MECHANICS OF MACHINERY

Transcription

1 MECHNICS OF MCHINERY (For B.E. Mechanical Engineering Students) s per New Revised Syllabus of PJ bdul Kalam Technological University Dr. S. Ramachandran, M.E., Ph.D., Dr..G. Mathew, PhD (NIT-Durgapur) Professor & HoD (Mechanical Engineering) Providence College of Engineering IRWLK PUBLICTIONS (Near ll India Radio) 80, Karneeshwarar Koil Street Mylapore, Chennai Ph.: , aishram2006@gmail.com, airwalk800@gmail.com

2 th First Edition: 8, July 2017 Price : ISBN: ISBN : and

3 Syllabus - Mechanics of Machinery Chapter 1: BSICS OF MECHNISMS Introduction to kinematics and mechanisms various mechanisms, kinematic diagrams, degree of freedom Grashof s criterion, inversions, coupler curves straight line mechanism exact approximate ckerman steering mechanism Hooke s joint Geneva mechanism mechanical advantage, transmission angle. Chapter 2: KINEMTIC NLYSIS Displacement, velocity and acceleration analysis relative motion relative velocity Instantaneous centre Kennedy s theorem Relative acceleration Coriolis acceleration graphical and analytical methods complex number methods computer oriented methods. Chapter 3: KINEMTICS OF CM Cams classification of cam and followers displacement diagrams, velocity and acceleration analysis of SHM, uniform velocity, uniform acceleration, cycloidal motion. Graphical cam profile synthesis, pressure angle nalysis of tangent cam with roller follower and circular cam with flat follower Introduction to polynomial cams. Chapter 4: GERS Gears terminology of spur gears law of Gearing involute spur gears involutometry contact ratio interference backlash gear standardization interchangability Non-standard gears centre distance modification long and short addendum system internal gears theory and details of bevel helical and worm gearing. Chapter 5: GER TRINS Gear trains simple and compound gear trains planetary gear trains differential solution of planetary gear train problems applications Chapter 6: KINEMTIC SYNTHESIS OF PLNR MECHNISM Kinematic synthesis (planar mechanisms) - Tasks of kinematic synthesis - Type, number and dimensional synthesis - Precision points Graphical synthesis for motion - Path and prescribed timing - Function generator 2 position and 3 position synthesis - Overlay method nalytical synthesis techniques Freudenstein s equation - Complex number methods - One case study in synthesis of mechanism.

4 Contents C.1 Contents Chapter 1. BSICS OF MECHNISMS 1.1 Introduction to Kinematics Machine Kinematic Link, Pair and Chain Kinematic Link (or) Element Types of links Slider - Crank Mechanism Kinematic Pair Types of kinematic pairs Kinematic Pairs-based on nature of relative motion Kinematic Pair-Based on nature of contact Kinematic Chain Introduction to Mechanism Number of Degrees of Freedom (or) Mobility Kutzbach Criterion (or) grubler s Equation Grubler s Criterion for Plane Mechanism Three Important Kinematic Chains: Four Bar Chain: Grashof s law for a four bar mechanism Kinematic inversions Inversions of four bar chain Beam engine: (Crank and lever mechanism) Coupling rod of a locomotive (Double crank mechanism) Watt s Indicator mechanism (Double lever mechanism) Single Slider Crank Chain Inversions of single slider crank chain Pendulum pump (or) Bull engine Oscillating cylinder engine

5 C.2 Mechanics of Machinery 3. Rotary internal combustion engine (or) Gnome engine Crank and slotted lever quick return motion mechanism Whitworth mechanism Double Slider Crank Chain Inversions of Double slider crank chain Elliptical trammel Scotch yoke mechanism Oldham s coupling Mechanical dvantage Transmission ngle Description of Common Mechanism Offset Slider crank mechanism as a quick return Mechanism Pantograph Straight line Generators Paucellier Mechanism Watt s Mechanism Steering mechanisms ckermann steering mechanism Hooke s Joint (or) Universal Coupling Toggle Mechanism Ratchets and escapements Indexing Mechanism Geneva mechanism Definition dvantages Disadvantages pplications

