Tool path generation for 5-axis flank milling of ruled surfaces with optimal cutter locations considering multiple geometric constraints

Ruled surfaces found in engineering parts are often blended with a constraint surface,like the blade surface and hub surface of a centrifugal impeller.It is significant to accurately machine these ruled surfaces in flank milling with interference-free and fairing tool path,while current models in fulfilling these goals are complex and rare.In this paper,a tool path planning method with optimal cutter locations(CLs)is proposed for 5-axis flank milling of ruled surfaces under multiple geometric constraints.To be specific,a concise three-point contact tool positioning model is firstly developed for a cylindrical cutter.Different tool orientations arise when varying the three contact positions and a tool orientation pool with acceptable cutter-surface deviation is constructed using a meta-heuristic algorithm.Fairing angular curves are derived from candidates in this pool,and then curve registration for cutter tip point on each determined tool axis is performed in respect of interference avoidance and geometric smoothness.On this basis,an adaptive interval determination model is developed for deviation control of interpolated cutter locations.This model is designed to be independent of the CL optimization process so that multiple CLs can be planned simultaneously with parallel computing technique.Finally,tests are performed on representative surfaces and the results show the method has advantages over previous meta-heuristic tool path planning approaches in both machining accuracy and computation time,and receives the best comprehensive performance compared to other multi-constrained methods when machining an impeller.

A Symbolic-Numeric Approach for Parametrizing Ruled Surfaces

This paper presents symbolic algorithms to determine whether a given surface(implicitly or parametrically defined)is a rational ruled surface and find a proper parametrization of the ruled surface.However,in practical applications,one has to deal with numerical objects that are given approximately,probably because they proceed from an exact data that has been perturbed under some previous measuring process or manipulation.For these numerical objects,the authors adapt the symbolic algorithms presented by means of the use of numerical techniques.The authors develop numeric algorithms that allow to determine ruled surfaces"close"to an input(not necessarily ruled)surface,and the distance between the input and the output surface is computed.

A Survey of the Representations of Rational Ruled Surfaces

The rational ruled surface is a typical modeling surface in computer aided geometric design.A rational ruled surface may have different representations with respective advantages and disadvantages.In this paper,the authors revisit the representations of ruled surfaces including the parametric form,algebraic form,homogenous form and Plucker form.Moreover,the transformations between these representations are proposed such as parametrization for an algebraic form,implicitization for a parametric form,proper reparametrization of an improper one and standardized reparametrization for a general parametrization.Based on these transformation algorithms,one can give a complete interchange graph for the different representations of a rational ruled surface.For rational surfaces given in algebraic form or parametric form not in the standard form of ruled surfaces,the characterization methods are recalled to identify the ruled surfaces from them.

Bonnet Ruled Surfaces

We consider the Bonnet ruled surfaces which admit only one non-trivial isometry that preserves the principal curvatures. We determine the Bonnet ruled surfaces whose generators and orthogonal trajectories form a special net called an A-net.

Existence Criteria of Blending Surfaces of Three Quadric Surfaces——Wu Wen-tsun's Formulae

Smoothly blending several implicit quadric surfaces with surfaces of the lowest degree is very useful in engineering technology.

RULED SURFACE MACHINING BY 4-AXES SIMULTANEOUS CONTROL WIRE-EDM

On the tasis of study in the mathematical model of 3-dimensional ruled surface (RS),this paper introduces a new concept of distance paramcter (DP) and also puts forward that themethod of modeling a RS depends on not only two boundary curves but also DP. According toabove theory, the formulas to calculate corresponding point coordinates to any kind of top and bot-tom profile of a workpiece and formulas to calcuate the maximum inclination angle of ruling linehave been obtained. Then a different top and bottom RS mathining method including profile withline-are combination as well as parametric curves has been achieved by 4-axes simultancous con-trol programming proposed.

CAD/CAM ALGORITHM OF RULED SURFACE BASED ON LINE GEOMETRY

A new concept of design and manufacturing of ruled surface based on line geometry is proposed. Some practical algorithm for CAD system is derived. Some problems in design and manufacturing of ruled surface can be solved by using the algorithm.

Notes on Curves at a Constant Distance from the Edge of Regression on a Curve in Galilean 3-Space G_(3)

In this paper,we define the curve rλ=r+λd at a constant distance from the edge of regression on a curve r(s)with arc length parameter s in Galilean 3-space.Here,d is a non-isotropic or isotropic vector defined as a vector tightly fastened to Frenet trihedron of the curve r(s)in 3-dimensional Galilean space.We build the Frenet frame{Tλ,Nλ,Bλ}of the constructed curve rλwith respect to two types of the vector d and we indicate the properties related to the curvatures of the curve rλ.Also,for the curve rλ,we give the conditions to be a circular helix.Furthermore,we discuss ruled surfaces of type A generated via the curve rλand the vector D which is defined as tangent of the curve rλin 3-dimensional Galilean space.The constructed ruled surfaces also appear in two ways.The first is constructed with the curve rλ(s)=r(s)+λT(s)and the non-isotropic vector D.The second is formed by the curve rλ=r(s)+λ2N+λ3B and the non-isotropic vector D.We calculate the distribution parameters of the constructed ruled surfaces and we show that the ruled surfaces are developable.Finally,we provide examples and visuals to back up our research.

Positioning method of a cylindrical cutter for ruled surface machining based on minimizing one-sided Hausdorff distance

Motivated by the definition of the machining errors induced by tool path planning methods, a mapping curve of the tool axis of a cylindrical cutter is constructed on the tool surface. The mapping curve is a typical one that can be used to express the closeness between the tool surface and the surface to be machined. A novel tool path planning method is proposed for flank or plunge milling ruled surfaces based on the minimization of the one-sided Hausdorff distance （HD） from the mapping curve to the surface to be machined. It is a nonlinear optimization problem in best uniform approximation （BUA） or Chebyshev sense. A mathematical programming model for computing the minimum one-sided HD is proposed. The linearization method of the programming model is provided and the final optimal solutions are obtained by simplex method. The effectiveness of the proposed BUA method is verified by two numerical examples and compared with the least squares （LS） and double point offset （DPO） methods. The variation in tool orientation induced by the optimization of the tool positions is also evaluated.

Proper Reparametrization of Rational Ruled Surface

In this paper, we present a proper reparametrization algorithm for rational ruled surfaces. That is, for an improper rational parametrization of a ruled surface, we construct a proper rational parametrization for the same surface. The algorithm consists of three steps. We first reparametrize the improper rational parametrization caused by improper supports. Then the improper rational parametrization is transformed to a new one which is proper in one of the parameters. Finally, the problem is reduced to the proper reparametrization of planar rational algebraic curves.

Blaschke Hypersurfaces with Symmetric Shape Operator

A Blaschke hypersurface admits S symmetry if and only if S（X, Y） = S（Y, X）. We prove that the shape operator has only one eigenvalue. And such Blaschke surfaces are classified as affine spheres or ruled surfaces.