## Couple Rotational Motion with Gears

A gear set consists of two or more meshed gears rotating together at some specified gear ratios. By convention, Simscape™ Driveline™ gear ratios are constant. The gear ratios determine how angular velocity and torque are transferred from one driveline component to another.

### Gear Coupling Rules

Ideal gears mesh and rotate together at a point of contact without frictional loss or slippage.

The simplest gear coupling consists of two circular gear wheels of radii
*r*_{1} and
*r*_{2}, spinning with angular velocities
*ω*_{1} and
*ω*_{2}, respectively, and lying in the
same plane. Their connected shafts are parallel and carry torques
*τ*_{1} and
*τ*_{2}. The *gear
ratio* of gear 2 to gear 1 is the ratio of their respective radii: *g*_{12} =
*r*_{2}/*r*_{1}. The power transferred along either shaft is *ω*·*τ*.

The gear coupling is often specified in terms of the number of gear teeth on each
gear, *N*_{1} and
*N*_{2}. The gear ratio of gear 2 to gear 1
is then *g*_{12} =
*N*_{2}/*N*_{1}
=
*r*_{2}/*r*_{1}.

The fundamental conditions on the simple gear coupling of rotational motion are *ω*_{2}/*ω*_{1}
= ±1/*g*_{12} and *τ*_{2}/*τ*_{1}
= ±*g*_{12}. That is, the ratio of angular velocities is the reciprocal of the
ratio of radii, while the ratio of torques is the ratio of radii. The transferred
power, being the product of angular velocity and torque, is the same on either
shaft.

The choice of signs indicates that the gears can spin in the same or in opposite directions. If the gears are external to one another (rotating together on their respective outside surfaces), they rotate in opposite directions. If the gears are internal to one another (rotating together with the outside of the smaller gear meshing with inside of the larger gear), they rotate in the same direction.

**Caution**

Gear ratios in driveline model blocks must be strictly positive. Vanishing or negative gear ratios cause Simscape Driveline simulation to stop with an error at model initialization. If you need to reverse the relative rotation direction of a shaft connected to a gear, you can change the direction in the gear block property inspector.

### Generalized Gear Coupling Rules

If you are coupling gears that are not constant in radii, not lying in the same plane, or not circular, you need the general ideal gear coupling conditions.

The general velocity constraint requires that the linear velocities of the gears
at the point of contact are the same. This constraint is a vector condition on the
angular velocities *ω*_{1} and
*ω*_{2} and the radius
vectors *r*_{1} and *r*_{2}: *ω*_{1}✗*r*_{1} = *ω*_{2} ✗ *r*_{2}. The alternative form in terms of the number of gear teeth is
equivalent to this linear velocity constraint. For the gear teeth to mesh, the
number of teeth per unit length of gear circumference must be the same on the two
gears.

The general torque condition arises from the force equilibrium at the point of
contact. If there is no linear motion of the whole gear assembly, the forces at
contact ** F** must be equal and opposite. The ratio of
torques is then:

|*τ*_{2}|/|*τ*_{1}| = |*r*_{2} ✗ ** F**|/|

*r*_{1}✗

**|**

*F*The power transferred along either shaft is conserved across ideal gear couplings:

*ω*_{2}·(*r*_{2} ✗ ** F**) =

*ω*_{1}·(

*r*_{1}✗

**)**

*F*