Cassini Oval Research Articles

Evaluation of Maize Seed Sieving Performance using Sieves with CassiniOval Shape Openings

The process of preliminary cleaning is a determining factor affecting the standard qualities of the seeds. The grain mass that undergoes primary processing contains significant amounts of impurities, the amount of which is strictly regulated by standards. Sieves with rectangular and round openings are the most widely used for separating maize seeds. In this study, the feasibility and efficiency of using sieves with openings in the Cassini oval shape were proposed and substantiated. Three factors subject to variation during the experiments were selected, each having a crucial influence on the process parameters: rotational speed (n, rpm), specific material feed (Q, kg), and angle of sieve inclination (α, deg.). A regression equation was developed, and the response surfaces of the dependencies of defining characteristics of the process of separation of maize seeds on sieves with openings in the Cassini oval shape were compiled and analyzed. The study showed that the rotational speed has a less significant influence on the mass of the undersize of sieved maize than the inclination angle and feeding rate. A maximum grain passage (2.9 kg) into the openings in the Cassini oval shape is obtained at a rotational speed of 290.35 min-1 and a grain feed rate of 3.06 kg min-1.

Bimodal Droplet-Based Electricity Generation Through Semi Cassini Oval Dynamic Morphology Control.

Droplet-based electricity generator (DEG) is the promising energy harvesting technology applicable in versatile scenarios. Despite numerous optimizations in DEG's materials and structures, few has paid attention to the droplet dynamics and morphology control. Here the droplet's spread-retraction dynamics and the resultant semi Cassini oval (SCO) morphology are reported, characterized by convex at both ends and concave in the middle. The lifted top electrode (LTE) is designed to respond to the dynamic SCO morphology in two steps: 1) LTE first contacts the leading edge of the spreading droplet, generating a negative voltage peak; 2) LTE second contacts the trailing edge of the retracting droplet, generating a positive voltage peak. In this way, bimodal electrical output is obtained, which exhibits 25% increase in peak-to-peak voltage and 33% increase in average power compared to the traditional DEG with only one voltage peak. These results prove that incorporating the droplet dynamics and morphology control into DEG design uplifts the energy harvesting limit from individual droplet. The proposed LTE-DEG inspires alternative route for DEG power enhancement by maximizing energy utilization of individual droplet from the perspective of droplet dynamic morphology design.

Geometric properties of integrable Kepler and Hooke billiards with conic section boundaries

We study the geometry of reflection of a massive point-like particle at conic section boundaries. Thereby the particle is subjected to a central force associated with either a Kepler or Hooke potential. The conic section is assumed to have a focus at the Kepler center, or have its center at the Hookian center respectively. When the particle hits the boundary it is ideally reflected according to the law of reflection. These systems are known to be integrable.We describe the consecutive billiard orbits in terms of their foci. We show that the second foci of these orbits always lie on a circle in the Kepler case. In the Hooke case, we show that the foci of the orbits lie on a Cassini oval. For both systems we analyze the envelope of the directrices of the orbits as well.

Hydroforming and buckling of a longan-shaped pressure vessel with longitudinal tension rods

ABSTRACT The hydroforming process of a longan-shaped multi-segmented preform with longitudinal tension rods and the buckling behaviour of a hydroformed longan-shaped pressure vessel were numerically and experimentally investigated. The shape of the longan followed the Cassini oval equation, with a shape index of 0.1. A preform with a nominal size of 400 mm for the major axis, 396 mm for the minor axis and a wall thickness of 1 mm was fabricated. Two longitudinal tension rods with a radius of 3 mm and a length of 420 mm were employed in the multi-segmented preform, which was the first to use longitudinal tension rods in the preform. The numerical analysis and experimental results showed that the excessive deformation of the polar plates was significantly reduced. Simultaneously, the overall geometric accuracy of the hydroformed pressure vessel was significantly improved. The results of the analytical, experimental and numerical approaches are in good agreement with one another.

Cassini-oval description of atomic binding: New insights into the quantitative relationship between hardness coefficient and bond energy energy.

