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Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using The 1 / 23 Flathead Valley Community College Spring 2016 Introduction Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using The 2 / 23 Interpreting the takeoff of a V35B Beachcraft Bonanza by utilizing Newton’s Second Law, net force equals the rate change of linear momentum. Aviation Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using The 3 / 23 Figure: Wrights’ first powered airplane flew at Kitty Hawk, North Carolina, on December 17, 1903, making a 12-second flight, traveling 36 m (120 ft), with Orville piloting. Aviation Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Figure: SR-71 Blackbird Thrust Lift Normal Force Using The 4 / 23 http://www.globalsecurity.org/intell/systems/sr-71.htm Newtons Second Law Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using The 5 / 23 Newtons 2nd law states the rate change of a objects linear momentum, p is equal to the net force, F on a object. d (p) = ΣF orces dt A objects linear momentum equals the product of its mass, m and its velocity, V. d (mV ) = ΣF orces dt dV dm V +m = ΣF orces dt dt Free Body Diagrams Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using The 6 / 23 AFTER LIFT-OFF ON RUNWAY Forces Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis T hrust = ρ πD2 (Ve − V ) 4 Lif t = FL = Cl ρV 2 Aw 2 Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Drag = FD = rVa2 W eight = W = mg F riction = FF = µk N Lift Normal Force Using The 7 / 23 N ormal = N = W + FL Thrust Impeding Gravity – The take-off of a Beachcraft Bonanza V35B T hrust = ρ πD2 (Ve − V ) 4 William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using The 8 / 23 https://www.grc.nasa.gov/www/k-12/airplane/propth.html Lift Impeding Gravity – The take-off of a Beachcraft Bonanza V35B Lif t = FL = Cl ρV 2 Aw 2 William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using The 9 / 23 http://www.grc.nasa.gov/WWW/K-12/airplane/newton3.html Normal Force Impeding Gravity – The take-off of a Beachcraft Bonanza V35B ON RUNWAY William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces From ΣFy = 0 Thrust Lift Normal Force Using 10 /The 23 N ormal = N = W + FL The Equation General Impeding Gravity – The take-off of a Beachcraft Bonanza V35B m William Alexander Pardis ∂ 2~s ~ +W ~ = F~T + F~L + F~D + F~F + N ∂t2 m ∂ 2~s ~ +D ~ + F~ + N ~ +W ~ = T~ + L ∂t2 Introduction X-Direction Aviation Aviation d2 ~ x dt2 Newtons Second Law = 1 m h i ~ cosθ − Lsinθ ~ T~ − D − F~ Free Body Diagrams Y-Direction Forces Thrust Lift Normal Force Using 11 /The 23 d2 ~ y dt2 = 1 m h i ~ ~ ~ −W ~ Lcosθ + T~ − D)sinθ +N The Equation Complex Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction V˙x = 1 m h gM RL gM ρo 1− LY Vx 2 i T o RL (Ve − Vx ) − rVx 2 cosθ − Cl Ap ρo 1 − LY Aw sinθ − µk N T 2 o Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using 12 /The 23 V˙y = 1 m h ρo Cl gM 1− LY To 2 RL Vx 2 gM i RL Aw cosθ+ Ap ρo 1− LY (Ve −Vy )−rVx 2 )sinθ−W +N To A Typical Take-off Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using 13 /The 23 Necessary takeoff perimeters are dependent on things like terrain, destination, and head/tailwinds advantageous to flight Full throttle is used throughout the take-off but, a normally aspirated aircraft will lose horse power with altitude (about %65 at 10,000ft) due to decreased air density. The fixed angle of attack of attack V35B Beachcraft Bonanza on runway is 3◦ from the horizon. During ascent a AOA of 10◦ is maintained until cruise altitude is achieved, then returned to an AOA of 3◦ for cruise. First Model Equations Impeding Gravity – The take-off of a Beachcraft Bonanza V35B X-Direction William Alexander Pardis Introduction Ẍ = h 1 Ap ρ 2 m 2 (Ẋ 2 − Vx ) − rx Ẋ − 2 Cl ρẊ2 Aw Aviation Aviation Newtons Second Law Y-Direction Free Body Diagrams Forces Thrust Lift Normal Force Using 14 /The 23 Ÿ = 1 m h i ρẊ 2 2 Cl 2 Aw − ry Ẋ + N − W i First Model Results Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using 15 /The 23 Second Model Air Density Impeding Gravity – The take-off of a Beachcraft Bonanza V35B ρo ≡ sea level standard atmospheric pressure William Alexander Pardis g ≡ earth-surface gravitation acceleration Introduction Aviation To ≡ sea level standard temperature L ≡ temperature lapse rate R ≡ ideal (universal) gas constant Aviation Newtons Second Law Free Body Diagrams Forces Thrust M ≡ molar mass of dry air gM LY RL AirDensity ≡ ρ = ρo 1 − To Lift Normal Force Using 16 /The 23 only valid inside the troposphere Second Model Equations Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction X-Direction Ẍ = 1 m h Ap ρ o gM RL 1− LY To (Ẋ − V~x2 )−rx Ẋ 2 −µk N 2 2 i Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using 17 /The 23 Y-Direction h ρo 1 Ÿ = m Cl gM RL 1− LY To Ẋ 2 2 2 Aw − ry Ẋ + N − W i Second Model Results Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using 18 /The 23 Second Model Results Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using 19 /The 23 LONG TERM BEHAVIOR Angle of Attack Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using 20 /The 23 http://www.somersf1.c of-attack-aoa-f1-technical-terms.html P Factor Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using 21 /The 23 Rigid Body Motion Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using 22 /The 23 Thanks You Impeding Gravity – The take-off of a Beachcraft Bonanza V35B William Alexander Pardis Introduction THANKS Flathead Valley Community College Aviation Aviation Newtons Second Law Free Body Diagrams Forces Thrust Lift Normal Force Using 23 /The 23 Don Hickethier Team Differential Equations Class
