Cycloidal Curves Research Articles

An Innovative STEAM-Based Method for Teaching Cycloidal Curves in Engineering Higher Education

Robotics offers innovative possibilities at all levels of education, which should be considered when planning the teaching process for certain engineering mathematics topics in higher education. This paper introduces a new teaching–learning technique that utilizes STEAM-based methods to explore cycloidal curves for Computer Science Engineering BSc students. Traditional frontal teaching has been enhanced with methods addressing the generational needs of students, including problem-based learning, STEAM integration, and project-based learning. We developed a methodological model that merges traditional teaching advantages with practical modern approaches suitable for Generation Z. The four-stage model for learning cycloidal curves employs various didactical approaches, utilizing different learning techniques at each stage to create an engaging and effective learning experience. A vital component of this model is the use of Desmos dynamic geometry software to create animations alongside educational robotics to aid visualization. We conducted quantitative studies with 98 first-year Computer Science Engineering students using a quasi-experimental research design to evaluate the new teaching technique’s effectiveness. Results from the Mann–Whitney U test indicated that the experimental group significantly outperformed the control group. Additionally, the Kruskal–Wallis test confirmed that the four-stage model for learning cycloidal curves enhances learning achievement for all students, regardless of their prior knowledge.

A Study on New Roulette and Special Forms of Cycloid and Laithoidal Curves

A Study on New Roulette and Special Forms of Cycloid and Laithoidal Curves

Повышение эффективности процессов дробления с использованием новой конструкции валковой дробилки

In order to increase the efficiency of crushing and grinding processes, technological and technical-economic characteristics of roller crushers, numerous scientific research works are carried out to substantiate the modernization of the main workers organs of crushing machines, innovative materials and methods of hardening the surfaces of the main working organs are used. The article provides a rationale for the operational characteristics of a new roller crusher with cycloidal movements and determines its design geometric parameters. The complex cycloidal movement of the rolls ensures the lowest possible energy intensity of the crushing process. The working body of the crusher uses the properties of mutually bending cycloidal curves and bodies of constant width, which made it possible to significantly reduce the weight and dimensions of the machine. The planetary-rotor motion of cycloidal-shaped rolls and their design features make it possible to reduce the minimum gap between the rolls, and the width of the unloading gap remains constant, which ensures the maximum size of the finished crushing product and the yield of cube-shaped crushed stone

Animation of cycloid and spiral curves in companion with instantaneous center of rotation and radius of curvature

ABSTRACT In this paper, the animations for the 2D cycloid and the 3D spiral curves are done. The trajectories of instantaneous rotation center and the corresponding radius of curvature are given. We prove that the trajectory of the instantaneous center of rotation is also a cycloid. For a 3D spiral curve, the two radii and the two instantaneous centers of rotation for the spiral curve are also given. It is interesting to find that the two parameters in the Frenet equation have the same meaning of radius of curvature but in different planes. In a similar way of the 2D experience, we also confirm that the trajectory of the instantaneous center of rotation for a spiral curve is also a spiral curve. An example is also given to discuss the Puyuma express incident, a major accident in 2018. The curve of rail is interpolated and the radius of curvature is determined. Discussions on the radius of rail curve and the speed of train for the failure are done. Finally, the animation is implemented by using the MATLAB and the Mathematica software. Not only theoretical derivation for the curvature of a curve but also its real application to rail engineering is proposed.

Development of Power-Assist Device for a Manual Wheelchair Using Cycloidal Reducer

This paper presents the design process and driving performance test results of a power-assist module to which a cycloidal reducer is applied in order to convert a manual wheelchair into an electric wheelchair. The types of electrification modules currently used to electrify manual wheelchairs include front-mounted, rear-mounted, and powered wheels. These assist devices are either difficult to carry and transport independently or require excellent hand dexterity to operate. To overcome this problem, a cycloidal reducer with no pin roller, and a novel cycloidal curve were designed to develop a small and easy-to-handle power-assist module that was tested by installing this reducer to a manual wheelchair. As a result of the test, the maximum speed of the wheelchair was 6 km/h, the maximum slope that this wheelchair can climb is 20%, and 0.358 Ah was consumed while the wheelchair moved 360 m in the current consumption test. This showed that it is possible to develop a small-sized power-assist module. In addition, the user can easily electrify the manual wheelchair by adding a small weight without replacing the manual wheel. The power-assist module consists of a DC servo motor, cycloidal reducer, battery, and joystick.

