Swept Surface Research Articles

Designing a helical knife for a shredding drum using a sweep surface

The object of this paper is a helical blade in a shredding drum from a sweep surface. Such drums are used in harvesters for crushing plant mass. If the flat blades are installed on the drum, cutting of the plant mass occurs simultaneously along the entire length of the blade. This could cause a pulsating dynamic load. If a flat knife with a straight blade is installed at an angle to the axis of the drum, then the distances from it to the points of the blade will be different, as well as the cutting conditions along the blade. The elliptical shape enables the same distance from the axis of rotation to the points of the blade, but this does not solve the problem. Many short flat knives with a straight blade can be mounted on the drum, placing them in such a way that the time between the individual knives is minimized. However, all these disadvantages can be eliminated by a helical knife with a blade in the form of a helical line. The design of a helical knife from an unfolding helicoid has been considered. In differential geometry, the bending of unfolded surfaces of zero thickness is considered. Bending of the workpiece into a finished product occurs with minimal plastic deformations, the magnitude of which depends on the thickness of the workpiece sheet. The methods of differential geometry of unfolding surfaces were applied to the analytical description of the surface of the helical knife. The parametric equations of the unfolding helicoid were derived according to the given structural parameters of the knife in space and on the plane. That has made it possible to mathematically describe the contour lines that cut the knife from the surface and on its sweep. Formulae for calculating a flat workpiece through the structural parameters of the knife have been derived. Thus, with the specified structural parameters of the knife R=0.25 m, τ=20°, φ=65°, according to the resulting formula, we find the radius of the knife blade on a flat workpiece: R0=4.8 m.

Balancing Rotation Minimizing Frames with Additional Objectives

AbstractWhen moving along 3D curves, one may require local coordinate frames for visited points, such as for animating virtual cameras, controlling robotic motion, or constructing sweep surfaces. Often, consecutive coordinate frames should be similar, avoiding sharp twists. Previous work achieved this goal by using various methods to approximate rotation minimizing frames (RMFs) with respect to a curve's tangent. In this work, we use Householder transformations to construct preliminary tangent‐aligned coordinate frames and then optimize these initial frames under the constraint that they remain tangent‐aligned. This optimization minimizes the weighted sum of squared distances between selected vectors within the new frames and fixed vectors outside them (such as the axes of previous frames). By selecting different vectors for this objective function, we reproduce existing RMF approximation methods and modify them to consider additional objectives beyond rotation minimization. We also provide some example computer graphics use cases for this new frame tracking.

Sweeping Surfaces Due to Conjugate Bishop Frame in 3-Dimensional Lie Group

In this work, we present a new Bishop frame for the conjugate curve of a curve in the 3-dimensional Lie group G3. With the help of this frame, we derive a parametric representation for a sweeping surface and show that the parametric curves on this surface are curvature lines. We then examine the local singularities and convexity of this sweeping surface and establish the sufficient and necessary conditions for it to be a developable ruled surface. Additionally, we provide detailed explanations and examples of its applications.

Designing and Studying Darboux Sweeping Surface in Isotropic Space I^1_3

This research aims to study Darboux sweeping surface in isotropic space I1 3 . We went through the geometric characteristics of sweeping surfaces in I1 3 . The first and second fundamental forms of the sweeping surface were evaluated. Furthermore, we investigate the mean and Gaussian curvature of the sweeping surface. We also show that the parametric curves on these surfaces are non-geodesic and non-asymptotic. Then, we derive the necessary and sufficient conditions for the sweeping surface to become a developable sweeping surface, minimal sweeping surface, and Weingarten surface. Finally, an example to illustrate the application of the results is introduced.

Epoxy-based reflective adhesive for lamination of single-sided metallized film over glass fabric destined for its use as reflective layer in fire proximity clothing

The genre of the present work is related to adhesives, which has been realized as an integral component in the fabrication of outermost reflective laminates, which are destined for their use in fire proximity clothing. This aspect has not received much attention from researchers and has motivated us to investigate the potential of extensively available diglycidyl ether of bisphenol-A (DGEBA)-based epoxy formulation as a laminating adhesive, where polyamide containing amine has been employed as a curing agent. The work is intended to adhere single–sided aluminized polyester film over glass fabric of satin weave pattern. It is to be noted that the single-sided metallized polyester film based reflective laminate loses its potential to reflect the radiant heat after abrasion damage. In view of the same, aluminium powder at 1, 3, and 5 wt.% loading has been incorporated in the optically clear layer of epoxy adhesive, with an aim to render reflective properties, which is expected to contribute in reflecting the radiant heat, after the abrasive damage. The influence of incorporating aluminium powder on the characteristics of adhesive has been investigated using techniques such as differential scanning calorimetry (DSC), temperature sweep rheological experiments, thermogravimetric analysis (TGA), and surface energy measurements. Further, the peel-adhesion strength of the formulations has been determined as per ASTM D1876 standard. The heat transmission in the developed reflective laminates on exposure to radiant heat has been studied. In addition, the heat transmission on exposure to flame has been investigated in the form of a multi-layer component assembly, and the charred regions have been analyzed using scanning electron microscopy (SEM) as well as energy dispersive X-ray (EDX) analysis.

