{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# EoMs for Translation\n", "\n", "With the expression for the absolute acceleration now obtained, we can apply Newton's second law,\n", "\n", "$$\\frac{d\\left(m\\vec{V}\\right)}{dt}_{abs}=\\sum\\vec{F}_{ext}$$ (eq:N2_EOM)\n", "\n", "Where the $()_{abs}$ refers to *absolute acceleration*, defined in an inertially-fixed reference frame - which, for our purposes, we may treat Earth axes as being. As we have shown, we have defined forces in *body axes*, which is moving with respect to Earth axes.\n", "\n", "When we have accelerations and translations defined in a non-inertial reference frame, we must use Eqn {eq}eq:coriolis1 to determine *absolute accelerations*.\n", "\n", "$$\\vec{a}_{b_{abs}} = \\left.\\frac{\\text{d}\\vec{V}_b}{\\text{d}t}\\right|_{Oxyz} + \\vec{\\omega_b}\\times\\vec{V}_b$$\n", "\n", "where\n", "\n", "\\begin{align}\\vec{V}_b&=\\begin{bmatrix}U\\\\V\\\\W\\end{bmatrix}\\\\\n", " \\vec{\\omega}_b&=\\begin{bmatrix}P\\\\Q\\\\R\\end{bmatrix}\\end{align}\n", " \n", "so we have\n", "\n", "\\begin{aligned}\n", " \\vec{a}_{b_{abs}} &=\\begin{bmatrix}\\dot{U}\\\\\\dot{V}\\\\\\dot{W}\\end{bmatrix} + \\left|\\begin{matrix}i & j & k \\\\ P & Q & R \\\\U & V & W\\end{matrix}\\right|\\\\\n", " &= \\begin{bmatrix} \\dot{U} + Q\\,W - R\\,V \\\\ \\dot{V} + R\\,U-P\\,W\\\\\\dot{W}+P\\,V-Q\\,U\\end{bmatrix}\\end{aligned}\n", "\n", "Thus we have defined our **absolute acceleration terms in body axes**, which means we can define the LHS of Equation {eq}eq:N2_EOM.\n", "\n", "$$\\frac{d\\left(m\\vec{V}\\right)}{dt}_{abs}=m\\begin{bmatrix} \\dot{U} + Q\\,W - R\\,V \\\\ \\dot{V} + R\\,U-P\\,W\\\\\\dot{W}+P\\,V-Q\\,U\\end{bmatrix}_b$$\n", "\n", "The RHS of Equation {eq}eq:N2_EOM are the sum of *gravitational/weight*, *aerodynamic*, and *propulsive* forces defined in body axes.\n", "\n", "$$\\sum{\\vec{F}_b} = \\vec{F}_{G_b} + \\vec{F}_{A_b} + \\vec{F}_{P_b}$$\n", "\n", "We have already expressed the aerodynamic and gravitational forces in body axes, Equations {eq}aero_body and {eq}gforce_body, respectively. We now define the propulsive forces.\n", "\n", "We presume the aircraft has one or more propulsors providing thrust, $T$, along a vector defined in the $x/z$ plane, at angle $\\theta_T$ to $x$. We include an additional term for sidewash effects due to propulsion, $F_{Ty}$:\n", "\n", "$$\\vec{F}_{T_b}=\\begin{bmatrix} F_{Tx} \\\\F_{Ty} \\\\F_{Tz}\\end{bmatrix}=\\begin{bmatrix} T\\cdot\\cos\\theta_T \\\\F_{Ty} \\\\-T\\cdot\\sin\\theta_T\\end{bmatrix}$$\n", "\n", "Thus we can write out equations of motion for translation:\n", "\n", "$$m\\begin{bmatrix} \\dot{U} + Q\\,W - R\\,V \\\\ \\dot{V} + R\\,U-P\\,W\\\\\\dot{W}+P\\,V-Q\\,U\\end{bmatrix}=\\begin{matrix} -mg\\sin\\theta - D\\cos\\alpha + L\\sin\\alpha + T\\cos\\theta_T\\\\mg\\sin\\phi\\cos\\theta + F_{A_Y} + F_{T_Y}\\\\mg\\cos\\phi\\cos\\theta - D\\sin\\alpha - L\\cos\\alpha - T\\sin\\theta_T\\end{matrix}$$ (eq:translationalEoM)\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "Which is a pretty big deal to have derived. Equations {eq}eq:translationalEoM describe the response of the aircraft translational acceleration to the aerodynamic forces and vice versa.\n", "\n", "This isn't enough to describe flight, though - for that, we require the ability to describe rotation and attitude." ] }, { "cell_type": "markdown", "metadata": {}, "source": [] }, { "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": "3.8.3" } }, "nbformat": 4, "nbformat_minor": 4 }