Rack & Pinion

A rack and pinion gear connecting translational and rotational motion, with adjustable pinion radius and friction losses.

rack pinion

Description

The Rack & Pinion unit is a rack & pinion transmission that converts translational and rotational motion.

The reciprocating gear causes the pinion (port P) and rack (port R) to rotate and move respectively in a fixed ratio that you specify. With the Rack direction parameter, you can select whether the rack axis moves in the positive or negative direction while the pinion gear rotates in the positive direction.

Model variables

Gear ratio

Angular speed of the pinion shaft

Travelling speed of the rack

Effective radius of the pinion

Number of teeth on the pinion

Distance between the teeth of the rack

Electromagnetic torque of the pinion shaft

Force on the rack

Total loss force

Friction force

Torque transmission efficiency

Threshold power

Viscous friction coefficient for pinion shaft

Viscous friction coefficient for the rack movement

Ideal gear limit and gear ratio

The rack and pinion transmission imposes one kinematic constraint on two coupled axes:

The transmission ratio is:

Two degrees of freedom are reduced to one independent degree of freedom. The notation of a direct transfer gear pair is: (1, 2) = (P, R).

Torque transfer is realised as follows:

at this in the ideal case.

Non-ideal transmission constraint

In the non-ideal case .

In a non-ideal gear-rail pair (P, R) the angular velocity and geometric constraints remain unchanged. But the transmitted torque, force and power are reduced due to the following factors:

Coulomb friction between tooth surfaces at P and R, characterised by constant efficiency, .
Viscous coupling of cardan shafts with bearings, parameterised by viscous friction coefficients, .

Clutch efficiency

Efficiency clutch between pinion and rack and pinion is only fully active if the transmitted power exceeds a threshold power.

If the power is less than the threshold, the actual efficiency automatically equalises to one at zero speed.

The efficiency is assumed to be the same for both forward and reverse power flow.

Viscous friction force

The viscous friction coefficients and determine the torque and viscous friction force experienced by the rack and pinion gear due to lubricated non-ideal bearings. The viscous friction torque at the pinion axis is . The viscous friction force on the rack movement is .

Thermal model

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

Assumptions and limitations

The inertia of the gears is negligible.
Gears are treated as rigid components.
Coulomb friction slows down the modelling.

Ports

Conserving

# P — gear
`rotational mechanics

Details

A non-directional port associated with a gear.

Program usage name

pinion_flange

# R — rail

Details

A non-directional port associated with the rail.

Program usage name

rack_flange

# H — heat flux
`heat

Details

A non-directional port associated with heat flow.

The heat port allows modelling the heat flow between the unit and the connected network.

Dependencies

To enable this port, set Friction model to `Temperature-dependent efficiency'.

Program usage name

thermal_port

Parameters

Main

# Parameterize by — method of rack and pinion parameterization
Pinion radius | Tooth parameters

Details

The parameterization method of the rack and pinion gear, given as:

Pinion radius - the gear ratio is determined by the effective radius of the pinion.
Tooth parameters - the transmission ratio is determined by the number of teeth on the pinion and the distance between the rack teeth.

Values	`Pinion radius` \| `Tooth parameters`
Default value	`Pinion radius`
Program usage name	`parameterization`
Evaluatable	No

# Pinion radius — effective gear radius
m | cm | ft | in | km | mi | mm | um | yd

Details

Effective gear radius . The value must be greater than zero.

Dependencies

To use this parameter, set the Parameterise by parameter to Pinion radius.

Units	`m` \| `cm` \| `ft` \| `in` \| `km` \| `mi` \| `mm` \| `um` \| `yd`
Default value	`100.0 mm`
Program usage name	`pinion_radius`
Evaluatable	Yes

# Number of pinion teeth — gear tooth count

Details

Number of teeth on the pinion . The value must be greater than zero.

Dependencies

To use this parameter, set the Parameterise by parameter to Tooth parameters.

Default value	`20`
Program usage name	`teeth_count`
Evaluatable	Yes

# Rack tooth spacing — rack tooth spacing
m | cm | ft | in | km | mi | mm | um | yd

Details

The distance between the teeth on the rail . The value must be greater than zero.

Dependencies

To use this parameter, set the Parameterise by parameter to Tooth parameters.

