def __init__(self, **kwargs):
Afuel = Variable('\\bar{A}_{fuel, max}', '-', 'Non-dim. fuel area')
AR = Variable('AR_w', '-', 'Wing aspect ratio')
CDp = Variable('C_{D_{p_w}}', '-',
'Wing parasitic drag coefficient')
CDw = Variable('C_{D_w}', '-', 'Drag coefficient, wing')
CLaw = Variable('C_{L_{aw}}', '-', 'Lift curve slope, wing')
CLw = Variable('C_{L_w}', '-', 'Lift coefficient, wing')
CLwmax = Variable('C_{L_{wmax}}', '-', 'Max lift coefficient, wing')
D = Variable('D_{wing}', 'N', 'Wing drag')
Lmax = Variable('L_{max_{w}}', 'N', 'Maximum load')
Lw = Variable('L_w', 'N', 'Wing lift')
M = Variable('M', '-', 'Cruise Mach number')
Re = Variable('Re_w', '-', 'Cruise Reynolds number (wing)')
Sw = Variable('S_w', 'm^2', 'Wing area')
Vinf = Variable('V_{\\infty}', 'm/s', 'Freestream velocity')
Vfuel = Variable('V_{fuel, max}', 'm^3', 'Available fuel volume')
Vne = Variable('V_{ne}', 'm/s', 'Never exceed velocity')
W = Variable('W', 'N', 'Aircraft weight')
W0 = Variable('W_0', 'N', 'Weight excluding wing')
Wfuel = Variable('W_{fuel}', 'N', 'Fuel weight')
Wfuelmax= Variable('W_{fuel,max}', 'N', 'Max fuel weight')
Ww = Variable('W_{wing}', 'N', 'Wing weight')
a = Variable('a', 'm/s', 'Speed of sound (35,000 ft)')
alpha = Variable('\\alpha_w', '-', 'Wing angle of attack')
amax = Variable('\\alpha_{max,w}', '-', 'Max angle of attack')
b = Variable('b_w', 'm', 'Wing span')
cosL = Variable('\\cos(\\Lambda)', '-',
'Cosine of quarter-chord sweep angle')
croot = Variable('c_{root}', 'm', 'Wing root chord')
ctip = Variable('c_{tip}', 'm', 'Wing tip chord')
cwma = Variable('\\bar{c}_w', 'm',
'Mean aerodynamic chord (wing)')
e = Variable('e', '-', 'Oswald efficiency factor')
eta = Variable('\\eta_w', '-',
'Lift efficiency (diff b/w sectional, actual lift)')
fl = Variable('f(\\lambda_w)', '-',
'Empirical efficiency function of taper')
g = Variable('g', 'm/s^2', 'Gravitational acceleration')
mu = Variable('\\mu', 'N*s/m^2', 'Dynamic viscosity (35,000 ft)')
p = Variable('p_w', '-', 'Substituted variable = 1 + 2*taper')
q = Variable('q_w', '-', 'Substituted variable = 1 + taper')
rho = Variable('\\rho', 'kg/m^3', 'Air density (cruise)')
rho0 = Variable('\\rho_0', 'kg/m^3', 'Air density (0 ft)')
rhofuel = Variable('\\rho_{fuel}', 'kg/m^3', 'Density of fuel')
tanL = Variable('\\tan(\\Lambda)', '-',
'Tangent of quarter-chord sweep angle')
taper = Variable('\\lambda', '-', 'Wing taper ratio')
tau = Variable('\\tau_w', '-', 'Wing thickness/chord ratio')
tcap = Variable('t_{cap}' ,'-', 'Non-dim. spar cap thickness')
tweb = Variable('t_{web}', '-', 'Non-dim. shear web thickness')
w = Variable('w', 0.5, '-', 'Wingbox-width-to-chord ratio')
#xw = Variable('x_w', 'm', 'Position of wing aerodynamic center')
ymac = Variable('y_{\\bar{c}_w}', 'm',
'Spanwise location of mean aerodynamic chord')
objective = D
with SignomialsEnabled():
constraints = [
Lw == 0.5*rho*Vinf**2*Sw*CLw,
p >= 1 + 2*taper,
2*q >= 1 + p,
ymac == (b/3)*q/p,
TCS([(2./3)*(1+taper+taper**2)*croot/q <= cwma],
reltol=1E-2),
taper == ctip/croot,
