Analysis and Synthesis of Mechanisms Presentation

Analysis and Synthesis of Mechanisms

1. Subject: Analysis and Synthesis of Mechanisms (2 nd Year 1 st Semester ) [ Module-1 ] PRODUCTION ENGINEERING DEPARTMENT JADAVPUR UNIVERSITY Kolkata-700032 Prepared by- Dr. Kingshuk Mandal

2. Mechanics Statics Dynamics Kinematics Kinetics INTRODUCTION Kinematics deals with motion only or how an object moves through space without reference to any associated force. Kinetics deals with forces and motion only and reveals how forces affect motion. Analysis- Study in details. Synthesis- Combination of different mechanisms so that it can be used in applied form. Mechanism- Any process that converts input to desired output.

3. FLOW CHART OF MACHINE Rigid/Resistant Body Kinematic Link Kinematic Pair Kinematic Chain Mechanism Machine Rigid/Resistant Body A Resistant body is a body which is not a rigid body but acts like a rigid body whiles its functioning in the machine.

4. Kinematic Link or Element A kinematic link or element is defined as the part of a machine that has a relative motion with regard to another part of the same machine . A kinematic pair  consists of minimum two elements , typically called the driver and the follower , that are connected in a way that allows motion or constrains specific types of relative motion. Kinematic Pair

5. Kinematic Chain A kinematic chain is an assembly of rigid bodies connected by joints to provide constrained motion.

6. Mechanism A mechanism is a combination of rigid or restraining bodies so shaped and connected that they move upon each other with definite relative motion.

7. Machine A machine is a combination of rigid or resistant bodies, formed and connected in such a way that they move with definite relative motions with each other and transmit force also. A machine has two functions : (i) transmitting definite relative motion, and (ii) transmitting forces. Milling Shaping Drilling

8. KINEMATIC PAIRS Kinematic Pairs Lower Pair Higher Pair Wrapping Pair The degree of freedom (DOF ) of a kinematic pair is given by the number of independent coordinates required to completely specify the relative movement. These coordinates are commonly known as pair variables . Degree of Freedom (DOF) Lower Pairs A pair is said to be a lower pair when the connection between two elements is through the area of contact or surface contact . In this section, commonly-used six types of lower pairs will be discussed-

9. 1. Revolute or Turning Pair (Hinged Joint): A revolute pair is shown in Fig. (a). It is seen that this connection allows only a relative rotation between elements-1 and elements-2 , which can be expressed by a single coordinate . Thus, a revolute pair has a single degree of freedom .  

10. 2. Prismatic Pair: As shown in Fig. (b), a prismatic pair allows only a relative translation between elements-1 and elements-2 , which can be expressed by a single coordinate S, and thus it possesses one degree of freedom . It should be noted that in such a pair it is only the direction of relative motion which is important. The location of a prismatic pair has no relevance to the kinematics of a system.

11. 3 . Screw Pair: As shown in Fig. (c), a screw pair also has one degree of freedom since the relative movement between element-1 and element-2 can be expressed by a single coordinate or S. These two coordinates are related, that is , Where L is the lead of the thread.  

12. 4 . Cylindrical Pair: As shown in Fig. ( d), a cylindrical pair has two degrees of freedom because it allows both rotation and translation, parallel to the axis of rotation, between the connected elements. These relative movements can be expressed by two independent coordinates and S, respectively .  

13. 5. Spherical Pair ( Spheric Pair): A ball-and-socket joint, as shown in Fig. (e), forms a spherical ( spheric ) pair. This connection has three degrees of freedom since the complete description of the relative movement between the connected elements needs three independent coordinates. Two of these coordinates α and β are required to specify the position of the axis OA and the third coordinate describes the rotation about the axis OA.  

14. 6 . Planar Pair: A planar pair, shown in Fig. ( f), has three degrees of freedom . Two coordinates x and y describe the relative translation in the x-y plane and the third describes the relative rotation about the z-axis .  

15. HIGHER PAIRS A higher pair is defined as one in which the connection between two elements has only a point or line contact . A point contact takes place in ball bearings or between the teeth of skew-helical gears. A line contact is observed in roller bearings , between the teeth of most gears, and in cam-follower elements. Following figure shows point contact and line contact in higher pairs.

16. HIGHER PAIRS (ANIMATION)

17. WRAPPING PAIRS Wrapping pairs comprise belts, chains , and such other devices. Following figure shows a belt-driven pulley and chain-sprocket arrangement.

