Rack and pinion (Simplified)#

block — Block appearance in REPEAT VISION#

A simplified rack and pinion transmission models a rack-and-pinion gear system that converts linear motion into rotary motion and vice versa. The transmission imposes a relationship between the pinion (p) and the rack (r), such that the former rotates and the latter moves linearly with a fixed, specified gear ratio.

When the On - positive rack displacement switch is turned on, you can set a positive rack offset with a positive wheel rotation.

The simplified rack and pinion transmission imposes one kinematic constraint on the associated axes (1) :

(1)#\[\omega_{к} = R_{рк} \cdot v_{р}\]

where:

\(R_{рк}\) – gear ratio;

\(\omega_{w}\) – angular velocity of the gear wheel;

\(v_{r}\) – speed of the rack.

Gear ratio (2) :

(2)#\[R_{рк} = \frac{1}{r_{к}} = \frac{\omega_{к}}{v_{р}}\]

where \(r_{w}\) is the radius of the gear wheel, [m].

Two degrees of freedom are reduced to one independent one.

The transfer of torque/force is described by the expression (3) :

(3)#\[R_{рк} \cdot M_{к} + F_{р} - F_{потери} = 0\]

where:

\(M_{w}\) – moment on the gear wheel;

\(F_{r}\) – force on the rack;

\(F_{losses}\) – power losses during power transmission. \(F_{losses}\) = 0 without taking into account losses.

1. Accounting for losses#

In a real transmission, mechanical losses occur between the rack and pinion—for example, due to tooth friction and lubrication. In the model, these are taken into account through the power transfer coefficient (Efficiency) \(\eta_{рш}\), which reduces the amount of power transmitted.

When the Calculate losses \(F_{losses} eq\) 0 property is enabled, losses are calculated relative to Efficiency. For a more detailed description of loss calculation, please refer to the Loss Calculation page.

The value of Efficiency for the forward (wheel -> rack) and reverse power flow (rack -> wheel) is the same.

Note

The full calculation of the coupling Efficiency occurs if the transmitted power exceeds the specified threshold value \(p_{prg}\). Otherwise, for numerical stability, the Efficiency is assumed to be equal to unity.

Restrictions:

• The inertia of the components is negligible;

• The components are considered absolutely rigid.

2. Properties#

Property	Default Value	Description	Units of measurement
Pinion radius	0.1	-	m
On - positive rack displacement; Off - negative rack displacement	Off	This setting defines the sign convention between the gear’s angular velocity and the rack’s linear velocity.	-
Calculate losses	Off	-	-
Efficiency, available when `Calculate losses` - `On`	0.8	-	-
Power threshold, available when `Calculate losses` - `On`	0.001	-	W

3. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Pinion	Input	-	-	Rotational motion
Rack	Output	-	-	Translational motion

In order to see the connection capabilities of the block, you need to go to page Description of block ports