Involute Of Circle Research Articles

Chiral spiral cyclic twins. II. A two-parameter family of cyclic twins composed of discrete circle involute spirals.

A mathematical toy model of chiral spiral cyclic twins is presented, describing a family of deterministically generated aperiodic point sets. Its individual members depend solely on a chosen pair of integer parameters, a modulus m and a multiplier μ. By means of their specific parameterization they comprise local features of both periodic and aperiodic crystals. In particular, chiral spiral cyclic twins are composed of discrete variants of continuous curves known as circle involutes, each discrete spiral being generated from an integer inclination sequence. The geometry of circle involutes does not only provide for a constant orthogonal separation distance between adjacent spiral branches but also yields an approximate delineation of the intrinsically periodic twin domains as well as a single aperiodic core domain interconnecting them. Apart from its mathematical description and analysis, e.g. concerning its circle packing densities, the toy model is studied in association with the crystallography and crystal chemistry of α-uranium and CrB-type crystal structures.

Extended-depth-of-focus wavefront design frompseudo-umbilical space curves.

Designing extended-depth-of-focus wavefronts is required in multiple optical applications. Caustic location and structure analysis offer a powerful tool for designing such wavefronts. An intrinsic limitation of designing extended-depth-of-focus wavefronts is that any smooth surface, with a non-constant mean curvature, unavoidably introduces a separation between caustic sheets, which is proportional to the ratio of change of the mean curvature along a curve embedded in the wavefront. We present a method to obtain extended-depth-of-focus wavefronts where the mean curvature variation ratio is reduced thanks to using a long circle-involute space curve effectively filling the wavefront surface. Additionally, we present a variant of the method in which the wavefront is modified within a small tubular neighborhood of the circle involute in order to partially meet the umbilical condition along that tubular region. Finally, we provide some numerical results showing the potential of our method in an application example.

Constructing geometrical models of spherical analogs of the involute of a circle and cycloid

The common properties of images on a plane and a sphere are considered in the scientific works by scientists-designers of spherical mechanisms. This is due to the fact that the plane and the sphere share common geometric parameters. They include constancy at all points of the Gaussian curve, which has a zero value for a plane and a positive value for a sphere. Figures belonging to them can slide freely on both surfaces. With unlimited growth of the radius of the sphere, its limited section approaches the plane, and the spherical shape transforms into a plane. Thus, a loxodrome that crosses all meridians at a constant angle is transformed into a logarithmic spiral that intersects at a constant angle the radius vectors that come from the pole. The tooth profile of cylindrical gears is outlined by the involute of a circle. A spherical involute is used for the corresponding bevel gears. Other spherical curves are also known, which are analogs of flat ones. The formation of a cycloid and an involute of a circle are associated with the mutual rolling of a line segment with each of these figures. If the segment is fixed and the circle rolls along it, then the point of the circle describes the cycloid. In the case of a stationary circle along which a segment is rolled, the point of the segment will execute the involute. To move to the spherical analogs of these curves, it is necessary to replace the circle with a cone, and the straight line with a plane. The spherical prototype of the cycloid will be the trajectory of the point of the base of the cone, which rolls along the plane, that is, along the sweep of the cone. The sweep of a cone is a sector, the radius of the limiting circle of which is equal to the generating cone. If this sweep, like a section of a plane, is rolled around a fixed cone, when its top coincides with the center of the sector, then the point of the limiting radius of the sector will execute a spherical involute. This paper analytically implements these two motions and reports the parametric equations of the spherical analogs of the circle involute and the cycloid

Construction and Analysis of Variable Wall Thickness Profile of Double Scroll Teeth of Scroll Compressor

Based on the changing trend of the geometric parameters of the scroll profile of the involute of circle with variable radii, a new type of double scroll teeth profile of scroll compressor composed of involute of circle with variable radii is constructed. The construction process of the baseline is discussed, and the baseline equations are deduced. According to the principle of normal equidistance, the geometric model of variable wall thickness double scroll teeth is established, and the suction volume, tooth thickness, and scroll area of double scroll teeth are compared with those of the double scroll teeth composed of ordinary involute of circle. The results show that: compared with ordinary double scroll teeth with constant thickness, the scroll diameter when two scrolls mesh is reduced and the revolutions of orbiting scroll required in the working process are reduced and it is easy to select better scroll profile through adjusting the changing coefficient of the base circle radius k, which provides a reference for scroll machinery using environment-friendly working fluids.

