def zetaDotSubMat(r, psi, rDot, psiDot, v_a, omega_a, psi_G2a, xi, xiDot, iLin, jLin, nBeam, nSection):
"""@brief create submatrix(3,N_states) for the linearized velocity at any
point on the surface, zeta, defined by beam DoFs R and Psi, and the
cross-sectional coordinate xi.
@param r Position vector on beam, defined in the a-frame.
@param psi CRV of orientation of beam cross-section.
@param rDot Velocity of beam at r, defined in the a-frame.
@param psiDot Time rate of change of psi.
@param v_a Velocity of a-frame, defined in the a-frame.
@param omega_a Angular vel of a-frame, defined in a-frame.
@param psi_G2a CRV of orientation of a-frame relative to earth.
@param xi Cross-sectional coordinate, defined in B-frame.
@param xiDot Time rate of change of xi, defined in B-frame.
@param iLin index of xi on the sectional DoF used for the linear analysis.
@param jLin index of R on the beam DoF used for the linear analysis.
@param nBeam Number of points on the beam axis.
@param nSection Number of points in the section definition.
@returns subMat Linear transformation from FBD + sectional DoFs to surface
velocity (G-frame) at point r_0 + C_Ga*(R + C(Psi)*xi).
@details This routine calculates the matrix corresponding to any point
on the beam axis and corresponding cross-section.
Interpolation of the primary beam and section DoFs may pre-
or post- multiply this matrix in an aeroelastic analysis.
"""
# Total states on RHS of matrix
nStates = 2 * 6 * nBeam + 9 + 2 * 3 * nSection
# Initialise subMat
subMat = np.zeros((3, nStates))
# term 1: \delta(C^{Ga}v_a)
cGa = xbl.Psi2TransMat(psi_G2a)
skewVa = xbl.Skew(v_a)
tangGa = xbl.Tangential(psi_G2a)
# term for \delta \psi_{Ga}
subMat[:, 2 * 6 * nBeam + 6 : 2 * 6 * nBeam + 9 : 1] += -np.dot(cGa, np.dot(skewVa, tangGa))
# term for \delta v_a
subMat[:, 2 * 6 * nBeam : 2 * 6 * nBeam + 3 : 1] += cGa
# term 2: \delta(C^{Ga} \tilde{\Omega_a_a} R_a)
skewOmAa = xbl.Skew(omega_a)
skewRa = xbl.Skew(r)
# term for \delta \psi_{Ga}
subMat[:, 2 * 6 * nBeam + 6 : 2 * 6 * nBeam + 9 : 1] += -np.dot(cGa, np.dot(xbl.Skew(np.dot(skewOmAa, r)), tangGa))
# term for \delta \Omega_a_a
subMat[:, 2 * 6 * nBeam + 3 : 2 * 6 * nBeam + 6 : 1] += -np.dot(cGa, skewRa)
# term for \delta R_a
subMat[:, 6 * jLin : 6 * jLin + 3 : 1] += np.dot(cGa, skewOmAa)
# term 3: \delta(C^{Ga}\tilde{\Omega_a_a}C^{aB}\xi_B)
cAb = xbl.Psi2TransMat(psi)
skewXi = xbl.Skew(xi)
tangAb = xbl.Tangential(psi)
# term for \delta \psi_{Ga}
subMat[:, 2 * 6 * nBeam + 6 : 2 * 6 * nBeam + 9 : 1] += -np.dot(
cGa, np.dot(xbl.Skew(np.dot(skewOmAa, np.dot(cAb, xi))), tangGa)
)
# term for \delta \Omega_a_a
subMat[:, 2 * 6 * nBeam + 3 : 2 * 6 * nBeam + 6 : 1] += -np.dot(cGa, xbl.Skew(np.dot(cAb, xi)))
# term for \delta \psi_{aB}
subMat[:, 6 * jLin + 3 : 6 * jLin + 6 : 1] += -np.dot(cGa, np.dot(skewOmAa, np.dot(cAb, np.dot(skewXi, tangAb))))
# term for \delta xi_B
subMat[:, 2 * 6 * nBeam + 9 + 3 * iLin : 2 * 6 * nBeam + 9 + 3 * iLin + 3 : 1] += np.dot(cGa, np.dot(skewOmAa, cAb))
# term 4: \delta(C^{Ga}\dot{R}_a)
skewRdot = xbl.Skew(rDot)
