introduction to mechanism synthesis - union …rbb.union.edu/courses/mer312/lectures/mer312 l01...
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INTRODUCTION TO
MECHANISM SYNTHESIS
Mechanics Place in Science
Mechanisms and Structures
Number Synthesis
Paradoxes and Isomers
Transformations and Inversions
Grashof’s Law
MER312: Dynamics of Mechanisms 1
The Ultimate Goal is to
Synthesize Machine Elements
MER312: Dynamics of Mechanisms 2
Manufacturing
Materials
Mechanics
Thermo-Fluids
Controls
Statics
Dynamics
Strength of
Martials
Kinetics
KinematicsADV. DYN. &
KINEMATICS
Advanced
Strength Machine
Design
MECHANISM
DESIGN/SYNTHESIS
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Historical Perspective of
Kinematics
MER312: Dynamics of Mechanisms 3
4000-3000 BC
Mesopotamia
• Lever
• Inclined Plane
Wedge
28 BC
Marcus Vitruvius
• De Architecture
Earliest Writing
on the Subject
• Moving Heavy
Objects
1st Century AD
Hero of Alexandria
• Named Components
• Wedge, Lever, Screw,
Windlass, & Pulley
1588
Ramelli
• Arteficiose Machine
• Describes Each Machine
of the Time
Without recognition of
similarity of components
287-212 BC
Archimedes of Syracuse
• Archimede’s Screw
• Claw of Archimedes
• Heat Ray
• Equilibrium of Planes
Description of lever
1724-1739
Jacob Leupold
• First to Recognize
Machine Components
• 9 Volume Series of
Books Published
1736-1742
Euler
• Mechanica Sive Motus
Scienta Analytice Exposita
• Analytical Treatment of Mechanisms
• Planar Motion
1817
James Watt & Oliver Evans
• Straight-Line Linkage
for Steam Engine
• Coupler Link Motion in 4-Bar
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Historical Perspective of
Kinematics
MER312: Dynamics of Mechanisms 4
Early 1800’s
Gaspard Monge
• L’Ecole Polytechnic in Paris
• Inventor of Descriptive
Geometry
• Course in Elements
of Machines
Early 1811
Jean Nicolas Pierre Hachette
• L’Ecole Polytechnic in Paris
• First Mechanisms Book
1834
Andre-Marie Ampere
• L’Ecole Polytechnic
• Essai sur la Philosophie
de Science
• First to use term Cinématique
1841
Robert Willis
• University of Cambridge, England
• Mechanism Synthesis
1876
Alexander Kennedy
• Theoretische Kinematik
• Translated to English
1875
Franz Reuleaux
• Theoretische Kinematik
• Mechanism Synthesis
Europe and
Australia
1940’s
Interest Builds
In US
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Historical Perspective of
Kinematics
MER312: Dynamics of Mechanisms 5RBB
Machine de Marly (1684)
• Created to pump water to the
Gardens in the Palace of
Versailles
• Delivered water to aqueduct 533 ft
above river
• Pumps driven by parallelograph
linkages
Bachannan Paddle Wheel
(1813)
Machines/Kinetics and
Mechanisms/Kinematics
MER312: Dynamics of Mechanisms 6
• Machines: A combination of resistant bodies so arranged
that by their means the mechanical forces of the nature
can be compelled to do work accompanied by certain
determinate motions.
• MECHANISMS: An assemblage of resistant bodies,
connected by movable joints, to form a closed kinematic
chain with one link fixed and having the purpose of
transforming motion.
• Structures: An assemblage of resistant bodies
connected by joints (or not) that do no work, and do not
transfer motion. It is intended to be rigid.
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Synthesis of Several Mechanisms
will be Considered
MER312: Dynamics of Mechanisms 7
Linkages
Gears
CAMs
Belts, Pulleys,
And Chains
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Mechanisms are Synthesized to
Produce Various Types of Motion
MER312: Dynamics of Mechanisms 8
PLANAR MOTION: All motion contained to one
geometric Plane or Parallel Planes.
TRANSLATION
ROTATION
COMBINATION
• Rectilinear Translation: All Points of
the body move in parallel straight
line paths.
