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Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 1
Theory of Machines
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Syllabus and Course Outline
2
SAT 09:30 – 11:00 Q412
MON 09:30 – 11:00 Q412
Faculty of EngineeringDepartment of Mechanical Engineering
EMEC 3302, Theory of Machines
Instructor: Dr. Anwar Abu-ZarifaOffice: IT Building, Room: I413 Tel: 2821eMail: aabuzarifa@iugaza.edu.psWebsite: http://site.iugaza.edu.ps/abuzarifaOffice Hrs: see my website
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 3
Text Book: R. L. Norton, Design of Machinery “An Introduction to the Synthesis and Analysis of Mechanisms and Machines”, McGraw Hill Higher Education; 3rd edition
Reference Books:
§ John J. Uicker, Gordon R. Pennock, Joseph E. Shigley, Theory of Machines and Mechanisms
§ R.S. Khurmi, J.K. Gupta,Theory of Machines§ Thomas Bevan, The Theory of Machines § The Theory of Machines by Robert Ferrier McKay § Engineering Drawing And Design, Jensen ect., McGraw-Hill Science, 7th
Edition, 2007§ Mechanical Design of Machine Elements and Machines, Collins ect., Wiley,
2 Edition, 2009
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 4
Grading:Attendance 5%Design Project 25%Midterm 30%Final exam 40%
Course Description:
The course provides students with instruction in the fundamentals of theory ofmachines. The Theory of Machines and Mechanisms provides the foundationfor the study of displacements, velocities, accelerations, and static anddynamic forces required for the proper design of mechanical linkages, cams,and geared systems.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 5
Course Objectives:
Students combine theory, graphical and analytical skills to understand the Engineering Design. Upon successful completion of the course, the student will be able:
§ To develop the ability to analyze and understand the dynamic(position, velocity, acceleration, force and torque) characteristics ofmechanisms such as linkages and cams.
§ To develop the ability to systematically design and optimizemechanisms to perform a specified task.
§ To increase the ability of students to effectively present written,oral, and graphical solutions to design problems.
§ To increase the ability of students to work cooperatively on teamsin the development of mechanism designs.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 6
Chapter 1Introduction
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Definitions
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The subject Theory of Machines may be defined as that branch ofEngineering-science, which deals with the study of relative motionbetween the various parts of a machine, and forces which act onthem. The knowledge of this subject is very essential for anengineer in designing the various parts of a machine.
Kinematics: The study of motion without regard to forces
More particularly, kinematics is the study of position, displacement, rotation, speed, velocity, and acceleration.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 8
Kinetics: The study of forces on systems in motion
A mechanism: is a device that transforms motion to some desirable patternand typically develops very low forces and transmits little power.
A machine: typically contains mechanisms that are designed to providesignificant forces and transmit significant power.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Application of Kinematics
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Any machine or device that moves contains one or more kinematic elements suchAs linkages, … gears…. belts and chains.
Bicycle is a simple example of a kinematic system that contains a chain drive to provide Torque.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 10
An Automobile contains many more examples of kin-systems…
the transmission is full of gears….
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 11
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 12
Chapter 2DEGREES OF FREEDOM (MOBILITY)
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 13
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.
• A mechanical system’s mobility (M) can be classified according to the number of degrees of freedom (DOF).
• DOF is defined with respect to a selected frame of reference (ground).
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 14
Ø Rigid body in a plane has 3 DOF: x,y,zØ Rigid body in 3D-space has 6 DOF, 3 translations & 3
rotations àthree lengths (x, y, z), plus three angles (θ, φ, ρ).
Ø The pencil in these examples represents a rigid body, or link.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 15
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.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 16
Excavator
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 17
Slider-Crank Mechanism
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 18
Links, joints, and kinematic chains
Linkage design:§ Linkages are the basic building blocks of all mechanisms§ All common forms of mechanisms (cams, gears, belts, chains)
are in fact variations on a common theme of linkages.• Linkages are made up of links and joints.
