def test_shift():
A = np.arange(1, 5)
assert np.allclose(utils.shift(A, 0), A)
assert np.allclose(utils.shift(A, 2, fill=0), [3, 4, 0, 0])
assert np.allclose(utils.shift(A, -2, fill=0), [0, 0, 1, 2])
assert np.allclose(utils.shift(A, 2), [3, 4, 4, 4])
assert np.allclose(utils.shift(A, -2), [1, 1, 1, 2])
def test_shift():
A = np.arange(1, 5)
assert np.allclose(utils.shift(A, 0), A)
assert np.allclose(utils.shift(A, 2, fill=0), [3, 4, 0, 0])
assert np.allclose(utils.shift(A, -2, fill=0), [0, 0, 1, 2])
assert np.allclose(utils.shift(A, 2), [3, 4, 4, 4])
assert np.allclose(utils.shift(A, -2), [1, 1, 1, 2])
def second_order_force_spectrum(w, Tc, S_wave):
"""Calculate the 2nd order force spectrum defined by
$$ S_i^{(2)}(\omega) =
8 \int S_{\eta\eta}(\mu) S_{\eta\eta}(\mu + \omega) \,
\left| T_i^{\mathrm{c}}(\mu, \mu + \omega) \right|^2 \mathrm{d}\mu $$
but extended to give the cross-spectrum S_ij
"""
Ndof = Tc.shape[1]
# First make the matrix SS_kl, where l is the freq and k is the offset
# SS_kl = S(w_l) * S(w_l + w_k)
SS = np.tile(S_wave, (len(w), 1))
for k in range(len(w)):
SS[k, :] *= shift(S_wave, k, fill=0)
# Next use the Newman approximation to make a similar matrix TTc
TTc = np.tile(Tc, (len(w), 1, 1)) # copy downwards new 0 axis
for k in range(len(w)):
for i in range(Ndof):
TTc[k, :, i] += shift(Tc[:, i], k)
TTc /= 2 # average
# Now loop through each DOF to make the 2nd order spectrum
S2 = np.zeros((len(w), Ndof, Ndof))
for i in range(Ndof):
for j in range(Ndof):
integrand = SS * TTc[:, :, i] * TTc[:, :, j]
S2[:, i, j] = 8 * integrate.simps(integrand, w, axis=1)
return S2
def second_order_force_spectrum(w, Tc, S_wave):
"""Calculate the 2nd order force spectrum defined by
$$ S_i^{(2)}(\omega) =
8 \int S_{\eta\eta}(\mu) S_{\eta\eta}(\mu + \omega) \,
\left| T_i^{\mathrm{c}}(\mu, \mu + \omega) \right|^2 \mathrm{d}\mu $$
but extended to give the cross-spectrum S_ij
"""
Ndof = Tc.shape[1]
# First make the matrix SS_kl, where l is the freq and k is the offset
# SS_kl = S(w_l) * S(w_l + w_k)
SS = np.tile(S_wave, (len(w), 1))
for k in range(len(w)):
SS[k, :] *= shift(S_wave, k, fill=0)
# Next use the Newman approximation to make a similar matrix TTc
TTc = np.tile(Tc, (len(w), 1, 1)) # copy downwards new 0 axis
for k in range(len(w)):
for i in range(Ndof):
TTc[k, :, i] += shift(Tc[:, i], k)
TTc /= 2 # average
# Now loop through each DOF to make the 2nd order spectrum
S2 = np.zeros((len(w), Ndof, Ndof))
for i in range(Ndof):
for j in range(Ndof):
integrand = SS * TTc[:, :, i] * TTc[:, :, j]
S2[:, i, j] = 8 * integrate.simps(integrand, w, axis=1)
return S2