def test_te_exp_minus1(self):
"""Test Taylor expansion of e^x - 1"""
x = Variable('x')
self.assertEqual(te_exp_minus1(x, 1), x)
self.assertEqual(te_exp_minus1(x, 3), x + x**2/2. + x**3/6.)
self.assertRaises(ValueError, te_exp_minus1, x, 0)
# make sure x was not modified
self.assertEqual(x, Variable('x'))
# try for VectorVariable too
y = VectorVariable(3, 'y')
self.assertEqual(te_exp_minus1(y, 1), y)
self.assertEqual(te_exp_minus1(y, 3), y + y**2/2. + y**3/6.)
self.assertRaises(ValueError, te_exp_minus1, y, 0)
# make sure y was not modified
self.assertEqual(y, VectorVariable(3, 'y'))
def test_te_exp_minus1(self):
"""Test Taylor expansion of e^x - 1"""
x = Variable('x')
self.assertEqual(te_exp_minus1(x, 1), x)
self.assertEqual(te_exp_minus1(x, 3), x + x**2 / 2. + x**3 / 6.)
self.assertRaises(ValueError, te_exp_minus1, x, 0)
# make sure x was not modified
self.assertEqual(x, Variable('x'))
# try for VectorVariable too
y = VectorVariable(3, 'y')
self.assertEqual(te_exp_minus1(y, 1), y)
self.assertEqual(te_exp_minus1(y, 3), y + y**2 / 2. + y**3 / 6.)
self.assertRaises(ValueError, te_exp_minus1, y, 0)
# make sure y was not modified
self.assertEqual(y, VectorVariable(3, 'y'))
def setup(self, aircraft, state, **kwargs):
self.aircraft = aircraft
self.aircraftP = aircraft.dynamic(state)
self.wingP = self.aircraftP.wingP
self.fuseP = self.aircraftP.fuseP
self.engineP = self.aircraftP.engineP
#variable definitions
z_bre = Variable('z_{bre}', '-', 'Breguet Parameter')
Rng = Variable('Rng', 'nautical_miles', 'Cruise Segment Range')
constraints = []
constraints.extend([
#steady level flight constraint on D
self.aircraftP['D'] == aircraft['numeng'] * self.engineP['F'],
#taylor series expansion to get the weight term
TCS([self.aircraftP['W_{burn}']/self.aircraftP['W_{end}'] >=
te_exp_minus1(z_bre, nterm=3)]),
#breguet range eqn
TCS([z_bre >= (self.engineP['TSFC'] * self.aircraftP['thr']*
self.aircraftP['D']) / self.aircraftP['W_{avg}']]),
#time
self.aircraftP['thr'] * state['V'] == Rng,
])
return constraints, self.aircraftP
def setup(self, goods=None, bads=None, taylor_order=6):
goods = goods if goods else {}
goods.update({1 / k: 1 / v for k, v in bads.items()})
N = check_values_length(goods)
exp_S = VectorVariable(N, "e^{S}")
VectorVariable(N, "S")
self.cost = 1
constraints = [[exp_S >= 1, exp_S.prod() == np.e]]
for monomial, options in goods.items():
m_nd = Variable("|%s|" %
monomial.latex(excluded=["models", "units"]))
exp_m = Variable("e^{%s}" % m_nd.latex(excluded=["models"]))
self.cost /= monomial
if hasattr(options, "units"):
monomial = monomial / (1 * options.units)
options = options.magnitude
options_scale = options.mean()
options /= options_scale
constraints.append([
m_nd == monomial / options_scale,
(exp_S**options).prod() == exp_m,
exp_m >= 1 + te_exp_minus1(m_nd, taylor_order),
])
return constraints
def setup(self, aircraft, state, **kwargs):
self.aircraft = aircraft
self.aircraftP = AircraftP(aircraft, state)
self.wingP = self.aircraftP.wingP
self.fuseP = self.aircraftP.fuseP
