Theory of Machines. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

Transcription

1 Theory of Machines Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

2 Syllabus and Course Outline Faculty of Engineering Department of Mechanical Engineering EMEC 3302, Theory of Machines Instructor: Dr. Anwar Abu-Zarifa Office: IT Building, Room: I413 Tel: Website: Office Hrs: see my website SAT 09:30 11:00 Q412 MON 09:30 11:00 Q412 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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

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 of machines. The Theory of Machines and Mechanisms provides the foundation for the study of displacements, velocities, accelerations, and static and dynamic 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

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 of mechanisms such as linkages and cams. To develop the ability to systematically design and optimize mechanisms to perform aspecified task. Toincrease the ability of students to effectively present written, oral, and graphical solutions to design problems. Toincrease the ability of students to work cooperatively on teams in the development of mechanism designs. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

6 Chapter 1 Introduction Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

7 Definitions The subject Theory of Machines may be defined as that branch of Engineering-science, which deals with the study of relative motion between the various parts of amachine, and forces which act on them. The knowledge of this subject is very essential for an engineer in designing the various parts of amachine. 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

8 Kinetics: The study of forces on systems in motion Amechanism: is adevice that transforms motion to some desirable pattern and typically develops very low forces and transmits little power. A machine: typically contains mechanisms that are designed to provide significant forces and transmit significant power. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

9 Application of Kinematics Any machine or device that moves contains one or more kinematic elements such As 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

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

11 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

12 Chapter 2 DEGREES OF FREEDOM (MOBILITY) Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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

14 Rigid body in a plane has 3 DOF: x,y,z Rigid body in 3D-space has 6 DOF, 3 translations & 3 rotations three lengths (x, y, z), plus three angles (θ, φ, ρ). The pencil in these examples represents a rigid body, or link. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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

16 Excavator Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

17 Slider-Crank Mechanism Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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

19 1. Binary link: 2 nodes 2. Ternary link: 3 nodes 3. Quaternary link: 4 nodes Joint: connection between two or more links (at their nodes) which allows motion; (Joints also called kinematic pairs) Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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

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 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

22 lower pair to describe joints with surface contact The six lower pairs Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

23 The half joint is also called a roll-slide joint because 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

24 Terminology of Joints A joint (also called kinematic pair) is aconnection between two or more links at their nodes, which may allow motion between the links. A lower pair is a joint with surface contact; a higher pair is a joint with point or line contact. A full joint has one degree of freedom; a half joint has two degrees of freedom. Full joints are lower pairs; half-joints are higher pairs and allow both rotation and translation (roll-slide). A form-closed joint is one in which the links are kept together form by its geometry; a force-closed joint requires some external force to keep the links together. Joint order is the number of links joined minus one (e.g. 1 st order means two links). Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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 with respect 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

26 crank mechanism Elements: 0: Ground (Casing, Frame) 1: Rocker 2: Coupler 3: Crank Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

27 The Ground Link 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. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

28 Drawing kinematic Diagrams Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

29 Determining Degrees of Freedom Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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-3G Where: L: number of links J: number of full joints G: number of grounded links Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

31 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 2J ( ) Kutzbach s modification of Gruebler s equation Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

32 The Cylindrical (cylindric) joint - two degrees of freedom It permits both angular rotation and an independent sliding motion (C joint) Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

33 The Spherical (spheric) - Three degree of freedom It 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

34 Example Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

35 Example Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

36 Gruebler s Equation Gruebler s equation can be used to determine the mobility of planar mechanisms. Gruebler sequation Link 1 3 DOF L = 2 J = 1 G = 1 DOF = 1 DOF L J G = mobility = number of links = number of revolute joints or prismatic joints = number of grounded links 1 DOF DOF (M) = 3*L 2* J 3 *G = 3 (L-1) 2 * J Link 2 3 DOF Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering. 2012

