Tính toán thiết kế bánh răng

Tài liệu tính toán thiết kế bánh răng - tiếng anh

1.0 INTRODUCTION

2.0 BASIC GEOMETRY OF SPUR GEARS

2.1 Basic Spur Gear Geometry

2.2 The Law of Gearing

2.3 The Involute Curve

5.1 Generation of the Helical Tooth

5.2 Fundamental of Helical Teeth

5.3 Helical Gear Relationships

5.4 Equivalent Spur Gear

5.5 Pressure Angle

5.6 Importance of Normal Plane Geometry

5.7 Helical Tooth Proportions

5.8 Parallel Shaft Helical Gear Meshes

5.9 Crossed Helical Gear Meshes

5.9.1 Helix Angle and Hands

5.9.2 Pitch

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7.1 Development of the Internal Gear

7.2 Tooth Parts of Internal Gear

7.3 Tooth Thickness Measurement

7.4 Features of Internal Gears

8.0 WORM MESH

8.1 Worm Mesh Geometry

8.2 Worm Tooth Proportions

8.3 Number of Threads

8.4 Worm and Wormgear Calculations

8.4.1 Pitch Diameters, Lead and Lead Angle

8.4.2 Center Distance of Mesh

8.5 Velocity Ratio

9.0 BEVEL GEARING

9.1 Development and Geometry of Bevel Gears

9.2 Bevel Gear Tooth Proportions

9.3 Velocity Ratio

9.4 Forms of Bevel Teeth

10.0 GEAR TYPE EVALUATION

11.0 CRITERIA OF GEAR QUALITY

11.1 Basic Gear Formats

11.2 Tooth Thickness and Backlash

11.3 Position Error (or Transmission Error)

11.4 AGMA Quality Classes

11.5 Comparison With Previous AGMA and International Standards

12.0 CALCULATION OF GEAR PERFORMANCE CRITERIA

12.1 Backlash in a Single Mesh

12.2 Transmission Error

12.3 Integrated Position Error

12.4 Control of Backlash

12.5 Control of Transmission Error

13.0 GEAR STRENGTH AND DURABILITY

13.1 Bending Tooth Strength

13.2 Dynamic Strength

13.3 Surface Durability

13.4 AGMA Strength and Durability Ratings

T57T57T57 T58 T58T59T60T61 T61T62T62T62T63T63T64 T64T66T66T67T68 T68T70T70T73T73 T76T77T77T78T78 T78T82T88T88T22

Catalog D190

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14.3 Die Cast Alloys

14.4 Sintered Powder Metal

16.1 Lubrication of Power Gears

16.2 Lubrication of Instrument Gears

18.3 Other Inspection Equipment

18.4 Inspection of Fine-Pitch Gears

18.5 Significance of Inspection and Its Implementation

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19.0 GEARS, METRIC

19.1 Basic Definitions

19.2 Metric Design Equations

19.3 Metric Tooth Standards

19.4 Use of Strength Formulas

19.5 Metric Gear Standards

19.5.1 USA Metric Gear Standards

19.5.2 Foreign Metric Gear Standards

20.0 DESIGN OF PLASTIC MOLDED GEARS

20.1 General Characteristics of Plastic Gears

20.2 Properties of Plastic Gear Materials

20.3 Pressure Angles

20.4 Diametral Pitch

20.5 Design Equations for Plastic Spur, Bevel, Helical and Worm Gears

20.5.1 General Considerations

20.5.2 Bending Stress - Spur Gears

20.5.3 Surface Durability for Spur and Helical Gears

20.5.4 Design Procedure - Spur Gears

20.5.5 Design Procedure Helical Gears

20.5.6 Design Procedure - Bevel Gears

20.5.7 Design Procedure - Worm Gears

20.6 Operating Temperature

20.7 Eftect of Part Shrinkage on Gear Design

20.8 Design Specifications

20.9 Backlash

20.10 Environment and Tolerances

20.11 Avoiding Stress Concentration

20.12 Metal Inserts

20.13 Attachment of Plastic Gears to Shafts

20.14 Lubrication

20.15 Inspection

20.16 Molded vs Cut Plastic Gears

20.17 Elimination of Gear Noise

20.18 Mold Construction

20.19 Conclusion

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1.0 INTRODUCTION

This section presents a technical coverage of gear fundamentals It is intended as a broad coverage written in a manner that iseasy to follow and to understand by anyone interested in knowing how gear systems function Since gearing involves specialtycomponents it is expected that not all designers and engineers possess or have been exposed to all aspects of this subjectHowever, for proper use of gear components and design of gear systems it is essential to have a minimum understanding of gearbasics 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, thismaterial 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 spur-gear 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 spur-gear 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 angularvelocities 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 Ageometric relationship can be derived (1,7)* for the form of the tooth profiles to provide cojugate action, which is summarized asthe 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 afixed point on the line-of-centers 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

