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Friction in contact between rotating bodies
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_{C}, results in a constant torque at any velocity. The viscous friction, T_{V}, opposes motion with the torque 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, T_{brk}. The friction is approximated with the following equations:
$$T=\left({T}_{C}+\left({T}_{brk}-{T}_{C}\right)\xb7\mathrm{exp}\left(-{c}_{v}\left|\omega \right|\right)\right)sign\left(\omega \right)+f\omega $$
$$\omega ={\omega}_{R}-{\omega}_{C}$$
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_{th}, where ω_{th} is the velocity threshold. It has been proven experimentally that the velocity threshold in the range between 10^{-3} and 10^{-5} rad/s is a good compromise between the accuracy and computational robustness and effectiveness. Notice that friction torque computed with this approximation does not actually stop relative motion when an acting torque drops below breakaway friction level. The bodies will creep relative to each other at a very small velocity proportional to acting torque.
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, 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, which is the friction that opposes rotation with a constant torque at any velocity. The default value is 20 N*m.
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).
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/$$\omega $$_{min, }where $$\omega $$_{min} is the relative velocity at which friction torque has its minimum value. By default, c_{v} is set to 10 rad/s, which corresponds to a minimum friction at velocity of about 0.4 s/rad.
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 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 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
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