Sweeping Surfaces according to Type-3 Bishop Frames in Euclidean 3-Space

The aim of this work is to investigate sweeping surfaces and their local singularities due to type-3 Bishop frames in Euclidean 3-space, E3. A sweeping surface a is surface that is designed from a section curve positioned along a path, which acts as the vertebral column or spine curve, and it has symmetrical characteristics. In this work, we have specified a sweeping surface and have examined its geometry and singularity. Thereafter, we deduced the circumstances required for this surface to be a developable surface. In great detail, we concentrated on the fundamental discussion on whether the resulting developable surface is a cylindrical, cone or tangent surface. Meanwhile, examples are detailed to explain the applications of the notional outcomes.

Investigation on Isotropic Bezier Sweeping Surface "IBSS" with Bishop Frame

Investigation on Isotropic Bezier Sweeping Surface "IBSS" with Bishop Frame

Sweeping surface due to rotation minimizing Darboux frame in Euclidean 3-space $ \mathbb{E}^{3} $

<abstract><p>In this paper, we address a new version of Darboux frame using a common tangent vector field to a surface along a curve and call this frame the rotation minimizing Darboux frame (RMDF). Then, we give the parametric equation due to the RMDF frame for a sweeping surface and show that the parametric curves on this surface are curvature lines. Consequently, necessary and sufficient conditions for sweeping surfaces to be developable ruled surfaces are derived. Also, we analyze the conditions when the resulting developable surface is a cylinder, cone or tangential surface. We also provide some examples to illustrate the main results.</p></abstract>

Effect of carbon nanotubes on fatigue cracking of asphalt mixtures modified by styrene-ethylene/propylene-styrene

ABSTRACT Fatigue cracking is a significant cause of failure in flexible pavements at moderate temperatures. Neat bitumen cannot properly perform at all temperatures and environmental conditions due to the increasing traffic volume. Consequently, this study examined the simultaneous usage of carbon nanotubes (CNTs) and the styrene-ethylene/propylene-styrene (SEPS) polymer as bitumen modifiers. The linear amplitude sweep (LAS) test and surface free energy (SFE) theory were used to determine the rheological characteristics and thermodynamic parameters of neat and modified bitumens, respectively. Using the SEPS nanocomposite up to 6% increased the fatigue life and moisture damage resistance of the asphalt mixtures by improving thermodynamic parameters such as adhesive free energy in dry and wet conditions. According to the LAS results, the modified bitumen outperformed the neat bitumen in terms of fatigue life under different strain levels. The fatigue life of the asphalt mixtures also decreased as the temperature increased from 10 to 20°C. However, for the mixtures containing the SEPS nanocomposite, the reduction in fatigue life was less noticeable due to the lower temperature sensitivity of the modified bitumen. The mixtures containing 6% SEPS nanocomposite demonstrated the highest performance.

Generalized Riemann-Liouville fractional Bézier curve and its applications in engineering surface

Modeling of free-form curves and surfaces is vital in the manufacturing industry and engineering. Curves and surfaces that have flexible shapes and adjustable lengths and sizes are a necessity to fulfill the needs of the manufacturing industry. Hence, researchers develop numerous aesthetic Bézier curves and surfaces to have such flexibility and adjustability. In this paper, the generalized Riemann–Liouville fractional Bézier curves and surfaces are proposed in the modeling of complex surfaces. The generalized Riemann–Liouville fractional Bézier curves and surfaces have two outstanding parameters: shape and fractional parameters. Shape parameters are utilized to change the shape of the curves and surfaces without altering the control points hence, adding flexibility in controlling the shape. While fractional parameters are utilized in curves and surfaces’ length and size adjustability. By adjusting the sizes of surfaces via fractional parameters, surfaces with optimal sizes can be modeled. In this paper, five types of engineering surfaces will be modeled, namely ruled surface, swept surface, swung surface, rotation surface, and coons patch. The geometric effects of the implementation of shape and fractional parameters to these engineering surfaces will also be analyzed, thus proving the generalized Riemann–Liouville fractional Bézier surfaces is an excellent tool in designing complex surfaces.