Units	`m` \| `cm` \| `ft` \| `in` \| `km` \| `mi` \| `mm` \| `um` \| `yd`
Default value	`5.0 mm`
Program usage name	`spacing`
Evaluatable	Yes

# Rack direction — Orientation of the rack as the pinion rotates
Negative for positive pinion rotation | Positive for positive pinion rotation

Details

Select whether the rail axis moves in the positive or negative direction when the pinion is rotating in the positive direction.

Values	`Negative for positive pinion rotation` \| `Positive for positive pinion rotation`
Default value	`Positive for positive pinion rotation`
Program usage name	`rack_directionality`
Evaluatable	No

Meshing Losses

# Friction model — friction losses due to non-ideal coupling of gear and rack teeth
No meshing losses - Suitable for HIL simulation | Constant efficiency | Temperature-dependent efficiency

Details

The friction loss is given as:

No meshing losses - Suitable for HIL simulation - perfect meshing of gears;
Constant efficiency - torque transmission between rack and pinion is reduced due to friction;
`Temperature-dependent efficiency' - torque transmission is determined based on efficiency and temperature data provided by the user.

Values	`No meshing losses - Suitable for HIL simulation` \| `Constant efficiency` \| `Temperature-dependent efficiency`
Default value	`No meshing losses - Suitable for HIL simulation`
Program usage name	`friction_model`
Evaluatable	No

# Efficiency — torque transmission efficiency

Details

Torque transmission efficiency at rack-and-pinion meshing, the same for forward and reverse power flows. Must be greater than zero but less than or equal to one.

Dependencies

To use this parameter, set the Friction model parameter to Constant efficiency.

Default value	`0.8`
Program usage name	`efficiency_const`
Evaluatable	Yes

Details

An array of temperatures used to construct a one-dimensional temperature-efficiency table. The array values should increase from left to right. The temperature array must be of the same dimensionality as the Efficiency parameter array.

Dependencies

To use this parameter, set the Friction model parameter to `Temperature-dependent efficiency'.

Units	`K` \| `degC` \| `degF` \| `degR` \| `deltaK` \| `deltadegC` \| `deltadegF` \| `deltadegR`
Default value	`[280.0, 300.0, 320.0] K`
Program usage name	`temperature_vector`
Evaluatable	Yes

# Efficiency — array of efficiencies

Details

An array of component efficiencies used to construct a one-dimensional table of temperature efficiencies. The array values are the efficiencies at the temperatures in the Temperature array. Both arrays must be of the same dimensionality.

Dependencies

To use this parameter, set the Friction model parameter to `Temperature-dependent efficiency'.

Default value	`[0.95, 0.90, 0.85]`
Program usage name	`efficiency_vector`
Evaluatable	Yes

# Rack power threshold — power threshold below which numerical smoothing is applied
W | GW | MW | kW | mW | uW | HP_DIN

Details

The power threshold value above which the full efficiency factor applies. Below this value, the hyperbolic tangent function smooths the efficiency factor, reducing the efficiency loss to zero when no power is transmitted.

Units	`W` \| `GW` \| `MW` \| `kW` \| `mW` \| `uW` \| `HP_DIN`
Default value	`0.001 W`
Program usage name	`power_threshold`
Evaluatable	Yes

Viscous Losses

# Pinion rotational viscous friction coefficient — gear viscous friction coefficient
N*m/(rad/s) | ft*lbf/(rad/s)

Details

Viscous friction coefficient for pinion shaft.

Units	`Nm/(rad/s)` \| `ftlbf/(rad/s)`
Default value	`0.0 N*m/(rad/s)`
Program usage name	`pinion_viscous_coefficient`
Evaluatable	Yes

# Rack translational viscous friction coefficient — lath viscous friction coefficient
kg/s | N*s/m | N/(m/s) | lbf/(ft/s) | lbf/(in/s)

Details

Viscous friction coefficient for the rack and pinion mechanism.

Units	`kg/s` \| `N*s/m` \| `N/(m/s)` \| `lbf/(ft/s)` \| `lbf/(in/s)`
Default value	`0.0 N/(m/s)`
Program usage name	`rack_viscous_coefficient`
Evaluatable	Yes

Thermal Port

# Thermal mass — heat capacity
J/K | kJ/K

Details

The heat energy required to change the temperature of a component by one degree. The greater the heat capacity, the more resistant the component is to temperature change.

Units	`J/K` \| `kJ/K`
Default value	`50.0 J/K`
Program usage name	`thermal_mass`
Evaluatable	Yes