Rack & Pinion

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

blockType: Engee1DMechanical.Transmission.Gears.RotationalTranslational.RackAndPinion

Path in the library:

/Physical Modeling/1D Mechanical/Gears/Rotational-Translational/Rack & Pinion

Description

The Rack & Pinion block represents rack and pinion transmission, which converts translational and rotational motion.

The rotational-translational transmission causes the gear (port P) and the rack (port R) to rotate and move respectively in the fixed ratio that you specify. Using the Rack direction parameter, you can choose whether the rack axis will move in the positive or negative direction, while the gear rotates in the positive direction.

Model variables

Gear ratio

Angular velocity of the gear shaft

The speed of the rail movement

Effective gear radius

The number of teeth on the gear

The distance between the teeth of the rail

Electromagnetic torque of the gear shaft

Effort on the rail

Total loss power

Friction force

Torque transmission efficiency

Threshold power

Coefficient of viscous friction for gear shaft

Coefficient of viscous friction for rail movement

Ideal transmission limitation and gear ratio

Rack and pinion transmission imposes one kinematic limitation on two connected axes:

The gear ratio is:

Two degrees of freedom are reduced to one independent degree of freedom. The symbol for a pair of forward gears is (1, 2) = (P, R).

The transmission of torque is carried out as follows:

at the same time in the ideal case.

An imperfect transfer restriction

In an imperfect case .

In an imperfect gear-rack 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 on P and R, characterized by constant efficiency, .
Viscous coupling of cardan shafts with bearings, parameterized by viscous friction coefficients, .

Coupling efficiency

Effectiveness The coupling between the gear and the rack is fully active only if the transmitted power exceeds the threshold power.

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

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

Viscous friction force

Coefficients of viscous friction and The torque and the force of viscous friction experienced by the rack and pinion due to lubricated imperfect bearings are determined. The torque of viscous friction on the gear axis is . The force of viscous friction during the movement of the rail is .

The thermal model

You can simulate the effects of heat flow and temperature changes by turning on an additional heat port H. To turn on the thermal port, set the Friction model parameter to `Temperature-dependent efficiency'.

Assumptions and limitations

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

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