• Links: rigid member having nodes• Node: attachment points
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 19
1. Binary link: 2 nodes2. Ternary link: 3 nodes3. Quaternary link: 4 nodes
Joint: connection between two or more links (at theirnodes) which allows motion;
(Joints also called kinematic pairs)
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 20
Joint Classification
Joints can be classified in several ways:1.By the type of contact between the elements, line, point, or surface.2.By the number of degrees of freedom allowed at the joint.3.By the type of physical closure of the joint: either force or form closed.4.By the number of links joined (order of the joint).
A more useful means to classify joints (pairs) is by the number of degrees of freedom that they allow between the two elements joined.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 21
A joint with more than one freedom may also be a higher pair
• 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
Joint order = number of links-1
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lower pair to describe joints with surface contact
The six lower pairs
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 23
The half joint is also called a roll-slide jointbecause it allows both rolling and sliding
Form closed (closed by geometry) or Force closed
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 24
ü A joint (also called kinematic pair) is a connection between two ormore links at their nodes, which may allow motion between the links.
ü A lower pair is a joint with surface contact; a higher pair is a joint withpoint or line contact.
ü A full joint has one degree of freedom; a half joint has two degreesof freedom. Full joints are lower pairs; half-joints are higher pairs andallow both rotation and translation (roll-slide).
ü A form-closed joint is one in which the links are kept together form byits geometry; a force-closed joint requires some external force tokeep the links together.
ü Joint order is the number of links joined minus one (e.g. 1st ordermeans two links).
Terminology of Joints
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 25
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: any link or links that are fixed, nonmoving withrespect to the 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
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 26
Elements:0: Ground (Casing, Frame)1: Rocker2: Coupler3: Crank
crank mechanism
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 27
§ When studying mechanisms it is very helpful to establish a fixed reference frame by assigning one of the links as “ground”.
§ The motion of all other links are described with respect to the ground link.
§ For example, a fourbar mechanism often looks like a 3-bar mechanism, where the first “bar” is simply the ground link.
The “Ground” Link
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 28
Drawing kinematic Diagrams
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 29
Determining Degrees of Freedom
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 30
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
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 31
Determining DOF’s
• Gruebler’s equation for planar mechanisms
• 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= − −Kutzbach’s modification of Gruebler’s equation
M= 3L-2J-3G
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 32
The Cylindrical (cylindric) joint - two degrees of freedomIt permits both angular rotation and an independent sliding motion (C joint)
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 33
The Spherical (spheric) - Three degree of freedomIt permits rotational motion about all three axes, a ball-and-socket joint (S joint)
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 34
Example
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 35
Example
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Gruebler’s Equation
Gruebler’s Equation
DOF = mobilityL = number of linksJ = number of revolute joints or
prismatic jointsG = number of grounded links
DOF (M) = 3*L – 2* J – 3 *G= 3 (L-1) – 2 * J
L = 2J = 1G = 1
DOF = 1
Gruebler’s equation can be used to determine the mobility of planar mechanisms.
Link 13 DOF
Link 23 DOF
1 DOF
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Mobility of Vise Grip Pliers
L = 5J = 4 (revolute)J = 1 (screw)G = 1 (your hand)
DOF = 3*5 - 2*5 - 1*3 = 2
1
23
4
1
2
3
4
This example applies Gruebler’s equation to the determine the mobility of a vise grip plier.
5
Each revolute joint removes two DOF.The screw joint removes two DOF.
The mobility of the plier is two. Link 3 can be moved relative link1 when you squeeze your hand and the jaw opening is controlled by rotating link 5.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 38
Punch Press
Slider-Crank Mechanism
As designated in the figure, there are four links link 1, link 2, link 3 and link 4. Link 1 acts as a crank. Link 2 acts as connecting link, link 3 is the slider and link 4 is ground.
Joint Number Formed between links Joint type
1 Link 4 and Link 1 Revolute (or Pin)
2 Link 1 and Link 2 Revolute (or Pin)
3 Link 2 and Link 3 Revolute (or Pin)
4 Link 3 and Link 4Translatio
nal or (Slider)
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 39
Mechanisms and Structures
§ If DOF > 0, the assembly of links is a mechanism and will exhibit relative motion
§ If DOF = 0, the assembly of links is a structure and no motion is possible.
