design of gears

Mechanical Engineering Dept. CEME NUST 1

Ch-6: Design of Gears


Introduction

Design of Gears

In precision machines, in which a definite velocity ratio is of importance (as in

watch mechanism), the only positive drive is by gears or toothed wheels

A Gear or Cogwheel is a rotating machine part having cut teeth, or cogs,

which mesh with another toothed part in order to transmit torque

Gear drive is also provided, when the distance between the Driver and the

Follower is very small

Friction Wheels

Motion and power transmitted by gears is

kinematically equivalent to that transmitted by

Frictional Wheels or discs

o Wheel B will be rotated by Wheel A so long as the tangential force exerted by the Wheel A does not exceed

the maximum frictional resistance between the two wheels

o when tangential force (P) exceeds the frictional resistance (F), slipping will take place between the two wheels


Introduction

Design of Gears

Introduction

Design of Gears

Friction Wheelscontd--

To avoid slipping, number of Projections (teeth) are provided on the periphery of

the wheel A which will fit into the corresponding Recesses on the periphery of

the wheel B

Friction Wheel with the teeth cut on it is known as Gear or Toothed Wheel


Introduction

Design of Gears

In any pair of gears, smaller one is called Pinion and the larger one is called

Gear immaterial of which is driving the other

When Pinion is Driver, it results in step down drive in which the output speed

decreases and the torque increases

when Gear is driver, it results in step up drive in

which output speed increases and the torque

decreases


Introduction

Design of Gears

Advantages of Gear Drives

o Transmits exact velocity ratio, without slipping which is not possible in case of belts, ropes etc.

o may be used to transmit wide range of power, i.e. 1/10th of HP to several thousands of HP

o may be used for small central distances of shafts

o high efficiency

o compact layout

Disadvantages of Gear Drives

o Manufacturing of gears require special tools and equipment, therefore it is costlier than other drives

o Error in cutting teeth may cause vibrations and noise during operation

o Requires suitable lubricant and reliable method of applying it

o Not suitable for large distances because the drive may become bulky

o Gear Drives can be used when precise timing is required


Classification of Gears

Design of Gears

1. According to the position of axes of the shafts

Axes of the two shafts between which the motion is to be transmitted, may be

(a) Parallel

(b) Intersecting

(c) Non-intersecting and non-parallel

Spur Gears/Spur Gearing

o Gears have teeth parallel to the axis of the wheel

o two parallel and co-planar shafts connected by Spur Gears



Design of Gears

1. According to the position of axes of the

shaftscontd--

Helical Gears/Spur Gearing

o Teeth are inclined to the axis

o Compared to spur gears, these are not as noisy, because of the more gradual engagement of the teeth during meshing

o Inclined Tooth also develops Thrust Loads and Bending Couples, which are not present with spur gearing

o Double Helical Gear balance out the end thrusts that are induced in single helical gears when transmitting load

o double helical gears are known as Herringbone Gears

o for the same width, their teeth are longer than spur gears and have higher load carrying

capacity

o manufacturing difficulty makes them costlier than single helical gears



Design of Gears

1. According to the position of axes of the shaftscontd--

Bevel Gears/Bevel Gearing

o have teeth formed on conical surfaces and are used mostly for transmitting motion between

intersecting shafts

two non-parallel or

intersecting, but

coplaner shafts

connected by gears

o complicated both form and fabrication limits achievement of precision



Design of Gears

1. According to the position of axes of the shaftscontd--

Bevel Gears/Bevel Gearingcontd--

o Skew Bevel Gears / Spiral Gear have curved teeth at an angle allowing tooth contact to be

gradual and smooth

Worms and Worm Gears

o Used to transmit rotary motion between nonparallel and nonintersecting shafts

o Worm resembles a screw

o Worm is usually meshed with a Spur or a Helical Gear, which is called the Worm Wheel



