【正文】 ecessary toothbending fatigue resistance and mesh stiffness. Thedirect gear design approach allows selection of anyfillet profile (parabola, ellipsis, cubic spline, etc.)that would best satisfy those conditions. This profileis not necessarily the trochoid formed by the rack orshaper generating process. Tool geometry definition is the next step indirect gear design. This will depend on the actualmanufacturing method. For plastic and metal gearmolding, gear extrusion, and powder metal gearprocessing, the entire gear geometry—includingcorrection for shrinkage—will be directly appliedto the tool cavity. For cutting tools (hobs, shapercutters), the reverse generating approach “gearforms tool” can be applied. In this case, the tooling pitch and profile (pressure) angle are selectedto provide the best cutting conditions. Area of Existence of Involute GearsFigure 5 shows an area of existence for a pinionand gear with certain numbers of teeth z1, z2, andproportional top land thicknesses ma1, ma2(Ref. 2).Unlike the zone shown in Figure 1, the area of existence in Figure 5 contains all possible gear binations and is not limited to restrictions imposed by agenerating rack. This area can be shown in proportional base tooth thicknesses mb1– mb2coordinatesor other parameters describing the angular distancebetween two involute flanks of the pinion and gearteeth, like αa1– αa2or ν1– ν2. A sample of the areaFigure 4—The angular shift of the transverse sections for a helical gear.Figure 5—The area of existence for the gear pairz1= 14, z2= 28.Figure 6—Involute gear meshes: 6a, at point A of Figure 5 (αwmax= 176。This paper presents an alternative method ofanalysis and design of spur and helical involutegears. IntroductionModern gear design is generally based onstandard tools. This makes gear design quite simple (almost like selecting fasteners), economical,and available for everyone, reducing toolingexpenses and inventory. At the same time, it iswell known that universal standard tools providegears with less than optimum performance and—in some cases—do not allow for finding acceptable gear solutions. Application specifics, including low noise and vibration, high density ofpower transmission (lighter weight, smaller size)and others, require gears with nonstandard parameters. That’s why, for example, aviation geartransmissions use tool profiles with custom proportions, such as pressure angle, addendum, andwhole depth. The following considerations makeapplication of nonstandard gears suitable andcostefficient:? CNC cutting machines and CMM gear inspection equipment make production of nonstandardgears as easy as production of standard ones. ? Cost of the custom cutting tool is not muchhigher than that of the cutting tool for standardgears and can be amortized if production quantityis large enough. ? The custom gear performance advantage makesa product more petitive and justifies largertooling inventory, especially in mass production. ? Gear grinding is adaptable to custom toothshapes. ? Metal and plastic gear molding cost largely doesnot depend on tooth shape. This article presents the direct gear designmethod, which separates gear geometry definitionfrom tool selection, to achieve the best possible performance for a particular product and application.The direct design approach that is monlyused for most parts of mechanisms and machines(for example, cams, linkages, pressor or turbine blades, etc.) determines their profiles according to the operating conditions and desired performance. Ancient engineers used the sameDirect Gear Design for Spurand Helical Involute GearsAlexander L. Kapelevich and Roderick E. KleissNomenclaturebwface width in the meshdaoutside circle diameter, mm (in.)dbbase circle diameter, mm (in.)d?tip circle diameter, mm (in.)maproportional top land tooth thickness mbproportional base tooth thicknesspbbase pitch, mm (in.)Satop land tooth thickness, mm (in.)Sbbase tooth thickness, mm (in.)u gear ratioz number of teethαaoutside circle profile angle, degreesαpprofile angle in the bottom contact point, degreesαwoperating pressure angle, degreesβbbase circle helix angle, degreesεαcontact ratioεβaxial contact ratioν tip circle profile angle, degreesSubscript1pinion2gearDr. Alexander L.Kapelevich is an owner of the consulting firm AKGears ofShoreview, MN, and principal engineer for KleissGears Inc. of Centerville,MN. He has more than 20years of experience indevelopment of aviationand mercial gear transmissions in Russia and theUnited States.Roderick E. Kleiss,professional engineer, isowner and president ofKleiss Gears Inc. His pany engineers and manufactures high precision,plastic molded gears usingthe direct gear designapproach. Figure 1—The zone allowed by the standard 20176。rack for a gear pair z1= 14, z2= 28. ? ? GEAR TECHNOLOGY ? SEPTEMBER/OCTOBER 2020 29Page 30 35 8/1/02 12:53 PM Page 3030 SEPTEMBER/OCTOBER 2020 ? GEAR TECHNOLOGY ? ? approach for gear design, developing the toothshape first and then figuring out a way to get it.During the technological revolution in the 19thcentury, the highly productive gear generat