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"# 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"
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"\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."
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