Bevel Gears Derived from pitch surfaces which are

frustums of cones

Elements of the pitch cones intersect at the point of intersection of the axes of rotation

This is the condition for proper rolling

(a) Cones can roll together without sliding

(b) Pure rolling possible at one point only

Must have sliding at all other points of contact

Bevel Gears (Contd…) Bevel gears are commonly used to connect shafts intersecting at 90º

Miter gears have equal pitch cone angles (Γ = 45º) Cones are identical in this case

A crown gear has its pitch angle 90º Crown gear corresponds to a rack in spur gearing

(a) Miter Gears (b) Bevel Gears with Σ= 90º Crown Gear

Bevel Gears (Contd…) Involute teeth for a spur gear can be

generated by the edge of a plane, as the plane rolls on a base cylinder Involutes lie on the corresponding planes

Similar analysis for a bevel gear shows that a true section of the resulting involute lies on the surface of a sphere Cannot be accurately represented on a

plane surface In practice, most bevel gears are generated

so that they are conjugate to crown gear teeth with flat sides

The true shape of the bevel gear tooth is obtained by taking a spherical section through the tooth, where the centre of the sphere is at the common apex

Tredgold’s Approximation

Tredgold’s Approximation is used to represent the tooth profiles approximately It is based on the fact that a cone tangent

to the sphere at the pitch point will closely approximate the surface of the sphere for a short distance either side of the pitch point

This cone, known as the back cone can then be developed as a plane surface and an equivalent spur gear tooth system can be drawn

The form of teeth formed based on Tredgold’s Approximation depends upon the slant height of the back cone (not on the radius R) Equivalent pitch radius, Re = R / cos Γ

Tredgold’s Approximation (Contd…)

The equivalent number of teeth ze is,

The action of the bevel gears will be the same as that of the equivalent spur gears Since the equivalent number of teeth is

always greater than the actual number of teeth, a given pair of bevel gears will have a larger contact ratio and will run more smoothly than a pair of spur gears with the same number of teeth

Γ=

Γ==

coscos22 z

mR

mRz e

e

Equivalent Spur Gears

Bevel Gear Nomenclature Bevel gears are not

interchangeable. They are made in pairs.

Pitch diameters are measured at the large, or heel ends of the teeth.

20º teeth are most widely used. Min. number of teeth≈13.

Pitch and root cones intersect at the pitch apex, or shaft intersection, while the face cone does not. Face cone of each gear is

turned parallel to the root cone of the other. This gives a constant clearance.

tan Γ2 = z2 / z3

Γ3 = Σ−Γ2

Cone distance = L ; Face = b ; Pitch diameter = d2 or d3

Force Analysis of Bevel Gears Tangential force, Wt , is

assumed to act at the mean pitch diameter

Radial force is given by,

Axial force is,

Γ= costanφtr WW

Γ= sintanφta WW

Design of Bevel Gears Design is based on Lewis equation for beam strength modified

as,

Face width is generally taken as, b is usually taken close to but not greater than L/3

y corresponds to the equivalent number of teeth

The factor Cv is taken as,

for cut teeth, and for generated teeth

The general design rule is,

mL

bLybFb

−

= πσ 0

34LbL

≤≤

V+66

V+6.56.5

Fb Cv ≥ Wt

Dynamic Load

Dynamic load can be calculated as, Fd = Ft + Fi , where,

with usual notations (as for spur gears)

C corresponds to that for spur gears

For safe design, Fd ≤ Fb

where, Fb = σ0 b y π m (L-b)/L

( )t

ti FbCV

FbCVF++

+=

2121

Wear Strength

Wear strength can be approximated to be

, with usual notations

Q = 2 z2e / (z1e + z2e) , z1e , z2e are equivalent tooth numbers on pinion and gear, respectively

Γp is the pitch angle of the pinion

K is the same as that for spur gears

For safe design, Fd ≤ Fw

p

pw

KQbDF

Γ=

cos75.0

Example – 1

Design a pair of bevel gears to transmit 9 kW between two shafts whose axes intersect at 90°. The pinion rotates at 1200 r.p.m. and the speed ratio is 3. The pinion has 20 teeth and is made of steel with an allowable bending stress, σ0 of 85 MPa. The gear is made of cast iron with σ0 = 55 MPa. Tooth profile is 20° involute. Also check the design for dynamic load and wear.