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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 formshave 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 straight-sided, 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 isimagined that a point on a string, which is pulled taut in a fixed direction, projects its trace onto a plane that rotates with thebase circle See Figure 1.2 The base cylinder, or base circle as referred to in gear literature, fully defines the form of the involuteand 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 onebase 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 line-of-centers It a second winding/unwinding taut string iswound 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 line-of-action in gear vernacular Where thisline crosses the line-of-centers 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

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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 aretwo 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 thepitch-circle 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:

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 straight-sided teeth forinvolute 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 havebeen 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 eachgear, Table 1.3 lists tooth proportions common to a given diametral pitch, as well as the diameter of a measuring wire Table 1.6lists pitch diameters and the over-wires measurement as a function of tooth number (which ranges from 18 to 218) and variousdiametral pitches, including most of the preferred fine pitches Both tables are for 20° pressure-angle 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 basicstandards, 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 bycontacting the headquarters office at 1500 King Street; Suite 201; Alexandria, VA 22314 (Phone: 703-684-0211)

TABLE 1.1 TOOTH PROPORTIONS OF BASIC RACK FOR

STANDARD INVOLUTE GEAR SYSTEMS

Tooth Parameter

Symbol in Rack Fig 1.6

14-1/2º Full Depth involute System

20º Full Depth involute System

20º Coarse-Pitch involute Spur Gears

20º Fine-Pitch involute System

8 Basic Circular Tooth

Thickness on Pitch Line

ANSI & AGMA14-1/2°

1/P1.157/P2.157/P2/P0.157/P

1 5708/P 1-1/3 x not specified B6.1201.02

ANSI20°

1/P1.157/P2.157/P2/P0.157/P1.5708/P 1-112 X not specified B6.1

AGMA20°

1.000/P1.250/P2.250/P2.000/P0250/P

π/2P 0.300/P not specified 201.02

ANSI & AGMA20°1.000/P1.200/P + 0.0022.200/P + 0.0022.000/P0.200/P + 0.0021.5708/P not standardizednot specified B6.7207.06

TABLE 1.2 PREFERRED DIAMETRAL PITCHES

Class Pitch

Coarse

1/21246810

Class Pitch

Coarse

Medium-12141618

Class Pitch

Fine

2024324864728096120128

.06545.03272.0478.0208.0270.0062

.04909.02454.0364.0156.0208.0051

.04363.02182.0326.0139.0187.0048

.03927.01963.0295.0125.0170.0045

.03272.01638.0249.0104.0145.0041

.02618.01309.0203.0083.0120.0037

.01571.00765.0130.0050.0080.0030Note: 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

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TABLE 1.4 SELECTED LIST OF AGMA STANDARDS

General AGMA 390AGMA

2000-A88

Gear Classification HandbookGear Classification And Inspection Handbookspurs And

Helicals AGMA 201AGMA 207 Tooth portions For Coarse-Pitch Involute Spur GearsTooth Proportions For Fine-Pitch Involute Spur Gears And Helical Gears

Non-Spur

AGMA2005-B88AGMA 203AGMA 374

Design-Manual For Bevel Gears

Fine-Pitch On-Center Face Gears For 20° Involute Spur PinionsDesign For Fine-Pitch Worm Gearing

