Toothed gears

• Toothed gears are used for transmission of power from one shaft to another shaft.

• Used when the distance between the driving and driven shafts is relatively small (compared and driven shafts is relatively small (compared to belts and chains).

• Used to transmit power from one rotating shaft to another and shaft may be parallel, intersecting, or skew under the following conditions:

• Used to transmit power from one rotating shaft to another and shaft may be parallel, intersecting, or skew under the following conditions:– Distance between the axes of the connecting shafts is

short

– Speed of the shaft is low and the belt drive is not recommendedrecommended

– Speed ratio of connecting shafts is to maintained constant

– Torque to be transmitted is high

– For different speed ratios

Two discs for power transmission

On account of friction between the surfaces of

discs in contact, power was transmitted. Later

teeth were formed below and above the surfaces

in contact to avoid the slip during larger power

transmissions.

With acknowledgements…

Types of gears

• Spur gears:

– to connect parallel shafts

– teeth are parallel to the axis of gears

– Used as sliding gears for speed change mechanisms

in gear boxes.in gear boxes.

– Noisy at high speeds

• Helical gears

– Used in the same way as spur gears

– Teeth cut on the periphery of the disc are of helical or

screw form

– Mating gears must have same helix angle but opposite

hand

– Tooth loading produces axial thrust

– Operate with less noise and vibration than spur gear

• Double helical gears– have both left-hand and right-hand helical teeth

– double helical form is used to balance the thrust forces

• Bevel gears– To connect shafts whose axes are intersecting and coplanar.

– equivalent to frusta of cones with their apices meeting at a point.

– Common type is that where teeth are radial to the point of – Common type is that where teeth are radial to the point of intersection of the shaft axes of apices and such gears are called straight bevel gears.

– Shaft angle may be from 0 to 180 degrees but most commonly encountered is 90 degrees

– Bevel gears having 90 degrees and giving equal speed are known as miter gears

– Teeth are smaller at the front than at the rear end.

– The teeth of bevel gears can also be cut in a curved manner to produce spiral bevel gears, which produce smoother and quieter operation than straight cut bevels.

• Spiral gears

– To connect two non-intersecting and non-coplanar shafts

– Kinematically equivalent to hyperboloids of revolution. Thus, there is a point contact between two gears.

• Worm gears:

– To connect skew shafts as spiral gears

– Worm exactly resembles one of a pair of spiral gears, but – Worm exactly resembles one of a pair of spiral gears, but the wheel is throated and has concave teeth.

– Normally, the two shafts are at right angles to each other

– Will tolerate large loads and high speed ratios.

– Meshes are self locking.

– The sliding velocity of the worm gear is high compared to other types of gears.

• Rack:

– Racks are straight gears that are used to convert

rotational motion to translational motion by

means of a gear mesh.

With acknowledgements…

Basic terms• Pitch circle diameter=d

• Pitch point

• Circular pitch, Pc=πd/T

• Diametral pitch, Pd=T/d

• Module, m=d/T

• Addendum circle

• Addendum: height of the tooth above the pitch circle = one module

• Dedendum circle

• Dedendum: the depth of a tooth below the pitch circle= 1.157 module• Dedendum: the depth of a tooth below the pitch circle= 1.157 module

• Clearance: Difference between dedendum and addendum of a tooth

• Pressure angle: the angle between the common normal at the point of contact and the common tangent at pitch point. The force transmitted acts along this normal. Its component along the pitch circle is the useful power component and it will be maximum when the pressure angle is minimum.

• Path of contact: it is the locus of the point of contact of two teeth from the beginning of engagement to the end of engagement.

• Arc of contact: It is the locus of a point on the pitch circle, from the beginning of engagement to the end of engagement of a pair of teeth in mesh.

pitch circle

Common tangent at

pitch point PA

BP

�

pitch circlecommon

normal

G H

Condition for correct gearing

P

QZ

12

• For correct gearing, the common normal at the point of contact would divide the line joining the centres in the inverse ratio of their angular velocities.

• If the velocity ratio is to be constant, Z must be a fixed point. So, the pitch circles drawn with radii ZP and ZQ will have constant velocity ratio as Z is pitch point.

• Hence, the condition for correct gearing is that the common normal at the point of contact between a pair of gear teeth in mesh must pass through the pitch point.