Nomograms for maximum pressure angles in radial cams with follower eccentricity for cycloidal and harmonic motion curves

For an adequate design of radial cams with translating roller followers, it is necessary to maintain the constraint that maximum pressure angle cannot exceed the empirically accepted value of 30°. The direct calculation of this parameter is difficult and it is usually sought to reduce it by trial and error. Nomograms that exist so far only calculated this angle for cams with no eccentricity between follower and cam. This research presents two nomograms with this additional parameter, which allows for greater design freedom using all available variables. The charts are for cycloidal and harmonic motion curves, and can be used for their full and half curves. The study discusses the advantages of using nomograms and the methods to obtain low values of maximum pressure angle by satisfactorily combining available parameters. Although nomograms are no longer a widely used tool in the industry, they still have didactic functions in textbooks and could be useful for preliminary analysis in engineering projects.

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.

A remarkable property of cycloidal curves

The purpose of this theoretical work is to establish a connection between the most important properties of plane curves: cycloids and sinusoids. For this, a drawing mechanism is considered, which simultaneously draws a sinusoid and two cycloids. Based on the results obtained using this mechanical method of obtaining curves, the following important, previously unknown, theoretical facts are established. Firstly, new in theoretical terms is that the sinusoid is not represented as a graph of a trigonometric function, but as a locus of points equidistant from the current points of two cycloids: an ordinary and another cycloid congruent to the original one, inverted and shifted along the axis by half a period. Secondly, the line passing through the current points of these cycloids is nothing like a normal to the resulting sinusoid. This property greatly simplifies the graphical construction of such a normal. And, finally, a simple trigonometric relationship was established between the angle of rotation of the generating circle and the angle of deviation of the normal from the vertical.

Algorithm for screws profiling of a conical trilobed compressor - analytical solution

Screw compressors, as rotary machinery, are capable of working with high compression ratios for gas compression. They transform the mechanical work done by an electric motor or turbine into the potential energy of a gas, by reducing its volume and increasing the pressure. The essential elements of these compressors are the two helical rotors with lobes, mounted in a housing. Existing rotors in the construction of compressors are working bodies whose composite profiles include circle arcs, line segments and cycloidal curves, while also often including non-analytical curves. Such geometry of profiles gives the gear processes superior characteristics, in the sense of reducing friction and improving sealing conditions. This paper analyses, by analytical approach, the possibility of constructing a multilobated conical screw compressor with a number of lobes Z, Z + 1, of the male and female rotor. The parametric equations of the contained worm and the kinematics of the working process of the conical screw compressor shall be defined. The theory of surface wrapping comprises analytical methods of study, which were developed on the basis of the fundamental theorems on wrapping of known surfaces in differential geometry and the kinematics laws on rotor movements. Thus, this paper proposes an algorithm based on the approach to the problems analyzed under the “virtual focal points theorem” between the complex surfaces of the conical surface lobes, which constitute the active lobes of the male and female rotors. Normal to active surfaces are defined, and also the speed in relative movement, in the “gear” process of the conical screw compressor. Regarding rotors in conical screw compressors, the circular pitch of the teeth is constant. Also, the screw pitch of the conical compressor rotors is a constant value. In these conditions, the rotors of the screw compressors are acknowledged as active surfaces, conical screw surfaces with constant axial screw pitch. The cross section (in the normal plane to the driving rotor axis) determines a plane gear. This defines the specific winding condition, and the shape of the conical screw surface of the female rotor.

Analytical solutions and computational nomograms for maximum pressure angle for cam mechanisms for full and half cycloidal and harmonic motion curves

High pressure angles in cams with roller followers still pose as a problem in cam projects. The main objective of this work is to deduce the maximum pressure angle equations for full and half cycloidal and harmonic motion curves for translating roller followers in radial cams and create nomograms that can calculate this parameter by simply connecting two points by a straight line in a chart. The nomograms were made for the mentioned curves via an open source software and it is discussed the limitations of the given charts and the advantages of using the current method. Additionally, the paper reinforces the need for authors to make it clear that another well known and frequently used nomogram cannot be used as is for half curves. Although use of nomograms has fallen into disuse, it still has applications in textbooks and quick preliminary analysis in real life projects.