TCS([Sw <= b*(croot + ctip)/2], reltol=1E-2), # [SP]
# DATCOM formula (Mach number makes it SP)
TCS([(AR/eta)**2*(1 + tanL**2 - M**2) + 8*pi*AR/CLaw
<= (2*pi*AR/CLaw)**2]),
CLw == CLaw*alpha,
alpha <= amax,
# Drag
D == 0.5*rho*Vinf**2*Sw*CDw,
CDw >= CDp + CLw**2/(pi*e*AR),
Re == rho*Vinf*cwma/mu,
1 >= (2.56*CLw**5.88/(Re**1.54*tau**3.32*CDp**2.62)
+ 3.8e-9*tau**6.23/(CLw**0.92*Re**1.38*CDp**9.57)
+ 2.2e-3*Re**0.14*tau**0.033/(CLw**0.01*CDp**0.73)
+ 6.14e-6*CLw**6.53/(Re**0.99*tau**0.52*CDp**5.19)
+ 1.19e4*CLw**9.78*tau**1.76/(Re*CDp**0.91)),
# 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=1E-2),
TCS([e*(1 + fl*AR) <= 1]),
taper >= 0.2, # TODO
Lmax == 0.5*rho0*Vne**2*Sw*CLwmax,
# Fuel volume [TASOPT doc]
Afuel <= w*0.92*tau,
# GP approx of the signomial constraint:
# Afuel <= (w - 2*tweb)*(0.92*tau - 2*tcap),
Vfuel <= croot**2 * (b/6) * (1+taper+taper**2)*cosL,
Wfuel <= rhofuel*Afuel*Vfuel*g,
]
standalone_constraints = [W >= W0 + Ww + Wfuel,
Lw == W,
M == Vinf/a,
]
self.standalone_constraints = standalone_constraints
wb = WingBox()
wb.subinplace({'A': AR,
'b': b,
'L_{max}': Lmax,
'p': p,
'q': q,
'S': Sw,
'taper': taper,
'\\tau': tau,
'W_{struct}': Ww})
lc = LinkedConstraintSet([constraints, wb])
CostedConstraintSet.__init__(self, objective, lc, **kwargs)
def __init__(self, **kwargs):
ARh = Variable('AR_h', '-', 'Horizontal tail aspect ratio')
ARw = Variable('AR_w', '-', 'Wing aspect ratio')
CD0h = Variable('C_{D_{0_h}}', '-',
'Horizontal tail parasitic drag coefficient')
CDh = Variable('C_{D_h}', '-', 'Horizontal tail drag coefficient')
CLah = Variable('C_{L_{ah}}', '-', 'Lift curve slope (htail)')
CLah0 = Variable('C_{L_{ah_0}}', '-',
'Isolated lift curve slope (htail)')
CLaw = Variable('C_{L_{aw}}', '-', 'Lift curve slope (wing)')
CLh = Variable('C_{L_h}', '-', 'Lift coefficient (htail)')
CLhmax = Variable('C_{L_{hmax}}', '-', 'Max lift coefficient')
CLw = Variable('C_{L_w}', '-', 'Lift coefficient (wing)')
Cmac = Variable('|C_{m_{ac}}|', '-', # Absolute value of CMwing
'Moment coefficient about aerodynamic centre (wing)')
Cmfu = Variable('C_{m_{fuse}}', '-', 'Moment coefficient (fuselage)')
D = Variable('D_{ht}', 'N', 'Horizontal tail drag')
Kf = Variable('K_f', '-',
'Empirical factor for fuselage-wing interference')
Lh = Variable('L_h', 'N', 'Horizontal tail downforce')
Lmax = Variable('L_{{max}_h}', 'N', 'Maximum load')
M = Variable('M', '-', 'Cruise Mach number')
Rec = Variable('Re_{c_h}', '-',
'Cruise Reynolds number (Horizontal tail)')
SM = Variable('S.M.', '-', 'Stability margin')
SMmin = Variable('S.M._{min}', '-', 'Minimum stability margin')
Sh = Variable('S_h', 'm^2', 'Horizontal tail area')
Sw = Variable('S_w', 'm^2', 'Wing area')
Vinf = Variable('V_{\\infty}', 'm/s', 'Freestream velocity')
Vne = Variable('V_{ne}', 'm/s', 'Never exceed velocity')
W = Variable('W_{ht}', 'N', 'Horizontal tail weight')
a = Variable('a', 'm/s', 'Speed of sound (35,000 ft)')
alpha = Variable('\\alpha', '-', 'Horizontal tail angle of attack')
amax = Variable('\\alpha_{max,h}', '-', 'Max angle of attack, htail')
