## 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 r1 and r2, spinning with angular velocities ω1 and ω2, respectively, and lying in the same plane. Their connected shafts are parallel and carry torques τ1 and τ2. The gear ratio of gear 2 to gear 1 is the ratio of their respective radii: g12 = r2/r1. 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, N1 and N2. The gear ratio of gear 2 to gear 1 is then g12 = N2/N1 = r2/r1.

The fundamental conditions on the simple gear coupling of rotational motion are ω2/ω1 = ±1/g12 and τ2/τ1 = ±g12. 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 ω1 and ω2 and the radius vectors r1 and r2: ω1r1 = ω2r2. 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:

|τ2|/|τ1| = |r2F|/|r1F|

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

ω2·(r2F) = ω1·(r1F)