def ltfree(lK, Rg, alpha, beta, delta, omegat):
ltx = np.cos(alpha) * np.cos(np.radians(omegat) + delta) + np.cos(
beta) * lK - np.cos(np.radians(omegat)) * Rg
lty = np.sin(np.radians(omegat) + delta) - np.sin(np.radians(omegat)) * Rg
ltz = -np.sin(alpha) * np.cos(np.radians(omegat) +
delta) + np.sin(beta) * lK
return np.sqrt(ltx * ltx + lty * lty + ltz * ltz)
def Ftry(lK, Rg, alpha, betas, beta, delta, omegat, ot):
cb = np.cos(beta)
sb = np.sin(beta)
ca = np.cos(alpha)
sa = np.sin(alpha)
cbs = np.cos(betas)
sbs = np.sin(betas)
cot = np.cos(np.radians(omegat) + ot)
sot = np.sin(np.radians(omegat) + ot)
cotd = np.cos(np.radians(omegat) + ot + delta)
sotd = np.sin(np.radians(omegat) + ot + delta)
#
ekxx = ca * cbs
ekxy = ca * sbs
ekxz = -sa
#
ekyx = -sbs
ekyy = cbs
ekyz = 0
#
ekzx = sa * cbs
ekzy = sa * sbs
ekzz = ca
#
ltx = ekxx * cotd + ekyx * sotd + cb * lK - cot * Rg
lty = ekxy * cotd + ekyy * sotd - sot * Rg
ltz = ekxz * cotd + sb * lK
#
return (ltx * ekyx + lty * ekyy + ltz * ekyz) / (ltx * ekzx + lty * ekzy +
ltz * ekzz)
def Mgrz(lK, Rg, alpha, betas, beta, delta, omegat, ot):
cb = np.cos(beta)
sb = np.sin(beta)
ca = np.cos(alpha)
sa = np.sin(alpha)
cbs = np.cos(betas)
sbs = np.sin(betas)
cot = np.cos(np.radians(omegat) + ot)
sot = np.sin(np.radians(omegat) + ot)
cotd = np.cos(np.radians(omegat) + ot + delta)
sotd = np.sin(np.radians(omegat) + ot + delta)
#
ekxx = ca * cbs
ekxy = ca * sbs
ekxz = -sa
#
ekyx = -sbs
ekyy = cbs
ekyz = 0
#
ekzx = sa * cbs
ekzy = sa * sbs
ekzz = ca
#
eaxx = ekxx * cotd + ekyx * sotd
eaxy = ekxy * cotd + ekyy * sotd
eaxz = ekxz * cotd
#
ltx = eaxx + cb * lK - cot * Rg
lty = eaxy - sot * Rg
ltz = eaxz + sb * lK
#
vcz = cot * lty - sot * ltx
#
return vcz / (ltx * ekzx + lty * ekzy + ltz * ekzz)
def rB(R, omegat):
cot = np.cos(np.radians(omegat))
sot = np.sin(np.radians(omegat))
rBx = cot * R
rBy = sot * R
rBz = 0
return np.array([rBx, rBy, rBz])
def rA(lK, R, alpha, betas, beta, delta, omegat):
cb = np.cos(beta)
sb = np.sin(beta)
ca = np.cos(alpha)
sa = np.sin(alpha)
cbs = np.cos(betas)
sbs = np.sin(betas)
cotd = np.cos(np.radians(omegat) + delta)
sotd = np.sin(np.radians(omegat) + delta)
rAx = (ca * cbs * cotd - sbs * sotd) * R + cb * lK
rAy = (ca * sbs * cotd + cbs * sotd) * R
rAz = (-sa * cotd) * R + sb * lK
return np.array([rAx, rAy, rAz])
def arc(r0, R, e, n, phi0, phi):
e1 = e / np.sqrt(np.sum(e * e)) # normalize
en = n / np.sqrt(np.sum(n * n)) # normalize
ip = np.argmax(phi > phi0) # find end index
e2 = np.cross(en, e1)
cp = np.cos(np.radians(phi[:ip]))
sp = np.sin(np.radians(phi[:ip]))
# r = cp*e1+sp*e2
r = np.zeros((3, ip))
r[0, :] = r0[0] + R * (cp * e1[0] + sp * e2[0])
r[1, :] = r0[1] + R * (cp * e1[1] + sp * e2[1])
