AUT301 Theory of Machines Module 1 PDF

AUT301 THEORY OF MACHINES MODULE- 1  SYLLABUS Planar Mechanisms - Terminology and definitions. Kinematic pairs, kinematic chains, and degrees of freedom of mechanisms. Kinematic Inversion of 4 bar and slider crank chains. Condition for correct steering. Davis Steering...

AUT301 THEORY OF MACHINES MODULE- 1  SYLLABUS Planar Mechanisms - Terminology and definitions. Kinematic pairs, kinematic chains, and degrees of freedom of mechanisms. Kinematic Inversion of 4 bar and slider crank chains. Condition for correct steering. Davis Steering gear and Ackerman Steering gear (No numerical questions). Brakes-Shoe brakes, Band Brakes, and Internal expanding brakes.  INTRODUCTION Mechanics of Machines: Branch of engineering which deals with the relative motion and forces between various machine elements. Classfication of Mechanics : 1. Statics: Deals with the forces and its effects on machine parts while the latter is at rest. 2. Dynamics: Deals with the forces and the effect of forces on machine components when they are on motion. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… Dynamics 1. Kinematics: Deals with relative motion without considering the forces. 2. Kinetics: Deals with the forces, which are formed due to the combined action of mass and motion of machine elements. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… Mechanisms: A mechanism is a combination of a number of bodies assembled in such a way that the motion of one causes constrained and predictable motion of others. Examples: Type writer, wiper mechanism of automobiles etc. Machine: A machine is a mechanism or combination of mechanisms which, apart from imparting definite motion to its parts, also transmits and modifies the available form of energy in to some useful work. Examples: Internal combustion engine, Automobile transmission, Lathe, Shaper etc. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Structure It is an assemblage of a number of resistant bodies (known as members) having no relative motion between them and meant for carrying loads having straining action. A railway bridge, a roof truss, machine frames etc., are the examples of a structure. Difference Between a Machine and a Structure The parts of a machine move relative to one another, whereas the members of a structure do not move relative to one another. A machine transforms the available energy into some useful work, whereas in a structure no energy is transformed into useful work. The links of a machine may transmit both power and motion, while the members of a structure transmit forces only. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Comparison between Mechanism, Machine & structure MECHANISM MACHINE STRUCTURE There is relative motion Relative motion exists There is no relative motion between the parts of a between parts of a machine. between the members of a mechanism structure. It is rigid as a whole. A mechanism modifies and A machine consists of one or A structure does not transmit transmits motion. more mechanisms and hence motion. transmits motion. A mechanism does not A machine modifies energy A structure does not do work. transmit forces and does not or do some work It only transmits forces. do work Only the centre lines of parts The cross sections of parts Cross sections of members of need be considered in also need be considered in a structure need be analysis and design of a analysis and design of a considered since they mechanism. machine. transmit forces. Mechanisms are dealt with in Machines are dealt with in Structures are dealt with in kinematics. kinetics. statics. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Kinematic Link or Element Each part of a machine, which moves relative to some other part, is known as a kinematic link (or simply link) or element. A link or element need not to be a rigid body, but it must be a resistant body. Resistant body: A body is said to be a resistant body if it is capable of transmitting the required forces with negligible deformation. Thus a kinematic link should have the following two characteristics: It should have relative motion. It must be a resistant body. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  CLASSIFICATION OF LINKS I. DEPENDING ON NUMBER OF JOINTS 1. Singular Link: - A link which is connected to only one other link. 2. Binary Link:- A link which is connected to two other links. 3. Ternary Link:- A link which is connected to three other links SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  II. DEPENDING ON CONNECTION BETWEEN DRIVER & FOLLOWER TO TRANSMIT MOTION 1. Rigid link:- A rigid link is one which does not undergo any deformation while transmitting motion. Strictly speaking, rigid links do not exist. However, as the deformation of a connecting rod, crank etc. of a reciprocating steam engine is not appreciable, they can be considered as rigid links. 