#variable definitions
z_bre = Variable('z_{bre}', '-', 'Breguet Parameter')
Rng = Variable('Rng', 'nautical_miles', 'Cruise Segment Range')
constraints = []
constraints.extend([
#taylor series expansion to get the weight term
TCS([
self.aircraftP['W_{burn}'] / self.aircraftP['W_{end}'] >=
te_exp_minus1(z_bre, nterm=3)
]),
#time
self.aircraftP['thr'] * state['V'] == Rng,
self.aircraftP['W_{burn}'] == self.aircraft.engine['TSFC'][2:] *
self.aircraft.engine['F'][:2:] * self.aircraftP['thr']
])
return constraints, self.aircraftP
def setup(self, perf):
z_bre = Variable("z_{bre}", "-", "Breguet coefficient")
t = Variable("t", "days", "Time per flight segment")
f_fueloil = Variable("f_{(fuel/oil)}", 0.98, "-", "Fuel-oil fraction")
Wfuel = Variable("W_{fuel}", "lbf", "Segment-fuel weight")
g = Variable("g", 9.81, "m/s^2", "gravitational acceleration")
constraints = [
TCS([
z_bre >= (perf["P_{total}"] * t * perf["BSFC"] * g /
(perf["W_{end}"] * perf["W_{start}"])**0.5)
]), f_fueloil * Wfuel / perf["W_{end}"] >= te_exp_minus1(z_bre, 3),
perf["W_{start}"] >= perf["W_{end}"] + Wfuel
]
return constraints
def setup(self, Wstart, Wend, perf):
z_bre = Variable("z_{bre}", "-", "Breguet coefficient")
Wfuel = Variable("W_{fuel}", "lbf", "segment-fuel weight")
g = Variable("g", 9.81, "m/s^2", "gravitational acceleration")
R = Variable("R", "nautical_miles", "range")
rhofuel = Variable("\\rho_{JetA}", 6.75, "lb/gallon",
"Jet A fuel density")
constraints = [
TCS([
z_bre >= (R * perf["\\dot{m}"] * rhofuel * g / perf["V"] /
(Wend * Wstart)**0.5)
]), Wfuel / Wend >= te_exp_minus1(z_bre, 3), Wstart >= Wend + Wfuel
]
return constraints
def setup(self, aircraft, state, **kwargs):
self.aircraft = aircraft
self.aircraftP = AircraftP(aircraft, state)
self.wingP = self.aircraftP.wingP
self.fuseP = self.aircraftP.fuseP
self.engineP = self.aircraftP.engineP
#variable definitions
z_bre = Variable('z_{bre}', '-', 'Breguet Parameter')
Rng = Variable('Rng', 'nautical_miles', 'Cruise Segment Range')
constraints = []
constraints.extend([
#steady level flight constraint on D
self.aircraftP['D'] == aircraft['numeng'] * self.engineP['thrust'],
#taylor series expansion to get the weight term
TCS([
self.aircraftP['W_{burn}'] / self.aircraftP['W_{end}'] >=
te_exp_minus1(z_bre, nterm=3)
]),
#breguet range eqn
# old version -- possibly unneeded numeng
# TCS([z_bre >= (self.aircraft['numeng'] * self.engineP['TSFC'] * self.aircraftP['thr']*
# self.aircraftP['D']) / self.aircraftP['W_{avg}']]),
# new version -- needs to be thought through carefully
# seems correct to me - I switched T to D below (steady level flight) but fogot
#about the Negn term
TCS([
z_bre >= (self.engineP['TSFC'] * self.aircraftP['thr'] *
self.aircraftP['D']) / self.aircraftP['W_{avg}']
]),
#time
self.aircraftP['thr'] * state['V'] == Rng,
])
return constraints, self.aircraftP
def setup(self):
# Fixed Parameters
LD_max = Variable('\\left(\\frac{L}{D}\\right)_{max}', 15, '-',
'Maximum lift-to-drag ratio')
MTOW = Variable('MTOW', 10000, 'lbf', 'Max takeoff weight')
