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"Angles (geometry)"
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Degree Spectra of Relations on a Cone
Let \\mathcal A be a mathematical structure with an additional relation R. The author is interested in the degree spectrum of R, either among computable copies of \\mathcal A when (\\mathcal A,R) is a \"natural\" structure, or (to make this rigorous) among copies of (\\mathcal A,R) computable in a large degree d. He introduces the partial order of degree spectra on a cone and begin the study of these objects. Using a result of Harizanov--that, assuming an effectiveness condition on \\mathcal A and R, if R is not intrinsically computable, then its degree spectrum contains all c.e. degrees--the author shows that there is a minimal non-trivial degree spectrum on a cone, consisting of the c.e. degrees.
eBook
Multi-objective integrated optimization of tool geometry angles and cutting parameters for machining time and energy consumption in NC milling
The manufacturing industry has a large volume and a wide range of energy consumption. It is the main body of energy consumption in the industrial field. In recent years, a great challenge has been posed to the manufacturing industry to improve sustainable development for low energy consumption and high efficiency. The energy efficiency of manufacturing systems has become an important research hotspot in the sustainable development of the world’s manufacturing. Aiming at the problem of low energy efficiency in the current machining process, this paper considers the relationship between tool geometric angles and energy consumption, and on the basis of the current energy consumption model establishes an energy consumption model with independent variables including cutting parameters and tool geometric angles. In order to achieve energy-saving and high efficiency, a multi-objective optimization model was established with cutting parameters and tool geometric angles as optimization variables, combined with actual machine tool and process parameter constraints. The model is solved by the elitist non-dominated sorting genetic algorithm (NSGA-II), and an optimized combination of cutting parameters and tool geometric angle is obtained. Through the comparison experiments conducted with the processing methods that are respectively based on the experience and simply optimize the cutting parameters in the actual milling test, the practivity and validity of the method proposed are verified. The test results show that integrated optimization of cutting parameters and tool geometric angles can reduce energy consumption by 8.77% at least.
Journal Article
Study on high precision calculation method of geometric parameters of blade profile
Two-dimensional blade profile design is the basis of three-dimensional blade modeling. The important parameters reflecting the geometric characteristics of the blade profile are chord length, maximum thickness, inlet and outlet metal angle, installation angle, etc. These parameters are the basis of blade profile design and performance analysis, and it is of great significance to calculate these parameters accurately. Based on the original blade profile data, this paper puts forward a high-precision blade profile parameter extraction method and obtains detailed geometric parameters, including small circle, middle arc and maximum thickness of the blade profile, which lays a foundation for blade profile design and analysis. It is verified that the proposed method has high accuracy and good engineering applicability.
Journal Article
A polyhedron comparison theorem for 3-manifolds with positive scalar curvature
The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by Gromov. For a large collections of polyhedra with interior non-negative scalar curvature and mean convex faces, we prove the dihedral angles along its edges cannot be everywhere less or equal than those of the corresponding Euclidean model, unless it is isometric to a flat polyhedron.
Journal Article
Hydrogel muscles powering reconfigurable micro-metastructures with wide-spectrum programmability
Stimuli-responsive geometric transformations endow metamaterials with dynamic properties and functionalities. However, using existing transformation mechanisms to program a single geometry to transform into diverse final configurations remains challenging, imposing crucial design restrictions on achieving versatile functionalities. Here, we present a programmable strategy for wide-spectrum reconfigurable micro-metastructures using linearly responsive transparent hydrogels as artificial muscles. Actuated by the hydrogel, the transformation of micro-metastructures arises from the collaborative buckling of their building blocks. Rationally designing the three-dimensional printing parameters and geometry features of the metastructures enables their locally isotropic or anisotropic deformation, allowing controllable wide-spectrum pattern transformation with programmable chirality and optical anisotropy. This reconfiguration mechanism can be applied to various materials with a wide range of mechanical properties. Our strategy enables a thermally reconfigurable printed metalattice with pixel-by-pixel mapping of different printing powers and angles for displaying or hiding complex information, providing opportunities for encryption, miniature robotics, photonics and phononics applications.It is difficult to program a single stimuli-responsive geometry to transform into diverse final configurations in a systematic manner. Here, linearly responsive transparent hydrogels are developed to create micro-metastructures with wide-spectrum thermal reconfigurability.
