Practical parametric geometry for aircraft design 26 May, 2015 J. Philip Barnes “Regenosoar” regen-electric aircraft rendered with Blender 3D open-source graphics software and 100% math.
Download ReportTranscript Practical parametric geometry for aircraft design 26 May, 2015 J. Philip Barnes “Regenosoar” regen-electric aircraft rendered with Blender 3D open-source graphics software and 100% math.
Practical parametric geometry for aircraft design
26 May, 2015 J. Philip Barnes
“Regenosoar” regen-electric aircraft rendered with Blender 3D open-source graphics software and 100% math modeled with parametric equations via Blender’s integrated Python programming language
J. Philip Barnes www.HowFliesTheAlbatross.com
Abstract
Practical parametric geometry for aircraft design
J. Philip Barnes, Technical Fellow, Pelican Aero Group Theory and application of practical methods for aircraft geometry parameterization and visualization are described. The methods, characterizing the surface geometry of complete aircraft, wings, fuselages, ducts, and new or existing airfoils, include fidelities ranging from “rapid visualization” to “high fidelity.
” We apply two integrated programming and visualization platforms. The first is
EXCEL
with its resident and
Visual Basic Python
and the second is
Blender 3D
(open-source) programming language. In all cases, we characterize Cartesian coordinates (x,y,z) with parametric coordinates (u,v).
For “rapid synthesis,” we introduce modified trigonometric functions capable of quickly approximating an airfoil, wing, or fuselage with just a handful of parameters. We also introduce a “
cubic quadrant
” method for for fuselage cross section design. For “good fidelity” modeling of new or existing airfoils, we introduce a “
parametric Fourier series
” method satisfying specified leading edge radius, max&min vertical coordinates, upper&lower afterbody slopes, and aft edge thickness. A “fine tuning” parameter allows further subtle adjustments.
Upper and lower surfaces can also be modeled separately for greater control.
For “high fidelity,” we describe the theory and application of the
cubic spline
which is unique in its class by passing through, not just near, all specified points while preserving C2 continuity. Although the cubic spline not C3 continuous, we show that airfoil surface velocity distributions remain smooth with cubic-spline parameterization of the airfoil geometry. We also apply the cubic spline to characterize wings and fuselages. Core algorithms and code blocks are listed or otherwise made available to ensure ready access to the methods.
J. Philip Barnes www.HowFliesTheAlbatross.com
Presentation Contents ~ Practical parametric geometry
Air vehicle Applications Objectives &Rationale EXCEL/VB Blender/Python Cubic spline Theory & App.
Airfoil Geom.
Fourier-series
J. Philip Barnes www.HowFliesTheAlbatross.com
“Rapid vis” Trigonometric
Blender 3D rendering of python-programmed geometry Python window Rendering window
J. Philip Barnes www.HowFliesTheAlbatross.com
Getting started: EXCEL as a scientific spreadsheet
• Purpose (typical):
• Read input and/or data from spreadsheet • Edit & run algorithm; generate new data • Write to spreadsheet cells & plot results • Copy all data & plots as new sheet; re-run
• One-time setup:
1) EXCEL Options ~ Formulas ~ R1C1 ...