Design of Gears

2. According to the Peripheral Velocity of the Gears

(a) Low Velocity

(b) Medium Velocity

(c) High Velocity

velocity less than 3 m/s

velocity b/w 3-15 m/s

velocity greater than 15m/s

3. According to Type of Gearing

External Gearing

o Gears of the two shafts mesh externally with each other

o larger of these two wheels is called Spur Wheel or Gear and smaller wheel is called

Pinion

o motion of the two wheels is always Unlike



Design of Gears

3. According to Type of Gearing

Internal Gearing

o Gears of the two shafts mesh internally with each other

o larger of these two wheels is called Annular Wheel and smaller wheel is called Pinion

Rack and Pinion

o comprises a pair of gears which convert rotational motion into linear motion

o circular gear "The Pinion" engages teeth on a linear

gear bar "The Rack"


o Spur Gears multiply speed or force. o Bevel Gears change vertical movement

into horizontal movement.

o Worm Gears change the direction of horizontal movement.

o Rack and Pinion Gears change rotation into back-and-forth motion.


Nomenclature

Design of Gears

Pitch Circle: Theoretical circle upon which all calculations are usually based

Pitch Circle Diameter: diameter of the pitch circle

Size of the gear is usually specified by Pitch Circle Diameter


Nomenclature

Design of Gears

Addendum: Radial distance of a tooth from the pitch circle to the top of the

tooth

Dedendum: Radial distance of a tooth from the pitch circle to the bottom of the

tooth


Nomenclature

Design of Gears

Addendum Circle: Circle drawn through the top of the teeth and is concentric

with the pitch circle

Dedendum Circle: Circle drawn through the bottom of the teeth and is

concentric with the pitch circle


Nomenclature

Design of Gears

Clearance: Radial distance from the top of the tooth to the bottom of the tooth,

in a meshing gear. A circle passing through the top of the meshing gear is

known as Clearance Circle

Total depth: It is the radial distance between the addendum and the dedendum

circle of a gear, i.e. sum of the addendum and dedendum

17 Mechanical Engineering Dept. CEME NUST

Nomenclature

Design of Gears

Working Depth: Radial distance from the addendum circle to the clearance

circle. It is equal to the sum of the addendum of the two meshing gears

Tooth Thickness: Width of the tooth measured along the pitch circle

Tooth space / width of space: space between the two adjacent teeth measured

along the pitch circle


Nomenclature

Design of Gears

Backlash: difference between tooth space and tooth thickness, as measured on

the pitch circle Face of the tooth: Surface of the tooth above the pitch surface

Top land: It is the surface of the top of the tooth

Face of the tooth: Surface of the tooth above the pitch surface


Design of Gears

Flank of the tooth: It is the surface of the tooth below the pitch surface

Face Width: It is the width of the gear tooth measured parallel to its axis

Nomenclature


Nomenclature

Design of Gears

Circular Pitch (pc): distance, measured on the pitch circle, from a point on one

tooth to a corresponding point on an adjacent tooth

It is equal to the sum of the Tooth Thickness and the Width of Space

pc = D/T D = Diameter of the pitch circle, and

T or Z or N = Number of teeth on the wheel.

two gears will mesh together correctly, if two

wheels have the same circular pitch

D1 and D2 are the diameters of the two

meshing gears having the teeth T1 and T2

21 Mechanical Engineering Dept. CEME NUST

Nomenclature

Design of Gears

Diametral Pitch (pd): Ratio of number of teeth to the pitch circle diameter in

millimetres T = Number of teeth

D = Pitch circle diameter

Module (m): Ratio of the pitch circle diameter in millimeters to the number of

teeth m = D / T

Pc . Pd =


Nomenclature

Design of Gears

Variation of Tooth Size with

Diametral Pitch


Nomenclature

Design of Gears

Actual Size of the Gear Tooth for Different Diametral Pitches


Nomenclature

Design of Gears

Standard Diametral Pitches

Standard Modules (mm)


