Chapter 2

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MENG 372Mechanical Systems

Spring 2011

Dr. Mustafa ArafaAmerican University in Cairo

Mechanical Engineering Department

mharafa@aucegypt.edu

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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