bht = Variable('b_{ht}', 'm', 'Horizontal tail span')
chma = Variable('\\bar{c}_{ht}', 'm', 'Mean aerodynamic chord (ht)')
croot = Variable('c_{root_h}', 'm', 'Horizontal tail root chord')
ctip = Variable('c_{tip_h}', 'm', 'Horizontal tail tip chord')
cwma = Variable('\\bar{c}_w', 'm',
'Mean aerodynamic chord (wing)')
dxlead = Variable('\\Delta x_{{lead}_h}', 'm',
'Distance from CG to horizontal tail leading edge')
dxtrail = Variable('\\Delta x_{{trail}_h}', 'm',
'Distance from CG to horizontal tail trailing edge')
dxw = Variable('\\Delta x_w', 'm',
'Distance from aerodynamic centre to CG')
e = Variable('e_h', '-', 'Oswald efficiency factor')
eta = Variable('\\eta_h', '-',
("Lift efficiency (diff between sectional and "
"actual lift)"))
etaht = Variable('\\eta_{ht}', '-', 'Tail efficiency')
fl = Variable(r"f(\lambda_h)", '-',
'Empirical efficiency function of taper')
lfuse = Variable('l_{fuse}', 'm', 'Fuselage length')
lht = Variable('l_{ht}', 'm', 'Horizontal tail moment arm')
mu = Variable(r'\mu', 'N*s/m^2', 'Dynamic viscosity (35,000 ft)')
p = Variable('p_{ht}', '-', 'Substituted variable = 1 + 2*taper')
q = Variable('q_{ht}', '-', 'Substituted variable = 1 + taper')
rho = Variable(r'\rho', 'kg/m^3', 'Air density (cruise)')
rho0 = Variable(r'\rho_0', 'kg/m^3', 'Air density (0 ft)')
tanLh = Variable(r'\tan(\Lambda_{ht})', '-',
'tangent of horizontal tail sweep')
taper = Variable(r'\lambda_h', '-', 'Horizontal tail taper ratio')
tau = Variable(r'\tau_h', '-',
'Horizontal tail thickness/chord ratio')
wf = Variable('w_{fuse}', 'm', 'Fuselage width')
xcg = Variable('x_{CG}', 'm', 'CG location')
xcght = Variable('x_{CG_{ht}}', 'm', 'Horizontal tail CG location')
xw = Variable('x_w', 'm', 'Position of wing aerodynamic center')
ymac = Variable('y_{\\bar{c}_{ht}}', 'm',
'Spanwise location of mean aerodynamic chord')
objective = D + 0.1*W
with SignomialsEnabled():
# Stability from UMich AE-481 course notes
constraints = [TCS([SM + dxw/cwma + Kf*wf**2*lfuse/(CLaw*Sw*cwma)
<= CLah*Sh*lht/(CLaw*Sw*cwma)]),
SM >= SMmin,
# Trim from UMich AE-481 course notes
TCS([CLh*Sh*lht/(Sw*cwma) + Cmac >=
CLw*dxw/cwma + Cmfu], reltol=0.02),
Lh == 0.5*rho*Vinf**2*Sh*CLh,
# Moment arm and geometry -- same as for vtail
TCS([dxlead + croot <= dxtrail]),
TCS([xcg + dxtrail <= lfuse], reltol=0.002),
TCS([dxlead + ymac*tanLh + 0.25*chma >= lht],
reltol=1e-2), # [SP]
p >= 1 + 2*taper,
2*q >= 1 + p,
ymac == (bht/3)*q/p,
TCS([(2./3)*(1 + taper + taper**2)*croot/q >=
chma]), # [SP]
taper == ctip/croot,
TCS([Sh <= bht*(croot + ctip)/2]), # [SP]
# DATCOM formula (Mach number makes it SP)
TCS([(ARh/eta)**2*(1+tanLh**2-M**2) + 8*pi*ARh/CLah0
<= (2*pi*ARh/CLah0)**2]),
# Loss of tail effectiveness due to wing downwash
CLah + (2*CLaw/(pi*ARw))*etaht*CLah0 <= CLah0*etaht,
CLh == CLah*alpha,
alpha <= amax,
# K_f as f(wing position) -- (fitted posynomial)
# from from UMich AE-481 course notes Table 9.1
Kf >= (1.5012*(xw/lfuse)**2 +
0.538*(xw/lfuse) +
0.0331),
TCS([xw >= xcg + dxw]),
# Drag
D == 0.5*rho*Vinf**2*Sh*CDh,