Incremental sliding-mode control and allocation for morphing-wing aircraft fast manoeuvring

This paper presents a novel robust controller for fast manoeuvring of a morphing aircraft to enhance the longitudinal manoeuvrability by combining a nonlinear disturbance observer (NDO) and incremental sliding-mode control (NDOISMC). Based on multi-body dynamics theory, a longitudinal nonlinear model of a symmetric variable sweep and variable span morphing aircraft is established. The NDO is introduced to estimate model uncertainties due to difficulties in precise dynamic modelling during fast morphing. In addition, the NDOISMC algorithm is proposed to generate virtual control inputs, which does not require an accurate model of the aircraft. The NDOISMC algorithm is proven semi-globally uniformly ultimately bounded (SGUUB) by using Lyapunov theory in the presence of uncertainties whose first derivatives are bounded. Furthermore, virtual control inputs are allocated to variable sweep, variable span, and aerodynamic control surfaces using second-order cone programming (SOCP). Finally, numerical simulations demonstrate the effectiveness of the proposed method. The results reveal that the control burden of the aerodynamic control surfaces is reduced with the help of a morphing wing in the manoeuvring flight. Moreover, superior control performance for fast manoeuvring of a morphing aircraft is achieved by using NDOISMC compared with existing methods under model uncertainties.

Singularity Properties of Timelike Sweeping Surface in Minkowski 3-Space

In this paper, we give the parametric equation of the Bishop frame for a timelike sweeping surface with a unit speed timelike curve in Minkowski 3-space. We introduce a new geometric invariant to explain the geometric properties and local singularities of this timelike surface. We derive the sufficient and necessary conditions for this timelike surface to be a timelike developable ruled surface. Afterwards, we take advantage of singularity theory to give the classification of singularities of this timelike developable surface. Furthermore, we give some representative examples to show the applications of the theoretical results.

Cutting Simulation in Unity 3D Using Position Based Dynamics with Various Refinement Levels

Augmented and Virtual Reality-based surgical simulations have become some of the fastest-developing areas, due to the recent technological advances and changes, in surgical education. Cutting simulation is a crucial part of the virtual surgery simulation in which an incision operation is performed. It is a complex process that includes three main tasks: soft body simulation, collision detection and handling, and topological deformation of the soft body. In this paper, considering the content developer’s convenience, the deformable object simulation, using position-based dynamics (PBD), was applied in the Unity 3D environment. The proposed algorithm for fast collision detection and handling between the cutting tool and the deformable object uses a sweep surface. In case of incision, the algorithm updates the mesh topology by deleting intersected triangles, re-triangulation, and refinement. In the refinement part, the boundary edges threshold was used to match the resolution of new triangles to the existing mesh triangles. Additionally, current research is focused on triangle surface meshes, which help to reduce the computational costs of the topology modifications. It was found that the algorithm can successfully handle arbitrary cuts, keeping the framerate within interactive and, in some cases, in the real-time.

A novel method to predict surface topography in robotic milling of directional plexiglas considering cutter dynamical displacement

Recently, industrial robots have been applied to the machining of non-metallic materials because of the advantages of low cutting force requirements, low cost, and high flexibility, etc. Due to the limited static/dynamic stiffness of robot joints and links, the accuracy of robotic machining, especially the surface topography, is more susceptible to the cutting forces and dynamic vibrations, compared with CNC machining. To provide clear insights into the formation process of surface topography in robotic machining of oriented Plexiglas and choose reasonably the machining parameters, a method to predict the surface topography in the robotic machining of oriented plexiglas is presented, considering fully the tool dynamic vibrations. In this method, the dynamic intersection behavior in the cutting zones of the robotic machining process is first modeled, which is then used to determine the dynamic displacements. The gained vibration displacements are integrated into the sweep surfaces of the cutter cutting edges by changing the theoretical equations of cutting edges, in order to more accurately calculate the machining scallop points in the presence of tool vibrations. After that, between the discrete sweep surfaces and the workpiece, a mapping-based intersecting method is proposed to compute the intersections with the lowest scallop heights, and these intersections are used to construct the 3D graph, which can represent the topography of the resulting machined surface. Finally, the computer simulation and machining experiments are conducted, and it is shown that the predicted and physically measured surface topography have a good agreement.

