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

Gears/Rotational-Translational

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.

The block models the effects of heat flow and temperature change
through an optional thermal port. To expose the thermal port, right-click
the block and select **Simscape** > **Block choices** > **Show thermal
port**. Exposing the thermal port causes
new parameters specific to thermal modeling to appear in the block
dialog box.

**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.**Pinion radius**Effective radius of the pinion

*r*_{P}. Must be greater than zero. The default is`100`

.From the drop-down list, choose units. The default is millimeters (

`mm`

).

`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`

.

Parameters for meshing and friction losses vary with the block variant chosen—one with a thermal port for thermal modeling and one without it.

**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)`

).

**Thermal mass**Thermal energy required to change the component temperature by a single degree. The greater the thermal mass, the more resistant the component is to temperature change. The default value is

`50`

J/K.**Initial temperature**Component temperature at the start of simulation. The initial temperature influences the starting meshing or friction losses by altering the component efficiency according to an efficiency vector that you specify. The default value is

`300`

K.

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 assumed negligible.

Gears are treated as rigid components.

Coulomb friction slows down simulation. See Adjust Model Fidelity.

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.

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