def reactions(phi1,
omg1,
alp1,
ms,
Is,
M5_ext,
g=-9.807j,
AB=0.15,
AC=0.1,
CD=0.15,
DF=0.4,
AG=0.3):
rA, rB, rC, rD, rF, rG = pos(phi1, AB, AC, CD, DF, AG)
M5_ext = -np.sign(vel(phi1, omg1, AB, AC, CD, DF, AG)[-1]) * M5_ext
a2, a3_T, a5_T, a4, aF, aG, alp3, alp5 = acc(phi1, omg1, alp1, AB, AC, CD,
DF, AG)
aGs = np.array([a2 / 2, a2, (a4 + aF) / 2, a4, aG / 2])
Gs, FGs = ms * g, ms * aGs
# find F45, F05 with matrix solve AX=B
# moment equation, force equation, dot product of F45 and rD = 0
A = dA.mat([
dA.moment(rD - rG / 2),
dA.moment(rA - rG / 2), [1, 0, 1, 0], [0, 1, 0, 1],
[np.real(rD), np.imag(rD), 0, 0]
], (4, 4))
b = dA.mat([[Is[-1] * alp5 - M5_ext], kA.toList(FGs[-1] - Gs[-1]), [0]], 4)
F45, F05 = kA.toComplexes(np.linalg.solve(A, b))
# find F34
F34 = FGs[-2] - (Gs[-2] + (-F45))
# find F03, F23 with matrix solve AX=B
# moment equation, force equation, dot product of F23 and rF-rC = 0
rG3 = (rD + rF) / 2
A = dA.mat([
dA.moment(rC - rG3),
dA.moment(rB - rG3), [1, 0, 1, 0], [0, 1, 0, 1],
[0, 0, np.real(rB - rG3), np.imag(rB - rG3)]
], (4, 4))
b = dA.mat([[Is[2] * alp3 - dA.toCross(rD - rG3, -F34)],
kA.toList(FGs[2] - Gs[2] - (-F34)), [0]], 4)
F03, F23 = kA.toComplexes(np.linalg.solve(A, b))
# find F12, F01
F12 = FGs[1] - Gs[1] - (-F23)
F01 = FGs[0] - Gs[0] - (-F12)
M_eq = Is[0] * alp1 - dA.toCross(rB / 2, -F12) - dA.toCross(-rB / 2, F01)
return F05, F45, F34, F03, F23, F12, F01, M_eq
def reactions(phi1,
omg1,
alp1,
ms,
Is,
F_ext,
g=-10j,
variation=0,
AB=1,
BC=1):
rB, rC, rBC = pos(phi1, AB, BC)[variation]
aB, aC, alp2 = acc(phi1, omg1, alp1, variation, AB, BC)
a_Cs = np.array([aB / 2, (aB + aC) / 2, aC])
vC = vel(phi1, omg1, variation, AB, BC)[1]
coeffs = -ms * g + ms * a_Cs
coeffs[2] -= np.sign(-vC) * F_ext
A = [[1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 0, 0, 0],
[0, -1, 0, 1, 0, 0, 0, 0], [0, 0, -1, 0, 1, 0, 0, 0],
[0, 0, 0, -1, 0, 1, 0, 0], [0, 0, 0, 0, -1, 0, 1, 0],
[0, 0, 0, 0, 0, -1, 0, 1],
[
0, 0, -np.imag(rBC) / 2,
np.real(rBC) / 2, -np.imag(rBC) / 2,
np.real(rBC) / 2, 0, 0
]]
b = [
0,
np.real(-np.sign(-vC) * F_ext + coeffs[2]),
np.imag(-np.sign(-vC) * F_ext + coeffs[2]),
np.real(coeffs[1]),
np.imag(coeffs[1]),
np.real(coeffs[0]),
np.imag(coeffs[0]), Is[1] * alp2
]
res = np.linalg.solve(A, b)
F30 = res[0] + res[1] * 1j
F23 = res[2] + res[3] * 1j
F12 = res[4] + res[5] * 1j