2) Trust Ctr. ~ settings ~ macro ~ enable & trust 3) Toolbar ~ more... ~ all ... ~ Visual Basic ~ Add 4) Set VB editor window to float on spreadsheet
• Typical operations:
1) Type in the column headers, i.e. t, x, y, z 2) VB ~ insert ~ module ~
Type
: sub example 3) Enter or edit code ~ save file as *.xlsm
4) Click
run
icon (note: module stays with the file) 5) Highlight applicable columns & plot the results 6) New case: Copy sheet, revise inputs, repeat 4)
Powerful Parametrics for Airfoil Geometry
J. Philip Barnes June, 2015
U W Cubic Spline W
J. Philip Barnes www.HowFliesTheAlbatross.com
Airfoil parametric geometry
•
Objectives and Applications
– Closely match/smooth existing airfoils – Geometric design of new airfoils – Option: modest-fidelity rapid vizualization •
Three methods herein
– Trigonometric (“Rapid viz”) – Fourier Series (good fidelity) – Parametric cubic spline (high fidelity) •
Common approach
– One or two parametric surfaces – Set LE radius, 1-to-3 midpoints, aft slope – – X(W) parametric, 0 ≤ W ≤ 1 , front to back Z(U) Fourier, or Z(W) polynomial or spline – – “Fine tuning” via one or more aux. params. EXCEL files included herein, each method
J. Philip Barnes www.HowFliesTheAlbatross.com
“Rapid viz” airfoil shaping: Hybrid Cartesian & trig. functions
X = 1 - sin( p u) ; Z = c sin(2 p u)
1. simple wave, Z(u)
D Z = c sin 2 (2 p u)
4. add camber
X = 1 - (1 g ) sin( p u) + g sin(3 p u)
2. reshape X(u)
D Z = c sin (X 3 p )
5. lower negative cusp
D Z = c sin (X 3 p ) D Z = c sin (X 3 p )
3. add aftbody cusp 6. opposite-sign cusps
J. Philip Barnes www.HowFliesTheAlbatross.com
“Higher” fidelity ~ Parametric Fourier-series airfoil
U W • • • • • • • • • • • • • • Fourier Series terms z(u) Best used for one curve Z(U), not two Z(W) Add 8 sinusoidal terms plus aft-edge width Single L.E. rad.(R), max/min (X,Z) , two aft ( b ) Use upper & lower fine-tune parameters ( g ) Continuous in all derivatives Solve eight eqns. for Fourier amplitudes Satisfy end slopes (dW/dZ) & max/min Compact “airfoil-sharing” formula Airfoil construction sequence: U = 0 to 1 ; W = if(U < 0.5, 1 - 2U, 2U - 1) g = g b + ( g X = 1 - (1 t g g b ) cos ) cos( p 2 ( p U/2) W/2) – g cos(3 p W/2) Z = S m=1 to 8 {a m sin(m p U)} + Z a (1-2U) W 1 X
Parameterization for X(W)
g
“fine-tune” parameter
0 0 W 1 Z 0 Fourier Series
First 4 terms of the series
U
J. Philip Barnes www.HowFliesTheAlbatross.com
Parametric Fourier-Series airfoil
~ NLF(1)-0416 ~ match
Fwd fine tuning, g inputs: 0.20
0.35000
0.17245
0.09366
0.30000
0.16
Upper g u 0.37500
0.12403
0.08208
0.25000
Z(X) Z(X)
0.070
0.15
0.40000
0.08190
0.06715
0.20000
Fourier Series Lower g L 0.42500
0.04734
0.05008
0.15000
0.14
0.130
0.10
0.45000
0.02153
0.03232
0.10000
Target Airfoil 0.47500
0.00548
0.01527
0.05000
0.05
NLF(1)-0416 specifications L.E. rad., R = r/c 0.0180
0.50000
0.00000
0.00000
0.00000
0.12
0.00
0.52500
0.00560
-0.01290
0.05000
-0.05
-0.10
0.55000
0.57500
0.60000
0.62500
0.10
Upp. max. position, Xu 0.3000
Low. min. position, XL 0.3500
-0.15
0.0
1.0
0.1
0.2
0.3
0.4
0.5
0.6
0.02243
0.05031
0.08864
0.13645
-0.02337
-0.03177
-0.03863
-0.04443
0.10000
0.15000
0.20000
0.25000
0.08
Upp. max. elevation, Zu 0.1045
Low. min. elevation, ZL -0.0555
Upp. aft slope, b u, deg 13.00
Low. aft slope, b L, deg -10.00
Half trailing-edge, Za 0.0030
0.8
0.6
0.4
