Editing Gear (section)

== Ideal gear model ==

For basic analysis purposes, each gear can be idealized as a perfectly [[rigid body]] that, in normal operation, turns around a ''rotation axis'' that is fixed in space, without sliding along it. Thus, each point of the gear can move only along a circle that is perpendicular to its axis and centered on it.  At any moment ''t'', all points of the gear will be rotating around that axis with the same [[angular speed]] ''ω''(''t''), in the same sense. The speed need not be constant over time.

The ''action surface'' of the gear consists of all points of its surface that, in normal operation, may contact the matching gear with positive [[pressure]]. All other parts of the surface are irrelevant (except that they cannot be crossed by any part of the matching gear).  In a gear with ''N'' teeth, the working surface has ''N''-fold [[rotational symmetry]] about the axis, meaning that it is [[Congruence (geometry)|congruent]] with itself when the gear rotates by {{sfrac|1|''N''}} of a turn.  

If the gear is meant to transmit or receive torque with a definite sense only (clockwise or counterclockwise with respect to some reference viewpoint), the action surface consists of ''N'' separate patches, the ''tooth faces''; which have the same shape and are positioned in the same way relative to the axis, spaced {{sfrac|1|''N''}} turn apart.  

If the torque on each gear may have both senses, the action surface will have two sets of ''N'' tooth faces; each set will be effective only while the torque has one specific sense, and the two sets can be analyzed independently of the other.  However, in this case the gear usually has also "flip over" symmetry, so that the two sets of tooth faces are congruent after the gear is flipped.  This arrangement ensures that the two gears are firmly locked together, at all times, with no [[backlash (engineering)|backlash]].

During operation, each point ''p'' of each tooth face will at some moment contact a tooth face of the matching gear at some point ''q'' of one of its tooth faces.  At that moment and at those points, the two faces must have the same perpendicular direction but opposite orientation. But since the two gears are rotating around different axes, the points ''p'' and ''q'' are moving along different circles; therefore, the contact cannot last more than one instant, and ''p'' will then either slide across the other face, or stop contacting it altogether.  

On the other hand, at any given moment there is at least one such pair of contact points; usually more than one, even a whole line or surface of contact.

Actual gears deviate from this model in many ways: they are not perfectly rigid, their mounting does not ensure that the rotation axis will be perfectly fixed in space, the teeth may have slightly different shapes and spacing, the tooth faces are not perfectly smooth, and so on.  Yet, these deviations from the ideal model can be ignored for a basic analysis of the operation of a gear set.