forked from convexengineering/SPaircraft
-
Notifications
You must be signed in to change notification settings - Fork 0
/
horizontal_tail.py
498 lines (460 loc) · 21.1 KB
/
horizontal_tail.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
"""D8 horizontal tail model linked with a simple flight profile"""
from __future__ import absolute_import
from numpy import pi, tan
import numpy as np
from gpkit import Variable, Model, units, SignomialsEnabled, Vectorize, SignomialEquality
from gpkit.tools import te_exp_minus1
from gpkit.constraints.tight import Tight as TCS
from wingbox import WingBox
class HorizontalTailNoStruct(Model):
"""
horizontal tail model from Philippe's thesis
as a performance model without the wing box
SKIP VERIFICATION
References:
[1] TASOPT code
[2] http://adg.stanford.edu/aa241/stability/staticstability.HTml
This model does not include the effects of wing downwash on tail
effectiveness.
"""
def setup(self):
#variables
p = Variable('p_{ht}', '-', 'Substituted variable = 1 + 2*taper')
q = Variable('q_{ht}', '-', 'Substituted variable = 1 + taper')
etaht = Variable('\\eta_{ht}', '-', 'Tail efficiency')
tanLh = Variable('\\tan(\\Lambda_{ht})', '-',
'tangent of horizontal tail sweep')
taper = Variable('\lambda_{ht}', '-', 'Horizontal tail taper ratio')
tau = Variable('\\tau_{ht}', '-',
'Horizontal tail thickness/chord ratio')
xcght = Variable('x_{CG_{ht}}', 'm', 'Horizontal tail CG location')
ymac = Variable('y_{\\bar{c}_{ht}}', 'm',
'Spanwise location of mean aerodynamic chord')
dxlead = Variable('\\Delta x_{lead_{ht}}', 'm',
'Distance from CG to horizontal tail leading edge')
dxtrail = Variable('\\Delta x_{trail_{ht}}', 'm',
'Distance from CG to horizontal tail trailing edge')
lht = Variable('l_{ht}', 'm', 'Horizontal tail moment arm')
ARht = Variable('AR_{ht}', '-', 'Horizontal tail aspect ratio')
amax = Variable('\\alpha_{ht,max}', '-', 'Max angle of attack, htail')
e = Variable('e_{ht}', '-', 'Oswald efficiency factor')
Sh = Variable('S_{ht}', 'm^2', 'Horizontal tail area')
bht = Variable('b_{ht}', 'm', 'Horizontal tail span')
chma = Variable('\\bar{c}_{ht}', 'm', 'Mean aerodynamic chord (ht)')
croot = Variable('c_{root_{ht}}', 'm', 'Horizontal tail root chord')
ctip = Variable('c_{tip_{ht}}', 'm', 'Horizontal tail tip chord')
Lmax = Variable('L_{ht_{max}}', 'N', 'Maximum load')
fl = Variable(r"f(\lambda_{ht})", '-',
'Empirical efficiency function of taper')
CLhmax = Variable('C_{L_{ht,max}}', '-', 'Max lift coefficient')
CLfCG = Variable('C_{L_{ht,fCG}}', '-', 'HT CL During Max Forward CG')
#new variables
Vh = Variable('V_{ht}', '-', 'Horizontal Tail Volume Coefficient')
mrat = Variable('m_{ratio}', '-', 'Wing to Tail Lift Slope Ratio')
#variable just for the D8
cattach = Variable('c_{attach}', 'm', 'HT Chord Where it is Mounted to the VT')
#constraints
constraints = []
with SignomialsEnabled():
constraints.extend([