• Rotation: Each point the body
remains a constant distance from a
fixed axis that is perpendicular to
the plane of motion.
• Rotation and Translation:
Combination of the above two.
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Curvilinear Translation a
Special Case of Translation
MER312: Dynamics of Mechanisms 9
Curvilinear Translation: The paths of the points are
identical curves parallel to a fixed plane.
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Non-Planar Motion Can Also
Be Generated By Mechanisms
MER312: Dynamics of Mechanisms 10
• Helical Motion: each point of the body has motion
of rotation about a fixed axis and at the same time
has translation parallel to the axis.
• Spherical Motion: each point of the body has
motion about a fixed point while remaining at a
constant distance from it.
• Spatial Motion: the body moves with rotations
about three non-parallel axes and translates in
three independent directions.
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Cycle, Period,
and Phase of Motion
MER312: Dynamics of Mechanisms 11
• Cycle: When the parts of a
mechanism have passed through
all the possible positions they can
assume after starting from some
simultaneous set of relative
positions and have returned to
their original relative positions.
• Period: The time required for a
cycle of motion.
• Phase: The simultaneous relative
position of a mechanism at a
given instant during a cycle.
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A Link Is A Rigid Body Having
Two or More Nodes
12MER312: Dynamics of Mechanisms
Unary Binary Ternary Quaternary Pentagonal
Nodes/Pairing Elements: Points at which links can be attached. The
order of the link is determined by the attachments used.
Joints/Kinematic Pairs: Allows relative motion between links.
Joint Classes: a kinematic pair is of the jth class if it diminishes the
relative motion of linked bodies by j Degrees of Freedom (DoF)
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• Open Kinematic Chain: A chain with one or more
open loops.
• Closed Kinematic Chain: A chain that forms one or
more closed loops.
• Simple-Closed Chain: Chain consisting of entirely
binary links and is closed.
• Compound Closed Chain: Chain including other than
binary links that is closed.
Kinematic Chains are Formed
by Connecting Links with Pairs
MER312: Dynamics of Mechanisms 13
R
R
R
R
P
R
R
R
R
R
R
RR
B
B
R
R
R
R
R
R
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Joint Classification,
Kinematic Pairs
Type of contact between elements
Line
Point
Surface
Degrees of Freedom Allowed
Type of Physical Closure
Force
Form
Number of Links Joined (Order)MER312: Dynamics of Mechanisms 14
Higher Pairs
Lower Pairs
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Joint Closure Classified as
Lower Pairs and Higher Pairs
MER312: Dynamics of Mechanisms 15
Form Closed, Rotating FULL Pin Joint Form Closed, Translating FULL Slider Joint
Force Closed,
Link against a plane HALF JointForm Closed,
Pin in Slot HALF Joint
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Degrees of Freedom or Mobility
The number of inputs needed to
provide in order to create a
predictable output
The number of independent
coordinates required to define its
position
MER312: Dynamics of Mechanisms 16
RR
R
R
P
R
R
R R
R
R
R R
B
B
RR
R
R
R
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1 DOF, Class I Kinematic Pairs
as Defined by Reuleaux
MER312: Dynamics of Mechanisms 17
Revolute (R) Prismatic (P) Helical (H)
• Lower Pair Contact
• Plainer & 3D Joint
• 1 DOF -
• Lower Pair Contact
• Plainer & 3D Joint
• 1 DOF - s
• Lower Pair Contact
• Plainer & 3D Joint
• 1 DOF
• Input - , Output – s
• Input - s, Output -
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2 DOF, Class II Kinematic Pairs
as Defined by Reuleaux
MER312: Dynamics of Mechanisms 18
Cam (Ca) Cylinder (C) Slotted Spherical (Sl)
