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11. Gear Design

11. Gears

Objectives

· · ·

Understand basic principles of gearing. Understand gear trains and how to calculate ratios. Recognize different gearing systems and relative advantages and disadvantages between them. Understand geometry of different gears and their dimensional properties. properties. Recognize different principles of gearing. Recognize the unorthodox ways gears can be used in different motion motion systems.

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Introduction

Gears are the most common means used for power transmission They can be applied between two shafts which are

Parallel Collinear Perpendicular and intersecting Perpendicular and nonintersecting Inclined at any arbitrary angle

Introduction

Gears are made to high precision Purchased from gear manufacturers rather than made in house However it is necessary to design for a specific application so that proper selection can be made Used to be called toothed wheels dating back to 2600 b.c.

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An 18th Century Application of Gears for Powering Textile Machinery

http://www.efunda.com/DesignStandards/gears/gears_history.cfm

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11.2 Types of Gears

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Gear Types

Spur gears

Internal gears

Gear Parameters

Number of teeth Form of teeth Size of teeth Face Width of teeth Style and dimensions of gear blank Design of the hub of the gear Degree of precision required Means of attaching the gear to the shaft Means of locating the gear axially on the shaft

Most common form Used for parallel shafts Suitable for low to medium speed application Relatively high ratios can be achieved (< 7) Steel, brass, bronze, cast iron, and plastics Can also be made from sheet metal

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Spur gear nomenclature

Gear Types

Helical gears Teeth are at an angle Used for parallel shafts Teeth engage gradually reducing shocks

Kalpakjian · Schmid Manufacturing Engineering 11 and Technology, Prentice Hall

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Helical Gears

Helical Gear

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Helical Gear Characteristics

Helix angle 7 to 23 degrees More power Larger speeds More smooth and quiet operation Used in automobiles Helix angle must be the same for both the mating gears Produces axial thrust which is a disadvantage

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Herringbone Gears

Two helical gears with opposing helical angles side-by-side side- byAxial thrust gets cancelled

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Herringbone Gears

Herringbone Gear

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Herringbone Gear Machining

Gear Types

Bevel gears They have conical shape

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Bevel Gears (Miter gears)

For one-to-one ratio Used to change the direction

Bevel Gears

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Bevel Gears

Gear Types

Worm gears For large speed reductions between two perpendicular and non-intersecting shafts nonDriver called worm looks like a thread

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Rack and pinion

A rack is a gear whose pitch diameter is infinite, resulting in a straight line pitch circle. Involute of a very large base circle approaches a straight line Used to convert rotary motion to straight line motion Used in machine tools

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Fig. 11.7 Rack and pinion

Rack and pinion

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Internal spur gear

Provides more compact drives compared to external gears They provide large contact ratio Relatively less sliding and hence less wear compare to external gears

Internal spur gear

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Internal spur gear

Internally Meshing Spur Gears

Figure 14.14 Internally meshing spur gears.

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©1998 McGraw-Hill, Hamrock, Jacobson and Schmid

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Gear Assemblies

Identified based on the input and output shaft positions

Parallel shaft Spur gears Helical gears Perpendicular shaft Other types

Fig. 11-9 Velocity Ratio 11-

Bevel and Miter Rack-and-pinion Rack- andgears Cross-helix CrossWorm gears

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Vr =

Ng Np

=

Dg Dp

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Velocity Ratio

Velocity ratio is defined as the ratio of rotational speed of the input gear to that of the output gear

Vr = Ng Np = Dg Dp

Velocity Ratio

Vr = Velocity ratio Vr = N p = D p Np = Number of teeth on pinion Ng = Number of teeth on gear Dp = Pitch diameter of pinion Dg = Pitch diameter of gear

Ng Dg

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Example Problem 11-1: Velocity Ratios and Gear Trains

· For the set of four gears shown below, calculate output speed, output torque, and horsepower for both input and output conditions and overall velocity ratio:

Example Problem 11-1: Velocity Ratios and Gear Trains

(cont'd.)

(11-1) Vr = N2 N4 · N1 N3

Vr = - Output speed:

60 60 9 · = 20 20 1

n4 =

n1 Vr

3600 rpm · - Output torque:

1 = 400 rpm 9

T4 = T1 Vr T4 = 200 in-lb · 9 = 1800 in-lb 1

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Example Problem 11-1: Velocity Ratios and Gear Trains

(cont'd.)

- Input horsepower: (2-6) hp = Tn 63,000

Example Problem 11-2: Velocity Ratios and Gear Trains

· For the gear train shown below, determine the train value, output speed, output direction, output torque, and output power.

hp =

200 in-lb 3600 rpm 63,000 hp = 11.4

- Output horsepower: hp = 1800 in-lb 400 rpm 63,000 hp = 11.4

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Example Problem 11-2: Velocity Ratios and Gear Trains

(cont'd.) ­ Train value:

Example Problem 11-2: Velocity Ratios and Gear Trains

(cont'd.)

- Direction:

Vr = Vr =

NB N N · D · E NA NC ND 65 60 · = 9.75 / 1 20 20

(11-1)

·

If:

Gear A ­ clockwise Gear B ­ counterclockwise Gear C ­ counterclockwise Gear D ­ clockwise Gear E ­ counterclockwise

Idler cancels out and has no effect on overall train value.

­ Output speed:

nE =

NA Vr

=

3000 rpm = 307.7 rpm 9.75 / 1

- Output power: P = TnE = 97.5 Nm 307.5 rev min min 60 sec J or W s

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­ Output torque:

TE = T A Vr TE = 10 Nm (9.75 / 1) = 97.5 Nm

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P = 1571 Nm/sec or P = 1.57 kW

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Spur Gears

Pinion Gears

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Internal Gears

Spur gear geometries

Pitch circle: is the imaginary circle on which most gear calculations are made. When two gears meet their pitch circles are tangent to each other Pitch diameter (Dp) and pitch radius (r): These are (D the diameter and radius of the pitch circle. Pitch point: The point on the imaginary line joining the centers of the two meshing gears where the pitch circle touch

D-d 2

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Pitch circle

Figure 14.1 Spur gear drive.

