Cycloidal Curves Research Articles

Design of Rotor for Internal Gear Pump Using Cycloid and Circular-Arc Curves

In the case of internal gear pumps, the eccentricity of the outer rotor, which resembles a circular lobe, must be limited to a certain value in order to avoid the formation of cusps and loops; furthermore, the tip width of the inner rotor, which has a hypocycloid curve and an epicycloid curve, should not be allowed to exceed the limit value. In this study, we suggest that the tip width of the inner rotor be controlled by inserting a circular-arc curve between the hypocycloid and epicycloid curves. We also suggest that the outer rotor be designed using the closed-form equation for the inner rotor and the width correction coefficient. Thus, it is possible to design a gerotor for which there is no upper limit on the eccentricity, as in this case, undercut is prevented and there is no restriction on the tip width. We also develop an automated program for rotor design and calculation of the flow rate and flow rate irregularity. We demonstrate the superior performance of the gerotor developed in this study by analyzing the internal fluid flow using a commercial computational fluid dynamics (CFD)-code.

Performance of eccentric-cycloid engagement with change in the interaxial distance: Modification of tooth configuration

The influence of the interaxial distance of the gears on the performance of an eccentric-cycloid engagement is studied by computer simulation. With wide variation in the interaxial distance, there is little shift in the contact point of the profiles, which remains at the front of the cycloid curves describing the tooth shape. This greatly simplifies gear manufacture for such engagements, since the surface area to be finished is considerably reduced.

Modeling of Current Distribution on Smooth and Columnar Platinum Structures

Studying the growth and stability of anisotropic or isotropic disordered surfaces in electrodeposition is of importance in catalytic electrochemistry. In some cases, the metallic nature of the electrode defines the topography and roughness, which are also controlled by the experimental time and applied external potential. Because of the experimental restrictions in conventional electrochemical techniques and ex situ electron microscopies, a theoretical model of the surface geometry could aid in understanding the electrodeposition process and current distributions. In spite of applying a complex theory such as dynamic scaling method or perturbation theories, the resolution of mixed mass-/charge-transfer equations (tertiary distribution) for the electrodeposition process would give reliable information. One of the main problems with this type of distribution is the mathematics when solving the spatial n-dimensional differential equations. Use of a primary current distribution is proposed here to simplify the differential equations; however it limits wide application of the first assumption. Distributions of concentration profile, current density, and electrode potential are presented here as a function of the distance normal to the surface for the cases of smooth and rough platinum growth. In the particular case of columnar surfaces, cycloid curves are used to model the electrode, from which the concentration profile is presented in a parameterized form after solving a first-type curvilinear integral. The concentration contour results in a combination of a trigonometric inverse function and a linear distribution leading to a negative concavity curve. The calculation of the current density and electrode potential contours also show trigonometric shapes exhibiting forbidden imaginary values only at the minimal values of the trochoid curve.

The Geometry of the Snail Ball

SummaryThe snail ball is a device that rolls down an inclined plane, but very slowly, repeatedly coming to a stop and staying motionless for several seconds. The interior of the ball is hollow, with a smaller solid ball inside it, surrounded by a very viscous fluid. We show how to model the stop-and-start motion by analyzing the cycloidal curve that would correspond to the motion of the center of gravity as the ball rolls down an inclined plane.

Tooth Contact Analysis of Longitudinal Cycloidal-Crowned Helical Gears With Circular Arc Teeth

Abstract This paper proposes a new type of double-crowned helical gear that can be continuously cut on a modern Cartesian-type hypoid generator with two face-hobbing head cutters and circular-arc cutter blades. The gear tooth flank is double crowned with a cycloidal curve in the longitudinal direction and a circular arc in the profile direction. To gauge the sensitivity of the transmission errors and contact patterns resulting from various assembly errors, this paper applies a tooth contact analysis technique and presents several numerical examples that show the benefit of the proposed double-crowned helical gear set. In contrast to a conventional helical involute gear, the tooth bearing and transmission error of the proposed gear set are both controllable and insensitive to gear-set assembly error.

