Minimum number of teeth on pinion to avoid interference formula

GearAssignment/Mechanical

Chapter Outline

13–1 Types of Gears 666

13–2 Nomenclature 667

13–3 Conjugate Action 669

13–4 Involute Properties 670

13–5 Fundamentals 670

13–6 Contact Ratio 676

13–7 Interference 677

13–8 The Forming of Gear Teeth 679

13–9 Straight Bevel Gears 682

13–10 Parallel Helical Gears 683

13–11 Worm Gears 687

13–12 Tooth Systems 688

13–13 Gear Trains 690

13–14 Force Analysis—Spur Gearing 697

13–15 Force Analysis—Bevel Gearing 701

13–16 Force Analysis—Helical Gearing 704

13–17 Force Analysis—Worm Gearing 706

Gears—General13

665

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666 Mechanical Engineering Design

This chapter addresses gear geometry, the kinematic relations, and the forces transmit- ted by the four principal types of gears: spur, helical, bevel, and worm gears. The forces transmitted between meshing gears supply torsional moments to shafts for motion and power transmission and create forces and moments that affect the shaft and its bearings. The next two chapters will address stress, strength, safety, and reli- ability of the four types of gears.

13–1 Types of Gears Spur gears, illustrated in Fig. 13–1, have teeth parallel to the axis of rotation and are used to transmit motion from one shaft to another, parallel, shaft. Of all types, the spur gear is the simplest and, for this reason, will be used to develop the primary kinematic relationships of the tooth form. Helical gears, shown in Fig. 13–2, have teeth inclined to the axis of rotation. Helical gears can be used for the same applications as spur gears and, when so used, are not as noisy, because of the more gradual engagement of the teeth during meshing. The inclined tooth also develops thrust loads and bending couples, which are not present with spur gearing. Sometimes helical gears are used to transmit motion between nonparallel shafts. Bevel gears, shown in Fig. 13–3, have teeth formed on conical surfaces and are used mostly for transmitting motion between intersecting shafts. The figure actually illustrates straight-tooth bevel gears. Spiral bevel gears are cut so the tooth is no longer straight, but forms a circular arc. Hypoid gears are quite similar to spiral bevel gears except that the shafts are offset and nonintersecting.

Figure 13–1 Spur gears are used to transmit rotary motion between parallel shafts.

Figure 13–2 Helical gears are used to transmit motion between parallel or nonparallel shafts.

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Gears—General 667

Worms and worm gears, shown in Fig. 13–4, represent the fourth basic gear type. As shown, the worm resembles a screw. The direction of rotation of the worm gear, also called the worm wheel, depends upon the direction of rotation of the worm and upon whether the worm teeth are cut right-hand or left-hand. Worm gearsets are also made so that the teeth of one or both wrap partly around the other. Such sets are called single- enveloping and double-enveloping worm gearsets. Worm gearsets are mostly used when the speed ratios of the two shafts are quite high, say, 3 or more.

13–2 Nomenclature The terminology of spur-gear teeth is illustrated in Fig. 13–5. The pitch circle is a theoretical circle upon which all calculations are usually based; its diameter is the pitch diameter. The pitch circles of a pair of mating gears are tangent to each other. A pinion is the smaller of two mating gears. The larger is often called the gear. The circular pitch p is the distance, measured on the pitch circle, from a point on one tooth to a corresponding point on an adjacent tooth. Thus the circular pitch is equal to the sum of the tooth thickness and the width of space.

Figure 13–3 Bevel gears are used to transmit rotary motion between intersecting shafts.

Figure 13–4 Worm gearsets are used to transmit rotary motion between nonparallel and nonintersecting shafts.