Curves of teeth• In practice the following geometrical

curves satisfy the condition for correct gearing.– Cycloid and its variants

– Involute

A cycloid is the locus of a point on the circumference of a circle, which rolls without slipping on a fixed straight line.line.

An epicycloid is the locus of a point on the circumference of a circle, which rolls without slipping outsideanother circle of finite radius.

An hypocycloid is the locus of a point on the circumference of a circle, which rolls without slipping insideanother circle of finite radius.

Involute curve

• An involute is the locus of a point on a straight line which rolls on the circumference of a circle without slipping.

• The circle on which the straight line rolls is called as base circle.

• The important property of an involute which makes it suitable for teeth of gears is that a normal to the involuteat any point is a tangent to its base circle.

involute

• Advantages:

• centre distance for a pair of involute gears can be varied within limits without changing the velocity ratio.

• The pressure angle remains constant which is necessary for smooth running and less wear of gears.

• easy to manufacture than cycloidal teeth.

• Disadvantages: the interference occurs with pinions having smaller number of teeth.

Base circle

Characteristics of involute teeth• As stated, a tangent to the base

circle is a normal to the involute.

• Hence, a common tangent to the

two base circles, is a common

normal to the involutes at the

point of contact on the teeth

profile of the gears in mesh.

�

�

Common

tangent

Or

Common profile of the gears in mesh.

• Thus the path of contact is a

straight line .

• So, the pressure angle for involute

gears is constant. This angle lies

between 14.5o to 22.5o, the

normal value being 20o.

�

Base circle

Pitch circle

tangent

O1

R

Common

normal

Interference in involute gears

Dedendum

Base circle

Dedendum circle

Base circle

Dedendum

circle

•The relative positions of dedendum and base circle vary with

number of teeth, diametral pitch and pressure angle as regards

base circle being smaller or larger than the dedendum circle.

•For the gears having small number of teeth, the dedendum circle

is small in diameter than its base circle as shown in the figure.

Interference in involute gears…

• The relative positions of dedendum and base circle vary with number of teeth, diametral pitch and pressure angle as regards base circle being smaller or larger than the dedendum circle.

• For the gears having small number of teeth, the dedendum circle is small in diameter than its base circle as shown in the figure.circle as shown in the figure.

• In such case, the tip of a tooth of the mating gear digs into the portion lying between the base circle and the dedendum circle and for this reason the teeth have to be undercut. This phenomenon is known as interference.

• Interference can be avoided if addenda circles cut the common tangent to the two base circles within points of

�

�

Common

tangent

Or

Common

Q

within points of tangency.

• Under limiting conditions, the addenda will pass through the points of tangency, P or Q.

�

Base circle

Pitch circle

tangent

O1R

Common

normal P

• Minimum number of teeth on wheel to avoid interference for the given values of gear ratio G, pressure angle � and the addendum coefficient aw. G =T/t

Minimum number of teeth on pinion for involute rack for

addendum coefficient of rack ar and pressure angle �

Path of contact

The length of path of contact is AB

r – radius of pitch circle of pinion

R – radius of pitch circle of wheel

r – radius of addendum

�

�

Common

tangent

Or

Common

B

A

P

Addendum

circle

Addendum

circle

ra – radius of addendum circle of pinion

Ra- radius of addendum circle of wheel

� - Pressure angle

�

Base circle

Pitch circle

O1R

Common

normalA

Arc of contact

Common tangent at pitch

point P

A

B

P�

pitch circle

pitch circle

G H

KJ

pitch circle

common

normal

Length of arc of contact GH is given by

K

• Number of pairs of teeth in contact

For continuous transmission of motion, at least • For continuous transmission of motion, at least one pair of teeth of mating gears must be in contact. Therefore, number of teeth in contact must be greater than unity.

CLARIFICATIONS

1. sir the law of gearing state that the passage of e

meshing of gears must pass through the pitch point ,

but in the practice the meshing of the gears will be

the point contact at every instant of the mating of

the gears from the beginning and the end of the the gears from the beginning and the end of the

engagement which violates the law of gearing

• SirIf crank and connecting rod is at '0' degrees or 360 degrees , crank and connecting rod distance is equal , the angular acceleration is minimum I think so, Is it right when crank and connecting rod is at 0 or 360 degrees , if the crank and connecting rod distance is not same . In this case also, the value of angular acceleration is minimum or slightly greater than minimum value.Is this statement is rightIs this statement is right

• θ =0 f=2rω2