0.2
0.0
0.0
0.15
X(u)
0.1
0.2
0.3
0.4
0.5
0.6
0.67500
0.70000
0.72500
0.75000
0.77500
0.80000
0.82500
0.85000
0.87500
0.90000
0.92500
0.95000
1.00000
0.25506
0.32250
0.39288
0.46431
0.53503
0.60345
0.66831
0.72871
0.78419
0.83469
0.88060
0.92269
1.00000
-0.05316
-0.05529
-0.05499
-0.05163
-0.04499
-0.03545
-0.02405
-0.01231
-0.00201
0.00529
0.00857
0.00767
-0.00300
0.35000
0.40000
0.45000
0.50000
0.55000
0.60000
0.65000
0.70000
0.75000
0.80000
0.85000
0.90000
1.00000
0.06
0.04
0.02
0.00
-0.02
0.10
Z(u)
specifications Fourier Series -0.04
RUN
0.05
0.00
-0.06
Trailing edge notes
No airfoil can have zero trailing-edge thickness; nor should it. Assume 0.001 aft-edge thickness, unless input otherwise -0.05
-0.10
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-0.08
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Upper and Lower 2nd Derivatives, d 2 Z/dW 2
0.6
Upper & Lower 1st Derivatives, dZ/dW Vs. W
5.0
Fourier Coefficients
2.2794E-02 6.9443E-02 1.8405E-02 -1.6762E-02 1.3916E-03 -4.7482E-03 5.7805E-03 -8.0577E-05 0.4
0.2
4.0
3.0
2.0
1.0
0 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 0.0
0 -1.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-0.2
0.9
-2.0
-0.4
-3.0
-4.0
-0.6
1.0
1
J. Philip Barnes www.HowFliesTheAlbatross.com
Parametric Fourier-Series airfoil
~ PCS-001 ~ new design
Fwd fine tuning, g inputs: 0.20
0.35000
0.23585
0.12542
0.30000
0.18
Upper g u 0.37500
0.16765
0.10526
0.25000
Z(X) Z(X)
0.200
0.15
0.40000
0.10905
0.08159
0.20000
Fourier Series Lower g L 0.42500
0.06190
0.05708
0.15000
0.10
0.16
0.100
0.45000
0.02757
0.03422
0.10000
Target Airfoil PCS-001 0.47500
0.00686
0.01486
0.05000
L.E. rad., R = r/c 0.0115
0.05
0.50000
0.00000
0.00000
0.00000
specifications 0.00
0.52500
0.00667
-0.01022
0.05000
0.14
-0.05
-0.10
0.55000
0.57500
0.60000
0.62500
0.12
Upp. max. position, Xu 0.4140
Low. min. position, XL 0.3500
-0.15
0.0
1.0
0.1
0.2
0.3
0.4
0.5
0.6
0.02606
0.05695
0.09782
0.14694
-0.01639
-0.01953
-0.02075
-0.02101
0.10000
0.15000
0.20000
0.25000
0.10
Upp. max. elevation, Zu 0.1465
Low. min. elevation, ZL -0.0210
Upp. aft slope, b u, deg 7.00
Low. aft slope, b L, deg 4.30
Half trailing-edge, Za 0.0005
0.8
0.6
0.4
0.2
0.0
0.0
0.25
X(u)
0.1
0.2
0.3
0.4
0.5
0.6
0.67500
0.70000
0.72500
0.75000
0.77500
0.80000
0.82500
0.85000
0.87500
0.90000
0.92500
0.95000
1.00000
0.26270
0.32583
0.39037
0.45502
0.51875
0.58079
0.64064
0.69803
0.75295
0.80552
0.85606
0.90498
1.00000
-0.02096
-0.02099
-0.02097
-0.02076
-0.02025
-0.01941
-0.01826
-0.01680
-0.01501
-0.01286
-0.01030
-0.00732
-0.00050
0.35000
0.40000
0.45000
0.50000
0.55000
0.60000
0.65000
0.70000
0.75000
0.80000
0.85000
0.90000
1.00000
0.08
0.06
0.04
0.02
0.20
Z(u)
specifications Fourier Series 0.15
0.00
0.10
0.05
-0.02
RUN
0.00
-0.05
-0.10
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
-0.04
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Upper and Lower 2nd Derivatives, d 2 Z/dW 2
0.6
Upper & Lower 1st Derivatives, dZ/dW Vs. W
5.0
4.0
0.4
3.0
2.0
0.2
0 0 -0.2
-0.4
-0.6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 1.0
0.0
0 -1.0
-2.0
0.1
0.2
0.3
0.4
-3.0
-4.0
J. Philip Barnes www.HowFliesTheAlbatross.com
0.5
0.6
0.7
0.8
0.9
1.0
1
Cubic spline
~ Parametric u(t) or Cartesian y(x) • Get smooth curve passing through (1_to_n) points • VB array dim. (n) elements: 0_to_n ~ ignore 0
th
elem.
