Taibah University Mechanical Engineering

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2 Chapter 2 Kinematics Fundamentals 1. Introduction 2. Degrees of Freedom 3. Types of Motion 4. Links, Joints, and Kinematic Chains 5. Determining Degree of Freedom Degree of Freedom in Planar Mechanisms Degree of Freedom in Spatial Mechanisms 6. Mechanisms and Structures 7. Number Synthesis 8. Paradoxes 9. Isomers 10. Linkage Transformation 11. Intermittent Motion 12. Inversion 13. The Grashof Condition Classification of the Four-bar Linkage 14. Linkages of More Than Four Bars Geared Five-bar Linkages Six-bar Linkages Grashof-type Rotatability Criteria for Higher-order Linkages 15. Springs as Links 16. Practical Considerations Pin Joints versus Sliders and Half Joints Cantilever versus Straddle Mount Short Links Bearing Ratio Linkages versus Cams 17. Motor and Drives Electric Motors Air and Hydraulic Motors Air and Hydraulic Cylinders Solenoids

3 Mechanism-Machine-Structure Mechanism: Is an assemblage of many connected rigid bodies, formed and connected so that they move with definite relative motions with respect to one another Its function is to transform/transfer motion/load. The motion of one part controls those of the remaining parts. The main idea is to achieve a desired motion. Machine: An assemblage of parts that transmit forces, motion and energy in a predetermined manner Its function is to transform energy into work. Terms: Force, torque, work, and power. Structure: Combination of resisting bodies, connected by joints, but its purpose is not to do work or to transform motion.

2.1 Degree of Freedom or Mobility Degrees of Freedom (DOF):the number of independent parameters that are needed to uniquely define its position in space at any instant of

4 2.1 Degree of Freedom or Mobility Degrees of Freedom (DOF):the number of independent parameters that are needed to uniquely define its position in space at any instant of time (w.r.tframe of reference) Rigid body in a plane has 3 DOF. (x, y, θ) Rigid body in space has 6 DOF(3 translation, 3 rotation) Try to identify these DOF of an object in space.

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

6 2.2 Types of Motion Pure Translation: All points on the body describe parallel (curvilinear or rectilinear) paths.

7 2.2 Types of Motion Complex Motion:A simultaneous combination of rotation and translation Translation and rotation represent independent motions of the body. Each can exist without the other.

8 2.3 Links Link: It is a rigid body with at least two nodes. Links are the building blocks of all mechanisms. Node:A point of attachment to another link. Binary link two nodes Ternary link three nodes Quaternary link four nodes

9 Link Classification Ground: fixed body w.r.t. frame of reference Crank: pivoted to ground, makes complete revolution Rocker: pivoted to ground, has oscillatory motion Coupler: link has complex motion, not attached to ground

10 Coupler?

11 Joints Joint:A connection between two or more links (at their nodes), which allows some motion, or potential motion, between the connected links. Joints: are called kinematic pairs can be classified as: By the type of contact between the elements, line, point, or surface. By the number of degrees of freedom allowed at the joint. By the type of physical closure of the joint: either force or form closed. By the number of links joined(order of the joint).

12 1. By Type of Contact Reuleaux s Classification Lower Pair:Surface Contact, Pin surrounded by a hole, block on a surface (slider), Revolute (R), Prismatic (P), Cylindrical (C), Screw (H), Higher Pair: Line and Point Contact Gear joint, Cam etc

13 Six Types of Lower Pair

14 2. Joint Nomenclature Full joint, lower pairs-single degree of freedom Half joint, higher pair-two degrees of freedom

15 2. Number of DOF Full: 1-DOF or lower pair Rotating revolute (R) Translating Prismatic (P) Threaded nut Half: 2 DOF, Higher Pair Paradoxically called as Half Joint Roll slide joint Slippage Occurs

16 Ex. Joint Classification: Identify Full Joint & Half Joint

17 3. Formed and Forced

18 4. Joint Order Joint order = number of links-1

19 Kinematic chains, Mechanisms, Machines Kinematic Chain: An assemblage of links and joints, interconnected in a way to provide a controlled output motion in response to a supplied input motion. Mechanism: A kinematic chain in which at least one link has been "grounded," or attached, to the frame of reference (which itself may be in motion). Machine: A combination of resistant bodies arranged to compel the mechanical forces of nature to do work accompanied by determinate motions.

