Rack and pinion gear#

Rack and pinion gear block is designed to convert translational motion into rotational motion. This gearing consists of a gear wheel with radius r and a rack, the main parameter of which is the lifting angle \(\alpha\). A negative angle value forms the left thread and a positive angle value forms the right thread.

The input port receives the speed signal in [m/s], the output port receives the angular speed in [rad/s] (see Fig. 1 below). The output port transmits the moment in [N \(\cdot\) m].

The stiffness and damping of the contact surface are necessary to calculate the normal force acting on the threads. Static friction and sliding friction factors shall be specified to account for friction.

In this unit, the “Integrator with reset” friction model is implemented, in which the magnitude of the static friction force depends on the boundary distance of static friction, which is specified in the unit properties.

The resultant force consists of normal and tangent components. Normal force depends on relative speed and distance traveled (1) :

(1)#\[F_{норм.}=d \cdot v_{отн.} + k\cdot \int v_{отн}dt\]

where:

d - damping, [N \(\cdot\) s/m];

k - hardness, [N/m];

v\(_{rel}\) - relative speed, [m/s].

Initially, the transmission is in rest mode, a static friction force is formed at a non-zero relative speed (2) :

(2)#\[F_{статич.}=c_{статич.}\cdot F_{норм.}\]

In static friction, the tangential force depends on the distance boundary s\(_{boundary}\) (3) :

(3)#\[F_{кас.}=\frac {F_{статич.}}{s_{граница}}\cdot s + k_{статич.} \cdot v_{отн.кас.}\]

where:

\(k_{static.}\) - stiffness at static friction, [N/(m/s)];

\(s_{boundary}\) - boundary distance, at overcoming of which sliding mode occurs, [m];

\(c_{static.}\) - coefficient of static friction, [r.u.];

\(v_{rel.tang.}\) - relative speed parallel to screw thread turn surface, [m/s].

In sliding mode, friction is calculated relative to the normal force using the expression (4) :

(4)#\[F_{скольж.}=c_{скольж.}\cdot F_{норм.}\]

The Stribeck effect allows a smooth transition between static and dynamic friction. Turn on the Account Stribeck effect toggle switch to take this effect into account. With this configuration, the friction force becomes exponential and is determined by the expression (5) :

(5)#\[Ф=(F_{скольж.}+(F_{статич.}-F_{скольж.})\cdot e^{(-3\cdot |v_{отн.кас.}|/ c_{Штрибек})})\]

The force on the rail is calculated using the expression (6):

(6)#\[F_1=F_{норм.}\cdot cos(\alpha)-F_{кас.}\cdot sin(\alpha)\]

The torque on the wheel is calculated using the expression (7) :

(7)#\[T_2=R\cdot (F_{норм.}\cdot cos(\alpha)-F_{кас.}\cdot sin(\alpha))\]

visual_demonstration — Fig. 2 Rack elevation angle#

1. Properties#

Property	Default value	Description	Units of measurement
Pin radius	0.01	-	m
Helix angle	2.0	-	deg
Contact stiffness	1000.0	-	N/m
Contact damping	1000.0	-	N/(m/s)
Coulumb friciton coefficient	0.01	-	r.u.
Stiction coefficient	0.02	-	-
Stiction viscous friction	1000	-	N/(m/s)
Stiction displacement threshold	0.001	-	m
Account Stribeck effect	Off	-	-
Stribeck constant, available when `Account Stribeck effect` - `On`	0.001	-	m/s
Reverse motion dircetion	Off	On(TRUE) - conversion of rotational motion into translational motion; Off(FALSE) - conversion of translational motion into rotational motion.	-

2. Inputs and outputs#

Parameters	Input/Output	Description	Units of measurement	Port type
Velocity	Input	Available when the property `Reverse motion dircetion` - `Off`	m/s	Translational motion
Angular velocity	Input	Available when the property `Reverse motion dircetion` - `On`	rad/s	Rotational motion
Torque	Output	Available when the property `Reverse motion dircetion` - `Off`	N \(\cdot\) m	Rotational motion
Force	Output	Available when the property `Reverse motion dircetion` - `On`	N	Translational motion

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