mechanical invention through computation mechanism basics
TRANSCRIPT
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MIT Class 6.S080 (AUS) Mechanical Invention through Computation Mechanism Basics
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Design Principles Structure and Mechanism
Structure: Force is resisted
Mechanism:
Force flows into movement
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Design Principles Structure and Mechanism
Structural
Resistance
Kinematic deflection Structural Deflection
(elastic or Inelastic)
Possible Responses to Applied Force
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Mechanism paradigms
Synthesize a motion path
Synthesize a form change
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Definitions
• Kinematics: the study of the motion of bodies without reference to mass or force
• Links: considered as rigid bodies • Kinematic pair: a connection between two bodies that imposes
constraints on their relative movement. (also referred to as a mechanical joint)
• Ground: static point of reference • Degree of freedom (DOF): of a mechanical system is the
number of independent parameters that define its configuration.
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Links types
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Kinematic pairs
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Historic Mechanisms
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Straight- line linkages (James Watt)
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Straight- line linkages (Richard Roberts)
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Straight- line linkages (Tchebicheff)
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Peaucellier Linkage
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Straight- line linkages (Peaucellier)
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Straight- line linkages (Kempe)
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4-bar linkage types
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Kinematic inversions
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Four-bar linkage examples
Parallel 4-bar
Anti-parallel 4-bar
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Gruebler’s equation N = Number of Links (including
ground link) P = Number of Joints (pivot connections between links) • Each link has 3
degrees of freedom • Each pivot subtracts 2
degree of freedom DOF = 3 (N-1) - 2P
dof = 3 X 4 = 12
dof = 3 X (4-1) – (2 X 4) = 1
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Examples
N = 4 P = 4 DOF = 3X(4-1)-(2X4)=1
N = 3 P = 3 DOF = 3X(3-1)-(2X3)=0
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Examples
N = 5 P = 5 DOF = 2
Geared connection removes one degree of freedom DOF = 1
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Examples
N = 6 P = 7 DOF = 3X(6-1)-(2X7)=1
N = 8 P = 10 DOF = 3X(8-1)-(2X10)=1
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Relation of DOF to special geometries
N = 5 P = 6 DOF = 3X(5-1)-(2X6) = 0
Agrees with Gruebler’s equation (doesn’t move)
Doesn’t agree with Gruebler’s equation (moves)
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Graph of linkages DOF=3(N-1)-2P
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Scissor Linkages
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Scissor mechanisms
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Historic examples of scissor mechanisms
Emilio Pinero
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Historic examples of scissor mechanisms
Emilio Pinero
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Examples of scissor mechanisms
Sergio Pellegrino
Felix Escrig
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Curvature of scissor mechanisms Off-center connection point => structures of variable curvature
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Scissor Types
Parallel / symmetric
Angulated
Parallel / asymmetric
No curvature
Constant curvature
Variable curvature
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Angulated scissors Provides invariant angle during deployment
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Scissor mechanism: demonstration Parallel / Symmetric
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Scissor mechanism: demonstration Off-center connection
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Scissor mechanism: demonstration Angulated
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Angulated link: geometric construction
Redraw links as 2 triangles
Rotate triangles to angle of curvature
Redraw angulated links
2 straight links
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Angulated link: geometric construction
Parametric
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Tong linkage
Hinged rhombs
Tong linkage
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Hinged rhombs – transforming between configurations
straight
arc
circle
ellipse
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Arc - geometric construction
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Circle - geometric construction
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Ring linkages
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Ring linkages
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Ellipse - geometric construction
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Unequal rhombs
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Unequal rhombs
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Constructing expanding polygons
Perpendicular bisectors
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Irregular polygon – geometric construction
All rhombs are similar (same angles, different
sizes)
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Degrees of freedom of a tong linkage
Number of pivots for a tong linkage: P = 3N/2 -2 DOF = 3 X (N-1) – 2P = 3N – 3 – (3N – 4) = 1
Number of pivots for a closed tong linkage: P = 3N/2 DOF = 3 X (N-1) – 2P = 3N – 3 – 3N = - 3
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N = 8 P = 12 DOF = - 3
DOF=3(N-1)-2P
Spatial interpretation of Gruebler’s equation
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Unequal rhombs with crossing connection
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Unequal rhombs with crossing connection
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Polygon linkages with fixed centers
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circular linkage with fixed center (four spokes)
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Circular linkages with fixed center
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Polygon linkages with fixed centers Construction for off-center fixed point
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Polygon linkages with fixed centers Polygon linkages – off-center fixed point
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Polygon linkages with fixed centers Polygon linkages – off-center fixed point
expanded
folded
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Polygon linkages with fixed centers Polygon linkages – off-center fixed point
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Polygon linkages – off-center fixed point
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Ring linkages
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Degrees of freedom (Graver formulation) Each point (pivot) has 2 degrees of freedom Each link subtracts 1 degree of freedom J = Number of joints (2D points in the plane) R = Number of links (not including ground link)
DOF = 2J - R
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Examples
DOF = 2J - R
J = 2 R = 3 DOF = (2X2)-(3X1)=1
J = 3 R = 5 DOF = (2X3)-(5X1)=1
J = 4 R = 7 DOF = (2X4)-(7X1)=1
J = 1 R = 2 DOF = (2X1)-(1X2)=0
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Book
• www.gfalop.org • Published
August 2007 • Cambridge
University Press
• Coming soon in Japanese
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Configuration Space configuration point
Figure 2.1
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Rigidity
flexible
rigid rigid
rigid
Figure 4.1
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Rigidity Depends on Configuration
rigid flexible
Figure 4.2
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Generic Rigidity
generically rigid generically flexible
Figure 4.3
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Generic Rigidity
generically rigid generically flexible
rarely flexible
rarely rigid
Figure 4.4
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Laman’s Theorem [1970]
• Generically rigid in 2D if and only if you can remove some extra bars to produce a minimal graph with 2 𝐽 − 3 bars total, and at most 2 𝑘 − 3 bars between
every subset of 𝑘 joints
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Laman’s Theorem [1970]
• Generically rigid in 2D if and only if you can remove some extra bars to produce a minimal graph with 2 𝐽 − 3 bars total, and at most 2 𝑘 − 3 bars between
every subset of 𝑘 joints