20 Kinematics Chain Links joined together for motion

21 Closed Chain If every link in the chain is connected to two or more links then the chain form one or more closed loops If the link form a closed loops, it is called closed kinematic chain If NOT, the chain is said to be open kinematics chain

22 Link Classification Links are classified as follows: Ground fixed w.r.t. reference frame Crank pivoted to ground, makes complete revolution Rocker pivoted to ground, has oscillatory motion Coupler -link has complex motion, not attached to ground

23 Planer and Spatial Mechanisms In a planar mechanisms, all of the relative motions of the rigid bodies are in one plane or in parallel planes Motion of such mechanism is called Coplanar If there is any relative motion that is not in the same plane or in parallel planes, the mechanism is called the spatial mechanism.

24 2.4 Determining Degree of Freedom The definition of the degrees of freedom of a mechanism is: The number of degrees of freedom of a mechanism is also called the mobility of the device

25 Planar Mechanisms

26 Degrees of Freedom Cont. 3 Degrees of Freedom (x, y, θ) 1 Degree of Freedom (θ) lower pair subtracts 2 degrees of freedom 1 Degree of Freedom (x) lower pair subtracts 2 degrees of freedom 2 Degrees of Freedom (x, θ) sliding contact subtracts 1 degree of freedom 0 Degrees of Freedom fixed to ground subtracts 3 degrees of freedom

27 Degrees of Freedom Cont. Different values of DOF mean different things. DOF = 0 Structure DOF < 0 Statically indeterminate structure DOF 1 Mobility is allowed

28 2.4 Determining Degree of Freedom For simple mechanisms calculating DOF is simple Open Mechanism DOF=3 Closed Mechanism DOF=1

29 Determining Degree of Freedom Gruebler s equation for planar mechanisms M =3L-2J-3G Where M =degree of freedom or mobility L = number of links J = number of joints (half joints count as 0.5) G = number of grounded links =1 This equation can be modified to be M =3(L-1)-2J

30 Determining Degree of Freedom Kutzbach s equation for planar mechanisms M = 3(L-1)-2J 1 -J 2 Where M = degree of freedom or mobility L = number of links J 1 = number of full joints 2 J = number of half joints For Spatial Mechanisms M = 6(L-1) -5J 1-4J 2-3J 3-2J 4 -J 5

31 Determining DOF (Examples) M=3(L-1)-2J 1 -J 2 Open Mechanism Closed Mechanism

32 Determining DOF(Examples) M=3(L-1)-2J 1 -J 2 L = 8 J = 10 DOF = 1

33 Determining DOF (Examples) M=3(L-1)-2J 1 -J 2 L = 6 J = 7.5 DOF DOF = 0

34 2.5 Mechanisms and Structures Mechanism : DOF > 0 Structure : DOF =0 Preloaded Structure: DOF<0, may require force to assemble statically indeterminate

35 Quiz: Count number of links?

36 Spring as Link

37 2.6 Number Synthesis Reading Assignment Determination of the number and order of links and joints necessary to produce motion of a particular DOF Link Order: here means # of node per link i.e. binary, ternary, quaternary Book gives details

38 Number Synthesis Book gives details Total Links Binary Ternary Quaternary Pentagonal Hexagonal

39 2.7 Paradoxes Greublercriterion does not include geometry, so it can give wrong prediction Agree Greubler Equation DOF = 0 Usually when things are the same Gears: DOF = 1 Disagree Greubler Equation DOF = 1

40 2.8 Isomers A Greek word means having equal parts Refers to valid ways to assemble different types of links Only one valid fourbar isomer Two valid sixbar isomers Third one fails DOF test, as the DOF is not distributed over the linkage.