TSFC = Variable('TSFC', 0.307, 'lb/lbf/hr',
'Thrust specific fuel consumption')
V_max = Variable('V_{max}', 420, 'knots', 'Maximum velocity')
W_e = Variable('W_{e}', 7000, 'lbf', 'Operating empty weight')
# Constants
g = Variable('g', 9.81, 'm/s^2', 'Gravitational acceleration')
# Free Variables
LD = Variable('\\frac{L}{D}', '-', 'Lift-to-drag ratio')
R = Variable('R', 'nautical_miles', 'Range')
V = Variable('V', 'knots', 'Velocity')
W_fuel = Variable('W_{fuel}', 'lbf', 'Fuel weight')
W_init = Variable('W_{init}', 'lbf', 'Initial gross weight')
z_bre = Variable('z_{bre}', '-', 'Breguet parameter')
# Model
objective = 1/R # Maximize range
constraints = [# Aircraft and fuel weight
W_init >= W_e + W_fuel,
# Performance constraints
W_init <= MTOW,
LD <= LD_max,
V <= V_max,
# Breguet range
R <= z_bre*LD*V/(TSFC*g),
# Taylor series expansion of exp(z_bre) - 1
W_fuel/W_e >= te_exp_minus1(z_bre, nterm=3)
]
return objective, constraints
def __init__(self, **kwargs):
T_tp = 216.65
k = GRAVITATIONAL_ACCEL/(GAS_CONSTANT*T_tp)
p11 = Variable('p_{11}', 22630, 'Pa', 'Pressure at 11 km')
objective = 1/rho # maximize density
constraints = [h >= 11*units.km,
h <= 20*units.km,
# Temperature is constant in the tropopause
T == T_tp,
# Pressure-altitude relation, using taylor series exp
TCS([np.exp(k*11000)*p11/p >=
1 + te_exp_minus1(g/(R*T)*h, 15)], reltol=1E-4),
# Ideal gas law
rho == p/(R*T),
]
su = Sutherland()
lc = su.link(constraints)
Model.__init__(self, objective, lc, **kwargs)
def __init__(self):
CL = Variable('C_L', '-', 'Lift coefficient')
CLmax = Variable('C_{L_{max}}', '-', 'Max lift coefficient')
CD = Variable('C_D', '-', 'Drag coefficient')
D = Variable('D', 'N', 'Total aircraft drag (cruise)')
Dfuse = Variable('D_{fuse}', 'N', 'Fuselage drag')
Dht = Variable('D_{ht}', 'N', 'Horizontal tail drag')
Dvt = Variable('D_{vt}', 'N', 'Vertical tail drag')
Dwing = Variable('D_{wing}', 'N', 'Wing drag')
LD = Variable('\\frac{L}{D}', '-', 'Lift/drag ratio')
Lh = Variable('L_h', 'N', 'Horizontal tail downforce')
Lw = Variable('L_w', 'N', 'Wing lift')
M = Variable('M', '-', 'Cruise Mach number')
R = Variable('Range', 'nautical_miles', 'Range')
Sw = Variable('S_w', 'm**2', 'Wing reference area')
Te = Variable('T_e', 'N', 'Engine thrust at takeoff')
TSFC = Variable('c_T', 'lb/lbf/hr',
'Thrust specific fuel consumption')
V = Variable('V_{\\infty}', 'm/s', 'Cruise velocity')
VTO = Variable('V_{TO}', 'm/s', 'Takeoff speed')
W = Variable('W', 'N', 'Total aircraft weight')
Weng = Variable('W_{eng}', 'N', 'Engine weight')
Wfuel = Variable('W_{fuel}', 'N', 'Fuel weight')
Wfuse = Variable('W_{fuse}', 'N', 'Fuselage weight')
Wht = Variable('W_{ht}', 'N', 'Horizontal tail weight')
Wlg = Variable('W_{lg}', 'N', 'Landing gear weight')
Wpay = Variable('W_{pay}', 'N', 'Payload weight')
Wvt = Variable('W_{vt}', 'N', 'Vertical tail weight')
Wwing = Variable('W_{wing}', 'N', 'Wing weight')
Wzf = Variable('W_{zf}', 'N', 'Zero fuel weight')
a = Variable('a', 'm/s', 'Speed of sound (35,000 ft)')