# term for \delta \psi_{Ga}
subMat[:, 2 * 6 * nBeam + 6 : 2 * 6 * nBeam + 9 : 1] += -np.dot(cGa, np.dot(skewRdot, tangGa))
# term for \delta \dot{R}_a
subMat[:, 6 * nBeam + 6 * jLin : 6 * nBeam + 6 * jLin + 3 : 1] += cGa
# term 5: \delta(C^{Ga}C^{aB}\dot{\xi}_B)
skewXiDot = xbl.Skew(xiDot)
# term for \delta \psi_{Ga}
subMat[:, 2 * 6 * nBeam + 6 : 2 * 6 * nBeam + 9 : 1] += -np.dot(cGa, np.dot(xbl.Skew(np.dot(cAb, xiDot)), tangGa))
# term for \delta \psi_{aB}
subMat[:, 6 * jLin + 3 : 6 * jLin + 6 : 1] += -np.dot(cGa, np.dot(cAb, np.dot(skewXiDot, tangAb)))
# term for \delta \dot{\xi}_B
subMat[:, 2 * 6 * nBeam + 9 + 3 * nSection + 3 * iLin : 2 * 6 * nBeam + 9 + 3 * nSection + 3 * iLin + 3] += np.dot(
cGa, cAb
)
# term 6: \delta(C^{Ga}C^{aB}\tilde{T(\psi_{aB})\dot{\psi}_{aB}\xi_B)
# Note: eqn. rearranged to \delta(-C^{Ga}C^{aB}\tilde{\xi}T(\psi_{aB})\dot{\psi}_{aB})
# term for \delta \psi_{Ga}
subMat[:, 2 * 6 * nBeam + 6 : 2 * 6 * nBeam + 9 : 1] += np.dot(
cGa, np.dot(xbl.Skew(np.dot(cAb, np.dot(skewXi, np.dot(tangAb, psiDot)))), tangGa)
)
# term for \delta \psi_{aB}
subMat[:, 6 * jLin + 3 : 6 * jLin + 6 : 1] += np.dot(
cGa, np.dot(cAb, np.dot(xbl.Skew(np.dot(skewXi, np.dot(tangAb, psiDot))), tangAb))
)
# term for \delta \xi_B
subMat[:, 2 * 6 * nBeam + 9 + 3 * iLin : 2 * 6 * nBeam + 9 + 3 * iLin + 3 : 1] += np.dot(
cGa, np.dot(cAb, xbl.Skew(np.dot(tangAb, psiDot)))
)
# term for \delta T(\psi_{aB})\dot{\psi_{aB}}
subMat[:, 6 * jLin + 3 : 6 * jLin + 6 : 1] += -np.dot(cGa, np.dot(cAb, np.dot(skewXi, xbl.a1(psi, psiDot))))
# term for \delta \dot{\psi}_{aB}
subMat[:, 6 * nBeam + 6 * jLin + 3 : 6 * nBeam + 6 * jLin + 6 : 1] += -np.dot(
cGa, np.dot(cAb, np.dot(skewXi, tangAb))
)
return subMat
def zetaDotSubMat(r, psi, rDot, psiDot, v_a, omega_a, psi_G2a, xi, xiDot, iLin,
jLin, nBeam, nSection):
"""@brief create submatrix(3,N_states) for the linearized velocity at any
point on the surface, zeta, defined by beam DoFs R and Psi, and the
cross-sectional coordinate xi.
@param r Position vector on beam, defined in the a-frame.
@param psi CRV of orientation of beam cross-section.
@param rDot Velocity of beam at r, defined in the a-frame.
@param psiDot Time rate of change of psi.
@param v_a Velocity of a-frame, defined in the a-frame.
@param omega_a Angular vel of a-frame, defined in a-frame.
@param psi_G2a CRV of orientation of a-frame relative to earth.
@param xi Cross-sectional coordinate, defined in B-frame.
@param xiDot Time rate of change of xi, defined in B-frame.
@param iLin index of xi on the sectional DoF used for the linear analysis.
@param jLin index of R on the beam DoF used for the linear analysis.
@param nBeam Number of points on the beam axis.
@param nSection Number of points in the section definition.
@returns subMat Linear transformation from FBD + sectional DoFs to surface
velocity (G-frame) at point r_0 + C_Ga*(R + C(Psi)*xi).
@details This routine calculates the matrix corresponding to any point
on the beam axis and corresponding cross-section.
Interpolation of the primary beam and section DoFs may pre-
or post- multiply this matrix in an aeroelastic analysis.