Nomenclature

Design of Gears

Fillet radius: Small radius that connects the profile of the tooth to the root circle


Condition for Constant Velocity Ratio of GearsLaw of Gearing

Design of Gears

Consider portions of the two teeth, one on the wheel

1 (or pinion) and the other on the wheel 2

Q is the point of contact

TT is the common tangent

MN is the common normal to the curves at Q

O1M and O2N are perpendicular to MN

point Q moves in the direction QC, when

considered as a point on wheel 1, with velocity v1

point Q moves in the direction QD, when

considered as a point on wheel 2, with velocity v2

If teeth are to remain in contact, then

components of velocities v1 and v2 along the

common normal MN must be equal



Design of Gears

from similar triangles O1MP and O2NP

Angular Velocity Ratio is inversely proportional to the

ratio of the distance of P from centers O1 and O2



Design of Gears

To have a constant angular Velocity Ratio for all

positions of the wheels, P must be the fixed point (called Pitch Point) for the two wheels

common normal at the point of contact between a pair of teeth must always pass through the pitch point

also known as Law of Gearing

This amounts to the following relationship:

velocity ratio is equal to the inverse ratio of the

diameters of pitch circles


Gear Profiles

Design of Gears

Profiles which satisfy the Law Of Gearing are called Conjugate Profiles, these are:

(a) Involute (b) cycloidal (c) Circular arc or Novikov


Gear Profiles

Design of Gears

Involute Gear Tooth Profile

Involute is the path generated by the end of a thread as it unwinds from a Reel

o imagine a reel with thread wound in the clockwise direction

o Tie a knot at the end of the thread

o Keeping the reel stationary, pull the thread and unwind it to position B0, B1,.

o knot now moves from C0 to C1.

o By Connecting these points C0 to C4 by a smooth curve, obtained profile is an involute

Involute gear

tooth profile

appearance after

generation


Gear Profiles

Design of Gears

Involute Gear Tooth Profilecontd--

o Involute gears are economical to make because the cutters used to make the gears are straight

o Variation in center distance does not affect the Velocity Ratio

o Pressure angle remains constant throughout the engagements which results in smooth running


Gear Profiles

Design of Gears

Cycloidal Gear Tooth Profile

Cycloid is the locus of a point or

curve traced by a point on the

circumference of a circle when it

rolls on a straight line without

slipping

If circle rolls on the outside of

another circle or inside of

another circle gives rise to

Epicycloid and Hypocycloid

respectively


Gear Profiles

Design of Gears

Cycloidal Gear Tooth ProfileContd--

o Two generating circles roll on the pitch circle to trace the cycloidal tooth profile

o inside circle traces the "Flank" of the gear tooth

o outside circle traces the "Face" of the gear tooth

o Have longer life as the contact is mostly rolling

o Less sliding friction and wear

Advantages

o Cycloidal Tooth is generally stronger than involute tooth owing to spreading

flanks in contrast to radial flanks of an

involute tooth

o Extensively used in watches, clocks where strength and interference are

prime considerations, used in crusher

drives in sugar mills


Gear Terminologies during meshing

Design of Gears

Pitch Point: point of tangency of the pitch circles of a pair of mating gears

Common Tangent: The line tangent to the pitch circle at the pitch point

Line of action: A line normal to a pair of mating tooth profiles at their point of

contact.



Design of Gears

Pressure angle : Angle between the common normal at the point of tooth contact and the common tangent to the pitch circles

o Standard values are 14.5, 20 and 25 degrees

o 14.5 were common as the cosine is larger for a smaller angle, providing more power transmission and less pressure on the bearing

o however, teeth with Smaller Pressure Angles are weaker



Design of Gears

Pressure angle contd--

o All mating gears must be of the same pressure angle to mesh properly

o 20-degree tooth is wider at the base and consequently is stronger than the 14.5-degree tooth form

o 20-degree tooth form has a greater factor of safety in strength, runs smoother, wears longer, and is no more expensive to manufacture than the 14.5-degree tooth form


Example 7.1

Design of Gears

A gear set consists of a 16 tooth pinion driving a 40 teeth gear. The diametral

pitch is 2. The addendum and addendum are 1/pd and 1.25/pd respectively.

