Center Of Curvature Research Articles

A Note on Equivalent Linkages of Direct-Contact Mechanisms

In this paper, the inequivalence of the direct-contact mechanisms and their equivalent four-bar linkages in jerk analysis is discussed. Kinematic analyses for three classical types of direct-contact mechanisms consisting of: (a) higher pairs with permanently invariant curvature centers, (b) higher pairs with suddenly changed curvature, and (c) higher pairs with continuously varying curvature are performed, respectively, through their representative case studies. The analyzed results show that the equivalent four-bar linkage cannot give a correct value of jerk for most situations in the three case studies. Subsequently, the concept of “equivalent six-bar linkage” for direct-contact mechanisms is proposed in order to discuss the infeasibility of the equivalent four-bar linkage for jerk analysis. It is found that the suddenly changed or continuously varying curvature of the higher pairs is not considered in sudden or continuous link-length variations of the equivalent four-bar linkage, which further leads to inconsistency between the angular accelerations of the coupler and the contact normal, and finally results in the infeasibility of the equivalent four-bar linkage for jerk analysis of most direct-contact mechanisms. It is also found that the concept of equivalent six-bar linkage could be applied to evaluate more higher-order time derivatives for most direct-contact mechanisms. The presented case studies and discussion can give demonstrations for understanding the inequivalence of the direct-contact mechanisms and their equivalent four-bar linkages in the aspect of jerk analysis.

Dual Waverider to Integrate External and Internal Flows

This paper proposes a method of integration of internal and external flow called the dual waverider. A three-dimensional shock structure is designed to simultaneously satisfy the theories of internal and external waveriders. By using the infinite radius of curvature and omnidirectional center of curvature of a planar shock wave, external flows are made to transition smoothly to internal flows in the cross section. Moreover, a compression efficiency factor (CEF) is proposed to integrate internal and external flows in sections along the flow. Different values of the CEF represent internal/planar/external flows. An integration methodology for the dual waverider is then proposed by combining the two flows. By using this design, various structures of the dual waverider are obtained for different conditions. The theory of the dual waverider is also improved to develop one with double internal passages. Three-dimensional numerical simulations were performed on different configurations of the dual waverider to verify the correctness of the proposed design. The results showed good agreement with those obtained during the design process. Research on the aerodynamic characteristics of the configuration indicates that the dual waverider can integrate internal and external flows and can maintain a high lift-to-drag ratio.

Analysis of high speed bearing based on virtual rods model

The traditional high-speed ball bearing analysis is based on the quasi-static model proposed by Jones-Harris (J-H). This study gives another bearing analysis model based on the relative position of the ball center and the centers of curvature of the raceway grooves, described by the virtual rods. Specifically, the load and deformation of the virtual rods are determined according to the principle of force transmissibility, the translation theorem of a force, and elastic-contact deformation between steel ball bearings and raceway grooves. And the iterative variables such as the axial, radial and angular deformations are gradually introduced into the calculation model to solve the force and kinematic parameters under pure axial load, bidirectional and tri-directional combined loads. The correctness of the virtual rods model is verified by the contact angle for axial and bidirectional load. The influence of gyroscopic moment and raceway grooves control parameter on the actual contact angle under tridirectional loading conditions are calculated and analyzed respectively. The proposed model uses fewer equations and lower-dimension parameters than those used by the traditional highspeed ball bearing analysis method, improves the solution rate of the equations.

Optical beam shift as a vectorial pointer of curved-path geodesics: an evolution-operator perspective.

When set to travel along a curved path, e.g., in a bending-waveguide setting, an optical beam tends to re-adjust its position, shifting away from the center of path curvature. This shift is highly sensitive to the spatial profile of the refractive index, providing a vectorial pointer for curved-path geodesics and bending-induced optical tunneling. An evolution-operator analysis of this effect extends an analogy with a time-evolution-operator treatment of quantum dynamics and suggests the routes whereby the ability of an optical beam to sense curved-path geodesics can be understood in terms of the pertinent evolution operators, path integrals, and imaginary-time/path theorems.

Three-dimensional curvature mismatch of the acetabular radius to the femoral head radius is increased in borderline dysplastic hips.