# Moment arm and geometry -- same as for vtail
TCS([dxlead + ymac * tanLh + 0.25 * chma >= lht], reltol=1e-2), # [SP]
TCS([dxlead + croot <= dxtrail]),
p >= 1 + 2*taper,
2*q >= 1 + p,
ymac == (bht/3)*q/p,
SignomialEquality((2./3)*(1 + taper + taper**2)*croot/q,
chma),
taper == ctip/croot,
SignomialEquality(Sh, bht*(croot + ctip)/2),
# Oswald efficiency
# Nita, Scholz,
# "Estimating the Oswald factor from basic
# aircraft geometrical parameters"
TCS([fl >= (0.0524*taper**4 - 0.15*taper**3
+ 0.1659*taper**2
- 0.0706*taper + 0.0119)], reltol=0.2),
# NOTE: slightly slack
TCS([e*(1 + fl*ARht) <= 1]),
ARht == bht**2/Sh,
taper >= 0.2, # TODO: make less arbitrary
taper <= 1,
])
return constraints
class HorizontalTailPerformance(Model):
"""
Horizontal tail performance model
SKIP VERIFICATION
ARGUMENTS
---------
fitDrag: True = use Martin's tail drag fits, False = use the TASOPT tail drag model
"""
def setup(self, ht, state, fitDrag):
self.HT = ht
#variables
D = Variable('D_{ht}', 'N', 'Horizontal tail drag')
Lh = Variable('L_{ht}', 'N', 'Horizontal tail downforce')
Rec = Variable('Re_{c_h}', '-',
'Cruise Reynolds number (Horizontal tail)')
CLah = Variable('C_{L_{\\alpha,ht}}', '-', 'Lift curve slope (htail)')
CLah0 = Variable('C_{L_{\\alpha,ht_0}}', '-',
'Isolated lift curve slope (htail)')
CLh = Variable('C_{L_{ht}}', '-', 'Lift coefficient (htail)')
CDh = Variable('C_{D_{ht}}', '-', 'Horizontal tail drag coefficient')
CD0h = Variable('C_{D_{0,ht}}', '-',
'Horizontal tail parasitic drag coefficient')
alphah = Variable('\\alpha_{ht}', '-', 'Horizontal tail angle of attack')
constraints = []
with SignomialsEnabled():
constraints.extend([
Lh == 0.5*state['\\rho']*state['V']**2*self.HT['S_{ht}']*CLh,
# Angle of attack and lift slope constraints
CLh == CLah*alphah,
alphah <= self.HT['\\alpha_{ht,max}'],
# Currently using TAT to approximate
CLah0 == 2*3.14,
# Drag
D == 0.5*state['\\rho']*state['V']**2*self.HT['S_{ht}']*CDh,
CDh >= CD0h + CLh**2/(pi*self.HT['e_{ht}']*self.HT['AR_{ht}']),
#cruise Reynolds number
Rec == state['\\rho']*state['V']*self.HT['\\bar{c}_{ht}']/state['\\mu'],
])
if fitDrag:
constraints.extend([
#Martin's TASOPT tail drag fit
CD0h**6.48983 >= (5.28751e-20 * (Rec)**0.900672 * (self.HT['\\tau_{ht}'])**0.912222 * (state['M'])**8.64547
+ 1.67605e-28 * (Rec)**0.350958 * (self.HT['\\tau_{ht}'])**6.29187 * (state['M'])**10.2559
+ 7.09757e-25 * (Rec)**1.39489 * (self.HT['\\tau_{ht}'])**1.96239 * (state['M'])**0.567066
+ 3.73076e-14 * (Rec)**-2.57406 * (self.HT['\\tau_{ht}'])**3.12793 * (state['M'])**0.448159
+ 1.44343e-12 * (Rec)**-3.91046 * (self.HT['\\tau_{ht}'])**4.66279 * (state['M'])**7.68852)
#Philippe thesis drag fit
## CD0h**0.125 >= 0.19*(self.HT['\\tau_{ht}'])**0.0075 *(Rec)**0.0017
## + 1.83e+04*(self.HT['\\tau_{ht}'])**3.54*(Rec)**-0.494
## + 0.118*(self.HT['\\tau_{ht}'])**0.0082 *(Rec)**0.00165
## + 0.198*(self.HT['\\tau_{ht}'])**0.00774*(Rec)**0.00168,
])
else:
#HT drag constraints in AircraftP
None
return constraints
