Involute Curve Research Articles

On Involute and Evolute Curves of Spacelike Curve with a Spacelike Principal Normal in Minkowski 3-Space

On Involute and Evolute Curves of Spacelike Curve with a Spacelike Principal Normal in Minkowski 3-Space

S1101-3-4 歯車の歯当たりとかみ合い評価法について : 可展円筒歯車歯面の図式展開について(伝動装置の基礎と応用セッション3)

Tooth bearing patterns give us very useful information about gear engagement situations. However, the detail analysis of gear engagement using actual contact patterns is not enough up to now, because the actual patterns are normally captured by a still camera as projected images. In the case of the involute helicoid surface, it is possible to develop it onto the plane geometrically. It is more convenient to compare the actual patterns with simulated ones on the plane. In this paper, geometrical developable surfaces are classified in the first place, and the involute helicoid surface is developed onto the plane geometrically with the help of the geodesic curvatures of tooth profiles and tooth leads. It is found that the lead circular helices are developed as the concentric circles on a plane, and the involute tooth profile is developed as an involute curve which has a certain diameter of a base circle.

Discrete Log-Aesthetic Filter

AbstractThe aesthetic curves include the logarithmic (equiangular) spiral, clothoid, and involute curves. Although most of them are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them and they are expected to be utilized for practical use of industrial and graphical design. However, the curvature of a log-aesthetic curve segment must be monotonically increasing or decreasing and a strong constraint as a function of arc length is imposed on its curvature. For geometrical design, it is desirable not to impose strong constraints on the designer’s activity, to let him/her design freely and to embed the properties of the log-aesthetic curves for complicated curves with both increasing and decreasing curvature. Hence we develop discrete filters named discrete log-aesthetic filters to fair a sequence of points with noises and to fit it locally to log-aesthetic curves. Furthermore we extend them for surfaces.

Discrete Log-Aesthetic Filter

AbstractThe aesthetic curves include the logarithmic (equiangular) spiral, clothoid, and involute curves. Although most of them are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them and they are expected to be utilized for practical use of industrial and graphical design. However, the curvature of a log-aesthetic curve segment must be monotonically increasing or decreasing and a strong constraint as a function of arc length is imposed on its curvature. For geometrical design, it is desirable not to impose strong constraints on the designer’s activity, to let him/her design freely and to embed the properties of the log-aesthetic curves for complicated curves with both increasing and decreasing curvature. Hence we develop discrete filters named discrete log-aesthetic filters to fair a sequence of points with noises and to fit it locally to log-aesthetic curves. Furthermore we extend them for surfaces.

Kinematic flowrate characteristics of external spur gear pumps using an exact closed solution

This paper investigates flow characteristics of external spur gear pumps using a flowrate formula derived in closed form. Nondimensional flowrates and fluctuation coefficients of gear pumps are discussed. By using the formula, flowrates can be calculated accurately and efficiently. Firstly, using the involute equation in polar form, an analytic solution of the arc area enveloped by the involute curve is derived. Then, the formula representing pump flowrate is deduced using a control volume approach. During the process, the volume exiting the control region is divided into two parts which, respectively belong to involutes and to the line of contact resulting from movement of the meshing point with the rotation of gear pairs. Next, flowrates and fluctuation coefficients of spur gear pumps both with and without relief grooves are calculated and compared. Results show the essential role of relief grooves in reducing flowrate fluctuation. Finally, influences of design parameters which include tooth number, module, pressure angle, and addendum coefficient on pump flowrate behavior are discussed.

An Optimization Approach to the Displacement Volumes for External Spur Gear Pumps

To increase volume displacement is always concerning in designing external gear pumps (EGPs). Therefore, an approach of optimization analysis intending to enhance their displacement capability is proposed in this study. Through that, design parameters of the spur gears in the pumps are systematically resulted to achieve their optimal volume displacements. Parametrically, a CAD model to visualize the designed gear of the optimal design is also created. Firstly, the study derives tooth profile equations by using the coordinate transformation and equation of meshing for gears on a rack cutter profile. Then, an analytic formula represented the enveloping area by an involute curve is derived. Next, volumes of output and trapped backflow of the EGP are achieved. Therefore the net output volume can be calculated accurately and efficiently. After that, the optimization analysis to maximize the volume displacement is performed. Through that, optimal design parameters for the pumps are achieved under assigned constraints for considerations of design and manufacturing. Additionally, influences of module, pressure angle, and addendum correction factor for the gears in the pumps on their displacement capability are also investigated. Additionally, flowrate fluctuation characteristics under different pressure angles of gears are finally discussed.