§ If DOF < 0,then the assembly is a preloaded structure, no motion is possible, and in general stresses are present.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 40
Paradoxes
• Greubler criterion does not include geometry, so it can give wrong prediction
• We must use inspection
E-quintetL=5J=6G=1M=3*5-2*6-3*1=0
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 41
Rolling cylinders even without slip (The joint between the two wheels can bepostulated to allow no slip, provided that sufficient friction is available) is anexample in which the ground link is exactly the same length as the sum of twoother links.If no slip occurs, then this is a one-freedom, or full, joint that allows only relative angular motion (Δθ) between the wheels.With that assumption, there are 3 links and 3 full joints,The equation predicts DOF = 0 (L=3,J1=3), but the mechanism has DOF = 1.
Others paradoxes exist, so the designermust not apply the equation blindly.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 42
Chapter 3Linkage
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
History
• Leonardo da Vinci (1452, 1519), Codex Madrid I. • Industrial Revolution was the boom age of linkages: cloth
making, power conversion, speed regulation, mechanical computation, typewriting and machining
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Linkages Today
§ In many applications linkages have been replaced by electronics.
§ Still linkages can have a cost advantage over electronic solutions: Couple different outputs by a mechanism rather than using one motor per output and electronics to achieve the coupling.
§ Current applications: Sports Equipment, Automotive (HVAC modules), Precision Machinery (Compliant Mechanisms), Medical Devices
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Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 45
Mechanical linkages are usually designed to transform a given input force and movement into a desired output force and movement.
Transmission System
Gear Linkage
consistent translationàlinear transfer function
Inconsistent translationnon-linear transfer function
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 46
transfer function
consistent translationàlinear transfer function
Gearbox
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 47
crank drive = Linkage Inconsistent translationnon-linear transfer function
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 48
The pushing movement of the piston (crank mechanism) is transferred into a swinging movement of the shovel.
Bagger
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 49
Fourbar MechanismØTwobar has -1 degrees of freedom
(preloads structure)ØThreebar has 0 degrees of freedom
(structure)ØFourbar has 1 degree of freedom§ The fourbar linkage is the simplest
possible pin-jointed mechanism for single degree of freedom controlled motion
One link is grounded in each case
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 50
§ The fourbar linkage is the simplest possible pin-jointed mechanism for controlled motion with one degree of freedom.
§ Changing the relative lengths of the links can create a wide variety of motions.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 51
4-Bar Nomenclature
• Ground Link• Links pivoted to ground:
– Crank– Rocker
• Coupler
Ground Link
Coupler
Link 1, length d
Pivot 02 Pivot 04
A
B
CrankRocker
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 52
Linkages of more than 4 bars
• Provide more complex motion• See Watt’s sixbar and Stephenson’s sixbar mechanisms in the textbook
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 53
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
Ø If S+L ≤ P+Q the linkage is Grashof :at least one link is capable of making a complete revolutionØOtherwise the linkage is non-Grashof : no link is
capable of making a complete revolution
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 54
I. If S + L < P + Q (Class I), the linkage is Grashof and at least one link will be capable of making a full revolution with respect to ground.
II. If S + L > P + Q (Class II), the linkage is non-Grashof and no link will be capable of making a full revolution with respect to any other link.
III. If S + L = P + Q (Class III), the linkage is special-case Grashof and although at least one link will be capable of making a full revolution.
Grashof-Type Rotatability Criteria for Higher-Order Linkages
Rotatability is defined as the ability of at least one link in a kinematic chain to make a full revolution with respect to the other links and defines the chain as Class I, II or III.
Revolvability refers to a specific link in a chain and indicates that it is one of the links that can rotate.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 55
Crank-Slider
The crank-slider (right) is a transformation of the fourbar crankrocker, by replacing the revolute joint at the rocker pivot by ajoint, maintaining the same one degree of freedom.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 56
Cam Follower
§ A cam follower is a mechanism that appears to have only two moving links (apart from ground), but it has 1 DOF.