Pressure angle is 20o.

(a) Compute the circular pitch, the central distance and the radii of base circles.

(a) In mounting these gears the central distance was incorrectly made larger. Compute the new values of the pressure angle and the pitch circle

diameters.


Tooth Systems

Design of Gears

Tooth System is a standard that specifies the relationships involving addendum,

dedendum, working depth, tooth thickness, and pressure angle

Standard and Commonly Used Tooth Systems for Spur Gears

Standardized by the American Gear Manufacturers Association (AGMA).

Standard makes it easy for design, production, quality assurance, replacement etc


Tooth Systems

Design of Gears


Contact Ratio

Design of Gears

Tooth contact begins and ends at the intersections of the two addendum circles

with the pressure line

o initial contact occurs at a and final contact at b

Tooth profiles drawn through these points intersect the pitch circle at A and B

o Distance AP is called the arc of approach qa

o Distance PB, the arc of recess qr


Contact Ratio

Design of Gears

o qa +qr = qt = Arc of Action

If arc of action is exactly equal to the circular pitch, i.e, qt = p, one tooth and its space will occupy the entire arc AB

when a tooth is just beginning contact at a, the previous tooth is simultaneously ending its contact at b


Contact Ratio

Design of Gears

Contact Ratio

Design of Gears

If arc of action is greater than the circular pitch, but not very much greater, say,

qt = 1.2p

one pair of teeth is just entering contact at a, another pair, already in contact, will not yet have reached b

Thus, for a short period of time, there will be two teeth in contact, one in the vicinity of A and another near B


Contact Ratio

Design of Gears

Term Contact Ratio mc is:

a number that indicates the average number of pairs of teeth in contact

Gears should not generally be designed having contact ratios less than about 1.20

A value less than 1 means that at times the teeth are not in contact (bad).


Design Considerations for a Gear Drive

Design of Gears

Following data is usually given for the design of Gear Drive:

1. power to be transmitted

2. speed of the driving gear

3. speed of the driven gear or the velocity ratio

4. center distance

Requirements must be met in the design of a Gear Drive:

o gear teeth should have sufficient strength under static loading or dynamic loading during normal running conditions to avoid failure

o gear teeth should have wear characteristics

o use of space and material should be economical

o alignment of the gears and deflections of the shafts must be considered to prevent their effect on the performance of the gears

o lubrication of the gears must be satisfactory


Spur Gear Design

Design of Gears

Spur Gear Tooth Force Analysis

Normal Force F can be resolved into two components

o Radial component Fr which does no work but tends to push the gears apart

o Tangential Force Ft which does transmit the power

Pitch Line Velocity V, in meters per second:

d: Pitch Diameter of the Gear (mm)

Power Transmitted:

N: Rotating speed in rpm

W: Power Transmitted in kW


Spur Gear Design

Design of Gears

Spur Gear Tooth Stresses

Two primary failure modes for gears are:

1) Tooth Breakage - from excessive bending stress

2) Surface Pitting/Wear - from excessive contact stress

In both cases, we are interested in the tooth load, which we got from the torque, T.

we compute the tangential force on the teeth as Wt = T/r = 2T/D

Lewis Equation for Tooth Bending Stress

Lewis considered gear tooth as a cantilever beam with static normal force W

applied at the tip


1. Full load is applied to the tip of a single tooth in static condition.

2. Radial component is negligible.

3. Load is distributed uniformly across the full face width.

4. Forces due to tooth sliding friction are negligible.

5. Stress concentration in the tooth fillet is negligible

Spur Gear Design

Design of Gears

Spur Gear Tooth Stressescontd--

Lewis Equation for Tooth Bending Stresscontd--

o Section BC is the section of maximum stress or the

critical section

o maximum value of the bending stress at the

section BC

Assumptions made in the derivation are:


Spur Gear Design

Design of Gears



M = Maximum bending moment at the critical section BC = WT h

WT = Tangential load acting at the tooth,

h = Length of the tooth

y = Half the thickness of the tooth (t) at critical section BC = t / 2

I = Moment of inertia about the center line of the

tooth = b.t3/12

b = Width of gear face


Spur Gear Design

Design of Gears



t and h are variables depending upon the size of tooth (i.e. Circular Pitch) and its profile

Let; t = x pc , and h = k pc ; where x and k are constants

Let , another constant

WT= w bYm

y is the Lewis Form Factor

(Y = y) Y is the Modified Lewis Form Factor


Spur Gear Design

Design of Gears



o y is a function of Tooth Shape (but not size) and vary with the number of teeth in the gear

As

if Gear is enlarged, distances t, h and pc

will each increase proportionately y will remain unchanged


Spur Gear - graph for Modified Lewis Form Factor

Spur Gear Design

Design of Gears



WT= w bYm

w= WT/(bYm)

Since Y is in denominator, bending stresses w are higher for the 14 pressure angle

teeth, and for fewer number of teeth

Stresses are lower for Stub form teeth than for full Involutes


Spur Gear Design

Design of Gears


Drawbacks of Lewis Equation

o Tooth Load in practice is not static, It is dynamic and is influenced by pitch line velocity

o whole load is carried by Single Tooth is not correct. Normally load is shared by teeth since contact ratio is near to 1.5

o Greatest force exerted at the tip of the tooth is not true as the load is shared by teeth. It is exerted much below the tip when single pair contact occurs

o Stress Concentration effect at the fillet is not considered


Spur Gear Design

Design of Gears

Permissible Working Stress for Gear Teeth in the Lewis Equation

permissible working stress (w) in the Lewis Equation depends upon the

material for which an allowable static stress (o) may be determined

Allowable Static Stress is the stress at the Elastic Limit of the material

To account for the Dynamic Effects which become more severe as the pitch

line velocity increases, the value of w is reduced

According to the Barth Formula, the permissible working stress:

if a pair of gears failed at 500 lbf tangential load at zero velocity and at 250 lbf at velocity V1, then a velocity factor, designated Cv, of 2 was specified for the gears at

velocity V1

o = Allowable static stress

Cv = Velocity factor


Spur Gear Design

Design of Gears

Permissible Working Stress for Gear Teeth in the Lewis Equationcontd--

v is the pitch line velocity in m/sec


Spur Gear Design

Design of Gears

Dynamic Tooth Load

Dynamic Loads are due to the

1. Inaccuracies of tooth spacing

2. Irregularities in tooth profiles

3. Deflections of teeth under load

WD = WT + WI WD = Total dynamic load WT = Steady load due to transmitted torque

WI = Increment load due to dynamic action

increment load (WI) depends upon the pitch line velocity, the face width,

material of the gears, the accuracy of cut and the tangential load

From Buckingham Equation:

WD = Total dynamic load in N,

WT = Steady transmitted load in N,

v = Pitch line velocity in m/s,

b = Face width of gears in mm, and

C = A deformation or dynamic factor in N/mm


From Buckingham Equation:

Spur Gear Design

Design of Gears

Dynamic Tooth Load

K = A factor depending upon the form of the teeth.

= 0.107, for 14.5 full depth involute system.

= 0.111, for 20 full depth involute system.

= 0.115 for 20 stub system.

EP = Young's modulus for the material of the pinion in N/mm2.

EG = Young's modulus for the material of gear in N/mm2.

e = Tooth error action in mm.