Simulation Research on Mechanical Performances of Several Kinds of Cultivating Components with Different Soil-engaging Surfaces

How to optimize the geometric shape of cultivating components to reduce the working resistance, thus enhancing working efficiency, is always paid much attention by humans beings. The mechanical properties and their discrepancies in different kinds of cultivating components with different soil-engaging surface directrix, after introducing the concept of longitudinal/depth (L/B) ratio, are analyzed and compared with finite element method. The directrix for each cultivating component’s soil-engaging surface is respectively composed of straight line, parabola, cycloid, circular arc and bionic curves including the soil-engaging surface contour line of the claw toes of house mouse, field mouse, mole cricket and cock. Results indicate that the soil-engaging surface directrix and longitudinal/depth ratio are two important factors to affect the working resistance. The curved cultivating components generally possess superior cultivating performance against the straight-line cultivating components with small L/B values (0L/B1.0). Optimal mechanical performance of the curved cultivating components can be obtained by way of bionic optimization. However, the bionic curves used in the design of soil-engaging surface should be researched respectively for their complexity. A parabolic cultivating component, even in a broader range of longitudinal/depth ratio (0L/B1.5), is possible to obtain good mechanical properties of cultivation. A cultivating component of circular soil-engaging surface is slightly poor in resistance reduction. A cycloid is suitable to the design of cultivating component’s soil-engaging surface with a bit big L/B value (L/B≈1.2). The results of this research would have important reference values in the optimization design of cultivating component’s soil-engaging surface for reducing resistance.

Numerically stable algorithm for cycloidal splines

We propose a knot insertion algorithm for splines that are piecewisely in L{1, x, sin x, cos x}. Since an ECC-system on [0, 2π] in this case does not exist, we construct a CCC-system by choosing the appropriate measures in the canonical representation. In this way, a B-basis can be constructed in much the same way as for weighted and tension splines. Thus we develop a corner cutting algorithm for lower order cycloidal curves , though a straightforward generalization to higher order curves, where ECC-systems exist, is more complex. The important feature of the algorithm is high numerical stability and simple implementation.

130 サイクロイド振子型動吸振器に関する研究

The dynamic absorber has been widely used for suppressing vibration in structures. In practical application, the optimal tuned absorbers are designed and used in terms of natural frequency and additional damping so that resultant frequency response becomes fully reduced. However, damping performance of ordinal pendulum system may be influenced by levels of excitation because of intrinsic nonlinearity in system which leads to slight change in natural frequency. The present paper deals with a study of new type pendulum dynamic absorber. It is known that pendulum moving along cycloidal curve strictly keeps uniform period. Both analytical and experimental investigations show that a strict optimal damping condition can be satisfied in cycloid pendulum damper, in conjunction with better damping performance compared with normal pendulum.

Study of Cycloidal Pendulum Dynamic Vibration Absorber

The dynamic absorber has been widely used for suppressing vibration in structures. In practical application, the optimal tuned absorbers are designed and used in terms of natural frequency and additional damping so that resultant frequency response may become fully reduced. Among various type of dynamic absorbers, the pendulum type is preferable due to its simple constitution and performance reliability. However, damping performance of ordinal pendulum system may be influenced by levels of excitation because of intrinsic nonlinearity in system which leads to slight change in natural frequency. The present paper deals with a study of new type pendulum dynamic absorber. It is known that pendulum moving along cycloidal curve strictly keeks uniform period. Both analytical and experimental investigations show that a strict optimal damping condition can be satisfied in cycloidal pendulum damper, in conjunction with better damping performance compared with normal pendulum.

Geometric analysis and tooth profiling of a three-lobe helical rotor of the Roots blower

This paper proposes a novel three-lobe helical rotor with the cross section as a combination of concave and convex arcs and cycloidal curves for the Roots blower. Based on the geometric analysis, the mathematical model of this new tooth profile is presented, leading to a presentation of a helical tooth profile and a comparative study between the three-lobe helical tooth and a conventional straight tooth. The study reveals the advantage of the new tooth design that creates greater air flow and reduces the peak of the blower pressure. The geometric study leads to the investigation of machining the profile of the new tooth. The cutter trajectory is revealed and the contact characteristics between the cutter and the work piece are identified. By identifying the cutter distance and the cutter mounting angle, the paper characterizes the relationship between these machining parameters that governs the machining of the novel tooth. The simulation and tooth machining are provided in the end of the paper.

