O C T O B E R 0 4 2 0 1 4

M A C H I N E E L E M E N T S

Submitted by: Beduya, Franbeau Rhaz Buendia, Kimberly Mae

Lambating, Kenneth Vincent Mayol, Hannah Mae Tura, Fabriene Mae Vasnani, Neelesh N.

BSIE - 3

Submitted to: Engr. Van Vergara

ME 312 Sat 12-3pm Instructor Mechanical Engineering Dept

University of San Carlos

. . . . . .

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

1

TABLE OF CONTENTS

I. INTRODUCTION 2 - 4

A Brief History 2

What is a Bevel Gear? 2

A Bevel gear is beveled 2

Types of Bevel Gear 3

Pros and Cons 3

Common Applications and Uses 4

II. DEFINITION OF TERMS 5 6

Definitions 5

Nomenclature Figure 6

III. DETAILS OF THE TOPIC 7 - 12

Types of Bevel Gears 7 8

Based on Tooth race 7

Based on Pitch angles 8

Determining the direction sense 9

Calculations 10 11

Table 12

IV. SAMPLE PROBLEMS 13 14

V. SOURCES AND REFERENCES 15

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

2

I. INTRODUCTION

A Brief History

Gears were used by mankind since early 400 B.C. They were first used

by the Chinese to carry out tasks related to transferring power through

rotations, similar to their purpose in the modern times.

One of the first applications was the Chinese South-Pointing Chariot

a vehicle built on two wheels which bore a movable indicator that

always pointed South no matter how the chariot turned, without the

use of magnets.

The essence is old, but technology has changed and evolved gears a lot. The 18th century industrial

revolution was the cause of this breakthrough. Alongside many new designs such as the involute curve,

the bevel gear was born.

What is a Bevel gear?

Bevel gears are useful when the direction of a shaft's rotation needs to be

changed. They are usually mounted on shafts that are 90 degrees apart,

but can be designed to work at other angles as well.

The picture on the right shows how bevel gears are used to change

direction. They change the transmission of the power or torque from one

plane to another, which is different from conventional gears that transfer

rotations within a single two dimensional plane.

A Bevel gear is Beveled

As the term bevel would suggest, the face of a bevel gear is angled. Typically, a bevel gear is conical

in shape, with the teeth running along the side of the cone, and the angle of the sides can be varied,

depending on the application for the gear.

Conical/Frustum shape

Beveled face (where

Teeth are placed)

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

3

Types of Bevel Gears

There are 4 commonly known types of bevel gears: Straight bevel gear, Zerol bevel gear, Spiral bevel

gear, and lastly the Hypoid bevel gear. These will be discussed in detail later on.

Straight bevel gear Zerol bevel gear Spiral bevel gear Hypoid bevel gear

Another type of bevel gear is the Miter gear. Miter gears are a type of bevel

gears that have equal numbers of teeth. Miter gears are useful for transmitting

rotational motion at a 90 degree angle with a 1:1 ratio. Geometrically, a miter

bevel gears teeth can be straight, zerol, or even spiral in shape.

Pros and Cons of the Bevel Gear

Advantages:

The main advantage of Bevel gears is that the gears make it possible to change the operating

angle.Through this, aside from transmitting direction alone, you can increase or decrease the torque,

force and speed being transmitted by changing or tweaking the ratio of the teeth.

Secondly, they can be made out of a variety of different materials. Metal ones are common in clocks

as well as in hand tools such as the handheld drill drivers. However, they can also be made out of

plastic or nylon, and they are more common in electronic machines like toys as well as other

components. Gears made with non-metallic materials give noiseless operation at higher speeds.

Disadvantages:

One disadvantage of the Bevel gear is that one wheel of such gear can only work with its

complementary wheel, and no other. Another downside is that bevel gears need extreme precision in

mounting. Lastly, the shaft bearings need to be made to support relatively heavier loads or forces.

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

4

Common Applications and Uses

1. Hand drill - Bevel gears are used as the main mechanism here. As the handle of the drill is turned

in a vertical direction, the bevel gears change the rotation of the chuck to a horizontal rotation.

