def equilibrium(self):
# Build equilibrium equations
try:
import numpy as np
except ImportError:
FreeCAD.Console.PrintMessage(
"numpy is not installed on your system\n")
raise ImportError("numpy not installed")
# Initialization of structures. All three axes are handled separately so everything is 3-fold
# dictionaries of (location : outer force/moment) with reverse sign, which means that the segment functions for the section force and section moment
# created from them will have signs as by the convention in
# http://www.umwelt-campus.de/ucb/fileadmin/users/90_t.preussler/dokumente/Skripte/TEMECH/TMI/Ebene_Balkenstatik.pdf (page 10)
# (see also example on page 19)
forces = [{0.0: 0.0}, {0.0: 0.0}, {0.0: 0.0}]
moments = [{0.0: 0.0}, {0.0: 0.0}, {0.0: 0.0}]
# Boundary conditions for shaft bending line
tangents = [[], [], []] # Tangents to shaft bending line
translations = [[], [], []] # Shaft displacement
# Variable names, e.g. Fx, Mz. Because the system must be exactly determined, not more than two independent variables for each
# force/moment per axis are possible (if there are more no solution is calculated)
variableNames = [[""], [""], [""]]
# # dictionary of (variableName : location) giving the x-coordinate at which the force/moment represented by the variable acts on the shaft
locations = {}
# Coefficients of the equilibrium equations in the form a = b * F1 + c * F2 and d = e * M1 + f * M2
# LHS (variables a1, a2, a3, d3) initialized to zero
coefficientsF = [[0], [0], [0]]
coefficientsM = [[0], [0], [0]]
for i in range(len(self.segments)):
cType = self.segments[i].constraintType
constraint = self.segments[i].constraint
if cType == "Fixed":
# Fixed segment
if i == 0:
# At beginning of shaft
location = 0
elif i == len(self.segments) - 1:
# At end of shaft
location = self.getLengthTo(len(
self.segments)) / 1000.0 # convert to meters
else:
# TODO: Better error message
FreeCAD.Console.PrintMessage(
"Fixed constraint must be at beginning or end of shaft\n"
)
return
for ax in range(3):
# Create a new reaction force
variableNames[ax].append("%s%u" % (self.Fstr[ax], i))
coefficientsF[ax].append(1)
# Register location of reaction force
locations["%s%u" % (self.Fstr[ax], i)] = location
# Boundary conditions for the translations
tangents[ax].append((location, 0.0))
translations[ax].append((location, 0.0))
coefficientsM[0].append(
0) # Reaction force contributes no moment around x axis
coefficientsM[1].append(
location
) # Reaction force contributes a positive moment around z axis
coefficientsM[2].append(
-location
) # Reaction force contributes a negative moment around y axis
for ax in range(3):
# Create a new reaction moment
variableNames[ax].append("%s%u" % (self.Mstr[ax], i))
coefficientsF[ax].append(0)
coefficientsM[ax].append(1)
locations["%s%u" % (self.Mstr[ax], i)] = location
elif cType == "Force":
# Static force (currently force on midpoint of segment only)
force = constraint.DirectionVector.multiply(constraint.Force)
# TODO: Extract value of the location from geometry
location = (self.getLengthTo(i) +
self.segments[i].length / 2.0) / 1000.0
# The force itself
for ax in range(3):
if abs(force[ax]) > 0.0:
coefficientsF[ax][0] = coefficientsF[ax][0] - force[
ax] # neg. because this coefficient is on the LHS of the equilibrium equation
self.addTo(
forces[ax], location, -force[ax]
) # neg. to fulfill the convention mentioned above
# Moments created by the force (by definition no moment is created by the force in x-direction)
if abs(force[1]) > 0.0:
coefficientsM[1][0] = coefficientsM[1][
0] - force[1] * location # moment around z-axis
self.addTo(moments[1], location, 0)
if abs(force[2]) > 0.0:
coefficientsM[2][0] = coefficientsM[2][
0] + force[2] * location # moment around y-axis
self.addTo(moments[2], location,
0) # No outer moment acts here!