• 1
st
& 2
nd
derivative Continuity (3
rd
is not continuous) • Independently control L/R-end slope or 2
nd
derivative • Internal-node continuity yields tri-diagonal system • End constraints are applied in first and last rows • Parametric x(t) ; v
“velocity”; a
“acceleration”
x i-1 v dx/dt a ≡ d 2 x/dt 2 i cubic parabolic linear i+1 x 1 2 3 n t t t t : 0 0 0 0 : 1 p 2 0 ...
0 q p 2 3 ...
0 0 0 r q 0 ...
...
0 2 0 3 p i 0 0 0 r 3 0 ...
q p ...
0 i 0 ...
n 2 0 0 0 r i 0 q p 0 ...
n ...
0 2 n 1 0 r n 0 2 q n 1 0 r n 0 0 0 0 0 1 1 a a 1 2 : a i 1 a i : a i a n 1 a n 1 : s s 0 s : s n i 2 3 s n 0 1 2
• Set ends; Solve linear EQs. for internal-knot accelerations (a)
Parametric cubic spline
~ Various end constraints
“Stiff” ends “Flexible” ends “Flat” ends
Parametric cubic spline airfoil
• • • • • • • • • • • • • • • Cubic spline(s) pass through all set points Wider design space including “unusual” Match 0 th , 1 st , 2 nd derivatives, ea. node Discontinuous 3 rd derivative Input LE rad.(R), 3 pts. (X,Z) , aft slope ( b ) g can be varied but is normally fixed (0.1) Solves 5 eqns. spline-knot 2 nd derivatives Gauss-Seidel in lieu of Gaussian Diag.
3 midpoints versus single midpoint Any position, not necessarily max/min Less compact “airfoil-sharing package” U = 0 to 1 ; W = if(U < 0.5, 1 - 2U, 2U - 1) X = 1 - (1 g ) cos( p W/2) – g cos(3 p W/2) EXCEL solves for cubic splines, Z(W) Package: sol’n data block & interpolator U W W Cubic Spline 1 X g 0 0 W 1
-
Z + 0 0 W 1
J. Philip Barnes www.HowFliesTheAlbatross.com
Parametric cubic spline airfoil
Sample Gauss-Seidel convergence J. Philip Barnes www.HowFliesTheAlbatross.com
Parametric cubic spline airfoil ~ 13-point match
Parametric cubic spline (blue) closely matches target (white points)
J. Philip Barnes www.HowFliesTheAlbatross.com
Parametric cubic-spline airfoil
~ NLF(1)-0416 ~ 9-pt match
Upp. L.E. rad., Ru=r/c 0.20
0.14286
0.70700
0.06143
0.0130
Z(X)
0.15306
0.68289
0.06538
0.69388
Z(X)
Low. L.E. rad., RL=r/c 0.15
W 0.16327
0.65830
0.06926
0.67347
0.0130
0.17347
0.63324
0.07304
0.65306
0.10
0.18367
0.60774
0.07671
0.19388
0.58182
0.08025
0.61224
fwd upper, Xfu 0.1500
fwd lower, XfL 0.1500
0.05
0.00
-0.05
NLF(1)-0416 0.20408
0.21429
0.55555
0.52896
0.08363
0.08683
0.22449
0.23469
0.50211
0.47507
0.08984
0.09263
0.59184
0.57143
0.08
0.53061
W fwd upper, Zfu 0.0900
fwd lower, ZfL -0.0480
-0.10
-0.15
0.0
1.00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.24490
0.25510
0.27551
0.28571
0.44791
0.42072
0.36654