Two-Stage Guidance Law With Constrained Impact via Circle Involute

The mission of simultaneous attack and warhead lethality enhancement requires the constraints on impact time and angle in the guidance design. To this end, a two-stage guidance law considering constrained impact as well as the robustness against external disturbances and heading errors is proposed. The first stage is based on a circle involute geometrical rule that guarantees the desired velocity direction at the involute's end. With this goal achieved, the involute guidance is switched to the guidance of nullifying line-of-sight rate such that near-zero maneuver in the second stage is ensured. To achieve both impact time and angle constraints, a simple parameter determination approach is provided to find explicit solutions for the desired two-stage trajectory. Meanwhile, a robust two-stage guidance law is constructed to implement the geometrical rule, guarantee terminal constraints, and ensure robustness. The proposed technique is easy to implement, in that it does not involve model linearization, time-to-go estimation, and numerical optimization routine. Additionally, the achievable range of different constraints is analyzed considering practical issues such as initial launch angles, initial line-of-sight angles, and acceleration limits. Extensive simulations are carried out in various engagement scenarios under uncertainties and disturbances to validate the effectiveness and robustness of the proposed guidance law.

Influence of Profile Parameters on Radial Leakage Line Length of Variable Tooth Thickness Scroll Compressor

In order to accurately calculate the radial length of leak line of scroll compressor of variable tooth thickness, with three sections of circle involute type of scroll line as the research object, according to the type line equation to obtain the geometric model of variable tooth thickness radial leakage line length calculation model was set up, numerical solution, through accurate profile parameters was studied (Base circle radius, Connection point, Turning radius, etc.) on the influence of the length of leak line of change. Results show that the model could accurately describe the radial length of leak line of instantaneous change rule, and got the type line parameter change on the influence of the radial length of leak line of law, and the best control the length of leak line type line parameters, providing theoretical basis for reducing radial leakage scroll compressor, at the same time, the model also applies to other type line leak line length calculation.

Log-aesthetic curves and their relation to fluid flow patterns in terms of streamlines☆

Abstract The log-aesthetic curve (LAC) is a family of aesthetic curves with linear logarithmic curvature graphs (LCGs). It encompasses well-known aesthetic curves such as clothoid, logarithmic spiral, and circle involute. LAC has been playing a pivotal role in aesthetic design. However, its application for functional design is an uncharted territory, e.g. the relationship between LAC and fluid flow patterns may aid in designing better ship hulls and breakwaters. We address this problem by elucidating the relationship between LAC and flow patterns in terms of streamlines at a steady state. We discussed how LAC pathlines form under the influence of pressure gradient via Euler's equation and how LAC streamlines are formed in a special case. LCG gradient ($\alpha $) for implicit and explicit functions is derived, and it is proven that the LCG gradient at the inflection points of explicit functions is always 0 when its third derivative is nonzero. Due to the complexity of the parametric representation of LAC, it is almost impossible to derive the general representation of LAC streamlines. We address this by analyzing the streamlines formed by incompressible flow around an airfoil-like obstacle generated with LAC having various shapes, ${\alpha _r} = \ \{ { - 20,{\rm{\ }} - 5,{\rm{\ }} - 1,{\rm{\ }} - 0.5,{\rm{\ }} - 0.15,{\rm{\ }}0,{\rm{\ }}1,{\rm{\ }}2,{\rm{\ }}3,{\rm{\ }}4,{\rm{\ }}20} \}$, and simulating the streamlines using FreeFem++ reaching a steady state. We found that the LCG gradient of the resultant streamlines is close to that of a clothoid. When the obstacle shape is almost the same as that of a circle ($\alpha \ = \ 20$), the streamlines adjacent to the obstacles have numerous curvature extrema despite nearing steady state. The flow speed variation is the lowest for $\alpha \ = \ - 1.43$ and gets higher as $\alpha$ is increased or decreased from $\alpha \ = \ - 1.43$.

A Quasi-dimensional Numerical Investigation of the Scroll Expander of an Organic Rankine Cycle Unit

Scroll compressors are often used in air conditioning and refrigeration systems thanks to their high efficiency, low noise and vibrations, light weight and high reliability. Scrolls are also widely used as positive displacement expanders in small-scale power generation systems, such as Organic Rankine Cycles (ORCs). In recent years, the research has focused on the development of mathematical models that help to predict the scroll expander performances under different operating conditions. In this work, a quasi-dimensional model of a scroll expander of an Organic Rankine Cycle unit is presented. Such an expander consists of two identical circle involute spirals, with appropriate starting segments. Specifically, the model is able to design the scroll expander geometry with either a “circular cutter” or a “perfect mesh profile” (PMP) approach, which are two of the most common scroll geometry starting segments. As regards the thermo-fluid dynamic aspects, specific sub-models have been used to account for radial and axial leakage, wall heat transfer, intake and exhaust of the working fluid. The model has been validated with available experimental data in order to assess its accuracy and, at the same time, to calibrate the implemented sub-models. The influence of some geometrical parameters on the expander performances has been assessed. In particular, a study of the influence of the wrap geometry is presented. The results show that the circular cutter approach returns better performances at the expenses of a higher mass consumption. Nevertheless, the circular cutter modification returns a higher specific work. Lastly, the influence of one of the PMP parameters on performances has been assessed. The results show a fairly strong dependence of both mechanical power and specific work, suggesting that the global optimization of all geometric parameters of the scroll expander may radically improve its performances.