F01 = res[6] + res[7] * 1j
M_equilibrium = Is[0] * alp1 - dA.toCross(rB / 2, -F12) - dA.toCross(
-rB / 2, F01)
return F30, F23, F12, F01, M_equilibrium
def dyad(phi1, omg1, alp1, ms, Is, F_ext, g=-10j, variation=0, AB=1, BC=1):
rB, rC, rBC = pos(phi1, AB, BC)[variation]
aB, aC, alp2 = acc(phi1, omg1, alp1, variation, AB, BC)
a_Cs = np.array([aB / 2, (aB + aC) / 2, aC])
Gs = ms * g
vC = vel(phi1, omg1, variation, AB, BC)[1]
F_ext *= -np.sign(vC)
F12x = np.real((ms * a_Cs)[2] + (ms * a_Cs)[1]) - F_ext
F12y = (Is[1] * alp2 + dA.toCross(rBC / 2, (ms * a_Cs)[1]) +
np.imag(rBC) * F12x - dA.toCross(rBC / 2, Gs[1])) / np.real(rBC)
F12 = F12x + F12y * 1j
F03 = (ms * a_Cs)[2] + (ms * a_Cs)[1] - (Gs[1] + Gs[2] + F12 + F_ext)
F23 = (ms * a_Cs)[1] - (Gs[1] + F12)
F01 = (ms * a_Cs)[0] - (Gs[0] + -F12)
M_equilibrium = Is[0] * alp1 - dA.toCross(rB / 2, -F12) - dA.toCross(
-rB / 2, F01)
return F03, F23, F12, F01, M_equilibrium
def reactions(theta1,
omg1,
alp1,
ms,
Is,
M_ext,
AB,
AD,
BC,
CD,
g=-9.807j,
variation=0):
rB, rC = pos(theta1, AB, AD, BC, CD)[variation]
aB, aC, alp2, alp3 = acc(theta1, omg1, alp1, AB, AD, BC, CD, variation)
aGs = np.array([aB / 2, (aB + aC) / 2, aC / 2])
M_ext *= -np.sign(vel(theta1, omg1, AB, AD, BC, CD, variation)[3])
Gs, FGs = ms * g, ms * aGs
# find F03, F12 using dyad method (link 2-3)
# equilibrium equations,
A = dA.mat([[1, 0, 1, 0], [0, 1, 0, 1],
dA.moment(AD - rC), [0, 0], [0, 0],
dA.moment(rB - rC)], (4, 4))
b = dA.mat([
kA.toList(FGs[1] + FGs[2] - Gs[1] - Gs[2]),
[
Is[2] * alp3 + dA.toCross((AD - rC) / 2, FGs[2] - Gs[2]) - M_ext,
Is[1] * alp2 + dA.toCross((rB - rC) / 2, FGs[1] - Gs[1])
]
], 4)
F03, F12 = kA.toComplexes(np.linalg.solve(A, b))
# find F23, F01
F23 = FGs[2] - Gs[2] - F03
F01 = FGs[0] - Gs[0] - (-F12)
# find drive moment
M_eq = Is[0] * alp1 - (dA.toCross(rB / 2, -F12) + dA.toCross(-rB / 2, F01))
return F03, F23, F12, F01, M_eq
def reactions(phi1, case, ms, Is, F_ext, g=-9.807j):
rB, rC, rD, rE, rF = pos(phi1, case)
aB, aC, aE, aF, alp2, alp3, alp4 = acc(phi1, case)
F = F_ext * -np.sign(np.real(vel(phi1, case)[1]))
Gs, FGs = ms * g, ms * np.array(
[aB / 2, (aB + aC) / 2, (aC + aE) / 2, (aE + aF) / 2, aF])
# find F05, F34 using dyad method
A = dA.mat([[1, 0, 1, 0], [0, 1, 0, 1], [0, 0],
dA.moment(rE - rF), [1, 0, 0, 0]], (4, 4))
b = dA.mat([
kA.toList(FGs[3] + FGs[4] - Gs[3] - Gs[4] - F),
