Gears P1
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Gears P1
This section presents a technical coverage of gear fundamentals. It is intended as a broad coverage written in a manner that is easy to follow and to understand by anyone interested in knowing how gear systems function.
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 1.0 INTRODUCTION T25 2.0 BASIC GEOMETRY OF SPUR GEARS 2.1 Basic Spur Gear Geometry T25 2.2 The Law of Gearing T25 2.3 The Involute Curve T27 2.4 Pitch Circles T27 2.5 Pitch T28 2.5.1 Circular Pitch T28 2.5.2 Diametral Pitch T28 2.5.3 Relation of Pitches T28 3.0 GEAR TOOTH FORMS AND STANDARDS 3.1 Preferred Pitches T29 3.2 Design Tables T29 3.3 AGMA Standards T29 4.0 INVOLUTOMETRY 4.1.1 Gear Nomenclature T31 4.1.2 Symbols T37 4.2 Pitch Diameter and Center Distance T37 4.3 Velocity Ratio T38 4.4 Pressure Angle T38 4.5 Tooth Thickness 138 4.6 Measurement OverPins T39 4.7 Contact Ratio 144 4.8 Undercutting 144 4.9 Enlarged Pinions 145 4.10 Backlash Calculation 145 4.11 Summary of Gear Mesh Fundamentals T48 5.0 HELICAL GEARS 5.1 Generation of the Helical Tooth T52 5.2 Fundamental of Helical Teeth T53 5.3 Helical Gear Relationships T53 5.4 Equivalent Spur Gear T54 5.5 Pressure Angle T54 5.6 Importance of Normal Plane Geometry T54 5.7 Helical Tooth Proportions T55 5.8 Parallel Shaft Helical Gear Meshes T55 5.8.1 Helix Angle 155 5.8.2 Pitch Diameter T55 5.8.3 Center Distance T55 5.8.4 Contact Ratio T55 5.8.5 Involute Interference 156 5.9 Crossed Helical Gear Meshes 156 5.9.1 Helix Angle and Hands T56 5.9.2 Pitch 156 T21
 Catalog D190 5.9.3 Center Distance T57 5.9.4 Velocity Ratio T57 5.10 Axial Thrust of Helical Gears T57 6.0 RACKS T58 7.0 INTERNAL GEARS 7.1 Development of the Internal Gear T58 7.2 Tooth Parts of Internal Gear T59 7.3 Tooth Thickness Measurement T60 7.4 Features of Internal Gears T61 8.0 WORM MESH 8.1 Worm Mesh Geometry T61 8.2 Worm Tooth Proportions T62 8.3 Number of Threads T62 8.4 Worm and Wormgear Calculations T62 8.4.1 Pitch Diameters, Lead and Lead Angle T63 8.4.2 Center Distance of Mesh T63 8.5 Velocity Ratio T64 9.0 BEVEL GEARING 9.1 Development and Geometry of Bevel Gears T64 9.2 Bevel Gear Tooth Proportions T66 9.3 Velocity Ratio T66 9.4 Forms of Bevel Teeth T67 10.0 GEAR TYPE EVALUATION T68 11.0 CRITERIA OF GEAR QUALITY 11.1 Basic Gear Formats T68 11.2 Tooth Thickness and Backlash T70 11.3 Position Error (or Transmission Error) T70 11.4 AGMA Quality Classes T73 11.5 Comparison With Previous AGMA and International Standards T73 12.0 CALCULATION OF GEAR PERFORMANCE CRITERIA 12.1 Backlash in a Single Mesh T76 12.2 Transmission Error T77 12.3 Integrated Position Error T77 12.4 Control of Backlash T78 12.5 Control of Transmission Error T78 13.0 GEAR STRENGTH AND DURABILITY 13.1 Bending Tooth Strength T78 13.2 Dynamic Strength T82 13.3 Surface Durability T88 13.4 AGMA Strength and Durability Ratings T88 T22 file:///C/A3/D190/HTML/D190T22.htm [9/27/2000 4:11:52 PM]
 14.0 GEAR MATERIALS 14.1 Ferrous Metals T91 14.1.1 Cast Iron T91 14.1.2 Steel T91 14.2 Non Ferrous Metals T92 14.2.1 Aluminum T92 14.2.2 Bronzes T92 14.3 Die Cast Alloys T92 14.4 Sintered Powder Metal T92 14.5 Plastics T92 14.6 Applications and General Comments T99 15.0 FINISH COATINGS 15.1 Anodize T99 15.2 Chromate Coatings T100 15.3 Passivation T100 15.4 Platings T100 15.5 Special Coatings T100 15.6 Application of Coatings T100 16.0 LUBRICATION 16.1 Lubrication of Power Gears T101 16.2 Lubrication of Instrument Gears T101 16.3 Oil Lubricants T101 16.4 Grease T103 16.5 Solid Lubricants T103 16.6 Typical Lubricants T103 17.0 GEAR FABRICATION 17.1 Generation of Gear Teeth T105 17.1.1 Rack Generation T105 17.1.2 Hob Generation T105 17.1.3 Gear Shaper Generation T105 17.1.4 Top Generating T106 17.2 Gear Grinding T106 17.3 Plastic Gears T107 18.0 GEAR INSPECTION 18.1 VariableCenterDistance Testers T107 18.1.1 Total Composite Error T107 18.1.2 Gear Size T107 18.1.3 Advantages and Limitations of VariableCenterDistance Testers... T107 18.2 OverPins Gaging T108 18.3 Other Inspection Equipment T108 18.4 Inspection of FinePitch Gears T108 18.5 Significance of Inspection and Its Implementation T108 T23
