Smooth quasi-developable surfaces bounded by smooth curves

Computing a quasi-developable strip surface bounded by design curves finds wide industrial applications. Existing methods compute discrete surfaces composed of developable lines connecting sampling points on input curves which are not adequate for generating smooth quasi-developable surfaces. We propose the first method which is capable of exploring the full solution space of continuous input curves to compute a smooth quasi-developable ruled surface with as large developability as possible. The resulting surface is exactly bounded by the input smooth curves and is guaranteed to have no self-intersections. The main contribution is a variational approach to compute a continuous mapping of parameters of input curves by minimizing a function evaluating surface developability. Moreover, we also present an algorithm to represent a resulting surface as a B-spline surface when input curves are B-spline curves.Comment: 18 page

A design tool for globally developable discrete architectural surfaces using Ricci flow

This paper presents an approach for the design of discrete architectural surfaces that are globally developable; that is, having zero Gaussian curvature at every interior node. This kind of architectural surface is particularly suitable for fast fabrication at a low cost, since their curved geometry can be developed into a plane. This highly non-linear design problem is broken down into two sub-problems: (1) find the member lengths of a triangular mesh that lead to zero Gaussian curvature, by employing the discrete surface Ricci flow developed in the field of discrete differential geometry; (2) realize the final geometry by solving an optimization problem, subject to the constraints on member lengths as well as the given boundary. It is demonstrated by the numerical examples that both of these two sub-problems can be solved with small computational costs and sufficient accuracy. In addition, the Ricci flow algorithm has an attractive feature-the final design is conformal to the initial one. Conformality could result in higher structural performance, because the shape of each panel is kept as close as possible to its initial design, suppressing possible distortion of the panels. This paper further presents an improved circle packing scheme implemented in the discrete surface Ricci flow to achieve better conformality, while keeping its simplicity in algorithm implementation as in the existing Thurston's scheme

Screw rotor manufacturing via 5-axis flank CNC machining using conical tools

We propose a new method for 5-axis flank computer numerically controlled (CNC) machining of screw rotors using conical tools. The flanks of screw rotors consist of helical surfaces, which predetermines the motion of the milling tool and reduces the search space for tool positioning to only 4-parametric family, which allows a quick search for good initial positions of a given conical tool. We initialize the search by looking at second order line contact between the tool and the helical flank of the rotor. Several positions of the tool are found, covering major part of the flank of the rotor, followed by global optimization that further reduces the tool-surface error and makes sure that there are no gaps between neighboring sweeps of the tool. We demonstrate our approach on several benchmark screw rotors, showing that our approach meets fine industrial tolerances with only few sweeps of the tool.RYC-2017-2264

Geometry and tool motion planning for curvature adapted CNC machining

CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow one to adapt the tool motion optimally to the surface to be produced. We aim at a high-quality surface finish, thereby reducing the need for hard-to-control post-machining processes such as grinding and polishing. Our work is based on a careful geometric analysis of curvature-adapted machining via so-called second order line contact between tool and target surface. On the geometric side, this leads to a new continuous transition between “dual” classical results in surface theory concerning osculating circles of surface curves and oscu- lating cones of tangentially circumscribed developable surfaces. Practically, it serves as an effective basis for tool motion planning. Unlike previous approaches to curvature-adapted machining, we solve locally optimal tool positioning and motion planning within a single optimization framework and achieve curvature adaptation even for convex surfaces. This is possible with a toroidal cutter that contains a negatively curved cutting area. The effectiveness of our approach is verified at hand of digital models, simulations and machined parts, including a comparison to results generated with commercial software

曲線折り動作のモデル化と可視化に関する研究

筑波大学 (University of Tsukuba)201

Optimal development of doubly curved surfaces,

Abstract This paper presents algorithms for optimal development (flattening) of a smooth continuous curved surface embedded in three-dimensional space into a planar shape. The development process is modeled by in-plane strain (stretching) from the curved surface to its planar development. The distribution of the appropriate minimum strain field is obtained by solving a constrained nonlinear programming problem. Based on the strain distribution and the coefficients of the first fundamental form of the curved surface, another unconstrained nonlinear programming problem is solved to obtain the optimal developed planar shape. The convergence and complexity properties of our algorithms are analyzed theoretically and numerically. Examples show the effectiveness of the algorithms

Folding of bistable composite tape-springs

Bistable composite tape-spring technologies have great potential in application to aircraft landing gears, in order to reduce weight, complexity and maintenance compared to the conventional lock-link assemblies. To investigate their implementation, the first area of interest is the ‘‘ploy’’ region, which corresponds to the transitional state between the folded and the extended configurations. We devise a simple ‘‘free’’ bending system with minimal constraints to study the folding nature of tape-spring structures in general. A finite element (FE) model is also established and calibrated using experimental data; a theoretical model is developed to provide further insights. The typical folding process consists of linear bending, torsional buckling, localisation and then folding; the shape of the central fold is developable; the ploy region is dominated by axial strains and transverse curvature changes. Here, we achieve a good agreement between experiments, simulation and theoretical analysis