class HorizontalTail(Model):
"""
horiziontal tail model from Philippe's thesis
as a performance model without the wing box
SKIP VERIFICATION
References:
[1] TASOPT code
[2] http://adg.stanford.edu/aa241/stability/staticstability.HTml
"""
def setup(self):
self.HTns = HorizontalTailNoStruct()
self.wb = WingBox(self.HTns, "horizontal_tail")
#HT system weight variable
Wht = Variable('W_{ht}', 'N', 'HT System Weight')
fht = Variable('f_{ht}' ,'-', 'Rudder etc. fractional weight')
#margin and sensitivity
Cht = Variable('C_{ht}', 1, '-', 'HT Weight Margin and Sensitivity Factor')
#variables only used for the TASOPT tail drag formulation
cdfh = Variable('c_{d_{fh}}', '-', 'VT friction drag coefficient')
cdph = Variable('c_{d_{ph}}', '-', 'VT pressure drag coefficient')
coslamcube = Variable('\\cos(\\Lambda_{ht})^3', '-', 'Cosine of tail sweep cubed')
constraints = []
with SignomialsEnabled():
constraints.append([
self.wb['L_{ht_{rect}}'] >= self.HTns['L_{ht_{max}}']/2.*self.HTns['c_{tip_{ht}}']*self.HTns['b_{ht}']/self.HTns['S_{ht}'],
self.wb['L_{ht_{tri}}'] >= self.HTns['L_{ht_{max}}']/4.*(1-self.wb['taper'])*self.HTns['c_{root_{ht}}']*self.HTns['b_{ht}']/self.HTns['S_{ht}'], #[SP]
Wht >= Cht*(self.wb['W_{struct}'] + self.wb['W_{struct}'] * fht),
])
return self.HTns, self.wb, constraints
def dynamic(self, state, fitDrag):
""""
creates a horizontal tail performance model
"""
return HorizontalTailPerformance(self, state, fitDrag)
## Old code not maintained. Can be used for subsystem testing.
##if __name__ == '__main__':
## plot = True
##
## #build required submodels
## aircraft = Aircraft()
##
## substitutions = {
#### 'V_{stall}': 120,
## 'R_{req}': 500, #('sweep', np.linspace(500,2000,4)),
## 'CruiseAlt': 30000, #('sweep', np.linspace(20000,40000,4)),
## 'numeng': 2,
## 'W_{pass}': 91 * 9.81,
## 'n_{pass}': 150,
## 'pax_{area}': 1,
## 'e': .9,
## 'b_{max}': 60,
##
## #HT subs
## 'C_{L_{ht,max}}': 2.5,
## '\\tan(\\Lambda_{ht})': tan(30*pi/180),
## 'w_{fuse}': 6,
## 'c_{m_{w}}': 1,
## 'C_{L_{max}}': 2,
## '\\alpha_{ht,max}': 2.5,
## 'SM_{min}': 0.5,
## '\\Delta x_{CG}': 4,
##
#### 'x_{CG}': [17, 18],
## #think about how to constrain this
## 'x_w': 19,
## 'mac': 2,
## }
## mission = Mission(aircraft)
## m = Model(mission['W_{f_{total}}'], [aircraft, mission], substitutions)
## sol = m.localsolve(solver='mosek', verbosity = 4)
##
##import matplotlib.pyplot as plt
##from gpkit.small_scripts import mag
##from simple_ac_imports_no_engine import Wing, Fuselage, Engine, CruiseP, ClimbP, FlightState, CruiseSegment, ClimbSegment
