MENG 372
Mechanical Systems
Spring 2011
Dr. Mustafa Arafa
American University in Cairo
Mechanical Engineering Department
mharafa@aucegypt.edu
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Course Information
Course goals:
• Analyze & design planar mechanisms
• Analyze forces, velocities & accelerations in machines
• Use computers for the above
Textbook: Design of Machinery, R.Norton, McGraw-Hill, 3rd ed., 2004.
Computer usage: Working Model, MATLAB
Grading: attendance 5%; homework 10%; quizzes 5%; mid-term
exams 30%; projects 25%; final exam 25%
Lecture notes: will be posted my website. I will communicate with you
on BlackBoard. Additional material will also be covered on the board.
Please print out the notes beforehand & bring them to class.
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MENG 372
Chapter 2
Kinematics Fundamentals
All figures taken from Design of Machinery, 3rd ed. Robert Norton 2003
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2.1 Degrees of Freedom (DOF) or
Mobility
• DOF: Number of independent parameters
(measurements) needed to uniquely define
position of a system in space at any instant of
time.
Rigid body in a plane has 3
DOF: x,y,q
Rigid body in space has 6
DOF (3 translations & 3
rotations)
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2.2 Types of Motion
• Pure rotation: the body possesses one point (center
of rotation) that has no motion with respect to the
“stationary” frame of reference. All other points
move in circular arcs.
• Pure translation: all points on the body describe
parallel (curvilinear or rectilinear) paths.
• Complex motion: a simultaneous combination of
rotation and translation.
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Backhoe Excavator
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Slider-Crank Mechanism
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2.3 Links, joints, and kinematic chains
• Links: rigid member having nodes
• Node: attachment points
– Binary link: 2 nodes
– Ternary link: 3 nodes
– Quaternary link: 4 nodes
• Joint: connection between two or more links
(at their nodes) which allows motion
Classified by type of contact, number of DOF,
type of physical closure, or number of links joined
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Joint Classification
• Type of contact: line, point,
surface
• Number of DOF: full
joint=1DOF, half joint=2DOF
• Form closed (closed by
geometry) or Force closed
(needs an external force to
keep it closed)
• Joint order = number of links-1
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Types of joints
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Kinematic chains, mechanisms,
machines, link classification
•
•
•
•
Kinematic chain: links joined together for motion
Mechanism: grounded kinematic chain
Machine: mechanism designed to do work
Link classification:
 Ground: fixed w.r.t. reference frame
 Crank: pivoted to ground, makes complete
revolutions
 Rocker: pivoted to ground, has oscillatory motion
 Coupler: link has complex motion, not attached to
ground
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Determining Degrees of Freedom
• For simple mechanisms calculating DOF is
simple
Open Mechanism
DOF=3
Closed Mechanism
DOF=1
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Determining Degrees of Freedom
Two unconnected links: 6 DOF
(each link has 3 DOF)
When connected by a full joint: 4 DOF
(each full joint eliminates 2 DOF)
Gruebler’s equation for planar mechanisms: DOF = 3L-2J-3G
Where:
L: number of links
J: number of full joints
G: number of grounded links
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2.4 Determining DOF’s
• Gruebler’s equation for planar mechanisms
M=3L-2J-3G
• Where
M = degree of freedom or mobility
L = number of links
J = number of full joints (half joints count as 0.5)
G = number of grounded links =1
M  3 L  1  2 J
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Example
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Example
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2.5 Mechanisms and Structures
• Mechanism: DOF>0
• Structure: DOF=0
• Preloaded Structure –
DOF<0, may require
force to assemble
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2.7 Paradoxes
• Greubler criterion does not include geometry, so it
can give wrong prediction
• We must use inspection
E-quintet
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2.10 Intermittent Motion
• Series of Motions and Dwells
• Dwell: no output motion with input motion
• Examples: Geneva Mechanism, Linear
Geneva Mechanism, Ratchet and Pawl
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Geneva Mechanism
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Linear Geneva Mechanism
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Ratchet and Pawl
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Fourbar Mechanism
Twobar has -1 degrees of freedom
(preloads structure)
Threebar has 0 degrees of freedom
(structure)
Fourbar has 1 degree of freedom
The fourbar linkage is the simplest
possible pin-jointed mechanism for
single degree of freedom controlled
motion
-1
0
1
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4-Bar Nomenclature
• Ground Link
• Links pivoted to ground:
– Crank
– Rocker
• Coupler
B
Coupler
A
Rocker
Crank
Link 1, length d
Ground Link
Pivot 02
Pivot 04
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Where would you see 4-bar mechanisms?
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Sheet Metal Shear (Mechanical Workshop)
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Sheet Metal Shear (Mechanical Workshop)
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Door Mechanism (ACMV Lab)
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Door Mechanism (ACMV Lab)
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Backhoe Excavator
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Brake of a Wheelchair
Folding sofa
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Honda Accord trunk
Chevy Cobalt
Garage door
Desk Lamp
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Inversions
• Created by attaching different links to ground
• Different behavior for different inversions
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Inversions of a 4-Bar Mechanism
Crank-rocker
Crank-rocker
Crank-crank
Rocker-rocker
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2.12 The Grashof Condition
• Grashof condition predicts behavior of linkage based
only on length of links
S=length of shortest link
L=length of longest link
P,Q=length of two remaining links
 If S+L ≤ P+Q the linkage is Grashof :at least one link
is capable of making a complete revolution
 Otherwise the linkage is non-Grashof : no link is
capable of making a complete revolution
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For S+L<P+Q
• Crank-rocker if either link adjacent to shortest is grounded
• Double crank if shortest link is grounded
• Double rocker if link opposite to shortest is grounded
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For S+L>P+Q
• All inversions will be double rockers
• No link can fully rotate
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For S+L=P+Q (Special case Grashof)
• All inversions will be double cranks or crank rockers
• Linkage can form parallelogram or antiparallelogram
• Often used to keep coupler parallel (drafting
machine)
Parallelogram form
Deltoid form
Anti parallelogram
form
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Problems with Special Grashof
• All inversions have change points twice
per revolution of input crank when all links
become collinear
• Behavior at change points is indeterminate
• If used in continuous machine, must have
some mechanism to “carry through”
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2.13 Linkages of more than 4 bars
5-bar 2DOF
Geared 5-bar 1DOF
• Provide more complex motion
• See Watt’s sixbar and Stephenson’s sixbar
mechanisms in the textbook
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Linkages of more than 4 bars
Volvo 740 Hood
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Volvo 740 Hood
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Animation using Working Model ®
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Cabinet Hinge
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2.15 Compliant Mechanisms
• Compliant “link” capable of significant deflection acts
like a joint
• Also called a “living hinge”
• Advantage: simplicity, no assembly, little friction
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More Examples: Front End Loader
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Drum Brake
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