In this paper, the research hypothesis of conservation of hardness coefficient is put forward, and a mathematical formula for describing and analyzing the relationship between hardness coefficient and bond energy is established in this model. The binding process of two heteronuclear atoms can be represented by Cassini oval in dynamic form, every molecular state corresponds to one of these graphs; then the critical phenomena of molecular deformation are discovered and the calculated potential energy at the critical point is consistent with the experimental dissociation energy of molecules (R is 0.99999, P < 0.0001).

Suzuki type $\mathcal{Z}_{c}$-contraction mappings and the fixed-figure problem

Geometric approaches are important for the study of some real-life problems. In metric fixed point theory, a recent problem called fixed-figure problem is the investigation of the existence of self-mapping which remain invariant at each points of a certain geometric figure (e.g. a circle, an ellipse and a Cassini curve) in the space. This problem is well studied in the domain of the extension of this line of research in the context of fixed circle, fixed disc, fixed ellipse, fixed Cassini curve and so on. In this paper, we introduce the concept of a Suzuki type $\mathcal{Z}_c$-contraction. We deal with the fixed-figure problem by means of the notions of a $\mathcal{Z}_c$-contraction and a Suzuki type $\mathcal{Z}_c$-contraction. We derive new fixed-figure results for the fixed ellipse and fixed Cassini curve cases by means of these notions. Also fixed disc and fixed circle results given for Suzuki type $\mathcal{Z}_c$-contraction. There are couple of illustration related to the obtained theoretical results.

Attachment of various-shaped polystyrene microplastics to silica surfaces: Experimental validation of the equivalent Cassini oval extended DLVO model

Microplastics (MPs) vary in shape and surface characteristics in the environment. The attachment of MPs to surfaces can be studied using the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. However, this theory does not account for the shape MPs. Therefore, we investigated the attachment of spherical, pear-shaped, and peanut-shaped polystyrene MPs to quartz sand in NaCl and CaCl2 solutions using batch tests. The attachment of MPs to quartz sand was quantified using the attachment efficiency (alpha). Subsequently, alpha behaviors were interpreted using energy barriers (EBs) and interaction minima obtained from extended DLVO calculations, which were performed using an equivalent sphere model (ESM) and a newly developed equivalent Cassini model (ECM) to account for the shape of the MPs. The ESM failed to interpret the alpha behavior of the three MP shapes because it predicted high EBs and shallow minima. The alpha values for spherical MPs (0.62–1.00 in NaCl and 0.48–0.96 in CaCl2) were higher than those for pear- and peanut-shaped MPs (0.01–0.63 in NaCl and 0.02–0.46 in CaCl2, and 0.01–0.59 in NaCl and 0.02–0.40 in CaCl2, respectively). Conversely, the ECM could interpret the alpha behavior of pear- and peanut-shaped MPs either by changes in EBs or interaction minima as a function of orientation angles and electrolyte ionic strength. Therefore, the particle shape must be considered to improve the attachment analyses.

The effect of magnetic field on the heat transfer in the porous medium octagonal cavity with Cassini oval barriers

This article investigates the effect of Rayleigh numbers, Hartmann numbers, and magnetic fields on nanofluid flow behavior and heat transfer. The flow behavior has been analyzed based on the contours and values of the flow function and heat transfer based on the temperature distribution contours and local and average Nusselt number values. Since the effects of the magnetic field are an important parameter in many technical and engineering applications such as cooling of nuclear reactors, extraction of geothermal energy, and induction casting of metals, in this study, its effect on aluminum oxide nanoparticle, Al2O3, and water, H2O, has been analyzed. The governing equations of nanofluid flow, continuity, momentum, and energy equations have been solved using the Galerkin finite element method and coded and simulated using FlexPDE software. The results obtained for both octagonal cavities containing horizontal and vertical Cassini oval barriers with specific boundary conditions show that with the Rayleigh number and nanoparticle volume fraction being constant, the value of the maximum stream function decreases with the increase of the Hartmann number, and when the Hartmann number and nanoparticle volume fraction is constant, with increasing Rayleigh number, its value increases significantly.