## class Aircraft(Model):
## "Aircraft class"
## def setup(self):
## #create submodels
## self.fuse = Fuselage()
## self.wing = Wing()
## self.engine = Engine()
## self.HT = HorizontalTail()
##
## #variable definitions
## numeng = Variable('numeng', '-', 'Number of Engines')
## Vne = Variable('V_{ne}', 144, 'm/s', 'Never exceed velocity')
## rho0 = Variable('\\rho_0', 1.225, 'kg/m^3', 'Air density (0 ft)')
##
## SMmin = Variable('SM_{min}', '-', 'Minimum Static Margin')
## dxCG = Variable('\\Delta x_{CG}', 'm', 'Max CG Travel Range')
##
## constraints = []
##
## constraints.extend([
## numeng == numeng, #need numeng in the model
## ])
##
## self.components = [self.fuse, self.wing, self.engine, self.HT]
##
## return self.components, constraints
##
## def dynamic(self, state):
## """
## creates an aircraft climb performance model, given a state
## """
## return AircraftP(self, state)
##
## def climb_dynamic(self, state):
## """
## creates an aircraft climb performance model, given a state
## """
## return ClimbP(self, state)
##
## def cruise_dynamic(self, state):
## """
## creates an aircraft cruise performance model, given a state
## """
## return CruiseP(self, state)
##
##class AircraftP(Model):
## """
## aircraft performance models superclass, contains constraints true for
## all flight segments
## """
## def setup(self, aircraft, state):
## #make submodels
## self.aircraft = aircraft
## self.wingP = aircraft.wing.dynamic(state)
## self.fuseP = aircraft.fuse.dynamic(state)
## self.engineP = aircraft.engine.dynamic(state)
## self.HTP = aircraft.HT.dynamic(aircraft.fuse, aircraft.wing, state)
##
## self.Pmodels = [self.wingP, self.fuseP, self.engineP, self.HTP]
##
## #variable definitions
## Vstall = Variable('V_{stall}', 'knots', 'Aircraft Stall Speed')
## D = Variable('D', 'N', 'Total Aircraft Drag')
## W_avg = Variable('W_{avg}', 'N', 'Geometric Average of Segment Start and End Weight')
## W_start = Variable('W_{start}', 'N', 'Segment Start Weight')
## W_end = Variable('W_{end}', 'N', 'Segment End Weight')
## W_burn = Variable('W_{burn}', 'N', 'Segment Fuel Burn Weight')
## WLoadmax = Variable('W_{Load_{max}}', 'N/m^2', 'Max Wing Loading')
## WLoad = Variable('W_{Load}', 'N/m^2', 'Wing Loading')
## t = Variable('tmin', 'min', 'Segment Flight Time in Minutes')
## thours = Variable('thr', 'hour', 'Segment Flight Time in Hours')
##
## xAC = Variable('x_{AC}', 'm', 'Aerodynamic Center Location')
## xCG = Variable('x_{CG}', 'm', 'CG location')
##
##
## constraints = []
## with SignomialsEnabled():
## constraints.extend([
## #speed must be greater than stall speed
## state['V'] >= Vstall,
##
##
## #Figure out how to delete
## Vstall == 120*units('kts'),
## WLoadmax == 6664 * units('N/m^2'),
##
## #compute the drag
## TCS([D >= self.wingP['D_{wing}'] + self.fuseP['D_{fuse}'] + self.HTP['D_{ht}']]),
##
## #constraint CL and compute the wing loading
## W_avg == .5*self.wingP['C_{L}']*self.aircraft['S']*state.atm['\\rho']*state['V']**2,
## WLoad == .5*self.wingP['C_{L}']*self.aircraft['S']*state.atm['\\rho']*state['V']**2/self.aircraft.wing['S'],
##
## #set average weight equal to the geometric avg of start and end weight
## W_avg == (W_start * W_end)**.5,
##
## #constrain the max wing loading
## WLoad <= WLoadmax,
##
## #compute fuel burn from TSFC
## W_burn == aircraft['numeng']*self.engineP['TSFC'] * thours * self.engineP['F'],
##
## #time unit conversion
## t == thours,
##
## #make lift equal weight --> small angle approx in climb
## self.wingP['L_{wing}'] >= W_avg,
## ])
##
## return self.Pmodels, constraints
##
##class Mission(Model):