 19.0 GEARS, METRIC 19.1 Basic Definitions T109 19.2 Metric Design Equations T122 19.3 Metric Tooth Standards T124 19.4 Use of Strength Formulas T125 19.5 Metric Gear Standards T126 19.5.1 USA Metric Gear Standards T126 19.5.2 Foreign Metric Gear Standards T126 20.0 DESIGN OF PLASTIC MOLDED GEARS 20.1 General Characteristics of Plastic Gears T131 20.2 Properties of Plastic Gear Materials T132 20.3 Pressure Angles T139 20.4 Diametral Pitch T139 20.5 Design Equations for Plastic Spur, Bevel, Helical and Worm Gears T139 20.5.1 General Considerations T139 20.5.2 Bending Stress  Spur Gears T140 20.5.3 Surface Durability for Spur and Helical Gears T141 20.5.4 Design Procedure  Spur Gears T143 20.5.5 Design Procedure Helical Gears T146 20.5.6 Design Procedure  Bevel Gears T146 20.5.7 Design Procedure  Worm Gears T147 20.6 Operating Temperature T147 20.7 Eftect of Part Shrinkage on Gear Design T147 20.8 Design Specifications T150 20.9 Backlash T150 20.10 Environment and Tolerances T150 20.11 Avoiding Stress Concentration T150 20.12 Metal Inserts T151 20.13 Attachment of Plastic Gears to Shafts T151 20.14 Lubrication T152 20.15 Inspection T152 20.16 Molded vs Cut Plastic Gears T152 20.17 Elimination of Gear Noise T153 20.18 Mold Construction T153 20.19 Conclusion T158 T24
 1.0 INTRODUCTION This section presents a technical coverage of gear fundamentals. It is intended as a broad coverage written in a manner that is easy to follow and to understand by anyone interested in knowing how gear systems function. Since gearing involves specialty components it is expected that not all designers and engineers possess or have been exposed to all aspects of this subject However, for proper use of gear components and design of gear systems it is essential to have a minimum understanding of gear basics and a reference source for details. For those to whom this is their first encounter with gear components, it is suggested this section be read in the order presented so as to obtain a logical development of the subject. Subsequently, and for those already familiar with gears, this material can be used selectively in random access as a design reference. 2.0 BASIC GEOMETRY OF SPUR GEARS The fundamentals of gearing are illustrated through the spurgear tooth, both because it is the simplest, and hence most comprehensible, and because it is the form most widely used, particularly in instruments and control systems. 2.1 Basic Spur Gear Geometry The basic geometry and nomenclature of a spurgear mesh is shown in Figure 1.1. The essential features of a gear mesh are: 1. center distance 2. the pitch circle diameters (or pitch diameters) 3. size of teeth (or pitch) 4. number of teeth 5. pressure angle of the contacting involutes Details of these items along with their interdependence and definitions are covered in subsequent paragraphs. 2.2 The Law of Gearing A primary requirement of gears is the constancy of angular velocities or proportionality of position transmission, Precision instruments require positioning fidelity. High speed and/or high power gear trains also require transmission at constant angular velocities in order to avoid severe dynamic problems. Constant velocity (i.e. constant ratio) motion transmission is defined as “conjugate action” of the gear tooth profiles. A geometric relationship can be derived (1,7)* for the form of the tooth profiles to provide cojugate action, which is summarized as the Law of Gearing as follows: “A common normal to the tooth profiles at their point of contact must, in all positions of the contacting teeth, pass through a fixed point on the lineofcenters called the pitch point.” Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate Curves. ___________ *Numbers in parenthesis refer to references at end of text. T25
 T26