Maximum allowable tooth error in action (e) depends upon the pitch line

velocity (v) and the class of cut of the gears. Table 28.6 Book R.S. Khurmi

Static Tooth Load

Wear Tooth Load


Spur Gear Design

Design of Gears

Design Procedure for Spur Gears

Step-1: designed tangential tooth load is obtained from the power transmitted and

the pitch line velocity by using the following relation:

WT = Permissible tangential tooth load in N

P = Power transmitted in watts

v = Pitch line velocity in m/sec

D = Pitch circle diameter in meters

m = Module in metres

T = Number of teeth

N = Speed in r.p.m.

CS = Service factor


Spur Gear Design

Design of Gears

Design Procedure for Spur Gearscontd--

Step-2: Apply the Lewis equation


Spur Gear Design

Design of Gears

Design Procedure for Spur Gearscontd--

o Lewis Equation is applied only to the weaker of the two wheels (i.e. pinion or gear)

o When pinion and the gear are made of same material, then pinion is the weaker

o When the pinion and the gear are made of different materials, then Lewis Equation is used to that wheel for which (w y) or (o y) is less.

o Face width (b) may be taken as 3pc to 4pc (or 9.5 m to 12.5 m) for cut teeth and 2pc to 3pc (or 6.5m to 9.5m) for cast teeth

Step-3: Calculate the dynamic load (WD) on the tooth by using Buckingham

Equation


Example 7.2

Design of Gears

Spur Gear Design

A pair of straight teeth spur gear is to transmit 135 kW at pinion speed of

300 r.p.m. The velocity ratio is 1:3. The allowable stress for pinion and

gear materials are 1000 N/cm2 and 1200 N/cm2, respectively. The pinion

has 15 teeth and its breadth is 14 times the module. Assuming 20o full

depth involute system, determine the module, the face width and pitch

circle diameters of both pinion and gear, taking into account the effect of

the dynamic load


Example 7.3

Design of Gears

Spur Gear Design

A reciprocating compressor is to be connected to an electric motor with

the help of spur gears. The distance between the shafts is to be 500 mm.

The speed of the electric motor is 900 r.p.m. and the speed of the

compressor shaft is desired to be 200 r.p.m. The torque, to be transmitted

is 5000 N-m. Taking starting torque as 25% more than the normal torque,

determine :

1. Module and face width of the gears using 20 degrees stub teeth, and

2. Number of teeth and pitch circle diameter of each gear. Assume

suitable values of velocity factor and Lewis factor.


Design of Gears

Gear Trains

Gear Train is any collection of two or more Meshing Gears

Simple Gear Trains

Each shaft carries only one gear

Expression for Trains Velocity Ratio:

o For Simple (Series) Train, numerical effects of all gears except the first and last cancel out, i.e., Train Ratio is just the ratio of first gear

over the last

o Intermediate gears are the Idlers, which changes only the sign of the overall Train Ratio

o If all gears in a Train are External and Even in numbers, Output direction will be opposite to that of the input

o If all gears in a Train are Odd in numbers, Output direction will be in the same direction as the input

o Simplest form of gear train is usually limited to a ratio of about 10:1


Design of Gears

Gear Trains

Compound Gear Trains

Gear Train in which at least one shaft carries more than one gear

o Parallel or Series-parallel arrangement

o Expression for Trains Velocity Ratio:

Generalized Expression for Trains

Velocity Ratio:

o Intermediate Ratios do not cancel each other and overall train ratio is

the product of the ratios of parallel

gear sets

Pitch Diameters can be used in Eq.

mV is + if last gear rotates in the same

sense as the first, and -ve if last rotates

in opposite sense


Final Exam Pattern

Total Marks: 100

Subjective

Numerical Problems,

Derivations,

Short Questions

Ch: 1, 2, 4 20-30 Marks

Ch: 5 20-30 Marks

Ch: 6, 7 50 Marks

Max. Time: 3 hrs

Formula sheet, standard tables etc. will be given within the Question Paper

Students are not allowed to bring any Formula Sheet/Table with them

design of gears

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