g = Variable('g', 9.81, 'm/s^2', 'Gravitational acceleration')
lr = Variable('l_r', 5000, 'ft', 'Runway length')
rho = Variable('\\rho', 'kg/m^3', 'Air density (cruise)')
rho0 = Variable('\\rho_0', 'kg/m^3', 'Air density (sea level)')
xCG = Variable('x_{CG}', 'm', 'x-location of CG')
xCGeng = Variable('x_{CG_{eng}}', 'm', 'x-location of engine CG')
xCGfu = Variable('x_{CG_{fu}}', 'm', 'x-location of fuselage CG')
xCGht = Variable('x_{CG_{ht}}', 'm', 'x-location of htail CG')
xCGlg = Variable('x_{CG_{lg}}', 'm', 'x-location of landing gear CG')
xCGvt = Variable('x_{CG_{vt}}', 'm', 'x-location of vtail CG')
xCGwing = Variable('x_{CG_{wing}}', 'm', 'x-location of wing CG')
xTO = Variable('x_{TO}', 'm', 'Takeoff distance')
xi = Variable('\\xi', '-', 'Takeoff parameter')
xw = Variable('x_w', 'm', 'x-location of wing aerodynamic center')
y = Variable('y', '-', 'Takeoff parameter')
z_bre = Variable('z_{bre}', '-', 'Breguet parameter')
with SignomialsEnabled():
objective = Wfuel
# High level constraints
hlc = [
# Drag and weight buildup
D >= Dvt + Dfuse + Dwing + Dht,
Wzf >= Wvt + Wfuse + Wlg + Wwing + Wht + Weng + Wpay,
W >= Wzf + Wfuel,
# Range equation for a jet
V == M*a,
D == 0.5*rho*V**2*Sw*CD,
W == 0.5*rho*V**2*Sw*CL,
LD == CL/CD,
Lw >= W, # TODO: add Lh
R <= z_bre*LD*V/(TSFC*g),
Wfuel/Wzf >= te_exp_minus1(z_bre, nterm=3),
# CG relationships
TCS([xCG*W >= Wvt*xCGvt + Wfuse*xCGfu + Wlg*xCGlg
+ Wwing*xCGwing + Wht*xCGht + Weng*xCGeng
+ Wfuel*xCGwing + Wpay*xCGfu],
reltol=1E-2, raiseerror=False),
xw == xCGwing,
xCGeng == xCGwing,
#Takeoff relationships
xi >= 0.5*rho0*VTO**2*Sw*CD/Te,
4*g*xTO*Te/(W*VTO**2) >= 1 + y,
1 >= 0.0464*xi**2.73/y**2.88 + 1.044*xi**0.296/y**0.049,
VTO == 1.2*(2*W/(rho0*Sw*CLmax))**0.5,
xTO <= lr,
]
# Subsystem models
vt = VerticalTail.coupled737()
fu = Fuselage.coupled737()
lg = LandingGear.coupled737()
ht = HorizontalTail.coupled737()
wi = Wing.coupled737()
wb = WingBox()
substitutions = {
'C_{L_{max}}': 2.5,
'M': 0.78,
'Range': 3000,
'c_T': 0.3,
'W_{eng}': 10000,
'a': 297,
}
lc = LinkedConstraintSet([hlc, vt, fu, lg, ht, wi],
exclude=[vk.name for vk in wb.varkeys
if not vk.value])
Model.__init__(self, objective,
lc,
substitutions)
# Constants
g = Variable('g', 9.81, 'm/s^2', 'Gravitational acceleration')
rho = Variable('\\rho', 0.4, 'kg/m^3', 'Air density at 33,000 ft')
rho_sl = Variable('\\rho_{SL}', 1.225, 'kg/m^3', 'Air density at sea level')
# Model
objective = 1 / R # Maximize range
constraints = [ # Weight buildup
W >= W_e + W_pay + W_fuel,
W_e >= f_str * W,
# Breguet range
R <= z_bre * CL / CD * V / (TSFC * g),
# Taylor series expansion of exp(z_bre) - 1
W_fuel / W_e >= te_exp_minus1(z_bre, nterm=3),
# Wing geometry
AR == b**2 / S,
b <= bmax,
# Steady level flight
W == 0.5 * rho * V**2 * S * CL,
T == 0.5 * rho * V**2 * S * CD,
T <= T_max,
# Drag buildup
CD >= CD0 + CL**2 / (np.pi * e * AR),
# Speed limited by compressible drag rise
M == V / a,
+ W_electrical
+ W_avionics
+ W_furnishings
+ W_air_conditioning
+ W_handling_gear
+ W_anti_ice
+ W_military_cargo_handling_system