Whether borderline hip dysplasia is pathologic remains unclear. In order to evaluate the three-dimensional joint congruity, this study sought to answer the question: are borderline dysplastic hip curvature mismatch and eccentricity between the acetabulum and the femoral head different from dysplastic or control hips three-dimensionally? The 113 hips, categorized as: dysplastic (LCEA ≤ 20°), 47 hips; borderline (20° ≤ LCEA < 25°), 32 hips; and control (25° ≤ LCEA < 35°), 34 hips; were evaluated. Three-dimensional (3D) femoral and coxal bone models were reconstructed from CT images. Using a custom-written Visual C++ routine, the femoral head and acetabular radii of curvature, and the femoral head and the acetabular curvature center were calculated. Then the ratio of the acetabular radius to the femoral head radius (3D curvature mismatch ratio), and the distance between the acetabular curvature center and the femoral head center (3D center discrepancy distance) were calculated. These indices were compared statistically among the three groups using Tukey’s post hoc test. The mean 3D curvature mismatch ratio in the borderline (1.13 ± 0.05) was smaller than in the dysplasia (1.23 ± 0.08, p < 0.001), and larger than in the control (1.07 ± 0.02, p < 0.001). The mean 3D center discrepancy distance in the borderline (3.2 ± 1.4 mm) was smaller than in the dysplasia (4.8 ± 2.3, p < 0.001) and larger than in the control (1.6 ± 0.7, p < 0.001). These results demonstrated that three-dimensional congruity of the borderline dysplastic hip is impaired, but its incongruity is not as severe as in dysplastic hips. The 3D curvature mismatch ratio and the 3D center discrepancy distance can be valuable signs of joint congruity in patients with borderline dysplasia. However, future studies are necessary to clarify any associations between curvature mismatch and pathogenesis of osteoarthritis in borderline dysplasia.

High-precision autofocus using a double wedge splitter

We present an autofocus technique that combines focusing technology by using a double wedge splitter and computer image processing technology in order to achieve high-precision focusing. The focusing technique we propose uses a cooperation target (a ring object) as an imaging object. The ring image is divided into two half-ring images by using a double wedge splitter, which extracts their edge information for curve fitting to obtain the center of curvature of the two half rings so we can calculate the deviation distance between the two half-ring images. The control system uses the deviation distance as feedback, and the real-time closed loop adjusts the relative position of the secondary imaging system and the double wedge splitter so that the deviation distance is the smallest and the defocusing amount is also the smallest. In this paper, we propose the optical system model, analyze the algorithm flow in detail, and verify the feasibility of the focusing method by experiments. The results show that this focusing method has the advantages of high focusing accuracy and easy operation. The defocusing amount of this technology is within 10 µm.

Defining the Position of the Right Wall of the Esophageal Hiatus to Identify the Circumferential Distribution of Small Lesions of the Lower Esophagus.

The reliability of methods for identifying the circumferential position of small lower esophageal lesions is unknown. We prospectively investigated a new method that presents lesion positions as times on a clock face. Eighty-seven patients were consecutively examined by endoscopy. After observing the esophagus, an endoscope was inserted into the stomach and fixed, and the greater curvature folds at the upper gastric corpus were set as horizontal on the endoscope monitor display. The scope was retrogressed into the lower esophagus. At this point, the right wall at the hiatus is at the 3 o'clock position (R-line). The scope was then retrogressed from the gastric angle to the cardia along the center of the lesser curvature in the retroflexed view to obtain the LC-line (the center of the lesser curvature at the cardia). The LC-line in the esophageal hiatus in the frontal view was then identified, and the angle between the R- and LC-lines (R-LC) was measured. After excluding 7 patients with hernias >2 cm and 3 with esophageal stenosis, data from 77 patients were analyzed. The R-LC angle ranged from -38° to +35°. The mean R-LC angle was -0.3°± 15.9°, and its 95% confidence interval was [-4.0°, 3.3°] within [-15°, + 15°]. When indicating lesion locations as times on a clock face, there was an error of ±30 min (±15°); therefore, R- and LC-lines were shown to be identical on an equivalence test. This new method allows the circumferential position of small lower esophageal lesions to be reliably represented as a clock face.

Technical Note: Benchmarking automated eye tracking and human detection for motion monitoring in ocular proton therapy

Ocular proton therapy is an effective therapeutic option for patients affected with uveal melanomas. An optical eye-tracking system (ETS) aiming at noninvasive motion monitoring was developed and tested in a clinical scenario. The ETS estimates eye position and orientation at 25 frames per second using the three-dimensional position of pupil and cornea curvature centers identified, in the treatment room, through stereoscopic optical imaging and infrared eye illumination. Its capabilities for automatic detection of eye motion were retrospectively evaluated on 60 treatment fractions. Then, the ETS performance was benchmarked against the clinical standard based on visual control and manual beam interruption. Eye-tracking system detected eye position successfully in 97% of all available frames. Eye-tracking system-based eye monitoring during therapy guarantees quicker response to involuntary eye motions than manual beam interruptions and avoids unnecessary beam interruptions. Eye-tracking system shows promise for on-line monitoring of eye motion. Its introduction in the clinical workflow will guarantee a swifter treatment course for the patient and the clinical personnel.

