You can view and download this file on Github: flexibleRotor3Dtest.py
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
# This is an EXUDYN example
#
# Details: Flexible rotor test using two rigid bodies connected by 4 springs (corotating)
# This test shows the unstable behavior if inner damping is larger than outer damping
#
# Author: Johannes Gerstmayr
# Date: 2019-12-05
#
# Copyright:This file is part of Exudyn. Exudyn is free software. You can redistribute it and/or modify it under the terms of the Exudyn license. See 'LICENSE.txt' for more details.
#
#+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
import exudyn as exu
from exudyn.itemInterface import *
from exudyn.utilities import * #includes itemInterface and rigidBodyUtilities
import exudyn.graphics as graphics #only import if it does not conflict
import time
import numpy as np
SC = exu.SystemContainer()
mbs = SC.AddSystem()
print('EXUDYN version='+exu.GetVersionString())
useGraphics = True
L=1 #total rotor axis length
m = 1 #mass of one disc in kg
r = 0.5 #radius for disc mass distribution
lRotor = 0.1 #length of one half rotor disk
k = 800 #stiffness of (all/both) springs in rotor in N/m
Jxx = 0.5*m*r**2 #polar moment of inertia
Jyyzz = 0.25*m*r**2 + 1/12.*m*lRotor**2 #moment of inertia for y and z axes
omega0=np.sqrt(2*k/(2*m)) #linear system; without flexibility of rotor
#case 1: external damping: D0=0.002, D0int=0
#case 2: external damping with small internal damping: D0=0.002, D0int=0.001
#case 3: external damping with larger internal damping: D0=0.002, D0int=0.1
#case 4: no external damping with small internal damping: D0=0, D0int=0.001
attr = 'g-' #color in plot
D0 = 0.002 #0.002 default; dimensionless damping
D0int = 0.001*200 #*200 #default 0.001; dimensionless damping (not fully); value > 0.08 gives instability
d = 2*omega0*D0*(2*m) #damping constant for external damping in N/(m/s)
kInt = 4*800 #stiffness of (all/both) springs in rotor in N/m
omega0int = np.sqrt(kInt/m)
dInt = 2*omega0int*D0int*m #damping constant in N/(m/s)
f0 = 0*omega0/(2*np.pi) #frequency start (Hz)
f1 = 2.*omega0/(2*np.pi) #frequency end (Hz)
torque = 0.5 #driving torque; Nm
eps = 2e-3 #excentricity of mass in y-direction
omegaInitial = 0*4*omega0 #initial rotation speed in rad/s
print('resonance frequency (rad/s)= '+str(omega0))
tEnd = 80 #end time of simulation
steps = 10000 #number of steps
#draw RGB-frame at origin
p=[0,0,0]
lFrame = 0.8
tFrame = 0.01
backgroundX = graphics.Cylinder(p,[lFrame,0,0],tFrame,[0.9,0.3,0.3,1],12)
backgroundY = graphics.Cylinder(p,[0,lFrame,0],tFrame,[0.3,0.9,0.3,1],12)
backgroundZ = graphics.Cylinder(p,[0,0,lFrame],tFrame,[0.3,0.3,0.9,1],12)
#mbs.AddObject(ObjectGround(referencePosition= [0,0,0], visualization=VObjectGround(graphicsData= [backgroundX, backgroundY, backgroundZ])))
#rotor is rotating around x-axis
ep0 = eulerParameters0 #no rotation
ep_t0 = AngularVelocity2EulerParameters_t([omegaInitial,0,0], ep0)
print(ep_t0)
p0 = [-lRotor*0.5,eps,0] #reference position
p1 = [ lRotor*0.5,eps,0] #reference position
v0 = [0.,0.,0.] #initial translational velocity
#node for Rigid2D body: px, py, phi:
n0=mbs.AddNode(RigidEP(referenceCoordinates = p0+ep0, initialVelocities=v0+list(ep_t0)))
n1=mbs.AddNode(RigidEP(referenceCoordinates = p1+ep0, initialVelocities=v0+list(ep_t0)))
#ground nodes
nGround0=mbs.AddNode(NodePointGround(referenceCoordinates = [-L/2,0,0]))
nGround1=mbs.AddNode(NodePointGround(referenceCoordinates = [ L/2,0,0]))
#add mass point (this is a 3D object with 3 coordinates):
transl = 0.9 #<1 gives transparent object
gRotor0 = graphics.Cylinder([-lRotor*0.5,0,0],[lRotor,0,0],r,[0.3,0.3,0.9,transl],32)
gRotor1 = graphics.Cylinder([-lRotor*0.5,0,0],[lRotor,0,0],r,[0.9,0.3,0.3,transl],32)
gRotor0Axis = graphics.Cylinder([-L*0.5+0.5*lRotor,0,0],[L*0.5,0,0],r*0.05,[0.3,0.3,0.9,1],16)
gRotor1Axis = graphics.Cylinder([-0.5*lRotor,0,0],[L*0.5,0,0],r*0.05,[0.3,0.3,0.9,1],16)
