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     "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))."
     ]
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