Circular-Arc Radial Cams with One Connection Arc

This paper deals with the geometric and kinematic analysis of the circular-arc profile cams with one connection arc. If the maximum lift of the follower is required, it is shown that it is possible to connect two circular-arcs – that are defined by center of curvature and radius – through a single circular-arc, and the connection points result. But, if the connection points of two given circular-arcs of the cam profile are required, at least two circular-arcs are needed to connect them. The specific equations for the geometric analysis for one circular-arc profile are described. Also, for two mechanisms, one with straight-line profile and another with one circular-arc connection profile, the geometric and kinematic analysis and simulations of the movements using SolidWorks and ADAMS programs are presented

Cauchy's Work on Integral Geometry, Centers of Curvature, and Other Applications of Infinitesimals

Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry where infinitesimals are used neither as variable quantities nor as sequences but rather as numbers. He also applied infinitesimals in an 1832 article on integral geometry, similarly as numbers. We explore these and other applications of Cauchy's infinitesimals as used in his textbooks and research articles. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis and geometry. We demonstrate the viability of Cauchy's infinitesimal techniques in fields as diverse as geometric probability, differential geometry, elasticity, Dirac delta functions, continuity and convergence. Keywords: Cauchy--Crofton formula; center of curvature; continuity; infinitesimals; integral geometry; limite; standard part; de Prony; Poisson

The lever rounding mechanisms

Mechanisms with complex movement of a curved executive organ rounding around a fixed set line, in the simplest case – direct line or circular arc – can be combined with guiding mechanisms in the analysis and synthesis tasks, if one assumes the center of curvature of the generator of the executive organ to be a guiding point. Moreover, the length of the rounding segment and the parameters of deviation can be also defined through the parameters of the guiding mechanism, whereby sliding in a reciprocal rounding depends on the generator’s curve radius. Unlike the less applied guiding mechanisms lever rounding mechanisms can find a broad application as executive organs in pressure-shaping machines (e.g. forging, molding, cutting, chipping, clamping and compacting), stepping and finned-type propelling devices and other unique mechanisms with a common feature of indifference of the structure and integration of qualities that are non-compatible at a simple movement of an executive organ.

The growth behavior of columnar grains in a TiAl alloy during directional induction heat treatments

In the process of DHT, the curving grain boundary will move towards curvature center of grain under the action of interface tension.

Determining the width of the workpiece under bending conditions with stretching of the stamped part

Aims of research. This paper deals with the problem of determining the radius of the neutral surface of the bending strip at the same time the action of tensile stresses. Methods. In the development of technological processes of sheet bending and the calculation of the initial billet (sweep), most often considered a flat stress state and preference is given to the theory of pure bending, providing uniform curvature of the individual layers of metal under the action of the bending moment and neglecting the action of transverse tensile forces. However, in reality, bending is always accompanied by sliding of the metal along the deforming tool, causing the appearance of friction forces and creating a complex loading in the form of plastic bending with stretching in the metal, which will additionally displace the neutral surface, forcing the experienced one to adjust the technology of manufacturing bent parts. Provides an overview and analysis of the most famous works dedicated to the topic, specifying the distribution of the contact stresses from the inner surface to the center of curvature and developed the theory of complex bending, which allows to specify the radius of the neutral surface in conditions of stretching of the flexible strip of the additional tangential stresses. Results. The results obtained will allow to simulate the stress state of the metal in the development of technological processes of sheet stamping and, in particular, to assess the specific braking force during the bending (straightening) of the strip in the process of pulling through the thresholds and brake edges of the die tooling to eliminate possible defects in the elongated parts and to clarify the dimensions of the workpiece and the finished profile.

Dynamic characteristic analysis and shaking table test for a curved surface isolated structure

To reduce the displacement of an isolation layer, an innovative curved surface isolated structure (CSIS) system was put forward by using the curved surface characteristics of the isolation layer and the weight of the structure. Firstly, a simplified analysis model with multiple isolators considering the rotation of the upper structure around the center of curvature of the isolation layer and the oscillation around the center of the isolation layer was established. Subsequently, two CSIS models, with 4 degrees inclined isolators and 8 degrees inclined isolators, separately, were produced and a shaking table test was conducted. For comparison purpose, a normal isolated structure (NIS) model and a non-isolated structure (NS) model were also tested. The dynamic characteristics, acceleration and displacement responses of different models are compared and discussed. Finally, finite element models of the CSIS were established by using the software ABAQUS and the numerical simulation results were compared with the test results. The results show that the displacement responses of a CSIS are reduced and the CSIS can return to the initial position faster than the NIS; the acceleration response of a CSIS is slightly increased relative to the NIS, but significantly reduced compared to the NS; the acceleration time-history curves of each floor and the displacement time-history curves of the isolation layer for numerical simulation results show good agreement with experimental data.