+ Nen * Wen
)
]
# Weights
constraints += [
W_maximum_gross >= W_operating_empty + Wc + W_fuel,
zbre >= R * (SFC/V) * (D/L),
W_fuel >= W_operating_empty*te_exp_minus1(zbre,4),
W_landing_gross >= .7 * W_maximum_gross
]
# # Performance Model
constraints += [
L == 1./2 * rho * V**2 * CL * S,
L == W_maximum_gross,
D == 1./2 * rho * V**2 * CD * S,
S >= Saft + Scab + Sw ,
]
# Aerodynamic Model
constraints += [
Re == rho*V*chord/mu,
M == V/a,
def setup(self):
constraints = []
# Steady level flight relations
CD = Variable('C_D', '-', 'Drag coefficient')
CL = Variable('C_L', '-', 'Lift coefficient')
P_shaft = Variable('P_{shaft}', 'W', 'Shaft power')
S = Variable('S', 'm^2', 'Wing reference area')
V = Variable('V', 'm/s', 'Cruise velocity')
W = Variable('W', 'lbf', 'Aircraft weight')
eta_prop = Variable(r'\eta_{prop}', 0.7, '-', 'Propulsive efficiency')
rho = Variable(r'\rho', 'kg/m^3')
constraints.extend([P_shaft == V*W*CD/CL/eta_prop, # eta*P = D*V
W == 0.5*rho*V**2*CL*S])
# Aerodynamics model
Cd0 = Variable('C_{d0}', 0.01, '-', "non-wing drag coefficient")
CLmax = Variable('C_{L-max}', 1.5, '-', 'maximum lift coefficient')
e = Variable('e', 0.9, '-', "spanwise efficiency")
A = Variable('A', 20, '-', "aspect ratio")
b = Variable('b', 'ft', 'span')
mu = Variable(r'\mu', 1.5e-5, 'N*s/m^2', "dynamic viscosity")
Re = Variable("Re", '-', "Reynolds number")
Cf = Variable("C_f", "-", "wing skin friction coefficient")
Kwing = Variable("K_{wing}", 1.3, "-", "wing form factor")
cl_16 = Variable("cl_{16}", 0.0001, "-", "profile stall coefficient")
constraints.extend([CD >= Cd0 + 2*Cf*Kwing + CL**2/(pi*e*A) + cl_16*CL**16,
b**2 == S*A,
CL <= CLmax,
Re == rho*V/mu*(S/A)**0.5,
Cf >= 0.074/Re**0.2])
# Engine Weight Model
W_eng = Variable('W_{eng}', 'lbf', 'Engine weight')
W_engmin = Variable('W_{min}', 1.3, 'lbf', 'min engine weight')
W_engmax = Variable('W_{max}', 275, 'lbf', 'max engine weight')
eta_t = Variable('\\eta_t', 0.75, '-', 'percent throttle')
eng_cnst = Variable('eng_{cnst}', 0.0011, '-',
'engine constant based off of power weight model')
constraints.extend([W_eng >= W_engmin,
W_eng <= W_engmax,
W_eng >= P_shaft*eng_cnst/eta_t * units('lbf/watt')])
# Weight model
W_airframe = Variable('W_{airframe}', 'lbf', 'Airframe weight')
W_pay = Variable(r'W_{pay}', 4, 'lbf', 'Payload weight')
W_fuel = Variable('W_{fuel}', 'lbf', 'Fuel Weight')
W_zfw = Variable('W_{zfw}', 'lbf', 'Zero fuel weight')
wl = Variable('wl', 'lbf/ft^2', 'wing loading')
f_airframe = Variable('f_{airframe}', 0.3, '-',
'Airframe weight fraction')
g = Variable('g', 9.81, 'm/s^2', 'Gravitational acceleration')
constraints.extend([W_airframe >= W*f_airframe,
W_zfw >= W_airframe + W_eng + W_pay,
wl == W/S,
W >= W_pay + W_eng + W_airframe + W_fuel])
# Breguet Range
z_bre = Variable("z_bre", "-", "breguet coefficient")
h_fuel = Variable("h_{fuel}", 42e6, "J/kg", "heat of combustion")
eta_0 = Variable("\\eta_0", 0.2, "-", "overall efficiency")
t = Variable('t', 7, 'days', 'time on station')
constraints.extend([z_bre >= g*t*V*CD/(h_fuel*eta_0*CL),