CDh >= CD0h + CLh**2/(pi*e*ARh),
# same drag model as vtail
CD0h**0.125 >= 0.19*(tau)**0.0075 *(Rec)**0.0017
+ 1.83e+04*(tau)**3.54*(Rec)**-0.494
+ 0.118*(tau)**0.0082 *(Rec)**0.00165
+ 0.198*(tau)**0.00774*(Rec)**0.00168,
Rec == rho*Vinf*chma/mu,
# 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*ARh) <= 1]),
taper >= 0.2, # TODO: make less arbitrary
Lmax == 0.5*rho0*Vne**2*Sh*CLhmax,
]
standalone_constraints = [M == Vinf/a,
]
CG_constraint = [TCS([xcght >= xcg+(dxlead+dxtrail)/2]),
xcght <= lfuse
]
self.standalone_constraints = standalone_constraints
self.CG_constraint = CG_constraint
wb = WingBox()
wb.subinplace({'A': ARh,
'b': bht,
'L_{max}': Lmax,
'p': p,
'q': q,
'S': Sh,
'taper': taper,
r'\tau': tau,
'W_{struct}': W})
lc = LinkedConstraintSet([constraints, wb])
CostedConstraintSet.__init__(self, objective, lc, **kwargs)
def __init__(self, **kwargs):
Afan = Variable('A_{fan}', 'm^2', 'Engine reference area')
Avt = Variable('A_{vt}', '-', 'Vertical tail aspect ratio')
CDvis = Variable('C_{D_{vis}}', '-', 'Viscous drag coefficient')
CDwm = Variable('C_{D_{wm}}', '-', 'Windmill drag coefficient')
CLvmax = Variable('C_{L_{vmax}}', '-', 'Max lift coefficient')
CLvt = Variable('C_{L_{vt}}', '-', 'Vertical tail lift coefficient')
Dvt = Variable('D_{vt}', 'N', 'Vertical tail viscous drag, cruise')
Dwm = Variable('D_{wm}', 'N', 'Engine out windmill drag')
Lmax = Variable('L_{max_{vt}}', 'N',
'Maximum load for structural sizing') # fuselage
Lvmax = Variable('L_{v_{max}}', 'N',
'Maximum load for structural sizing')
Lvt = Variable('L_{vt}', 'N', 'Vertical tail lift in engine out')
Rec = Variable('Re_{vt}', '-', 'Vertical tail reynolds number, cruise')
S = Variable('S', 'm^2', 'Vertical tail reference area (full)')
Svt = Variable('S_{vt}', 'm^2', 'Vertical tail reference area (half)')
Te = Variable('T_e', 'N', 'Thrust per engine at takeoff')
V1 = Variable('V_1', 'm/s', 'Minimum takeoff velocity')
Vinf = Variable('V_{\\infty}', 'm/s', 'Cruise velocity')
Vne = Variable('V_{ne}', 'm/s', 'Never exceed velocity')
Wstruct= Variable('W_{struct}', 'N', 'Full span weight')
Wvt = Variable('W_{vt}', 'N', 'Vertical tail weight')
b = Variable('b', 'm', 'Vertical tail full span')
bvt = Variable('b_{vt}', 'm', 'Vertical tail half span')
clvt = Variable('c_{l_{vt}}', '-',
'Sectional lift force coefficient (engine out)')
cma = Variable('\\bar{c}_{vt}', 'm', 'Vertical tail mean aero chord')
croot = Variable('c_{root_{vt}}', 'm', 'Vertical tail root chord')
ctip = Variable('c_{tip_{vt}}', 'm', 'Vertical tail tip chord')
cvt = Variable('c_{vt}', 'm', 'Vertical tail root chord') # fuselage
dfan = Variable('d_{fan}', 'm', 'Fan diameter')
dxlead = Variable('\\Delta x_{lead_v}', 'm',
'Distance from CG to vertical tail leading edge')
dxtrail= Variable('\\Delta x_{trail_v}', 'm',
'Distance from CG to vertical tail trailing edge')
e = Variable('e_v', '-', 'Span efficiency of vertical tail')
lfuse = Variable('l_{fuse}', 'm', 'Length of fuselage')
lvt = Variable('l_{vt}', 'm', 'Vertical tail moment arm')