0.33974
0.09519
0.09747
0.10104
0.10219
0.51020
0.46939
0.44898
0.42857
mid upper, Xmu 0.4500
mid lower, XmL 0.4500
mid upper, Zmu 0.0950
mid lower, ZmL -0.0520
aft upper, Xau 0.8000
aft lower, XaL 0.8000
0.75
0.50
0.25
0.00
0.0
0.15
X(u)
Upper Lower 0.29592
0.30612
0.31633
0.32653
0.33673
0.34694
0.35714
0.31326
0.28721
0.26168
0.23678
0.21261
0.18927
0.16688
0.36735
0.37755
0.14552
1 W 0 W 0.12530
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.38776
0.8
0.40816
0.10631
0.9
0.07238
0.10286
0.10298
0.10249
0.10133
0.09945
0.09679
0.09330
0.38776
0.36735
0.32653
0.30612
0.28571
0.24490
0.22449
0.18367
spline Target Airfoil specifications 0.10
Z(u)
0.41837
specifications 0.05760
0.04438
0.08891
1 0.08362
0.07757
1.0
0.06371
0.05618
0.04844
0.16327
0.14286
aft upper, Zau 0.0450
aft lower, ZaL 0.0000
Upp. aft slope, b u, deg 14.00
Low. aft slope, b L, deg -14.00
Half trailing-edge, Za 0.0030
Public Domain
J. Philip Barnes
Phil's web site 0.05
0.00
-0.05
-0.10
0.0
0.6
0.4
0.2
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.44898
0.45918
0.46939
0.48980
0.51020
0.53061
0.54082
0.55102
0.56122
0.57143
0.58163
0.59184
0.60204
0.61224
0.03279
0.02288
0.01470
0.00829
0.00369
0.00092
0.00000
0.00092
0.00369
0.00829
0.01470
0.02288
0.03279
0.04438
0.05760
0.07238
0.08864
0.10631
0.04063
0.03288
0.02534
0.01814
0.01142
0.00533
-0.00481
-0.00945
-0.01389
-0.01815
-0.02221
-0.02609
-0.02976
-0.03323
-0.03650
-0.03956
-0.04241
0.10204
0.08163
0.06122
-0.06
0.02041
0.00000
-0.08
0.00000
0.0
0.02041
0.04082
0.08163
0.12245
0.16327
2.0
0.18367
0.22449
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Upper and Lower 2nd Derivatives, d 2 Z/dW 2
0.8
0.9
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
RUN
-0.2
-0.4
-0.6
0.63265
0.64286
0.65306
0.66327
0.67347
0.68367
0.69388
0.14552
0.16688
0.18927
0.21261
0.23678
0.26168
0.28721
-0.04747
-0.04967
-0.05163
-0.05333
-0.05473
-0.05580
-0.05654
0 0.28571
0.32653
0.36735
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1
J. Philip Barnes www.HowFliesTheAlbatross.com
Parametric cubic-spline airfoil
~ PCS-001 ~ new design
X = 1 - (1 ) cos ( W/2) cos (3 W/2) Upp. L.E. rad., Ru=r/c 0.20
0.14286
0.70700
0.09871
0.0120
Z(X)
W 0.15306
0.68289
0.10702
0.69388
Z(X)
Low. L.E. rad., RL=r/c 0.15
0.16327
0.65830
0.11486
0.67347
spline 0.0120
0.10
0.17347
0.63324
0.12209
0.65306
0.18367
0.60774
0.12856
Target Airfoil PCS-001 fwd upper, Xfu 0.05
0.19388
0.58182
0.13415
0.61224
specifications 0.1500
0.20408
0.55555
0.13872
