def evol_dphre(P):
mss = [-0.1, 0, 0.2, 1]
alphas = np.arange(ut.rad_of_deg(-10), ut.rad_of_deg(20), ut.rad_of_deg(1))
km = 0.1
for ms in mss:
P.set_mass_and_static_margin(km, ms)
dphrs = []
dphrs = np.array([(P.ms * P.CLa * (alpha - P.a0) - P.Cm0) / P.Cmd
for alpha in alphas])
plt.plot(ut.deg_of_rad(alphas), ut.deg_of_rad(dphrs))
legends = ['ms {}'.format(ms) for ms in mss]
ut.decorate(plt.gca(),
"Evolution de DPHRe en fonction de l'incidence",
'Incidence',
'Delta de braquage équilibré',
legend=legends)
plt.figure()
Vts = [0.9, 1, 1.1]
for Vt in Vts:
P.Vt = Vt
P.Cmd = -P.Vt * P.CLat
dphrs = np.array([(P.ms * P.CLa * (alpha - P.a0) - P.Cm0) / P.Cmd
for alpha in alphas])
plt.plot(ut.deg_of_rad(alphas), ut.deg_of_rad(dphrs))
legends = ['Vt {}'.format(Vt) for Vt in Vts]
ut.decorate(plt.gca(),
"Evolution de DPHRe en fonction de l'incidence",
'Incidence',
'Delta de braquage équilibré',
legend=legends)
plt.show()
def polaire_equi(P):
q = 0
mss = [0.2, 1]
alphas = np.arange(ut.rad_of_deg(-10), ut.rad_of_deg(20), ut.rad_of_deg(1))
km = 0.1
for ms in mss:
P.set_mass_and_static_margin(km, ms)
dphrs = []
dphrs = np.array([(P.ms * P.CLa * (alpha - P.a0) - P.Cm0) / P.Cmd
for alpha in alphas])
CLe = []
CLe = [
dyn.get_aero_coefs(dyn.va_of_mach(0.7, 7000), alpha, q, dphr, P)[0]
for alpha, dphr in zip(alphas, dphrs)
]
Cxe = [
dyn.get_aero_coefs(dyn.va_of_mach(0.7, 7000), alpha, q, dphr, P)[1]
for alpha, dphr in zip(alphas, dphrs)
]
finesse = [CL / Cx for Cx, CL in zip(Cxe, CLe)]
fmax = np.max(finesse)
idmax = np.argmax(finesse)
print(fmax)
plt.plot([0, Cxe[idmax]], [0, CLe[idmax]])
plt.plot(Cxe, CLe)
label1 = patches.Patch(color='blue', label='fmax (ms = 0.2)')
label2 = patches.Patch(color='green', label='Polaire (ms = 0.2)')
label3 = patches.Patch(color='red', label='fmax (ms = 1)')
label4 = patches.Patch(color='cyan', label='Polaire (ms = 1)')
plt.legend(loc='upper left', handles=[label1, label2, label3, label4])
ut.decorate(plt.gca(), "Evolution de la polaire équilibrée",
'Coefficient de trainée équilibrée',
'Coefficient de portance équilibrée')
plt.show
def run_trajecotry(time, h, Ma, ms, km, Wh=2):
Xe, Ue = get_trim(aircraft, h, Ma, ms, km)
fmt = r'Trim: h={:.1f}, Ma={:.1f}, ms={:.1f}, km={:.1f} -> $\alpha$={:.1f}, $\delta_{{PHR}}$={:.1f}, $\delta_{{th}}$={:.1f}'
print fmt.format(h, Ma, ms, km, ut.rad_of_deg(Xe[dyn.s_a]), ut.rad_of_deg(Ue[dyn.i_dm]), Ue[dyn.i_dth])
X = NonLinearTraj(aircraft, h, Ma, ms, km, Wh, time)
Xlin = LinearTraj(aircraft, h, Ma, ms, km, Wh, time)
figure = dyn.plot(time, X)
dyn.plot(time, Xlin, figure=figure)
def plot_CL(P, filename=None):
alphas = np.linspace(ut.rad_of_deg(-10), ut.rad_of_deg(20), 30)
dms = np.linspace(ut.rad_of_deg(20), ut.rad_of_deg(-30), 3)
figure = ut.prepare_fig(None, u'Coefficient de Portance {}'.format(P.name))
for dm in dms:
plt.plot(ut.deg_of_rad(alphas), CL(P, alphas, dm))
ut.decorate(plt.gca(), u'Coefficient de Portance', r'$\alpha$ en degres', '$C_L$')
plt.legend(['$\delta _{{PHR}} = ${:.1f}'.format(ut.deg_of_rad(dm)) for dm in dms], loc='best')
if filename<> None: plt.savefig(filename, dpi=160)
def plot_Cm(P, filename=None):
alphas = np.linspace(ut.rad_of_deg(-10), ut.rad_of_deg(20), 30)
mss = np.array([-0.1, 0, 0.2, 1])
figure = ut.prepare_fig(None, u'Coefficient du moment de tangage {}'.format(P.name))
for ms1 in mss:
plt.plot(ut.deg_of_rad(alphas), Cm(P, alphas, ms1))
ut.decorate(plt.gca(), u'Coefficient du moment de tangage', r'$\alpha$ en degres', '$C_m$')
plt.legend(['ms = {}'.format(ms) for ms in mss], loc='best')
if filename<> None: plt.savefig(filename, dpi=160)
def plot_dphr_e(P, filename=None):
alphas = np.linspace(ut.rad_of_deg(-10), ut.rad_of_deg(20), 30)
mss = [-0.1, 0., 0.2, 1.]