r[2, :] = r0[2] + R * (cp * e1[2] + sp * e2[2])
return r
def cosgammag(lK, Rg, beta, delta, omegat):
alpha = 0.5*np.pi-beta
cb = np.cos(beta)
sb = np.sin(beta)
cot = np.cos(np.radians(omegat))
sot = np.sin(np.radians(omegat))
cotd = np.cos(np.radians(omegat)+delta)
sotd = np.sin(np.radians(omegat)+delta)
ltx = np.cos(alpha)*cotd + cb*lK - cot*Rg
lty = sotd - sot*Rg
ltz = -np.sin(alpha)*cotd + sb*lK
Rwx = cot*Rg
Rwy = sot*Rg
Rwz = 0
lt = np.sqrt(ltx*ltx+lty*lty+ltz*ltz)
return (Rwx*lty-Rwy*ltx)/(lt*Rg)
def vt(lK, Rg, beta, delta, omegat):
alpha = 0.5*np.pi-beta
cb = np.cos(beta)
sb = np.sin(beta)
ca = np.cos(alpha)
sa = np.sin(alpha)
cot = np.cos(np.radians(omegat))
sot = np.sin(np.radians(omegat))
cotd = np.cos(np.radians(omegat)+delta)
sotd = np.sin(np.radians(omegat)+delta)
ltx = ca*cotd + cb*lK - cot*Rg
lty = sotd - sot*Rg
ltz = -sa*cotd + sb*lK
lt = np.sqrt(ltx*ltx+lty*lty+ltz*ltz)
dltxdt = -ca*sotd + sot*Rg
dltydt = cotd - cot*Rg
dltzdt = sa*sotd
return (ltx*dltxdt+lty*dltydt+ltz*dltzdt)/lt
def cosgammak(lK, Rg, beta, delta, omegat):
alpha = 0.5*np.pi-beta
cb = np.cos(beta)
sb = np.sin(beta)
ca = np.cos(alpha)
sa = np.sin(alpha)
cot = np.cos(np.radians(omegat))
sot = np.sin(np.radians(omegat))
cotd = np.cos(np.radians(omegat)+delta)
sotd = np.sin(np.radians(omegat)+delta)
Rax = ca*cotd
Ray = sotd
Raz = -sa*cotd
ltx = Rax + cb*lK - cot*Rg
lty = Ray - sot*Rg
ltz = Raz + sb*lK
lt = np.sqrt(ltx*ltx+lty*lty+ltz*ltz)
return (sa*(Ray*ltz-Raz*lty) + ca*(Rax*lty-Ray*ltx))/lt
def Marz(lK, Rg, alpha, betas, beta, delta, omegat, ot):
cb = np.cos(beta)
sb = np.sin(beta)
ca = np.cos(alpha)
sa = np.sin(alpha)
cbs = np.cos(betas)
sbs = np.sin(betas)
cot = np.cos(np.radians(omegat) + ot)
sot = np.sin(np.radians(omegat) + ot)
cotd = np.cos(np.radians(omegat) + ot + delta)
sotd = np.sin(np.radians(omegat) + ot + delta)
#
ekxx = ca * cbs
ekxy = ca * sbs
ekxz = -sa
#
ekyx = -sbs
ekyy = cbs
ekyz = 0
#
ekzx = sa * cbs
ekzy = sa * sbs
ekzz = ca
#
eaxx = ekxx * cotd + ekyx * sotd
eaxy = ekxy * cotd + ekyy * sotd
eaxz = ekxz * cotd
#
ltx = eaxx + cb * lK - cot * Rg
lty = eaxy - sot * Rg
ltz = eaxz + sb * lK
#
vcx = eaxy * ltz - eaxz * lty
vcy = eaxz * ltx - eaxx * ltz
vcz = eaxx * lty - eaxy * ltx
#
d = ltx * ekzx + lty * ekzy + ltz * ekzz
#
return np.where(d > dmin, (vcx * ekzx + vcy * ekzy + vcz * ekzz) / d, 0.0)
def gammak2(lK, Rg, beta, delta, omegat):
alpha = 0.5 * np.pi - beta
cb = np.cos(beta)
sb = np.sin(beta)
cot = np.cos(np.radians(omegat))
sot = np.sin(np.radians(omegat))
cotd = np.cos(np.radians(omegat) + delta)
sotd = np.sin(np.radians(omegat) + delta)
cotdt = np.cos(np.radians(omegat + 90) + delta)
sotdt = np.sin(np.radians(omegat + 90) + delta)
ltx = np.cos(alpha) * cotd + cb * lK - cot * Rg