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668 Mechanical Engineering Design

The module m is the ratio of the pitch diameter to the number of teeth. The custom- ary unit of length used is the millimeter. The module is the index of tooth size in SI. The diametral pitch P is the ratio of the number of teeth on the gear to the pitch diameter. Thus, it is the reciprocal of the module. Since diametral pitch is used only with U.S. units, it is expressed as teeth per inch. The addendum a is the radial distance between the top land and the pitch circle. The dedendum b is the radial distance from the bottom land to the pitch circle. The whole depth ht is the sum of the addendum and the dedendum. The clearance circle is a circle that is tangent to the addendum circle of the mat- ing gear. The clearance c is the amount by which the dedendum in a given gear exceeds the addendum of its mating gear. The backlash is the amount by which the width of a tooth space exceeds the thickness of the engaging tooth measured on the pitch circles. You should prove for yourself the validity of the following useful relations:

P 5 N d

(13–1)

m 5 d N

(13–2)

p 5 pd N

5 pm (13–3)

pP 5 p (13–4)

where P 5 diametral pitch, teeth per inch

N 5 number of teeth

d 5 pitch diameter, in or mm

m 5 module, mm

p 5 circular pitch, in or mm

Figure 13–5 Nomenclature of spur-gear teeth.

Addendum

Dedendum

Clearance

Bo tto

m lan

d

Fillet radius

Dedendum circle

Clearance circle

Tooth thickness

Fa ce

w idt

h

Width of space

Face

Top land

Addendum circle

Pitch circle

Flank Circular pitch

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Gears—General 669

13–3 Conjugate Action The following discussion assumes the teeth to be perfectly formed, perfectly smooth, and absolutely rigid. Such an assumption is, of course, unrealistic, because the appli- cation of forces will cause deflections. Mating gear teeth acting against each other to produce rotary motion are similar to cams. When the tooth profiles, or cams, are designed so as to produce a constant angular-velocity ratio during meshing, these are said to have conjugate action. In theory, at least, it is possible arbitrarily to select any profile for one tooth and then to find a profile for the meshing tooth that will give conjugate action. One of these solutions is the involute profile, which, with few exceptions, is in universal use for gear teeth and is the only one with which we will be concerned. When one curved surface pushes against another (Fig. 13–6), the point of contact occurs where the two surfaces are tangent to each other (point c), and the forces at any instant are directed along the common normal ab to the two curves. The line ab, representing the direction of action of the forces, is called the line of action. The line of action will intersect the line of centers O-O at some point P. The angular-velocity ratio between the two arms is inversely proportional to their radii to the point P. Circles drawn through point P from each center are called pitch circles, and the radius of each circle is called the pitch radius. Point P is called the pitch point. Figure 13–6 is useful in making another observation. A pair of gears is really a pair of cams that act through a small arc and, before running off the involute contour, are replaced by another identical pair of cams. The cams can run in either direction and are configured to transmit a constant angular-velocity ratio. If involute curves are used, the gears tolerate changes in center-to-center distance with no variation in con- stant angular-velocity ratio. Furthermore, the rack profiles are straight-flanked, making primary tooling simpler. To transmit motion at a constant angular-velocity ratio, the pitch point must remain fixed; that is, all the lines of action for every instantaneous point of contact must pass through the same point P. In the case of the involute profile, it will be shown that all points of contact occur on the same straight line ab, that all normals to the tooth profiles at the point of contact coincide with the line ab, and, thus, that these profiles transmit uniform rotary motion.

O

B

rB

rA

b

c

a

A O

P

Figure 13–6 Cam A and follower B in contact. When the contacting surfaces are involute profiles, the ensuing conjugate action produces a constant angular-velocity ratio.