18. PLANE AND SPACE MECHANISMS If all points of a mechanism move in parallel planes , then it is defined as a plane mechanism . A simple plane mechanism is shown in following Fig . ( a), where all points move in parallel planes. A space mechanism is one in which all points of the mechanism do not move in parallel planes . A very common example of a space mechanism, known as Hooke's joint , is shown in Fig. (b). The major part of this course will be devoted to a study of the characteristics of only the plane mechanisms.

19. 4-Bar Mechanism Hook’s Joint (Universal Joint) ANIMATIONS

20. KINEMATIC DIAGRAMS To facilitate the study of a real-life mechanism , it is generally represented by a kinematic diagram . Such a diagram depicts the essential kinematic features of the mechanism . A prismatic pair can always be thought of as the limit of a revolute pair .

21. The curved slider (the connection between links 1 and 4) in Fig. (b) remains a revolute pair, as the pair variable (to describe relative motion) is still represented by an angular movement . If the radius of curvature p of the curved slider shown in Fig. ( b) becomes infinitely large, the pair variable transforms to linear displacement (from angular movement) and the revolute pair R 4 transforms to a prismatic pair . LIMIT AND DISGUISE OF REVOLUTE PAIRS

22. We can see that a slider-crank mechanism is obtained as a limit of the four-bar linkage in which one hinge point goes to infinity, thus transforming one revolute pair to a prismatic pair.

23. KINEMATIC INVERSION The process of fixing different links of the same kinematic chain to produce distinct mechanisms is called kinematic inversion . R elative motions of the links of the mechanisms produced remain unchanged . Consider the simplest kinematic chain, i.e., a chain consisting of four binary links and four revolute pairs. See following Fig. (a):

24. The four dissimilar mechanisms that can be obtained by the four different inversions of the chain are shown in Fig. (b).

25. Let us consider a kinematic chain with four binary links, three revolute pairs, and one prismatic pair, shown in Fig. (a). Fixing links 1, 2, and 3 in turn, we get three different mechanisms shown in Fig. (b). By fixing link 4, the mechanism produced is a hand pump where the connecting rod acts as the driving member . It should be noted that a kinematic inversion may change the appearance of a mechanism . R-R-R-P Chain (3R1P Chain)

26. R-R-R-P Chain (3R1P Chain)

27. R-R-R-P Chain (3R1P Chain) Hand Pump Mechanism

28. R-R-P-P Chain (2R2P Chain) Scotch-Yoke Mechanism: This time, in a kinematic chain with four binary links and T wo Revolute and T wo P rismatic pairs placed in the order stated (See the following Figure). In this kinematic chain, the constant rotation of C rank-2 produces a harmonic translation of Y oke-4 . Such a mechanism is known as scotch yoke. F our binary links are- 1- Fixed link 2- Crank 3- Sliding block 4- Yoke Four kinematic pairs are- R l - Revolute pair (Between links 1 and 2) R 2 - Revolute pair (Between links 2 and 3) P 3 - Prismatic pair (Between links 3 and 4) P 4 - Prismatic pair (Between links 4 and 1) Figure:- Scotch-Yoke M echanism

29. Scotch-Yoke M echanism Oldham’s Coupling ANIMATIONS

30. If in the chain shown in Fig . below link-2 is fixed , we obtain a mechanism, known as Oldham's coupling. It is used for transmitting constant angular velocity between two parallel but eccentric shafts . The representative kinematic scheme for this mechanism is shown in Fig. bellow. It should be noted that Stotch - Yoke and Oldham’s Coupling are the same but for the fixed link. OLDHAM'S COUPLING Figure:- Oldham’s Coupling

31. An inversion of the same chain yields yet another common mechanism called Elliptic Trammel , shown in Fig. in which L ink-4 is fixed. Any point D on L ink-2 describes an ellipse as it moves . This can be proved by considering the coordinates of the point D in the xy -system, which are- = AD = …. …. (1) = CD = …. …. (2) Squaring and adding ( Eq n 1) and ( 1Eq n 2 ), we get- Which is the equation for an ellipse. The midpoint B of AC will obviously describe a circle .   ELLIPTIC TRAMMEL Figure:- Elliptic Trammel

32. ANIMATIONS Elliptic Trammel

33. EQUIVALENT LINKAGES Very often a mechanism with higher pairs can be replaced by an equivalent mechanism with lower pairs . The relative motion between links 2 and 3 consists of rolling, coupled with a certain amount of sliding. A and B are the centres of curvature of surfaces 2 and 3, respectively, at the point P. The instantaneous equivalent lower-pair mechanism shown in following figure calls for an additional link AB, and the higher pair is replaced by two revolute pairs at A and B.

