Trim 2D. Holly Lewis University of Colorado Center for Aerospace Structures April 29, 2004

Transcription

1 rim D Holly Lewis University of Colorado Center for Aerospace Structures April 9 004

2 Overview rimming an Aircraft in D Forces and Moments AERO-rimD Assumptions Capabilities Results Conclusions Future wor

3 Forces acting on an Aircraft in Flight L is Lift D is Drag M is Pitch Moment is hrust W is weight is angle of attac θ is flap deflection J is Jacobian

4 An Aircraft Straight-line Climging γ is the climb angle V is the velocity vector Horizontal Plane is the ground which we assume is flat; flat is o-ay due to size of airplane compared to earth

5 Forces acting on an Aircraft in Climb

6 What is rimming? rimming is finding the equilibrium of an aircraft w.r.t. a coordinate frame that traverses the aircraft s flight path and is generally attached to the center of gravity. rimming is necessary to eep the aircraft controllable for the pilot and to eep the aircraft in a stable flight pattern.

7 Basic Physical Assumptions for D rim D: longitudinal motions only Forces: Lift and Drag Moments: Pitching Flaps deflect the same on both wings Symmetric Aircraft

8 Body axis system of the Aircraft

9 Lift-Drag axis system for Aircraft

10 Calculating Lift and Drag Forces from Body Forces F Zb F X b otal Aerodynamic Force on Aircraft in Z b Body axis otal Aerodynamic Force on Aircraft in X b Body axis L = ( ) sin( ) F Z cos F b X b D = F Z sin F b ( ) cos( ) X b

11 Lift-Drag axis system for Aircraft

12 Calculating Pitching Moment from Body Forces and Moments Calculate moment about the CG x cg y cg z cg is the position of the CG in the Body frame M OYb is the moment about the Y b Body axis M z F CG Z = M CG Y = M O Y b b cg X b x cg F Z b

13 Including hrust

14 Including Weight

15 My rim Model Uses Newton s Method for finding roots for Systems of Equations Good to have nearly linear functions Must reformulate for equilibrium as: F X ( ) W sin( ) D = cos γ ( ) W cos( ) L FY = sin γ ( ) x cos( ) y M M Z = sin

16 Linear Variation of Lift and Moment Lift vs. Angle of Attac Control Surface Deflection = 0 Lift vs. Control Surface Deflection Factor Alpha =.8 o Lift Angle of Attac (degrees) Lift Control Surface Deflection Factor Moment vs. Angle of Attac Control Surface Deflection = 0 Moment vs. Control Surface Deflection Factor Alpha =.8 o Moment Angle of Attac (degrees) Moment Control Surface Deflection Factor

17 Variation of Drag Drag is non-linear w.r.t angle of attac and w.r.t flap angle but it is nearly linear Drag vs. Angle of Attac Control Surface Deflection = 0 Drag vs. Control Surface Deflection Factor Alpha =.8 o Drag Angle of Attac (degrees) Drag Control Surface Deflection Factor

18 Matrix Formulation For Newton s ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = γ γ cos sin cos sin sin cos M y x L W D W P P P P ( ) ( ) 0 sin cos = D W γ ( ) ( ) 0 cos sin = L W γ ( ) ( ) 0 cos sin = M y x

19 Jacobian Definition ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = P P P P P P P P P J

20 Matrix Formulation For Newton s ( ) [ ] ( ) [ ] P a J = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = y x M M y x L L D D J cos sin sin cos sin cos cos sin

21 Approximating Derivatives in Jacobian D D D L L L M M M D D D L L L M M M

22 Results: vs. Magnitude of Residual for x System Initially [ ] = [ ] Recomputing the Derivatives Magnitude of Residua ( 95.85) Aero ( ) ( 60.99) Aero ( ) ( ) ( ) Aero ( 6.58) Aero ( ) 4 Iteration of Newton's method

23 x Results Case : Initial = o = = N Case : Initial = o = 0. = 500 N Case : Initial = o = 0.5 = 000 N Case 4: Initial =.7848 o = -0.8 = 500 N Constant Derivatives # Iterations Solution Vector Magnitude of Residual Case [ ] [ ] Case [ ] 0.48 [ ] Case [ ] [ ] Case 4 4 [ ] [ ]

24 x Results Case : Initial = o = = N Case : Initial = o = 0. = 500 N Case : Initial = o = 0.5 = 000 N Case 4: Initial =.7848 o = -0.8 = 500 N Updating Derivatives # Iterations Solution Vector Magnitude of Residual Case [ ] [ ] Case [ ] 0.48 [ ] Case [ ] [ ] Case 4 [ ] [ ]

25 x Results Case : Initial = o = = N Case : Initial = o = = N Case : Initial = o = = 500 N Constant Derivatives # Iterations Solution Vector Magnitude of Residual Case [ ] [ ] 0.59 Case 4 [ ] [ ] Case 4 [ ] [ ] 0.75

26 x Results Case : Initial = o = = N Case : Initial = o = = N Case : Initial = o = = 500 N Case 4: Initial =.7848 o = = 500 N Constant Derivatives # Iterations Alpha Solution Magnitude of Residual Case Case Case Case

27 x Results Case : Initial = o = = N Case : Initial = o = = N Case : Initial = o = = 500 N Case 4: Initial =.7848 o = = 500 N Updating Derivatives # Iterations Alpha Solution Magnitude of Residual Case Case Case Case

28 Conclusions AERO-rimD for low Mach number <0.8 produces results which indicate: It wors for or parameters in the trim solution. YEAH!! and aeroelastic are very close YEAH!! Should not be very different for low Mach numbers because the aeroelastic affects are small. Updating the derivatives in the Jacobian does not mae a significant difference.

29 For the Future Extend to D Include lateral-directional motions and trimming parameters i.e. include rudder alieron elevator etc. Develop and test AERO-rimD for other aircraft models besides Langley Fighter est AERO-rimD for large range of Mach numbers Mae the input and output more user friendly

30 References Burden Richard L. and J. Douglas Faires. Numerical Analysis 6th ed. Broos/Cole Publishing Company; Pacific Grove CA: 997. Farhat Charbel and Bruno Koobus. Aero-FD/FD: A User s Manual. University of Colorado Center for Aerospace Structures; Boulder Colorado: 00. Farhat Charbel. RCfem and FEM. University of Colorado Center for Aerospace Structures; Boulder Colorado: 00. Geuzaine Philippe. Aero-F Manual: he documentation for Aero-F Version.0 ed 0.. University of Colorado Center for Aerospace Structures; Boulder Colorado: 00. Lewis Holly. rim D: Reference material for AERO-rimD. University of Colorado Center for Aerospace Structures; Boulder Colorado: 004. Rosam Jan. Airplane Flight Dynamics and Automatic Flight Controls: PartI Design Analysis and Research Corporation; Lawrence KS: 998.