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Session #52D Mechanisms: Mobility,
Kinematic Analysis & Synthesis
Courtesy of Design Simulation Technologies, Inc. Used with permission.
Dan Frey
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Today’s Agenda• Collect assignment #2• Begin mechanisms
– Mobility / degrees of freedom– Some basic mechanisms– Some basic synthesis and analysis techniques– MathCad preliminaries– Homogeneous Transformation Matrices– Solving systems of equations– Kinematic analysis– Mechanism synthesis
• Distribute & discuss assignment #3• Distribute graded assignment #1
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2D Mechanisms
• Where are they found?• What do they do?
Courtesy of Design Simulation Technologies, Inc. Used with permission.
http://workingmodel.design-simulation.com/
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2D Mechanisms
• Where are they found?• What do they do?
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2D Mechanisms
• Where are they found?• What do they do?
Courtesy of Design Simulation Technologies, Inc. Used with permission.
http://workingmodel.design-simulation.com/
![Page 6: Session #5 2D Mechanisms: Mobility, Kinematic Analysis & Synthesis · 2019. 9. 12. · • Kutzbach criterion (to find the DOF) • Grübler criterion (to have a single DOF) F=3(n-1)-2j](https://reader036.vdocuments.us/reader036/viewer/2022071407/60fe1f3cacfa8a3c841daa1f/html5/thumbnails/6.jpg)
Two Goals in our Study of 2D Mechanisms
• Analysis – given a mechanism, predict its behavior– DOF, position, velocity, acceleration
• Synthesis– given a desired behavior, develop a mechanism
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Degrees of Freedom
• DOF = the number of independent parameters required to completely define the entity’s positioning
• If a body is constrained to a plane, how many DOF does it possess?
• If a pin joint is added?• If a sliding joint is added?• If it rolls without slipping?
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Mobility Criteria
• Kutzbach criterion (to find the DOF)
• Grübler criterion (to have a single DOF)
F=3(n-1)-2j
DOF # of bodies # of joints
2j-3n+4=0
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Concept Question• How many degrees of freedom does this
mechanism possess?
Ground1)1 DOF2)2 DOF3)3 DOF4)0 DOF
Pin joints
Clamprigidly
Pin joints
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Concept Question• How many degrees of freedom does this
mechanism possess?
1)1 DOF2)2 DOF3)3 DOF4)0 DOF
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Degrees of Freedom• How many degrees of freedom does this
mechanism possess?
We’ll cover 3D mechanisms later
Figure removed for copyright reasons.
Rear suspension of a Honda Accord
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The 4 Bar Mechanism• Use working Model to arrange a
mechanism in which neither link can rotate a full turn
• Use working Model to arrange a mechanism in which rotating the input link a full turn causes the output link to:– Oscillate– Rotate a full turn
• Same direction• Opposite direction
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Parallelogram Linkage
• This object is connected to links that are parallel and of equal length
• What can we say (qualitatively) about the motion of the body (bce)?
b
e
c
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3 Position Synthesis• Say we want a
mechanism to guide a body in a prescribed way
• Pick 3 positions• Pick two
attachment points• The 4 bar
mechanism can be constructed graphically
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Concept Question• If you do not specify the attachment point,
how many positions can you specify and still generally retain the capability to synthesize a mechanism?
1)32)43)54)>5
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Slider-Crank Mechanism• Kinematically the same (in the limit) as a 4 bar
mechanism with a very long output crank
Courtesy of Design Simulation Technologies, Inc. Used with permission.
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Inversion
• Take a mechanism• Change which link is “ground”• A new mechanism results
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Velocity Analysis
• This object is translating and is NOT rotating
• The velocity if point “b” is indicated by a green arrow
• What do we know about the velocity of point “e”?
b
c
e
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Velocity Analysis
• This object is translating and ALSO rotating about “b” at rate ω
• The velocity if point “b” is indicated by a green arrow
• Can we construct the velocity of vector of point “e”?
b
e
c
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Velocity Analysisb
e
• This object is a rigid body• The velocity if point “b” is
indicated by a green arrow• The velocity of point “e” is
indicated by a blue arrow• Can the vector “e” take any
value, or are there some restrictions on values it might have?
c
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Homogeneous Transformation Matrices
• How can we mathematically express the motions of a rigid body?– Translations– Rotations– Both
• Matrix multiplication will do the trick
body'
cos θ( )
sin θ( )
0
0
sin θ( )−
cos θ( )
0
0
0
0
1
0
δ x
δ y
0
1
⎛⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎠
body⋅:=
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MathCad: Preliminaries
• Assigning a value to a scalar variable• Assigning a value to a vector or matrix• Defining a function• Displaying results
– Numerically– Graphically
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MathCad: Useful Capabilities
• Taking Derivatives• Integration• Symbolic manipulation• Solving non-linear equations• Solving blocks of non-linear equations
including constraints• Minimizing / maximizing under constraints• Making animations• Processing images
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HTM MathCad Demonstrator
• See HTM_DEMO_2D v2.xmcd
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A Challenge• Can you modify the HTM Demonstrator to
simulate the operation of a 4 Bar mechanism?– Specify input and output crank geometry– Specify an input crank angle– Solve for an HTM that satisfies the physical
constraints created by the cranks
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Assignment #3
• Do some problems from the handout• Analyze the kinematics of a robot’s leg
mechanism• Take some measurements of
– Motion– Force
• Synthesize a mechanism (your choice)
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Next Steps
• Start Assignment #3• Lab tomorrow, Friday 24 FEB• Next Class Tues 28 FEB