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Kinematics Fundamentals
Lecture 2
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04/18/23 Robert L. Norton, WPI 2
Joints ReduceSystem DOF
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04/18/23 Robert L. Norton, WPI 3
Determining MobilityGrubler & Kutzbach Equations
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04/18/23 Robert L. Norton, WPI 4
Mechanisms and Structures
Fourbar Linkage Delta Triplet (Truss)
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04/18/23 Robert L. Norton, WPI 5
Applying Mobility Equations - 1
M = 3 (L-1) –2(J1) – J2 = 3(8-1) –2(10) – 0 = 1
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04/18/23 Robert L. Norton, WPI 6
Applying Mobility Equations - 2
M = 3 (L-1) –2(J1) – J2 = 3(6-1) –2(7) – 1 = 0
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04/18/23 Robert L. Norton, WPI 7
Gruebler Paradoxes
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04/18/23 Robert L. Norton, WPI 8
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04/18/23 Robert L. Norton, WPI 9
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04/18/23 Robert L. Norton, WPI 10
Possible Mechanismsfrom “Number Synthesis”
Some combinations will make valid “Isomers,” and some will not.
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04/18/23 Robert L. Norton, WPI 11
Isomers
C4H10
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04/18/23 Robert L. Norton, WPI 12
Invalid Isomers
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04/18/23 Robert L. Norton, WPI 13
Grashof Condition
Demonstration
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LINKAGE TRANSFORMATION
• Revolute joints in any loop can be replaced by prismatic joints with no change in DOF of the mechanism, provided that at least two revolute joints remain in the loop.
• Any full joint can be replaced by a half joint, but this will increase the DOF by one.
• Removal of a link will reduce the DOF by one.
04/18/23 Robert L. Norton, WPI 14
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LINKAGE TRANSFORMATION
• The combination of rules 2 and 3 above will keep the original DOF unchanged.
• Any ternary or higher-order link can be partially "shrunk" to a lower-order link by coalescing nodes. This will create a multiple joint but will not change the DOF of the mechanism.
04/18/23 Robert L. Norton, WPI 15
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LINKAGE TRANSFORMATION
04/18/23 Robert L. Norton, WPI 16
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LINKAGE TRANSFORMATION
04/18/23 Robert L. Norton, WPI 17
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LINKAGE TRANSFORMATION
04/18/23 Robert L. Norton, WPI 18
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LINKAGE TRANSFORMATION
04/18/23 Robert L. Norton, WPI 19
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INTERMITTENT MOTION
04/18/23 Robert L. Norton, WPI 20
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04/18/23 Robert L. Norton, WPI 21
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04/18/23 Robert L. Norton, WPI 22
Inversions
• Inversions result from grounding different links in the chain.
• So, there are as many inversions as links.• Not all inversions will have unique kinds of
motion.• For example, a Grashof Fourbar has only 3
distinct inversions, 2 crank-rockers, 1 double-crank, and 1 double-rocker.
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04/18/23 Robert L. Norton, WPI 23
Inversions of the Grashof FourbarS + L < P + Q
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04/18/23 Robert L. Norton, WPI 24
Inversions of the Non-Grashof FourbarS + L > P + Q
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04/18/23 Robert L. Norton, WPI 25
Special-Case Grashof LinkageS + L = P + Q
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04/18/23 Robert L. Norton, WPI 26
Inversions of the Slider Crank
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04/18/23 Robert L. Norton, WPI 27
Inversions of the Sixbar Linkage
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04/18/23 Robert L. Norton, WPI 28
Geared Fivebar Linkage
M = 3(5 – 1) – 2(5) – 0 = 2 M = 3(5 – 1) – 2(5) – 1 = 1
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04/18/23 Robert L. Norton, WPI 29
Compliant Linkages
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04/18/23 Robert L. Norton, WPI 30
Find Its Degree of Freedom
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04/18/23 Robert L. Norton, WPI 31
Find Their Grashof Conditions