2. Flexible link:- A flexible link is one which partly deformed in a manner not to affect the transmission of motion. For example, belts, ropes, chains and wires are flexible links and transmit tensile forces only. 3. Fluid link:- A fluid link is one which formed by having a fluid in a receptacle and the motion is transmitted through the fluid by pressure or compression only, as in the case of hydraulic presses, jacks and brakes. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Types of Constraint Motion 1. Completely constrained motion: When the motion between a pair is limited to a definite direction irrespective of the direction of force applied, then the motion is said to be a completely constrained motion. For example, the piston and cylinder form a pair and the motion of the piston is limited to a definite direction (i.e. it will only reciprocate) relative to the cylinder irrespective of the direction of motion of the crank. The motion of a square bar in a square hole and the motion of a shaft with collars at each end in a circular hole are examples of completely constrained motion. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 2. Incompletely constrained motion: When the motion between a pair can take place in more than one direction, then the motion is called an incompletely constrained motion. The change in the direction of impressed force may alter the direction of relative motion between the pair. A circular bar or shaft in a circular hole is an example of an incompletely constrained motion as it may either rotate or slide in a hole. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 3. Successfully constrained motion: When the motion between the elements, forming a pair, is such that the constrained motion is not completed by itself, but by some other means, then the motion is said to be successfully constrained motion. Consider a shaft in a foot-step bearing. The shaft may rotate in a bearing or it may move upwards. This is a case of incompletely constrained motion. But if the load is placed on the shaft to prevent axial upward movement of the shaft, then the motion of the pair is said to be successfully constrained motion. The motion of an I.C. engine valve (these are kept on their seat by a spring) and the piston reciprocating inside an engine cylinder are also the examples of successfully constrained motion. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Kinematic Pairs The two links or elements of a machine, when in contact with each other, are said to form a pair. If the relative motion between them is completely or successfully constrained, the pair is known as kinematic pair. In the above given Slider crank mechanism, Link 2 rotates relative to link 1 and constitutes a revolute or turning pair. Similarly, links 2, 3 and 3, 4 constitute turning pairs. Link 4 (Slider) reciprocates relative to link 1 and its a sliding pair. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Classification of Kinematic Pairs 1. According to the type of relative motion between the elements (a) Prismatic or Sliding pair When the two elements of a pair are connected in such a way that one can only slide relative to the other, the pair is known as a sliding pair. A sliding pair has a completely constrained motion. The piston and cylinder, cross-head and guides of a reciprocating steam engine, ram and its guides in shaper, tail stock on the lathe bed etc. are the examples of a sliding pair. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Contd… (b) Revolute or Turning pair When the two elements of a pair are connected in such a way that one can only turn or revolve about a fixed axis of another link, the pair is known as turning pair. A turning pair also has a completely constrained motion. A shaft with collars at both ends fitted into a circular hole, the crankshaft in a journal bearing in an engine, lathe spindle supported in head stock, cycle wheels turning over their axles etc. are the examples of a turning pair. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Contd… (c) Rolling pair When the two elements of a pair are connected in such a way that one rolls over another fixed link, the pair is known as rolling pair. Ball and roller bearings are examples of rolling pair. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Contd… (d) Screw pair When the two elements of a pair are connected in such a way that one element can turn about the other by screw threads, the pair is known as screw pair. The lead screw of a lathe with nut, and bolt with a nut are examples of a screw pair. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Contd… (e) Spherical pair When the two elements of a pair are connected in such a way that one element (with spherical shape) turns or swivels about the other fixed element, the pair formed is called a spherical pair. The ball and socket joint, attachment of a car mirror, pen stand etc., are the examples of a spherical pair. This pair has three degrees of freedom. