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1 MENG 372 Mechanical Systems Spring 2011 Dr. Mustafa Arafa American University in Cairo Mechanical Engineering Department

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2 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, 3 rd ed., 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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3 All figures taken from Design of Machinery, 3 rd ed. Robert Norton 2003 MENG 372 Chapter 2 Kinematics Fundamentals

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4 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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5 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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6 Backhoe Excavator

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7 Slider-Crank Mechanism

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8 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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9 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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10 Types of joints

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11 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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12 Determining Degrees of Freedom For simple mechanisms calculating DOF is simple Closed Mechanism DOF=1 Open Mechanism DOF=3

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13 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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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

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15 Example

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16 Example

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Mechanisms and Structures Mechanism: DOF>0 Structure: DOF=0 Preloaded Structure – DOF<0, may require force to assemble

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Paradoxes Greubler criterion does not include geometry, so it can give wrong prediction We must use inspection E-quintet

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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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20 Geneva Mechanism

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21 Linear Geneva Mechanism

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22 Ratchet and Pawl

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23 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

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24 4-Bar Nomenclature Ground Link Links pivoted to ground: –Crank –Rocker Coupler Ground Link Coupler Link 2, length a Link 1, length d Link 3, length b Link 4, length c Pivot 0 2 Pivot 0 4 A B Crank Rocker

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25 Where would you see 4-bar mechanisms?

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26 Sheet Metal Shear (Mechanical Workshop)

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27 Sheet Metal Shear (Mechanical Workshop)

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28 Door Mechanism (ACMV Lab)

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29 Door Mechanism (ACMV Lab)

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30 Backhoe Excavator

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31 Brake of a WheelchairFolding sofa

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32 Honda Accord trunk Garage door Desk Lamp Chevy Cobalt

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33 Inversions Created by attaching different links to ground Different behavior for different inversions

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34 Inversions of a 4-Bar Mechanism Crank-rocker Crank-crank Rocker-rocker

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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

37 36 For S+L

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37 For S+L>P+Q All inversions will be double rockers No link can fully rotate

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38 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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39 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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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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41 Linkages of more than 4 bars Volvo 740 Hood

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42 Volvo 740 Hood

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43 Animation using Working Model ®

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44 Cabinet Hinge

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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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46 More Examples: Front End Loader

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47 Drum Brake

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