41 Fourbar Isomer Only way to construct a fourbar isomer is to have one binary link next to another binary link.

42 Watt s Sixbar Isomer One way to construct a sixbar isomer is to have the two ternary links attached.

43 Stephenson s Sixbar Isomer One way to construct a sixbar isomer is to have the two ternary links separated.

44 Invalid Sixbar Isomer This is an invalid isomer as the DOF is not distributed through the mechanism

45 Invalid Sixbar Isomer This is an invalid isomer as the DOF is not distributed through the mechanism This is a structure Effective link

46 2.10 Intermittent Motion It is a series of Motionsand Dwells Dwell: a period of time with no output motion while there is an input motion Examples: Geneva Mechanism, Linear Geneva Mechanism, Ratchet and Pawl

47 Geneva Mechanism

48 Linear Geneva Mechanism

49 Linear Geneva Mechanism

50 Ratchet and Pawl

51 2.11 Inversion Reading Assignment Created by grounding a different link in a kinematic chain Different behavior for different inversions

52 Three (3) Stephenson 6-bar inversions

53 Two (2) Watt s 6-bar inversions

54 2.12 Grashof Condition Fourbar linkage is simplest linkage with 1DOF

55 2.12 Grashof Condition Fourbar linkage is simplest linkage with 1-DOF Grashof condition predicts behavior of linkage based only on links length S = length of shortest link L = length of longest link P,Q=length of other two links If S + L P + Qthe linkage is Grashof with at least one link capable of making a complete rotation Otherwise the linkage is non-grashof with no link capable of making a complete rotation relative to ground

56 For case of S +L <P + Q Ground link adjacent to shortest => crank-rocker Ground shortest link => double crank Ground link opposite shortest link Grashof double rocker with shortest link capable of making a complete rotation

57 For the case of S +L >P +Q All inversions will be double rockers

58 For the case of S +L =P +Q According to the book, all inversions will be double cranks or crank rockers (true if S =P, L =Q) Indeterminate point when links are aligned (change points) Parallelogram form Deltoid form Anti parallelogram form

59 Barker s Complete Classification Type s+l vs p+q Inversion Class Barker s Designation Code Also Known as 1 < L 1 =s=ground I-1 Grashof crank-crank-crank GCCC double-crank 2 < L 2 =s=input I-2 Grashof crank-rocker-rocker GCRR crank-rocker 3 < L 3 =s=coupler I-3 Grashof rocker-crank-rocker GRCR double-rocker 4 < L 4 =s=output I-4 Grashof rocker-rocker-crank GRRC rocker-crank 5 > L 1 =l=ground II-1 Class 1 rocker-rocker-rocker RRR1 Triple-rocker 6 > L 2 =l= input II-2 Class 2 rocker-rocker-rocker RRR2 Triple-rocker 7 > L 3 =l= coupler II-3 Class 3 rocker-rocker-rocker RRR3 Triple-rocker 8 > L 4 =l= output II-4 Class 4 rocker-rocker-rocker RRR4 Triple-rocker 9 = L 1 =s=ground III-1 Change point crank-crank-crank SCCC SC double-crank 10 = L 2 =s=input III-2 Change point crank-rocker-rocker SCRR SC crank-rocker 11 = L 3 =s=coupler III-3 Change point rocker-crank-rocker SRCR SC double-rocker 12 = L 4 =s=output III-4 Change point rocker-rocker-crank SRRC SC rocker-crank Parallelogram or 13 = Two equal pairs III-5 Double change point S2X deltoid 14 = Taibah L University 1 =L 2 =L 3 =L 4 III-6 Triple change point S3X Square

60 2.13 Linkages of more than 4 bars 5-bar 2DOF Geared 5-bar 1DOF Provides for more complex motion Watt s sixbar 2 fourbar linkages in series Stephenson s sixbar 2 fourbar linkages in parallel

61 2.14 Springs as links Spring can act as a link to counteract static loads. Springs remove a degree of freedom (1 more equation) if they provided the right amount of force. Examples: car hood, desk arm lamp, garage door

62 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

63 2.17 Practical Consideration Revolute (Pin) Joints: Easy to build perfect pin joint.

64 Bearings

65 2.17 Practical Consideration

66 Eccentric crank

67 2.18 Motors and Drives Unless manually operated, a mechanism will require some type of driver device to provide the input motion and energy. Grashof linkage, slider-crank, and cam-follower, requires a continuous rotary input motion. Electric motors and gasoline/diesel engines are widely used to produce rotary input motion. Compressed air and pressurized hydraulic fluid are also used to power air and hydraulic motors. If the input motion is translation, as is common in earthmoving equipment, then a hydraulic or pneumatic cylinder is usually needed.

68 Electric Motors: The main electrical configuration division is: AC and DC. ACand DCrefer to alternating current and direct current The universal motor is designed to run on either ACor DC. Functional classificationsof electric motors: are gearmotors, servomotors, and stepping motors.