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, eitherstraight 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 line-of-action Correspcndng sides of involutegear teeth are parallel curves, and the base pitch is the constant and fundamental distance between them along a commonnormal 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 pitchcircles As actually indicated by measuring devices, backlash may be

*Portions of this section are repented with permission from the Barber-Colman Co., Rockford, Ml

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determined variously in the transverse, normal, or axial planes, and either in the direction of the pit circles or on the

line-of-action 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 circular-thickness arc (Figure1.10)

CHORDAL THICKNESS (tc) is the length of the chord subtending a circular-thickness arc (Figure 1.10)

CIRCULAR PITCH (pc) is the distance along the pitch circle or pitch line between corresponding profiles of adjacent teeth (Figure1.1)

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)

CLEARANCE-OPERATING (c) is the amount by which the dedendum in a given gear exceeds addendum of its mating gear (Figure1.1)

CONTACT RATIO (Spur) is the ratio of the length-of-action 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 rootcircle (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 relationbetween 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

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 transverseplane is the involute of a circle

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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)

LENGTH-OF-ACTION (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

LINE-OF-ACTION is the path of contact in involute gears It is the straight line passing through the pitch point and tangent to thebase circles (Figure 1.12)

LONG- AND SHORT-ADDENDUM 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) ornearest 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 helixbetween 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 ofrotation

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 theimaginary 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 bedetermined directly from the center distance and the number of teeth by proportionality Operating pitch diameter is the pitchdiameter 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 line-of-centers Also, forinvolute gears, it is at the intersection of the line-of-action 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

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PRESSURE ANGLE (φ), for involute teeth, is the angle between the line-of-action and a line tangent to the pitch circle at the pitchpoint Standard pressure angles are established in connection with standard gear-tooth 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 pressureangle 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 geartooth (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 withlow 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 plusclearance (Figure 1.1)

WORKING DEPTH (hk) is the depth of engagement of two gears; that is, the sum of their addenda

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4.1.2 Symbols

The symbols used in this section are summarized below.This is consistent with most gear literature and the publications of AGMAand ANSI

SYMBOL NOMENCLATURE & DEFINITION

B backlash, linear measure alongpitch circle a addendum

BLA backlash, linear measurealong line-of-action b dedendum

∆ change in center distance dw pin diameter, for over-pinsmeasurement

L length, general; also lead of worm pc 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

Pdn normal diametral pitch t tooth thickness, and forgeneral use for tolerance

pt horsepower, transmitted yc Lewis factor, circular pitch

R pitch radius, gear or general use γ pitch angle, bevel gear

Y Lewis factor, diametral pitch φο operating pressure angle

Z mesh velocity ratio ψ helix angle (Wb = base helix angle;operating helix angle)

ω angular velocityinvφ 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 line-of-centers calledthe 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 pitch-circle dimensions are related as follows:

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° pressureangle gears have base circles much nearer to the roots of teeth than 20° gears It is for this reason that 14 ½° gears encountergreater undercutting problems than 20° gears This is further elaborated on in section 4.8

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 involute-function tables have been computed and are available in the references An abridged liting

is given in Table 1.5

4.6 Measurement Over-Pins

Often tooth thickness is measured indirectly by gaging over pins which are placed in diametrically opposed tooth spaces, or thenearest to it for odd numbered gear teeth This is pictured in Figure 1.15

For a specified tooth thickness the over-pins measurement, M, is calculated as follows:

For an even number of teeth:

TABLE 1.5 INVOLUTE FUNCTONS

Inv θ = tan θ - θ for values of θ from 10º to 40º

0.001910.002530.003280.004170.00520

0.002020.002670.003440.004360.00543

0.002140.002810.003620.004570.00566

0.002260.002960.003790.004760.0059015

0.006400.007790.009350.011130.01314

0.006670.008090.009690.011420.01357

0.006940.008390.010040.011910.01400

0.007210.008700.010390.012310.0144420

0.015370.017860.020630.023680.02705

0.015850.018400.021220.024330.02776

0.016340.018940.021820.024990.02849

0.016830.019490.022420.025660.0292225

0.030740.034780.039200.044020.04924

0.031520.035630.040130.045030.05034

0.032320.036500.041080.046060.05146

0.033130.037390.042040.047100.0526030

0.054920.061080.067730.074930.08270

0.056120.162370.069130.076440.08432

0.057330.063680.070550.077970.08597

0.058560.065020.071990.079520.0876535

0.091060.100080.109780.120200.121410.14344

0.092810.010960.111800.122380.133750.14595

0.094590.103880.113860.124590.136120.14850

0.096390.105820.115940.126830.138530.15108T40

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or specially designed non-standard external spur gears.

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tooth numbers below a critical value are automatically undercut in the generating process The limiting number of teeth in a gearmeshing with a rack is given by the expression:

Nc = 2 (19)

sin²φ

This indicates the minimum number of teeth free of undercutting decreases with increasing Pressure angle For 14½º thevalue of Nc is 32, and for 20° it is 18 Thus, 200 pressure angle gears with low numbers of teeth have the advantage of muchless undercutting and, therefore, are both stronger and smoother acting

4.9 Enlarged Pinions

Undercutting of pinion teeth is undesirable because of losses of strength, contact ratio and smoothness of

action The seventy of these faults depends upon how far below N, the tooth number is Undercutting for the first few numbers issmall and in many applications its adverse effects can be neglected

For very small numbers of teeth, such as ten and smaller, and for

high-precision applications, undercutting should be avoided This is achieved by

pinion enlargement (or correction as often termed), wherein the pinion teeth, still

generated with a standard cutter, are shifted radially ourward to form a full

involute tooth free of undercut The tooth is enlarged both radially and

circumferentially Comparison of a tooth form before and after enlargement is

shown in Figure 1.18

The details of enlarged pinion design, mating gear design and, in general,

profile-shifted gears is a large and involved subject beyond the scope of this

writing References 1, 3, 5 and 6 offer additional information For measurement

and inspection Figure 1.18 Comparison of such gears, in particular, consult

distance — all are factors to consider in the specification of the amount of backlash On the other hand, excessive backlash isobjectionable, particularly if the drive is frequently reversing or if there is an overrunning load The amount of backlash must not

be excessive for the requirements of the job, but it should be sufficient so that machining costs are not higher than necessary

In order to obtain the amount of backlash desired, it is necessary to decrease tooth thickness (see Figure 1.19) This

decrease must almost always be greater than the desired backlash because of

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the errors in manufacturing and assembling Since the amount of the decrease in tooth thickness depends upon the accuracy ofmachining, the allowance for a specified backlash will vary according to the manufacturing conditions.

It is customary to make half of the allowance for backlash on the tooth thickness of each gear of a pair, although there areexceptions For example, on pinions having very low numbers of teeth, it is desirable to provide all of the allowance on themating gear so as not to weaken the pinion teeth

In spur and helical gearing, backlash allowance is usually obtained by sinking the hob deeper into the blank than the

theoretically standard depth Further, it is true that any increase or decrease in center distance of two gears in any mesh willcause an increase or decrease in backlash Thus, this is an alternate way of designing backlash into the system

In the following we give the fundamental equations for the determination of backlash in a single gear mesh For the

determination of backlash in gear trains, it is necessary to sum the backlash of each mated gear pair However, to obtain thetotal backlash for a series of meshes it is necessary to take into account the gear ratio of each mesh relative to a chosen

reference shaft in the gear train For details see Reference 5

Backlash is defined in Figure 1.20a as the excess thickness of tooth space over the thickness of the mating tooth There aretwo basic ways in which backlash arises: Tooth thickness is below the zero-backlash value; and the operating center distance isgreater than the zero-backlash value

If the tooth thickness of either or both mating gears is less than the zero-backlash value, the amount of backlash

introduced in the mesh is simply this numerical difference:

B = Tstd - Tact = ∆T (20)

where:

B = linear backlash measured along the pitch circle (Figure 1.20b)

Tstd = no backlash tooth thickness on the operating-pitch circle, which is the standard teeth

thickness for ideal gears

Tact = actual tooth thickness

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When the center distance is increased by a relatively small amount, ∆C, a backlash space develops

between mating teeth, as in Figure 1.21 The relationship between center distance increase and linear

backlash, BLA, along the line of action, is:

Hence, an approximate relationship between center distance change and change in backlash is:

∆C= 1.933 ∆B for 14½° pressure-angle gears (22b)

∆C= 1.374 ∆B for 20° pressure-angle gears (22c)

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Although these are approximate relationships they are adequate for most uses Their derivation, limitations, and correctionfactors are detailed in Reference 5.

Note that backlash due to center distance opening is dependent upon the tangent function of the pressure angle Thus, 20°gears have 41% more backlash than 14½º gears, and this constitutes one of the few advantages of the lower pressure angle Equations 22 are a useful relationship, particularly for converting to angular backlash Also for fine-pitch gears the use offeeler gages for measurement is impractical, whereas an indicator at the pitch line gives a direct measure The two linear

backlashes are related by:

BLA (23)

B = _

cos φ

The angular backlash at the gear shaft is usually the critical factor in the gear application As seen

from Figure 1.20a this is related to the gear’s pitch radius as follows:

4.11 Summary of Gear Mesh Fundamentals

The basic geometric relationships of gears and meshed pairs given in the above sections are summarized in Table 1.7

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TABLE 1.7 SUMMARY OF FUNDAMENTALS

SPUR GEARS

Pd πCircular Pitch Diametral pitch or number ofteeth and pitch diameter Pc = π = π D

Pd NDiametral pitch Circular pitch or number ofteeth and pitch diameter Pd = π = N

Pc D

Pc Outside diameter Pitch and pitch diameter orpitch and number of teeth Do =D + 2 = N+2

Pd Pd

Base circle diameter Pitch diameter and pressure angle Db=D cos φ

Base pitch Circular pitch and pressure angle Pb = Pc cos φ

Tooth thickness at

2 2N

Center distance Pitch diameters Or numberof teeth and pitch C=D1+D2=N1+N2=Pc(N1+N2)

2 2Pd 2πContact ratio Outside radii, base radii, centerdistance and pressure angle mp = (Ro²-Rb²)½+(ro²-rb²)½-C sin φ

Pc cos φBacklash (linear) From change in center distance B = 2 (∆C) tan φ

Backlash (linear) From change in tooth thickness B = ∆T

Backlash (linear)

along line of acvon Linear backlash along pitch cirde BLA = B cos φ

D Minimum number of

teeth for no undercutting Pressure angle N = 2 sin² φ

Dedendum root diameter ( DR )Pitch diameter and b = ½(D-DR)

TABLE 1.7 CONT - SUMMARY OF FUNDAMENTALS

HELICAL GEARING

Normal circular pitch Transverse circular pitch Pcn = Pc cos ψ

Normal diametral pitch Transverse diametral pitch Pdn = Pd

cos ψ

sin ψNormal pressure angle Transverse pressure angle tan φn = tan φ cos ψ

Pitch diameter Number of teeth and pitch Pd Pdn cos D = N = N ψ

Center distance

(parallel shafts) Number of teeth and pitch C = N1 + N2

2 Pdn cos ψ Center distance

(crossed shafts) Number of teeth and pitch C = 1 ( N1 + N2 )

2 Pdn cos ψ1 cos ψ2Shaft angle

(Crssed shafts) Helix angles of 2 mated gears θ = ψ1 + ψ2

Addendum Pitch; or outside and pitchdiameters a = 0.5 ( Do - D ) = 1

PdDedendum Pitch diameter and rootdiameter (DR) b = 0.5 ( D - DR )