6 Contents C.3 Chapter 2 KINEMTIC NLYSIS 2.1 Introduction Displacement Representation of Vector quantities Scalar and Vector quantities ddition of vectors Subtraction of vectors Types of Motions Linear Displacement, Velocity & cceleration nalysis ngular Displacement, Velocity & cceleration Relative Velocity Relative velocity of two moving bodies moving in straight line: Relative velocity of two moving bodies moving in inclined direction Relative motion Key for Construction of the Velocity Polygons Velocity polygon Graphical method - Solved Problems Relative cceleration Solved Problems on Relative cceleration Graphical Method nalytical Method for Relative Velocity and cceleration Instantaneous Centre Method Kennedy Theorem Locating Instantaneous Centers Coriolis cceleration Coriolis component of acceleration Computer Oriented Methods

7 C.4 Mechanics of Machinery Chapter 3 KINEMTICS OF CM 3.1 Introduction Classification of Cam and Followers ccording to Cam Shape ccording to Shape of the follower ccording to the surface in contact (Shape of the follower) ccording to the motion of the follower ccording to the path of motion of the follower Terms used in Radial Cams Motion of the Follower Displacement Diagram, Velocity and cceleration nalysis when the follower Moves with uniform Velocity Displacement Diagram, Velocity and cceleration nalysis when the follower Moves with SHM (Simple Harmonic Motions) Maximum Velocity and Maximum cceleration Displacement Diagram, Velocity and cceleration nalysis when the Follower Moves with Uniform cceleration and Retardation Displacement Diagram, Velocity and cceleration nalysis when the Follower Moves with Cycloidal Motion Graphical Cam Profile with Knife Edge Follower Graphical Cam Profile with Roller Follower Graphical Cam Profile with Flat Faced Follower Cam Profile with Oscillating Follower Graphical Cam Profile with Cycloidal Motion nalysis of Tangent Cam with Roller Follower nalysis of Circular rc Cam with Flat Faced Follower: Polynomial Cams

8 Contents C.5 Chapter 4 GERS 4.1 Introduction dvantages and Limitations of Gear Drive Classification of Gears Based on position of axes of the shaft Based on type of gearing Terminology of Spur Gears nd Definitions Gear Materials Law of Gearing Involute Spur Gears Involutometry Velocity of sliding Length of path of contact Length of arc of contact Contact ratio (or) Number of pairs of teeth in contact Important Formulae Interference in Involute Gears Minimum Number of Teeth on Pinion to void Interference Minimum Number of Teeth on Wheel to void Interference Undercutting Backlash Gear Standardisation and Interchangeability Interchangeability Gear Standardisation dvantages of standard gears and interchangeability Non-standard Gears Long and short addendum system Purpose of Non-Standard gears

9 C.6 Mechanics of Machinery Centre Distance Modification Bevel Gears Types of bevel gears Bevel gear nomenclature Helical Gears pplications Worm and Worm Wheel Introduction Terms used in worm gearing Efficiency of worm gearing Chapter 5 GER TRINS 5.1 Introduction Classification Simple Gear Train Velocity ratio (or) Speed ratio : Compound Gear Train Epicyclic Gear Train (or) Planetry Gear Train pplications of Epicyclic gear trains Velocity Ratio for Epicyclic gear train Solution of Planetary Gear Train Problems: Differential Gear Final Drive using Differential gears Chapter 6 KINEMTIC SYNTHESIS OF PLNR MECHNISM 6.1 Introduction Tasks of Kinematic Synthesis Type Synthesis Number Synthesis Degrees of freedom DOF

10 Contents C Dimensional Synthesis Path Generation Function Generation Motion Generation (or) Rigid body guidance Precision Points (ccuracy Points) Chebyshev s spacing of precision points Graphical Synthesis for Motion Generation ngular Relationships for Function Generation Coupler Curves Equation of Coupler Curves for 4 bar Mechanism nalytical Synthesis Techniques nalytical Synthesis Techniques Problems Velocity and cceleration Synthesis using complex Number Methods Dead points Graphical Synthesis Two position synthesis - Overlay method - Problems Three position synthesis - solved problems Case study - Quick Return Mechanism