elif cType == "Bearing":
location = constraint.BasePoint.x / 1000.0 # TODO: This assumes that the shaft feature starts with the first segment at (0,0,0) and its axis corresponds to the x-axis
# Bearing reaction forces. TODO: the bearing is assumed to not induce any reaction moments
start = (0 if constraint.AxialFree == False else 1)
for ax in range(start, 3):
variableNames[ax].append("%s%u" % (self.Fstr[ax], i))
coefficientsF[ax].append(1)
locations["%s%u" % (self.Fstr[ax], i)] = location
# Boundary condition
translations[ax].append((location, 0.0))
if constraint.AxialFree == False:
coefficientsM[0].append(
0
) # Reaction force contributes no moment around x axis
coefficientsM[1].append(
location
) # Reaction force contributes a positive moment around z axis
coefficientsM[2].append(
-location
) # Reaction force contributes a negative moment around y axis
elif cType == "Gear":
force = constraint.DirectionVector.multiply(constraint.Force)
location = constraint.BasePoint.x / 1000.0
lever = [
0, constraint.Diameter / 2.0 / 1000.0 *
math.sin(constraint.ForceAngle / 180.0 * math.pi),
constraint.Diameter / 2.0 / 1000.0 *
math.cos(constraint.ForceAngle / 180.0 * math.pi)
]
# Effect of the gear force
for ax in range(3):
if abs(force[ax]) > 0.0:
# Effect of the force
coefficientsF[ax][0] = coefficientsF[ax][0] - force[ax]
self.addTo(forces[ax], location, -force[ax])
# Moments created by the force (by definition no moment is created by the force in x-direction)
if abs(force[1]) > 0.0:
coefficientsM[1][0] = coefficientsM[1][
0] - force[1] * location # moment around z-axis
self.addTo(moments[1], location, 0)
if abs(force[2]) > 0.0:
coefficientsM[2][0] = coefficientsM[2][
0] + force[2] * location # moment around y-axis
self.addTo(moments[2], location,
0) # No outer moment acts here!
# Moments created by the force and lever
if abs(force[0]) > 0.0:
momenty = force[0] * lever[2]
momentz = force[0] * lever[1]
coefficientsM[1][0] = coefficientsM[1][
0] + momentz # moment around z-axis
self.addTo(moments[1], location, momentz)
coefficientsM[2][0] = coefficientsM[2][
0] - momenty # moment around y-axis
self.addTo(moments[2], location, -momenty)
if abs(force[1]) > 0.0:
moment = force[1] * lever[2]
coefficientsM[0][0] = coefficientsM[0][0] + moment
self.addTo(moments[0], location, moment)
if abs(force[2]) > 0.0:
moment = force[2] * lever[1]
coefficientsM[0][0] = coefficientsM[0][0] - moment
self.addTo(moments[0], location, -moment)
elif cType == "Pulley":
forceAngle1 = (constraint.ForceAngle + constraint.BeltAngle +
90.0) / 180.0 * math.pi
forceAngle2 = (constraint.ForceAngle - constraint.BeltAngle +
90.0) / 180.0 * math.pi
#FreeCAD.Console.PrintMessage("BeltForce1: %f, BeltForce2: %f\n" % (constraint.BeltForce1, constraint.BeltForce2))
#FreeCAD.Console.PrintMessage("Angle1: %f, Angle2: %f\n" % (forceAngle1, forceAngle2))
force = [
0, -constraint.BeltForce1 * math.sin(forceAngle1) -
constraint.BeltForce2 * math.sin(forceAngle2),
constraint.BeltForce1 * math.cos(forceAngle1) +
constraint.BeltForce2 * math.cos(forceAngle2)
]
location = constraint.BasePoint.x / 1000.0
# Effect of the pulley forces
for ax in range(3):
if abs(force[ax]) > 0.0:
# Effect of the force
coefficientsF[ax][0] = coefficientsF[ax][0] - force[ax]
self.addTo(forces[ax], location, -force[ax])
# Moments created by the force (by definition no moment is created by the force in x-direction)
if abs(force[1]) > 0.0:
coefficientsM[1][0] = coefficientsM[1][
0] - force[1] * location # moment around z-axis
self.addTo(moments[1], location, 0)
if abs(force[2]) > 0.0:
coefficientsM[2][0] = coefficientsM[2][
0] + force[2] * location # moment around y-axis
self.addTo(moments[2], location,
0) # No outer moment acts here!