Effect of the load location along the involute curve of spur gears on the applied stress at the fillet radius

AbstractIn this study, attempts are carried out to determine the amount of stress at the fillet radius region of spur gears when the applied load location changes, along the involute curve, from the top surface to the bottom. For this purpose the photoelastic method and numerical MSC/NASTRAN software are used. The gears with pressure angle (ϕ) 20o, and 25o were prepared from photoelastic material type PLM‐4B in this study. Practical calibration is used to determine the fringe order value of this material. Four different modules (m); 6, 10, 14, and 20 mm were prepared for two different numbers of teeth (N); 18 and 26, with different face widths (B1, B2, B3); 10, 17, 25.4 mm respectively. Four load values were applied on each tooth at five to six different load locations along the involutes curve profile. In order to accomplish the comparison between the results for different methods, the same sample dimensions and parameters were prepared again for the MSC/NASTRAN software.The results showed that the maximum applied stress at the fillet radius occur when the applied load location is on the top land of a tooth, and then that amount is decreased when the applied load positions change toward the bottom land. The results of the NASTRAN method showed that the applied stress at the fillet radius would be minimum when the loading point locates between the pitch circle and dedendum circle, in particular around 1.5 times module (1.5 m) of the total tooth height which equals 2.25 module, and then the applied stresses are increased again. However, in the photoelastic method the applied stresses were decreased continuously to the bottom land. The reasons behind such results can be attributed to the type of failure theories that can be used in NASTRAN software for characterization the applied stresses, i.e. considering types of applied stresses into account, such as bending, direct compressive, and shear stresses. Moreover, in order to compare the applied stress values at fillet regions, obtained by theoretical and practical approaches, four different standard mathematical equations are used for stress calculation at fillet radius of spur gears to show the difference of the parameters and variables that can affect the applied stress results.

Tooth Form Evaluation Using Ball Artifact Development of a Measuring Instrument of a Ball Center Distance Traceable to National Standard of Length

AIST (National Institute of Advanced Industrial Science and Technology) has proposed a new calibration method of gear measuring instrument using a ball artifact consisting of two master balls and an optical flat base (DBA). We measure a circular arc instead of involute tooth form. Difference of the distance between the arc of the master ball and involute curve can be calculated theoretically. For the calculation, center distance between two balls is required and the value of measurement uncertainty is less than 0.1 'm and also should be traceable to national standard of length. This paper reports on a new method and an originally developed instrument to measure the center distance between two balls using laser interferometer. The expanded uncertainty in measurement of a center distance of 44 mm was 21 nm.

21014 ディスク型SOFCの流路形状に関する流体力学的検討(流体工学(2),一般講演,学術講演)

The velocity fields between two parallel disks which simulate the disk-type solid oxide fuel cell (SOFC), were measured by using particle image velocimetry (PIV). The experiments have been performed for two kinds of disk channels to investigate the effects of channel geometries on flow behavior. In the fuel channel with four linear-shape current collectors, a working gas was decelerated along the flow direction and hence optimum flow conditions were restricted. However, by adopting the involute curve as a current collector shape, uniform velocity distributions were realized in the fuel channel except for near its outer rim. On the other hands, in the air channel having four injection holes, non-uniform flow distribution is enlarged with increase of flow rate and local high velocity regions were observed at a high flow rate. We confirmed that the increase of the hole cross section area and the number of injection holes led to the uniform velocity distributions.

Approximation of involute curves for CAD-system processing

In numerous instances, accurate algorithms for approximating the original geometry is required. One typical example is a circle involute curve which represents the underlying geometry behind a gear tooth. The circle involute curves are by definition transcendental and cannot be expressed by algebraic equations, and hence it cannot be directly incorporated into commercial CAD systems. In this paper, an approximation algorithm for circle involute curves in terms of polynomial functions is developed. The circle involute curve is approximated using a Chebyshev approximation formula (Press et al. in Numerical recipes, Cambridge University Press, Cambridge, 1988), which enables us to represent the involute in terms of polynomials, and hence as a Bezier curve. In comparison with the current B-spline approximation algorithms for circle involute curves, the proposed method is found to be more accurate and compact, and induces fewer oscillations.