Applications of the Symmetrical Structures of Cassini Ovals

One of the geometric figures that has symmetry properties is the Cassini oval. The Cassini oval is a curve defined as the locus of points in the plane such that the product of the distances to two fixed points is constant. Cassini ovals are named after the astronomer Giovanni Domenico Cassini, who studied them in 1680. Today, the geometric properties of Cassini ovals are used in many fields: analytical geometry, nuclear physics, radiolocation, and industrial applications. The bistatic radar uses Cassini ovals to detect various targets in radiolocation. Until now, there have been no studies on the clustering capabilities of Cassini ovals. As a novelty, it is hypothesized that clustering possibilities could be used for Cassini ovals. In this article, a study of the capabilities of Cassini ovals in radiolocation was carried out, and their suitability for clustering purposes was shown.

A Simplified Method for Calculating Surface Area of Mammalian Erythrocytes.

Knowledge of the geometric quantities of the erythrocyte is useful in several physiological studies, both for zoologists and veterinarians. While the diameter and volume (MCV) are easily obtained from observations of blood smears and complete blood count, respectively, the thickness and surface area are instead much more difficult to measure. The precise description of the erythrocyte geometry is given by the equation of the oval of Cassini, but the formulas deriving from it are very complex, comprising elliptic integrals. In this article, three solids are proposed as models approximating the erythrocyte: sphere, cylinder and a spheroid with concave caps. The volumes and surface areas obtained with these models are compared to those effectively measured. The spheroid with concave caps gives the best approximation and can be used as a simple model to determine the erythrocyte surface area. With this model, a simple method that allows one to estimate the surface area by knowing only the diameter and MCV is proposed.

Power system small-disturbance stability analysis and control design: A characteristic locus method

A characteristic locus method developed from the celebrated generalized Nyquist criterion is proposed for power systems small-disturbance stability analysis and control design in this paper. First, a loop transformation is introduced to re-formulate the familiar Heffron-Phillips model. The new model has a feedback control structure so that the generalized Nyquist criterion is applicable when concluding the stability of the closed-loop system. Subsequently, the relationship between generalized Nyquist curve and characteristic loci is utilized to acquire a new stability margin. The effectiveness of the stability margin is demonstrated using Cassini Oval Theorem. An analytical expression of the stability margin is derived as well. It is convenient to quantify the effects of control loops on stability margin by applying this expression. Moreover, the analytical relationship between stability margin and frequency responses of controllers provides information for the design of the controllers. The utility of the proposed method is confirmed through case studies.

Computing the zeros of the Szegö kernel for doubly connected regions using conformal mapping

&lt;abstract&gt;&lt;p&gt;An explicit formula for the zero of the Szegö kernel for an annulus region is well-known. There exists a transformation formula for the Szegö kernel from a doubly connected region onto an annulus. Based on conformal mapping, we derive an analytical formula for the zeros of the Szegö kernel for a general doubly connected region with smooth boundaries. Special cases are the explicit formulas for the zeros of the Szegö kernel for doubly connected regions bounded by circles, limacons, ellipses, and ovals of Cassini. For a general doubly connected region with smooth boundaries, the zero of the Szegö kernel must be computed numerically. This paper describes the application of conformal mapping via integral equation with the generalized Neumann kernel for computing the zeros of the Szegö kernel for smooth doubly connected regions. Some numerical examples and comparisons are also presented. It is shown that the conformal mapping approach also yields good accuracy for a narrow region or region with boundaries that are close to each other.&lt;/p&gt;&lt;/abstract&gt;

Dieless bulging and nonlinear buckling of longan-shaped pressure hull

In this study, the dieless bulging and nonlinear buckling behaviours of a stainless longan-shaped pressure hull were investigated. According to the Cassini oval equation, the longan shape has a shape index of 0.1. The lengths of the major and minor axes of the designed longan-shaped pressure hull were approximately 400 and 396 mm, respectively. The wall thickness of this pressure hull was 1 mm. The Huber–von Mises stress and first yielding load of the designed longan-shaped shell and its inscribed polyhedral preform were investigated analytically and experimentally. Moreover, the dieless bulging of the designed longan-shaped pressure hull was investigated numerically and experimentally by using the nonlinear finite element method and a hydrostatic test, respectively. The analytical, experimental, and numerical results exhibited good agreement with each other.