"""
# Total states on RHS of matrix
nStates = 2 * 6 * nBeam + 9 + 2 * 3 * nSection
# Initialise subMat
subMat = np.zeros((3, nStates))
# term 1: \delta(C^{Ga}v_a)
cGa = xbl.Psi2TransMat(psi_G2a)
skewVa = xbl.Skew(v_a)
tangGa = xbl.Tangential(psi_G2a)
# term for \delta \psi_{Ga}
subMat[:, 2 * 6 * nBeam + 6:2 * 6 * nBeam +
9:1] += -np.dot(cGa, np.dot(skewVa, tangGa))
# term for \delta v_a
subMat[:, 2 * 6 * nBeam:2 * 6 * nBeam + 3:1] += cGa
# term 2: \delta(C^{Ga} \tilde{\Omega_a_a} R_a)
skewOmAa = xbl.Skew(omega_a)
skewRa = xbl.Skew(r)
# term for \delta \psi_{Ga}
subMat[:, 2 * 6 * nBeam + 6:2 * 6 * nBeam +
9:1] += -np.dot(cGa, np.dot(xbl.Skew(np.dot(skewOmAa, r)), tangGa))
# term for \delta \Omega_a_a
subMat[:, 2 * 6 * nBeam + 3:2 * 6 * nBeam + 6:1] += -np.dot(cGa, skewRa)
# term for \delta R_a
subMat[:, 6 * jLin:6 * jLin + 3:1] += np.dot(cGa, skewOmAa)
# term 3: \delta(C^{Ga}\tilde{\Omega_a_a}C^{aB}\xi_B)
cAb = xbl.Psi2TransMat(psi)
skewXi = xbl.Skew(xi)
tangAb = xbl.Tangential(psi)
# term for \delta \psi_{Ga}
subMat[:, 2 * 6 * nBeam + 6:2 * 6 * nBeam + 9:1] += -np.dot(
cGa, np.dot(xbl.Skew(np.dot(skewOmAa, np.dot(cAb, xi))), tangGa))
# term for \delta \Omega_a_a
subMat[:, 2 * 6 * nBeam + 3:2 * 6 * nBeam +
6:1] += -np.dot(cGa, xbl.Skew(np.dot(cAb, xi)))
# term for \delta \psi_{aB}
subMat[:, 6 * jLin + 3:6 * jLin + 6:1] += -np.dot(
cGa, np.dot(skewOmAa, np.dot(cAb, np.dot(skewXi, tangAb))))
# term for \delta xi_B
subMat[:, 2 * 6 * nBeam + 9 + 3 * iLin:2 * 6 * nBeam + 9 + 3 * iLin +
3:1] += np.dot(cGa, np.dot(skewOmAa, cAb))
# term 4: \delta(C^{Ga}\dot{R}_a)
skewRdot = xbl.Skew(rDot)
# term for \delta \psi_{Ga}
subMat[:, 2 * 6 * nBeam + 6:2 * 6 * nBeam +
9:1] += -np.dot(cGa, np.dot(skewRdot, tangGa))
# term for \delta \dot{R}_a
subMat[:, 6 * nBeam + 6 * jLin:6 * nBeam + 6 * jLin + 3:1] += cGa
# term 5: \delta(C^{Ga}C^{aB}\dot{\xi}_B)
skewXiDot = xbl.Skew(xiDot)
# term for \delta \psi_{Ga}
subMat[:, 2 * 6 * nBeam + 6:2 * 6 * nBeam +
9:1] += -np.dot(cGa, np.dot(xbl.Skew(np.dot(cAb, xiDot)), tangGa))
# term for \delta \psi_{aB}
subMat[:, 6 * jLin + 3:6 * jLin +
6:1] += -np.dot(cGa, np.dot(cAb, np.dot(skewXiDot, tangAb)))
# term for \delta \dot{\xi}_B
subMat[:, 2 * 6 * nBeam + 9 + 3 * nSection + 3 * iLin:2 * 6 * nBeam + 9 +
3 * nSection + 3 * iLin + 3] += np.dot(cGa, cAb)
# term 6: \delta(C^{Ga}C^{aB}\tilde{T(\psi_{aB})\dot{\psi}_{aB}\xi_B)
# Note: eqn. rearranged to \delta(-C^{Ga}C^{aB}\tilde{\xi}T(\psi_{aB})\dot{\psi}_{aB})
# term for \delta \psi_{Ga}
subMat[:, 2 * 6 * nBeam + 6:2 * 6 * nBeam + 9:1] += np.dot(
cGa,
np.dot(xbl.Skew(np.dot(cAb, np.dot(skewXi, np.dot(tangAb, psiDot)))),
tangGa))
# term for \delta \psi_{aB}
subMat[:, 6 * jLin + 3:6 * jLin + 6:1] += np.dot(
cGa,
np.dot(
cAb,
np.dot(xbl.Skew(np.dot(skewXi, np.dot(tangAb, psiDot))), tangAb)))
# term for \delta \xi_B
subMat[:, 2 * 6 * nBeam + 9 + 3 * iLin:2 * 6 * nBeam + 9 + 3 * iLin +
3:1] += np.dot(cGa, np.dot(cAb, xbl.Skew(np.dot(tangAb, psiDot))))
# term for \delta T(\psi_{aB})\dot{\psi_{aB}}
subMat[:, 6 * jLin + 3:6 * jLin + 6:1] += -np.dot(
cGa, np.dot(cAb, np.dot(skewXi, xbl.a1(psi, psiDot))))
# term for \delta \dot{\psi}_{aB}
subMat[:, 6 * nBeam + 6 * jLin + 3:6 * nBeam + 6 * jLin +
6:1] += -np.dot(cGa, np.dot(cAb, np.dot(skewXi, tangAb)))
return subMat