Time-Like Sweeping Surfaces with a Bishop Frame in the Minkowski 3-Space E 1 3

In this paper, we consider a time-like sweeping surface and its local singularities with Bishop frame in Minkowski 3-space E 1 3 . Subsequently, we derive the sufficient and necessary conditions for this surface to be spacelike developable ruled surface. In particular, we mainly focus on the study for the resulting spacelike developable surface is a cylinder, cone or tangent surface. Finally, some representative curves are chosen to construct the corresponding spacelike developable surfaces.

Sweeping Surfaces in theThree-Dimensional Lie Group

This paper investigated the rotation minimizing frames that are related to the space curves and the sweeping surfaces that are traced by these frames in the three-dimensional Lie group. Then, the sufficient and necessary conditions for the sweeping surface to be a developable ruled surface were obtained. In particular, we mostly focused on the study of the resulting developable surface is a cylinder, cone, or tangent surface. Meanwhile, to support the results in the paper, some illustrative examples are presented.

On the Timelike Sweeping Surfaces and Singularities in Minkowski 3-Space E 1 3

The Bishop frame or rotation minimizing frame (RMF) is an alternative approach to define a moving frame that is well defined even when the curve has vanished second derivative, and it has been widely used in the areas of computer graphics, engineering, and biology. The main aim of this paper is using the RMF for classification of singularity type of timelike sweeping surface and Bishop spherical Darboux image which is mightily concerning a unit speed spacelike curve with timelike binormal vector in E 1 3 .

Tri-Dexel Based Cutter-Workpiece Engagement Determination For Robotic Machining Simulator

In the spirit of digital twins, computer simulations is a crucial aspect for a technology. Within the frame of virtual manufacturing, and more precisely robotic machining simulation, this article aims to present a simulator developed in C++ for test purposes. It computes the machining forces and the update of the workpiece for 5-axis operations. The proposed method is based on the modelling of workpiece using multi-dexels with scalable resolution and a disk modelled tool with triangle-mesh surfaces. Chip thickness computation approaches are based on the interference between the dexel network with the surface swept by cutting edges either modelled as quadratic surfaces or planar swept surfaces coupled with a quadratic interpolation in the tool space, in the view of reducing computation time. Simulation results of both methods are compared with benchmark operations from the literature in terms of machining forces and with each other in terms of computation performances.

Irregular Symmetrical Object Designed By Using Lambda Miu B-Spline Degree Four

In Computer Aided Geometry Design (CAGD), B-splines curves are piecewise polynomial parametric curves that play an important role. CAGD involves the interpolation and approximation curves and surfaces. CAGD has been widely used which brings good impact of computers to industries in manufacturing. There are many improved methods in the B-spline curve such as extended cubic B-spline, trigonometric B-spline, quasi trigonometric B-spline, and λμ-B-spline. Each of the methods has its behaviour and advantage. In this paper, λμ-B-spline was used to be implemented in generating irregular symmetrical objects. λμ-B-spline has a shape parameter that can change the global shape by manipulating the value of the shape parameter. The bottle has been chosen as an irregular symmetrical object. The 2-dimensional symmetrical curves of Bottle design were formed by using λμ-B-spline degree 4. The curves designed are dependent on the shape parameter which can be adjusted. Then, the curves generated were revolved using the Sweep Surface method to form 3-dimensional objects. Every object has its volume and this research focused on the numerical method which was Simpson’s 3/8 to compute the volume. The volumes obtained were compared to the actual volume to determine the best shape parameter used. The results show that the λμ-B-spline curve with a shape parameter of 1 is the best shape parameter in designing symmetrical irregular objects with the desired volume.

Lines of Curvature for Log Aesthetic Surfaces Characteristics Investigation

Lines of curvatures (LoCs) are curves on a surface that are derived from the first and second fundamental forms, and have been used for shaping various types of surface. In this paper, we investigated the LoCs of two types of log aesthetic (LA) surfaces; i.e., LA surfaces of revolution and LA swept surfaces. These surfaces are generated with log aesthetic curves (LAC) which comprise various families of curves governed by α. First, since it is impossible to derive the LoCs analytically, we have implemented the LoC computation numerically using the Central Processing Unit (CPU) and General Processing Unit (GPU). The results showed a significant speed up with the latter. Next, we investigated the curvature distributions of the derived LoCs using a Logarithmic Curvature Graph (LCG). In conclusion, the LoCs of LA surface of revolutions are indeed the duplicates of their original profile curves. However, the LoCs of LA swept surfaces are LACs of different shapes. The exception to this is when this type of surface possesses LoCs in the form of circle involutes.