11 Basics of Mechanisms 1.1 Chapter 1 BSICS OF MECHNISMS Introduction to kinematics and mechanisms various mechanisms, kinematic diagrams, degree of freedom Grashof s criterion, inversions, coupler curves straight line mechanism exact approximate ckerman steering mechanism Hooke s joint Geneva mechanism mechanical advantage, transmission angle. 1.1 INTRODUCTION TO KINEMTICS Kinematics of Machinery (or) simply kinematics is a branch of engineering science which deals with the study of reltive motion between variuos parts of machine without considering forces. Note: Dynamics of machines deal with the relative motion between the parts by considering forces Machine machine is an apparatus for applying mechanical power consisting of a number of interrelated parts, each having a definite function. machine consists of so many links. Machine is an assemblage of rigid bodies that transmits and/or transforms forces, motion and energy in a predetermined manner, to do work. Examples: (a) Scooter is a machine which converts the available energy of petrol into useful mechanical work. (b) dynamometer used in bi-cycle is a machine which converts mechanical energy into electrical energy which powers the head lamp of the bi-cycle. (c) ll the machine tools like lathe, shaper, planar etc., are machines as they convert the electrical energy into useful work like turning, thread cutting, shaping etc., Resistant body: body is said to be resistant if it is capable of transmitting the required force with negligible deformation. These bodies are the parts of the machines which are used for transmitting motion and forces. Practically there is some deformatin in the body while transmitting the motion or force. link need not necessarily be a rigid body but it must be a resistant body. Examples: Springs and belts are not rigid links but they are resistant bodies.

12 Mechanics of Machinery Kerala structure is an assemblage of number of resistant bodies having no relative motion between them and structures are meant for taking up loads. In structure, there will not be any relative motion between the links. Forces are transmitted through links but there are no motion. Link Link Link -1 Fig. 1.1 Structure Difference between Machine and Structure Sl.No. Machine Structure 1. There is relative motion between the parts/ members of machine. 2. Machine converts available energy into useful work. 3. Members of machine are meant to transmit motion and forces. 4. Machine modifies and transmits the mechanical work. 5. Examples are Car, Lathe, Scooter, Shaping machines, Press etc. There is no relative motion between the parts/ members of structure. Structure does not convert available energy into useful work. Members of structure are meant to-take up loads only. Structure modifies and transmits forces only Examples are Bridges, Frames, trusses, buildings etc.,

13 Basics of Mechanisms KINEMTIC LINK, PIR ND CHIN Three important terms to be considered are 1. Kinematic link (or) Element 2. Kinematic pair 3. Kinematic chain 1.3 KINEMTIC LINK (OR) ELEMENT It is a resistant body (or) assembly of resistant bodies of a machine connecting other parts of the machine with relative motion between them Types of links The type of links available in order to transmit motion are: (a) Rigid Link rigid link is one which does not undergo any deformation while transmitting motion. Practically rigid links does not exist. However, the deformation in connecting rod, crank shaft, piston etc., are not appreciable, so these can be considered as rigid links. (b) Flexible link flexible link is one which undergoes partial deformation without affecting the transfer of motion. Examples: Ropes, belts, chains, springs etc., are flexible to transfer motions. (c) Fluid link fluid link is a link which has a fluid in the container and motion is transmitted through the fluid by pressure or compression only as in case of hydraulic presses, jacks, brakes etc., Slider - Crank Mechanism Consider the slider-crank mechanism. (reciprocating engine mechanism). Link 1: Engine head, cylinder and crankshaft bearing constitute link 1. Since this link does not have any movement, it is considered as fixed link. Link 2: The crankshaft and flywheel constitute the link 2. They rotate about a fixed axis. ie. They are subjected to rotary motion.

14 Mechanics of Machinery Kerala Link 3: It is a connecting rod which connects the crankshaft and piston. It has rotary as well as translatory motion. ie. It has general plane motion. (Combination of rotation and translation). Link 4: The piston and gudgeon pin constitute the link 4. It has reciprocating motion. (Translation motion). So all links constitute a machine. These links have relative motions with one another. 1.4 KINEMTIC PIR joint of two links (elements) that permits relative motion is called a pair. Consider the slider crank mechanism. Refer Fig 1.2. is a turning pair joining fixed link 1 (crankshaft bearing) and link 2 (crank shaft). Connecting rod B Crank shaft Piston C Link3 Link2 Fly wheel Link4 D Frame & cylinder(fixed) Link1 Fig.1.2 B is a turning pair joining link 2 (crankshaft) and link 3 (connecting rod) C is a turning pair joining link 3 (connecting rod) and link 4 (piston) D is a sliding pair joining link 4 (piston) and fixed link 1 (frame and cylinder) Types of kinematic pairs The kinematic pairs are classified based on the 1. Nature of relative motion between the links. 2. Nature of contact between the links. 3. Nature of the mechanical arrangement for complete (or) successful constraint between the elements.

