Chapter 13site.iugaza.edu.ps/mhaiba/files/2013/09/CH-13-Gears-General-2015.pdf · Example 13–1 A...

Chapter 13

Gears

General

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Mohammad Suliman Abuhaiba, Ph.D., PE1

2nd Exam

Wednesday 2/12/2015

Chapter 8

12/25/2015 12:27 PM

Mohammad Suliman Abuhaiba, Ph.D., PE

2

Chapter Outline

1. Types of Gears

2. Nomenclature

3. Conjugate Action

4. Involute Properties

5. Contact Ratio

6. Interference

7. Forming of Gear Teeth

8. Straight Bevel Gears

9. Parallel Helical Gears

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10. Worm Gears

11. Tooth Systems

12. Gear Trains

13. Force Analysis—Spur

Gearing

14. Force Analysis—Bevel

Gearing

15. Force Analysis—Helical

Gearing

16. Force Analysis—Worm

Gearing

Example 13–1

A gear set consists of a 16-tooth pinion driving a

40-tooth gear. The diametral pitch is 2, and the

addendum & 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

Circular pitch & pitch

diameter are calculated

in the same manner as

for spur gears

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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.

Shape of 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 GearsFigure 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, smallest pinion

tooth count is

Largest gear with a specified pinion:

Smallest pinion that can be run with a rack:

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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 SystemsTable 13–3: Tooth Proportions for 20° Straight Bevel-Gear Teeth

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13–12 Tooth SystemsTable 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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Figure 13–26: Thrust, rotation, and hand relations

for crossed helical gears

13–13 Gear Trains

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Right Hand

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crossed helical gears



13–13 Gear Trains

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Right Hand

13–13 Gear Trains



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Left Hand

13–13 Gear Trains



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Left Hand

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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Figure 13–33

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Figure 13–33

V = pitch-line

velocity, fpm

d = gear diameter, in

n = gear speed, rpm

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English SI

Wt = transmitted load lbf kN

H = power hp kW

V = pitch-line velocity ft/min NA

d = gear diameter in mm

n = speed rpm 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 shown rotates at 600 rpm in the direction

shown and transmits 5hp 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 13–812/25/2015 12:27 PM


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Example 13–812/25/2015 12:27 PM


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13–16 Force

Analysis Helical

Gearing

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Example 13–9

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. Make a 3-D sketch of the motor shaft &

pinion, and show the forces acting on the pinion

and the bearing reactions at A & B. The thrust

should be taken out at A.

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Example 13–912/25/2015 12:27 PM


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Example 13–912/25/2015 12:27 PM


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Chapter 13site.iugaza.edu.ps/mhaiba/files/2013/09/CH-13-Gears-General-2015.pdf · Example 13–1 A...

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