0.59184
0.00
fwd lower, XfL 0.21429
0.52896
0.14227
0.57143
0.12
0.1500
-0.05
W 0.22449
0.50211
0.14485
0.23469
0.47507
0.14649
0.53061
-0.10
fwd upper, Zfu 0.1000
fwd lower, ZfL 0.24490
0.25510
0.44791
0.42072
0.14722
0.14708
0.51020
-0.15
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.46939
-0.0204
0.27551
0.36654
0.14433
0.44898
1.00
0.28571
0.33974
0.14179
0.42857
mid upper, Xmu 0.5800
mid lower, XmL 0.4500
0.75
0.50
X(u)
Upper Lower 0.29592
0.30612
0.31633
0.32653
0.33673
0.31326
0.28721
0.26168
0.23678
0.21261
0.13852
0.13456
0.12995
0.12471
0.11889
0.38776
0.36735
0.32653
mid upper, Zmu 0.1345
mid lower, ZmL -0.0200
0.25
0.34694
0.35714
0.18927
0.16688
0.11251
0.10563
0.30612
0.28571
aft upper, Xau 0.8000
aft lower, XaL 0.8000
0.00
0.0
0.15
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.37755
0.38776
0.8
0.40816
0.14552
0.12530
0.10631
0.9
0.07238
0.09047
0.08236
1.0
0.06554
0.24490
0.22449
0.18367
0.10
Z(u)
0.41837
specifications 0.05760
0.04438
0.05704
0.04861
0.16327
0.14286
aft upper, Zau 0.0630
aft lower, ZaL -0.0130
Upp. aft slope, b u, deg 8.00
Low. aft slope, b L, deg 3.50
Half trailing-edge, Za 0.0010
Public Domain
J. Philip Barnes
Phil's web site 0.05
0.00
-0.05
-0.10
0.0
0.6
0.4
0.2
0 0.44898
0.45918
0.03279
0.02288
0.01470
0.46939
0.50000
0.00829
0.47959
0.48980
0.00369
Aft upper Aft lower 0.00092
0.00000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.51020
0.53061
0.54082
0.55102
0.56122
0.57143
0.58163
0.59184
0.60204
0.61224
0.00092
0.00369
0.00829
0.01470
0.02288
0.03279
0.04438
0.05760
0.07238
0.08864
0.10631
0.04035
0.03236
0.02475
0.01760
0.01103
0.00513
0.00000
-0.00436
-0.00805
-0.01113
-0.01365
-0.01567
-0.01724
-0.01842
-0.01927
-0.01983
-0.02017
-0.02034
0.10204
0.08163
0.06122
-0.02
0.02041
0.00000
-0.04
0.00000
0.0
0.02041
0.04082
0.08163
0.12245
0.16327
2.0
0.18367
0.22449
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Upper and Lower 2nd Derivatives, d 2 Z/dW 2
0.8
0.9
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
RUN
-0.2
-0.4
-0.6
0.63265
0.64286
0.65306
0.66327
0.67347
0.68367
0.69388
0.14552
0.16688
0.18927
0.21261
0.23678
0.26168
0.28721
-0.02040
-0.02039
-0.02039
-0.02040
-0.02040
-0.02040
-0.02039
-1.0
0 0.28571
0.32653
0.36735
0.38776
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
J. Philip Barnes www.HowFliesTheAlbatross.com
1.0
1
Laminar airfoil study
~ integrated geometric/aero design
f Theodorsen Angle ( f ) Parametric cubic spline • Velocity ratio Discontinuous 3rd-deriv.
