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Friction in contact between moving bodies
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_{C}, results in a constant force at any velocity. The viscous friction, F_{V}, opposes motion with the force directly proportional to the relative velocity. The sum of the Coulomb and Stribeck frictions at the vicinity of zero velocity is often referred to as the breakaway friction, F_{brk}. The friction is approximated with the following equations:
$$F=\left({F}_{C}+\left({F}_{brk}-{F}_{C}\right)\xb7\mathrm{exp}\left(-{c}_{v}\left|v\right|\right)\right)sign\left(v\right)+fv$$
$$v={v}_{R}-{v}_{C}$$
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 v_{th} is the velocity threshold. It has been proven experimentally that the velocity threshold in the range between 10^{-4} and 10^{-6} m/s is a good compromise between the accuracy and computational robustness and effectiveness. Notice that friction force computed with this approximation does not actually stop relative motion when an acting force drops below breakaway friction level. The bodies will creep relative to each other at a very small velocity proportional to acting force.
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, 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, which is the friction that opposes motion with a constant force at any velocity. The default value is 20 N.
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).
The parameter sets the value of coefficient c_{v}, 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 3/c_{v}, and 98% at velocity 4/c_{v}, which makes it possible to develop an approximate relationship c_{v} ~= 4/v_{min, }where v_{min} is the relative velocity at which friction force has its minimum value. By default, c_{v} is set to 10 s/m, which corresponds to a minimum friction at velocity of about 0.4 m/s.
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 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.