Resistance to cyclic fatigue of reciprocating instruments determined at body temperature and phase transformation analysis.

The purpose of this study was to compare cyclic fatigue (CF) resistance of reciprocating instruments at body temperature and relate the findings to their martensitic transformation temperatures. Contemporary nickel-titanium (NiTi) reciprocating instruments WaveOne Primary, WaveOne Gold Primary and EdgeFile X1 (n=20 each and #25 tip diameter) were tested for CF resistance at body temperature (37±1°C). Instruments were actioned according to manufacturer guidelines until fracture occurred in a simulated canal (angle of curvature=60°, radius of curvature=3mm and centre of curvature=5mm from the tip). Time to fracture was recorded, and data analysed using Weibull analysis. Two instruments of each were tested using differential scanning calorimetry (DSC) to assess phase transformation temperatures. Reciprocating instruments manufactured with new alloys seem to be safer to CF than those manufactured with traditional M-Wire at body temperature. Martensitic transformation temperatures seem not to relate with fatigue behaviour for reciprocating motions.

Whispering gallery waves localized near circular isobaths in shallow water.

Sound propagation in an area of shallow sea with circular isobaths and the bottom depth increasing towards their curvature center is considered. Such an area could be part of a bay, a lagoon, or a lake with bowl-like bottom relief. The possible formation of whispering gallery waves localized in the vicinity of curved isobaths is considered. Simple criteria for the formation of such whispering gallery due to horizontal refraction are derived; their transverse modal structure is described analytically in terms of Bessel functions and using the Wentzel-Kramers-Brillouin-Jeffreys approximation. The corresponding azimuthal wavenumbers and radial mode functions are computed. Dispersion of such waves is also investigated, and the possibility of their experimental observation is discussed.

Testing lenses and reflecting concave surfaces using a knife-edge interferometer.

In a previous paper by Cebrian et al. [Opt. Eng.53, 092006 (2014)OPEGAR0091-328610.1117/1.OE.53.9.092006], the knife-edge interferometer (KEI) was used for testing lenses with a collimated beam illuminating the lens. In this paper, a 2f scheme is used to test lenses, eliminating the use of a collimation beam, and testing, for the first time, concave reflecting (spherical and parabolic) surfaces at the center of curvature with KEI.

Influence of Convex and Concave Curvatures in a Coastal Dike Line on Wave Run-up

Due to climatic change and the increased usage of coastal areas, there is an increasing risk of dike failures along the coasts worldwide. Wave run-up plays a key role in the planning and design of a coastal structure. Coastal engineers use empirical equations for the determination of wave run-up. These formulae generally include the influence of various hydraulic, geometrical and structural parameters, but neglect the effect of the curvature of coastal dikes on wave run-up and overtopping. The scope of this research is to find the effects of the dike curvature on wave run-up for regular wave attack by employing numerical model studies for various dike-opening angles and comparing it with physical model test results. A numerical simulation is carried out using DualSPHysics, a mesh-less model and OpenFOAM, a mesh-based model. A new influence factor is introduced to determine the influence of curvature along a dike line. For convexly curved dikes (αd = 210° to 270°) under perpendicular wave attack, a higher wave run-up was observed for larger opening angles at the center of curvature whereas for concavely curved dikes (αd = 90° to 150°) under perpendicular wave attack, wave run-up increases at the center of curvature as the opening angle decreases. This research aims to contribute a more precise analysis and understanding the influence of the curvature in a dike line and thus ensuring a higher level of protection in the future development of coastal structures.

Inequalities for curvature integrals in Euclidean plane

Let γ be a closed strictly convex curve in the Euclidean plane mathbb{R}^{2} with length L and enclosing an area A, and tilde{A}_{1} denote the oriented area of the domain enclosed by the locus of curvature centers of γ. Pan and Xu conjectured that there exists a best constant C such that \t\t\tL2−4πA≤C|A˜1|,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} L^{2}-4\\pi A\\leq C \\vert \\tilde{A}_{1} \\vert , \\end{aligned}$$ \\end{document} with equality if and only if γ is a circle. In this paper, we give an affirmative answer to this question. Moreover, instead of working with the domain enclosed by the locus of curvature centers we consider the domain enclosed by the locus of width centers of γ, and we obtain some new reverse isoperimetric inequalities.

Roll-over analysis of a new design carbon fiber prosthetic foot

Many researchers adopted the quasi static roll over testing in the calculation of a prototype prosthetic feet roll over shape. In this work, the prototype of a new carbon fiber prosthetic foot was tested for roll over characteristic. An inverted pendulum like apparatus has been built to achieve the required test, and the prosthesis roll over shape calculated. Roll over characteristics were estimated as: effective radius of curvature about 202mm, center of curvature of 26.55mm, and the forward travel against shank angle was plotted.