When two gears make contact at the Pitch Point, the areas of contact on the face of each gear will have the same instantaneous tangential velocity Vt.

The smaller gear (the pinion) has a smaller radius but a higher angular velocity, that will match the larger gear radius but lower angular velocity.

Hence there is no sliding between the involutes at the Pitch Point.

P

Gears their designs & uses

Andrei Lozzi 2012

P

As the point of contact between the gear teeth moves away from the Pitch Point P the tangential velocities of the points on each gear will change in magnitudes and direction Vt1 & Vt2.

Since instantaneously the two points shared the same location in space there must have been an additional relative velocity Vs that when added (or subtracted) to one of those tangential velocity would give the other.

This additional velocity is parallel to the involutes at the point of contact and hence represents pure sliding.

The further from the pitch point the greater will be the sliding

Tooth size, strength & friction

A tooth may be viewed as a short beam subjected to bending, on the surface of which, the normal stress due to bending will vary inversely with the square of the tooth thickness. Bigger teeth are therefore stronger, but bigger i.e. is longer teeth, while in contact with each other, will undergo relatively more sliding.

We may conclude that although big teeth will be disproportionally stronger than smaller teeth, they will be less efficient due to frictional losses. The most efficient practical spur gears may have about 2% loss, the worse 5%.

Regions of High stress

There are 4 regions of relatively high stress in each gear pair, 2 in the pinion (the smaller dia gear) and 2 in the gear (larger dia gear).

To have a balanced designed gear pair, the factor of safety should be very nearly the same at these 4 locations. The highest stresses are assumed to take place when the gears make contact at the pitch point, only for high quality gears. For average gears, the highest stresses are calculated, when the teeth first make contact, at their tip.

Each variable shown here is a function of other variables.

There are 2 sets of such equations one for each gear

In a gear box there are many such sets, and in an industrial plant there may be many such boxes

Typical industrial multi stage gear box with large shaft diameters and broad face widths. For gear reduction of 1:6 or more the gear box will be lighter and usually cheaper if it incorporates 2 or more stages.

A computer controlled digitizing machine, measuring the surface of a helical gear.

CCD machines can have programs which guide the sensor tip around any profile, or just move until the sensor touched something.

These machines though quite dear are the only practical means of determining the accuracy of a tooth profile

A coal pulveriser as used in modern coal fired power station. A very high final gear ratio is used here because of the convenience of being able to use the pulveriser drum as the final gear stage

The pulveriser drive includes a traditional gear box just after the drive motor.

Another example of a supposedly rare high ratio gear pair. Here used to drive a deep cable drum on an off-shore oil rig service vessel. The power comes from a hydraulic low speed high torque motor, seen at left.

A herringbone or double helical gear box driving the propeller on a commercial vessel. The opposing helical gears balance the axial thrust generated by each gear. The cost of making double gears must payoff in reduced bearing loads and gearbox mass

A herringbone gear set driving the propeller on a naval vessel. Note the difference in teeth sizes to the previous gears, indicating quieter more efficient operation, but certainly not cheaper construction.

A typical heavy duty shaft mounted gear box set. Such installation may be capable of transmitting 2 MW. The weight of the assembly is used to partly balance the torque being transmitted. Note the separate lubrication system needed to provide high pressure temperature controlled oil.

A rare example of hot generated large gear wheels.

The tools steel wheels at right and left, are impressed on the circular gear blank in the centre, while rotating to form a finished gear with favorable grain flow and very fine surface finish

It is possible to have too few a number of teeth on a gear. For any particular pressure angle, If teeth number become too small, when the teeth are cut there will discontinuities in the generation of the involute profile, either done by a rack or hob.

In the last few decades cycloidal teeth have almost fallen out of use, possibly because of the difficulty in generating them. Today with accurate NC machine they may return, because they allow the use of very few teeth, that is strong teeth and high gear ratios in single gear pair. Note that Rootes blowers and screw compressors using 2, 3 or 4 teeth are cycloidal in profile. Cycloids are also used in mechanical clocks.

Rootes blowers (gear wheels) and their casing from a GM uniflow engine as used by main line locomotives.

Note that educated well-bred engineers refer to ‘inter-coolers’ as the heat exchangers that are used in between supercharging stages, not for the exchangers used between a compressor and the engine, those are after-coolers !