The bevel gears in a hand drill have the added advantage of increasing the speed of rotation of

the chuck and this makes it possible to drill a range of materials.

2. Differential Drives - Bevel gears are also used in differential drives. Differentials are devices found

in automobiles. They allow the wheels of an automobile to rotate at different speeds. This is

important when turning corners as the inner wheel needs to travel a shorter distance. Without

differentials, the inner wheel might spin or the outer wheel might be dragged which may result

to a strained or unpredictable handling of the vehicle as well as damage to the tires or roads.

3. Mechanical Shutter doors - bevel gears are used to roll up the door, or to turn and open it.

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

5

II. DEFINITION OF TERMS

It is important to understand the following terms discussed in this section before going into the

calculations and technicalities of the bevel gear.

1. Diametral Pitch is referred to as the ratio of the number of teeth to the pitch diameter of the

back cone in a bevel gear.

2. Pinion a gear with a small number of teeth, especially one engaging with a larger gear like the

spur or rack gear. Bevel gears also have pinions.

3. Module It is used to designate tooth size in the metric system.

Technical Terminologies

4. Apex point of intersection of the gear axes. It is obtained by drawing imaginary lines

perpendicular to the bevel gears.

5. Pitch Cone the geometric shape of bevel gears based on equivalent rolling contact.

6. Cone Distance the slant height of the pitch cone.

7. Face Cone the cone formed by the elements passing through the top of the teeth and the apex.

8. Root Cone the cone formed by the elements passing through the bottom of the teeth and the

apex.

9. Face Width a distance that should not exceed one-third of the cone distance. A width of a tooth.

10. Face Angle the angle between the top of the teeth and the gear axis.

11. Pitch Angle the angle between an element of the face cone and the axis of the gear.

12. Root Angle the angle between the bottom of the tooth space and the gear axis.

13. Addendum the distance from an element on the pitch cone to an element on the face cone,

measure on the outside of the tooth.

14. Dedendum the distance from an element on the pitch cone to an element on the root cone,

measured on the outside of the tooth.

15. Addendum Angle the angle between an element on the pitch cone and an element on the face

cone.

16. Dedendum Angle the angle between the element on the pitch cone and an element on the root

cone.

17. Pitch Diameter the diameter of the base of the pitch cone.

18. Inside Pitch Diameter the pitch diameter measured on the inside of the tooth.

19. Outside Diameter the pitch diameter measured on the outside of the tooth.

20. Back Cone the cone formed by elements perpendicular to the pitch cone elements at the outside

of the teeth.

21. Back Cone Radius the length of a back cone element.

22. Clearance - the small distance in the tooth gap of the gear which is subtracted from the

dedendum to get the addendum.

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

6

Nomenclature

View with both pinion and gear:

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

7

III. DETAILS OF THE TOPIC

Four Types of Bevel gear According to Tooth Race

1. Straight Bevel Gear a bevel gear whose teeth are straight but their sides

are tapered so that they would intersect the axis at a common point (pitch

cone apex) if extended inward.

Features: Easy to manufacture, and provides high reduction rations: up

to 1:5

Applications: Machine tools, printing presses, differential units.

2. Zerol Bevel Gear a bevel gear whose teeth is curved but lie in the same

direction as the teeth of straight bevel gears. Spiral angle is zero.

Features: These gears normally have a pressure angle of 20 deg. The

minimum number of teeth on the pinion is 14. The design of Zerol

gears is relatively specialized and they are manufactured using special

"Gleason" machine tools..

Applications: Essentially used in applications similar to spiral or straight

bevel gear, but with higher accuracy and speed.

3. Spiral Bevel Gear a bevel gear having curved oblique teeth on which

contact begins gradually and continues smoothly from end to end.

Features: High strength and durability, reduced gear noise, but involves

difficulties in manufacutring.

Applications: Vehicles, final reduction gearing for ships, suitable for high

speed and heavy load drives.

