Tooth design

Forces acting on Spur Gears

T= Force transmitted due to torque (torque/pitch radius)
P= Actual Force
S=Force tending to cause seperation of wheel and pinion

Forces all act in the same plain as the pitch circle surface diametral plane. The force that must be transmitted by the gearing is that related to the power developed in the turbine P=2πnT hence for the same power the torque is inversely related to the speed of transmission.
This resultantforce P = Tcosθ is found on both driving and driven teeth.
Straight ( spur ) gear teeth meshing is accompanied by impact as the load is transmitted from tooth to tooth. No more than 1.2 to 1.4 teeth are in mesh at any one time.

Forces acting on Helical Gears

For helical gears the force triangle is inclined to the diametral plane. An additional component acts along the shaft.
It is normal, for large gear sets, to have a second attached wheel with teeth angled opposite to the first to cancewl out this component.
As the pitch circle is now in the form of an ellipse it is now necessary to resolve the angles in the normal and diametral plane to find a new pressure angle so the forces can be resolved in the diametral plane.
This can be shown to give the formula

θ' = tan^-1(tanθ / Cos α)

θ' = Pressure angle in diametral plane
θ = Pressure angle in normal plane
α = angle of helix

As cosα is less than 1 then θ' is always greater than θ hence the actual loading on a tooth is increased slightly for the transmission of the smae force.

The angle of helix given to helical gears (about 30^o) is to ensure that one end of a tooth engages before its preceding teeth has disengaged. In this way several teeth may be in mesh and smooth transfer of load is allowed. The axial loading caused by this type of mesh is countered by having back to back opposite hand teeth.

Due to unbalanced axial loads caused by irregularities in the manufacturing process and wear, the gear teeth tend to shuttle and flexible coupling arrangements must be able to cope.

Gear Bearing Load

The forc P must be carried by the bearings. Additionally the weight of the gear wheel must be carried. By resolving the force triangle the resultant magnitude and direction of force may be calculated. The bearing split on some gearbox designs are angled to be at right angles to the resultant force direction under full load. Oil supply holes are provided well away from the direction of load. Relief and oil channels are provided to carry the oil to the load point. The length to diameter ratio is approx. 2/3
Fo the main gear wheel which may have more than one pinion a polygon of forces must be resolved at the wheel centre to determine resultant ahead and astern load on the bearings

Construction of Primary Wheel

The wheel cenrte is forged integral with the shaft. Wheel is stiifened by a number of axial steel tubes welded to the side plate. This type of construction is resistant to vibration.
No key is fitted

Tip relief

Some early methods of gear cutting led to a lack of uniformity between the start and end of the helix. Teeth relief is given to prevent shock loading caused by this. Some teeth relief is also given to reduce loading and prevent subsequent breakdown of the oil film. Too much tip relief reduces the effective depth to a point where the number of teeth in contact is reduced. Also due to the distortion of the Torque twist and bending due to the tooth load and bearing reactions the load tends to be thrown towards the outer edge of the tooth. Hence, the ends of the teeth are chamfered to 30^o both from tip to root but also the tooth width is reduced by chamfer to about half root width .

Tooth cutting process

The gear teeth are cut in a separate room which is kept at constant temperature. They are hobbled, then they are shaved ( a scrapper takes off very fine slivers and is free to follow the tooth form )

The ends and tip of the teeth are relieved.

Pinion and wheel are arranged so as not to be as multiples of each other e.g. if ratio 10:250 was required the designer would use 10:251 so that there where many revolutions before two teeth repeated a mesh

Involute shape

Often described as the form a the end of a taut string on a drum follows when it is unwound.

This form gives a strong root section, improving the resistance to bending whilst being able to tolerate a degree of misalignment.

Nomenclature

It can be seen that increasing the distance between the centres of the gears will not change the gearing ratio but will change the pressure angle.

The pitch circle can be related to the diameter of a drum with no gear teeth when related to gear speeds and hence gear ratio's.

n1/n2 = d1/d2 = N1/N2

n = speed of rotation
d = diameter of pitch circle
N = Number of teeth

Naming convention for gear tooth

Geartooth nomenclature

The pressure angles are normally kept between 14.5 - 20^o . Too high and it tends to produce sharp pointed teeth of increased pitch. Too fine and it tends to produce undercutting.

The root circle must be at a radius greater than the base circle for the tooth shape to fall on the involute curve. By definition, any undercutting below the base line cannot be of the involute shape i.e. the involute curve is generated from the base circles of both gears. For pinion of the all addendum form the root circle is pushed out to the base circle so all of the tooth is of the involute shape. The mating teeth are then all addendum. the teeth engage with pure rolling action at the pitch circle and are only in contact during the arc of recess with the relative sliding in one definite direction over the whole tooth

The addendum and deddendum for the pinion and wheel are made different to give the clearance. This allow oil to become entrapped flow around and out giving a cooling effect. Also it allows debris to be washed out. The provision of the clearance also allows fillets to be introduced into the base of the teeth without causing interference.

Modified teeth ( not normal nowadays)

The pinion, being subjected to the highest stress fluctuations is more likely to fail. Hence the pinion may be given a positive Addendum Modification to increase the thickness of the root thereby reducing bending stresses. This is especially seen on pinions with a small number of teeth to avoid undercutting If the pinion was made with all addendum, the arc of contact would be reduced and the wheel would require all deddendum teeth profile. This gives a very thick root form for the pinion , this is particularly seen on nested gears.

BACKLASH

Angular movement
Expansion
Flexibility within gear set

MATERIALS IN USE

Up until 20 yrs ago 'through hardened' materials were widely used and still are but less frequently. These are carbon steel wheel rims and nickel steel pinions.

Surface strength,i.e. resistance to pitting and flaking. This has found to increase with tensile strength but only to a point with fatigue strength.
Tooth bending fatigue strength i.e. the ability to resist fracture at or about the root due to the cyclical application of loads
The ability to resist scuffing and scoring during short term lubrication failure, and a resistance to wear.

The ideal was to have both wheel and pinion carburised then machined to remove imperfections caused by the carburising process. Here materials are held at 900'C in carbon rich atmosphere. However this is expensive and difficult to carry out on large wheel rims.

Heat treatment is carried out after the hardening process.

Balance is to use the hard on soft principle, after hobbing and shaving only the pinions are hardened by nitriding (the shaft is heated in an atmosphere containing free hydrogen, created by heating ammonia to 500'C)
Nitriding created little distortion hence making grinding unnecessary.

graph showing the different loading capacities for various hardeneing processes

Providing proper design and manufacture and adequate lubrication the surface should tend to work harden and provide a polished surface. As the surface improves so does friction and wear. Pinion may be 3% nickel, chromium, molybdenum steel. Wheel may be 0.4-0.5 % carbon steel