Journal Article
High-resolution tomographic volumetric additive manufacturing
In tomographic volumetric additive manufacturing, an entire three-dimensional object is simultaneously solidified by irradiating a liquid photopolymer volume from multiple angles with dynamic light patterns. Though tomographic additive manufacturing has the potential to produce complex parts with a higher throughput and a wider range of printable materials than layer-by-layer additive manufacturing, its resolution currently remains limited to 300 µm. Here, we show that a low-étendue illumination system enables the production of high-resolution features. We further demonstrate an integrated feedback system to accurately control the photopolymerization kinetics over the entire build volume and improve the geometric fidelity of the object solidification. Hard and soft centimeter-scale parts are produced in less than 30 seconds with 80 µm positive and 500 µm negative features, thus demonstrating that tomographic additive manufacturing is potentially suitable for the ultrafast fabrication of advanced and functional constructs.
Tomographic additive manufacturing produces complex parts with a wide range of printable materials but remains limited in terms of resolution. Here, the authors tune the étendue of the light source and accurately control the photopolymerization kinetics using an integrated feedback system, leading to the fabrication of high resolution features.
Journal Article
Programming shape using kirigami tessellations
Kirigami tessellations, regular planar patterns formed by partially cutting flat, thin sheets, allow compact shapes to morph into open structures with rich geometries and unusual material properties. However, geometric and topological constraints make the design of such structures challenging. Here we pose and solve the inverse problem of determining the number, size and orientation of cuts that enables the deployment of a closed, compact regular kirigami tessellation to conform approximately to any prescribed target shape in two or three dimensions. We first identify the constraints on the lengths and angles of generalized kirigami tessellations that guarantee that their reconfigured face geometries can be contracted from a non-trivial deployed shape to a compact, non-overlapping planar cut pattern. We then encode these conditions into a flexible constrained optimization framework to obtain generalized kirigami patterns derived from various periodic tesselations of the plane that can be deployed into a wide variety of prescribed shapes. A simple mechanical analysis of the resulting structure allows us to determine and control the stability of the deployed state and control the deployment path. Finally, we fabricate physical models that deploy in two and three dimensions to validate this inverse design approach. Altogether, our approach, combining geometry, topology and optimization, highlights the potential for generalized kirigami tessellations as building blocks for shape-morphing mechanical metamaterials.
Journal Article
Optimal design of the toggle mechanism in servo presses
To augment the stretching capabilities of servo presses, an optimal transmission mechanism design is imperative. Firstly, a kinematic model of the servo press transmission system was established by employing the closed-vector formulation. A normalized composite fitness function, integrating the mechanism’s mechanical advantage, transmission angle, and slider velocity fluctuation as optimization criteria, was constructed to quantify the overall performance. Subsequently, a genetic algorithm was implemented to execute the optimization design under geometric, kinematic, transmission angle, and stroke-speed ratio constraints. The findings indicate that the optimized transmission mechanism reduces the maximum speed during the working stroke by 54.40%, increases the slider displacement during the working stage by 2.27%, and increases the proportion of the working stroke by 96.01%, effectively improving the overall forging performance of the press.
Journal Article
Configurable phonon polaritons in twisted α-MoO3
Moiré engineering is being intensively investigated as a method to tune the electronic, magnetic and optical properties of twisted van der Waals materials. Advances in moiré engineering stem from the formation of peculiar moiré superlattices at small, specific twist angles. Here we report configurable nanoscale light–matter waves—phonon polaritons—by twisting stacked α-phase molybdenum trioxide (α-MoO
3
) slabs over a broad range of twist angles from 0° to 90°. Our combined experimental and theoretical results reveal a variety of polariton wavefront geometries and topological transitions as a function of the twist angle. In contrast to the origin of the modified electronic band structure in moiré superlattices, the polariton twisting configuration is attributed to the electromagnetic interaction of highly anisotropic hyperbolic polaritons in stacked α-MoO
3
slabs. These results indicate twisted α-MoO
3
to be a promising platform for nanophotonic devices with tunable functionalities.
Infrared nanoimaging of phonon polaritons in twisted α-phase molybdenum trioxide bilayers reveals tunable wavefront geometries and topological transitions over a broad range of twist angles, offering a configurable platform for nanophotonic applications.
Journal Article