 2.3 The Involute Curve There are almost an infinite number of curves that can be developed to satisfy the law of gearing, and many different curve forms have been tried in the past. Modem gearing (except for clock gears) based on involute teeth. This is due to three major advantages of the involute curve: 1. Conjugate action is independent of changes in center distance. 2. The form of the basic rack tooth is straightsided, and therefore is relatively simple and can be accurately made; as a generating tool ft imparts high accuracy to the cut gear tooth. 3. One cutter can generate all gear tooth numbers of the same pitch. The involute curve is most easily understood as the trace of a point at the end of a taut string that unwinds from a cylinder. It is imagined that a point on a string, which is pulled taut in a fixed direction, projects its trace onto a plane that rotates with the base circle. See Figure 1.2. The base cylinder, or base circle as referred to in gear literature, fully defines the form of the involute and in a gear it is an inherent parameter, though invisible. The development and action of mating teeth can be visualized by imagining the taut string as being unwound from one base circle and wound on to the other, as shown in Figure 1.3a Thus, a single point on the string simultaneously traces an involute on each base circles rotating plane. This pair of involutes is conjugate, since at all points of contact the common normal is the common tangent which passes through a fixed point on the lineofcenters. It a second winding/unwinding taut string is wound around the base circles in the opposite direction, Figure 1 .3b, oppositely curved involutes are generted which can accommodate motion reversal. When the involute pairs are properly spaced the result is the involute gear tooth, Figure 1.3c. 2.4 Pitch Circles Referring to Figure 1.4 the tangent to the two base circles is the line of contact, or lineofaction in gear vernacular. Where this line crosses the lineofcenters establishes the pitch point, P. This in turn sets the size of the pitch circles, or as commonly called, the pitch diameters. The ratio of the pitch diameters gives the velocity ratio: Velocity ratio of gear 2 to gear 1 = Z = D1 (1) D2 T27
 2.5 Pitch Essential to prescribing gear geometry is the size, or spacing of the teeth along the pitch circle. This is termed pitch and there are two basic forms. 2.5.1 Circular pitch — A naturally conceived linear measure along the pitch circle of the tooth spacing. Referring to Figure 1.5 it is the linear distance (measured along the pitch circle ar between corresponding points of adjacent teeth. it is equal to the pitchcircle circumference divided by the number of teeth: pc = circular pitch = pitch circle circumference = Dπ (2) number of teeth N 2.5.2 Diametral pitch — A more popularly used pitch measure, although geometrically much less evident, is one that is a measure of the number of teeth per inch of pitch diameter. This is simply: expressed as: Pd = diametral pitch = N (3) D Diametral pitch is so commonly used with fine pitch gears that it is usually contracted simply to "pitch" and that it is diametral is implied. 2.5.3 Relation of pitches: From the geometry that defines the two pitches it can be shown that they are related by the product expression: Pd x Pe = π (4) This relationship is simple to remember and permits an easy transformation from one to the other. T28
 3.0 GEAR TOOTH FORMS AND STANDARDS involute gear tooth forms and standard tooth proportions are specified in terms of a basic rack which has straightsided teeth for involute systems. The American National Standards Institute (ANSI) and the American Gear Manufacturers Association (AGMA) have jointly established standards for the USA. Although a large number of tooth proportions and pressure angle standards have been formulated, only a few are currently active and widely used. Symbols for the basic rack are given in Figure 1.6 and pertinent standards for tooth proportions in Table 1.1. Note that data in Table 1.1 is based upon diametral pitch equal to one. To convert to another pitch divide by diametral pitch. 3.1 Preferred Pitches Although there are no standards for pitch choice a preference has developed among gear designers and producers. This is given in Table 1.2. Adherence to these pitches is very common in the fine pitch range but less so among the coarse pitches. 3.2 Design Tables For the preferred pitches it is helpful in gear design to have basic data available as a function of the number of teeth on each gear, Table 1.3 lists tooth proportions common to a given diametral pitch, as well as the diameter of a measuring wire. Table 1.6 lists pitch diameters and the overwires measurement as a function of tooth number (which ranges from 18 to 218) and various diametral pitches, including most of the preferred fine pitches. Both tables are for 20° pressureangle gears. 