# Worm and Gear Constraint

Kinematic constraint between worm and gear bodies with perpendicular non-intersecting rotation axes

Libraries:
Simscape / Multibody / Gears and Couplings / Gears

## Description

The Worm and Gear Constraint block represents a kinematic constraint between worm and gear bodies held at a right angle. The base frame port identifies the connection frame on the worm and the follower frame port identifies the connection frame on the gear. The rotation axes coincide with the connection frame z-axes. The worm and gear rotate at a fixed velocity ratio determined by the gear pitch radii or tooth-thread ratio.

The worm thread direction can follow either right-hand or left-hand conventions. The convention used determines the relative directions of the worm and gear rotational velocities. A right-hand convention causes the worm and gear to rotate in the same direction about the respective z-axes. A left-hand convention causes the worm and gear to rotate in opposite directions instead.

The block represents only the kinematic constraint characteristic to a worm-and-gear 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 worm and gear pitch circles. The pitch circles are imaginary circles concentric with the worm and gear bodies and tangent to the thread contact point. The pitch radii, labeled `RB` and `RF` in the figure, are the radii that the worm and gear would have if they were reduced to friction cylinders in mutual contact.

### Gear Assembly

Gear constraints occur in closed kinematic loops. The figure shows the closed-loop topology of a simple worm-and-gear model. Joint blocks connect the worm and gear bodies to a common fixture or carrier, defining the maximum degrees of freedom between them. A Worm and Gear Constraint block connects the worm and gear bodies, eliminating one degree of freedom and effectively coupling the worm and gear motions.

### 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 z-axes, denoted dB-F in the figure, must be equal to the distance between the gear centers.

• The translational offset between the base and follower frame origins along the follower frame z-axis, denoted ΔZF in the figure, must be zero.

Orientation Restrictions

• The z-axes of the base and follower frames must be perpendicular to each other. The z-axes are shown in blue in the figure.

• The cross product of the follower frame z-axis with the base frame z-axis must be a vector aimed from the follower frame origin to the base frame z-axis. The z-axes and their cross-product vector are shown in the figure. The cross product is defined as ${\stackrel{^}{z}}_{F}×{\stackrel{^}{z}}_{B}$.

## Ports

### Frame

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Connection frame on the worm body.

Connection frame on the gear body.

## Parameters

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Winding direction of the worm thread relative to the base frame z-axis. As viewed from the base frame origin, a right-hand thread is one that wraps around the base frame z-axis in a counterclockwise direction. A left-hand thread is one that wraps in a clockwise direction. This parameter determines the relative directions of motion of the worm and gear bodies.

Angle between the tangent to the worm thread and the plane perpendicular to the base frame z-axis. The lead angle impacts the gear rotation corresponding to a full worm revolution.

Parameterization for specifying the worm and gear geometries. You can specify the gear dimensions in terms of the distance between the gear centers or the individual gear radii.

Distance between the worm and gear centers. This distance must equal that enforced by rigid transforms, joints, and any other constraints located between the gear bodies and the common carrier body.

#### Dependencies

This parameter is enabled when the Specification Method parameter is set to ```Center Distance and Ratio```.

Ratio of gear teeth to worm threads, or starts. This ratio impacts the torque transmitted between the worm and gear.

#### Dependencies

This parameter is enabled when the Specification Method parameter is set to ```Center Distance and Ratio```.

Radius of the worm pitch circle. This is the distance between the worm rotation axis and the tooth-thread contact point. This parameter impacts the torque transmitted between the worm and gear.

#### Dependencies

This parameter is enabled when the Specification Method parameter is set to ```Pitch Circle Radius```.

Radius of the gear pitch circle. This is the distance between the gear rotation axis and the tooth-thread contact point. This parameter impacts the torque transmitted between the worm and gear.

#### Dependencies

This parameter is enabled when the Specification Method parameter is set to ```Pitch Circle Radius```.

## Version History

Introduced in R2016b