Gear Design

To avoid the slipping

Exact velocity ratio  Transmit large power  Used for small centre distances  High efficiency  Reliable service  Compact layout Require special tools and equipments to produce  Improper cutting of teeth produce vibration and noise  Lubrication is must

TERMINOLOGIES USED IN GEARS

Driver pinion

Driven gear wheel

Arc of contact: Path traced by a point on the pitch circle from the beginning to the end of the engagement of a given pair of teeth. It consists of two parts.

Arc of approach: Portion of the path of contact from the

beginning of engagement to the pitch point.

Arc of recess: Portion of the path of contact from the pitch point to the end of the engagement of a pair of teeth.

Line of Action

Arc of approach

Arc of recess

Spur gear

Double helical or Herringbone gear

Helical gear

Cross helical gear

Straight bevel gear

Worm and worm wheel

Spiral bevel gear

Rack & Pinion

Pressure Angle or Angle of Obliquity: Angle between common normal to two gear teeth at the point of contact (line of contact) and the common tangent at the

pitch point.

Standard values include 14.5, 20 and 25 degrees.

Backlash: It is the difference between tooth space and the tooth thickness as measured along pitch circle. Theoretically backlash should be zero. But in actual practice some backlash must be allowed to prevent jamming of the teeth due to the tooth errors and thermal expansion.

Module (m): Pitch diameter divided by number of teeth. The pitch diameter is usually specified in inches or millimeters;

It is a measure of the tooth strength. Higher the module bigger the size of the gear. More important, higher the module, wider the tooth at the base and larger the height of the tooth.

Main parameters to be designed:

Center Distance Module

Face width

Let Mt be the torque transmitted by the pinion Normal force on the tooth Fn = Torque / lever arm Radius of the base circle = ½ d1 cos α

1

Diameter d1 can be expressed in terms of center distance ‘a’ as

Substituting the value of d1 in Eq. 1

 Therefore, Fn is inversely proportional to the centre distance.

 As center distance decreases the normal force will increase and hence the surface compressive stress

increases.  Therefore the centre distance is limited by the permissible surface compressive stress of the

material of the pinion.

Let q be the unit load i.e. Load per unit face width b

Though the normal load is shared by more than one pair of teeth for conservative design it is assumed that that the full load is taken by one tooth. This can be taken as the allowance for tooth inaccuracy and deflection of the tooth under load.

STRESS CONCENTRATION:

Load is not uniformly distributed over the line of contact due to: Deformation of tooth Change of mutual position Inaccuracies in machining the gears, shafts and housing Assembly errors Let qmax be the maximum load

k is the load concentration factor

DYNAMIC LOAD:

Inaccuracies in contact due to inaccuracies of tools and tooth cutting machines lead to jerky motion of the gears. This causes angular accelerations and decelerations of the gears even though the angular

velocity is constant. Shock is also possible because of tooth deformation under load and change in pitch. This shock gives rise to an additional load on the tooth called dynamic load. Dynamic load depends on  the degree of accuracy of the teeth  stiffness of the teeth  surface hardness  pitch line velocity

Fd= Fn +Fi

SURFACE COMPRESSIVE STRESS When two cylindrical surfaces of radius of curvature ρ1 and ρ2 are in contact and subjected to unit load qmax, the surface stress is given by

Substituting the values of ρ1 and ρ2 in Eq. 2 equivalent radius of curvature

Substituting the values of q max and ρ in Eq. 2

IMPORTANT POINTS TO BE NOTED:

Minimum centre distance depends upon the surface compressive strength of the material

So induced surface compressive stress < Design surface compressive strength of the material

Minimum module depends upon the bending strength

So induced bending stress < Design bending strength of the material

Design surface compressive strength [σc] Surface strength is proportional to the hardness of the surface. σc α HB or RC

Therefore, σC = CB × HB N/cm2 = CR × RC N/cm2 CB and CR are constants depending on the material and heat treatment.

Also the design compressive strength depends on load conditions. Hence correction factor is introduced. Design surface compressive strength

[σC ] = CB HB Kcl in N/cm2 [σC] = CR HRC Kcl in N/cm2

Where Kcl is the life factor for surface compressive strength. 𝟏𝟎𝟕 𝑵

Kcl = Where N – number of fatigue cycles the pinion teeth has undergone in its life period of T hours. Number of fatigue cycles per hour = 60 n

Number of cycles in life period N = 60 n T

Design Bending Stress [σb] It depends on endurance limit, stress concentration factor at the root and life factor for bending.

[σb] =

𝝈−𝟏 𝒌𝒃𝒍 For gears having both directions of rotation 𝒏𝒌𝝈

[σb] =

𝝈−𝟏 𝒌𝒃𝒍 × 𝒏𝒌𝝈

1.4

For gears having one direction of rotation only

𝝈−𝟏 = endurance limit stress in bending 𝒌𝒃𝒍 = life factor in bending 𝒌𝝈 = stress concentration of fillet at the root n = factor of safety

Gear materials Commonly used materials  cast iron and steel For large power transmission and reduction in size  Alloy steel of Nickel, chromium & vanadium (with proper heat treatment to obtain sufficient surface strength) For corrosive environment  Brass and bronze

Non metallic materials  Laminated fabric, Bakelite and mica (to reduce noise)

Gear Failures Teeth breakage: due to fatigue Pitting: hard and smooth working surfaces of the teeth reduce the danger of pitting Surface abrasion: due to sliding of the teeth Seizure: surface of the teeth mesh so tightly together causes particles of softer material to break away from the teeth surface and groove it.

Law of gearing The common normal to the tooth profile at the point of contact should always pass through a fixed point, in order to obtain a constant velocity ratio.

Only involute and cycloidal curves satisfies the fundamental law of gearing.

Involute Profile

Cycloidal Profile

Helical Gears and Herringbone Gears They have teeth cut in the form of helix on their pitch cylinders. Teeth are not parallel to the axis of rotation.

More than one pair of teeth are in engagement. Runs smoothly because of the gradual engagement of teeth. Higher peripheral speeds are permissible in

helical gears.

Limitation: Axial thrust

By providing another helical gear of opposite hand, the axial thrust can be balanced. They are called as double helical or herringbone gears.

Helix angle is helical gears are in between 8° and 25°

Axial pitch = π m , where m is the axial module Normal pitch = π mn , where mn is the normal module

Cos β = π mn / π m Therefore,

Cos β = mn / m

Centre distance =

Forces acting on a Helical gear tooth

Virtual Number of Teeth