§ It has a fourbar equivalent if the coupler (Link 3) is viewed as a link of variable length.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 57
Practical Considerations
Pin Joints versus Sliders and Half Joints
A. Pin JointØ Easy to lubricate ( with hydrodynamic lubrication)Ø Can use relatively inexpensive bearings
B. SliderØ Requires carefully machined straight slot or rodØ Custom made bearingsØ Lubrication is difficult to maintain
§ There are many factors that need to be considered to create good-quality designs.
§ The choice of joint type can have a significant effect on the ability to provide good, clean lubrication over the lifetime of the machine.
pin joint is the clear winner
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 58
Sleeve or journal bearing, the geometry of pin-in-hole traps a lubricant film within its annular interface by capillary action and promotes a condition called hydrodynamic lubrication in which the parts are separated by a thin film of lubricant .
Seals can easily be provided at the ends of the hole, wrapped around the pin, to prevent loss of the lubricant.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 59
Relatively inexpensive ball and roller bearings are commercially available in a large variety of sizes for revolute joints.
Their rolling elements provide low-friction operation and good dimensional control.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 60
For revolute joints pivoted to ground, several commercially available bearing types, Pillow blocks and flange-mount bearings.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 61
MOTORS AND DRIVERS§ Unless manually operated, a mechanism will require some type of
driver device to provide the input motion and energy.
§ A motor is the logical choice to create the input.
§ Motors come in a wide variety of types. The most common energy source for a motor is electricity, but compressed air and pressurized hydraulic fluid are also used to power air and hydraulic motors.
Electrical MotorsØ ACØ DCØ ServoØ Stepping
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 62
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 63
Chapter 4Design of Linkage Systems
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 64
Engineering Design involves
1. Synthesis
2. Analysis
Design a mechanism to obtain a specified motion or force.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 65
Mechanism Synthesis
•Type Synthesis given the required performance, what type of mechanism is suitable? Linkages, gears, cam and follower, belt and pulley and chain and sprocket.
•Number Synthesis How many links should the mechanism have? How many degrees of freedom are desired?
deals with determining the length of all links, gear diameter, cam profile.
•Dimensional Synthesis
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 66
QUALITATIVE SYNTHESIS
• The creation of potential solutions in the absence of awell-defined algorithm which configures or predicts thesolution and also judge its quality.
• Several tools and techniques exist to assist you in thisprocess. The traditional tool is the drafting board, onwhich you layout, to scale, multiple orthographic viewsof the design, and investigate its motions by drawingarcs, showing multiple positions, and usingtransparent, movable overlays.
• Commercially available programs such as SolidWorkand Working Model allow rapid analysis of a proposedmechanical design. The process then becomes one ofqualitative design by successive analysis which isreally an iteration between synthesis and analysis.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 67
TYPE SYNTHESIS
• The definition of the proper type of mechanism bestsuited to the problem and is a form of qualitativesynthesis.
• This is perhaps the most difficult task for the student asit requires some experience and knowledge of thevarious types of mechanisms which exist and whichalso may be feasible from a performance andmanufacturing standpoint.
• An engineer can do, with one dollar, what any fool cando for ten dollars. Cost is always an importantconstraint in engineering design.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 68
DIMENSIONAL SYNTHESIS
• The determination of the proportions (lengths) of thelinks necessary to accomplish the desired motions andcan be a form of quantitative synthesis if an algorithmis defined for the particular problem, but can also be aform of qualitative synthesis if there are more variablesthan equations.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 69
MECHANISM SYNTHESIS: TWO APPROACHES
CAD program à SolidWorks
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 70
LIMITING CONDITIONS
• Once a potential solution is found, it must beevaluated for its quality. There are many criteria whichmay be applied. However, one does not want toexpend a great deal of time analyzing, in great detail,a design which can be shown to be inadequate bysome simple and quick evaluations.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 71
TOGGLE: One important test consist in to check that the linkage can infact reach all of the specified design positions without encountering alimit or toggle position, also called a stationary configuration.
The toggle positions are determined by the colinearity of two of the moving links.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 72
o TRANSMISSION ANGLE: The transmission angle μ isdefined as the angle between the output link and thecoupler.
o It is usually taken as the absolute value of the acute angle ofthe pair of angles at the intersection of the two links.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 73
o It is a measure of the quality of force transmission atthe joint.