lty = sotd - sot * Rg
ltz = -np.sin(alpha) * cotd + sb * lK
eyrx = np.cos(alpha) * cotdt
eyry = sotdt
eyrz = -np.sin(alpha) * cotdt
lt = np.sqrt(ltx * ltx + lty * lty + ltz * ltz)
return np.degrees(np.arccos((eyrx * ltx + eyry * lty + eyrz * ltz) / lt))
cot = np.cos(np.radians(omegat))
sot = np.sin(np.radians(omegat))
cotd = np.cos(np.radians(omegat)+delta)
sotd = np.sin(np.radians(omegat)+delta)
ltx = ca*cotd + cb*lK - cot*Rg
lty = sotd - sot*Rg
ltz = -sa*cotd + sb*lK
lt = np.sqrt(ltx*ltx+lty*lty+ltz*ltz)
dltxdt = -ca*sotd + sot*Rg
dltydt = cotd - cot*Rg
dltzdt = sa*sotd
return (ltx*dltxdt+lty*dltydt+ltz*dltzdt)/lt
f, (ax1, ax2) = plt.subplots(1, 2)
beta = np.radians(30)
delta = np.radians(0)
lK = 2
Rg = 1
omegat = np.arange(0, 360, 1)
ax1.plot(omegat, cosgammak(lK, Rg, beta, delta, omegat), 'r') # gammak
ax1.plot(omegat, Rg*cosgammag(lK, Rg, beta, delta, omegat), 'b') # Rg*gammag
ax1.plot(omegat, vt(lK, Rg, beta, delta, omegat), 'g')
ax1.plot(omegat, cosgammak(lK, Rg, beta, delta, omegat) -
Rg*cosgammag(lK, Rg, beta, delta, omegat) -
vt(lK, Rg, beta, delta, omegat), 'k--')
ax1.set_xlim(0,360)
ax1.set_ylim(-1,1)
ax1.set_xticks(np.arange(0,361,90))
ax1.set_yticks(np.arange(-1,1.1,0.5))
def ea(alpha, betas, delta, omegat):
cotd = np.cos(np.radians(omegat) + delta)
sotd = np.sin(np.radians(omegat) + delta)
T = ek(alpha, betas)
e = np.array([[cotd, -sotd, 0], [sotd, cotd, 0], [0, 0, 1]])
return T.dot(e)
# mean aerodynamic moment over one cycle
def Marzmean(lK, Rg, alpha, betas, beta, delta):
pos = np.array([0, 0.5 * np.pi, np.pi, 1.5 * np.pi])
omegat = np.arange(0, 360, 1)
alpha, betas = orientation(lK, Rg, alpha, betas, beta, delta, omegat, pos)
return alpha, np.mean(
Marzsum(lK, Rg, alpha, betas, beta, delta, omegat,
pos)), Ftrminmax(lK, Rg, alpha, betas, beta, delta, omegat,
pos)
vMarzmean = np.vectorize(Marzmean)
#f, (ax1, ax2) = plt.subplots(1, 2)
alpha = np.radians(60)
beta = np.radians(30)
betas = np.radians(0)
delta = np.radians(45)
N = 4
lK = np.arange(1.0, 5.2, 0.02) # 0.02 is for print quality
Rg = np.arange(0.0, 3.2, 0.02) # 0.02 is for print quality
x, y = np.meshgrid(lK, Rg)
a, z, f = vMarzmean(x, y, alpha, betas, beta, delta)
h = np.sin(beta) * lK - np.sin(a)
h1 = np.sin(beta) * lK - np.sin(a) * 1.5
h2 = np.sin(beta) * lK - np.sin(a) * 2.0
h3 = np.sin(beta) * lK - np.sin(a) * 2.5
# max tether length over cycle
def ltmax(lK, Rg, beta, delta):
omegat = np.arange(0, 360, 1)
return np.amax(lt(lK, Rg, beta, delta, omegat))
vltfreemin = np.vectorize(ltfreemin)
vltfreemax = np.vectorize(ltfreemax)
vltmin = np.vectorize(ltmin)
vltmax = np.vectorize(ltmax)
f, (ax1, ax2) = plt.subplots(1, 2)
lK = 2
Rg = 1
beta = np.radians(np.arange(0, 91, 1))
delta = np.radians(0)
