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      "# InertiaMsgsTest::SetPendulumInertia\n",
      "\n",
      "This documents the effect of moment of inertia on the expected natural frequency in the pendulum test.\n",
      "\n",
      "## Pendulum dimensions\n",
      "\n",
      "![Pendulum dimensions](pendulum.svg)\n",
      "\n",
      "A pendulum is illustrated with distance $L$ between the pin joint and center of mass.\n",
      "The pendulum is modeled as a box of mass $m$ with overall length $2L$ and width $L/5$.\n",
      "\n",
      "## Moment of inertia\n",
      "\n",
      "Computing the moment of inertia requires specifying a location on the body and an axis direction.\n",
      "In the following equation, the moment of inertia $I$ is computed\n",
      "at the center of mass along an axis parallel to the axis of rotation:\n",
      "\n",
      "$I = \\frac{m}{12} ((2L)^2 + (\\frac{L}{5})^2)$\n",
      "\n",
      "$I = mL^2 (\\frac{1}{3} + \\frac{1}{300})$\n",
      "\n",
      "$I = \\frac{101}{300} mL^2$\n",
      "\n",
      "## Natural frequency\n",
      "\n",
      "With gravity $g$ and pendulum angle $\\theta$, the equations of motion are given as:\n",
      "\n",
      "$(I + mL^2) \\ddot{\\theta} + mgL * sin(\\theta) = 0$\n",
      "\n",
      "Factoring out $mL^2$ and dividing by $mgL$,\n",
      "\n",
      "$\\frac{mL^2}{mgL} (\\frac{I}{mL^2} + 1) \\ddot{\\theta} + sin(\\theta) = 0$\n",
      "\n",
      "$\\frac{L}{g} (\\frac{I}{mL^2} + 1) \\ddot{\\theta} + sin(\\theta) = 0$\n",
      "\n",
      "With the value of $I$ computed above,\n",
      "\n",
      "$ \\frac{401}{300} \\frac{L}{g} \\ddot{\\theta} + sin(\\theta) = 0$\n",
      "\n",
      "Then when $\\theta$ is small, $sin(\\theta) \\approx \\theta$\n",
      "and the pendulum will have an approximately sinusoidal trajectory.\n",
      "The frequency $\\omega$ of the sinusoidal trajectory satisfies the following:\n",
      "\n",
      "$ \\frac{401}{300} \\frac{L}{g} \\omega^2 = 1$\n",
      "\n",
      "$ \\omega^2 = \\frac{300}{401} \\frac{g}{L} $\n",
      "\n",
      "The frequency $f$ in Hz is then:\n",
      "\n",
      "$ f = \\frac{1}{2\\pi} \\sqrt{\\frac{300}{401} \\frac{g}{L}}$\n",
      "\n",
      "## Modified natural frequency with larger inertia\n",
      "\n",
      "Suppose the inertia $I$ is artificially increased by a factor of $2$:\n",
      "\n",
      "$I = \\frac{101}{150} mL^2$\n",
      "\n",
      "Then the natural frequency changes as follows:\n",
      "\n",
      "$\\frac{L}{g} (\\frac{I}{mL^2} + 1) \\ddot{\\theta} + sin(\\theta) = 0$\n",
      "\n",
      "$ \\frac{251}{150} \\frac{L}{g} \\ddot{\\theta} + sin(\\theta) = 0$\n",
      "\n",
      "$ f = \\frac{1}{2\\pi} \\sqrt{\\frac{150}{251} \\frac{g}{L}}$\n",
      "\n"
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