Rack & Pinion

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

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  • Simscape / Driveline / Gears / Rotational- Translational

  • Rack & Pinion block

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

RRPRack-pinion gear ratio
ωPAngular velocity of the pinion shaft
vRTranslational velocity of the rack
rPEffective radius of the pinion
NPNumber of teeth on the pinion
xRRack tooth spacing
τPPinion shaft torque
FRRack force
FlossTotal loss force
FCoulFriction force
ηTorque transfer efficiency
pthPower threshold
μPViscous friction coefficient for the pinion shaft
μRViscous friction coefficient for the rack motion

Ideal Gear Constraint and Gear Ratio

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

ωP = RRPvR .(1)

The transmission ratio is:

RRP = 1 / rP = ωP / vN = ± 2π / NPvR .(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:

RRPτP + FR – Floss = 0 ,(3)

with Floss = 0 in the ideal case.

Nonideal Gear Constraint

In the nonideal case, Floss ≠ 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 μP and μR 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 –μPωP. The viscous friction force on the rack motion is –μRvR.

Thermal Model

You can model the effects of heat flow and temperature change by exposing an optional thermal port. To expose the port, in the Meshing Losses tab, set the Friction model parameter 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.

Dependencies

Variable settings are exposed only when, in the Meshing Losses settings, the Friction model parameter is set to Temperature-dependent efficiency.

Limitations

  • Gear inertia is assumed negligible.

  • Gears are treated as rigid components.

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

Ports

PortDescription
PRotational conserving port representing the pinion
RTranslational conserving port representing the rack
HThermal 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

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Main

Parameterization method of the rack and pinion gear.

  • 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.

Effective radius of the pinion rP. The value must be greater than zero.

Dependencies

To enable this parameter, set Parameterize by to Pinion radius.

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

Dependencies

To enable this parameter, set Parameterize by to Tooth parameters.

Spacing between the teeth on the rack xR. The value must be greater than zero.

Dependencies

To enable this parameter, set Parameterize by to Tooth parameters.

Choose whether the rack axis translates in a positive or negative direction when the pinion rotates in a positive direction.

Meshing Losses

  • No meshing losses — Suitable for HIL simulation — Gear meshing is ideal.

  • Constant efficiency — Transfer of torque between rack and pinion is reduced by friction.

  • Temperature-dependent efficiency — Torque transfer is determined from user-supplied data for efficiency and temperature.

Torque transfer efficiency η for the rack-pinion gear meshing, which is the same for the forward and reverse power flows. Must be greater than zero but less than, or equal to, one.

Dependencies

To enable this parameter, set Friction model to Constant efficiency.

Array of temperatures used to construct a 1-D temperature-efficiency lookup table. The array values must increase left to right. The temperature array must be the same size as the Efficiency array.

Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Array of component efficiencies used to construct a 1-D temperature-efficiency lookup table. The array values are the efficiencies at the temperatures in the Temperature array. The two arrays must be the same size.

Dependencies

To enable this parameter, set Friction model to Temperature-dependent efficiency.

Power threshold above which the full efficiency factor is in effect. Below this value, a hyperbolic tangent function smooths the efficiency factor, lowering the efficiency losses to zero when no power is transmitted.

Viscous Losses

Viscous friction coefficient μP for the pinion shaft.

Viscous friction coefficient μR for the rack motion.

Thermal Port

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.

Component temperature at the start of simulation. The initial temperature alters the component efficiency according to an efficiency vector that you specify, affecting the starting meshing or friction losses.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

See Also

Simscape Blocks

Topics

Introduced in R2011a

Simscape Driveline Documentation

Support