gears general -...

Chapter 13

Gears

General

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Dr. Mohammad Suliman Abuhaiba, PE 1

Chapter Outline

1. Types of Gears

2. Nomenclature

3. Conjugate Action

4. Involute Properties

5. Fundamentals

6. Contact Ratio

7. Interference

8. The Forming of Gear

Teeth

9. Straight Bevel Gears

10. Parallel Helical Gears

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Dr. Mohammad Suliman Abuhaiba, PE

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11. Worm Gears

12. Tooth Systems

13. Gear Trains

14. Force Analysis—Spur

Gearing

15. Force Analysis—Bevel

Gearing

16. Force Analysis—Helical

Gearing

17. Force Analysis—Worm

Gearing

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 1/P and 1.25/P,

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 ¼ in larger. Compute the

new values of the pressure angle and the

pitch-circle diameters.

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13–9 Straight Bevel Gears

Figure 13–20

Terminology of bevel

gears

Pitch of bevel gears is

measured at the large

end of tooth

Both the circular pitch

and the pitch diameter

are calculated in the

same manner as for

spur gears

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13–9 Straight Bevel Gears

Shape of the teeth,

when projected on back

cone, is the same as in

a spur gear having a

radius equal to the

back-cone distance rb.

The number of teeth in

this imaginary gear is

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13–10 Parallel Helical Gears

Used to transmit motion between parallel shafts

Helix angle is the same on each gear, one gear must

have a RH helix and the other a LH helix.

The shape of the tooth is an involute helicoid (Fig. 13–

21).

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spur-gear helical-gear

line contact extending

all the way across face

of tooth

Initial contact of teeth is a point

that extends into a line as teeth

come into more engagement.

line of contact is parallel

to axis of rotation

line is diagonal across face of

tooth.

gradual engagement of teeth and

smooth transfer of load from one

tooth to another that gives helical

gears the ability to transmit

heavy loads at high speeds.


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13–10 Parallel

Helical Gears

Figure 13–22: a portion of

top view of a helical rack

ab & cd: centerlines of two

adjacent helical teeth taken

on the same pitch plane

ψ = helix angle

ac = transverse circular pitch

pt in the plane of rotation

(circular pitch)

ae = normal circular pitch

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13–10 Parallel

Helical Gears

ad = axial pitch px

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13–10 Parallel

Helical Gears Figure 13–23

The oblique plane ab cuts out

an arc having a radius of

curvature of R.

when ψ = 0, radius of

curvature is R = D/2.

If ψ slowly increases from

zero to 90◦, R begins at a

value of D/2 and increases

until, when ψ = 90°, R = ∞.

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R = apparent pitch radius of a

helical gear tooth when viewed in

direction of the tooth elements.

A gear of same pitch and with

radius R will have a greater

number of teeth, because of

increased radius.

In helical-gear terminology this is

called the virtual number of teeth.

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Example 13–2

A stock helical gear has a normal pressure angle

of 20°, a helix angle of 25°, and a transverse

diametral pitch of 6 teeth/in, and has 18 teeth.

Find:

a. The pitch diameter

b. Transverse, normal, and axial pitches

c. The normal diametral pitch

d. The transverse pressure angle

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For a given gear ratio mG = NG/NP = m, the smallest

pinion tooth count is

The largest gear with a specified pinion is given by

The smallest pinion that can be run with a rack is

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13–11 Worm Gears

Helix angle on worm is

large, and that on gear

very small.

For a 90° shaft angle,

lead angle λ on worm =

helix angle ψG on gear

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Axial pitch px of worm = transverse circular pitch

pt of mating gear if shaft angle is 90°

Pitch diameter of gear = diameter measured on

a plane containing the worm axis,

Pitch diameter of worm should be selected so as

to fall into the range

13–11 Worm Gears

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Lead L & lead angle λ of worm have the

following relations:

13–11 Worm Gears

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13–12 Tooth Systems

Table 13–1: Standard & Commonly Used

Tooth Systems for Spur Gears

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Table 13–2: Tooth Sizes in General Uses

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13–12 Tooth Systems Table 13–3: Tooth Proportions for 20° Straight Bevel-Gear Teeth

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13–12 Tooth Systems Table 13–4: Standard Tooth Proportions for Helical Gears

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Table 13–5: Recommended Pressure Angles and

Tooth Depths for Worm Gearing

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13–13 Gear Trains

Figure 13–26: Thrust, rotation, and hand relations for

crossed helical gears

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13–13 Gear Trains

Figure 13–26: Thrust, rotation, and hand relations for


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13–13 Gear Trains Figure 13–26: Thrust, rotation, and hand relations for


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13–13 Gear Trains Figure 13–26: Thrust, rotation, and hand relations for


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Example 13–3

A gearbox is needed to provide a 30:1 (±1%)

increase in speed, while minimizing the overall

gearbox size. Specify appropriate teeth numbers

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Example 13–4

A gearbox is needed to provide an exact 30:1


gearbox size. Specify appropriate teeth numbers

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Example 13–5

A gearbox is needed to provide an exact 30:1


gearbox size. The input and output shafts should

be in-line. Specify appropriate teeth numbers

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13–14 Force Analysis—Spur

Gearing

Figure 13–32

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Gearing

Figure 13–33

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Gearing

Figure 13–33

V = pitch-line velocity,

fpm

d = gear diameter, in

n = gear speed, rpm

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13–14 Force Analysis—Spur Gearing

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English SI

Wt = transmitted load lbf kN

H = power hp kW

V = pitch-line velocity ft/min

d = gear diameter mm

n = speed rpm

Example 13–7

Pinion 2 in Fig. 13–34a runs at 1750 rpm and transmits 2.5

kW to idler gear 3. The teeth are cut on the 20° full-depth

system and have a module of m = 2.5 mm. Draw a FBD of

gear 3 and show all the forces that act upon it.

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13–15 Force Analysis—Bevel Gearing

rav = pitch radius

at midpoint of

tooth for the gear

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Example 13–8

The bevel pinion in Fig. 13–36a rotates at 600 rpm in the

direction shown and transmits 5 hp to the gear. The

mounting distances, the location of all bearings, and the

average pitch radii of the pinion and gear are shown in the

figure. For simplicity, the teeth have been replaced by pitch

cones. Bearings A and C should take the thrust loads. Find

the bearing forces on the gearshaft.

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Example 13–8 1/2/2015 2:04 PM


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Example 13–8 1/2/2015 2:04 PM


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13–16 Force

Analysis Helical

Gearing

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Example 13–9

In Fig. 13–38 a 1-hp electric motor runs at 1800 rpm in the

cw direction, as viewed from the positive x axis. Keyed to the

motor shaft is an 18-tooth helical pinion having a normal

pressure angle of 20°, a helix angle of 30°, and a normal

diametral pitch of 12 teeth/in. The hand of the helix is

shown in the figure. Make a three-dimensional sketch of the

motor shaft and pinion, and show the forces acting on the

pinion and the bearing reactions at A and B. The thrust

should be taken out at A.

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Example 13–9 1/2/2015 2:04 PM


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Example 13–9 1/2/2015 2:04 PM


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