figure = ut.prepare_fig(None, u'Équilibre {}'.format(P.name))
for ms in mss:
P.set_mass_and_static_margin(0.5, ms)
dmes = np.array([dphr_e(P, alpha) for alpha in alphas])
plt.plot(ut.deg_of_rad(alphas), ut.deg_of_rad(dmes))
ut.decorate(plt.gca(), r'$\delta_{PHR_e}$', r'$\alpha$ en degres', r'$\delta_{PHR_e}$ en degres',
['$ms = ${: .1f}'.format(ms) for ms in mss])
def plot_CLe(P):
alphas = np.linspace(ut.rad_of_deg(-10), ut.rad_of_deg(20), 30)
figure = ut.prepare_fig(None, u'Coefficient de portance équilibrée {}'.format(P.name))
sms = [0.2, 1]
for sm in sms:
P.set_mass_and_static_margin(0.5, sm)
dphres = [dphr_e(P, alpha) for alpha in alphas]
CLes = [CL(P, alpha, dphr) for alpha, dphr in zip(alphas, dphres)]
plt.plot(ut.deg_of_rad(alphas), CLes)
ut.decorate(plt.gca(), u'Coefficient de portance équilibrée', r'$\alpha$ en degres', r'$CL_e$',
['$ms = ${: .1f}'.format(sm) for sm in sms])
def plot_polar(P):
alphas = np.linspace(ut.rad_of_deg(-10), ut.rad_of_deg(20), 30)
figure = ut.prepare_fig(None, u'Polaire équilibrée{}'.format(P.name))
sms = [0.2, 1]
for sm in sms:
P.set_mass_and_static_margin(0.5, sm)
dphres = [dphr_e(P, alpha) for alpha in alphas]
CLes = [CL(P, alpha, dphr) for alpha, dphr in zip(alphas, dphres)]
CDes = [P.CD0 + P.ki * CL1**2 for CL1 in CLes]
plt.plot(CDes, CLes)
ut.decorate(plt.gca(), u'Polaire équilibrée', r'$CD_e$', r'$CL_e$',
['$ms = ${: .1f}'.format(sm) for sm in sms])
def __init__(self):
self.g = 9.81
self.m_k = 0.5
self.ms = 0.3 # static margin
# aero
self.a0 = ut.rad_of_deg(-2.) # zero lift angle
self.CD0 = 0.025 # zero drag coefficient
self.Cm0 = -0.59 # zero moment coefficient
self.CLq = 1.3
def __init__(self):
self.g = 9.81
self.m_k = 0.5
self.ms = 0.3 # static margin
# aero
self.a0 = ut.rad_of_deg(-2.) # zero lift angle
self.CD0 = 0.025 # zero drag coefficient
self.Cm0 = -0.59 # zero moment coefficient
self.CLq = 1.3
def trim(P, args=None):
va = args.get('va', 100.)
gamma = args.get('gamma', 0.)
h = args.get('h', 5000.)
wy, wz = 0, 0
def err_func((dm, dth, alpha)):
theta = gamma + alpha
U = np.array([dm, dth, wy, wz])
X = np.array([0., h, va, alpha, theta, 0])
L, D, M = get_aero_forces_and_moments(X, U, P)
F = propulsion_model(X, U, P)
cg, sg = math.cos(gamma), math.sin(gamma)
ca, sa = math.cos(alpha), math.sin(alpha)
return [(F*ca-D)/P.m-P.g*sg, -(L+F*sa)/P.m + P.g*cg, M]
p0 = [ut.rad_of_deg(0.), 0.5, ut.rad_of_deg(1.)]
