Rack and Pinion Constraint

Kinematic constraint between a translating rack body and a rotating pinion body

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  • Rack and Pinion Constraint block

Libraries:
Simscape / Multibody / Gears and Couplings / Gears

Description

The Rack and Pinion Constraint block represents a kinematic constraint between a translating rack body and a rotating pinion body. The base frame port identifies the connection frame on the pinion body and the follower frame port identifies the connection frame on the rack body. The pinion rotation axis and the rack translation axis coincide with the frame z-axes.

The block represents only the kinematic constraint characteristic to a rack-and-pinion system. Gear inertia and geometry are solid properties that you must specify using solid blocks. The gear constraint model is ideal. Backlash and gear losses due to Coulomb and viscous friction between teeth are ignored. You can, however, model viscous friction at joints by specifying damping coefficients in the joint blocks.

Gear Geometry

The rack-and-pinion constraint is parameterized in terms of the dimensions of the pinion pitch circle. The pitch circle is an imaginary circle concentric with the pinion body and tangent to the tooth contact point. The pitch radius, labeled RB in the figure, is the radius that the pinion would have if it was reduced to a friction cylinder in contact with a brick approximation of the rack.

Gear Assembly

Gear constraints occur in closed kinematic loops. The figure shows the closed-loop topology of a simple rack-and-pinion model. Joint blocks connect the rack and pinion bodies to a common fixture or carrier, defining the maximum degrees of freedom between them. A Rack and Pinion Constraint block connects the rack and pinion bodies.

Assembly Requirements

The block imposes special restrictions on the relative positions and orientations of the gear connection frames. The restrictions ensure that the gears assemble only at distances and angles suitable for meshing. The block enforces the restrictions during model assembly, when it first attempts to place the gears in mesh, but relies on the remainder of the model to keep the gears in mesh during simulation.

Position Restrictions

  • The distance between the base and follower frame origins along the follower frame y-axis must equal the pinion radius. This constraint ensures that the pitch point of the rack is at the proper distance from the rotation axis of the pinion.

  • The follower frame origin must lie on the xy plane of the base frame. This constraint ensures that the pitch point of the rack is coplanar with the pitch circle of the pinion.

Orientation Restrictions

  • The x-axis of the follower frame must be perpendicular to the xy plane of the base frame. This constraint ensures that the rack and pinion are coplanar, and therefore that their motion axes are perpendicular to each other.

Examples

Using the Rack-Pinion Block - Windshield Wiper Mechanism

Using the Rack-Pinion Block - Windshield Wiper Mechanism

A windshield wiper mechanism. The mechanism utilizes a rack and pinion to drive the wiper blades in a synchronized fashion. The rack is actuated using a scotch yoke coupling (modeled using a pin-slot joint) that converts the rotary motion of the motor to reciprocating motion of the rack. The rack and pinion arrangement converts the reciprocating linear motion of the rack into reciprocating angular motion of the wiper blades (which are rigidly attached to the pinion).

Ports

Frame

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Connection frame on the pinion body.

Connection frame on the rack body.

Output

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Physical signal port that outputs the constraint force. The force maintains the kinematic constraint between the pinion body and rack body. The output is a 3-D vector in units of force. The settings of the Resolution Frame and Direction parameters affect the output value.

Dependencies

To enable this port, under Constraint Force/Torque Sensing, select Force Vector.

Physical signal port that outputs the constraint torque. The torque maintains the kinematic constraint between the pinion body and rack body. The output is a 3-D vector in units of torque. The settings of the Resolution Frame and Direction parameters affect the output value.

Dependencies

To enable this port, under Constraint Force/Torque Sensing, select Torque Vector.

Parameters

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Radius of the pitch circle of the pinion body. The pitch circle is an imaginary circle concentric with the pinion body and tangent to the tooth contact point.

Constraint Force/Torque Sensing

Measurement direction, specified as one of these values:

  • Follower on Base — The block senses the force and torque that the rack body exerts on the pinion body.

  • Base on Follower — The block senses the force and torque that the pinion body exerts on the rack body.

Frame to use to resolve the measurements, specified as one of these values:

  • Base — The block resolves the measurements in the coordinates of the base frame.

  • Follower — The block resolves the measurements in the coordinates of the follower frame.

The block specifies the measurements with respect to the origin of the resolution frame.

Select this parameter to expose port f, which outputs the constraint force between the pinion body and rack body.

Select this parameter to expose port t, which outputs the constraint torque between the pinion body and rack body.

Extended Capabilities

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C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2013a

See Also

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