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_rad_stiff(wheel):
'Calculate radial stiffness.'
# Create a ModeMatrix model with 24 modes
mm = ModeMatrix(wheel, N=24)
# Calculate stiffness matrix
K = mm.K_rim(tension=True) + mm.K_spk(smeared_spokes=False, tension=True)
# Create a unit radial load pointing radially inwards at theta=0
F_ext = mm.F_ext(0., np.array([0., 1., 0., 0.]))
# Solve for the mode coefficients
dm = np.linalg.solve(K, F_ext)
return 1e-6 / mm.rim_def_rad(0., dm)[0]
def calc_rot_stiff(wheel):
'Calculate rotational (wind-up) stiffness.'
# Create a ModeMatrix model with 24 modes
mm = ModeMatrix(wheel, N=24)
# Calculate stiffness matrix
K = mm.K_rim(tension=True) + mm.K_spk(smeared_spokes=False, tension=True)
# Create a unit tangential load at theta=0
F_ext = mm.F_ext(0., np.array([0., 0., 1., 0.]))
# Solve for the mode coefficients
dm = np.linalg.solve(K, F_ext)
return 1e-3*np.pi/180*wheel.rim.radius / mm.rim_def_tan(0., dm)[0]
def calc_lat_stiff(wheel):
'Calculate lateral (side-load) stiffness.'
# Create a ModeMatrix model with 24 modes
mm = ModeMatrix(wheel, N=24)
# Calculate stiffness matrix
K = mm.K_rim(tension=True) + mm.K_spk(smeared_spokes=False, tension=True)
# Create a unit lateral load at theta=0
F_ext = mm.F_ext(0., np.array([1., 0., 0., 0.]))
# Solve for the mode coefficients
dm = np.linalg.solve(K, F_ext)
return 1e-3 / mm.rim_def_lat(0., dm)[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])
I_lat=200. / 69e9,
I_rad=100. / 69e9,
J_tor=25. / 26e9,
I_warp=0.0,
young_mod=69e9,
shear_mod=26e9)
wheel.lace_cross(n_spokes=36, n_cross=3, diameter=2.0e-3, young_mod=210e9)
# Create a ModeMatrix model with 24 modes
mm = ModeMatrix(wheel, N=24)
# Create a 500 Newton pointing radially inwards at theta=0
F_ext = mm.F_ext(0., np.array([0., 500., 0., 0.]))
# Calculate stiffness matrix
K = mm.K_rim(tension=False) + mm.K_spk(smeared_spokes=False, tension=False)
# Solve for the mode coefficients
dm = np.linalg.solve(K, F_ext)
# Get radial deflection
theta = np.linspace(-np.pi, np.pi, 100)
rad_def = mm.rim_def_rad(theta, dm)
# Calculate change in spoke tensions
dT = [
-s.EA / s.length * np.dot(s.n,
mm.B_theta(s.rim_pt[1], comps=[0, 1, 2]).dot(dm))
for s in wheel.spokes
]