lecture 9 10th march

8/12/2019 Lecture 9 10th March

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Bevel gears have teeth shapedgenerally like ordinary spur

gears, except that the toothsurfaces are made up of conicalelements.

The teeth may be straight or

spiral Spiral teeth engagegradually (starting at one side,as with helical gears), a featureenabling them to operate moresmoothly and quietly.

Except for hypoid gears bevelgears are mounted on shaftshaving intersecting axes. Theshaft axes are usually, butnot necessarily, perpendicular.



When the pitch angle is equal to 90 it is called crown bevel gear

These bevel gears are mounted on shafts which are intersecting at

an angle that is more than 90 . Crown gear is equivalent to rack.

The pitch angle of a bevel gear is the angle of the pitch cone. It is ameasure of the amount of taper in the gear.

Bevels gears are classified on the basis of pitch angle

When the pitch angl e is less than 90 it is called external bevel gear.

Miter gears: Identical bevel gears are mounted on shafts that are

intersecting at right angles, pitch angle of both pinion and gear are 45 .

Pinion and gear have same dimensions(adde, ded, pcd no of teeth and

module) and rotate at same speed

When the pitch angle is more than 90 it is called internal bevel

gear

Teeth of bevel gears are cut on the inside of the pitch cone. In

this case, the pitch angle of internal gear is more than 90 and the

apex point is one the back side of the teeth on that gear. Theseinternal gears are used in planetary gear trains



Straight bevel gears: Teeth elements are straight lines which converge into a common apex

point

Spiral bevel gears: Teeth elements are spiral curves which also converge into a common apex

pointSkew bevel gears: Shaft on which straight bevel gears are mounted are non parallel and non-

intersecting. Apex point of the pinion is offset with respect to that of gear

Hypoid gears: Similar to spiral bevel gears that are mounted on shafts which are non parallel and

non intersecting. Used in automobile differential which allow the drive shaft to place well belowthe centerline of the rear axle thereby lower the center of gravity of vehicle

Face gears: Consists of spur or helical pinion mating with a conjugate gear of disk form



The tooth dimensions are usually specified for thelarge end of the tooth. However, in calculatingbearing loads, central-section dimensions and forcesare used

The simplest type of bevel gear is the straight bevel gear. These gears are commonly used for transmitting power between intersecting shafts. Usually the shaft

angle is 90°, but it may be almost any angle. The gearsimpose both radial and thrust load on their bearing.

In contrast to the cylindrical pitch surfaces of spur and helical gears, the pitch surfaces of

meshing bevel gears are conical. When teeth are cut on the conical pitch surfaces they must be

tapered both in tooth thickness and height from a large tooth profile at one end to a smaller tooth

profile at the other end

Bevel gear teeth have profiles that closelyresemble an involute curve. The shape of astraight bevel gear tooth (in a section normal tothe tooth) closely approximates that of aninvolute spur gear



The practice of characterizing the size and shape of bevel gear teeth as those of an

imaginary spur gear appearing on the developed back cone corresponds to what is

known as Tredgold’s approximation

When intersecting shafts are

connected by gears, the pitch cones

(analogous to the pitch cylinders of

spur and helical gears) are tangent

along an element, with their apexes

at the intersection of the shafts. The

size and shape of the teeth are

defined at the large end, where they

intersect the back cones. Note that

the pitch cone and back cone

elements are perpendicular.



z’=2r b/m= z/cos

Virtual number of teeth z’= 2r b/m

where m is the module at the large end the tooth

If z is the actual number of teeth onthe bevel gear , then z= D/m

z’/z = 2r b/D

A line AB is drawn perpendicular to line OC

From triangle ABC

sin BCA= AB/AC

sin(90- )=(D/2)/ r b

r b= D/2cos



D p and D g are the pitch circle diameters of pinion andgear respectively. is the pitch angle of the pinion,while is the pitch angle of the gear. Line AB is

perpendicular to line OB. Consider triangle OAB

tan =AB/OB; (D p/2)/ (D g/2)= D p/Dg= mZ p/mZ g

tan = Z p/Zg tan = Z g/Z p + =90

The cone distance A O is given byOA=(AB 2+OB 2)1/2 = (D p

2+D g2)1/2



Figure shows the resolution of resultant tooth force F into its tangential(torqueproducing), radial (separating), and axial (thrust) components, designated Ft , Fr

, and Fa, respectively. Note that an auxiliary view is needed to show the true length ofthe vector representing resultant force F (which is normal to the tooth profile).Resultant force F is shown applied to the tooth at the pitch cone surface and midwayalong tooth width b. This conforms to the usual assumption that the load is uniformlydistributed along the tooth width, despite the fact that the tooth is largest



In force analysis, it is assumed that theresultant force between two meshingteeth of a pair of bevel gear isconcentrated at the midpoint along theface width of the tooth

Consider a plane ABCD shaded by dots

and fig b. From triangle BCDtan =BC/CD=P s/P tPs=Separating component

=Pressure anglePs=p t tan

Resultant force P shown by dotted line actsat the midpoint D of the face width of the

pinion. The resultant force has followingthree components

Pt= tangential componentPr =radial componentPa = axial or thrust component



The separating force P s is perpendicular to the pitch line OD, Therefore AD OD

Line FD is vertical ,while line OX is horizontal, Therefore, FD OX

There are two pairs of perpendicular lines and theirincluded angle should be equal. The angle betweenlines OD and OX should be equal to the angle

between the lines AD and FD. The angle betweenlines OD and OX is the pitch angle . Therefore theangle between lines AD and FD should be equal to the

pitch angle

Consider the plane DEAF, from triangle ADF

Pr =P scos

Pa=P ssin

The tangential component is determined from relation, P t= M t/r m

M t: Torque transmitted by gearsr m: radius of the pinion at midpoint along the face width

Since P s=P t tan

Pr = P t tan cos

Pa= P t tan sin



AC= D p/2

sin = AB/(b/2); AB=(b/2)sin

r m=[(D p/2)-(bsin /2)]

b : face width of the tooth

As seen in fig c, the radial component on the gear is equal to the axial

component on the pinion. Similarly the axial component on the gear is

equal to the radial component on the pinion

A two dimensional

representation of force

components is illustrated. In

figure b, the mean radius r m

where the resultant force acts

is given by

r m=AC-AB



In two dimensional representation of forces, very often (.) indicates a force that is perpendicular to the plane of paper and which is towards the observer. (x) indicates aforce that is perpendicular to the plane of paper and which is away from the observer

Tangential component(P t)

The direction of tangentialcomponent for driving gear isopposite to the direction of rotationThe direction of tangentialcomponent for driven gear is same as

the direction of rotationRadial component(P r )

The radial component on the pinionacts towards the centre of pinionThe radial component on gear acts

towards the centre of gear

Thrust components(P a):The thrust component on pinion is equal and opposite of the radial component onthe gearThe thrust component on gear is equal and opposite of the radial component on the

pinion

lecture 9 10th march

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