Improving Efficiency of Corn Seed Separation and Calibration Process

Abstract The paper analyzes the existing theoretical research of corn seed separation and calibration processes. The machines, tools and equipment that implement the process were studied. An innovative sieve with openings in the Cassini oval shape was proposed for the economical separation and calibration of corn seeds. The results of operation of the proposed sieve in the operating conditions of Kharkiv Feed Mill on BSH-100 serial separators were considered. The study of the effectiveness of sifting corn seeds through the holes of the sieves was carried out in partnership with the State University “Ukrainian Research Institute of Forecasting and Testing of Agricultural Production Techniques and Technologies named after Leonid Pohorily” (Kharkiv Branch). The use of sieves with openings in the Cassini oval shape, instead of classic circular base openings, was found to result in an increase in throughput capacity of the sieve and open area of up to 20%.

Automated Construction of Airfoils Lattice Based on Cassini Oval

Automated Construction of Airfoils Lattice Based on Cassini Oval

About the geometry of the Cassini oval, its non-convexity degree and $\varepsilon$-offset layer

The paper studies the geometry of a closed nonconvex smooth simply connected curve on a plane — of the Cassini oval, as well as the geometry of the $\varepsilon$-layer around the set whose boundary is the Cassini oval. Various analytical representations of the $\varepsilon$-layer boundary are formed, and special points of this boundary are described. The measure of nonconvexity $\alpha$ of a simply connected set whose boundary is the Cassini oval is determined, and the angular characteristic of nonconvexity of its $\varepsilon$-neighborhood is defined.

STEM PROJECT AS A MEANS OF LEARNING MODELING FOR PRE-SERVICE MATHEMATICS AND COMPUTER SCIENCE TEACHERS

Modern science operates with various methods, among which modeling is one of the most popular. The development of information technology allows the study of analogues (models) with the most significant characteristics of the real object. Modeling activities have been considered as useful teaching method in STEM education. Cloud services (like GeoGebra) are effective means for STEM education. The paper features a methodology of forming modeling skills based on STEM projects, which is grounded on modeling interesting curves of Analytic Geometry course. The content of the methodology is a course in Computer Modeling, which includes a module "STEM education and modeling". The module idea is based on the formation of skills required to model interesting curves (ellipse, hyperbola, parabola, conchoid of Nicomedes, limaçon of Pascal, strophoid, cissoid of Diocles, lemniscate of Bernoulli, Cassini oval, cycloidal curves, folium of Descartes, witch of Agnesi, logarithmic spiral). The methodology provides 4 steps (Step 1 – the teacher offers an example of a STEM project, which is discussed in class and solved by the teacher using GeoGebra; Step 2 – students are divided into groups of 3-4 people; Step 3 – the teacher offers a short STEM project (7-10 days), in which students model the curve; Step 4 – students offer their own STEM project (15-20 days), the solution of which is based on the modeling of an interesting curve). To test the effectiveness of the developed methodology, a pedagogical experiment was organized (2019-2021), which was joined by Master's students majoring in "Secondary Education (Mathematics)" and "Secondary Education (Computer Science)". Makarenko Sumy State Pedagogical University (Ukraine) was the experimental base. The effectiveness of the proposed methodology is proved by the sign test at the significant level of 0.05.

