Aircraft Flight Dynamics MAE331Lecture2.pdf8页

Point-Mass Dynamics and Aerodynamic Forces Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2008 ? Frames of reference ? Velocity and momentum ? Newton!s laws ? Introduction to Lift, Drag, and Thrust ? Simpli?ed longitudinal equations of motion Copyright 2008 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE331.html http://www.princeton.edu/~stengel/FlightDynamics.html Newtonian Frame of Reference ? Newtonian (Inertial) Frame of Reference ?C Unaccelerated Cartesian frame whose origin is referenced to an inertial (non-moving) frame ?C Right-hand rule ?C Origin can translate at constant linear velocity ?C Frame cannot be rotating with respect to inertial origin ? Translation = Linear motion ! r = x y z " # $ $ $ % & ' ' ' ? Position: 3 dimensions ? What is a non-moving frame? Approximations to Inertial Frame of Reference Depend Upon the Application Velocity and Momentum ? Velocity of a particle ! v = dx dt = ?Bx = ?Bx ?By ?Bz " # $ $ $ % & ' ' ' = vx vy vz " # $ $ $ % & ' ' ' ? Linear momentum of a particle ! p = mv = m vx vy vz " # $ $ $ % & ' ' ' where m = mass of particle Newton’s Laws of Motion: Dynamics of a Particle ? First Law ?C If no force acts on a particle, it remains at rest or continues to move in a straight line at constant velocity, as observed in an inertial reference frame -- Momentum is conserved ! d dt mv( ) = 0 ; mv t1 = mv t2 ! d dt mv( ) = m dv dt = F ; F = fx fy fz " # $ $ $ % & ' ' ' ( dv dt = 1 m F = 1 m I3F = 1/m 0 0 0 1/m 0 0 0 1/m " # $ $ $ % & ' ' ' fx fy fz " # $ $ $ % & ' ' ' ? Second Law ?C A particle of ?xed mass acted upon by a force changes velocity with an acceleration proportional to and in the direction of the force, as observed in an inertial reference frame; ?C The ratio of force to acceleration is the mass of the particle: F = m a ? Third Law ?C For every action, there is an equal and opposite reaction Equations of Motion for a Point Mass: Position and Velocity ! dv dt = ?Bv = ?Bvx ?Bvy ?Bvz " # $ $ $ % & ' ' ' = 1 m F = 1/m 0 0 0 1/m 0 0 0 1/m " # $ $ $ % & ' ' ' fx fy fz " # $ $ $ % & ' ' ' ! dr dt = ?Br = ?Bx ?By ?Bz " # $ $ $ % & ' ' ' = v = vx vy vz " # $ $ $ % & ' ' ' Equations of Motion for a Point Mass ! ?Bx(t) = dx(t) dt = f[x(t),F] ? Written as a single equation x " r v # $ % & ' ( = x y z vx vy vz # $ % % % % % % % & ' ( ( ( ( ( ( ( ? With Combined Equations for a Point Mass ! ?Bx ?By ?Bz ?Bvx ?Bvy ?Bvz " # $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' = vx vy vz fx /m fy /m fz /m " # $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' = 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 " # $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' x y z vx vy vz " # $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' + 0 0 0 0 0 0 0 0 0 1/m 0 0 0 1/m 0 0 0 1/m " # $ $ $ $ $ $ $ % & ' ' ' ' ' ' ' fx fy fz " # $ $ $ % & ' ' ' ! FI = fx fy fz " # $ $ $ % & ' ' ' I = Fgravity + Faerodynamics + Fthrust[ ]I Gravitational Force: Flat-Earth Approximation ? g is gravitational acceleration ? mg is gravitational force ? Independent of position ? z measured down ! u = mgf = m 0 0 go " # $ $ $ % & ' ' ' ; go = 9.807 m /s2 Aerodynamic Force ! FI = X Y Z " # $ $ $ % & ' ' 'I = CX CY CZ " # $ $ $ % & ' ' 'I 1 2 (V 2 S = CX CY CZ " # $ $ $ % & ' ' 'I q S ? Referenced to the Earth not the aircraft Inertial Frame Body-Axis Frame Velocity-Axis Frame ! FB = CX CY CZ " # $ $ $ % & ' ' 'B q S ! FV = CD CY CL " # $ $ $ % & ' ' ' q S ? Aligned with the aircraft axes ? Aligned with and perpendicular to the direction of motion Angles Between Reference Frames Velocity Orientation in Inertial Frame Body Orientation in Inertial Frame Velocity Orientation in Body Frame Angles Projected on the Unit Sphere ! " : angle of attack # : sideslip angle $ :vertical flight path angle % : horizontal flight path angle & : yaw angle ' : pitch angle ( : roll angle (about body x ) axis) μ :bank angle (about velocity vector) ? Origin is airplane!s center of mass Angles Related to an Aircraft V, !, " V, #, $ Lift and Drag are Oriented to the Velocity Vector ? Drag components sum to produce total drag ?C Skin friction ?C Base pressure differential ?C Shock-induced pressure differential (M > 1) ? Lift components sum to produce total lift ?C Pressure differential between upper and lower surfaces ?C Wing ?C Fuselage ?C Horizontal tail ! Lift = CL 1 2 "V 2 S # CL0 + $CL $% % & '( ) *+ 1 2 "V 2 S ! Drag = CD 1 2 "V 2 S # CD0 + $CL 2 [ ]1 2 "V 2 S Aerodynamic Lift ? Fast ?ow over top + slow ?ow over bottom = Mean ?ow + Circulation ? Speed difference proportional to angle of attack ? Kutta condition (stagnation points at leading and trailing edges) Chord Section Streamlines ! Lift = CL 1 2 "V 2 S # CLw + CL f + CLht( )1 2 "V 2 S # CL0 + $CL $% % & '( ) *+ 1 2 "V 2 S 2D vs. 3D Lift ? Inward ?ow over upper surface ? Outward ?ow over lower surface ? Bound vorticity of wing produces tip vortices Inward-Outward Flow Tip Vortices Identical Chord Sections In?nite vs. Finite Span
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