• Higher Pair Contact
• RP
• Plainer & 3D Joint
• 2 DOF - ,s
• Lower Pair Contact
• RP
• 3D Joint
• 2 DOF – s,
• Lower Pair Contact
• RR
• 3D Joint
• 2 DOF - ,
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3 DOF, Class III Kinematic Pairs
as Defined by Reuleaux
MER312: Dynamics of Mechanisms 19
Spherical (S) Spherical Slotted Cylinder (C) Plane Pair (Pl)
• Lower Pair Contact
• RRR
• 3D Joint
• 3 DOF: , ,
• Lower Pair Contact
• RRP
• 3D Joint
• 3 DOF: , , s
• Lower Pair Contact
• RRP
• 3D Joint
• 2 DOF - , , s
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4 DOF, Class IV Kinematic Pairs
as Defined by Reuleaux
MER312: Dynamics of Mechanisms 20
Spherical Grove (Sg) Cylindrical Plane Pair (Cp)
• Lower Pair Contact
• RRRP
• 3D Joint
• 4 DOF: , , , s
• Lower Pair Contact
• RRPP
• 3D Joint
• 4 DOF: , , s, t
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5 DOF, Class V Kinematic Pairs
as Defined by Reuleaux
MER312: Dynamics of Mechanisms 21
Spherical Plane (Sp)
• Lower Pair Contact
• RRRPP
• 3D Joint
• 4 DOF: , , , s, t
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Planar Mechanisms
Each link has 3 DoF when moving relative to a fixed link
n link planar mechanism (one link is considered FIXED) has 3(n-1) degrees of freedom before joints are connected
Connecting a revolute pair 1 DoF → 2 constraints
2 DoF → 1 constraint
Mobility of Mechanism Constraints of all joints
minus total DoF of unconnected links
MER312: Dynamics of Mechanisms 22RBB
Mobility Calculation
L- number of links
M- mobility of planar n-link mechanism
j1- number of 1 DoF pairs
j2- number of 2 DoF pairs
Kutzbach Criterion
M = 3·(L-1) - 2·j1 - j2
Grübler Criterion
M = 3·(L-1) - 2·j1
MER312: Dynamics of Mechanisms 23RBB
Mobility Criterion:
Kutzbach or Gruebler
M=1
Mechanism can be driven by a single input direction
M=2
Two separate input motions are necessary to produce constrained motion for the mechanism
Differential Mechanism
M=0
Motion is impossible and the mechanism is a structure
Exact Constraint
M=-1
Redundant constraint
Pre-Load
MER312: Dynamics of Mechanisms 24RBB
Slipping
MER312: Dynamics of Mechanisms 25
Example: Calculate the Mobility
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Example: Calculate the Mobility
MER312: Dynamics of Mechanisms 26RBB
MER312: Dynamics of Mechanisms 27
Example: Calculate the Mobility
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Order of a Joint is One Less
than the Number of Links Joined
MER312: Dynamics of Mechanisms 28
First order pin Joint Second order pin Joint
RBB
MER312: Dynamics of Mechanisms 29
Example: Calculate the Mobility
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MER312: Dynamics of Mechanisms 30
Example: Calculate the Mobility
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MER312: Dynamics of Mechanisms 31
Example: Calculate the Mobility
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Kutzback Criterion for Half Joints
Particular attention should be paid to the
contact between the wheel and the fixed
link
Slipping
MER312: Dynamics of Mechanisms 32RBB
MER312: Dynamics of Mechanisms 33
Example: Calculate the Mobility
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Example: Calculate the Mobility
MER312: Dynamics of Mechanisms 34RBB
Spatial Mechanism Mobility
Kutzbach Criterion
Where
j3- 3 Dof joints
j4- 4 Dof joints
j5- 5 Dof joints
1 2 3 4 56 ( 1) 5 4 3 2M L j j j j j
MER312: Dynamics of Mechanisms 35RBB
Mobility Paradoxes
Over-constrained Linkage with Redundant
Constraint
MER312: Dynamics of Mechanisms 36RBB
E-quintet,
Delta Triplet
M=0
E-quintet
M=1
Mobility Paradoxes
Over-constrained Linkage with Redundant
Constraint
MER312: Dynamics of Mechanisms 37
Mobility Paradoxes
Passive or Idle Degree of Freedom
MER312: Dynamics of Mechanisms 38RBB
Example: Calculate the Mobility
MER312: Dynamics of Mechanisms 39RBB