An Analytic Expression for the Inverse Involute

This article introduces new types of rational approximations of the inverse involute function, widely used in gear engineering, allowing the processing of this function with a very low error. This approximated function is appropriate for engineering applications, with a much reduced number of operations than previous formulae in the existing literature, and a very efficient computation. The proposed expressions avoid the use of iterative methods. The theoretical foundations of the approximation theory of rational functions, the Chebyshev and Jacobi polynomials that allow these approximations to be obtained, are presented in this work, and an adaptation of the Remez algorithm is also provided, which gets a null error at the origin. This way, approximations in ranges or degrees different from those presented here can be obtained. A rational approximation of the direct involute function is computed, which avoids the computation of the tangent function. Finally, the direct polar equation of the circle involute curve is approximated with some application examples.

Circle involute as an optimal scan path, minimizing acquisition time, in surface topography

A relevant problem in point-by-point scanning surface topography is to find scanning paths minimizing the overall measurement time. We establish a rigorous mathematical framework that allows us to state, for the first time, that circle involutes are the best candidates for such scanning paths maintaining a prescribed measurement accuracy condition. We provide a quantitative analysis comparing time savings achieved by circle involutes with respect to three widely used scanning paths: (1) Archimedean spiral; (2) equispaced concentric rings; (3) equispaced parallel rectilinear lines. While time reduction with respect to the last two is significant, for the first is not the case, which indirectly supports Archimedean spirals as a good choice too. We explain this fact by showing the close relationship between circle involutes and Archimedean spirals. Finally, we perform some numerical simulations comparing circle involutes and Archimedean spirals within a disk (quasi-planar surfaces), assuming constant linear velocity of the measurement device and constant sampling frequency.

3D Finite Element Approach to Simulate the Contact Pressure between Two Deformable Cylinders. Application to a Spur Gear

Knowing the stresses and pressures in the contact between two deformable solids is fundamental in order to optimize the strength and the lifetime of mechanical components such as bearings or gears. These constraints can be determined by the calculation (finite element method or Hertz theory) or by experimental methods such as photoelasticity. The objective of this study is to model and compute the stress field and contact pressure using 3D finite element software. The validation of obtained results is done by comparison with the classical results of the non linear Hertz theory between two deformable cylinders. An application to spur gears with a circle involute profile is done and also validate with the same Hertz theory.

Thermodynamic Model and Experimental Study of Oil-free Scroll Compressor

In order to study the performance characteristics of oil-free scroll compressor, this paper is based on the basic equation of circle involute profile, and uses the differential geometry theory to calculate the variation law of pressure with volume. Based on the basic law of thermodynamics, the thermodynamic model of the oil-free scroll compressor is established by considering the heat transfer model and the gas leakage model, considering the mass, energy conservation equation and gas state equation. The change of the mass flow rate of the gas in each chamber is obtained by solving the established model by using the improved Euler method. The experiment results show that with the increase of frequency, the temperature, the displacement and the power show a clear upward trend. The thermodynamic model has some guidance and reference for the development and performance analysis of oil-free scroll compressors.

A unified approach to construct generalized B-splines for isogeometric applications

Generalized B-splines have been employed as geometric modeling and numerical simulation tools for isogeometric analysis (IGA for short). However, the previous models used in IGA, such as trigonometric generalized B-splines or hyperbolic generalized B-splines, are not the unified mathematical representation of conics and polynomial parametric curves/surfaces. In this paper, a unified approach to construct the generalized non-uniform B-splines over the space spanned by {α(t), β(t), ξ(t), η(t), 1, t, · · ·, tn−4} is proposed, and the corresponding isogeometric analysis framework for PDE solving is also studied. Compared with the NURBS-IGA method, the proposed frameworks have several advantages such as high accuracy, easy-to-compute derivatives and integrals due to the non-rational form. Furthermore, with the proposed spline models, isogeometric analysis can be performed on the computational domain bounded by transcendental curves/surfaces, such as the involute of circle, the helix/helicoid, the catenary/catenoid and the cycloid. Several numerical examples for isogeometric heat conduction problems are presented to show the effectiveness of the proposed methods.

