Friction in contact between moving bodies

Mechanical Translational Elements

The Translational Friction block represents friction in contact between moving bodies. The friction force 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, *F _{S}*,
is the negatively sloped characteristics taking place at low velocities
(see [1]). The
Coulomb friction,

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

$$v={v}_{R}-{v}_{C}$$

where

F | Friction force |

F_{C} | Coulomb friction |

F_{brk} | Breakaway friction |

c_{v} | Coefficient |

v | Relative velocity |

v_{R,}v_{C} | Absolute 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 * v* =
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 force (see [1]). There are numerous
models of friction without discontinuity. The Translational
Friction block implements one of the simplest versions of
continuous friction models. The friction force-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
force is assumed to be linearly proportional to velocity, with the
proportionality coefficient *F _{brk}/v_{th}*,
where

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

If

*|v|*>=*v*,_{th}$$F=\left({F}_{C}+\left({F}_{brk}-{F}_{C}\right)\xb7\mathrm{exp}\left(-{c}_{v}\left|v\right|\right)\right)sign\left(v\right)+fv$$

If

*|v|*<*v*,_{th}$$F=v\frac{\left(f{v}_{th}+\left({F}_{C}+\left({F}_{brk}-{F}_{C}\right)\xb7\mathrm{exp}\left(-{c}_{v}{v}_{th}\right)\right)\right)}{{v}_{th}}$$

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 force from R to C.

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

`25`

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

`20`

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

`100`

N/(m/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`

/where`v`

_{min, }is the relative velocity at which friction force has its minimum value. By default,`v`

_{min}is set to`c`

_{v}`10`

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

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

`1e-6`

and`1e-4`

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

m/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 translational conserving port.

`C`

Mechanical translational conserving port.

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

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