def calc_tor_stiff(wheel,
theta=0.,
N=20,
smeared_spokes=True,
tension=True,
buckling=True,
coupling=True,
r0=False):
'Calculate torsional (wind-up) stiffness in [N/rad].'
mm = ModeMatrix(wheel, N=N)
F_ext = mm.F_ext(theta, [0., 0., 1., 0.])
d = np.zeros(F_ext.shape)
K = (mm.K_rim(tension=buckling, r0=r0) +
mm.K_spk(tension=tension, smeared_spokes=smeared_spokes))
if coupling:
d = np.linalg.solve(K, F_ext)
else:
ix_uc = mm.get_ix_uncoupled(dim='radial')
K = mm.get_K_uncoupled(K=K, dim='radial')
d[ix_uc] = np.linalg.solve(K, F_ext[ix_uc])
return wheel.rim.radius / mm.B_theta(theta).dot(d)[2]
def calc_lat_stiff(wheel,
theta=0.,
N=20,
smeared_spokes=True,
tension=True,
buckling=True,
coupling=False,
r0=False):
'Calculate lateral stiffness.'
mm = ModeMatrix(wheel, N=N)
F_ext = mm.F_ext(0., [1., 0., 0., 0.])
d = np.zeros(F_ext.shape)
K = (mm.K_rim(tension=buckling, r0=r0) +
mm.K_spk(tension=tension, smeared_spokes=smeared_spokes))
if coupling:
d = np.linalg.solve(K, F_ext)
else:
ix_uc = mm.get_ix_uncoupled(dim='lateral')
K = mm.get_K_uncoupled(K=K, dim='lateral')
d[ix_uc] = np.linalg.solve(K, F_ext[ix_uc])
return 1. / mm.B_theta(0.).dot(d)[0]
def calc_buckling_tension_modematrix(wheel,
smeared_spokes=False,
coupling=True,
r0=True,
N=24):
'Estimate buckling tension from condition number of stiffness matrix.'
mm = ModeMatrix(wheel, N=N)
K_matl = (mm.K_rim_matl(r0=r0) +
mm.K_spk(tension=False, smeared_spokes=smeared_spokes))
K_geom = (mm.K_rim_geom(r0=r0) -
mm.K_spk_geom(smeared_spokes=smeared_spokes))
# Solve generalized eigienvalue problem:
# (K_matl + T*K_geom)
if coupling:
w, v = eig(K_matl, K_geom)
else:
w, v = eig(mm.get_K_uncoupled(K_matl), mm.get_K_uncoupled(K_geom))
return np.min(np.real(w)[np.real(w) > 0])
def calc_tor_stiff(wheel, theta=0., N=20, smeared_spokes=True, tension=True, buckling=True, coupling=True, r0=False):
'Calculate torsional (wind-up) stiffness in [N/rad].'
mm = ModeMatrix(wheel, N=N)
F_ext = mm.F_ext(theta, [0., 0., 1., 0.])
d = np.zeros(F_ext.shape)
K = (mm.K_rim(tension=buckling, r0=r0) +
mm.K_spk(tension=tension, smeared_spokes=smeared_spokes))
if coupling:
d = np.linalg.solve(K, F_ext)
else:
ix_uc = mm.get_ix_uncoupled(dim='radial')
K = mm.get_K_uncoupled(K=K, dim='radial')
d[ix_uc] = np.linalg.solve(K, F_ext[ix_uc])
return wheel.rim.radius / mm.B_theta(theta).dot(d)[2]
def calc_lat_stiff(wheel, theta=0., N=20, smeared_spokes=True, tension=True, buckling=True, coupling=False, r0=False):
'Calculate lateral stiffness.'
mm = ModeMatrix(wheel, N=N)
F_ext = mm.F_ext(0., [1., 0., 0., 0.])
d = np.zeros(F_ext.shape)
K = (mm.K_rim(tension=buckling, r0=r0) +
mm.K_spk(tension=tension, smeared_spokes=smeared_spokes))
if coupling:
d = np.linalg.solve(K, F_ext)
else:
ix_uc = mm.get_ix_uncoupled(dim='lateral')
K = mm.get_K_uncoupled(K=K, dim='lateral')
d[ix_uc] = np.linalg.solve(K, F_ext[ix_uc])
return 1. / mm.B_theta(0.).dot(d)[0]
def calc_buckling_tension_modematrix(wheel, smeared_spokes=False, coupling=True, r0=True, N=24):
'Estimate buckling tension from condition number of stiffness matrix.'
mm = ModeMatrix(wheel, N=N)
K_matl = (mm.K_rim_matl(r0=r0) +
mm.K_spk(tension=False, smeared_spokes=smeared_spokes))
K_geom = (mm.K_rim_geom(r0=r0) -
mm.K_spk_geom(smeared_spokes=smeared_spokes))
# Solve generalized eigienvalue problem:
# (K_matl + T*K_geom)
if coupling:
w, v = eig(K_matl, K_geom)
else:
w, v = eig(mm.get_K_uncoupled(K_matl),
mm.get_K_uncoupled(K_geom))
return np.min(np.real(w)[np.real(w) > 0])