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 2. According to the type of contact between the elements (a) Lower pair When the two elements of a pair have a surface contact when relative motion takes place and the surface of one element slides over the surface of the other, the pair formed is known as lower pair. It will be seen that sliding pairs, turning pairs and screw pairs form lower pairs. (b) Higher pair When the two elements of a pair have a line or point contact when relative motion takes place and the motion between the two elements is partly turning and partly sliding, then the pair is known as higher pair. A pair of friction discs, toothed gearing, belt and rope drives, ball and roller bearings and cam and follower are the examples of higher pairs. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Lower Pair Higher Pair SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 3. According to the type of closure (a) Self closed pair When the two elements of a pair are connected together mechanically in such a way that only required kind of relative motion occurs, it is then known as self closed pair. The lower pairs are self closed pair. (b) Force - closed pair When the two elements of a pair are not connected mechanically but are kept in contact by the action of external forces, the pair is said to be a force-closed pair. The cam and follower is an example of force closed pair, as it is kept in contact by the forces exerted by spring and gravity. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Kinematic Chain When the kinematic pairs are coupled in such a way that the last link is joined to the first link to transmit definite motion (i.e. completely or successfully constrained motion), it is called a kinematic chain. For example, the crankshaft of an engine forms a kinematic pair with the bearings which are fixed in a pair, the connecting rod with the crank forms a second kinematic pair, the piston with the connecting rod forms a third pair and the piston with the cylinder forms a fourth pair. The total combination of these links is a kinematic chain. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY RECIPROCATING ENGINE SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… If each link is assumed to form two pairs with two adjacent links, then the relation between the number of pairs forming a kinematic chain and the number of links may be expressed in the form of an equation: l = 2P – 4 ------------------------------- (1) Another relation between the number of links and the number of joints which constitute a kinematic chain is given by the expression: --------------------------- (2) Eqn. (1) & (2) are applicable only to kinematic chains with lower pairs. In the case of higher pairs, each higher pair may be taken as equivalent to two lower pairs with an additional element or link. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… Consider the arrangement of three links AB, BC and CA with pin joints at A, B and C as shown in Figure. In this case, Number of links, l = 3 Number of pairs, p = 3 and number of joints, j = 3 From equation (1), l = 2P – 4 l=2x3–4 ;3=2 i.e, L.H.S > R.H.S Now from equation (2), j = 1.5 x 3 – 2 ; 3 = 2.5 ; L.H.S > R.H.S Since the left hand side is greater than the right hand side, it is not a kinematic chain and hence no relative motion is possible. Such type of chain is called locked chain and forms a rigid frame or structure which is used in bridges and trusses. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… Consider the arrangement of four links AB, BC, CD and DA as shown in Figure. ; ; Since the arrangement of four links, as shown in Fig, satisfy the equations (1) and (2), therefore it is a kinematic chain. If a definite displacement (say θ) is given to the link AD, keeping the link AB fixed, then the resulting displacements of the remaining two links BC and CD are also perfectly definite. Thus we see that in a four bar chain, the relative motion is completely constrained. Hence it may be called as a constrained kinematic chain, and it is the basis of all machines. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… l = 5, p = 5, j = 5 ; ; Since left hand side is less than right hand side, therefore it is not a kinematic chain. Such a type of chain is called unconstrained chain i.e. the relative motion is not completely constrained. This type of chain is of little practical importance. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Conditions of Kinematic Chain l = 2P – 4 LHS > RHS LOCKED CHAIN CONSTRAINED LHS = RHS KINEMATIC CHAIN UNCONSTRAINED LHS < RHS KINEMATIC CHAIN SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… Types of Joints 1. Binary Joint In order to determine the nature of chain, i.e. whether the chain is a locked chain (or structure) or kinematic chain or unconstrained chain, the following relation between the number of links and the number of binary joints, as given by A.W. Klein, may be used: j = Number of binary joints Kinematic chain (LHS = RHS) SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 2. Ternary Joint When three links are joined at the same connection, the joint is known as ternary joint. It is equivalent to two binary joints as one of the three links joined carry the pin for the other two links. It has three binary joints at A, B and D and two ternary joints at