37 Mobility of Vise Grip Pliers This example applies Gruebler s equation to the determine the mobility of a vise grip plier Each revolute joint removes two DOF. The screw joint removes two DOF. L = 5 J = 4 (revolute) J = 1 (screw) G = 1 (your hand) DOF = 3*5-2*5-1*3 = 2 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 1 Link 4 and Link 1 2 Link 1 and Link 2 3 Link 2 and Link 3 4 Link 3 and Link 4 Joint type Revolute (or Pin) Revolute (or Pin) Revolute (or Pin) Translatio nal or (Slider) Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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

40 Paradoxes Greubler criterion does not include geometry, so it can give wrong prediction We must use inspection L=5 J=6 G=1 M=3*5-2*6-3*1=0 E-quintet Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

41 Rolling cylinders even without slip (The joint between the two wheels can be postulated to allow no slip, provided that sufficient friction is available) is an example in which the ground link is exactly the same length as the sum of two otherlinks. 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, Theequationpredicts DOF =0(L=3, J1=3),butthemechanismhas DOF =1. Others paradoxes exist, so the designer must not apply the equation blindly. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

42 Chapter 3 Linkage Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

43 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

44 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 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

45 Mechanical linkages are usually designed to transform a given input force and movement into a desired output force and movement. Transmission System Gear consistent translation linear transfer function Linkage Inconsistent translation non-linear transfer function Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

46 Gearbox transfer function consistent translation linear transfer function Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

47 crank drive = Linkage Inconsistent translation non-linear transfer function Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

48 Bagger The pushing movement of the piston (crank mechanism) is transferred into a swinging movement of the shovel. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

49 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 One link is grounded in each case Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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

51 4-Bar Nomenclature Ground Link Links pivoted to ground: Crank Rocker Coupler A Crank Coupler Link 1, length d Ground Link Pivot 0 2 Pivot 0 4 B Rocker Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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

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 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 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

54 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. 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. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

55 Crank-Slider The crank-slider (right) is a transformation of the fourbar crank rocker, by replacing the revolute joint at the rocker pivot by a joint, maintaining the same one degree of freedom. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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

57 Practical Considerations There are many factors that need to be considered to create goodquality 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 Joints versus Sliders and Half Joints A. Pin Joint Easy to lubricate ( with hydrodynamic lubrication) Can use relatively inexpensive bearings B. Slider Requires carefully machined straight slot or rod Custom made bearings Lubrication is difficult to maintain pin joint is the clear winner Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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

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

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

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 Motors AC DC Servo Stepping Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

62 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

63 Chapter 4 Design of Linkage Systems Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

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

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? Dimensional Synthesis deals with determining the length of all links, gear diameter, cam profile. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

66 QUALITATIVE SYNTHESIS The creation of potential solutions in the absence of a well-defined algorithm which configures or predicts the solution and also judge its quality. Several tools and techniques exist to assist you in this process. The traditional tool is the drafting board, on which you layout, to scale, multiple orthographic views of the design, and investigate its motions by drawing arcs, showing multiple positions, and using transparent, movable overlays. Commercially available programs such as SolidWork and Working Model allow rapid analysis of aproposed mechanical design. The process then becomes one of qualitative design by successive analysis which is really an iteration between synthesis and analysis. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

67 TYPE SYNTHESIS The definition of the proper type of mechanism best suited to the problem and is a form of qualitative synthesis. This is perhaps the most difficult task for the student as it requires some experience and knowledge of the various types of mechanisms which exist and which also may be feasible from a performance and manufacturing standpoint. An engineer can do, with one dollar, what any fool can do for ten dollars. Cost is always an important constraint in engineering design. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

68 DIMENSIONAL SYNTHESIS The determination of the proportions (lengths) of the links necessary to accomplish the desired motions and can be aform of quantitative synthesis if an algorithm is defined for the particular problem, but can also be a form of qualitative synthesis if there are more variables than equations. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

69 MECHANISM SYNTHESIS: TWO APPROACHES CAD program SolidWorks Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

70 LIMITING CONDITIONS Once a potential solution is found, it must be evaluated for its quality. There are many criteria which may be applied. However, one does not want to expend agreat deal of time analyzing, in great detail, a design which can be shown to be inadequate by some simple and quick evaluations. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