ax1.plot(np.degrees(beta), vltmax(lK, Rg, beta, delta), 'r')
ax1.plot(np.degrees(beta), vltmin(lK, Rg, beta, delta), 'b')
#print 'ltmin & ltmax ', ltmin(lK, Rg, np.radians(30), delta), ltmax(lK, Rg, np.radians(30), delta)
delta = np.radians(30)
ax1.plot(np.degrees(beta), vltmax(lK, Rg, beta, delta), 'r')
ax1.plot(np.degrees(beta), vltmin(lK, Rg, beta, delta), 'b')
delta = np.radians(60)
ax1.plot(np.degrees(beta), vltmax(lK, Rg, beta, delta), 'r')
ax1.plot(np.degrees(beta), vltmin(lK, Rg, beta, delta), 'b')
delta = np.radians(90)
ax1.plot(np.degrees(beta), vltmax(lK, Rg, beta, delta), 'r')
ax1.plot(np.degrees(beta), vltmin(lK, Rg, beta, delta), 'b')
delta = np.radians(120)
#ax1.plot(np.degrees(beta), vltmax(lK, Rg, beta, delta), 'r')
return alpha, betas, Ma, Mg
def Marzmean0(lK, Rg, alpha, betas, beta, delta):
pos = np.array([0, 0.5 * np.pi, np.pi, 1.5 * np.pi])
omegat = np.arange(0, 360, 1)
return np.mean(Marzsum(lK, Rg, alpha, betas, beta, delta, omegat, pos))
vw = 12.
Rg = 25.
Rk = 20.
Rko = 35.
N = 4
lK = 100.
beta = np.radians(30)
omega = 2.05
delta = np.radians(45)
Pnom = 1.4e6
rho = 1.225
lKRk = lK / Rk
RgRk = Rg / Rk
vtip = omega * Rko
lam = omega * Rko / vw
Ma = Pnom / omega
S = np.pi * (Rko * Rko - Rk * Rk)
alpha = np.radians(60) # Initial value
betas = np.radians(0) # Initial value
a, b, c, d = Marzmean(lKRk, RgRk, alpha, betas, beta, delta)
def eb(omegat):
cot = np.cos(np.radians(omegat))
sot = np.sin(np.radians(omegat))
return np.array([[cot, -sot, 0], [sot, cot, 0], [0, 0, 1]])
Mg = np.mean(Marzsum(lK, Rg, a, b, beta, delta, omegat, pos))
return alpha, betas, Ma, Mg
def Marzmean0(lK, Rg, alpha, betas, beta, delta):
pos = np.array([0, 0.5 * np.pi, np.pi, 1.5 * np.pi])
omegat = np.arange(0, 360, 1)
return np.mean(Marzsum(lK, Rg, alpha, betas, beta, delta, omegat, pos))
vMarzmean = np.vectorize(Marzmean)
vMarzmean0 = np.vectorize(Marzmean0)
f, (ax1, ax2) = plt.subplots(1, 2)
alpha = np.radians(60) # 90 - beta | 62.3608549873
betas = np.radians(0) # 4.38381787735
lK = 2
Rg = 1
N = 4
delta = np.arange(0, 181, 1)
beta = np.radians(30)
a, b, Ma, Mg = vMarzmean(lK, Rg, alpha, betas, beta, np.radians(delta))
ax1.plot(delta, np.degrees(a), 'r', label=r"\$\alpha\$")
ax1.plot(delta, np.degrees(b), 'b', label=r"\$\betas\$")
ax2.plot(delta, Ma, 'r', label=r"\$\Mamean/(\Rk\Fakz)\$")
#ax2.plot(delta, vMarzmean0(lK, Rg, alpha, betas, beta, np.radians(delta)), 'b--')
beta = np.radians(45)
d = np.amax(Ftrysum(lK, Rg, alpha, betas, beta, delta, omegat, pos))
return (np.abs(b-a)+np.abs(d-c)+np.abs(a+b)+np.abs(c+d))
def fopt(f, lK, Rg, alpha, betas, beta, delta, omegat, pos):
def foptfixed(p):
return f(lK, Rg, p[0], p[1], beta, delta, omegat, pos)
return foptfixed