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670 Mechanical Engineering Design

13–4 Involute Properties An involute curve may be generated as shown in Fig. 13–7a. A partial flange B is attached to the cylinder A, around which is wrapped a cord def, which is held tight. Point b on the cord represents the tracing point, and as the cord is wrapped and unwrapped about the cylinder, point b will trace out the involute curve ac. The radius of the curvature of the involute varies continuously, being zero at point a and a maximum at point c. At point b the radius is equal to the distance be, since point b is instantaneously rotating about point e. Thus the generating line de is normal to the invo- lute at all points of intersection and, at the same time, is always tangent to the cylinder A. The circle on which the involute is generated is called the base circle. Let us now examine the involute profile to see how it satisfies the requirement for the transmission of uniform motion. In Fig. 13–7b, two gear blanks with fixed centers at O1 and O2 are shown having base circles whose respective radii are O1a and O2b. We now imagine that a cord is wound clockwise around the base circle of gear 1, pulled tight between points a and b, and wound counterclockwise around the base circle of gear 2. If, now, the base circles are rotated in different directions so as to keep the cord tight, a point g on the cord will trace out the involutes cd on gear 1 and ef on gear 2. The involutes are thus generated simultaneously by the tracing point. The tracing point, therefore, represents the point of contact, while the portion of the cord ab is the generating line. The point of contact moves along the generating line; the generating line does not change position, because it is always tangent to the base circles; and since the generating line is always normal to the involutes at the point of contact, the requirement for uniform motion is satisfied.

13–5 Fundamentals Among other things, it is necessary that you actually be able to draw the teeth on a pair of meshing gears. You should understand, however, that you are not doing this for manufacturing or shop purposes. Rather, we make drawings of gear teeth to obtain an understanding of the problems involved in the meshing of the mating teeth.

Figure 13–7 (a) Generation of an involute; (b) involute action.

+

+

+

Base circle

Pitch circle

O1

O2

c a

P e d

g

f

b

d

B

b

c

e

a

A

O

f

Pitch circle

Gear 1

Gear 2 Base circle

(a) (b)

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Gears—General 671

First, it is necessary to learn how to construct an involute curve. As shown in Fig. 13–8, divide the base circle into a number of equal parts, and construct radial lines OA0, OA1, OA2, etc. Beginning at A1, construct perpendiculars A1B1, A2B2, A3B3, etc. Then along A1B1 lay off the distance A1A0, along A2B2 lay off twice the distance A1A0, etc., producing points through which the involute curve can be constructed. To investigate the fundamentals of tooth action, let us proceed step by step through the process of constructing the teeth on a pair of gears. When two gears are in mesh, their pitch circles roll on one another without slip- ping. Designate the pitch radii as r1 and r2 and the angular velocities as v1 and v2, respectively. Then the pitch-line velocity is

V 5 0 r1v1 0 5 0 r2v2 0 Thus the relation between the radii on the angular velocities is

` v1 v2 ` 5 r2

r1 (13–5)

Suppose now we wish to design a speed reducer such that the input speed is 1800 rev/min and the output speed is 1200 rev/min. This is a ratio of 3:2; the gear pitch diameters would be in the same ratio, for example, a 4-in pinion driving a 6-in gear. The various dimensions found in gearing are always based on the pitch circles. Suppose we specify that an 18-tooth pinion is to mesh with a 30-tooth gear and that the diametral pitch of the gearset is to be 2 teeth per inch. Then, from Eq. (13–1), the pitch diameters of the pinion and gear are, respectively,

d1 5 N1 P

5 18 2

5 9 in d2 5 N2 P

5 30 2

5 15 in

The first step in drawing teeth on a pair of mating gears is shown in Fig. 13–9. The center distance is the sum of the pitch radii, in this case 12 in. So locate the pinion and gear centers O1 and O2, 12 in apart. Then construct the pitch circles of radii r1 and r2. These are tangent at P, the pitch point. Next draw line ab, the common tangent, through the pitch point. We now designate gear 1 as the driver, and since it is rotating counterclockwise, we draw a line cd through point P at an angle f to the common tangent ab. The line cd has three names, all of which are in general use. It is called the pressure line, the generating line, and the line of action. It represents the direction in which the resultant force acts between the gears. The angle f is called the pressure angle, and it usually has values of 20 or 25°, though 1412° was once used.

Figure 13–8 Construction of an involute curve.

