Friction in contact between rotating bodies

Mechanical Rotational Elements

The Rotational Friction block represents friction in contact between rotating bodies. The friction torque is simulated as a function of relative velocity and is assumed to be the sum of Stribeck, Coulomb, and viscous components, as shown in the following figure.

The Stribeck friction, *T _{S}*,
is the negatively sloped characteristics taking place at low velocities
(see [1]). The
Coulomb friction,

$$T=\left({T}_{C}+\left({T}_{brk}-{T}_{C}\right)\xb7\mathrm{exp}\left(-{c}_{v}\left|\omega \right|\right)\right)sign\left(\omega \right)+f\omega $$

$$\omega ={\omega}_{R}-{\omega}_{C}$$

where

T | Friction torque |

T_{C} | Coulomb friction torque |

T_{brk} | Breakaway friction torque |

c_{v} | Coefficient |

ω | Relative velocity |

ω_{R},ω_{C} | Absolute angular velocities of terminals R and C, respectively |

f | Viscous friction coefficient |

The approximation above is too idealistic and has a substantial drawback. The characteristic is discontinuous at ω = 0, which creates considerable computational problems. It has been proven that the discontinuous friction model is a nonphysical simplification in the sense that the mechanical contact with distributed mass and compliance cannot exhibit an instantaneous change in torque (see [1]). There are numerous models of friction without discontinuity. The Rotational Friction block implements one of the simplest versions of continuous friction models. The friction torque-relative velocity characteristic of this approximation is shown in the following figure.

The discontinuity is eliminated by introducing a very small,
but finite, region in the zero velocity vicinity, within which friction
torque is assumed to be linearly proportional to velocity, with the
proportionality coefficient *T _{brk}/*ω

As a result of introducing the velocity threshold, the block equations are slightly modified:

If |ω| >= ω

,_{th}$$T=\left({T}_{C}+\left({T}_{brk}-{T}_{C}\right)\xb7\mathrm{exp}\left(-{c}_{v}\left|\omega \right|\right)\right)sign\left(\omega \right)+f\omega $$

If |ω| < ω

,_{th}$$T=\omega \frac{\left(f{\omega}_{th}+\left({T}_{C}+\left({T}_{brk}-{T}_{C}\right)\xb7\mathrm{exp}\left(-{c}_{v}{\omega}_{th}\right)\right)\right)}{{\omega}_{th}}$$

The block positive direction is from port R to port C. This means that if the port R velocity is greater than that of port C, the block transmits torque from R to C.

**Breakaway friction torque**Breakaway friction torque, which is the sum of the Coulomb and the static frictions. It must be greater than or equal to the Coulomb friction torque value. The default value is

`25`

N*m.**Coulomb friction torque**Coulomb friction torque, which is the friction that opposes rotation with a constant torque at any velocity. The default value is

`20`

N*m.**Viscous friction coefficient**Proportionality coefficient between the friction torque and the relative angular velocity. The parameter value must be greater than or equal to zero. The default value is

`0.001`

N*m/(rad/s).**Transition approximation coefficient**The parameter sets the value of coefficient

, which is used for the approximation of the transition between the static and the Coulomb frictions. Its value is assigned based on the following considerations: the static friction component reaches approximately 95% of its steady-state value at velocity`c`

_{v}`3`

/, and 98% at velocity`c`

_{v}`4`

/, which makes it possible to develop an approximate relationship`c`

_{v}~=`c`

_{v}`4`

/$$\omega $$where $$\omega $$_{min, }is the relative velocity at which friction torque has its minimum value. By default,_{min}is set to`c`

_{v}`10`

s/rad, which corresponds to a minimum friction at velocity of about`0.4`

rad/s.**Linear region velocity threshold**The parameter sets the small vicinity near zero velocity, within which friction torque is considered to be linearly proportional to the relative velocity. MathWorks recommends that you use values in the range between

`1e-5`

and`1e-3`

rad/s. The default value is`1e-4`

rad/s.

Use the **Variables** tab in the block dialog
box (or the **Variables** section in the block Property
Inspector) to set the priority and initial target values for the block
variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables.

The block has the following ports:

`R`

Mechanical rotational conserving port.

`C`

Mechanical rotational conserving port.

The Mechanical Rotational System with Stick-Slip Motion example illustrates the use of the Rotational Friction block in mechanical systems. The friction element is installed between the load and the velocity source, and there is a difference between the breakaway and the Coulomb frictions. As a result, stick-slip motion is developed in the regions of constant velocities.

[1] B. Armstrong, C.C. de Wit, *Friction Modeling
and Compensation*, The Control Handbook, CRC Press, 1995

Rotational Damper | Rotational Hard Stop | Rotational Spring

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