Study on the effects of tooth profile design parameters of rotor to performance of vacuum pump

The vacuum pump usually used traditional curves such as the circular, cycloidal curves and their combinations to construct tooth profile. However, to increase efficiency and design flexibility for the vacuum pump, a novel rotor tooth profile for Roots rotor of vacuum pumps is proposed. Which is named “CEIEC” tooth profile and orderly composed of five significant segments, a circular arc for tooth tip, an epicycloid curve with variable extension, an involute, an enveloped epicycloid curve and a conjugated circular arc for tooth root. A numerical example is presented to evaluate the performance indices for proposed vacuum pump, including the hermeticity coefficients of the rotor mesh gap and tip gap.

THE INFLUENCE OF ECCENTRICITY CHANGES ON POWER LOSSES IN CYCLOIDAL GEARING

Gearing is one of the most important rolling nodes in a cycloidal planetary transmission. It consists of gears whose outlines are created according to cycloidal curves and the rollers cooperating with them. The epicycloidal and hypocycloidal gearing can be distinguished. There are power losses during the transmission operation, due to the transmitted load and friction in the gearing. The paper presents an analytical method for determining losses in cycloidal gearing, taking into account the manufacturing deviations of the elements making this gearing. The analysis allowed determining the distribution of intertooth clearances and forces at the points of contact between the profiles and the rollers, and consequently the distribution of power losses in gearing. The distribution of intertooth forces was verified by elastooptic tests of the gears set with cycloidal gearing. The results of the calculations indicate that the influence of eccentricity changes on the distribution of intertooth forces is significant and also that the distribution of power losses in gearing is closely related to the distribution of intertooth forces and has a very similar character.

Wind turbine designs for urban applications: A case study of shrouded diffuser casing for turbines

The increased demand for renewable energy and the development of energy independent building designs have motivated significant research into the improvement of wind power technologies that target urban environments. However, the implementation of wind turbines in urban environments is still very limited. There have been some studies analyzing different designs of urban wind turbines either using computational fluid dynamics (CFD), wind tunnel tests or field data for existing or new turbine designs. This paper reviews the state-of-the-art of urban wind energy by examining the various types of urban wind turbine designs, with a view to understand their performance and the synergy between the turbines and the urban environments. It also considers a flanged diffuser shroud mechanism - a fluid machine, mounted on rooftop of buildings used as casing for small wind turbines to improve turbine performance by using mainly CFD. The diffuser shroud mechanism can draw the airflow over buildings utilizing its special features such as, cycloidal curve geometry at the inlet and a vortex generating flange at the outlet, to guide and accelerate the airflow inside. The performance of the fluid machine is optimized parametrically. The mechanism is modeled on a building rooftop in a real test site in Montreal, Canada with real statistical wind data. The CFD result confirms the functionality of the fluid machine to take advantage of the airflow over buildings in complex built-environments for wind power generation.

Parametric Design on Internal Gear of Cycloid Gear Pump with NX10.0

The parametric equation of the theoretical contour curve of cycloid gear was established in this paper. Using the NX10.0 software, parametric design of cycloidal gear profile curve was carried out, and the radius of curvature with the contour curve of cycloid gear was analyzed. The consistency between the simulation results and the theoretical design results was verified by assembly analysis, and the parametric drive of the cycloidal gear was verified simultaneously.