O

Base circle InvoluteA4

A3

A2

A1

A0 B1

B2

B3

B4

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672 Mechanical Engineering Design

Next, on each gear draw a circle tangent to the pressure line. These circles are the base circles. Since they are tangent to the pressure line, the pressure angle determines their size. As shown in Fig. 13–10, the radius of the base circle is

rb 5 r cos f (13–6)

where r is the pitch radius. Now generate an involute on each base circle as previously described and as shown in Fig. 13–9. This involute is to be used for one side of a gear tooth. It is not necessary to draw another curve in the reverse direction for the other side of the tooth, because we are going to use a template which can be turned over to obtain the other side. The addendum and dedendum distances for standard interchangeable teeth are, as we shall learn later, 1yP and 1.25yP, respectively. Therefore, for the pair of gears we are constructing,

a 5 1 P

5 1 2

5 0.500 in b 5 1.25

P 5

1.25 2

5 0.625 in

Using these distances, draw the addendum and dedendum circles on the pinion and on the gear as shown in Fig. 13–9. Next, using heavy drawing paper, or preferably, a sheet of 0.015- to 0.020-in clear plastic, cut a template for each involute, being careful to locate the gear centers prop- erly with respect to each involute. Figure 13–11 is a reproduction of the template used to create some of the illustrations for this book. Note that only one side of the tooth profile is formed on the template. To get the other side, turn the template over. For some problems you might wish to construct a template for the entire tooth.

Figure 13–9 Circles of a gear layout.

Base circle

+

+

Dedendum circle

Pitch circle Base circle

Involute

Addendum circles

Pitch circle

b

d

a

c

P

O1

O2

r1

r2

Dedendum circle

Involute

!1

!2

"

Figure 13–10 Base circle radius can be related to the pressure angle f and the pitch circle radius by rb 5 r cos f.

O

r

P

Pitch circle

Pressure line

Base circle

rb

"

"

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Gears—General 673

To draw a tooth, we must know the tooth thickness. From Eq. (13–4), the circular pitch is

p 5 p

P 5 p

2 5 1.57 in

Therefore, the tooth thickness is

t 5 p

2 5

1.57 2

5 0.785 in

measured on the pitch circle. Using this distance for the tooth thickness as well as the tooth space, draw as many teeth as desired, using the template, after the points have been marked on the pitch circle. In Fig. 13–12 only one tooth has been drawn on each gear. You may run into trouble in drawing these teeth if one of the base circles happens to be larger than the dedendum circle. The reason for this is that the involute begins at the base circle and is undefined below this circle. So, in drawing gear teeth, we usually draw a radial line for the profile below the base circle. The actual shape, however, will depend upon the kind of machine tool used to form the teeth in manufacture, that is, how the profile is generated. The portion of the tooth between the clearance circle and the dedendum circle includes the fillet. In this instance the clearance is

c 5 b 2 a 5 0.625 2 0.500 5 0.125 in

The construction is finished when these fillets have been drawn.

Figure 13–11 A template for drawing gear teeth.

21 O2

O1

Figure 13–12 Tooth action.

Angle of approach

P

Angle of recess

O2

O1

Pressure line

Dedendum circle Base circle Pitch circle Addendum circle

Angle of recess

Pinion (driver)

Addendum circle

Pitch circle

Base circle

Dedendum circle Gear

(driven)

a

b

Angle of approach

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674 Mechanical Engineering Design