69 Electric Motors: DC MOTORS: made in different electrical configurations, such as permanent magnet (PM), shunt-wound, series-wound, and compound-wound. The names refer to the manner in which the rotating armature coils are electrically connected to the stationary field coils: -in parallel (shunt), in series, or in combined series-parallel (compound). Permanent magnets replace the field coils in a PM motor. Each configuration provides different torque-speed characteristics. The torque-speed curve of a motor describes how it will respond to an applied load. The torque-speed curve predicts how the mechanical-electrical system will behave when the load varies dynamically with time.

71 AC MOTORS AC MOTORS: least expensive and have variety of torque-speed curves to suit various load applications. Limited to a few standard speeds that are a function of the AC line frequency (60 Hz in North America, 50 Hz elsewhere). The synchronous motor speed n s is a function of line frequency f and the number of magnetic poles p present in the rotor. n s =120f/p Synchronous motors "lock on" to the AC line frequency and run exactly at synchronous speed. These motors are used for clocks and timers. Non-synchronous AC motors have a small amount of slip which makes them lag the line frequency by about 3 to 10%.

73 AC MOTORS The most common AC motors have 4 poles, giving nonsynchronous no-load speeds of about 1725 rpm, which reflects slippage from the 60-Hz synchronous speed of 1800 rpm. The single-phase shaded pole and permanent split capacitor designs have a starting torque lower than their full-load torque. To boost the start torque, the split-phase and capacitor-start designs employ a separate starting circuit that is cut off by a centrifugal switch as the motor approaches operating speed.

74 Air and Hydraulic Motors Air and Hydraulic motors have more limited application than electric motors. They require the availability of a compressed air or hydraulic source. Both are less energy efficient than the direct electrical to mechanical conversion of electric motors.

75 Air and Hydraulic Motors Air motors widely used in factories and shops. A common example is the air impact wrench used in automotive repair shops. Although individual air motors and air cylinders are relatively inexpensive, these pneumatic systems are quite expensive when the cost of all the ancillary equipment is included.

76 Air Motors Air motors

77 Air and Hydraulic Motors Hydraulic motors are most often found within machines or systems such as construction equipment (cranes), aircraft, and ships, where high-pressure hydraulic fluid is provided for many purposes. Hydraulic systems are very expensive when the cost of all the ancillary equipment is included. These are linear actuators (piston in cylinder) which provide a limited stroke, straight-linemotion output from a pressurized fluid flow input of either compressed air or hydraulic fluid.

78 Air and Hydraulic Cylinders Both have high cost, low efficiency, problem in control, and complication factors as listed under their air and hydraulic motor equivalents above. A linear actuator, when subjected to a constant pressure fluid source, typical of most compressors, will respond with more nearly constant acceleration, which means its velocity will increase linearly with time. This can result in severe impact loads on the driven mechanism when the actuator comes to the end of its stroke at maximum velocity.

79 Air and Hydraulic Cylinders Servo valve control of the fluid flow, to slow the actuator at the end of its stroke, is possible but is quite expensive. The most common application of fluid power cylinders is in form and construction equipment such as tractors and bulldozers, where open loop (non servo) hydraulic cylinders actuate the bucket or blade through linkages. The cylinder and its piston become two of the links (slider and track) in a slider-crank mechanism.

80 Solenoids Electromechanical (AC or DC) linear actuators similar to air cylinders. They are energy inefficient, are limited to very short strokes (about 2 to 3 cm), develop a force which varies exponentially over the stroke, and deliver high impact loads. They are, inexpensive, reliable, and have very rapid response times. They cannot handle much power, and typically used as control or switching devices rather than as devices which do large amounts of work on a system. A common application of solenoids is in camera shutters, where a small solenoid is used to pull the latch and trip the shutter action when you push the button to take the picture. Its nearly instantaneous response is an asset in this application, and very little work is being done in tripping a latch. Another application is in electric door or trunk locking systems in automobiles, where the click of their impact can be clearly heard when you turn the key (or press the button) to lock or unlock the mechanism.

81 Assignment:

Theory of Machines Course # 1

Theory of Machines Course # 1 Ayman Nada Assistant Professor Jazan University, KSA. arobust@tedata.net.eg March 29, 2010 ii Sucess is not coming in a day 1 2 Chapter 1 INTRODUCTION 1.1 Introduction Mechanisms