On the involute and evolute curves of the spacelike curve with a spacelike binormal in Minkowski 3-space

In this study, we have generalized the involute and evolute curves of the spacelike curve α with a spacelike binormal in Minkowski 3- Space. Firstly, we have shown that, the length between the spacelike curve α and the timelike curve β is constant. Furthermore, the Frenet frame of the involute curve β has been found as depend on curvatures of the curve α .W e have determined the curve α is planar in which conditions. Secondly, we have found transformation matrix between the evolute curve β and the curve α. Finally, we have computed the curvatures of the evolute curve β. Mathematics Subject Classification: 53A04, 53B30, 53A35

3206 制御点入力による複合リズムを持つ美的曲線の生成(OS4-3 デジタルエンジニアリング-3)意匠曲線・曲面)

The aesthetic curves include the logarithmic (equiangular) spiral, clothoid, and involute curves. Although most of them are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them and they are expected to be utilized for practical use of industrial and graphical design. However, their input method proposed so far by use of three so-called control points can generate only an aesthetic curve segment with monotonic curvature variation and can not create a curve with the compounded-rhythm. Hence for four input control points with two specified α values, we propose a technique to generate a compounded-rhythm aesthetic curve.

I11 インボリュート曲線の最適円弧補間の簡易方法(I1 加工・生産システム1(マイクロ技術の高度化))

I11 インボリュート曲線の最適円弧補間の簡易方法(I1 加工・生産システム1(マイクロ技術の高度化))

Experimental and finite element analysis to predict the dimensional changes of a cold-forged spur gear

The dimensional tolerance of spur gears for automobiles is within the range of several micrometres. Therefore, to manufacture these gears by cold forging, dimensions should be controlled in the same range. The dimensional changes by the various forging variables should be analysed and compensated into the forging tool. Computer-aided engineering (CAE) would be appropriate for investigating the effects of variables. However, the realistic finite element modelling (FEM) techniques should be confirmed and used to predict the dimension of the forged part in the order of micrometres. As realistic modelling considers the elastic characteristics of both workpiece and die at unloading and ejecting stage, threedimensional analysis for spur gears may often encounter difficulties in solving time and convergence. In this study, dimensional changes for cold-forged parts of spur gears were investigated by experimental and FEM analysis, after modelling to obtain desirable FEM results had been studied by closed-die upsetting. The dimensional changes of involute curve for the spur gear can be analysed from the FEM results during the whole forging stage; and give an index of a modified involute curve of the forging tool for improving gear profile accuracy.

INVESTIGATION OF WRAP PROFILE MODIFICATION USING CIRCLE INVOLUTE FOR SCROLL COMPRESSOR

The condition of a pair of circle involutes being used as meshing profile for a scroll wrap is proposed. The wrap profile modification using circle involute curve is established, and by this method the perfect meshing profile can be obtained. The generation of this modified wrap is investigated, in addition, the variation relationship of modified wrap parameters, the equations of modification profile as well as the coordinates of important design points are derived. The effects of profile modification on the modified wrap's performances are discussed in detailed when the wrap parameters are changed. Compared with the wrap modified by using twin-circular arcs, this modified wrap improves wrap's meshing angle 10°~25°, increases wrap's strengthened angle 12°~30°, so that the pressure ratio and wrap head strength are improved. This modification method, by which the modified wrap has the higher performance, is more perfect than the present profile modification method. The study results can be applied in the calculation of profile modification directly.

3813 非円形歯車のCAD(S70-2 新しい設計支援技術(2),S70 新しい設計支援技術)

As a non-circular gear is equipped the motional characteristics of both cam and circular gear, it can be used for transmitting rotary motion at a variable rate in an instrument or a machine. This paper describes a CAD method for non-circular gears. It consists of: (1) Necessity and purposes of the research, (2) Mechanism of non-circular gears and the relationship between a shape and its transmission motion or velocity, (3) Design of teeth and pitch curves, such as an involute curve as a tooth curve, and an ellipse or a free-form curve of B-spline as a pitch curve defined by analytical geometry methods, (4) Calculation and adjustment of the number of teeth, (5) Distribution and trimming of teeth, (6) Creation of 3D models of the gears and (7) Discussions and conclusions.