of cubic spline does not disrupt smooth airflow • Pressure coefficient
Parametric fuselage section: “cubic quadrant” method
Four parameters per quadrant One cubic curve per quadrant J. Philip Barnes www.HowFliesTheAlbatross.com
Parametric Fuselage – cubic spline & trig. compared
cubic-spline basis Trig. functions provide 99% desired result with just 1% of computation
J. Philip Barnes www.HowFliesTheAlbatross.com
Parametric wing: cubic spline throughout
(EXCEL/VB)
symbol
c b o 0 b o 1 u Z=z/c Ev_ v description station, v = 0:1 → local chord upper tr. edge boattail angle lower tr. edge boattail angle c'clockwise from upper t.e., 0:1 foil vertical coord. (local) spline start/end edge constraints sparwise parameter, 0:1 v → chord, c b o 0 b o 1 u1 Z1 u2 Z2 u3 Z3 u4 Z4 u5 Z5 w o d o s Ev0 ↓ 0 0 0 0 0 0 0 0 0 0 0 0 0 0.0000
1.0000
7.0000
13.0000
0.0000
0.001
0.25
0.1
0.5
0 0.75
-0.05
1 -0.001
0.2000
0.6600
9.0000
11.0000
0.0000
0.001
0.25
0.1
0.5
0 0.75
-0.05
1 -0.001
0.4000
0.3800
9.9000
10.1000
0.0000
0.001
0.25
0.1
0.5
0 0.75
-0.05
1 -0.001
0.7500
0.1800
10.0000
10.0000
0.0000
0.001
0.25
0.1
0.5
0 0.75
-0.05
1 -0.001
1.0000
0.0200
10.0000
10.0000
0.0000
0.001
0.25
0.1
0.5
0 0.75
-0.05
1 -0.001
Ev1↓ 1 1 w o d o s xb yb zb g TBD washout (trailing-edge up) dihedral (local x-rotation) spar chord sta. (x-xLE)/c (local) spar backbone x (global coord.) spar backbone y spar backbone z 0.0:0.10 option moves Zmax aft spare parameter xb yb zb g 0 0 0 0 1 0 0 0 0 1 0.5
0 0 0.0900
1 0 1 0.35
0.2
0.034
0.0900
3.2
0 1 0.35
0.4
0.074
0.0900
6.1
0 1 0.46
0.75
0.09
0.0900
8 0 1 0.54
1 0.09
0.0900
1
Edit columns 4-10, open VB editor and click the Run icon
Phil Barnes, 08 Mar 2015
Summary
The table above represents one half wing.
RUN
Half-wing geometry is parametric with (u,v) using cubic splines, airfoil c'clockwise Vs. u, sparwise Vs. v Input one column per wing "sparwise" station, including the local airfoil as a column (5-points for now) Spline-edge integer constraints are [not] used for the airfoil ; set the boattail slopes (+ for typical foil) x/c for the airfoil is an output: x/c = 1-sin( p u), given (u) as an input. x/c is optionally modified with Airfoil "spline Left and Right" (lower t.e., upper t.e.) edge slopes (dz/du) are then given by p tan b Spline-edge constraints are used versus sparwise position for all other parameters, i.e. c(v), b (v),...
g .
Sparwise position (v) has an airfoil "backbone" point at xb,yb,zb (global) The spar backbone chordwise station s = (x-xLE)/c, nominally 0.25, is anywhere from 0.0-to-1.0
The airfoil is first translated such that its backbone is anchored to the backbone global position The airfoil is then "washed out" (trailing edge up), rotating about a local y-axis thru the backbone pt.
The airfoil is then rotated about a local x-axis thru the backbone point by the dihedral angle ( d ).