mu = Variable('\\mu', 'N*s/m^2', 'Dynamic viscosity (35,000 ft)')
mu0 = Variable('\\mu_0', 1.8E-5, 'N*s/m^2', 'Dynamic viscosity (SL)')
p = Variable('p_{vt}', '-', 'Substituted variable = 1 + 2*taper')
plamv = Variable('p_{\\lambda_v}', '-',
'Dummy variable = 1 + 2\\lambda') # fuselage
q = Variable('q_{vt}', '-', 'Substituted variable = 1 + taper')
rho0 = Variable('\\rho_{TO}', 'kg/m^3', 'Air density (SL))')
rho = Variable('\\rho', 'kg/m^3', 'Air density (cruise)')
tanL = Variable('\\tan(\\Lambda_{vt})', '-',
'Tangent of leading edge sweep (40 deg)')
taper = Variable('\\lambda_{vt}', '-', 'Vertical tail taper ratio')
tau = Variable('\\tau_{vt}', '-', 'Vertical tail thickness/chord ratio')
xCG = Variable('x_{CG}', 'm', 'x-location of CG')
xCGvt = Variable('x_{CG_{vt}}', 'm', 'x-location of tail CG')
y_eng = Variable('y_{eng}', 'm', 'Engine moment arm')
zmac = Variable('z_{\\bar{c}_{vt}}', 'm',
'Vertical location of mean aerodynamic chord')
with SignomialsEnabled():
objective = Dvt + 0.05*Wvt
constraints = [
Lvt*lvt >= Te*y_eng + Dwm*y_eng,
# Force moment balance for one engine out condition
# TASOPT 2.0 p45
TCS([dxlead + zmac*tanL + 0.25*cma >= lvt]), # [SP]
# Tail moment arm
Lvt == 0.5*rho0*V1**2*Svt*CLvt,
# Vertical tail force (y-direction) for engine out
Avt == bvt**2/Svt,
TCS([CLvt*(1 + clvt/(np.pi*e*Avt)) <= clvt]),
# Finite wing theory
# people.clarkson.edu/~pmarzocc/AE429/AE-429-4.pdf
# Valid because tail is untwisted and uncambered
# (lift curve slope passes through origin)
Dwm >= 0.5*rho0*V1**2*Afan*CDwm,
Afan >= np.pi*(dfan/2)**2,
# Drag of a windmilling engine
Svt <= bvt*(croot + ctip)/2, # [SP]
# Tail geometry relationship
TCS([dxtrail >= croot + dxlead]),
# Tail geometry constraint
lfuse >= dxtrail + xCG,
# Fuselage length constrains the tail trailing edge
p >= 1 + 2*taper,
2*q >= 1 + p,
zmac == (bvt/3)*q/p,
TCS([(2./3)*(1 + taper + taper**2)*croot/q >= cma]), # [SP]
taper == ctip/croot,
# Define vertical tail geometry
taper >= 0.27,
# TODO: Constrain taper by tip Reynolds number
# source: b737.org.uk
Dvt >= 0.5*rho*Vinf**2*Svt*CDvis,
CDvis**0.125 >= 0.19*(tau)**0.0075 *(Rec)**0.0017
+ 1.83e+04*(tau)**3.54*(Rec)**-0.494
+ 0.118*(tau)**0.0082 *(Rec)**0.00165
+ 0.198*(tau)**0.00774*(Rec)**0.00168,
# Vertical tail viscous drag in cruise
# Data fit from Xfoil
Rec == rho*Vinf*cma/mu,
# Cruise Reynolds number
S == Svt*2,
b == bvt*2,
Wvt == Wstruct/2,
Lmax == 2*Lvmax,
# Relate vertical tail geometry/weight to generic
# wing used in structural model
Lvmax == 0.5*rho0*Vne**2*Svt*CLvmax,
# Max load for structural sizing
]
# For linking to other models
linking_constraints = [plamv == p,
cvt == croot]
CG_constraint = [TCS([xCGvt >= xCG+(dxlead+dxtrail)/2],
raiseerror=False),
xCGvt <= lfuse]
self.linking_constraints = linking_constraints
self.CG_constraint = CG_constraint
# Incorporate the structural model
wb = WingBox()
wb.subinplace({'b': b,
'L_{max}': Lmax,
'p': p,
'q': q,
'S': S,
'taper': taper,
'\\tau': tau})
lc = LinkedConstraintSet([constraints, wb])
CostedConstraintSet.__init__(self, objective, lc)