# 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 `R`

and
_{B}`R`

in the figure, are the radii
that the worm and gear would have if they were reduced to friction cylinders in
mutual contact._{F}

### 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 d_{B-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 ΔZ_{F}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 $${\widehat{z}}_{F}\times {\widehat{z}}_{B}$$.

## Ports

### Frame

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2016b**