71 TOGGLE: One important test consist in to check that the linkage can in fact reach all of the specified design positions without encountering a limit 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

72 o o TRANSMISSION ANGLE: The transmission angle μ is defined as the angle between the output link and the coupler. It is usually taken as the absolute value of the acute angle of the pair of angles at the intersection of the two links. Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

73 o It is ameasure of the quality of force transmission at the 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

74 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

75 4.5 ALGEBRAIC POSITION ANALYSIS OF LINKAGES -Additional Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering. 2012

76 ALGEBRAIC POSITION ANALYSIS OF LINKAGES Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering. 2012

77 4.5 ALGEBRAIC POSITION ANALYSIS OF LINKAGES -Additional Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering. 2012

78 4.5 ALGEBRAIC POSITION ANALYSIS OF LINKAGES -Additional Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering. 2012

79 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

80 Graphical Synthesis Motion Generation Mechanism Two positions, coupler as the output 1. Draw the link AB in its two desired positions, A 1 B 1 anda 2 B 2 2. Connect A 1 to A 2 and B 1 to B 2. A 1 A 2 B 1 3. Draw two lines perpendicular to A 1 A 2 and B 1 B 2 at the midpoint (midnormals). O 2 B 2 4. Select two fixed pivot points, O 2 and O 4, anywhere on the two midnormals. O 4 5. Measure the length of all links, O 2 A = link 2, AB = link 3, O 4 B = link 4 and O 2 O 4 = link 1 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering. 2012

81 Graphical Synthesis Motion Generation Mechanism Three positions, coupler as the output Same procedure as for two positions. 1. Draw the link AB in three desired positions. A 1 A 2 B 1 2. Draw the midnormals to A 1 A 2 and A 2 A 3, the intersection locates the fixed pivot point O 2. Same for point B to obtain second pivot point O Check the accuracy of the mechanism, Grashof condition and the transmission angle. O 2 B 2 A 3 O 4 4. Change the second position of link AB to vary the locations of the fixed points B 3 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering. 2012

82 Graphical Synthesis Motion Generation Mechanism Two positions Grashof 4-Bar mechanism with rocker as the output 1. Draw the link CD in its two desired positions, C 1 D 1 and C 2 D 2 2. Connect C 1 to C 2 and D 1 to D 2 and draw two midnormals to C 1 C 2 and D 1 D 2 3. The intersection of the two midnormals is the fixed pivot point O Select point B 1 anywhere on link O 4 C 1 and locate B 2 so O 4 B 1 = O 4 B 2 5. Connect B 1 to B 2 and extend. Select any location on this line for fixed pivot point O Draw a circle with radius B 1 B 2 / 2, point A is the intersection of the circle with the B 1 B 2 extension. O 2 A 2 C 1 C 2 B 1 B 2 O 4 D 1 D 2 O 2 A = B 1 B 2 / 2 7. Measure the length of all links, O 2 A = link 2, AB = link 3, O 4 CD = link 4 and O 2 O 4 = link 1 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering. 2012

83 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

85 Coupler Curves 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 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

86 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

87 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

88 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

89 Chapter 5 Velocity Analysis Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

90 Velocity Definitions Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

91 Velocity of a point Position of Point P r R = PA pe iθ Velocity of Point P r V PA r = V = P pje = jθ &r R PA dθ dt = pωje jθ R PA as a complex number in polar form P is the scalar length J is the complex operator (constant) Link in pure rotation Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

92 Imaginary Vector r can be written as: [ cosθ sinθ] iθ re = r + i Multiplying by i gives: [ sinθ cosθ] iθ ire = r + i Multiplying by i rotates a vector 90 r sinθ r cosθ r Euler's formula r r sinθ θ cosθ Real Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

93 If point A is moving (Relative Velocity) Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

94 Velocity Analysis of a 4-Bar Linkage Given ω 2. Find ω 3 and ω 4 Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering

95 Analytical Velocity Analysis of Fourbar Linkage Numerical Example Dr. Anwar Abu-Zarifa. Islamic University of Gaza. Department of Mechanical Engineering. 2012