{ "metadata": { "name": "", "signature": "sha256:f218ebd237548c708e594d95d56a7506a1bb913f500d55dbe0580203d8b7ea12" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Torsional friction based on Hertzian contact" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is an attempt to document some equations used to derive equations for torsional friction based on Hertzian contact. It is drawn largely from Popov's [Contact Mechanics and Friction (2010)](https://dx.doi.org/10.1007/978-3-642-10803-7)." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Normal contact between rigid sphere and elastic half-space" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "It is shown by Popov in Section 5.1(a) that a rigid sphere (radius $R$) in contact with an elastic half-space (elastic modulus $E$, Poisson ratio $\\nu$) with a circular contact patch of radius $a$ generates a pressure distribution\n", "\n", "$p(r) = p_0 \\sqrt{1 - (\\frac{r}{a})^2}$\n", "\n", "where $r$ is the distance from the center of the contact patch and $0 \\le r \\le a$.\n", "The total normal force $F_z$ is also computed as\n", "\n", "$F_z = \\frac{2}{3} p_0 \\pi a^2$\n", "\n", "In section 5.2, the parameters $p_0$ and $a$ are solved in terms of other parameters and the maximum penetration depth $d$\n", "\n", "$a^2 = Rd$\n", "\n", "$E^* = \\frac{E}{1 - \\nu^2}$\n", "\n", "$p_0 = \\frac{2}{\\pi} E^* \\sqrt{\\frac{d}{R}}$\n", "\n", "so the normal force can be expressed as\n", "\n", "$F_z = \\frac{4}{3} E^* R^{0.5} d^{1.5}$" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Torsional contact between rigid sphere and elastic half-space" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In Section 8.2.4, the case of torsional tangential loading is presented for a rigid sphere (radius $R$) in contact with an elastic half-space (elastic modulus $E$, Poisson ratio $\\nu$) with a circular contact patch of radius $a$. That example uses a shear stress distribution corresponding to \"stiction\", a uniform rotation of all components of the contact patch.\n", "\n", "This analysis uses a different shear stress distribution that corresponds to the maximum torsional moment during sliding, with a direct relation to the normal pressure distribution." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The torsional loading at any point in the contact patch (specified in polar coordinates by radius $r$ and angle $\\phi$) consists of tangential forces directed perpendicular to the polar radius $r$ with stresses\n", "\n", "$\\sigma_{zx} = -\\tau(r) \\sin \\phi$\n", "\n", "$\\sigma_{zy} = \\tau(r) \\cos \\phi$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The stresses at a given position are combined into the vector\n", "\n", "$\\boldsymbol{\\sigma}_z = (\\sigma_{zx},\\sigma_{zy},0)^T$\n", "\n", "and with the incremental area at a given point defined as\n", "\n", "$dA = r dr d\\phi$\n", "\n", "the incremental force at a given point is\n", "\n", "$d\\textbf{F}_z = \\boldsymbol{\\sigma}_z dA$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The position vector\n", "\n", "$\\textbf{r} = (r \\cos \\phi,r \\sin \\phi,0)^T$\n", "\n", "is then used to compute the incremental torsional moment\n", "with respect to the center of the contact patch\n", "\n", "$dM_z = \\textbf{r} \\times d\\textbf{F}_z = r^2 \\tau(r) dr d\\phi$\n", "\n", "The full torsional moment is computed by integrating\n", "over the entire circular area of the contact patch:\n", "\n", "$M_z = \\int dM_z = \\int_0^{2\\pi}\\int_0^{a} r^2 \\tau(r) dr d\\phi$\n", "\n", "with the following simplification due to angular symmetry:\n", "\n", "$M_z = 2\\pi \\int_0^{a} r^2 \\tau(r) dr$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For sliding torsion, $\\tau(r)$ has a similar form to the\n", "normal pressure $p(r)$ generated by a rigid sphere\n", "contacting an elastic half-space (see the previous section):\n", "\n", "$\\tau(r) = \\tau_0 \\sqrt{1 - (\\frac{r}{a})^2}$\n", "\n", "The torsional moment integral then becomes:\n", "\n", "$M_z = 2\\pi \\tau_0 \\int_0^{a} r^2 \\sqrt{1 - (\\frac{r}{a})^2} dr$\n", "\n", "and is simplified with a substitution $s = \\frac{r}{a}$ and $dr = a ds$:\n", "\n", "$M_z = 2\\pi \\tau_0 \\int_0^{1} (as)^2 \\sqrt{1 - s^2} a ds$\n", "\n", "or\n", "\n", "$M_z = 2 a^3 \\pi \\tau_0 \\int_0^{1} s^2 \\sqrt{1 - s^2} ds$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "From [integrals.wolfram.com](http://integrals.wolfram.com/index.jsp?expr=x%5E2+Sqrt%5B1+-+x%5E2%5D&random=false):\n", "\n", "$\\int s^2 \\sqrt{1 - s^2} ds = \\frac{1}{8} (s \\sqrt{1-s^2} (2s^2 -1) + \\sin^{-1} s)$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The first term evaluates to $0$ at both $s=0$ and $s=1$, so only the second term needs to be considered. The torsional moment then evaluates to\n", "\n", "$M_z = 2 a^3 \\pi \\tau_0 \\frac{1}{8} (\\sin^{-1} 1 - \\sin^{-1} 0)$\n", "\n", "or\n", "\n", "$M_z = 2 a^3 \\pi \\tau_0 \\frac{1}{8} (\\pi / 2)$\n", "\n", "or\n", "\n", "$M_z = \\frac{\\pi}{8} a^3 \\pi \\tau_0$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The ratio between the torsional moment and normal force $M_z / F_z$ is then computed as" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$M_z = \\frac{\\pi}{8} a^3 \\pi \\tau_0$\n", "\n", "$F_z = \\frac{2}{3} p_0 \\pi a^2$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\frac{M_z}{F_z} = \\frac{3\\pi}{16} a \\frac{\\tau_0}{p_0}$\n", "\n", "and labelling the stress ratio $\\frac{\\tau_0}{p_0}$ as $\\mu$,\n", "\n", "$\\frac{M_z}{F_z} = \\frac{3\\pi}{16} a \\mu$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "This analysis indicates that the sliding torsional moment is equivalent\n", "to a tangential friction force acting at a radius of about 59% of the\n", "contact patch radius.\n", "It also matches a result reported in equation (10) of \"Soft Finger Model with Adaptive Contact Geometry for Grasping and Manipulation Tasks (2007)\" (DOI [10.1109/WHC.2007.103](http://dx.doi.org/10.1109/WHC.2007.103) and [pdf here](http://academiccommons.columbia.edu/download/fedora_content/download/ac:154103/CONTENT/04145178.pdf))." ] }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [] } ], "metadata": {} } ] }