C and E. Since one ternary joint is equivalent to two binary joints, therefore equivalent binary joints in a chain, are j = 3 + (2 × 2) = 7 h=0 Kinematic chain (LHS = RHS) SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 3. Quaternary joint. When four links are joined at the same connection, the joint is called a quaternary joint. It is equivalent to three binary joints. In general, when l number of links are joined at the same connection, the joint is equivalent to (l – 1) binary joints. ; h=0 Total Binary Joints = It has one binary joint at D, four ternary joints at A, B, E and F, and two quaternary joints at C and G. Locked chain (LHS > RHS) SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… If the link CG is removed, as shown in Fig, it has ten links and has one binary joint at D and six ternary joints at A, B, C, E, F and G. Therefore total number of binary joints are (1 + 2 × 6) = 13 It is a kinematic chain. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  MECHANISM When one of the links of a kinematic chain is fixed, the chain is known as mechanism. It may be used for transmitting or transforming motion e.g. wiper, typewriter etc. A mechanism with four links is known as simple mechanism, and the mechanism with more than four links is known as compound mechanism. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY FOUR BAR MECHANISM SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Slider Crank Mechanism SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  GRASHOF’S LAW For a planar four bar linkage, the sum of the shortest and largest link lengths cannot be greater than the sum of the remaining two link lengths if there is to be continuous relative motion between two members. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  GRASHOF’S LAW In a four bar mechanism the longest link has length L and the shortest link has length S, the lengths of other two links are P and Q. Then according to Grashof’s law, P Q S S +L ≤ P + Q L If this inequality is not satisfied, no link makes complete revolution to another. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… Some important concepts in link mechanisms are: Crank: A side link which revolves relative to the frame is called a crank. Rocker: Any link which does not revolve is called a rocker. Coupler Rocker / Follower Crank Frame Fig : A 4 bar Mechanism SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… Possible Combinations of 4 bar Mechanism Crank-rocker mechanism: In a four bar linkage, if the shorter side link revolves and the other one oscillates, it is called a crank-rocker mechanism. Double-crank mechanism: In a four bar linkage, if both of the side links revolve, it is called a double-crank mechanism. Double-rocker mechanism: In a four bar linkage, if both of the side links oscillates, it is called a double-rocker mechanism SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… Mechanisms satisfying this condition fall into the following three categories: Crank-rocker mechanism - The shortest link is the crank Double-crank mechanism - The shortest link is the frame Double-rocker mechanism - The shortest link is the coupler SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Kinematic Diagrams A kinematic diagram or kinematic scheme illustrates the connectivity of links and joints of a mechanism or machine rather than the dimensions or shape of the parts. Often links are presented as geometric objects, such as lines, triangles or squares, that support schematic versions of the joints of the mechanism or machine. A kinematic diagram is sometimes called a joint map or a skeleton diagram. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Kinematic Diagrams Simple Link Simple Link with a point Ternary Link / Complex Link SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Kinematic Diagrams Pin Joint Slider Joint Cam Joint Gear Joint SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Kinematic Diagrams 1 X 4 Y 3 2 Vice Plier or Vice Grips SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Kinematic Diagrams Frame D A 1 4 3 Crank X Follower Y B C 2Coupler SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Kinematic Diagrams 1 2 3 4 Punch mechanism. A skeleton representing the punch mechanism SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Kinematic Diagrams C X B E, F A G D SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Kinematic Diagrams Dump Truck SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Kinematic Diagrams SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  DEGREE OF FREEDOM(DOF) OF PLANAR MECHANISMS The degree of freedom of a body is equal to the number of independent coordinates required to specify the movement. An unconstrained rigid body in space can describe the following independent motions – Translational motions along three mutually perpendicular axes x, y and z(3 dof). – Rotational motions about these axes(3 dof). Thus a rigid body possesses six dof. Degrees of freedom = 6 – Number of restraints SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  MOBILITY OF MECHANISMS In a mechanism, one link is fixed. Therefore, – Number of movable links = N-1 – Number of degrees of freedom of (N-1) movable links before they are connected to any other link = 6(N-1) Let, – N = Total number of links in a mechanism – F = Degrees of freedom – P1 = Number of pairs having one degree of freedom – P2 = Number of pairs having two degrees of freedom and so on. Each pair having one degree of freedom imposes 5 restraints on the mechanism, and reduces its degrees of freedom by 5P1. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… Each pair having two degrees of freedom will impose 4 restraints, and reducing the degrees of freedom by 4P2. Similarly other pairs having 3, 4, 5 degrees of freedom reduce the degrees of freedom of the mechanism. Thus, F or n= 6 (N – 1) – 5P1 – 4P2 – 3P3 – 2P4 – 1P5 ------- (1) SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  KUTZBACH CRITERION Since most of the mechanisms are two dimensional, in which displacement is possible along two axes and rotation about one axis. Thus there are three general restraints (1 restraint displacement and 2 restraints for rotation). Therefore for a planar mechanism the equation reduces to, F or n = 3 (N – 1) – 2P1 – 1P2 This equation is called Kutzbach criterion for the movability of a mechanism having plane motion. F or ------------------- (2) SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… The mechanism, as shown in Fig(a), has three links and three binary joints, i.e. l = 3 and j = 3. ∴ n = 3 (3 – 1) – 2 × 3 = 0 The mechanism, as shown in Fig.(b), has four links and four binary joints, i.e. l = 4 and j = 4. ∴ n = 3 (4 – 1) – 2 × 4 = 1 SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Grubler’s Criterion The Grubler’s criterion applies to mechanisms with only single degree of freedom joints where the overall movability of the mechanism is unity. Substituting n = 1 and h = 0 in Kutzbach equation, we have 1 = 3 (l – 1) – 2 j or 3l – 2j – 4 = 0 SCMS SCHOOL OF ENGINEERING & TECHNOLOGY FIND THE MOBILITY OF THE SYSTEM L = Number of links =3 J1 = Number of 1 DOF joints =2 J2 = Number of 2 DOF joints / Higher pair =1 Mobility M = 3(L - 1) - 2J1 - J2 = 3(3 – 1) – 2x2 – 1 =1 SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  SPECIAL CASE – REDUNDANT DOF If certain links can be moved without causing motion in the rest of the links of the mechanism, it is called redundant degree of freedom (Fr). SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  FIND THE MOBILITY OF THE SYSTEM Considering redundancy – No slipping between the wheel & surface L = Number of links =4 J1 = Number of 1 DOF joints =3 J2 = Number of 2 DOF joints / Higher pair =1 Fr = 1 Mobility M = 3(L - 1) - 2J1 - J2 - Fr = 3(4 – 1) – 2x3 – 1 - 1 =1 SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  FIND THE MOBILITY OF THE SYSTEM When slipping occurs L = Number of links =4 J1 = Number of 1 DOF joints slipping =3 J2 = Number of 2 DOF joints =1 Mobility M = 3(L - 1) - 2J1 - J2 = 3(4 – 1) – 2x3 – 1 =2 SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  FIND THE MOBILITY OF THE SYSTEM Considering redundancy – No slipping between the wheels L = Number of links =4 J1 = Number of 1 DOF joints =3 J2 = Number of 2 DOF joints / Higher pair =1 Fr = 1 Mobility M = 3(L - 1) - 2J1 - J2 - Fr = 3(4 – 1) – 2x3 – 1 - 1 =1 SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  FIND THE MOBILITY OF THE SYSTEM When slipping occurs slipping L = Number of links =4 J1 = Number of 1 DOF joints =3 J2 = Number of 2 DOF joints / Higher pair =1 Mobility M = 3(L - 1) - 2J1 - J2 = 3(4 – 1) – 2x3 – 1 =2 SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  FIND THE MOBILITY OF THE SYSTEM Number of links, L = 7, Number of one DOF joints, J1 = 6 (pins) + 1 (slider) = 7, Number of two DOF joints, J2 = 1 (fork joint) DOF = 3(L – 1) – 2J1 – J2 = 3(7-1) – 2(7) – 1 = 3 Fork Joint Spring – No effect on mechanism DOF Three input sources are 1 needed to control the mechanism 4 3 2 Sprin 5 g Slider 1 6 7 SCMS SCHOOL OF ENGINEERING & TECHNOLOGY 1  Exceptions to the Gruebler’s Equation Two rollers in contact, no slipping L = 3, J1 = 2, J2 = 1 , Fr = 1 DOF = 3(3-1) - 2(2) – 1 - 1 = 0 Redundant support 3 4 2 5 L = 5, J1 = 6, J2 = 0 DOF = 3(5-1) - 2(6) = 0 SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  INVERSION OF THE MECHANISM When one of links is fixed in a kinematic chain, it is called a mechanism. So we can obtain as many mechanisms as the number of links in a kinematic chain by fixing, in turn, different links in a kinematic chain. This method of obtaining different mechanisms by fixing different links in a kinematic chain is known as inversion of the mechanism. The part of a mechanism which initially moves with respect to the frame or fixed link is called driver and that part of the mechanism to which motion is transmitted is called follower. Most of the mechanisms are reversible, so that same link can play the role of a driver and follower at different times. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  TYPES OF KINEMATIC CHAINS The following three types of kinematic chains with four lower pairs are important from the subject point of view: 1. Four bar chain or quadric cyclic chain, 2. Single slider crank chain, and 3. Double slider crank chain. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  FOUR BAR CHAIN OR QUADRIC CYCLIC CHAIN It consists of four links, each of them forms a turning pair at A, B, C and D. The four links may be of different lengths. According to Grashof ’s law for a four bar mechanism, the sum of the shortest and longest link lengths should not be greater than the sum of the remaining two link lengths if there is to be continuous relative motion between the two links. In a four bar chain, one of the links, in particular the shortest link, will make a complete revolution relative to the other three links, if it satisfies the Grashof ’s law. Such a link is known as crank or driver. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… In Fig, AD (link 4) is a crank. The link BC (link 2) which makes a partial rotation or oscillates is known as lever or rocker or follower and the link CD (link 3) which connects the crank and lever is called connecting rod or coupler. The fixed link AB (link 1) is known as frame of the mechanism. When the crank (link 4) is the driver, the mechanism is transforming rotary motion into oscillating motion. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY INVERSIONS OF FOUR BAR CHAIN 1. c > b (Double crank mechanism) 2. b > a (Crank-lever mechanism) 3. Double lever mechanism SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  INVERSIONS OF FOUR BAR CHAIN 1. BEAM ENGINE (CRANK AND LEVER MECHANISM). In this mechanism, when the crank rotates about the fixed centre A, the lever oscillates about a fixed centre D. The end E of the lever CDE is connected to a piston rod which reciprocates due to the rotation of the crank. In other words, the purpose of this mechanism is to convert rotary motion into reciprocating motion. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 2. COUPLING ROD OF A LOCOMOTIVE (DOUBLE CRANK MECHANISM) In this mechanism, the links AD and BC (having equal length) act as cranks and are connected to the respective wheels. The link CD acts as a coupling rod and the link AB is fixed in order to maintain a constant centre to centre distance between them. This mechanism is meant for transmitting rotary motion from one wheel to the other wheel. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 3. WATT’S INDICATOR MECHANISM (DOUBLE LEVER MECHANISM) The tracing point P traces out an approximate straight line over certain positions of its movement. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  SINGLE SLIDER CRANK CHAIN It consists of one sliding pair and three turning pairs. This type of mechanism converts rotary motion into reciprocating motion and vice versa. In a single slider crank chain, as shown in Figure, the links 1 and 2, links 2 and 3, and links 3 and 4 form three turning pairs while the links 4 and 1 form a sliding pair. As the crank rotates, the cross-head reciprocates in the guides and thus the piston reciprocates in the cylinder SCMS SCHOOL OF ENGINEERING & TECHNOLOGY INVERSIONS OF SLIDER CRANK CHAIN SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 1. First Inversion: This inversion is obtained when link 1 is fixed and links 2 & 4 are made the crank & the slider respectively. Application: a) Reciprocating Steam Engine: Link 4 (piston) is the driver. b) Reciprocating Compressor. Link 2 (crank) is the driver. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 2. Second Inversion: This inversion is obtained when link 2 is fixed; link 3 along with the slider becomes crank and link 1 rotate about O along with the slider which also reciprocates on it. Applications: a) Whitworth quick-return mechanism b) Rotary internal combustion engine / Gnome engine SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 1. Whitworth Quick-Return Mechanism This mechanism is mostly used in shaping and slotting machines. When the driving crank CA moves from the position CA1 to CA2 (or the link DP from the position DP1 to DP2 ) through an angle α in the clockwise direction, the tool moves from the left hand end of its stroke to the right hand end through a distance 2PD. when the driving crank moves from the position CA2 to CA1 (or the link DP from DP2 to DP1 ) through an angle β in the clockwise direction, the tool moves back from right hand end of its stroke to the left hand end. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… Since the crank link CA rotates at uniform angular velocity therefore time taken during the cutting stroke (or forward stroke) is more than the time taken during the return stroke. The ratio between the time taken during the cutting and return strokes is given by SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 2. Rotary Internal Combustion Engine / Gnome Engine This mechanism is used in radial engines. It consists of seven cylinders in one plane and all revolves about fixed centre D. The crank 2 is fixed, connecting rod 4 rotates and the piston 3 reciprocates inside the cylinders forming link 1. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 3. Third Inversion: This inversion is obtained when the link 3 is fixed, the link 2 acts as a crank and link 4 oscillates. Application: a) Oscillating cylinder engine b) Crank & slotted lever mechanism SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 1. Oscillating Cylinder Engine It is used to convert reciprocating motion into rotary motion. In this mechanism link 3 is fixed, the crank 2 rotates, piston rod 1 reciprocates and cylinder 4 oscillates about A. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 2. Crank & Slotted Lever Mechanism This mechanism is used in shaping machines, slotting machines and in rotary internal combustion engine In this mechanism, the link AC (link 3) is fixed. The driving crank CB revolves with uniform angular speed about the fixed centre C. A sliding block attached to the crank pin at B slides along the slotted bar AP and thus causes AP to oscillate about the pivoted point A. A short link PR transmits the motion from AP to the ram which carries the tool and reciprocates along the line of stroke R1 R2. The line of stroke of the ram (i.e. R1R2 ) is perpendicular to AC produced. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Differences Whitworth Lever Mechanism Crank & Slotted Lever Mechanism SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 4. Fourth Inversion: This inversion is obtained when the link 4 is fixed, the link 3 oscillates about B on the link 4 and the end A of the link 2 is oscillates about B and the end O reciprocates along the fixed link 4. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 1. Pendulum Pump or Bull Engine. In this mechanism, the inversion is obtained by fixing the cylinder or link 4. In this case, when the crank (link 2) rotates, the connecting rod (link 3) oscillates about a pin pivoted to the fixed link 4 at A and the piston attached to the piston rod (link 1) reciprocates. The duplex pump which is used to supply feed water to boilers have two pistons attached to link 1, as shown in Fig. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  DOUBLE SLIDER CRANK CHAIN A kinematic chain which consists of two turning pairs and two sliding pairs is known as double slider crank chain. We see that the link 2 and link 1 form one turning pair and link 2 and link 3 form the second turning pair. The link 3 and link 4 form one sliding pair and link 1 and link 4 form the second sliding pair. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  INVERSIONS OF DOUBLE SLIDER CRANK CHAIN 1. Elliptical trammels. It is an instrument used for drawing ellipses. This inversion is obtained by fixing the slotted plate (link 4). The fixed plate or link 4 has two straight grooves cut in it, at right angles to each other. The link 1 and link 3, are known as sliders and form sliding pairs with link 4. The link AB (link 2) is a bar which forms turning pair with links 1 and 3. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  the co-ordinates of the point P on the link BA will be or Squaring and adding, SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 2. Scotch yoke mechanism This mechanism is used for converting rotary motion into a reciprocating motion. The inversion is obtained by fixing the link 1. In this mechanism, when the link 2 (which corresponds to crank) rotates about B as centre, the link 4 (which corresponds to a frame) reciprocates. The fixed link 1 guides the frame.. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… 3. Oldham’s coupling. An oldham's coupling is used for connecting two parallel shafts whose axes are at a small distance apart. The shafts are coupled in such a way that if one shaft rotates, the other shaft also rotates at the same speed. This inversion is obtained by fixing the link 2. The shafts to be connected have two flanges (link 1 and link 3) rigidly fastened at their ends by forging. The link 1 and link 3 form turning pairs with link 2. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd… A circular disc, have two tongues (i.e. diametrical projections) T1 and T2 on each face at right angles to each other. The tongues on the link 4 closely fit into the slots in the two flanges (link 1 and link 3). The link 4 can slide or reciprocate in the slots in the flanges. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  PANTOGRAPH A pantograph is an instrument used to reproduce to an enlarged or a reduced scale and as exactly as possible the path described by a given point. It consists of a jointed parallelogram as shown in Fig. It is made up of bars connected by turning pairs. The bars BA and BC are extended to O and E respectively, such that. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  EXACT STRAIGHT LINE MOTION MECHANISMS 1. PEAUCELLIER MECHANISM. 