# Torque
moment = constraint.Force * (1 if constraint.IsDriven is True
else -1)
coefficientsM[0][0] = coefficientsM[0][0] + moment
self.addTo(moments[0], location, moment)
areas = [None, None, None]
areamoments = [None, None, None]
bendingmoments = [None, None, None]
torquemoments = [None, None, None]
for ax in range(3):
FreeCAD.Console.PrintMessage("Axis: %u\n" % ax)
self.printEquilibrium(variableNames[ax], coefficientsF[ax])
self.printEquilibrium(variableNames[ax], coefficientsM[ax])
if len(coefficientsF[ax]) <= 1:
# Note: coefficientsF and coefficientsM always have the same length
FreeCAD.Console.PrintMessage(
"Matrix is singular, no solution possible\n")
self.parent.updateButtons(ax, False)
continue
# Handle special cases. Note that the code above should ensure that coefficientsF and coefficientsM always have same length
solution = [None, None]
if len(coefficientsF[ax]) == 2:
if coefficientsF[ax][1] != 0.0 and coefficientsF[ax][0] != 0.0:
solution[0] = coefficientsF[ax][0] / coefficientsF[ax][1]
if coefficientsM[ax][1] != 0.0 and coefficientsM[ax][0] != 0.0:
solution[1] = coefficientsM[ax][0] / coefficientsM[ax][1]
if abs(solution[0] - solution[1]) < 1E9:
FreeCAD.Console.PrintMessage(
"System is statically undetermined. No solution possible.\n"
)
self.parent.updateButtons(ax, False)
continue
else:
# Build matrix and vector for linear algebra solving algorithm
# TODO: This could easily be done manually... there are only 2 variables and 6 coefficients
A = np.array([coefficientsF[ax][1:], coefficientsM[ax][1:]])
b = np.array([coefficientsF[ax][0], coefficientsM[ax][0]])
try:
solution = np.linalg.solve(A, b) # A * solution = b
except np.linalg.linalg.LinAlgError, e:
FreeCAD.Console.PrintMessage(e.message)
FreeCAD.Console.PrintMessage(". No solution possible.\n")
self.parent.updateButtons(ax, False)
continue
# Complete dictionary of forces and moments with the two reaction forces that were calculated
for i in range(2):
if solution[i] is None:
continue
FreeCAD.Console.PrintMessage(
"Reaction force/moment: %s = %f\n" %
(variableNames[ax][i + 1], solution[i]))
if variableNames[ax][i + 1][0] == "M":
moments[ax][locations[variableNames[ax][i +
1]]] = -solution[i]
else:
forces[ax][locations[variableNames[ax][i +
1]]] = -solution[i]
FreeCAD.Console.PrintMessage(forces[ax])
FreeCAD.Console.PrintMessage("\n")
FreeCAD.Console.PrintMessage(moments[ax])
FreeCAD.Console.PrintMessage("\n")
# Forces
self.F[ax] = SegmentFunction(self.Fstr[ax])
self.F[ax].buildFromDict("x", forces[ax])
self.parent.updateButton(1, ax, not self.F[ax].isZero())
self.F[ax].output()
# Moments
if ax == 0:
self.M[0] = SegmentFunction(self.Mstr[0])
self.M[0].buildFromDict("x", moments[0])
elif ax == 1:
self.M[1] = self.F[1].integrated().negate()
self.M[1].name = self.Mstr[1]
self.M[1].addSegments(
moments[1]) # takes care of boundary conditions
elif ax == 2:
self.M[2] = self.F[2].integrated()
self.M[2].name = self.Mstr[2]
self.M[2].addSegments(
moments[2]) # takes care of boundary conditions
self.parent.updateButton(2, ax, not self.M[ax].isZero())
self.M[ax].output()
# Areas and area moments
location = 0.0
areas[ax] = IntervalFunction() # A [m²]
areamoments[ax] = IntervalFunction() # I [m⁴]
bendingmoments[ax] = IntervalFunction() # W_b [m³]
torquemoments[ax] = IntervalFunction() # W_t [m³]
for i in range(len(self.segments)):
od = self.segments[i].diameter / 1000.0
id = self.segments[i].innerdiameter / 1000.0
length = self.segments[i].length / 1000.0
areas[ax].addInterval(
location, length,
math.pi / 4.0 * (math.pow(od, 2.0) - math.pow(id, 2.0)))
areamoment = math.pi / 64.0 * (math.pow(od, 4.0) -
math.pow(id, 4.0))
areamoments[ax].addInterval(location, length, areamoment)
bendingmoments[ax].addInterval(location, length,
areamoment / (od / 2.0))
torquemoments[ax].addInterval(location, length,