def orientation(lK, Rg, alpha, betas, beta, delta, omegat, pos):
guess = [ alpha, betas ]
return fmin(fopt(Ftrminmax, lK, Rg, alpha, betas, beta, delta, omegat, pos), guess, xtol=0.00001, ftol=0.00001, maxiter=100)
f, (ax1, ax2) = plt.subplots(1, 2)
beta = np.radians(30)
alpha = np.radians(60) # 90 - beta | 62.3608549873
betas = np.radians(0) # 4.38381787735
delta = np.radians(30) # 30
lK = 2
Rg = 1
N = 4
pos = np.array([ 0, 0.5*np.pi, np.pi, 1.5*np.pi ])
omegat = np.arange(0, 360, 1)
a, b = orientation(lK, Rg, alpha, betas, beta, delta, omegat, pos)
print "alpha = ", np.degrees(a), " betas = ",np.degrees(b)
alpha = a
betas = b
ax1.plot(omegat, Ftrxsum(lK, Rg, alpha, betas, beta, delta, omegat, pos), 'r', label=r"\$\Fakx/\Fakz\$")
sotdt = np.sin(np.radians(omegat + 90) + delta)
ltx = np.cos(alpha) * cotd + cb * lK - cot * Rg
lty = sotd - sot * Rg
ltz = -np.sin(alpha) * cotd + sb * lK
eyrx = np.cos(alpha) * cotdt
eyry = sotdt
eyrz = -np.sin(alpha) * cotdt
lt = np.sqrt(ltx * ltx + lty * lty + ltz * ltz)
return np.degrees(np.arccos((eyrx * ltx + eyry * lty + eyrz * ltz) / lt))
f, (ax1, ax2) = plt.subplots(1, 2)
lK = 2
Rg = 1
beta = np.radians(30)
omegat = np.arange(0, 360, 1)
delta = np.radians(0)
ax1.plot(omegat, gammag(lK, Rg, beta, delta, omegat), 'r')
#ax1.plot(omegat, gammak(lK, Rg, beta, delta, omegat), 'b--')
#print "gammag @ 90deg = ", gammag(lK, Rg, beta, delta, 90)
#print "gammag @ 270deg = ", gammag(lK, Rg, beta, delta, 270)
#print "gammag_max = ", np.max(gammag(lK, Rg, beta, delta, omegat))
#print "gammag_min = ", np.min(gammag(lK, Rg, beta, delta, omegat))
#print "max @ = ", omegat[np.argmax(gammag(lK, Rg, beta, delta, omegat))]
#print "min @ = ", omegat[np.argmin(gammag(lK, Rg, beta, delta, omegat))]
#print "gammag_max = ", gammag(lK, Rg, beta, delta, omegat[np.argmax(gammag(lK, Rg, beta, delta, omegat))])
#print "gammag_min = ", gammag(lK, Rg, beta, delta, omegat[np.argmin(gammag(lK, Rg, beta, delta, omegat))])
delta = np.radians(30)
ax1.plot(omegat, gammag(lK, Rg, beta, delta, omegat), 'r')
#
# This is important for font substitution
mpl.rcParams['svg.fonttype'] = 'none'
def orthogonal_proj(zfront, zback):
a = (zfront + zback) / (zfront - zback)
b = -2 * (zfront * zback) / (zfront - zback)
# -0.0001 added for numerical stability as suggested in:
# http://stackoverflow.com/questions/23840756
return np.array([[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, a, b],
[0, 0, -0.0001, zback]])
# Propblem parameters
beta = np.radians(30)
alpha = np.radians(60) # 90 - beta
betas = np.radians(10)
delta = np.radians(30)
lK = 100
Rg = 25
Rk = 20
Rko = 35
le = 40
N = 4
ot0 = 40 # offset
ot = np.array([0, 90, 180, 270])
# Derived trigonometric data
omegat = np.arange(0, 361, 1)