3.3 AGMA Standards In the United States most gear standards have been developed and sponsored by the AGMA. They range from general and basic standards, such as those already mentioned for tooth form, to specialized standards. The list is very long and only a selected few, most pertinent to fine pitch gearing, are listed in Table 1.4. These and additional standards can be procured from the AGMA by contacting the headquarters office at 1500 King Street; Suite 201; Alexandria, VA 22314 (Phone: 7036840211). a = Addendum b = Dedendum c = Clearance hk = Working Depth ht = Whole Depth Pc = Circular Pitch rf = Fillet Radius t = circular Tooth Thickness φ = Pressure Angle Figure 1.6 Extract from AGMA 201.02 (ANSI B6.1 1968) T29
 TABLE 1.1 TOOTH PROPORTIONS OF BASIC RACK FOR STANDARD INVOLUTE GEAR SYSTEMS Symbol 141/2º 20º 20º 20º in Full Depth Full Depth CoarsePitch FinePitch Tooth Parameter Rack involute involute involute involute Fig. 1.6 System System Spur Gears System 1. System Sponsors −− ANSI & AGMA ANSI AGMA ANSI & AGMA 2. Pressure Angle φ 141/2° 20° 20° 20° 3. Addendum a 1/P 1/P 1.000/P 1.000/P 4. Dedendum b 1.157/P 1.157/P 1.250/P 1.200/P + 0.002 5. Whole Depth ht 2.157/P 2.157/P 2.250/P 2.200/P + 0.002 6. Working Depth hk 2/P 2/P 2.000/P 2.000/P 7. Clearance. C 0.157/P 0.157/P 0250/P 0.200/P + 0.002 8. Basic Circular Tooth t 1 5708/P 1.5708/P π/2P 1.5708/P Thickness on Pitch Line 9. Fillet Radius In rf 11/3 x 1112 X 0.300/P not standardized Basic Rack 10. Diametral Pitch Range  not specified not specified not specified not specified 11. Governing Standard: ANSI  B6.1 B6.1  B6.7 AGMA  201.02  201.02 207.06 TABLE 1.2 PREFERRED DIAMETRAL PITCHES Class Pitch Class Pitch Class Pitch Class Pitch 20 24 1/2 32 1 12 48 2 150 Medium 14 64 Coarse 4 Fine UltraFine 180 Coarse 16 72 6 200 18 80 8 96 10 120 128 TABLE 1.3 BASIC GEAR DATA FOR 20° P.A. FINEPITCH GEARS Diameter Pitch 32 48 64 72 80 96 120 200 Diameter of .0540 .0360 .0270 .0240 .0216 .0180 .0144 .0086 Measuring Wire* Circular Pitch .09817 .06545 .04909 .04363 .03927 .03272 .02618 .01571 Circular Thickness .04909 .03272 .02454 .02182 .01963 .01638 .01309 .00765 Whole Depth .0708 .0478 .0364 .0326 .0295 .0249 .0203 .0130 Addendum .0313 .0208 .0156 .0139 .0125 .0104 .0083 .0050 Dedendum .0395 .0270 .0208 .0187 .0170 .0145 .0120 .0080 clearance .0083 .0062 .0051 .0048 .0045 .0041 .0037 .0030 Note: Outside Diameter for N number of teeth equals the Pitch Diameter far (N+2) number at teeth. *For 1.7290 wire diameter basic wire system. T30
 TABLE 1.4 SELECTED LIST OF AGMA STANDARDS AGMA 390 Gear Classification Handbook General AGMA Gear Classification And Inspection Handbook 2000A88 spurs And AGMA 201 Tooth portions For CoarsePitch Involute Spur Gears Helicals AGMA 207 Tooth Proportions For FinePitch Involute Spur Gears And Helical Gears AGMA DesignManual For Bevel Gears 2005B88 NonSpur AGMA 203 FinePitch OnCenter Face Gears For 20° Involute Spur Pinions AGMA 374 Design For FinePitch Worm Gearing 4.0 INVOLUTOMETRY Basic calculations for gear systems are included in this section for ready reference in design. More advanced calculations are available in the listed references. 4.1.1 GEAR NOMENCLATURE* ACTIVE PROFILE is that part of the gear tooth profile which actually comes in contact with the profile of its mating tooth along the line of action. ADDENDUM (a) is the height by which a tooth projects beyond the pitch circle or pitch line; also, the radial distance between the pitch circle and the addendum circle (Figure 1.1); addendum can be defined as either nominal or operating. AXIAL PITCH (pa) is the circular pitch in the axial plane and in the pitch surface between corresponding sides of adjacent teeth, in helical gears and worms. The term axial pitch is preferred to the term linear pitch. (Figure 1.7) AXIAL PLANE of a pair of gears is the plane that contains the two axes. In a single gear, an axial plane may be any plane containing the axis and a given point. BASE DIAMETER (Db = gear, and db = pinion) is the diameter of the base