W_fuel/W_zfw >= te_exp_minus1(z_bre, 3)])
# Atmosphere model
h = Variable("h", "ft", "Altitude")
p_sl = Variable("p_{sl}", 101325, "Pa", "Pressure at sea level")
T_sl = Variable("T_{sl}", 288.15, "K", "Temperature at sea level")
L_atm = Variable("L_{atm}", 0.0065, "K/m", "Temperature lapse rate")
T_atm = Variable("T_{atm}", "K", "air temperature")
M_atm = Variable("M_{atm}", 0.0289644, "kg/mol",
"Molar mass of dry air")
R_atm = Variable("R_{atm}", 8.31447, "J/mol/K")
TH = (g*M_atm/R_atm/L_atm).value.magnitude # dimensionless
constraints.extend([h <= 20000*units.m, # Model valid to top of troposphere
T_sl >= T_atm + L_atm*h, # Temp decreases w/ altitude
rho <= p_sl*T_atm**(TH-1)*M_atm/R_atm/(T_sl**TH)])
# http://en.wikipedia.org/wiki/Density_of_air#Altitude
# station keeping requirement
footprint = Variable("d_{footprint}", 100, 'km',
"station keeping footprint diameter")
lu = Variable(r"\theta_{look-up}", 5, '-', "look up angle")
R_earth = Variable("R_{earth}", 6371, "km", "Radius of earth")
tan_lu = lu*pi/180. + (lu*pi/180.)**3/3. # Taylor series expansion
# approximate earth curvature penalty as distance^2/(2*Re)
constraints.extend([
h >= tan_lu*0.5*footprint + footprint**2/8./R_earth])
# wind speed model
V_wind = Variable('V_{wind}', 'm/s', 'wind speed')
wd_cnst = Variable('wd_{cnst}', [0.002, 0.0015], 'm/s/ft',
'wind speed constant predited by model')
wd_ln = Variable('wd_{ln}', [13.009, 8.845], 'm/s',
'linear wind speed variable')
h_min = Variable('h_{min}', 11800, 'ft', 'minimum height')
h_max = Variable('h_{max}', 20800, 'ft', 'maximum height')
constraints.extend([V_wind >= wd_cnst*h + wd_ln, # model predicting worse case scenario at 45 deg latitude
V >= V_wind,
h >= h_min,
h <= h_max])
objective = W
return objective, constraints
m_structures = VectorVariable(n_stages,"m_structures","kg")
m_dot = VectorVariable(n_stages,"m_dot","kg/s")
v_exhaust_effective = VectorVariable(n_stages,"v_exhaust_effective","m/s")
F_thrust = VectorVariable(n_stages,"F_thrust","N")
total_mass = VectorVariable(n_stages,"total_mass","kg")
m_total = Variable("m_total","kg")
g = Variable("g",9.8,"m/s/s")
constraints = []
for stage in range(n_stages):
constraints += [
Isp[stage] == v_exhaust_effective[stage]/g,
m_dot[stage] == F_thrust[stage]/(g*Isp[stage]),
z[stage] >= (dV[stage]+g*(m_fuel[stage]/m_dot[stage]))/v_exhaust_effective[stage],
theta_fuel[stage] >= te_exp_minus1(z[stage],5)
]
if stage == 0:
constraints+=[F_thrust[stage]/g >= total_mass[stage],
total_mass[stage] >= m_fuel[stage]+m_fuel[stage+1]+m_structures[stage]+m_structures[stage+1]+m_payload,
m_structures[stage] == 0.02*m_fuel[stage],
theta_fuel[stage] == m_fuel[stage]/total_mass[stage]]
if stage == 1:
constraints+=[F_thrust[stage]/g >= total_mass[stage],
total_mass[stage] >= m_fuel[stage]+m_structures[stage] + m_payload,
m_structures[stage] == 0.02*m_fuel[stage],
theta_fuel[stage] == m_fuel[stage]/total_mass[stage]]
constraints+=[
m_total >= m_structures[0] + m_fuel[0] + m_structures[1] + m_fuel[1] + m_payload]
