Velocity and Normal Force Dependent Friction

Velocity and Normal Force Dependent Friction#

This command is used to construct a VelNormalFrcDep friction model.

frictionModel VelNormalFrcDep $tag $aSlow $nSlow
        $aFast $nFast $alpha0 $alpha1 $alpha2 $maxMuFact
tag

unique friction model object tag
aSlow

constant for coefficient of friction at low velocity
nSlow

exponent for coefficient of friction at low velocity
aFast

constant for coefficient of friction at high velocity
nFast

exponent for coefficient of friction at high velocity
alpha0

constant rate parameter coefficient

alpha1

linear rate parameter coefficient

alpha2

quadratic rate parameter coefficient
maxMuFact

factor for determining the maximum coefficient of friction. This value prevents the friction coefficient from exceeding an unrealistic maximum value when the normal force becomes very small. The maximum friction coefficient is determined from μFast, for example μ ≤ $maxMuFac*μFast.

Theory#

  1. Define the friction coefficient at slow (μSlow) and fast (μFast) velocity [1] (Figure 3):

    μSlow = aSlow*N^(nSlow-1)
    μFast = aFast*N^(nFast-1)
    where `aSlow`, `aFast`, `nSlow` ≤ 1, `nFast` ≤ 1 are constants that determine the friction coefficient models. As the friction coefficients μSlow and μFast are unitless, the user must be careful to define the constants to coincide with the units of the model input data.

  2. The friction coefficient as a function of velocity is [2]:

    μ = μFast - (μFast-μSlow )*exp(-a*udot)

    where udot is the velocity at the sliding interface and a is a rate parameter.

  3. In this friction model, a is assumed to be dependent on axial force \(N\) through: $\( a = \alpha_0 + \alpha_1 N + \alpha_2 N^2 \)\( where \)\alpha_0\(, \)\alpha_1\( and \)\alpha_2$ are constants, with units of Time/Length, Time/Length/Force and Time/Length/Force2 respectively.

Special Cases#

  1. Constant friction coefficient:

    μ = const: `nSlow` = `nFast` = 1.0; `aSlow` = $aFast = μ; all other constants defining μ are arbitrary.

  2. Friction coefficient varies with velocity but is independent of vertical force:

    μ is independent of vertical force: $nSlow = $nFast = 1.0; $alpha0 = a (rate parameter); $alpha1 = $alpha2 = 0.

  3. Friction coefficient varies with vertical force but is independent of velocity:

    μ is independent of velocity: $nSlow = $nFast; $aSlow = $aFast; all other constants defining μ are arbitrary.

Examples#

set muSlow 0.12
set muFast 0.18
set nSlow 0.8
set nFast 0.7
set alpha0 25.0
set alpha1 0.0
set alpha2 0.0
frictionModel VelNormalFrcDep 1 [expr $muSlow/pow($W,$nSlow-1.0)] \
        $nSlow [expr $muFast/pow($W,$nFast-1.0)] $nFast $alpha0 $alpha1 $alpha2 3.0

References#

[1] Bowden F.P., Tabor D. (1964). "The friction and lubrication of solids - part II." Oxford University Press, London, Great Britain, 1964.

[2] Constantinou M.C., Mokha A., Reinhorn A. (1990). "Teflon bearings in base isolation. II: Modeling." Journal of Structural Engineering (ASCE) 1990; 116(2): 455-474

Code Developed by: Nhan D. Dao, University of Nevada - Reno. E-mail: nhan.unr@gmail.com

Contents