Transverse pressure

angle Base circle diameter andpitch circle diameter cos φt = Db / D

Pitch helix angle normal diametral pitch andNumber of teeth,

pitch diameter

cos ψ = N

Pn DLead Pitch diameter andpitch helix angle L = π D cos ψ

INVOLUTE GEAR PAIRS

ZA = (Ro²-Rb²)½ (ro²-rb²-C sin φr)½

SAPg = Zmax-(ro²-rb²)½

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TABLE 1.7 CONT - SUMMARY OF FUNDAMENTALS

WORM MESHES

Pitch diameter of worm Number of teeth and pitch dw = nw Pcn

p sin λPitch diameter of

π cos λLead angle Pitch, diameter, teeth λ = tan-1 nw = sin-1 nw Pcn

Pddw pdw

cos λNormal circular pitch Transverse pitch and lead angle Pcn = Pc cos λ

2Center distance Pitch, lead angle, teeth C = Pcn [ Ng + nw ]

2π cos λ sin λ

BEVEL GEARING

5.0 HELICAL GEARS

The helical gear differs from the spur gear in that its teeth are twisted along a helical path in the axial direction It resembles thespur gear in the plane of rotation, but in the axial direction it is as if there were a series of staggered spur gears See Figure 1.22.This design brings forth a number of different features relative to the spur gear, two of the most important being as follows:

1 tooth strength is improved because of the elongated helical wrap around

tooth base support

2 contact ratio is increased due to the axial tooth overlap Helical gears thus

tend to have greater load-carrying capactiy than spur gears of the same size

Spur gears, on the other hand, have a somewhat higher efficiency

Helical gears are used in two forms:

1 Parallel shaft applications, which is the largest usage

2 Crossed-helicals (or spiral gears) for connecting skew shafts, usually at tight

angles

5.1 Generation of the Helical Tooth

The helical tooth form is involute in the plane of rotation and can be developed in a manner similar to that of the spur gear.However, unlike the spur gear which can be viewed essentially as two dimensional, the helical gear must be portrayed in threedimensions to show changing axial features

Referring to Figure 1.23, there is a base cylinder from which a taut plane is unwrapped, analogous to the unwinding tautstring of the spur gear in Figure 12 On the plane there is a straight line AB, which when wrapped on the base cylinder has ahelical trace AoBo As the taut plane is unwrapped any point on the line AB can be visualized as tracing an involute from the basecylinder Thus, there is an infinite series of involutes generated by line AB, all alike, but displaced in phase along a helix on thebase cylinder

Again a concept analogous to the spur-gear tooth development is to imagine the taut plane being wound from one basecylinder on to another as the base cylinders rotate in opposite directions The result is the generation of a pair of conjugatehelical involutes If a reverse direction of rotation is assumed and a second tangent plane is arranged so that it crosses the first, acomplete involute helicoid tooth is formed

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5.2 Fundamental of Helical Teeth

In tho piano of rotation the helical gear tooth is involute and all of the relationships govorning spur gears apply to the helical.However, tho axial twist of the teeth introduces a holix anglo Since the helix angle varies from the base of the tooth to theoutside radnjs, the helix angle, w~ is detned as the angle between the tangent to the helicoidal tooth at the intersection of thepitch cylinder and the tooth profile, and an element of the pitch cylinder See Figure 1.24

The direction of the helical twist is designated as either left or right The direction is defined by the right-hand rule

5.3 Helical Gear Relationships

For helical gears there are two related pitches: one in the plane of rotation and the other in a plane normal to the tooth Inaddition there is an axial pitch These are defined and related as follows: Referring to Figure 1.25, the two circular pitches arerelated as follows:

Pcn = Pc cos ψ = normal circular pitch (25)

The normal circular pitch is less than the transverse or circular pitch in the plane of rotation, the ratio between the two beingequal to the cosine of the helix angle Consistent with this, the normal diametral pitch is greater than the transverse pitch:Pdn = Pd = normal diametral pitch (26)