Referring again to Fig. 13–12, the pinion with center at O1 is the driver and turns counterclockwise. The pressure, or generating, line is the same as the cord used in Fig. 13–7a to generate the involute, and contact occurs along this line. The initial contact will take place when the flank of the driver comes into contact with the tip of the driven tooth. This occurs at point a in Fig. 13–12, where the addendum circle of the driven gear crosses the pressure line. If we now construct tooth profiles through point a and draw radial lines from the intersections of these profiles with the pitch circles to the gear centers, we obtain the angle of approach for each gear. As the teeth go into mesh, the point of contact will slide up the side of the driving tooth so that the tip of the driver will be in contact just before contact ends. The final point of contact will therefore be where the addendum circle of the driver crosses the pressure line. This is point b in Fig. 13–12. By drawing another set of tooth profiles through b, we obtain the angle of recess for each gear in a manner similar to that of finding the angles of approach. The sum of the angle of approach and the angle of recess for either gear is called the angle of action. The line ab is called the line of action. We may imagine a rack as a spur gear having an infinitely large pitch diameter. Therefore, the rack has an infinite number of teeth and a base circle which is an infinite distance from the pitch point. The sides of involute teeth on a rack are straight lines making an angle to the line of centers equal to the pressure angle. Figure 13–13 shows an involute rack in mesh with a pinion. Corresponding sides on involute teeth are parallel curves; the base pitch is the constant and fundamental distance between them along a common normal as shown in Fig. 13–13. The base pitch is related to the circular pitch by the equation

pb 5 pc cos f (13–7)

where pb is the base pitch. Figure 13–14 shows a pinion in mesh with an internal, or ring, gear. Note that both of the gears now have their centers of rotation on the same side of the pitch point. Thus the positions of the addendum and dedendum circles with respect to the pitch circle are reversed; the addendum circle of the internal gear lies inside the pitch circle. Note, too, from Fig. 13–14, that the base circle of the internal gear lies inside the pitch circle near the addendum circle. Another interesting observation concerns the fact that the operating diameters of the pitch circles of a pair of meshing gears need not be the same as the respective design pitch diameters of the gears, though this is the way they have been constructed in Fig. 13–12. If we increase the center distance, we create two new operating pitch circles having larger diameters because they must be tangent to each other at the pitch

Figure 13–13 Involute-toothed pinion and rack.

Circular pitch

Base pitch

!

pc

pb

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Gears—General 675

point. Thus the pitch circles of gears really do not come into existence until a pair of gears are brought into mesh. Changing the center distance has no effect on the base circles, because these were used to generate the tooth profiles. Thus the base circle is basic to a gear. Increasing the center distance increases the pressure angle and decreases the length of the line of action, but the teeth are still conjugate, the requirement for uniform motion trans- mission is still satisfied, and the angular-velocity ratio has not changed.

Figure 13–14 Internal gear and pinion.

Pitch circle

Base circle

!2 Base circle

Pitch circle

Pressure line

Dedendum circle

Addendum circle

3

2

!3

O2

EXAMPLE 13–1 A gearset consists of a 16-tooth pinion driving a 40-tooth gear. The diametral pitch is 2, and the addendum and dedendum are 1yP and 1.25yP, respectively. The gears are cut using a pressure angle of 20°. (a) Compute the circular pitch, the center distance, and the radii of the base circles. (b) In mounting these gears, the center distance was incorrectly made 14 in larger. Compute the new values of the pressure angle and the pitch-circle diameters.

Solution

Answer (a) p 5 p

P 5 p

2 5 1.571 in

The pitch diameters of the pinion and gear are, respectively,

dP 5 NP P

5 16 2

5 8 in dG 5 NG P

5 40 2

5 20 in

Therefore the center distance is

Answer dP 1 dG

2 5

8 1 20 2

5 14 in

Since the teeth were cut on the 20° pressure angle, the base-circle radii are found to be, using rb 5 r cos f,

Answer rb(pinion) 5 8 2

cos 20° 5 3.759 in

Answer rb(gear) 5 20 2

cos 20° 5 9.397 in

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676 Mechanical Engineering Design

(b) Designating d9P and d9G as the new pitch-circle diameters, the 1 4-in increase in the

center distance requires that

d¿P 1 d¿G

2 5 14.250 (1)

Also, the velocity ratio does not change, and hence

d¿P d¿G

5 16 40

(2)