4. Hypoid Gear is similar to spiral bevel gears; except that the shaft axes are

offset, nonintersecting.

Features: The minimum number of teeth for speed ratios greater than

6:1 is eight although 6 teeth pinions can be used for ratios below 6:1.

Hypoid gears have pressure angles between 19 and 22 deg.

Applications: Automotive applications, applications that require smooth

and quiet operation.

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

8

Classification of Bevel Gears Based on Pitch Angles

1. Miter gears are bevel gears of the same size, have the same number of teeth connecting shafts

at right angle (90). They have speed ratio of one and a pitch angle of 45 in each gear. It is used to

change direction without changing speed since gear ratio is 1: 1.

2. Crown Gear a bevel gear in which the pitch angle of 90 and a plane for its pitch surface.

3. Internal Bevel Gear a bevel gear in which the pitch angle exceeds 90.

4. External Bevel Gear a bevel gear in which the pitch angle is less than 90.

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

9

Comparison of Intersecting and offset shaft (Hypoid)

Determining the direction sense

For those who have a difficult time imagining the direction sense, a general rule can be applied. Using

any hand (right or left), this direction sense can be determined. If the direction sense of either the

pinion or gear is obtained by pointing your hand inward (towards the apex), the direction sense of its

partner gear can be determined by pointing the hand outward (away from the apex). Conversely, if

the known direction sense of one of the gears is obtained by pointing the hand outward, the direction

sense for its partner gear can be obtained by pointing the hand inward. If one is outward, the other is

inward, and vice versa.

Inward Pitch Apex

Outward

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

10

Calculations

Just like the other types of gears, bevel gears also have geometric quantities like diametral pitch, number of

teeth, speed ratio, angular speed, etc.

The relationships between the geometric quantities and the speed for bevel gears are given as follows

The number of teeth can be found by multiplying the diametral pitch and the pitch diameter.

Np = P ( dp) , Ng = P ( dg )

Unit:

(i) = teeth

where:

dp = pitch diameter of pinion dg = pitch diameter of gear Pitch Diameter (gear) and = Number of teeth of pinion and gear respectively

= Diametral Pitch

Tangent of the pitch angle can be obtained by the ratio of the radiuses or the diameters of the pinion and gear. If we try to find the pitch angle of the gear for example:

tan( ) =

=

=

where:

= pitch angle of gear

rg and rp = gear and pinion pitch radiuses and = gear and pinion pitch diameters

opp = opposite side of the pitch angle

adj = adjacent side of the pitch angle

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

11

The virtual number of teeth in the spur gear that can

be visualized from the bevel gear.

Whichever yields a smaller value

should be chosen

Speed ratio: The speed ratio () can be found through a number of ways.

Since the: number of teeth is directionally proportional to the gear pitch diameter,

=

The smaller the diameter, the faster the gear rotates. This means that the angular speed and the

diameter are inversely proportional.

=

where = angular speed in (rad/unit time). We can also express this speed it terms of rpm.

Lastly, the tangent of the pitch angle from earlier can also be used to find the speed ratio.

tan( ) =

Combining all the quantities from above, the speed ratio () is:

=

=

=

= tan( ) = cot( )

Other quantities (based on 20 degree pressure angle the most common for bevel gears):

Addendum: Formative/Virtual Number of teeth

a = m = 1

N =

Whole depth: Cone distance:

ht = 0.002 + 2.188

P L = 2 + 2 or

1

22 + 2

ht = a + d Dedendum:

d = ht a or a + c

Clearance: Face Width:

c = ht - 2a or d - a b =

3 or b =

10

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

12

Example of Bevel gear dimension table (20 deg.)

Note: For values of different number of teeth and various pitch diameters, the tables on spur gear can

be also be used. For this topic however, we will only focus on 20 degree bevel gears.