## """
## mission class, links together all subclasses
## """
## def setup(self, aircraft):
## #define the number of each flight segment
## Nclimb = 2
## Ncruise = 2
##
## #Vectorize
## with Vectorize(Nclimb):
## climb = ClimbSegment(aircraft)
##
## with Vectorize(Ncruise):
## cruise = CruiseSegment(aircraft)
##
## #declare new variables
## W_ftotal = Variable('W_{f_{total}}', 'N', 'Total Fuel Weight')
## W_fclimb = Variable('W_{f_{climb}}', 'N', 'Fuel Weight Burned in Climb')
## W_fcruise = Variable('W_{f_{cruise}}', 'N', 'Fuel Weight Burned in Cruise')
## W_total = Variable('W_{total}', 'N', 'Total Aircraft Weight')
## CruiseAlt = Variable('CruiseAlt', 'ft', 'Cruise Altitude [feet]')
## ReqRng = Variable('R_{req}', 'nautical_miles', 'Required Cruise Range')
## W_dry = Variable('W_{dry}', 'N', 'Aircraft Dry Weight')
##
## h = climb['h']
## hftClimb = climb['hft']
## dhft = climb['dhft']
## hftCruise = cruise['hft']
##
## #make overall constraints
## constraints = []
##
## constraints.extend([
## #weight constraints
## TCS([aircraft['W_{e}'] + aircraft['W_{payload}'] + aircraft['numeng'] * aircraft['W_{engine}'] + aircraft['W_{wing}'] + aircraft.HT['W_{struct}'] <= W_dry]),
## TCS([W_ftotal + W_dry <= W_total]),
##
## climb['W_{start}'][0] == W_total,
## climb['W_{end}'][-1] == cruise['W_{start}'][0],
##
## # similar constraint 1
## TCS([climb['W_{start}'] >= climb['W_{end}'] + climb['W_{burn}']]),
## # similar constraint 2
## TCS([cruise['W_{start}'] >= cruise['W_{end}'] + cruise['W_{burn}']]),
##
## climb['W_{start}'][1:] == climb['W_{end}'][:-1],
## cruise['W_{start}'][1:] == cruise['W_{end}'][:-1],
##
## TCS([W_dry <= cruise['W_{end}'][-1]]),
##
## TCS([W_ftotal >= W_fclimb + W_fcruise]),
## TCS([W_fclimb >= sum(climb['W_{burn}'])]),
## TCS([W_fcruise >= sum(cruise['W_{burn}'])]),
##
## #altitude constraints
## hftCruise == CruiseAlt,
## TCS([hftClimb[1:Ncruise] >= hftClimb[:Ncruise-1] + dhft]),
## TCS([hftClimb[0] >= dhft[0]]),
## hftClimb[-1] <= hftCruise,
##
## #compute the dh
## dhft == hftCruise/Nclimb,
##
## #constrain the thrust
## climb.climbP['F'] <= 2 * max(cruise.cruiseP['F']),
##
## #set the range for each cruise segment, doesn't take credit for climb
## #down range disatnce covered
## cruise.cruiseP['Rng'] == ReqRng/(Ncruise),
##
## #set the TSFC
## climb['TSFC'] == .7*units('1/hr'),
## cruise['TSFC'] == .5*units('1/hr'),
##
## # climb['C_{L_{ht}}'] == 2*3.14*climb['\\alpha_{ht}],
## # cruise['C_{L_{ht}}'] == 2*3.14*cruise['\\alpha_{ht}],
## ])
##
## #Horizontal Tail Constraints
## with SignomialsEnabled():
## constraints.extend([
##
## # Trim condition for each flight segment