High Precision and High Efficiency Machining of the Scroll Compressor Pieces (Improvement of Machining Accuracy in Constant Cutting Point Control)

Scroll compressor components have spiral grooves known as scroll wraps. Since their shape is determined by an involute curve and their accuracy dominates the compressor performance, they are produced by precise contour milling using end mills. The progress in NC device for milling involute shape has enabled a feed speed control at a cutting point and a position control, keeping a cutting point constant relatively to the end mill center. Instead of the conventional method reported formerly which controls the feed speed at the center of the end mill, more precise milling process has been required, which is based on a new control. While milling the inside face of the curve, even if the feed speed is controlled so as to keep the projection cutting area constant, the cutting force increases in connection with the tool contact length. This increase in cutting force deteriorates the profile accuracy. Though controlling the radial depth of cut is one solution, the minimum depth of cut near the involute center is so small as the surface roughness, and the accuracy is not stable. In the present paper, the relationship among cutting area, tool contact length and cutting force is analyzed by the multiple linear regression analysis, and the cutting force was simulated. The result of the cutting force simulation showed a good coincidence with the experimental result in tendency. Using the result of calculation to control the feed speed at the cutting point so that the cutting force can be kept constant, it was possible to improve the profile accuracy from 8μm to 3μm.

Modelling, Implementation, and Manufacturing of Spiral Bevel Gears with Crown

The operational performance of gears in terms of smoothness, quietness, wear and life span is largely affected by how gear and pinion teeth make contact. To enhance the operational performance, a crown is often applied to a standard (nominal) gear tooth surface. Regardless of the presence of a crown, all the gears are machined by special types of machine tools, such as gear hobbers and shapers. This paper develops a tooth surface for a spiral bevel gear with a crown so that it can be machined by numerically controlled machine tools. Specifically, we derived: 1. A bi-parametric tooth surface model for a standard spiral bevel gear. 2. A crown model along the spiral curve direction. 3. A crown model along the involute curve direction. The developed algorithm was tested and implemented in a prototype software system called GearCAM. With the GearCAM system, a set of spiral bevel gear and pinion was machined using a 4-axis CNC milling machine to check the validity and effectiveness of the crowning method. Through various verifications, it is shown that the models developed for the standard and crown gears can be used as a means for design and verification of spiral bevel gears.

Computer-aided design and manufacturing of scroll compressors

Scroll type compressors have been becoming increasingly popular in refrigeration and air conditioning products due to their high efficiency, low noise, fewer components, low vibration, light weight and small size compared with other types of compressors. To design and manufacture scroll compressors, however, is not easy as their key parts have complex shapes with involute curves. A high degree of accuracy during assembly is required which brings difficulty into the manufacturing cycle. This paper presents the integrated approach of computer-aided design (CAD) and computer-aided manufacturing (CAM) for the design and manufacture of refrigeration scroll compressors. The approach is based on concurrent engineering and is implemented in Pro/ENGINEER software and Visual C++ programming. The work presented is focused on the integration of constructive solid geometry (CSG) solid modelling and CAM to effect the design of scroll compressors in particular. The involute scroll curve is created from equations. Designers can transfer the model to a manufacturing engineer for process planning, etc. The manufacturing engineer can also advise the designer if some revisions are needed and the model can even be evaluated through the Internet. Many experts can check the models and processes of design and manufacturing. Thus, solid modelling and machine tool path simulation can bridge the gap between design and manufacturing which is essential for concurrent engineering. With this approach a correct and precise solid model can be achieved, which leads to an optimal design in terms of quality, delivery time and cost. This work is essential for the manufacturing of the scroll components and their inspection on co-ordinate measuring machines (CMMs) during manufacture. This CAD/CAM system has a potential application to other complex mechanical components, as the approach is a generic one.

A study of the stress and strain distributions of first-pass conventional spinning under different roller-traces

In this paper, based on the establishment of a reasonable mechanics model for a typical draw-spinning, research on the first pass of the spinning process is carried out with an elasto-plastic FEM method. The distributions of the stress and strain are obtained under three types of roller-trace curves: straight lines, involute curves and quadratic curves. The distribution laws of the radial and tangential stress that are obtained can serve as a significant guide to determining the forming limits in multi-pass conventional spinning under different roller-traces. Comparison of the distributions of strain and stress of three roller-traces could provide a theoretical basis for selecting a reasonable roller-trace in multi-pass conventional spinning.