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MENG 372Mechanical Systems
Spring 2011
Dr. Mustafa ArafaAmerican University in Cairo
Mechanical Engineering Department
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Course InformationCourse goals: • Analyze & design planar mechanisms• Analyze forces, velocities & accelerations in machines• Use computers for the aboveTextbook: Design of Machinery, R.Norton, McGraw-Hill, 3rd ed., 2004.Computer usage: Working Model, MATLABGrading: 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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All figures taken from Design of Machinery, 3rd ed. Robert Norton 2003
MENG 372Chapter 2
Kinematics Fundamentals
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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,
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 motionClassified 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
Closed MechanismDOF=1
Open MechanismDOF=3
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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-3GWhere:L: number of linksJ: number of full jointsG: number of grounded links
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2.4 Determining DOF’s
• Gruebler’s equation for planar mechanismsM=3L-2J-3G
• WhereM = degree of freedom or mobilityL = number of linksJ = number of full joints (half joints count as 0.5)G = number of grounded links =1
3 1 2M L 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 freedomThe fourbar linkage is the simplest
possible pin-jointed mechanism for single degree of freedom controlled motion
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0
-1
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4-Bar Nomenclature
• Ground Link• Links pivoted to ground:
– Crank– Rocker
• CouplerGround Link
Coupler
Link
2, l
engt
h a
Link 1, length d
Link 3, length b
Link 4, length c
Pivot 02 Pivot 04
A
B
Crank
Rocker
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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
Garage door
Desk Lamp
Chevy Cobalt
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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
Anti parallelogram form
Deltoid 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 2DOFGeared 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