Solving Eqs. (1) and (2) simultaneously yields

Answer d¿P 5 8.143 in d¿G 5 20.357 in

Since rb 5 r cos f, using either the pinion or gear, the new pressure angle is

Answer f¿ 5 cos21 rb(pinion)

d¿Py2 5 cos21

3.759 8.143y2

5 22.59°

13–6 Contact Ratio The zone of action of meshing gear teeth is shown in Fig. 13–15. We recall that tooth contact begins and ends at the intersections of the two addendum circles with the pressure line. In Fig. 13–15 initial contact occurs at a and final contact at b. Tooth profiles drawn through these points intersect the pitch circle at A and B, respectively. As shown, the distance AP is called the arc of approach qa, and the distance PB, the arc of recess qr. The sum of these is the arc of action qt. Now, consider a situation in which the arc of action is exactly equal to the cir- cular pitch, that is, qt 5 p. This means that one tooth and its space will occupy the entire arc AB. In other words, when a tooth is just beginning contact at a, the previ- ous tooth is simultaneously ending its contact at b. Therefore, during the tooth action from a to b, there will be exactly one pair of teeth in contact. Next, consider a situation in which the arc of action is greater than the circular pitch, but not very much greater, say, qt < 1.2p. This means that when one pair of teeth is just entering contact at a, another pair, already in contact, will not yet have reached b. Thus,

Figure 13–15 Definition of contact ratio.

Lab Motion

A

a

b

B

Addendum circle

Pres sure

line

Pitch circle Addendum circle

Arc of approach qa

Arc of recess qr

P

!

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Gears—General 677

for a short period of time, there will be two teeth in contact, one in the vicinity of A and another near B. As the meshing proceeds, the pair near B must cease contact, leaving only a single pair of contacting teeth, until the procedure repeats itself. Because of the nature of this tooth action, either one or two pairs of teeth in contact, it is convenient to define the term contact ratio mc as

mc 5 qt p

(13–8)

a number that indicates the average number of pairs of teeth in contact. Note that this ratio is also equal to the length of the path of contact divided by the base pitch. Gears should not generally be designed having contact ratios less than about 1.20, because inaccuracies in mounting might reduce the contact ratio even more, increasing the possibility of impact between the teeth as well as an increase in the noise level. An easier way to obtain the contact ratio is to measure the line of action ab instead of the arc distance AB. Since ab in Fig. 13–15 is tangent to the base circle when extended, the base pitch pb must be used to calculate mc instead of the circular pitch as in Eq. (13–8). If the length of the line of action is Lab, the contact ratio is

mc 5 Lab

p cos f (13–9)

in which Eq. (13–7) was used for the base pitch.

13–7 Interference The contact of portions of tooth profiles that are not conjugate is called interference. Consider Fig. 13–16. Illustrated are two 16-tooth gears that have been cut to the now obsolete 1412° pressure angle. The driver, gear 2, turns clockwise. The initial and final points of contact are designated A and B, respectively, and are located on the pressure line. Now notice that the points of tangency of the pressure line with the base circles C and D are located inside of points A and B. Interference is present. The interference is explained as follows. Contact begins when the tip of the driven tooth contacts the flank of the driving tooth. In this case the flank of the driving tooth first makes contact with the driven tooth at point A, and this occurs before the involute portion of the driving tooth comes within range. In other words, contact is occurring below the base circle of gear 2 on the noninvolute portion of the flank. The actual effect is that the involute tip or face of the driven gear tends to dig out the noninvo- lute flank of the driver. In this example the same effect occurs again as the teeth leave contact. Contact should end at point D or before. Since it does not end until point B, the effect is for the tip of the driving tooth to dig out, or interfere with, the flank of the driven tooth. When gear teeth are produced by a generation process, interference is automati- cally eliminated because the cutting tool removes the interfering portion of the flank. This effect is called undercutting; if undercutting is at all pronounced, the undercut tooth is considerably weakened. Thus the effect of eliminating interference by a gen- eration process is merely to substitute another problem for the original one. The smallest number of teeth on a spur pinion and gear,1 one-to-one gear ratio, which can exist without interference is NP. This number of teeth for spur gears is