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

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.

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

**Parameterize by**Select how to parameterize the rack and pinion gear. The default is

`Pinion radius`.`Pinion radius`— Gear ratio is defined by the effective radius of the pinion.`Tooth parameters`— Gear ratio is defined by the number of teeth on the pinion gear and the rack tooth spacing. If you select this option, the panel changes from its default.

**Rack direction**Choose whether the rack axis translates in a positive or negative direction when the pinion rotates in a positive direction. The default is

`Positive for positive pinion rotation`.

**Friction model**Select how to implement friction losses from nonideal meshing of gear teeth. The default is

`No meshing losses`.`No meshing losses — Suitable for HIL simulation`— Gear meshing is ideal.`Constant efficiency`— Transfer of torque and force between pinion and rack is reduced by constant efficiency 0 <*η*≤ 1. If you select this option, the panel expands.

**Pinion rotational viscous friction coefficient**Viscous friction coefficient

*μ*_{P}for the pinion shaft. The default is`0`.From the drop-down list, choose units. The default is newton-meters/(radians/second) (

`N*m/(rad/s)`).**Rack translational viscous friction coefficient**Viscous friction coefficient

*μ*_{R}for the rack motion. The default is`0`.From the drop-down list, choose units. The default is newton/(meters/second) (

`N/(m/s)`).

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 |

v_{th} | Absolute translational velocity threshold |

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

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

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

*ω*_{P} = *R*_{RP}*v*_{R} .

The transmission ratio is:

*R*_{RP} = 1 / *r*_{P} = *ω*_{P} / *v*_{N} =
± 2*π* / *N*_{P}*v*_{R} .

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 ,

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

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*

The loss force has the general form:

*F*_{loss} = *F*_{Coul}·
tanh(4*v*_{R}/*v*_{th})
+ *μ*_{P}*ω*_{P}*R*_{RP} + *μ*_{R}*v*_{R}.

The hyperbolic tangent regularizes the sign change in the Coulomb friction force when the rack velocity changes sign.

Power Flow | Power Loss Condition | Output Driveshaft | Coulomb Friction Force F_{Coul} |
---|---|---|---|

Forward | ω_{P}τ_{P} > F_{R}v_{R} | Rack, v_{R} | R_{RP}·
|τ_{P}|· (1 – η) |

Reverse | ω_{P}τ_{P} ≤ F_{R}v_{R} | Pinion, ω_{P} | R_{RP}·
|τ_{P}|· (1 – η)
/ η |

The efficiency *η* of meshing between
pinion and rack is fully active only if the absolute value of the
rack velocity is greater than the velocity threshold *v*_{th}.

If the velocity 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.

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}.

Gear inertia is negligible. It does not impact gear dynamics.

Gears are rigid. They do not deform.

Coulomb friction slows down simulation. See Adjust Model Fidelity.

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