sol = scipy.optimize.root(err_func, p0, method='hybr')
dm, dth, alpha = sol.x
X, U = [0, h, va, alpha, gamma+alpha, 0], [dm, dth, 0, 0]
return X, U
def evol_Cm(P):
ms = [-0.1, 0, 0.2, 1]
dphr = 0
alphas = np.arange(ut.rad_of_deg(-10), ut.rad_of_deg(20), ut.rad_of_deg(1))
Cm_tab1 = []
Cm_tab2 = []
Cm_tab3 = []
Cm_tab4 = []
q = 0
for alpha in alphas:
Cm1 = P.Cm0 - ms[0] * P.CLa * (alpha -
P.a0) + P.Cmq * P.lt / dyn.va_of_mach(
0.7, 7000) * q + P.Cmd * dphr
Cm_tab1.append(Cm1)
Cm2 = P.Cm0 - ms[1] * P.CLa * (alpha -
P.a0) + P.Cmq * P.lt / dyn.va_of_mach(
0.7, 7000) * q + P.Cmd * dphr
Cm_tab2.append(Cm2)
Cm3 = P.Cm0 - ms[2] * P.CLa * (alpha -
P.a0) + P.Cmq * P.lt / dyn.va_of_mach(
0.7, 7000) * q + P.Cmd * dphr
Cm_tab3.append(Cm3)
Cm4 = P.Cm0 - ms[3] * P.CLa * (alpha -
P.a0) + P.Cmq * P.lt / dyn.va_of_mach(
0.7, 7000) * q + P.Cmd * dphr
Cm_tab4.append(Cm4)
plt.plot(ut.deg_of_rad(alphas), Cm_tab1)
plt.plot(ut.deg_of_rad(alphas), Cm_tab2)
plt.plot(ut.deg_of_rad(alphas), Cm_tab3)
plt.plot(ut.deg_of_rad(alphas), Cm_tab4)
label1 = patches.Patch(color='blue', label='ms = -0.1')
label2 = patches.Patch(color='green', label='ms = 0')
label3 = patches.Patch(color='red', label='ms = 0.2')
label4 = patches.Patch(color='cyan', label='ms = 1')
plt.legend(loc='upper right', handles=[label1, label2, label3, label4])
ut.decorate(plt.gca(),
title="Evolution du Cm en fonction de l'incidence",
xlab='Incidence',
ylab='Cm')
plt.show()
def trim(P, args=None):
va = args.get('va', 100.)
gamma = args.get('gamma', 0.)
h = args.get('h', 5000.)
wy, wz = 0, 0
def err_func(arg):
(dm, dth, alpha) = arg
theta = gamma + alpha
U = np.array([dm, dth, wy, wz])
X = np.array([0., h, va, alpha, theta, 0])
L, D, M = get_aero_forces_and_moments(X, U, P)
F = propulsion_model(X, U, P)
cg, sg = math.cos(gamma), math.sin(gamma)
ca, sa = math.cos(alpha), math.sin(alpha)
return [(F * ca - D) / P.m - P.g * sg, -(L + F * sa) / P.m + P.g * cg, M]
p0 = [ut.rad_of_deg(0.), 0.5, ut.rad_of_deg(1.)]
sol = scipy.optimize.root(err_func, p0, method='hybr')
dm, dth, alpha = sol.x
X, U = [0, h, va, alpha, gamma + alpha, 0], [dm, dth, 0, 0]
return X, U
def evol_CLe(P):
q = 0
mss = [0.2, 1]
alphas = np.arange(ut.rad_of_deg(-10), ut.rad_of_deg(20), ut.rad_of_deg(1))
km = 0.1
for ms in mss:
P.set_mass_and_static_margin(km, ms)
dphrs = []
dphrs = np.array([(P.ms * P.CLa * (alpha - P.a0) - P.Cm0) / P.Cmd
for alpha in alphas])
CLe = []
CLe = [
dyn.get_aero_coefs(dyn.va_of_mach(0.7, 7000), alpha, q, dphr, P)[0]
for alpha, dphr in zip(alphas, dphrs)
]
plt.plot(ut.deg_of_rad(alphas), CLe)
label1 = patches.Patch(color='blue', label='ms = 0.2')
label2 = patches.Patch(color='green', label='ms = 1')
plt.legend(loc='upper left', handles=[label1, label2])
ut.decorate(plt.gca(), "Evolution de CLe en fonction de l'incidence",
'Incidence', 'Coefficient de portance équilibrée')
plt.show
def evol_coeff_portance(P):
alphas = np.arange(ut.rad_of_deg(-10), ut.rad_of_deg(20), ut.rad_of_deg(1))
dphrs = [ut.rad_of_deg(-30), ut.rad_of_deg(20)]
q = 0
Cz_tab1 = []
Cz_tab2 = []
for alpha in alphas:
coeff_portance1 = dyn.get_aero_coefs(dyn.va_of_mach(0.7, 7000), alpha,
q, dphrs[0], P)
Cz_tab1.append(coeff_portance1[0])
coeff_portance2 = dyn.get_aero_coefs(dyn.va_of_mach(0.7, 7000), alpha,
q, dphrs[1], P)
Cz_tab2.append(coeff_portance2[0])