cylinder from which involute tooth surfaces, either straight or helical, are derived. (Figure 1.1); base radius (Rb = gear, rb = pinion) is one half of the base diameter. BASE PITCH (pb) in an involute gear is the pitch on the base circle or along the lineofaction. Correspcndng sides of involute gear teeth are parallel curves, and the base pitch is the constant and fundamental distance between them along a common normal in a plane of rotation. (Figure 1.8) BASIC RACK is a rack that is adopted as the basis for a system of interchangeable gears. BACKLASH (B) is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth on the pitch circles. As actually indicated by measuring devices, backlash may be ______________ *Portions of this section are repented with permission from the BarberColman Co., Rockford, Ml. T31
 determined variously in the transverse, normal, or axial planes, and either in the direction of the pit circles or on the lineofaction. Such measurements should be corrected to corresponding values a transverse pitch circles for general comparisons. (Figure 1.9) CENTER DISTANCE (C), Distance between axes of rotation of mating spur or helical gears. CHORDAL ADDENDUM (ac) is the height from the top of the tooth to the chord subtending the circularthickness arc. (Figure 1.10) CHORDAL THICKNESS (tc) is the length of the chord subtending a circularthickness arc. (Figure 1.10) CIRCULAR PITCH (pc) is the distance along the pitch circle or pitch line between corresponding profiles of adjacent teeth. (Figure 1.1) CIRCULAR THICKNESS (t) is the length of arc between the two sides of a gear tooth on the p4 circle, unless otherwise specified. (Figure 1.10) CLEARANCEOPERATING (c) is the amount by which the dedendum in a given gear exceeds addendum of its mating gear. (Figure 1.1) CONTACT RATIO (Spur) is the ratio of the lengthofaction to the base pitch. CONTACT RATIO (Helical) is the contact ratio in the plane of rotation plus a contact portion a tributted to the axial advance. DEDENDUM (b) is the depth of a tooth space below the pitch line; also, the radial distance beta, the pitch circle and the root circle. (Figure 1.1); dedendum can be defined as either nominal or operating. DIAMETRAL PITCH (Pd) is the ratio of the number of teeth to the number of inches in the pitch diameter. There is a fixed relation between diametral pitch (Pd) and circular pitch (pc): pc = π / Pd FACE WIDTH (F) is the length of the teeth in an axial plane. FILLET RADIUS (r,) is the radius of the fillet curve at the base of the gear tooth. In generated this radius is an approximate radius of curvature. (Figure 1.13) FULL DEPTH TEETH are those in which the working depth equals 2000" diametral pitch GENERATING RACK is a rack outline used to indicate tooth details and dimensions for the design of a hob to produce gears of a basic rack system. HELIX ANGLE (ψ) is the angle between any helix and an element of its cylinder. In helical gears a worms, it is at the pitch diameter unless otherwise specified. (Figure 1.7) INVOLUTE TEETH of spur gears, helical gears, and worms are those in which the active portion of the profile in the transverse plane is the involute of a circle. T32
 LEAD (L) is the axial advance of a helix for one complete turn, as in the threads of cylindrical worms and teeth of helical gears. (Figure 1.11) LENGTHOFACTION (ZA) is the distance on an involute line of action through which the point of contact moves during the action of the tooth profiles. (Figure 1.8) LEWIS FORM FACTOR (Y, diametral pitch; yc, circular pitch). Factor in determination of beam strength of gears. LINEOFACTION is the path of contact in involute gears. It is the straight line passing through the pitch point and tangent to the base circles. (Figure 1.12) LONG AND SHORTADDENDUM TEETH are those in which the addenda of two engaging gears are unequal. MEASUREMENT OVER PINS (M). Distance over two pins placed in diametrically opposed tooth spaces (even number of teeth) or nearest to it (odd number of teeth). NORMAL CIRCULAR PITCH, Pcn, is the circular pitch in the normal plane, and also the length of the arc along the normal helix between helical teeth or threads. (Figure 1.7) NORMAL CIRCULAR THICKNESS (tn) is the circular