# Rack and Pinion Constraint

Kinematic constraint for converting between rotation and translation

## Library

Gears and Couplings/Gears

## Description

This block represents a kinematic constraint between a rack and a pinion. The constraint converts rotation of the pinion into translation of the rack and vice-versa. The pinion, which connects to the base port frame, spins about the base Z axis. The rack, which connects to the follower port frame, translates along the follower Z axis.

Kinematic constraints in the remainder of the model must hold the base and follower frames at the correct distance and with the proper alignment. These constraints might be due to rigid transforms, joints, and model topology. Assembly conditions include:

• Base and follower Z axes must be mutually orthogonal. This condition ensures that the pinion rotation axis sits at a right angle to the rack translation axis. You can rotate frames using the Rigid Transform block.

• Base and follower frame origins must be apart by a distance equal to the Pinion Radius parameter. This condition ensures that the rack and pinion cogs are at the correct distance for engagement. You can translate frames using the Rigid Transform block.

The figure shows the distance and alignment of the base and follower frames in the zero configuration. This is the primary configuration that Simscape™ Multibody™ attempts to achieve during model assembly.

In the zero configuration, the pinion rotation angle and the rack translation distance are both zero. To achieve this configuration, Simscape Multibody:

• Aligns the base and follower Y axes.

• Positions the follower frame origin along the negative Y axis of the base frame.

Joint blocks provide the base and follower frames with the proper internal degrees of freedom. These degrees of freedom must support rotation about the base Z axis and translation along the follower Z axis. You can achieve these degrees of freedom using different joint block combinations. For example, you can connect the pinion to a Revolute Joint block and the rack to a Prismatic Joint block.

During simulation, a positive pinion rotation about the base Z axis corresponds to a positive rack translation along the follower Z axis. By definition, the translational velocity of the rack is equal to the tangential velocity at a point in the pinion pitch circle, an imaginary circle that intersects the rack and pinion cogs at the mutual contact point.

The figure shows the relative motion of the base and follower frames due to the rack and pinion constraint.

## Dialog Box and Parameters

Pinion Radius

Distance between the pinion center and pitch circle. This circle contains the instantaneous contact point between a pair of rack and pinion cogs. The pinion radius must equal the actual distance between the base and follower frames as specified by the remainder of the model. The default value is `10` cm.

## Ports

The block contains frame ports B and F, representing base and follower frames, respectively.

## See Also

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