plt.plot(ut.deg_of_rad(alphas), Cz_tab1, 'r')
plt.plot(ut.deg_of_rad(alphas), Cz_tab2, 'b')
label1 = patches.Patch(color='red', label='dPHR1 = -30°')
label2 = patches.Patch(color='blue', label='dPHR2 = 20°')
plt.legend(loc='upper left', handles=[label1, label2])
ut.decorate(plt.gca(),
title="Evolution du Cz en fonction de l'incidence",
xlab='Incidence',
ylab='Cz')
plt.show()
def plot_trims(aircraft, gamma=0., saturate=True):
'''
Affichage de trims sous forme de surfaces colorées
'''
vs = np.linspace(60, 350, 21, endpoint=True)
hs = np.linspace(1000, 14000, 21, endpoint=True)
trims = np.zeros((len(hs), len(vs), 2))
for i,v in enumerate(vs):
for j,h in enumerate(hs):
Xe, Ue = dyn.trim(aircraft, args = {'va': v, 'h': h, 'gamma': gamma})
trims[j, i] = Xe[dyn.s_a], tons_of_newton(dyn.propulsion_model([0, h, v, 0, 0, 0], [0, Ue[dyn.i_dth], 0, 0], aircraft))
if saturate:
alpha_c, T_c = trims[j,i][0], trims[j,i,1]
alpha_max = ut.rad_of_deg(15)
if alpha_c > alpha_max:
trims[j,i][0] = 1e16
trims[j,i][1] = 1e16
Tmax = tons_of_newton(dyn.propulsion_model([0, h, v, 0, 0, 0], [0, 1, 0, 0], aircraft))
if T_c > Tmax:
trims[j,i, 0] = 1e16
trims[j,i, 1] = 1e16
fig = plt.figure(figsize=(0.75*20.48, 0.5*10.24))
ax = plt.subplot(1,2,1)
thetas_deg = ut.deg_of_rad(trims[:, :, 0])
v = np.linspace(0, 15, 16, endpoint=True)
plt.contour(vs, hs, thetas_deg, v, linewidths=0.5, colors='k')
plt.contourf(vs, hs, thetas_deg, v, cmap=plt.cm.jet)
ax.set_title(u'$\\alpha$ (degrés)')
ax.xaxis.set_label_text('vitesse (m/s)')
ax.yaxis.set_label_text('altitude (m)')
plt.colorbar(ticks=v)
ax = plt.subplot(1,2,2)
T_tons = trims[:, :, 1] #/9.81/1000.
v = np.linspace(2, 10, 17, endpoint=True)
plt.contour(vs, hs, T_tons, v, linewidths=0.5, colors='k')
plt.contourf(vs, hs, T_tons, v, cmap=plt.cm.jet)
ax.set_title('thrust (tons)')
ax.xaxis.set_label_text('vitesse (m/s)')
ax.yaxis.set_label_text('altitude (m)')
plt.colorbar(ticks=v)
print("alpha16 = ", alpha16, " rad")
print("angle réglage PHR dphr16 = ", dphr16, " rad")
print("Gaz ou thrust dth16 = ", dth16)
print("CL16 = ", CL16)
# 4.2.2.4 Conclure quant à la technique de réglage de la vitesse de l’avion.
# >>>> pour augmenter la vitesse, il faut diminuer CL donc alphae donc augmenter algébriquement
# l'angle du PHR pour augmenter la portance ou diminuer la déportance.
# 4.2.2.5 - Utiliser la valeur de CL obtenue ainsi que les graphiques tracés lors de la première séance
# pour déterminer (approximativement) les valeurs de trim αe et δPHRe .
print(
"La valeur de alphae obtenue via le graphique de la séance 1 CLe=f(alphae) est ",
ut.rad_of_deg(6.7))
print(
"La valeur de dphre obtenue via le graphique dphre = f(alphae) de la séance 1 en utilisant la valeur de alphae précédente est -0.21rad"
)
# 4.2.2.6 - Comment obtenir la valeur de trim de la manette des gaz δth à partir de la polaire équilibrée ?
# connaissant CL16=CLe = 0.57 la polaire équilibrée donne CDe = 0.038 donc f = CLe/CDe donc D = mg/f ;
# on utilise F = Fmax * delta-th = D, i.e. poussee = trainee, pour calculer deltath
# le graphe de poussée donne Fmax = 50128N
# delta_th = D/Fmax
# par le calcul on delta_th = 0.67 ; par la méthode graphique on a delta_th=0.0.70
CLe = CL16
CDe = 0.038
f = CLe / CDe
# calcul du coef de poussée tel que F = coef * deltath