# Rack & Pinion

Rack and pinion gear coupling translational and rotational motion, with adjustable pinion radius and friction losses

**Libraries:**

Simscape /
Driveline /
Gears /
Rotational- Translational

## Description

The Rack & Pinion block represents rack and pinion gear that converts between
translational and rotational motion. The rotational-translational gear constrains the
pinion (P) and rack (R) to, respectively, rotate and translate together in a fixed ratio
that you specify. You can choose whether the rack axis translates in a positive or
negative direction, as the pinion rotates in a positive direction, by using the
**Rack direction** parameter.

### Model Variables

R_{RP} | Rack-pinion gear ratio |

ω_{P} | Angular velocity of the pinion shaft |

v_{R} | Translational velocity of the rack |

r_{P} | Effective radius of the pinion |

N_{P} | Number of teeth on the pinion |

x_{R} | Rack tooth spacing |

τ_{P} | Pinion shaft torque |

F_{R} | Rack force |

F_{loss} | Total loss force |

F_{Coul} | Friction force |

η | Torque transfer efficiency |

p_{th} | Power threshold |

μ_{P} | Viscous friction coefficient for the pinion shaft |

μ_{R} | Viscous friction coefficient for the rack motion |

### Ideal Gear Constraint and Gear Ratio

Rack & Pinion imposes one kinematic constraint on the two connected axes:

ω_{P} =
R_{RP}v_{R}
. | (1) |

The transmission ratio is:

R_{RP} = 1 /
r_{P} =
ω_{P} /
v_{N} = ± 2π
/
N_{P}v_{R}
. | (2) |

The two degrees of freedom are reduced to one independent degree of freedom. The forward-transfer gear pair convention is (1,2) = (P,R).

The torque-force transfer is:

R_{RP}τ_{P}
+ F_{R} –
F_{loss} = 0 , | (3) |

with *F*_{loss} = 0 in the ideal case.

### Nonideal Gear Constraint

In the nonideal case, *F*_{loss} ≠ 0. For general considerations on nonideal gear modeling, see Model Gears with Losses.

In a nonideal pinion-rack pair (P,R), the angular velocity and geometric constraints are unchanged. But the transferred torque, force, and power are reduced by:

Coulomb friction between teeth surfaces on P and R, characterized by constant efficiency

*η*Viscous coupling of driveshafts with bearings, parametrized by viscous friction coefficients

*μ*

### Meshing Efficiency

The efficiency *η* of meshing between pinion and rack is fully
active only if the transmitted power is greater than the power threshold.

If the power is less than the threshold, the actual efficiency is automatically regularized to unity at zero velocity.

Efficiency is assumed equal for both the forward and reverse power flow.

### Viscous Friction Force

The viscous friction coefficients *μ*_{P} and
*μ*_{R} control the viscous friction torque
and force experienced by the rack and pinion from lubricated, nonideal bearings. The
viscous friction torque on the pinion axis is
–*μ*_{P}*ω*_{P}.
The viscous friction force on the rack motion is
–*μ*_{R}*v*_{R}.

### Thermal Model

You can model
the effects of heat flow and temperature change by enabling the optional thermal port. To enable
the port, set **Friction model** to ```
Temperature-dependent
efficiency
```

.

### Hardware-in-the-Loop Simulation

For optimal performance of your real-time simulation, set the **Friction
model** to ```
No meshing losses - Suitable for HIL
simulation
```

on the **Meshing Losses** tab.

### Variables

Use the **Variables** settings to set the priority and initial target
values for the block variables before simulating. For more information, see Set Priority and Initial Target for Block Variables.

## Limitations

Gear inertia is assumed to be negligible.

Gears are treated as rigid components.

Coulomb friction slows down simulation. For more information, see Adjust Model Fidelity.

## Ports

Port | Description |
---|---|

P | Rotational conserving port representing the pinion |

R | Translational conserving port representing the rack |

H | Thermal conserving port for modeling heat transfer |

P is a rotational conserving port. R is a translational conserving port. They represent the pinion and the rack, respectively.

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2011a**