Variable thickness scroll compressor performance analysis—Part I: Geometric and thermodynamic modeling

In order to investigate the performance of variable thickness scroll compressors, a detail mathematical modeling based on energy and mass balances is established in this two-part. In part I, the geometric modeling and thermodynamic modeling are developed. The profile based on circle involute, high order curve, and arc is built up using the base line method. The volume of working chambers from suction to discharge is defined. Thereafter, the evolution and derivative of the working chamber volume with respect to the orbiting angle are discussed. The energy and the mass balance for working chamber are described. Suction gas heating, radial and flank leakage, heat transfer between the working fluid, scroll wraps and plates are considered in the thermodynamic modeling. The established geometric modeling and thermodynamic modeling can provide better understanding of the variable thickness scroll compressor working process. The dynamical modeling and model validation are reported in part II.

Variable thickness scroll compressor performance analysis—Part II: Dynamic modeling and model validation

In order to investigate the performance of variable thickness scroll compressor, a detail mathematical modeling based on energy and mass balances is established in this two-part. In part II, dynamic modeling and model validation are developed. Temperature, pressure, mass flow of working chambers, friction loss power of moving parts, efficiency, and shaft power are investigated by solving the mathematical modeling. The experimental rig for variable thickness scroll compressor based on involute of circle, high order curve and arc is set up. From the comparison of the simulated and measured data, it can be seen that the compressor model predicts the mass flow, discharge temperature, and shaft power very well. So the proposed mathematical modeling can accurately describe all the suction, compression, and discharge processes for variable thickness scroll compressor.

GS0504 ディスク型SOFC流路内の速度場に及ぼす集電体の透過率・気孔率の影響

Swirling flow phenomena in a model channel of a disk shape planar solid oxide fuel cell (SOFC) with porous wall were experimentally investigated using a laser Doppler velocimetry (LDV). In the case of applying circle involute shape current collectors made of porous metal, a reduction of a velocity close to their surface was reduced. Furthermore, a velocity component normal to the current collector had positive value due to the gas flow permeating porous metal, and discrepancy from the streamline along the involute curve was observed. These characteristics were remarkable with increase in permeability and porosity of the current collectors. This result suggests that an optimum range of permeability and porosity of the current collector exists for favorable flow fields in the disk shape SOFC.

Modeling of Top Scroll Profile Using Equidistant-Curve Approach for a Scroll Compressor

Scroll profile plays a key role in determining the performance of a scroll compressor. In this study geometric and dynamic characteristics of the scroll profile are analyzed to investigate the influence of its geometric continuity on property of a scroll compressor. Firstly, scroll profiles are created to redesign the geometry of scroll wrap by using the equidistant-curve approach on the basis of a generation line consisting of involute of circle and circular arc. Subsequently, the geometric and dynamic models of the scroll compressor are established. These models are related to parameters of the generation line of scroll profile and rotation angle of a moving scroll. Lastly, some simulation examples of second-order continuity (SOC) scroll profile are compared with first-order continuity (FOC) scroll profiles and some important conclusions are obtained. Results show that SOC scroll profile is superior to FOC profile in terms of volume ratio, stability of gas force, and possible leakage loss in a scroll compressor.

11009 透過性多孔壁を持つディスク型SOFC流路内の旋回流れ

Characteristics of a swirling flow in an experimental channel of a disk shape planar solid oxide fuel cell (SOFC) were experimentally investigated with the aid of a laser Doppler velocimetry (LDV). In the case of applying circle involute shape current collectors made of porous metal, a rapid reduction of a velocity close to their surface was reduced. Furthermore, a velocity component normal to the current collector has positive value due to the gas flow permeating porous metal. These phenomena led to the discrepancy from the streamline along the involute curve and caused the mild deceleration toward the channel exit. On the other hand, a velocity distribution along the normal to the current collector became uniform by using the current collectors made of permeable porous material.

Modeling of Incompressible Flow inside the Involute Channel

Based on the model of circle involute scroll channel, a following coordinates was built, mathematical model was established in the coordinates. On the basis of the four basic assumptions, equations of motion for the flow field inside the involute channel was described under the N-S equations, and the boundary conditions for the model was analyzed. This paper explores the flow field inside the involute channel.

Study on Scroll Profile Design Using Involute of Elliptic for Scroll Compressor

The scroll compressor has the conjugate motion of meshing by the fixed scroll and the planet scroll. Several kinds of geometrical curves can be adopted to form the scroll compressor profile such as involute of circle with variable and constant radius, algebraic spiral, and so on, to improve the performance of scroll compressor. This study proposes a new scroll profile design using elliptic involute and two-envelope method to create a new conjugate meshing scroll of variable cross section. The curve of the scroll section on both sides can be specified as an arbitrary mathematical function. First, the base scroll swept out by the central base circle. Next, the base circle translation path envelope of the planet scroll is moved by the basic involute curve. The base circle translation path envelope is the first envelope. The second envelope generated by the planet scroll conjugate profile is the fixed scroll. By using two-envelope method, the variety of conjugate scroll can be produce with the cross section and symmetrical variables for parametric design.