2. HART’S MECHANISM 3. SCOTT RUSSELL’S MECHANISM SCMS SCHOOL OF ENGINEERING & TECHNOLOGY 1. PEAUCELLIER MECHANISM. Number of links = 8 OA x OB = Constant SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  2. HART’S MECHANISM FC = DE & CD = EF The points O, A and B divide the links FC, CD and EF in the same ratio. the point B will trace a straight line perpendicular to the diameter OP produced. OA x OB = Constant Number of links = 6 SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  3. SCOTT RUSSELL’S MECHANISM (one sliding pair) A is the middle point of PQ and OA=AP=AQ. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  APPROXIMATE STRAIGHT LINE MOTION MECHANISMS Watt’s mechanism Modified Scott-Russell mechanism Grasshopper mechanism SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  WATT’S MECHANISM The tracing point P traces out an approximate straight line over certain positions of its movement, if SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  MODIFIED SCOTT-RUSSELL MECHANISM AP is not equal to AQ OA must be equal to SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  GRASSHOPPER MECHANISM Earlier used as a engine mechanism which gave long stroke with a very short crank. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Coupler Curve During motion of a linkage, any point attached to the plane of coupler generates some path with respect to the fixed link, this path is called a coupler curve SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Coupler Curve SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Coupler Curve SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  TRANSMISSION ANGLE The angle γ between the output link (DC) and the coupler (BC) is known as transmission angle. In the figure is AB is the input link. Min transmission angle, γ or μ= 450 Max torque transmitted at γ or μ= 900 Mechanism locks when γ or μ= 00 A measure of quality of design – Transmission angle SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  MECHANICAL ADVANTAGE Mechanical advantage of a mechanism is the ratio of the output force or torque to input force or torque at any instant. Mechanical Advantage = output torque/input torque Mechanical advantage of a four bar linkage is directly proportional to the sine of angle γ and inversely proportional to sine of angle β. Where γ is the transmission angle and β is the angle between input link and coupler. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  MECHANICAL ADVANTAGE When β = 0, the mechanical advantage will be infinite and the mechanism is said to be in toggle position. When mechanism moves, the angles as well as the mechanical advantage changes. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY STEERING GEAR MECHANISM SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  STEERING GEAR MECHANISM In order to avoid skidding, the two front wheels must turn about the same instantaneous centre I which lies on the axis of the back wheels. If the instantaneous centre of the two front wheels do not coincide with the instantaneous centre of the back wheels, the skidding on the front or back wheels will definitely take place, which will cause more wear and tear of the tyres. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  STEERING GEAR MECHANISM Thus, the condition for correct steering is that all the four wheels must turn about the same instantaneous centre. The axis of the inner wheel makes a larger turning angle θ than the angle Ø subtended by the axis of outer wheel. – Let a = Wheel track, – b = Wheel base, and – c = Distance between the pivots A and B of the front axle. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY STEERING GEAR MECHANISM SCMS SCHOOL OF ENGINEERING & TECHNOLOGY DAVIS STEERING GEAR MECHANISM SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  DAVIS STEERING GEAR MECHANISM Let a = Vertical distance between AB and CD b = Wheel base d = Horizontal distance between AC and AA’ & BD and BD’ c = Distance between the pivots A and B of the front axle x = Distance moved by AC to AC’ =CC’ = DD’ α = Angle of inclination of the links ,AC and BD, to the vertical. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY DAVIS STEERING GEAR MECHANISM SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd.. - θ -θ + θ - θ - - θ + + θ θ SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd.. - + θ - θ + θ- θ - θ θ - SCMS SCHOOL OF ENGINEERING & TECHNOLOGY Contd.. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY ACKERMAN STEERING GEAR MECHANISM SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  Three Positions for Correct Steering SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  ACKERMAN Vs DAVIS Steering Gear The Ackerman steering gear mechanism is much simpler than Davis gear. The difference between the Ackerman and Davis steering gears are: 1. The whole mechanism of the Ackerman steering gear is on back of the front wheels; whereas in Davis steering gear, it is in front of the wheels. 2. The Ackerman steering gear consists of turning pairs, whereas Davis steering gear consists of sliding members. 3. Davis steering gear mechanism meets condition for correct steering at all angles where as Ackerman steering gear meets only at three positions. SCMS SCHOOL OF ENGINEERING & TECHNOLOGY  N K YO U THA SCMS SCHOOL OF ENGINEERING & TECHNOLOGY