thickness in the normal plane. In helical gears. it is an arc of the normal helix, measured at the pitch radius. NORMAL DIAMETRAL PITCH (Pdn) is the diametral pitch as calculated in the normal plane. NORMAL PLANE is the plane normal to the tooth. For a helical gear this plane is inclined by the helix angle, ψ, to the plane of rotation. OUTSIDE DIAMETER (Do gear, and do = pinion) is the diameter of the addendum (outside) circle (Figure 1.1); the outside radius (Ro gear, ro pinion) is one half the outside diameter. PITCH CIRCLE is the curve of intersection of a pitch surface of revolution and a plane of rotation. According to theory, it is the imaginary circle that rolls without slip with a pitch circle of a mating gear. (Figure 1.1) PITCH CYLINDER is the imaginary cylinder in a gear that rolls without slipping on a pitch cylinder or pitch plane of another gear. PITCH DIAMETER (D = gear, d = pinion) is the diameter of the pitch circle. In parallel shaft gears, the pitch diameters can be determined directly from the center distance and the number of teeth by proportionality. Operating pitch diameter is the pitch diameter at which the gears operate. (Figure 1.1) The pitch radius (R = gear, r pinion) is one half the pitch diameter (Figure 11). PITCH POINT is the point of tangency of two pitch circles (or of a pitch circle and pitch line) and is on the lineofcenters. Also, for involute gears, it is at the intersection of the lineofaction and a straight line connecting the two gear centers. The pitch point of a tooth profile is at its intersection with the pitch circle. (Figure 1.1) PLANE OF ROTATION is any plane perpendicular to a gear axis. T33
 PRESSURE ANGLE (φ), for involute teeth, is the angle between the lineofaction and a line tangent to the pitch circle at the pitch point. Standard pressure angles are established in connection with standard geartooth proportions. (Figure 1.1) PRESSURE ANGLE — NORMAL (φn) is the pressure angle in the normal plane of a helical or spiral tooth PRESSURE ANGLE — OPERATING (φr) is determined by the specific center distance at which the gears operate. It is the pressure angle at the operating pitch diameter. STUB TEETH are those in which the working depth us less than 2.000” diametral pitch TIP RELIEF is an arbitrary modification of a tooth profile whereby a small amount of material is removed near the tip of the gear tooth. (Figure 1.13) TOOTH THICKNESS (T) Tooth thickness at pitch circle (circular or chordal — Figure 1.1). TRANSVERSE CIRCULAR PITCH (Pt) is the circular pitch in the transverse plane. (Figure 1.7) TRANSVERSE CIRCULAR THICKNESS (tt) is the circular thickness in the transverse plane. TRANSVERSE PLANE is the plane of rotation and, therefore, is necessarily perpendicular to the go axis. TRANSVERSE PRESSURE ANGLE (φt) is the pressure angle in the transverse plane. UNDERCUT is the loss of profile in the vicinity of involute start at the base circle due to tool cutter action in generating teeth with low numbers of teeth. Undercut may be deliberately introduced to facilitate finishing operations. (Figure 1.13) WHOLE DEPTH (ht) is the total depth of a tooth space, equal to addendum plus dedendurn, also equal to working depth plus clearance. (Figure 1.1) WORKING DEPTH (hk) is the depth of engagement of two gears; that is, the sum of their addenda. T34
 T35
 T36 T36
 4.1.2 Symbols The symbols used in this section are summarized below.This is consistent with most gear literature and the publications of AGMA and ANSI. SYMBOL NOMENCLATURE & DEFINITION backlash, linear measure along B a addendum pitch circle BLA backlash, linear measure b dedendum along lineofaction aB backlash in arc minutes c clearance C center distance d pitch diameter, pinion dw pin diameter, for overpins ∆ change in center distance measurement Co operating center distance e eccentricity Cstd standard center distance hk working depth D pitch diameter ht whole depth Db base circle diameter mp contact ratio Do outside diameter n number of teeth, pinion DR root diameter nw number of threads in worm F face width pa axialpitch K factor; general pb base pitch L length, general; also lead of worm pc circular pitch M measurement overpins pcn normal circular pitch N number of