[Is[3] * alp4 + dA.toCross((rE - rF) / 2, FGs[3] - Gs[3]), 0]
], 4)
F05, F34 = kA.toComplexes(np.linalg.solve(A, b))
# find F45
F45 = FGs[4] - Gs[4] - F05 - F
# find F03, F12 using dyad method, moment at C
rG3, rG2 = (rE + rC) / 2, (rB + rC) / 2
A = dA.mat([[1, 0, 1, 0], [0, 1, 0, 1],
dA.moment(rD - rC), [0, 0], [0, 0],
dA.moment(rB - rC)], (4, 4))
b = dA.mat([
kA.toList(FGs[2] + FGs[1] - Gs[2] - Gs[1] - (-F34)),
[
Is[2] * alp3 + dA.toCross(rG3 - rC, FGs[2] - Gs[2]) -
dA.toCross(rE - rC, -F34),
rB, rC, rD, rE, rF = pos(phi1, case)
aB, aD, aE, alp2, alp4, alp5 = acc(phi1, case)
M_ext *= -np.sign(vel(phi1, case)[-2])
aGs = np.array([aB / 2, (aB + aD) / 2, 0, (aE + aD) / 2, aE / 2])
Gs, FGs = ms * g, ms * aGs
# find F05, F34 using dyad method, moment at E
A = dA.mat([[1, 0, 1, 0], [0, 1, 0, 1],
dA.moment(rF - rE), [0, 0], [0, 0],
dA.moment(rD - rE)], (4, 4))
b = dA.mat([
kA.toList(FGs[3] + FGs[4] - Gs[3] - Gs[4]),
[
Is[4] * alp5 + dA.toCross((rF - rE) / 2, FGs[4] - Gs[4]) - M_ext,
Is[3] * alp4 + dA.toCross((rD - rE) / 2, FGs[3] - Gs[3])
]
], 4)
F05, F24 = kA.toComplexes(np.linalg.solve(A, b))
# find F45
F45 = FGs[4] - Gs[4] - F05
# find F03, F12 using dyad method, moment at C
rG2 = (rB + rD) / 2
A = dA.mat([[1, 0, 1, 0], [0, 1, 0, 1], [0, 0],
dA.moment(rB - rC),
kA.toList(rB - rD), [0, 0]], (4, 4))
b = dA.mat([
kA.toList(FGs[2] + FGs[1] - Gs[2] - Gs[1] - (-F24)),
def reactions(phi1, case, ms, Is, F_ext, g=-9.807j):
rB, rC, rD, rD_max, rE, rC_max = pos(phi1, case)
aB, aC, aE, aD_max, aC_max, alp2, a2_T = acc(phi1, case)
F = F_ext*-np.sign(vel(phi1, case)[2])
aGs = np.array([aB/2, (aD_max+aC)/2, 0, aC, (aE+aC_max)/2])
Gs, FGs = ms*g, ms*aGs
# find F05, F24 using dyad method, moment at C
rG5 = (rE+rC_max)/2
A = dA.mat([[1, 0, 1, 0],
[0, 1, 0, 1],
dA.moment(rE-rC), [0, 0],
[0, 0, 1, 0]], (4, 4))
b = dA.mat([kA.toList(FGs[3]+FGs[4]-Gs[3]-Gs[4]-F),
[dA.toCross(rG5-rC, FGs[4]-Gs[4]), 0]], 4)
F05, F24 = kA.toComplexes(np.linalg.solve(A, b))
# find F45
F45 = FGs[4] - Gs[4] - F05 - F
# find F32, F12 using dyad method, moment at D
rG2 = (rC+rD_max)/2
A = dA.mat([[1, 0, 1, 0],
[0, 1, 0, 1],
kA.toList(rB-rD), [0, 0],
[0, 0], dA.moment(rB-rD)], (4, 4))
b = dA.mat([kA.toList(FGs[1]-Gs[1]-(-F24)),
[0, dA.toCross(rG2-rD, FGs[1]-Gs[1]) - dA.toCross(rC-rD, -F24)]], 4)
F32, F12 = kA.toComplexes(np.linalg.solve(A, b))