Pose Estimation of 3D Free-Form Contours

In this article we discuss the 2D-3D pose estimation problem of 3D free-form contours. In our scenario we observe objects of any 3D shape in an image of a calibrated camera. Pose estimation means to estimate the relative position and orientation (containing a rotation and translation) of the 3D object to the reference camera system. The fusion of modeling free-form contours within the pose estimation problem is achieved by using the conformal geometric algebra. The conformal geometric algebra is a geometric algebra which models entities as stereographically projected entities in a homogeneous model. This leads to a linear description of kinematics on the one hand and projective geometry on the other hand. To model free-form contours in the conformal framework we use twists to model cycloidal curves as twist-depending functions and interpret n-times nested twist generated curves as functions generated by 3D Fourier descriptors. This means, we use the twist concept to apply a spectral domain representation of 3D contours within the pose estimation problem. We will show that twist representations of objects can be numerically efficient and easily be applied to the pose estimation problem. The pose problem itself is formalized as implicit problem and we gain constraint equations, which have to be fulfilled with respect to the unknown rigid body motion. Several experiments visualize the robustness and real-time performance of our algorithms.

Design of a milling cutter for a novel three-lobe arc-cycloidal helical rotor

This paper focuses on the design of a milling cutter for machining the arc-cycloidal helical rotor of a Roots blower. The milling cutter is developed on the basis of a novel rotor of a Roots blower which consists of both concave and convex arcs and a cycloidal curve in the transversal section. The geometric analysis of the transverse section and the helical tooth profile for the proposed rotor are presented. The paper reveals the contact characteristics between the cutter and the rotor blank being machined and derives the normal vectors of three different tooth sections. The paper further develops equations of the contact line for the conjugate motion of the cutter and the rotor blank and presents a design methodology of the milling cutter. The interference in the machining process is then investigated and numerical results are produced. The results from the analysis demonstrate that there is no interference in the machining using the developed cutter. The developed cutter is then demonstrated in the end of this paper in machining the novel arc-cycloidal helical rotor with three lobes.

Free-Form Pose Estimation by Using Twist Representations

In this article we discuss the 2D–3D pose estimation problem of 3D free-form contours. We observe objects of any 3D shape in an image of a calibrated camera. Pose estimation means estimating the relative position and orientation of the 3D object to the reference camera system. While cycloidal curves are derived as orbits of coupled twist transformations, we apply a spectral domain representation of 3D contours as an extension of cycloidal curves. Their Fourier descriptors are also related to twist representations. A twist is an element of se(3) and is a pair containing two 3D vectors. In a matrix representation, its exponential leads to an element of SE(3) and therefore to a rigid motion. We show that twist representations of objects can numerically efficiently and easily be applied to the free-form pose estimation problem. The pose problem itself is formalized as an implicit problem and we gain constraint equations, which have to be fulfilled with respect to the unknown rigid body motion.

An Enhanced DFM Model for Shaper Cutters

This paper presents a systematic CAD/CAM based geometric modelling technique for enhancing the manufacturing accuracy of general straight-tooth shaper cutters. The proposed modelling technique systematically computes and improves on the shape of general shaper cutters using a five-step procedure. In the first modelling step, the significant geometric parameters of shaper cutters which including tooth profile, rake and clearance angles, offsets, and backlash are estimated using criteria associated with the practical requirements and the theory of gearing, differential geometry, and motion theory. In the second modelling step, the precise cutting-edge curve model of the shaper cutter is constructed. The theoretical involute or cycloidal cutting-edge curve on the plane perpendicular to the motion axis of the shaper cutter is first computed and is then projected onto the conic surface of the shaper cutter which has both the rake and the clearance angles. In the third modelling step, the convoluted sectional profile of the disk-shaped generating cutter required for producing the cutting-edge profile is established using the theory of gearing and the relative motion relationship between the grinder and the cutter in the manufacturing process. In the fourth modelling step, a simple procedure to obtain the profile shift coefficient of the shaper cutters from various given process parameters is presented. Shaper cutters of various shapes can thus be designed and produced. In the last modelling step, a practical working example for producing the precise disk-shaped generating cutter and associated shaper cutter is presented to illustrate the effectiveness of the proposed modelling approach. The results of numerical modelling indicate that the proposed approach is systematic, accurate and reliable for producing shaper cutters of various shapes.

Harmonographs. II. Circular design

Devices that produce curves by the rotation of circles (or gears) about one another constitute one class of harmonographs. In the simplest case, a point on one of the circles will trace a cycloidal curve. A number of devices of varying degrees of complexity and which make use of gears or pulleys in their operation have been invented over the years to draw various curves. This paper offers a survey of some of these.