МОДЕЛЮВАННЯ ГЕОМЕТРІЇ РОТОРА ВАКУУМНОГО НАСОСА РУТС

Mathematical model for designing the surface geometry of the Roots pump rotor based on the Cassini oval principle was derived. The polar coordinate system was used, and the radius vector, the direction of which was set by the φ angle, characterizes the location of the point on the surface of the rotor. The distance of this point from the axis of rotor rotation was set by the calculated value of the ρ_R polar radius vector. The γ angle of rotors rotation characterizes their mutual orientation in the plane of rotation. Peculiarities of the choice of the a and b parameters that satisfy the shape of the rotor surface geometry are considered. An example of rotor geometry is given for rotor radius R = 50 mm, rotor rounding radius r = 20 mm, parameters a = 33.166 and b = 28. Rotor geometry depends on normalized parameters of a and b, which are constant for a given shape of the surface and constructive dimensions. A mathematical model of the usable cross-sectional area of the pump has been developed. The usable cross-sectional area of the pump was simulated by the geometry of the rotors. The area of the rotor was determined by the geometry of the surface, which was described by an elliptic integral of the 2nd kind. The usable cross-sectional area for the given parameters is modelled. The results of simulation in the form of graphical dependences are given. Parameters a and b must meet the condition of √2⁄2&lt;b⁄a&lt;1. Under such conditions, the geometry of the rotor surface will be a Cassini oval. The rotation of the two rotors against each other will be by rolling one surface over another.

МОДЕЛЮВАННЯ ГЕОМЕТРІЇ РОТОРА ВАКУУМНОГО НАСОСА РУТС

Mathematical model for designing the surface geometry of the Roots pump rotor based on the Cassini oval principle was derived. The polar coordinate system was used, and the radius vector, the direction of which was set by the φ angle, characterizes the location of the point on the surface of the rotor. The distance of this point from the axis of rotor rotation was set by the calculated value of the ρ_R polar radius vector. The γ angle of rotors rotation characterizes their mutual orientation in the plane of rotation. Peculiarities of the choice of the a and b parameters that satisfy the shape of the rotor surface geometry are considered. An example of rotor geometry is given for rotor radius R = 50 mm, rotor rounding radius r = 20 mm, parameters a = 33.166 and b = 28. Rotor geometry depends on normalized parameters of a and b, which are constant for a given shape of the surface and constructive dimensions. A mathematical model of the usable cross-sectional area of the pump has been developed. The usable cross-sectional area of the pump was simulated by the geometry of the rotors. The area of the rotor was determined by the geometry of the surface, which was described by an elliptic integral of the 2nd kind. The usable cross-sectional area for the given parameters is modelled. The results of simulation in the form of graphical dependences are given. Parameters a and b must meet the condition of √2⁄2&lt;b⁄a&lt;1. Under such conditions, the geometry of the rotor surface will be a Cassini oval. The rotation of the two rotors against each other will be by rolling one surface over another.

Determining the coordinates of the earthquake hypocenter with the simultaneous determination of seismic wave velocities

Objective. The purpose of the study is to determine the coordinates of the earthquake hypocenter with the simultaneous determination of the seismic wave velocity.Method. The study is based on the use of figures of the fourth and second order - the Cassini oval and hyperbola, figures of the second order - ellipse and hyperbola and combined methods that allow, at given seismic wave velocities, according to three seismic sensors, to determine the coordinates of the earthquake hypocenter. In this case, the velocities are assumed to be known a priori. According to the readings of the fourth seismic sensor, and for given seismic wave velocities, the distance from the fourth seismic sensor to the hypocenter is determined. Again, according to the coordinates of the fourth seismic sensor and the hypocenter coordinates calculated from the three seismic sensors, the distance between them is calculated. If the seismic wave velocities do not correspond to the true ones, a discrepancy will appear between the two calculated values of the distances from the hypocenter to the fourth seismic sensor. By varying the velocities of seismic waves when calculating the coordinates of the hypocenter, we achieve a minimum discrepancy.Result. The problem of determining the coordinates of the earthquake hypocenter with the simultaneous determination of seismic wave velocities is solved. The error distribution densities in determining the coordinates of the earthquake hypocenter and seismic wave velocities are obtained.Conclusion. The proposed method makes it possible to increase the accuracy of determining the coordinates of the earthquake source, as well as to refine the values of seismic wave velocities.