5.4 Equivalent Spur Gear

The true involute pitch and involute geometry of a helical gear is that in the plane of rotation However, in the normal plane, looking

at one tooth, there is a resemblance to an involute tooth of a pitch corresponding to the normal pitch However, the shape of thetooth corresponds to a spur gear of a larger number of teeth, the exact value depending on the magnitude of the helix angle

The geometric basis of deriving the number of teeth in this equivalent tooth

form spur gear is given in Figure 1.27 The result of the transposed geometry

is an equivalent number of teeth given as:

NV = N (28)

cos³ψ

This equivalent number is also called a virtual number because this spur

gear is imaginary The value of this number is its use in determining helical

tooth strength

5.5 Pressure Angle

Although strictly speaking, pressure angle exists only for a gear pair, a nominal pressure angle can be considered for an individualgear For the helical gear there is a normal pressure angle as well as the usual pressure angle in the plane of rotation Figure 1.28shows their relationship, which is expressed as:

tan φ = tan φn (29)

cos ψ

5.6 Importance of Normal Plane Geometry

Because of the nature of tooth generation with a rack-type hob, a single tool can generate helical gears at all helix angles as well asspur gears However, this means the normal pitch is the common denominator, and usually is taken as a standard value Since thetrue involute features are in the transverse plane, they will differ from the standard normal values Hence, there is a real need forrelating parameters in the two reference planes

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f

5.7 Helical Tooth Proportions

These follow the same standards as those for spur gears Addendum, dedendum, whole depth and clearance are the same

regardless of whothor measured in tho piano of rotation er the normal piano Pressure angle and pitch are usually specified asstandard values in tho normal plane, but there are times when they are specified standard in the transverse plane

5.8 Parallel Shaft Helical Gear Meshes

Fundamental information for the design of gear meshes is as follows:

5.8.1 Helix angle — Both gears of a meshed pair must have the same helix angle However, the

helix directions must be opposite, i.e., a left-hand mates with a right-hand helix

5.8.2 Pitch dIameter — This is given by the same expression as for spur gears, but if the normal

pitch is involved it is a function of the helix angle The expressions are:

Note that for standard parameters in the normal plane, the center distance will not be a standard value compared to

standard spur gears Further, by manipulating the helix angle (ψ) the center distance can be adjusted over a wide range ofvalues Conversely, it is possible

a to compensate for significant center distance changes (or erçors) without changing the speed ratio between parallel gearedshafts; and

b to alter the speed ratio between parallel geared shafts without changing center distance by manipulating helix angle along withtooth numbers

5.8.4 Contact Ratio — The contact ratio of helical gears is enhanced by the axial overlap of the teeth Thus, the contact ratio is

the sum of the transverse contact ratio, calculated in the same manner as for spur gears (equation 18), and a term involving theaxial pitch

(mp)total = (mp)trans + (mp)axial (32)

where

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(mp)trans = value per equation 18

(mp)axial = F = F tan ψ = F sin ψ

Pa Pc Pcn

and F = face width of gear

5.8.5 Involute interference — Helical gears cut with standard normal pressure angles can have considerably higher pressure

angles in the plane of rotation (see equation 29), depending on the helix angle Therefore, referring to equation 19, the minimumnumber of teeth without undercutting can be significantly reduced and helical gears having very low tooth numbers withoutundercutting are feasible

5.9 Crossed Helical Gear Meshes

These are also known as spiral and screw gears They are used for interconnecting skew shafts, such as in Figure 1.29 They can

be designed to connect shafts at any angle, but in most applications the shafts are at right angles

5.9.1 Helix angle and hands — The helix angles need not be the same However, their sum must equal the shaft

angle:

ψ1 + ψ2 = θ (33)

where:

ψ1, ψ2 = the respective helix angles of the two gears

θ = shaft angle (the acute angle between the two shafts when viewed in a direction parallel

ing a common perpendicular between the shafts)

Except for very small shaft angles, the helix hands are the same

5.9.2 Pitch — Because of the possibility of ditferent helix angles for the gear pair, the transverse pitches may not be the same.

However, the normal pitches must always be identical

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