2 * (areamoment / (od / 2.0)))
location += length
# Bending line
if ax > 0:
if len(tangents[ax]) + len(translations[ax]) == 2:
# TODO: Get Young's module from material type instead of using 210000 N/mm² = 2.1E12 N/m²
self.w[ax] = TranslationFunction(self.M[ax].negated(),
2.1E12, areamoments[ax],
tangents[ax],
translations[ax])
self.w[ax].name = self.wstr[ax]
self.parent.updateButton(3, ax, not self.w[ax].isZero())
else:
self.parent.updateButton(3, ax, False)
# Normal/shear stresses and torque/bending stresses
self.sigmaN[ax] = StressFunction(self.F[ax], areas[ax])
self.sigmaN[ax].name = self.sigmaNstr[ax]
self.parent.updateButton(4, ax, not self.sigmaN[ax].isZero())
if ax == 0:
self.sigmaB[ax] = StressFunction(self.M[ax], torquemoments[ax])
else:
self.sigmaB[ax] = StressFunction(self.M[ax],
bendingmoments[ax])
self.sigmaB[ax].name = self.sigmaBstr[ax]
self.parent.updateButton(5, ax, not self.sigmaB[ax].isZero())
def equilibrium(self):
# Build equilibrium equations
forces = {0.0:0.0} # dictionary of (location : outer force)
moments = {0.0:0.0} # dictionary of (location : outer moment)
variableNames = [""] # names of all variables
locations = {} # dictionary of (variableName : location)
coefficientsFy = [0] # force equilibrium equation
coefficientsMbz = [0] # moment equilibrium equation
for i in range(len(self.segments)):
lType = self.segments[i].loadType
load = -1 # -1 means unknown (just for debug printing)
location = -1
if lType == "Fixed":
# Fixed segment
if i == 0:
location = 0
variableNames.append("Fy%u" % i)
coefficientsFy.append(1)
coefficientsMbz.append(0)
variableNames.append("Mz%u" % i)
coefficientsFy.append(0)
coefficientsMbz.append(1) # Force does not contribute because location is zero
elif i == len(self.segments) - 1:
location = self.getLengthTo(len(self.segments)) / 1000
variableNames.append("Fy%u" % i)
coefficientsFy.append(1)
coefficientsMbz.append(location)
variableNames.append("Mz%u" % i)
coefficientsFy.append(0)
coefficientsMbz.append(1)
else:
# TODO: Better error message
FreeCAD.Console.PrintMessage("Fixed constraint must be at beginning or end of shaft\n")
return
locations["Fy%u" % i] = location
locations["Mz%u" % i] = location
elif lType == "Static":
# Static load (currently force only)
load = self.segments[i].loadSize
location = (self.getLengthTo(i) + self.segments[i].loadLocation) / 1000 # convert to meters
coefficientsFy[0] = coefficientsFy[0] - load
forces[location] = load
coefficientsMbz[0] = coefficientsMbz[0] - load * location
moments[location] = 0
#elif lType == "None":
# # No loads on segment
FreeCAD.Console.PrintMessage("Segment: %u, type: %s, load: %f, location: %f\n" % (i, lType, load, location))
self.printEquilibrium(variableNames, coefficientsFy)
self.printEquilibrium(variableNames, coefficientsMbz)
# Build matrix and vector for linear algebra solving algorithm
try:
import numpy as np
except ImportError:
FreeCAD.Console.PrintMessage("numpy is not installed on your system\n")
raise ImportError("numpy not installed")
if (len(coefficientsFy) < 3) or (len(coefficientsMbz) < 3):
return
A = np.array([coefficientsFy[1:], coefficientsMbz[1:]])
b = np.array([coefficientsFy[0], coefficientsMbz[0]])
solution = np.linalg.solve(A, b)
# Complete dictionary of forces and moments
if variableNames[1][0] == "F":
forces[locations[variableNames[1]]] = solution[0]
else:
moments[locations[variableNames[1]]] = solution[0]
if variableNames[2][0] == "F":
forces[locations[variableNames[2]]] = solution[1]
else:
moments[locations[variableNames[2]]] = solution[1]
FreeCAD.Console.PrintMessage(forces)
FreeCAD.Console.PrintMessage(moments)
self.Qy = SegmentFunction("Qy")
self.Qy.buildFromDict("x", forces)
self.Qy.output()
self.Mbz = self.Qy.integrated().negate()
self.Mbz.addSegments(moments) # takes care of boundary conditions
self.Mbz.name = "Mbz"
self.Mbz.output()