§ Radial component only increases friction at pivot O4.
§ Tangential (normal to Link 4) produces torque.– μ = 90o is optimal.– In design, keep μ > 40o
To promote smooth running and good force transmission.
Ideally, as close to 90° as possible
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Position analysis for Crank-Rocker mechanism
• The calculation of out-put angle
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
4.5 ALGEBRAIC POSITION ANALYSIS OF LINKAGES -Additional
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
ALGEBRAIC POSITION ANALYSIS OF LINKAGES
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
4.5 ALGEBRAIC POSITION ANALYSIS OF LINKAGES -Additional
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
4.5 ALGEBRAIC POSITION ANALYSIS OF LINKAGES -Additional
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
4.5 ALGEBRAIC POSITION ANALYSIS OF LINKAGES -Additional
Excel or other program
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
O2
O44. Select two fixed pivot points, O2and O4, anywhere on the two midnormals.
Graphical Synthesis –Motion Generation Mechanism
Two positions, coupler as the output
A1A2
B1
B2
1. Draw the link AB in its two desired positions, A1B1 and A2B2
5. Measure the length of all links,
O2A = link 2, AB = link 3,
O4B = link 4 and O2 O4 = link 1
2. Connect A1 to A2 and B1 to B2.
3. Draw two lines perpendicular to A1 A2 and B1B2 at the midpoint (midnormals).
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
O4O2
Graphical Synthesis – Motion Generation MechanismThree positions, coupler as the output
A1
A2
A3
B1
B2
B3
Same procedure as for two positions.
1. Draw the link AB in three desired positions.
2. Draw the midnormals to A1A2 and A2A3, the intersection locates the fixed pivot point O2. Same for point B to obtain second pivot point O4.
3. Check the accuracy of the mechanism, Grashof condition and the transmission angle.
4. Change the second position of link AB to vary the locations of the fixed points
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Graphical Synthesis –Motion Generation Mechanism
Two positions Grashof 4-Bar mechanism with rocker as the output
D1
C1C2
D2O2
5. Connect B1 to B2 and extend. Select any location on this line for fixed pivot point O2.
O2A = B1B2 / 2
7. Measure the length of all links, O2A = link 2, AB = link 3, O4CD = link 4 and O2 O4 = link 1
1. Draw the link CD in its two desired positions, C1D1 and C2D2
2. Connect C1 to C2 and D1 to D2 and draw two midnormals to C1C2 and D1D2
O4
3. The intersection of the two midnormals is the fixed pivot point O4.
B1 B2
4. Select point B1 anywhere on link O4C1 and locate B2 so O4B1= O4B2
A2
6. Draw a circle with radius B1 B2 / 2, point A is the intersection of the circle with the B1 B2 extension.
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
DIMENSIONAL SYNTHESIS - Solution
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 84
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Coupler Curves
85
q A coupler in a linkage in general has complex motion and provides the greatest variety of paths that can be traced.
q The Hrones and Nelson Atlas of Fourbar Coupler Curves is a good reference
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Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 87
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 88
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 89
Chapter 5Velocity Analysis
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Velocity
90
Definitions
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 91
θiPA peR =
r
Velocity of a point
Link in pure rotation
RPA as a complex number in polar formP is the scalar lengthJ is the complex operator (constant)
Position of Point P
Velocity of Point P
θθ ωθ jj
PAPPA
jepdtdpje
RVV
==
== &rrr
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 92
[ ]cos sinire r iθ θ θ= +
[ ]sin cosiire r iθ θ θ= − +
cosr θsinr θθ
r
Real
Imaginary
cosr θ
sinr θ
Vector r can be written as:
Multiplying by i gives:
Multiplying by i rotates a vector 90°
Euler's formula
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012 93
If point A is moving (Relative Velocity)
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Velocity Analysis of a 4-Bar Linkage
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Given ω2. Find ω3 and ω4
Dr. Anwar Abu-Zarifa . Islamic University of Gaza . Department of Mechanical Engineering . © 2012
Analytical Velocity Analysis of Fourbar Linkage
Numerical Example