and computational efficiency of the proposed heuristic method is also compared with earlier techniques in the literature. Computer interface implementation and illustrative examples are also presented in this paper

5-axis double-flank CNC machining of spiral bevel gears via custom-shaped milling tools -- Part I: modeling and simulation

A new category of 5-axis flank computer numerically controlled (CNC) machining, called \emph{double-flank}, is presented. Instead of using a predefined set of milling tools, we use the shape of the milling tool as a free parameter in our optimization-based approach and, for a given input free-form (NURBS) surface, compute a custom-shaped tool that admits highly-accurate machining. Aimed at curved narrow regions where the tool may have double tangential contact with the reference surface, like spiral bevel gears, the initial trajectory of the milling tool is estimated by fitting a ruled surface to the self-bisector of the reference surface. The shape of the tool and its motion then both undergo global optimization that seeks high approximation quality between the input free-form surface and its envelope approximation, fairness of the motion and the tool, and prevents overcutting. That is, our double-flank machining is meant for the semi-finishing stage and therefore the envelope of the motion is, by construction, penetration-free with the references surface. Our algorithm is validated by a commercial path-finding software and the prototype of the tool for a specific gear model is 3D printed.RYC-2017-22649 BERC 2014-201

Efficient 5-axis CNC trochoidal flank milling of 3D cavities using custom-shaped cutting tools

A novel method for trochoidal flank milling of 3D cavities bounded by free-form surfaces is proposed. Existing 3D trochoidal milling methods use on-market milling tools whose shape is typically cylindrical or conical, and is therefore not well-suited for meeting fine milling tolerances required for finishing of benchmark free-form surfaces like blades or blisks. In contrast, our variational framework incorporates the shape of the tool into the optimization cycle and looks not only for the trochoidal milling paths, but also for the shape of the tool itself. High precision quality is ensured by firstly designing flank milling paths for the side surfaces that delimit the motion space, in which the trochoidal milling paths are further computed. Additionally, the material removal rate is maximized with the cutter-workpiece engagement being constrained under a given tolerance. Our framework also supports multi-layer approach that is necessary to handle deep cavities. The ability and efficacy of the proposed method are demonstrated by several industrial benchmarks, showing that our approach meets fine machining tolerances using only a few trochoidal paths.RYC-2017-2264

Five-Axis Numerical Control Machining of the Tooth Flank of a Logarithmic Spiral Bevel Gear Pinion

In this paper, the production of a logarithmic spiral bevel gear prototype is illustrated by the manufacture of the gear pinion. Firstly, the conical gear body of a logarithmic spiral bevel gear pinion was shaped on a C6140A1 lathe. A kinematic model of a five-axis vertical machining centre DMG DMU40 monoBLOCK, with the position and orientation of each axis relative to the movement of the workpiece, was created. In addition, the processing coordinate transformation formula between the workpiece coordinate system and the cutter coordinate system was devised. The cutter location file was converted to the numerical control code of the DMG DMU40 monoBLOCK. Finally, the pinion of a logarithmic spiral bevel gear was machined on the DMG DMU40 monoBLOCK as a prototype to be used in further research of the logarithmic spiral bevel gear

Characterizing envelopes of moving rotational cones and applications in CNC machining

Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes results on local approximations of a surface by cones of revolution, which are expressed by contact order in the space of planes. To this purpose, the isotropic model of Laguerre geometry is used as there rotational cones correspond to curves (isotropic circles) and higher order contact is computed with respect to the image of the input surface in the isotropic model. Therefore, one studies curve-surface contact that is conceptually simpler than the surface-surface case. We show that, in a generic case, there exist at most six positions of a fixed rotational cone that have third order contact with the input surface. These results are themselves of interest in geometric computing, for example in cutter selection and positioning for flank CNC machining.RYC-2017-2264

Surface Design for Flank Milling

In this dissertation, a numerical method to design a curved surface for accurately flank milling with a general tool of revolution is presented. Instead of using the ruled surface as the design surface, the flank millable surface can better match the machined surface generated by flank milling techniques, and provide an effective tool to the designer to control the properties and the specifications of the design surface. A method using the least squares surface fitting to design the flank millable surface is first discussed. Grazing points on the envelope of the moving tool modeled by the grazing surface are used as the sample points and a NURBS surface is used to approximate the given grazing surface. The deviation between the grazing surface and the NURBS surface can be controlled by increasing the number of the control points. The computation process for this method is costly in time and effort. In engineering design, there is a need for fast and effortless methods to simplify the flank millable surface design procedure. A technique to approximate the grazing curve with NURBS at each tool position is developed. Based on the characteristics of the grazing surface and the geometries of the cutting tool, these NURBS representations at a few different tool positions, namely at the start, interior and end, are lofted to generate a NURBS surface. This NURBS surface represents the grazing surface and is treated as the design surface. Simulation results show that this design surface can accurately match the machined surface. The accuracy of the surface can be controlled by adding control points to the control net of the NURBS surface. A machining test on a 5-axis machine was done to verify the proposed flank millable surface design method. The machined surface was checked on a CMM and the obtained results were compared with the designed flank millable surface. The comparison results show that the machined surface closely matches the design surface. The proposed flank millable surface design method can be accurately used in the surface design