## TCS([cruise['x_{AC}']/aircraft.wing['mac'] <= aircraft.wing['c_{m_{w}}']/cruise['C_{L}'] + \
## cruise['x_{CG}']/aircraft.wing['mac'] + aircraft.HT['V_{ht}']*(cruise['C_{L_{ht}}']/cruise['C_{L}'])]),
## TCS([climb['x_{AC}']/aircraft.wing['mac'] <= aircraft.wing['c_{m_{w}}']/climb['C_{L}'] + \
## climb['x_{CG}']/aircraft.wing['mac'] + aircraft.HT['V_{ht}']*(climb['C_{L_{ht}}']/climb['C_{L}'])]),
##
##
## aircraft.HT['L_{ht_{max}}'] == 0.5*aircraft['\\rho_0']*aircraft['V_{ne}']**2*aircraft.HT['S_{ht}']*aircraft.HT['C_{L_{ht,max}}'],
## #compute mrat, is a signomial equality
## SignomialEquality(aircraft.HT['m_{ratio}']*(1+2/aircraft.wing['AR']), 1 + 2/aircraft.HT['AR_{ht}']),
##
## #tail volume coefficient
## aircraft.HT['V_{ht}'] == aircraft.HT['S_{ht}']*aircraft.HT['x_{CG_{ht}}']/(aircraft.wing['S']*aircraft.wing['mac']),
##
## #enforce max tail location is the end of the fuselage
## aircraft.HT['x_{CG_{ht}}'] <= aircraft.fuse['l_{fuse}'],
## aircraft.HT['l_{ht}'] >= aircraft.HT['x_{CG_{ht}}'] - cruise['x_{CG}'],
## aircraft.HT['l_{ht}'] >= aircraft.HT['x_{CG_{ht}}'] - climb['x_{CG}'],
##
## #Stability constraint, is a signomial
## TCS([aircraft['SM_{min}'] + aircraft['\\Delta x_{CG}']/aircraft.wing['mac'] + aircraft.wing['c_{m_{w}}']/aircraft.wing['C_{L_{max}}'] <= aircraft.HT['V_{ht}']*aircraft.HT['m_{ratio}'] + aircraft.HT['V_{ht}']*aircraft.HT['C_{L_{ht,max}}']/aircraft.wing['C_{L_{max}}']]),
##
## # TCS([aircraft.wing['x_w'] >= cruise['x_{CG}'] + cruise['\\Delta x_w']]),
## # TCS([aircraft.wing['x_w'] >= climb['x_{CG}'] + climb['\\Delta x_w']]),
##
##
## TCS([cruise['x_{CG}'] + cruise['\\Delta x_{trail_{ht}}'] <= aircraft.fuse['l_{fuse}']], reltol=0.002),
## TCS([climb['x_{CG}'] + climb['\\Delta x_{trail_{ht}}'] <= aircraft.fuse['l_{fuse}']], reltol=0.002),
##
## #compute the aerodynamic center location
## #TODO: this sets xAC to xW in a stupid and long winded way
## # TCS([climb['x_{AC}'] <= climb['x_{CG}'] + climb['\\Delta x_w'] ]),
## # TCS([cruise['x_{AC}'] <= cruise['x_{CG}'] + cruise['\\Delta x_w'] ]),
##
#### SignomialEquality(cruise['x_{ac}'],xcg + cruise['\\Delta x_w'] ),
#### SignomialEquality(climb['x_{ac}'],xcg + climb['\\Delta x_w'] ),
## TCS([aircraft.HT['x_{CG_{ht}}'] >= climb['x_{CG}'] + (climb['\\Delta x_{lead_{ht}}']+climb['\\Delta x_{trail_{ht}}'])/2]),
## TCS([aircraft.HT['x_{CG_{ht}}'] >= cruise['x_{CG}'] + (cruise['\\Delta x_{lead_{ht}}']+cruise['\\Delta x_{trail_{ht}}'])/2]),
## #---------------------------------------------------------#
##
## # Substitutions for xCG and xAC
## cruise['x_{CG}'] == 15*units('m'),
## climb['x_{CG}'] == 15*units('m'),
## cruise['x_{AC}'] == aircraft.wing['x_w'],
## climb['x_{AC}'] == aircraft.wing['x_w'],
##
## #compute the HT chord at its attachment point to the VT
## (aircraft.HT['b_{ht}']/aircraft.fuse['w_{fuse}'])*aircraft.HT['\lambda_{ht}']*aircraft.HT['c_{root_{ht}}'] == aircraft.HT['c_{attach}']
##
## ])
##
## return climb, cruise, constraints