An algorithm that translates intrinsic equations of curves into intrinsic procedures

The paper describes an algorithm which translates each intrinsic equation of a curve into an intrinsic turtle procedure to draw the same curve. This algorithm inputs an intrinsic equation (or function) of a plane curve of the type ρ = ρ (φ) (where rho is the curvature-radius and φ is the tangent-angle). It ‘outputs’ a new procedure whose ‘action’ part is a simple step of the form (in Logo or in C+ +, etc.): right 1 forward < ρ >. This algorithm can be represented by a general procedure Intrinsic_Graph that produces a graphical representation of the ‘input’ equation curve. Thus, we can use the intrinsic equation: ρ = sin (n φ) of any cycloidal curve (or any other curve) to draw it. The intrinsic algorithm demonstrated links classical mathematical topics with turtle geometry. It uses the turtle's fundamental property that it always looks in the direction of the tangent to the curve. Thus, it ‘directs’ the turtle to proceed in a ‘natural way’ along the curve and at the same time to draw it.

Design of Antibacklash Pin-Gearing

This paper draws attention to pin gearing for its antibacklash gear property. The gear tooth curve derived from a cycloidal curve furnishes the pin gearing with kinematic antibacklash, low pressure angle, and no interference among other features. The pins on one side of a centerline are contacting the forward faces of the mating tooth gear. Simultaneously the pins on the other side of the centerline are contacting the backward faces of the gear. The correct center distance can be maintained by applying a force directed to each other center for multiple contacts. A generating method of gear manufacture is also suggested. Contact and bending deformations as well as the gear stress are analyzed to help its engineering design.

Design of antibacklash pin-gearing

The paper suggests that pin gearing can be used for an antibacklash gear. The gear tooth curve derived from a cycloidal curve furnishes the pin gearing with kinematic antibacklash, low pressure angle, and no interference among other features. Pins on one side of a centerline are contacting the forward faces of a mating tooth gear. Simultaneously, pins on the other side of the centerline are contacting the backward faces of the gear. The correct center distance can be maintained in this multiple-contact mechanism by applying a force directed to each other center. A generating method of tooth-gear manufacture is also suggested. Contact and bending deformations as well as the gear stress are analyzed to help its design.

Estimation of capillary length density in skeletal muscle by unbiased stereological methods: I. Use of vertical slices of known thickness.

The unbiased estimation of the capillary length density in skeletal muscle tissue Lv(cap/mus) has been performed in this study applying a new stereological methodology based on the use of vertical slices and the intersections of cycloid test curves with capillaries in a three-dimensional space defined by systematically chosen fields of vision and the thickness of the sections. The following simple requirements must be fulfilled: selection of a fixed vertical axis in skeletal muscle, adequate systematic muscle sampling, obtention of vertical slices of constant and known thickness but indifferent in magnitude, superposition of a cycloid test system with the minor axis of cycloid curves positioned perpendicularly to the vertical axis, and counting the intersections between cycloid curves and capillaries. In our study, the vertical axis was defined as that which is parallel to the natural, major axis of the muscle where fibres and capillaries are arranged parallel to this axis. The muscle sampling was performed using the fractionator method, and 25 microns thick sections were chosen. The application of the equation for estimation of Lv(cap/mus) permits determination of an average of 1,480 mm of capillaries per mm3 of muscle tissue, knowing the number of intersections, section thickness, and the points hitting the muscle with a known ratio between cycloid test curve length to a test point. The estimation of Lv(cap/mus) is efficient, unbiasedly obtained, and no assumptions on the degree of capillary anisotropy are required.

Sensitivity Analysis of Plate Cam Mechanisms with a Roller Follower.

The sensitivities of the output displacement of plate cam mechanisms with a translating or oscillating roller follower are derived on the basis of the equations obtained by partially differentiating the closed loop equations with respect to the kinematic constants and the condition for the roller follower to be in contact with the plate cam. These expressions are very simple, and the velocity and acceleration sensitivities are derived by differentiating the displacement sensitivities with respect to the angular displacement of the plate cam (the input angle). Moreover, the sensitivities of plate cam mechanisms with a roller follower are compared, in which cam curves are the cycloidal curve, the modified sine curve, the modified constant velocity curve, and the modified trapezoidal curve.