constraints += [
W_operating_empty >=
(W_centerbody + W_afterbody + W_wing + W_vertical_tail +
W_main_landing_gear + W_nose_landing_gear + W_engine_contents +
W_nacelle_group + W_engine_controls + W_starter_penumatic +
W_fuel_system + W_flight_controls + W_APU_installed + W_instruments +
W_hydraulics + W_electrical + W_avionics + W_furnishings +
W_air_conditioning + W_handling_gear + W_anti_ice +
W_military_cargo_handling_system + Nen * Wen)
]
# Weights
constraints += [
W_maximum_gross >= W_operating_empty + Wc + W_fuel,
zbre >= R * (SFC / V) * (D / L),
W_fuel >= W_operating_empty * te_exp_minus1(zbre, 4),
W_landing_gross >= .7 * W_maximum_gross
]
# # Performance Model
constraints += [
L == 1. / 2 * rho * V**2 * CL * S,
L == W_maximum_gross,
D == 1. / 2 * rho * V**2 * CD * S,
S >= Saft + Scab + Sw,
]
# Aerodynamic Model
constraints += [
Re == rho * V * chord / mu,
M == V / a,
# Constants
g = Variable('g', 9.81, 'm/s^2', 'Gravitational acceleration')
rho = Variable('\\rho', 0.4, 'kg/m^3', 'Air density at 33,000 ft')
rho_sl = Variable('\\rho_{SL}', 1.225, 'kg/m^3', 'Air density at sea level')
# Model
objective = 1/R # Maximize range
constraints = [# Weight buildup
W >= W_e + W_pay + W_fuel,
W_e >= f_str*W,
# Breguet range
R <= z_bre*CL/CD*V/(TSFC*g),
# Taylor series expansion of exp(z_bre) - 1
W_fuel/W_e >= te_exp_minus1(z_bre, nterm=3),
# Wing geometry
AR == b**2/S,
b <= bmax,
# Steady level flight
W == 0.5*rho*V**2*S*CL,
T == 0.5*rho*V**2*S*CD,
T <= T_max,
# Drag buildup
CD >= CD0 + CL**2/(np.pi*e*AR),
# Speed limited by compressible drag rise
M == V/a,
def setup(self):
constraints = []
# Steady level flight relations
CD = Variable('C_D', '-', 'Drag coefficient')
CL = Variable('C_L', '-', 'Lift coefficient')
P_shaft = Variable('P_{shaft}', 'W', 'Shaft power')
S = Variable('S', 'm^2', 'Wing reference area')
V = Variable('V', 'm/s', 'Cruise velocity')
W = Variable('W', 'lbf', 'Aircraft weight')
eta_prop = Variable(r'\eta_{prop}', 0.7, '-', 'Propulsive efficiency')
rho = Variable(r'\rho', 'kg/m^3')
constraints.extend([
P_shaft == V * W * CD / CL / eta_prop, # eta*P = D*V
W == 0.5 * rho * V**2 * CL * S
])
# Aerodynamics model
Cd0 = Variable('C_{d0}', 0.01, '-', "non-wing drag coefficient")
CLmax = Variable('C_{L-max}', 1.5, '-', 'maximum lift coefficient')
e = Variable('e', 0.9, '-', "spanwise efficiency")
A = Variable('A', 20, '-', "aspect ratio")
b = Variable('b', 'ft', 'span')
mu = Variable(r'\mu', 1.5e-5, 'N*s/m^2', "dynamic viscosity")
Re = Variable("Re", '-', "Reynolds number")
Cf = Variable("C_f", "-", "wing skin friction coefficient")
Kwing = Variable("K_{wing}", 1.3, "-", "wing form factor")
cl_16 = Variable("cl_{16}", 0.0001, "-", "profile stall coefficient")
constraints.extend([
CD >= Cd0 + 2 * Cf * Kwing + CL**2 / (pi * e * A) + cl_16 * CL**16,
b**2 == S * A, CL <= CLmax, Re == rho * V / mu * (S / A)**0.5,
Cf >= 0.074 / Re**0.2
])
# Engine Weight Model
W_eng = Variable('W_{eng}', 'lbf', 'Engine weight')
W_engmin = Variable('W_{min}', 1.3, 'lbf', 'min engine weight')