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

13

IV. SAMPLE PROBLEMS

1. A 20 degree pressure angle straight bevel pinion having 20 teeth and a diametral pitch of 8

teeth/in drives a 42-tooth gear. Determine

a. The pitch diameters

b. The pitch angles

c. Face width

d. Clearance

a. Np = P ( dp) 20 = 8 ( dp ) dp = 2.5 in

Ng = P ( dg) 42 = 8 (dg ) dg = 5.25 in

b. tan( ) =

=

20

42 ; = 25. 5 , tan( ) =

=

42

20 ; = 64. 5

c. b = 1

3 =

1

3 x

1

22 + 2 =

1

3 x

1

22.52 + 5.252 = b = 0.969 in

d. c = ht - 2a = 0.002 + 2.188

P - 2

1

P = 0.002 +

2.188

8 - 2

1

8 = c = 0.026 in

2. Two cut cast steel miter gears have a face width of 52mm, a pitch diameter of 230mm, and a

module of 8. Compute the actual and formative numbers of teeth.

Actual: N = P ( d ) = 1

m ( d) =

1

8 (230mm) = 28.75 say 29 teeth

Formative: N =

=

29

45 = 41.02 say 41 teeth

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

14

3. A pair of 20 degree stub straight bevel gears drives the vertical drill spindle of a drilling machine. The

approximate speed reduction is 2.25 to 1. If the pinion has 24 teeth and a module of 4 mm/teeth ,

find the pitch diameters and the number of teeth in the gear. Also find the exact cone distance.

Np =

24mm =

4 dp = 96mm ; dg = dp(2.25) = 96(2.25)=dg = 216mm

Ng = 24 ( 2.25 ) = Ng = 54 teeth

L = 1

22 + 2 =

1

22162 + 962 = L = 118.2mm

4. From the previous problem, solve the addendum, dedendum, whole depth, and clearance of the

pinion.

Addendum: a = m = 4mm

Whole depth: ht = 0.002 + 2.188

P = 0.002 + 2.188m = 0.002 + 2.188(4)

ht = 8.754 mm

Dedendum: d = ht - a = 8.754mm 4mm = 4.754mm

Clearance: c = d a = 4.754mm 4mm = 0.754mm

5. A pinion of a 20 deg. bevel gear has a face width of 3 in and a pitch diameter of 5 in. How many

times more revolutions does this pinion make than its partner gear in one minute?

b = 1

3 3b = L L = 3(3) = 9 in >>> cone distance of pinion

We are interested in finding the speed ratio: using the pitch angle we have:

sin ( ) = rp

L =

2.5

9 ; = 16.128

speed ratio: tan ( 16.128) = 0.289 =

>>> Gear is 0.289 times slower than pinion

Therefore, the pinion makes 1

0.289 or 3.46 times more revolutions than the gear in one minute

M A C H I N E E L E M E N T S | T H E B E V E L G E A R

15

Sources and References

Online

http://science.howstuffworks.com/transport/engines-equipment/gear4.htm

http://nptel.ac.in/courses/IIT-MADRAS/Machine_Design_II/pdf/2_13.pdf

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

http://en.wikipedia.org/wiki/Bevel_gear#Types

http://EzineArticles.com/5674966

http://www.technologystudent.com/gears1/bevel1.htm

http://books.google.com

http://dictionary.reference.com

http://www.khkgears.co.jp/en/gear_technology/pdf/gear_guide_060817.pdf

http://www.ehow.com/info_8457201_uses-bevel-gear.html

http://www.roymech.co.uk/Useful_Tables/Drive/Bevel_Gears.html

http://www.cross-morse.co.uk/pdf/Bevel%20GearsEN.pdf

http://www.qtcgears.com/q410/pdf/techsec9.pdf

http://www3.nd.edu/~manufact/FME_pdf_files/FME3_Ch15.pdf

http://www.google.com/images

Print

Machine Design Reviewer by Jas Tordillo

Ugural, Ansel C. Mechanical Design, an integrated approach. McGraw Hill 2006

Elements of Mechanism by Daughtry

Norton, Robert. Fundamentals of Machine Design

Madsen, David A., and David P. Madsen. "Mechanisms: Linkages, Cams, Gears, and Bearings."

Engineering Drawing and Design. 5th ed. N.p.: Cengage Learning, 2012.

END