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.
Select how to parameterize the rack and pinion gear. The default
Pinion radius — Gear ratio
is defined by the effective radius of the pinion.
Effective radius of the pinion rP.
Must be greater than zero. The default is
From the drop-down list, choose units. The default is millimeters
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
Choose whether the rack axis translates in a positive or negative
direction when the pinion rotates in a positive direction. The default
Positive for positive pinion rotation.
Parameters for meshing and friction losses vary with the block variant chosen—one with a thermal port for thermal modeling and one without it.
Viscous friction coefficient μP for
the pinion shaft. The default is
From the drop-down list, choose units. The default is newton-meters/(radians/second)
Viscous friction coefficient μR for
the rack motion. The default is
From the drop-down list, choose units. The default is newton/(meters/second)
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
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
|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|
|Floss||Total loss force|
|η||Torque transfer efficiency|
|vth||Absolute translational velocity threshold|
|μP||Viscous friction coefficient for the pinion shaft|
|μR||Viscous friction coefficient for the rack motion|
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 + FR – Floss = 0 ,
with Floss = 0 in the ideal case.
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 Flow||Power Loss Condition||Output Driveshaft||Coulomb Friction Force FCoul|
|Forward||ωPτP > FRvR||Rack, vR||RRP· |τP|· (1 – η)|
|Reverse||ωPτP ≤ FRvR||Pinion, ωP||RRP· |τP|· (1 – η) / η|
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.
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.
|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.