Four Bar LinkageΒΆ

We will simulate a four bar linkage and introduce the command for handling kinematic loops like this. If you want to try it first, or look at the complete source code, see FourBarLinkage.py.

As usual, we start with importing PyMbs, creating a model system and adding various parameters. Here, they are the lengths, masses and interia of the bars to be connected:


world = MbsSystem([0, 0, -1])

# Parameter
l1 = world.addParam('l1', 1)

m2 = world.addParam('m2', 1)
l2 = world.addParam('l2', 4)
...

Now we create our additional three bars (world will act as the first bar) and add the coordinate systems \(A\) through \(D\) at the corners:

world.addFrame(name='CS_D', p=[l1, 0, 0])

bar2 = world.addBody(name='Bar2', mass=m2, cg=[l2/2, 0, 0],
                     inertia=diag([0, I2, I2]))
bar2.addFrame(name='CS_A')
bar2.addFrame(name='CS_B', p=[l2, 0, 0])

bar3 = world.addBody(name='Bar3', mass=m3, cg=[l3/2, 0, 0],
                     inertia=diag([0, I3, I3]))
bar3.addFrame(name='CS_B')
bar3.addFrame(name='CS_C', p=[l3, 0, 0])

bar4 = world.addBody(name='Bar4', mass=m4, cg=[l4/2, 0, 0],
                     inertia=diag([0, I4, I4]))
bar4.addFrame(name='CS_D')
bar4.addFrame(name='CS_C', p=[l4, 0, 0])

Then we can connect three of them using normal joints:

jB = world.addJoint(bar2.CS_B, bar3.CS_B, 'Ry')
jD = world.addJoint(world.CS_D, bar4.CS_D, 'Ry')

However, the last one (here its \(C\)) has to be connected using the command addLoop.FourBar to correctly handle the kinematic loop:

world.addLoop.FourBar(bar3.CS_C, bar4.CS_C, posture=1)

With the posture parameter, either 1 or -1, you can choose which solution is being used. Usually, this switches between the bars intersecting or not.

In the end, we add a line of each of the bars for visualisation, generate our equations of motion and start the GUI:

world.addVisualisation.Line(bar3, 4)
world.addVisualisation.Line(bar4, 5)

world.genEquations.Recursive()
world.show('FourBarLinkage')

The result should look roughly like this:

../_images/FourBarLinkage.gif

In the end, the complete source looks like this:

# -*- coding: utf-8 -*-
'''
This file is part of PyMbs.

PyMbs is free software: you can redistribute it and/or modify
it under the terms of the GNU Lesser General Public License as
published by the Free Software Foundation, either version 3 of
the License, or (at your option) any later version.

PyMbs is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU Lesser General Public License for more details.

You should have received a copy of the GNU Lesser General Public
License along with PyMbs.
If not, see <http://www.gnu.org/licenses/>.

Copyright 2011, 2012 Carsten Knoll, Christian Schubert,
                     Jens Frenkel, Sebastian Voigt
'''
# Warning: The source code of the examples is quoted in the documentation. If
# you change this file, you'll have to change the corresponding file in the
# documentation (see doc/examples).
from PyMbs.Input import *

world = MbsSystem([0, 0, -1])

# Parameter
l1 = world.addParam('l1', 1)

m2 = world.addParam('m2', 1)
l2 = world.addParam('l2', 4)
I2 = world.addParam('I2', m2/12*l2**2)

m3 = world.addParam('m3', 1)
l3 = world.addParam('l3', 4)
I3 = world.addParam('I3', m3/12*l3**2)

m4 = world.addParam('m4', 1)
l4 = world.addParam('l4', 5)
I4 = world.addParam('I4', m4/12*l4**2)

# Create Bodies with Coordinate Systems
world.addFrame(name='CS_A')
world.addFrame(name='CS_D', p=[l1, 0, 0])

bar2 = world.addBody(name='Bar2', mass=m2, cg=[l2/2, 0, 0],
                     inertia=diag([0, I2, I2]))
bar2.addFrame(name='CS_A')
bar2.addFrame(name='CS_B', p=[l2, 0, 0])

bar3 = world.addBody(name='Bar3', mass=m3, cg=[l3/2, 0, 0],
                     inertia=diag([0, I3, I3]))
bar3.addFrame(name='CS_B')
bar3.addFrame(name='CS_C', p=[l3, 0, 0])

bar4 = world.addBody(name='Bar4', mass=m4, cg=[l4/2, 0, 0],
                     inertia=diag([0, I4, I4]))
bar4.addFrame(name='CS_D')
bar4.addFrame(name='CS_C', p=[l4, 0, 0])

# Insert Joints
jA = world.addJoint(world.CS_A, bar2.CS_A, 'Ry', pi/2)
jB = world.addJoint(bar2.CS_B, bar3.CS_B, 'Ry')
jD = world.addJoint(world.CS_D, bar4.CS_D, 'Ry')

world.addLoop.FourBar(bar3.CS_C, bar4.CS_C, posture=1)

world.addVisualisation.Line(bar2, 4)
world.addVisualisation.Line(bar3, 4)
world.addVisualisation.Line(bar4, 5)

world.genEquations.Recursive()
world.show('FourBarLinkage')