gRotorCS = [backgroundX, backgroundY, backgroundZ]
rigid0 = mbs.AddObject(RigidBody(physicsMass=m, physicsInertia=[Jxx,Jyyzz,Jyyzz,0,0,0], nodeNumber = n0, visualization=VObjectRigidBody2D(graphicsData=[gRotor0, gRotor0Axis]+gRotorCS)))
rigid1 = mbs.AddObject(RigidBody(physicsMass=m, physicsInertia=[Jxx,Jyyzz,Jyyzz,0,0,0], nodeNumber = n1, visualization=VObjectRigidBody2D(graphicsData=[gRotor1, gRotor1Axis]+gRotorCS)))
#marker for ground (=fixed):
groundMarker0=mbs.AddMarker(MarkerNodePosition(nodeNumber= nGround0))
groundMarker1=mbs.AddMarker(MarkerNodePosition(nodeNumber= nGround1))
#marker for rotor axis and support:
rotorAxisMarker0 =mbs.AddMarker(MarkerBodyPosition(bodyNumber=rigid0, localPosition=[-0.5*L+0.5*lRotor,-eps,0]))
rotorAxisMarker1 =mbs.AddMarker(MarkerBodyPosition(bodyNumber=rigid1, localPosition=[ 0.5*L-0.5*lRotor,-eps,0]))
#++++++++++++++++++++++++++++++++++++
#supports:
mbs.AddObject(CartesianSpringDamper(markerNumbers=[groundMarker0, rotorAxisMarker0],
stiffness=[k,k,k], damping=[d, d, d]))
mbs.AddObject(CartesianSpringDamper(markerNumbers=[groundMarker1, rotorAxisMarker1],
stiffness=[0,k,k], damping=[0, d, d])) #do not constrain x-axis twice
#++++++++++++++++++++++++++++++++++++
#flexible rotor:
nSprings = 4
for i in range(nSprings):
#add corresponding markers
phi = 2*np.pi*i/nSprings
rSpring = 0.5
yPos = rSpring*np.sin(phi)
zPos = rSpring*np.cos(phi)
rotorM0 =mbs.AddMarker(MarkerBodyPosition(bodyNumber=rigid0, localPosition=[ 0.5*lRotor,yPos,zPos]))
rotorM1 =mbs.AddMarker(MarkerBodyPosition(bodyNumber=rigid1, localPosition=[-0.5*lRotor,yPos,zPos]))
mbs.AddObject(CartesianSpringDamper(markerNumbers=[rotorM0, rotorM1],
stiffness=[kInt,kInt,kInt], damping=[dInt, dInt, dInt]))
#coordinate markers for loads:
rotorMarkerUy=mbs.AddMarker(MarkerNodeCoordinate(nodeNumber= n1, coordinate=1))
rotorMarkerUz=mbs.AddMarker(MarkerNodeCoordinate(nodeNumber= n1, coordinate=2))
#add torque:
rotorRigidMarker =mbs.AddMarker(MarkerBodyRigid(bodyNumber=rigid0, localPosition=[0,0,0]))
mbs.AddLoad(Torque(markerNumber=rotorRigidMarker, loadVector=[torque,0,0]))
#print(mbs)
mbs.Assemble()
#mbs.systemData.Info()
simulationSettings = exu.SimulationSettings()
simulationSettings.solutionSettings.solutionWritePeriod = 1e-5 #output interval
simulationSettings.timeIntegration.numberOfSteps = steps
simulationSettings.timeIntegration.endTime = 30#tEnd
simulationSettings.timeIntegration.newton.useModifiedNewton=True
simulationSettings.timeIntegration.generalizedAlpha.useIndex2Constraints = True
simulationSettings.timeIntegration.generalizedAlpha.useNewmark = True
simulationSettings.timeIntegration.verboseMode = 1
simulationSettings.displayStatistics = True
simulationSettings.displayComputationTime = True
simulationSettings.linearSolverType = exu.LinearSolverType.EXUdense
simulationSettings.timeIntegration.generalizedAlpha.spectralRadius = 1
SC.visualizationSettings.general.useMultiThreadedRendering = False
if useGraphics:
exu.StartRenderer() #start graphics visualization
mbs.WaitForUserToContinue() #wait for pressing SPACE bar to continue
#start solver:
mbs.SolveDynamic(simulationSettings)
if useGraphics:
SC.WaitForRenderEngineStopFlag()#wait for pressing 'Q' to quit
exu.StopRenderer() #safely close rendering window!
#evaluate final (=current) output values
u = mbs.GetNodeOutput(n1, exu.OutputVariableType.AngularVelocity)
print('omega=',u)
##+++++++++++++++++++++++++++++++++++++++++++++++++++++
import matplotlib.pyplot as plt
import matplotlib.ticker as ticker
if useGraphics:
data = np.loadtxt('coordinatesSolution.txt', comments='#', delimiter=',')
n=steps
plt.rcParams.update({'font.size': 24})
plt.plot(data[:,0], data[:,3], 'r-') #numerical solution
ax=plt.gca() # get current axes
ax.grid(True, 'major', 'both')
ax.xaxis.set_major_locator(ticker.MaxNLocator(10))
ax.yaxis.set_major_locator(ticker.MaxNLocator(10))
plt.tight_layout()
plt.show()