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1.25
0.75
0.25
-0.25
-0.75
-1.25
-1 u x
zp(xp)
0 0.5
0.049734 0.399053
0.103766 0.283219
0.165539 0.137789
0.236968 -0.04563
0.318051 -0.25242
0.406877 -0.42884
0.5
-0.5
0.593123 -0.42884
0.681949 -0.25242
0.763032 -0.04564
0.834461 0.137787
0.896235 0.283217
0.950266 0.399052
1 0.499999
y 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 z xp zp v 0 c 1 b0 7 b1 13 0.0015
0.427531 0.210985 0.048455 0.968519 7.200982 12.79902
1.25
0.023936 0.351654 0.190995 0.094122 0.894321 7.663301 12.3367
y(x)
0.099866 -0.04623 0.065076
0.225
0.610098 9.238646 10.76135
0.088465 -0.26756 -0.04973 0.271468 0.530646 9.557754 10.44225
0.047867 -0.46766
-0.1841 0.321684 0.461477 9.760373 10.23963
0.0005
-0.54374
-0.2664 0.376489 0.401712 9.871555 10.12844
-0.0435
-0.25246 -0.16176
0.5
0.306187 9.976475 10.02352
-0.05047 -0.04341 -0.06294 0.566996 0.268707 9.998582 10.00142
-0.05302 0.124879 0.019427 0.635267 0.235607 10.00523 9.994773
-0.04648 0.249268 0.086731 0.702586 0.203931 10.00303 9.996974
-0.0005 0.427217 0.209342
0.825
0.136872 9.997489 10.00251
0 0.483562 0.048455 0.004313 0.380554 0.232849 0.918767 0.075956 9.997956 10.00204
0.049734 0.385768 0.048455 0.026299 0.306532 0.214557 0.952901 0.052623 9.998722 10.00128
0.165539 0.132667 0.048455 0.081408 0.092947 0.160205 1.000001
0.236968
-0.045
0.048455 0.099023 -0.07843 0.097187
0.318051 -0.24527 0.048455 0.08762 -0.28996 -0.01059
0.406877 -0.41609 0.048455 0.048145 -0.48009 -0.13695
0.5
-0.48496 0.048455 0.00225
-0.5521
-0.21505
0.593123
-0.416
0.048455 -0.02667 -0.46415 -0.19955
0.681949 -0.24512 0.048455
-0.5
0 -0.0402
0.5
-0.27423 -0.11924
1 0.834461 0.132812 0.048455 -0.04857 0.088442 0.050896
1.5
0.50
0.896235 0.273656 0.048455 -0.04193 0.208591 0.114697
0.950266 0.385825 0.048455 -0.02524 0.300795 0.172157
z(y)
0.483563 0.048455 0.002376 0.380288 0.231259
0.25
0.00
-0.25
0 0.446655 0.094122 0.01115 0.321105 0.248783
0.049734 0.356289 0.094122 0.031961 0.251539 0.232826
0.103766 0.252592 0.094122 0.056873 0.16695 0.214296
0.165539 0.122426 0.094122 0.082011 0.052472 0.183813
-1 0.236968 -0.04168 0.094122 0.097215 -0.10497 0.126737
0.318051 -0.22657 0.094122 0.08584 -0.29704 0.029879
0.406877 -0.38419 0.094122 0.048998 -0.46825
0.5
-0.0832
0.75
1 0.02
2 10 -0.75
-1.00
-1.25
-1 0.50
0.25
0.00
-0.25
-1 10
z(x)
-0.75
-0.75
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 u1 0 0 -0.5
-0.5
z1 0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
-0.25
-0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
u2 0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0 0 0.25
0.25
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
z2 0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
u3 0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.75
0.75
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 z3 0 0 1 1 0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
u4 0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
J. Philip Barnes www.HowFliesTheAlbatross.com
Application: Dynamic soaring in the jet stream
Energy From an Atmosphere in Motion - Dynamic Soaring and Regen-electric Flight Compared J. Philip Barnes www.HowFliesTheAlbatross.com 23
Application: Regen of electrical power in ridge lift
J. Philip Barnes www.HowFliesTheAlbatross.com
About the Author
Phil Barnes has a Master’s Degree in Aerospace Engineering from Cal Poly Pomona and BSME from the University of Arizona. He is a Principal Engineer and 34-year veteran of air vehicle and subsystems performance analysis at Northrop Grumman, where he presently supports both mature and advanced tactical aircraft programs. Author of several SAE and AIAA technical papers, and often invited to lecture at various universities, Phil is presently leading several Northrop Grumman-sponsored university research projects including an autonomous thermal soaring demonstration, passive bleed-and-blow airfoil wind-tunnel test, and application of Blender 3D software for flight simulation. This presentation includes highlights of one such collaboration (public domain) using EXCEL/Visual Basic and Blender 3D with its resident Python programming language to parameterize and visualize aircraft geometry. Outside of work, Phil is a leading expert on dynamic soaring, and he is pioneering the science of regen-electric flight.
J. Philip Barnes www.HowFliesTheAlbatross.com