Geodesic flows and the mother of all continued fractions

We extend the Series [The modular surface and continued fractions, J. London Math. Soc. (2) 31(1) (1985) 69–80] connection between the modular surface [Formula: see text], cutting sequences, and regular continued fractions to the slow converging Lehner and Farey continued fractions with digits [Formula: see text] and [Formula: see text] in the notation used for the Lehner continued fractions. We also introduce an alternative insertion and singularization algorithm for Farey expansions and other non-semiregular continued fractions, and an alternative dual expansion to the Farey expansions so that [Formula: see text] is invariant under the natural extension map.

Rigidity Regulation Approach for Geometric Tolerance Optimization in End Milling of Thin-walled Components

The paper presents a novel approach to improve geometric tolerances (flatness and cylindricity) by manipulating the rigidity among finishing and roughing cutting sequences during end milling of thin-walled components. The proposed approach considers the design configuration of the thin-walled component as an input and aims to determine semi-finished geometry such that the geometric tolerances are optimized while performing finish cutting sequence. The objective is accomplished by combining Mechanistic force model, Finite Element (FE) analysis based workpiece deflection model and Particle Swarm Optimization (PSO) technique to determine optimal disposition of material along the length of component thereby regulating rigidity. The algorithm has been validated by determining rigidity regulated semi-finished geometries for thin-walled components having straight, concave and convex configurations. The outcomes of the proposed algorithm are substantiated further by conducting a set of end milling experiments for each of these cases. The results of the proposed strategy are compared with a traditional approach considering no change in the rigidity of component along length of the cut. It is demonstrated that the proposed approach can effectively optimize geometric tolerances for thin-walled components during end milling operation.

Experimental investigation of shear cutting techniques for fiber-reinforced-plastics-metal-laminates

Hybrid structures made of fiber-reinforced plastics (FRP) and metals are currently in focus of research and industry to develop weight reduced and functional optimized components for lightweight solutions. Manufacturing processes were adapted and developed to produce components based on hybrid materials with high economic efficiency. The cutting process is used to pre-assemble the semi-finished products or to post-process the edges of consolidated parts. The mechanisms of damage edge behavior and possible cutting qualities on these parts are not investigated jet. To close this knowledge gap and to support the future application of hybrid FRP-Metal-Laminates different cutting procedures were studied. This paper shows the process related dependences on the failure behavior of two dimensional specimens. The failure modes are described via quality characteristics like surface roughness, trueness and precision of the cut as well as influences of aging processes. In the end optimized parameter for each process are shown and compared under technical and economic criteria for large scale production. In the scope of this work an experimental study of piercing of glass and carbon fiber reinforced thermoplastic with different steel and bonding agents at different cutting sequences were performed. It was shown that the cutting edge geometry significantly differs. Possible mechanical explanations of the dependencies were formulated. Also the accuracy of the cuts was evaluated which showed a higher accuracy for the steel component. The measurements on the surface roughness could not show any dependencies and relations.

Case Study and Numerical Simulation of Excavation beneath Existing Buildings

Excavation beneath existing buildings may cause the superstructure to tilt and crack, which seriously affects the normal use of the superstructure. Due to the new working conditions of excavation beneath existing buildings, related case reports are rare and limited. In the case of No. 3 section basement construction project of Ganshuixiang, we monitored the excavation construction by burying test instruments at the designated location. Afterwards, Plaxis 3D finite element software was used to establish an underpinning pile-cap-excavation model, which can analyze the influence of different pile cutting sequences on the bearing behavior of new basement structural pillars. By comparing the in situ measurement data with the finite element model, it can be concluded that when the excavation depth rises, the axial force of the underpinning pile gradually increases, and the pile skin friction is slowly exerted from top to bottom. Different cutting sequences will influence the bearing behavior of the structural pillar. Moreover, the pile cutting process also significantly impacts its bearing behavior and the settlement behavior of the superstructure. Compared with the clockwise pile cutting sequence, the symmetrical pile cutting is more advantageous. In the whole process of the storey adding and reconstruction, the superstructure settlement is related to the working condition of digging and adding layers. In the stage from soil excavation to the concrete curing period of the structural pillar, it increases slowly with time and tends to be stable in the concrete curing period. However, in the pile cutting stage, the superstructure settlement increases sharply, and after pile cutting, it becomes stable.

The conformal measures of a normal subgroup of a cocompact Fuchsian group

In this paper we study the conformal measures of a normal subgroup of a cocompact Fuchsian group. In particular, we relate the extremal conformal measures to the eigenmeasures of a suitable Ruelle operator. Using Ancona’s theorem, adapted to the Ruelle operator setting, we show that if the group of deck transformations G is hyperbolic then the extremal conformal measures and the hyperbolic boundary of G coincide. We then interpret these results in terms of the asymptotic behavior of cutting sequences of geodesics on a regular cover of a compact hyperbolic surface.

Biomechanical Analysis of the Individual Ligament Contributions to Syndesmotic Stability

Background: Biomechanical data and contributions to ankle joint stability have been previously reported for the individual distal tibiofibular ligaments. These results have not yet been validated based on recent anatomic descriptions or using current biomechanical testing devices. Methods: Eight matched-pair, lower leg specimens were tested using a dynamic, biaxial testing machine. The proximal tibiofibular joint and the medial and lateral ankle ligaments were left intact. After fixation, specimens were preconditioned and then biomechanically tested following sequential cutting of the tibiofibular ligaments to assess the individual ligamentous contributions to syndesmotic stability. Matched paired specimens were randomly divided into 1 of 2 cutting sequences: (1) anterior-to-posterior: intact, anterior inferior tibiofibular ligament (AITFL), interosseous tibiofibular ligament (ITFL), deep posterior inferior tibiofibular ligament (PITFL), superficial PITFL, and complete interosseous membrane; (2) posterior-to-anterior: intact, superficial PITFL, deep PITFL, ITFL, AITFL, and complete interosseous membrane. While under a 750-N axial compressive load, the foot was rotated to 15 degrees of external rotation and 10 degrees of internal rotation for each sectioned state. Torque (Nm), rotational position (degrees), and 3-dimensional data were recorded continuously throughout testing. Results: Testing of the intact ankle syndesmosis under simulated physiologic conditions revealed 4.3 degrees of fibular rotation in the axial plane and 3.3 mm of fibular translation in the sagittal plane. Significant increases in fibular sagittal translation and axial rotation were observed after syndesmotic injury, particularly after sectioning of the AITFL and superficial PITFL. Sequential sectioning of the syndesmotic ligaments resulted in significant reductions in resistance to both internal and external rotation. Isolated injuries to the AITFL resulted in the most substantial reduction of resistance to external rotation (average of 24%). However, resistance to internal rotation was not significantly diminished until the majority of the syndesmotic structures had been sectioned. Conclusion: The ligaments of the syndesmosis provide significant contributions to rotary stability of the distal tibiofibular joint within the physiologic range of motion. Clinical Relevance: This study defined normal motion of the syndesmosis and the biomechanical consequences of injury. The degree of instability was increased with each additional injured structure; however, isolated injuries to the AITFL alone may lead to significant external rotary instability.

Using Advanced CAM System in Modern Machining

Goal of this paper highlight characteristics and spectrum of machining cutting sequences that programming system SolidCAM support. The practical goal of this paper is defining post-processor and machine simulation for 3-axis CNC machine like a tool for verification modern tool-path and generation G-code that will be used for cutting real part.