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Text Reference: Figure 14.1, page 616 ©1998 McGraw-Hill, Hamrock, 48 Jacobson and Schmid

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Spur gear geometries

Addendum circle: It is the circle that bounds the outer ends of the teeth and whose center is at the center of the gear (Fig. 7.2). 7.2). Dedendum circle: It is the circle that bounds the bottoms of the teeth and whose center is at the center of the gear (Fig. 7.2). 7.2). Addendum (a): is the radial distance from the pitch circle to the outer end of the teeth. (Fig. 7.2). 7.2). Dedendum (b): is the radial distance from the pitch circle to the bottom of the teeth. (Fig. 7.2). 7.2).

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Spur gear geometries

Circular pitch (Pc): is the distance between corresponding points on adjacent teeth measured along the pitch circle (Fig. 7.2). 7.2). Diametral pitch (Pd): specifies the number of teeth per inch of pitch diameter. Tooth space: is the space between the adjacent teeth measured along the pitch circle (Fig. 7.2). 7.2). Tooth thickness: is the thickness of the tooth measured along the pitch circle (Fig. 7.2). 7.2).

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Spur gear geometries

Face width (W): is the length of the tooth measured parallel to the gear (Fig. 7.2). 7.2). Face: is the surface between the pitch circle and the top of the tooth (Fig. 7.2). 7.2). Flank: is the surface between the pitch circle and the bottom of the tooth (Fig. 7.2). 7.2). Pressure angle (): is the angle between the line of ( action and a line tangent to the two pitch circles at the pitch point. (Fig. 14.8 Hamrock). (Fig. Hamrock).

Figure 14.8 Pitch and base circles for pinion and gear as well as line of action and pressure angle.

rb = r cos

D b = D p cos

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©1998 McGraw-Hill, Hamrock, Jacobson and Schmid

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Spur gear geometries

Line of action: is the locus of all the points of contact between two meshing teeth from the time the teeth go into contact until they lose contact. Pinion: is the smaller of the two meshing gears. Backlash: is the difference (clearance) between the tooth thickness of one gear and the tooth space of the meshing gear measured along the pitch circle (Fig. 7.5). 7.5)

Gear terminology

Clearance (c): is the addendum minus dedendum. dedendum. Working depth: is the distance that one tooth of a meshing gear penetrates into the tooth space. Base circle: is an imaginary circle about which the tooth involute profile is developed. Fillet: is the radius that occurs where the flank of the tooth meets the dedendum circle. Module: replaces diametral pitch in metric system.

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Basic formulas for spur gears

Diametral pitch, Pd = Circular pitch, Pc = Addendum, a =

NP DP

Specifications for standard gear teeth

Item Full depth & pitches coarser than 20 20° 20° 25° 25° Full depth & pitches finer than 20 20° 20° 14½° full 14½ depth 14½° 14½

DP NP

Pressure angle Addendum (in.) Dedendum (in.)

1 Pd

1.0/ Pd 1.250/ Pd

1.0/ Pd 1.250/ Pd

1.0/ Pd 1.2/ Pd + 0.002

1/ Pd 1.157/ Pd

Dedendum, b = 1.250 Dedendum, Pd

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Basic formulas for spur gears

Clearance, c = b ­ a = Where

0.250 Pd

Basic formulas for spur gears

Center to center distance D + D pp Ng + Np CtoC = pg = 2 Pd 2 Where

Dpp = pitch diameter of pinion Np = number of teeth on the pinion Dpg = pitch diameter of gear Ng = number of teeth on the gear Pd = Diametral pitch

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Dp = pitch diameter of pinion Np = number of teeth on the pinion

It can be shown that

Pd × Pc =

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Metric System

Module (m) = 1 Pd

Metric System

Diametral pitch, Pd =

1 m

See Table 11.1 for equivalents Normally they are not converted

Circular pitch, Pc = m Addendum, a = m Dedendum, b = 1.25 m Dedendum,

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Inch units

A spur gear of the 14 ½ degree involute system has 32 teeth of diametral pitch 8. Find

The pitch diameter The circular pitch The outside diameter (addendum diameter)

Metric units

A spur gear of the 14 ½ degree involute system has a module of 8 mm and 35 teeth. Find

The pitch diameter The circular pitch The outside diameter (addendum diameter)

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Example Problem 11-3: Pressure Angle

· For the set of gears shown in Figure 11-17, if diametral pitch is 8, find the pitch diameter, circular pitch, and shaft center-to-center distance. · The pinion has 16 teeth and the gear has 32 teeth.

- Pitch diameter: (11-4) Dp = - Pinion: Dp = - Gear: Dp = - Circular pitch: Pc = 32 = 4 inches 8 (11-3) 16 = 2 inches 8 Np Ng or Pd Pd

Example Problem 11-3: Pressure Angle (cont'd.)

- Centerline distance: (11-2) Np + Ng Dp Dg + or C - C = C-C = 2 2 2Pd C-C = 16 + 32 2 (8)

C ­ C = 3 inches

Dp

Np = .393 inch

Pc =

2 in

16

· Circular pitch would be the same for both pinion and gear.

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Summary

To understand the gears one should be familiar with the gear terminology. Spur gears are most commonly used for transmission of power. Speed of mating gears is inversely proportional to the number of teeth. Mating gears should have the same diametral pitch. A number of gear manufacturing methods are available. Good gear design should take care of the power, speed, life and material properties.

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