34. FOUR-LINK PLANAR MECHANISMS Simplest form of all mechanisms. A large variety of motions is possible with different combinations of kinematic pairs and with different inversions of the chain. Four lower pairs are required to connect the four links in a 4-link mechanism. Two types of kinematic pairs in a plane kinematic chains: i. Revolute pairs (R ), and ii. Prismatic pairs (P) . The possible topologically different combinations are as follows: RRRR RRR P RR PP R P R P . There are four possible inversions for each kinematic chain, resulting in a wide variety of mechanisms .

35. Following figures show the four possible situations:

36. 4R MECHANISMS (RRRR) 4R kinematic chain is commonly known as four-bar mechanism . Geometric motion characteristics can be different for mechanisms resulting from different inversions . When one of the links of a 4R chain is grounded (Say , link-1 ), two hinges ( R-pairs i.e. Revolute Pair) are grounded. These are called fixed or ground hinges, denoted by generally O 2 and O 4 . The other two hinges, A and B, are called moving hinges . The points A and B are also termed as motion-transfer points . Links 2 and 4 execute a pure rotary motion.

37. Link 3 executes a generalized plane motion, being composed of both translation and rotation . Such a link is termed as a floating link . When a link hinged to the frame can make a complete rotation , it is called a crank . When a link can execute only an oscillatory motion is called rocker . One of the links hinged to the frame is chosen as the input link and the other as the output link or the follower . The connecting link (AB) is called the coupler . 4R MECHANISMS (RRRR)

38. Three different geometric motion characteristics are possible for a 4R mechanism Crank-Rocker Mechanism Double-Crank or a Drag-Link Mechanism Double Rocker Mechanism Following figures shows the three types of mechanisms : 4R MECHANISMS (RRRR)

39. RRR P MECHANISMS (3R 1P ) The different inversions of a 3R-1P kinematic chain lead to four different types of mechanism. The first one, obtained by fixing link-1 , is by far the most important and the mechanism is popularly known as slider-crank mechanism . All engines , compressors , presses are based on this mechanism. The path of the motion-transfer point on the slider does not pass through the fixed hinge of the crank (known as offset slider crank ). In most cases , the offset, e is equal to zero .

40. RR PP MECHANISMS (2R 2P ) With this chain, three distinct possibilities exist. L ink between revolute pairs can be grounded and one such mechanism based on this inversion is Oldham's coupling . When the link with one revolute pair and one prismatic pair is grounded, the inversion yields a Scotch-Yoke mechanism. L ast case (i.e., when the link with two prismatic pairs is grounded), an example is the Elliptic T rammel .

41. R P R P MECHANISMS There are not many mechanisms derived from this chain . Following figure shows a part (indicated by the circumscribing dashed lines) of the Davis automobile steering gear which can be easily identified as an inversion of RPRP chain A 3P1R chain is not meaningful as no relative rotation can take place among the links. When all the four lower pairs are prismatic pairs, the derived system is not a constrained mechanism.

42. MOBILITY AND RANGE OF MOVEMENT An arbitrary number of links connected by a number of kinematic pairs do not result in a mechanism . Some conditions must be satisfied for a system of interconnected links to serve as a useful mechanism. The foremost thing which has to be investigated is the mobility of a mechanism in terms of the number of degrees of freedom (F). The number of independent coordinates required to specify its configuration, i.e., the relative positions of all the links. Determine the number of DOF (F) of a mechanism in terms of the number of links (n) and the number of pair connections of a given type . This process is known as number synthesis .

43. Kutzbach Equation and Grubler's Criterion: Let n be the number of links in a mechanism. Out of which one is fixed. Let j be the number of simple hinges (connect two links ). Now (n - 1) links move in a plane. Once connect two links by a hinge , there cannot be any relative translation between them and only one coordinate is necessary to specify their relative orientation . Thus, two DOF (Translational ) are lost at every hinge and only one DOF (Rotational ) remains . So, the number of degrees of freedom of the mechanism is given by the following equation known as the Kutzbach equation : If F = 0 , we call the mechanism a structure and there is no relative motion between the links . If F = 1 , the mechanism is said to be constrained . Most mechanisms used in machinery are constrained.  

