{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Linear Aerodynamic Terms\n", "\n", "In addition to the terms expressed previously, linear expressions are required for the aerodynamic angles and the total flightspeed.\n", "\n", "## Angle of Attack\n", "\n", "Angle of attack is defined as\n", "\n", "$$\\alpha\\triangleq\\arctan\\frac{W}{U}$$\n", "\n", "which with the small perturbation theory is\n", "\n", "$$\\alpha=\\arctan\\frac{W_0+w}{U_0+u}$$\n", "\n", "in stability axes, $W_0=0$\n", "\n", "$$\\alpha=\\arctan\\frac{w}{U_0+u}$$\n", "\n", "and since $w$ is small\n", "\n", "$$\\alpha\\simeq\\frac{w}{U_0+u}$$\n", "\n", "the perturbational forward speed is much smaller than the trim forward speed and the **linear angle of attack is**:\n", "\n", "$$\\alpha=\\frac{w}{U_0}$$(eq:linearalpha)\n", "\n", "## Sideslip\n", "\n", "Sideslip is defined as\n", "\n", "$$\\beta\\triangleq\\arcsin\\frac{V}{V_f}$$\n", "\n", "where $V_f=\\sqrt{U^2+V^2+W^2}$. Looking at a linear expression for the total flightspeed:\n", "\n", "\\begin{align}V_f&=\\sqrt{\\left(U_0+u\\right)^2+\\left(V_0+v\\right)^2+\\left(W_0+w\\right)^2}\\\\\n", "&= \\sqrt{\\left(U_0+u\\right)^2+v^2+w^2}\\end{align}\n", "\n", "the trim $U_0$ is $\\gg$ all the perturbational terms so\n", "\n", "$$V_f\\simeq U_0$$\n", "\n", "giving the linear sideslip, subject to small $v$ as\n", "\n", "$$\\beta=\\frac{v}{U_0}$$" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.5" } }, "nbformat": 4, "nbformat_minor": 4 }