Rack & Pinion

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

Library

Simscape / Driveline / Gears / Rotational-Translational

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.

Thermal Modeling

You can model the effects of heat flow and temperature change through an optional thermal conserving port. By default, the thermal port is hidden. To expose the thermal port, right-click the block in your model and, from the context menu, select Simscape > Block choices. Select a variant that includes a thermal port. Specify the associated thermal parameters for the component.

Gear Model

Model Variables

RRP	Rack-pinion gear ratio
ωP	Angular velocity of the pinion shaft
vR	Translational velocity of the rack
rP	Effective radius of the pinion
NP	Number of teeth on the pinion
xR	Rack tooth spacing
τP	Pinion shaft torque
FR	Rack force
Floss	Total loss force
FCoul	Friction force
η	Torque transfer efficiency
pth	Power threshold
μP	Viscous friction coefficient for the pinion shaft
μR	Viscous friction coefficient for the rack motion

Ideal Gear Constraint and Gear Ratio

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

ω_P = R_RPv_R .

The transmission ratio is:

R_RP = 1 / r_P = ω_P / v_N = ± 2π / N_Pv_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_RPτ_P + F_R – F_loss = 0 ,

with F_loss = 0 in the ideal case.

Nonideal Gear Constraint

In the nonideal case, F_loss ≠ 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 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_Pω_P. The viscous friction force on the rack motion is –μ_Rv_R.

Limitations

• Gear inertia is assumed negligible.

• Gears are treated as rigid components.

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

Ports

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.

Parameters

Main

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.

   Tooth parameters

  Number of pinion teeth

  Number of teeth on the pinion N_P. Must be greater than zero. The default is 20.

  Rack tooth spacing

  Spacing between the teeth on the rack x_R. Must be greater than zero. The default is 5.

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

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

Friction model

Select how to implement friction losses from nonideal meshing of gear teeth. The default is No meshing losses.

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

• Constant efficiency — Transfer of torque and force between pinion and rack is reduced by constant efficiency 0 < η ≤ 1. If you select this option, the panel expands.

   Constant Efficiency

  Efficiency

  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. The default is 0.8.

  Rack power threshold

  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. The default value is 0.001 W.

 With Thermal Port

Temperature

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. The default array is [280 300 320] K.

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. The default array is [0.95 0.9 0.85].

Rack power threshold

Power threshold above which the full efficiency factor is in effect. A hyperbolic tangent function smooths the efficiency factor between zero at rest and the value provided by the temperature-efficiency lookup table at the power threshold. The default value is 0.001 W.

Viscous Losses

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 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 alters the component efficiency according to an efficiency vector that you specify, affecting the starting meshing or friction losses. The default value is 300 K.

Real-Time Simulation

Hardware-in-the-Loop Simulation

For optimal simulation performance, use the Meshing Losses > Friction model parameter default setting, No meshing losses - Suitable for HIL simulation.

Extended Capabilities

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