teeth, usually gear r pitch radius, pinion Nc critical number of teeth for no undercutting rb base circle radus, pinion Nv virtual number of teeth for helical gear rt fillet radius Pd diametral pitch ro outside radius, pinion Pdn tooth thickness, and for normal diametral pitch t general use for tolerance pt horsepower, transmitted yc Lewis factor, circular pitch R pitch radius, gear or general use γ pitch angle, bevel gear Rb base circle radius, gear θ rotation angle, general Ro outside radius, gear λ lead angle, worm gearing RT testing radius µ mean value T tooth thickness, gear v gear stage velocity ratio Wb beam tooth strength φ pressure angle Y Lewis factor, diametral pitch φο operating pressure angle helix angle (Wb = base helix angle; Z mesh velocity ratio ψ operating helix angle) ω angular velocity invφ involute function 4.2 Pitch Diameter and Center Distance As already mentioned in par. 2.4, the pitch diameters for a meshing gear pair are tangent at a point on the lineofcenters called the pitch point. See figure 1.4. The pitch point always divides the line of centers proportional to the number of teeth in each gear. Center distance = C = D1 + D2 = N1 + N2 (5) 2 2Pd
 and the pitchcircle dimensions are related as follows: D1 = R1 = N1 (6) D2 R2 N2 4.3 Velocity Ratio The gear ratio, or velocity ratio, can be obtained from several different parameters: Z = D1 = N1 = ω1 (7) D2 N2 ω2 The ratio, Z, in this equation is the ratio of the angular velocity of gear 2 to that of gear 1. 4.4 Pressure Angle The pressure angle is defined as the angle between the line ofaction (common tangent to the base circles in Figs. 1.3 and 1.4) and a perpendicular to the lineofcenters. See Figure 1.14. From the, geometry of these figures, it is obvious that the pressure angle varies (slightly) as the cen distance of a gear pair is altered. The base circle is related to the pressure angle and pitch dinmeter by the equation: Db = D cos φ where D and φ are the standard values or alternately, (8) Db = D cos φ where D and φ are the exact operating values. This basic formula shows that the larger the pressure angle the smaller the base circle. Thus, for standard gears, 14½° pressure angle gears have base circles much nearer to the roots of teeth than 20° gears. It is for this reason that 14 ½° gears encounter greater undercutting problems than 20° gears. This is further elaborated on in section 4.8. 4.5 Tooth Thickness This is measured along the pitch circle. For this reason it is specifically called the circular tooth thickness. This is shown in Figure 1.1. Tooth thickness is related to the pitch as follows: T = Pc = π (9) 2 2Pd T38
 The tooth thickness (T2) at a given radius, R2, from the center can be found from a known value (T1) and known pressure angle (θ1) at that radius (R1), as follows: T2 = T1 R2  2R2 2R2 (inv θ2  inv θ1) (10) R1 where: inv θ =tan θ  θ = involute function. To save computing time involutefunction tables have been computed and are available in the references. An abridged liting is given in Table 1.5. 4.6 Measurement OverPins Often tooth thickness is measured indirectly by gaging over pins which are placed in diametrically opposed tooth spaces, or the nearest to it for odd numbered gear teeth. This is pictured in Figure 1.15. For a specified tooth thickness the overpins measurement, M, is calculated as follows: For an even number of teeth: M = D cos θ + dw (11) cos θ1 For an odd number of teeth M = D cos θ cos 90º + dw (12) cos θ1 where the value of θ1 is obtained from inv θ1 = T + invθ + dw  π (13) D D cos θ Ν Tabulated values of overpins measurements for standard gears are given in Table 1.6. This provides a rapid means for calculating values of M, even for gears with slight departures trom standard tooth thicknesses. When tooth thickness is to be calculated from a known overpins measurement, M, the equations can be manipulated to yield: T = D ( π + inv θc  inv θ  dw ) (14) N D cos θ where: cos θc = D cos θ (15) 2Rc for an even number of teeth: Rc = M  dw (16) 2 and for an odd number of teeth: Rc = M  dw (17) 2 cos 90º N T39
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Mechanism Design  Enumeration of Kinematic Structures According to Function P1
19 p  58  4