Gear 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 30o) 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 30o 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.


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

Geartooth nomenclature

The pressure angles are normally kept between 14.5 - 20o . 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.


The backlash in a tooth is limited to oil film thickness and also to allow movement to alleviate problems caused by;

Angular movement


Flexibility within gear set


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.

The factors which govern the suitability of a material are;

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.

Advantages of Double helical Gears


The main advantage is that the double helical gear does not have end thrust However they do take more time to manufacture and are slightly heavier   

Brown Boveri Thrust Cone

This is a method of absorbing end thrust in single helical gears without resorting to large thrust bearings. This design is seen insmall steam turbine generator sets.

With the cone system there is a line of contact and a very large relative radius of curvature with a large oil entraining velocity of 220 ft/s .There is thus considerable axial resilience with the large radius of curvature, a small radial width of cone is sufficient to take the thrust



Comparative simplicity in grinding No gap, low helix angles - 15 '

Longer grinding times , normal gap normal helix angle - 30 '

Complete absence of pinion shuttling obviates the use of sliding couplings Apex wander due to different composite pitch errors cause shuttling.Gear tooth couplings do not respond because of the high frictional loads. Best compromise is to use axially flexible couplings. With highly accurate gear manufacture this effect is small

Axial thrust on primary high speed pinion unless taken by turbine thrust bearing can lead to high losses if flooded thrust pads are used. The use of brown boveri thrust cones can be used to overcome this problem.(see Below)

No axial thrust and no high speed thrust bearings required. Final reduction wheel located by propeller thrust bearing

Ball and roller bearings may be used to take end thrust

Quill shafts can be solidly coupled to primary wheels and secondary pinion. The helix angle on each being arranged to balance the axial thrusts.

Simple side bearings serve to locate the shafts . The axial thrust of the final reduction wheel being carried by the propeller thrust bearing.

Axial tilting moment on wheels generally negligible.

 No tilting moment

Small helix errors can be perfectly corrected . Allows tooth helix angle adjustment to negate bending , torsional and heating effects and hence balance loading across the teeth. Helix errors can be adjusted in a similar way, but not so perfectly as for single helical

Gear Tooth Angle Correction

When a pinion having a uniform meshing at no load, is torqued at one end, it bends and twists according to a known algebraic combined deflection ,the load distribution is proportional to the tooth deflection.

New load distributions can be calculated which can take into account alignment , bearing flexibility and thermal effects.

It can be shown that the tooth separation for double helix gears is less than that for single helix gears.

Temperature effects

Usually the pinion operates at a higher temperature than the wheel. The pinion will expand and hence the pitch will change. The change in axial pitch is most important as this wears the teeth at one end of the helix

With apex trailing , the teeth bear hard on the inner ends and with apex leading the teeth bear hard on the outer ends.

Apex trailing is advantageous as apex leading teeth tend to compound the effect of heat distortion into the torque distortion