W_engmax = Variable('W_{max}', 275, 'lbf', 'max engine weight')
eta_t = Variable('\\eta_t', 0.75, '-', 'percent throttle')
eng_cnst = Variable('eng_{cnst}', 0.0011, '-',
'engine constant based off of power weight model')
constraints.extend([
W_eng >= W_engmin, W_eng <= W_engmax,
W_eng >= P_shaft * eng_cnst / eta_t * units('lbf/watt')
])
# Weight model
W_airframe = Variable('W_{airframe}', 'lbf', 'Airframe weight')
W_pay = Variable(r'W_{pay}', 4, 'lbf', 'Payload weight')
W_fuel = Variable('W_{fuel}', 'lbf', 'Fuel Weight')
W_zfw = Variable('W_{zfw}', 'lbf', 'Zero fuel weight')
wl = Variable('wl', 'lbf/ft^2', 'wing loading')
f_airframe = Variable('f_{airframe}', 0.3, '-',
'Airframe weight fraction')
g = Variable('g', 9.81, 'm/s^2', 'Gravitational acceleration')
constraints.extend([
W_airframe >= W * f_airframe, W_zfw >= W_airframe + W_eng + W_pay,
wl == W / S, W >= W_pay + W_eng + W_airframe + W_fuel
])
# Breguet Range
z_bre = Variable("z_bre", "-", "breguet coefficient")
h_fuel = Variable("h_{fuel}", 42e6, "J/kg", "heat of combustion")
eta_0 = Variable("\\eta_0", 0.2, "-", "overall efficiency")
t = Variable('t', 7, 'days', 'time on station')
constraints.extend([
z_bre >= g * t * V * CD / (h_fuel * eta_0 * CL),
W_fuel / W_zfw >= te_exp_minus1(z_bre, 3)
])
# Atmosphere model
h = Variable("h", "ft", "Altitude")
p_sl = Variable("p_{sl}", 101325, "Pa", "Pressure at sea level")
T_sl = Variable("T_{sl}", 288.15, "K", "Temperature at sea level")
L_atm = Variable("L_{atm}", 0.0065, "K/m", "Temperature lapse rate")
T_atm = Variable("T_{atm}", "K", "air temperature")
M_atm = Variable("M_{atm}", 0.0289644, "kg/mol",
"Molar mass of dry air")
R_atm = Variable("R_{atm}", 8.31447, "J/mol/K")
TH = (g * M_atm / R_atm / L_atm).value.magnitude # dimensionless
constraints.extend([
h <= 20000 * units.m, # Model valid to top of troposphere
T_sl >= T_atm + L_atm * h, # Temp decreases w/ altitude
rho <= p_sl * T_atm**(TH - 1) * M_atm / R_atm / (T_sl**TH)
])
# http://en.wikipedia.org/wiki/Density_of_air#Altitude
# station keeping requirement
footprint = Variable("d_{footprint}", 100, 'km',
"station keeping footprint diameter")
lu = Variable(r"\theta_{look-up}", 5, '-', "look up angle")
R_earth = Variable("R_{earth}", 6371, "km", "Radius of earth")
tan_lu = lu * pi / 180. + (lu * pi /
180.)**3 / 3. # Taylor series expansion
# approximate earth curvature penalty as distance^2/(2*Re)
constraints.extend(
[h >= tan_lu * 0.5 * footprint + footprint**2 / 8. / R_earth])
# wind speed model
V_wind = Variable('V_{wind}', 'm/s', 'wind speed')
wd_cnst = Variable('wd_{cnst}', [0.002, 0.0015], 'm/s/ft',
'wind speed constant predited by model')
wd_ln = Variable('wd_{ln}', [13.009, 8.845], 'm/s',
'linear wind speed variable')
h_min = Variable('h_{min}', 11800, 'ft', 'minimum height')
h_max = Variable('h_{max}', 20800, 'ft', 'maximum height')
constraints.extend([
V_wind >= wd_cnst * h +
wd_ln, # model predicting worse case scenario at 45 deg latitude
V >= V_wind,
h >= h_min,
h <= h_max
])
objective = W
return objective, constraints