Rotor profile design and numerical analysis of 2–3 type multiphase twin-screw pumps

Increasing demands for high-performance handling of fluids in oil and gas as well as other applications require improvements of efficiency and reliability of screw pumps. Rotor profile plays the key role in the performance of such machines. This paper analyses difference in performance of 2–3 lobe combination of twin-screw pumps with different rotor profiles. A-type profile formed of involute–cycloid curves and D-type formed of cycloid curves are typical representatives for 2–3 type screw pumps. The investigation is performed by use of a full 3-D computational fluid dynamics analysis based on a single-domain structured moving mesh obtained by novel grid generation procedure. The real-time mass flow rate, rotor torque, pressure distribution and velocity field were obtained from 3D computational fluid dynamics calculations. The performance curves were produced for variable rotation speeds and variable discharge pressures. The computational fluid dynamics model was validated by comparing the simulation results of the A-type pump with the experimental data. In order to get the performance characteristics of D-type profile, two rotors with D-type profile were designed. The first has the same displacement volume as A-type while the second has the same lead and rotor length as A-type but different displacement volume. The comparison of results obtained with two rotor profiles gave an insight on the advantages and disadvantages of each of them.

Comparison Analysis and Verification on Spur and Helical Three-lobe Rotors with Novel Tooth Profile in Roots Blower

This paper proposes a novel three-lobe rotor tooth profile with the cross section of a combination of eccentric arcs and cycloidal curves for the Roots blower. Based on the geometric modelling of this new tooth profile, a helical tooth profile and a comparative study between the spur and the helical three lobes have been presented. The study reveals the advantages of the new helical tooth profile design which creates greater air flow and reduces the peak of the blower pressure. The tooth profile machining and testing are provided at the end of this paper.

Innovative Method of Determination of Speed of Interaction of Wheels with Rails

The purpose of work is adaptation of fundamental provisions of the theory and properties of cycloidal curves for determination of speed and speedups of conditional points of a cross-section of a bandage of a railway wheel.Results of studying and establishment of the gear of interaction of a railway wheel with a rail are given in work. Modeling is executed on the basis of the fundamental theory of kinematics of the movement of a circle.The offered method allows to define characteristic criteria of kinematics of interaction of a cross-section of a bandage of wheel pairs of the rolling stock with side sides of heads of rails. On the basis of mathematical provisions of trajectories of speeds and speedups of conditional points of a wheel profile gain further development of probe in elimination of debatable contradictions in the field of interaction of “wheels with rails”.

Cycloids on the Spherical Surface

Abstract The article describes the creation of the normal cycloidal curve by rotation of the point about the normal of the circle on the spherical surface and binormal cycloidal curve created by rotation of the point about the binormal of the previous cycloidal curve. Cycloidal cyclical surfaces are created by moving circles along the curves lying in the normal plane of the curves. Described cycloidal cyclical surfaces are projected on the spherical surface. Varying parameters of the curves are generated different ornaments on the spherical surface.

Pythagorean-hodograph cycloidal curves

AbstractIn the paper, Pythagorean-hodograph cycloidal curves as an extension of PH cubics are introduced. Their properties are examined and a constructive geometric characterization is established. Further, PHC curves are applied in the Hermite interpolation, with closed form solutions been determined. The asymptotic approximation order analysis carried out indicates clearly which interpolatory curve solution should be selected in practice. This makes the curves introduced here a useful practical tool, in particular in algorithms that guide CNC machines.

Evaluation of the mixing performance for one novel twin screw kneader with particle tracking

In this article, a novel continuous twin‐screw kneader was proposed. The end‐cross section of the screw rotor consists of convex arcs and cycloidal curves and the rotors profiles were presented. The mixing performance of the novel twin screw kneader was simulated using finite element method (FEM) combined with mesh superimposition technique (MST). Statistical analysis was carried out for flow field using particle tracking technique to research the effect of geometry parameters and working parameters on the mixing performance. To study the dispersive mixing performance, specifically the maximum shear rate, maximum shear stress, maximum mixing index, residence time distribution (RTD) and RTD density function of tracer particles, and dispersive mixing is evaluated using the mixing index in combination with the shear stress. The results show that the changes of centre distance between female and male rotor have little influence on dispersive mixing performance, the lead of rotor has little effect on maximum shear stress and maximum shear rate, while it has an obvious effect on mixing index, cumulative RTD, and RTD density function. The rotor speed has obvious influence on mixing performance, and average residence time of material decreases greatly and the mixing ability is weakened, while the self‐cleaning performance of rotor improved obviously with the increasing of rotor speed. POLYM. ENG. SCI., 54:2407–2419, 2014. © 2013 Society of Plastics Engineers