44. For constrained condition , putting F = 1, we have- The simple estimate of constrained movement expressed by (Eq. 2) is known as Grubler's criterion for plane mechanisms . To have a closed chain with simple hinges, we must have a minimum of three links with three hinges (Bellow Figure). Using (Eq. 1), we get-   It will be observed from Figure that there cannot be any relative movement between the links .

45. A higher pair has two degrees of freedom. The degrees of freedom of a mechanism having higher pairs can also be written as- Where , h is the number of higher pairs . One or more links of a mechanism may have a redundant degree of freedom . If a link can be moved without causing any movement in the rest of the mechanism, then the link is said to have a redundant degree of freedom . For example, consider the mechanisms shown in Figs (a) and (b) respectively. Link can slide and rotate without causing any movement in the rest of the mechanism and thus each represents one redundant degree of freedom  

46. The effective degrees of freedom of a mechanism can be expressed as- Where Fr is the number of redundant degrees of freedom. It is thus interesting to note that the mechanism in Fig. (a) is a locked system since its effective degree of freedom is zero .  

47. A system may possess one or more links which do not introduce any extra constraint. Such links are redundant . For example, consider the mechanism shown in Fig . (d). A little examination of the system reveals that the functions of the links AB and CD are identical , and therefore each of the links leads to the same constraint . Thus, when determining the degrees of freedom of the system, only one of these two constraints should be counted . Similarly, some of the constraints may not be independent and should not be counted .  

48. Minimum Number of Binary Links in a Constrained Mechanism with Simple Hinges In a mechanism, there cannot be any singular link. = number of binary links = number of ternary links = number of quaternary links and so on . Then, the total number of links is Each simple hinge consists of two elements as shown in Fig. (a) [1 + and 1 - or 2 + and 2 - ]. Thus , the total number of elements in the mechanism is- Where, j is the number of simple hinges. From Fig. (a), it is seen that a binary link has two elements (1 + and 2 + ).  

49. Similarly, a ternary link will have three elements, and so on. or, using (Eq.-7), we get- To satisfy Grubler's criterion- So, the minimum number of binary links is four , i.e., the four-bar linkage is the simplest mechanism .  

50. Maximum Number of Hinges on One Link in a Constrained Mechanism with n Links Assume all hinges to be simple. First of all find the minimum number of links n required for closure when one link has i hinges. The mechanism shown in Fig. (b) has one such link with i hinges. To each of these i hinges, we connect a link denoted by 1, 2, 3, ..., i . To transfer motion from one of the i links to the next, the number of motion-transfer links to be added will be ( i-1 ), which are numbered ( i+1 ), ( i+2 ), ( i+3 ) …. ( 2i-1 ). T he closure is obtained with a total number of 2i links- (i.e ., the link with i hinges)  

51. In other words, the maximum number of hinges which a link can possess is- If the total number of links n is 2i , the total, number of simple hinges will be- Now, from the Eq.-2, we get-  

52. Ranges of Motion of a Four-bar Linkage ( Grashof's Criterion) In a four-bar linkage- The link not connected to the frame is called the coupler. The two links hinged to the frame are called the crank and follower . The driving member is referred to as the crank The other member (Driven member) is the follower. The three different kinds of mechanisms that can be obtained from a four-bar linkage are- The double-crank or drag-link mechanism (Both the crank and the follower make a complete rotation ). The crank-rocker mechanism [Complete rotation of one link (crank) causes an oscillation of the other (rocker )]. The double-rocker mechanism [Driver and driven links only oscillate (as none of these two links can make a complete rotation )].

53. Let, l = length of the longest link, s = length of the shortest link, p , q = length of the other two links. ( l + s) < (p + q) In this situation, the linkage is known as Grashof's linkage. Its inversions are leads to- (a) a double-crank mechanism when s is the frame , (b) two different crank-rocker mechanisms when s is the crank and any one of the adjacent links is the frame , ( c) one double-rocker mechanism when s is the coupler (opposite to the frame). 2. (l + s) > (p + q) In this situation, all four inversions result in a double-rocker mechanism .

54. 3. (l + s ) = ( p + q) The four inversions result in mechanisms similar to those obtained when (l + s) < (p + q) , the only difference being that there will be instances here when the links become collinear. The situation ( l + s ) = ( p + q ) is also true when a linkage has two pairs of equal link lengths . This results in two special mechanisms, namely :- The parallelogram linkage [in which the equal links are not adjacent]. The deltoid linkage [in which the equal links are adjacent. when any of the longer link-1 is fixed, two crank-rocker mechanisms are obtained. when any of the shorter links s is fixed, two double-crank mechanisms result. This is known as the Galloway mechanism .