University of Florida/Egm6321/f09.team1.gzc/Mtg34

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page34-1 Template:Font Find angle of attack and thrust to reach a given altitude inTemplate:Font Template:Font Bryson & Denham 1962 Template:Font: min altitude to reach target with a bunt maneuver (inverted loop) page34-2 $V=velocityofP(airplane=pt)$ $T=thrust//airplaceaxis$ $\alpha =\mathrm{\angle }(V,T)=angleofattack$ $\gamma =\mathrm{\angle }(x,colorblueV)=anglebetweenhonzxaxisandveloctity$ $D=axialaeroforce//T$ $(parall.airplaneaxis)$ page34-3 $L=Tramsv.aero.force\mathrm{\perp }T$ $W=mg=airplaneweight.$ $Equationofmotion(EOM)$ $colorblueControlinput:\underset{timedependent}{\underbrace{T(t)},\alpha (t)}$ $statevar,:x(t),y(t)v(t),r(t)$ $\{\begin{array}{cc}& D=\frac{L}{2}{C}_d\rho V^2{S}_{reft}\\ & L=\frac{L}{2}{C}_l\rho V^2{S}_{reft}\end{array}$ $C_d=drgcoefficient|\rho =airdensity$ $C_l=liftcoefficient|{\mathrm{\S }}_{ref}=ref.areaofairplane$ page34-4 $C_d=A_1{\alpha }^2+A_2\alpha +A_3(1)$ $(A_1,A_2,A_3)=curvefittingcoefficient$ $C_l=B_1\alpha +B_2(2)$ $(B_1,B_2)=curvefittingcoefficient$ $\rho =C_1{h}^2+C_{2}h+C_3(3)$ $(C_1,C_2,C_3)=curvefittingcoefficient$ Template:Font S & Z 2007 $\underset{\_}{Kinematics:}\frac{dx}{dt}=V_x=Vcosr(4)$ $\frac{dy}{dt}=V_y=Vsinr(5)$ $\underset{\_}{Kinematics:}Eulerequations:\frac{d}{dt}\underset{\_}{P}=\sum _i{\underset{\_}{F}}_i(6)$ page34-5 $\underset{co{s}^{2}t}{\underbrace{{\left(\frac{x}{a}\right)}^2}}+\underset{si{n}^{2}t}{\underbrace{{\left(\frac{y}{b}\right)}^2}}=1ellipse$ $<dl=[d{x}^2+d{y}^2{]}^{1/2}/math><math>Eccentricity:e={(1-\frac{b^2}{a^2})}^{1/2}$ $C=\int dl=a\underset{t=0}{\overset{2\pi }{\int }}[1-e^{2}co{s}^{2}t{]}^{1/2}dt$ $=41\underset{\alpha =0}{\overset{\frac{\pi }{2}}{\int }}[1-e^{2}si{n}^2\alpha {]}^{\frac{1}{2}}d\alpha =4aE(e)$