Rack & Pinion

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.

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.

Dialog Box and Parameters


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 rP. 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.

     Tooth parameters

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.

Meshing Losses

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

 Without Thermal Port

 With Thermal Port

Viscous Losses

Pinion rotational viscous friction coefficient

Viscous friction coefficient μP 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 μR 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 Port

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.

Gear Model

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
vthAbsolute translational velocity 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 .

The transmission ratio is:

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

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 + FRFloss = 0 ,

with Floss = 0 in the ideal case.

Nonideal Gear Constraint

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:

Floss = FCoul· tanh(4vR/vth) + μPωPRRP + μRvR.

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

Power FlowPower Loss ConditionOutput DriveshaftCoulomb Friction Force FCoul
ForwardωPτP > FRvRRack, vRRRP· |τP|· (1 – η)
ReverseωPτPFRvRPinion, ωPRRP· |τP|· (1 – η) / η

Meshing Efficiency

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 vth.

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.

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.


  • Gear inertia is assumed negligible.

  • Gears are treated as rigid components.

  • Coulomb friction slows down simulation. See Adjust Model Fidelity.


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.

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