15 Basics of Mechanisms 1.5 Based on the nature of relative motion between the links, the kinematic pairs are classified as follows. 1. Sliding pair 2. Turning pair 3. Cylindrical pair 4. Rolling pair 5. Spherical pair 6. Helical pair (or) Screw pair Based on nature of contact between the links, the kinematic pairs are classified as follows. 1. Lower pairs 2. Higher pairs. Based on the nature of mechanical constraint, the kinematic pairs are classified as follows. 1. Closed pair 2. Unclosed pair Kinematic Pairs-based on nature of relative motion 1. Sliding pair rectangular bar (Link ) sliding in a rectangular hole B (link B) is considered as sliding pair. This bar can move linearly but will not rotate. So it has only one degree B of freedom. Since it has sliding motion only, it is considered as completely constrained motion. Fig.1.3 Sliding Pair In a slider crank mechanism, the piston and cylinder form a sliding pair, since there is a sliding motion of piston surface relative to cylinder surface. Here the piston will not rotate in the cylinder since it is connected with connecting rod by gudgeon pin. So it is called successfully constrained motion.

16 Mechanics of Machinery Kerala Piston Connecting rod C Link3 B Crank shaft Fly wheel Link2 Sliding Pair D Link4 Frame & cylinder Link1 Fig Turning pair It is also called Revolute pair (or) Hinged pair. If a link has only rotary motion relative to another link, then these links form a turning pair. It allows only rotary motion. So it has single degree of freedom. i.e., this rotational motion can be expressed in terms of only. In slider crank mechanism,,b and C are the turning pairs. The following turning pair shows a shaft with two collars and a bearing B in which it rotates. Turning pair B Here & B form a turning pair Fig. 1.5 Turning pair Since it has only rotary motion and no linear motion, it is considered as completely constrained motion. 3. Cylindrical pair If the collar is removed in a turning pair, then the shaft has two motions-translation as well as rotation. This pair is called cylindrical pair. It has two degrees of freedom. These motions have no relationship with each

17 Basics of Mechanisms 1.7 B Fig. 1.6 Cylindrical pair other. Since the shaft can move linearly (or) rotate, this motion is known as incompletely constrained motion. 4. Rolling pair wheel and surface on which it rolls form a rolling pair at the line of contact. Consider the belt and pulley. The connection between the belt surface and pulley surface constitutes a rolling pair. Rolling wheel Pulley Belt Pulley Floor Fig. 1.7 Rolling Pair 5. Spherical pair ball and a socket joint form a spherical pair. Here is a ball element and B is a socket element. y Link x B Link B Ball and Socket joint Fig. 1.8 Spherical Pair z

18 Mechanics of Machinery Kerala This pair has three degrees of freedom. The coordinates, and are needed to describe the relative motion of link with respect to B. 6. Screw Pair (or) Helical Pair If a screw is rotating and moving inside a nut, then it is known as screw pair. Here both rotational and linear motions of relative to B occur. Even then, it is considered as completely constrained motion. Because, a specified amount of rotation of relative to B makes a proportional amount of axial motion (linear motion) of relative to B. B Screw Pair or Helical Pair Fig Kinematic Pair-Based on nature of contact 1. Lower pair pair having surface contact or area contact between the two elements while in motion, is called a lower pair. The relative motion in lower pair is only turning or sliding. In slider-crank mechanisms, a pairs, B, C and D are lower pairs. 2. Higher pair If a pair has a line contact between the two elements while in motion, it is called a higher pair. Consider Fig In this slider crank mechanism, the piston is replaced by a sphere. The cylinder and sphere have line contact. The motion between the sphere and the cylinder is sliding as well as turning. So it is a complicated motion. The cam and followers (Fig 1.10 B), tooth gears, ball bearings and roller bearings are the examples of higher pair.

19 Basics of Mechanisms 1.9 Connecting rod Crankshaft Flywheel Follower Sphere Link3 B Sliding & turning Pair C ylinder () Cam Fig (B) 1.5 KINEMTIC CHIN If the last link is joined to first link to transmit definite motion, then it is known as kinematic chain. It is a combination of kinematic pairs and the relative motion between the links is completely (or) successfully constrained. Load Shaft Foot step bearing Fig. 1.11

20 Mechanics of Machinery Kerala If a number of links are connected in space so that relative motion of one link with respect to another link follows a law, then the chain is called kinematic chain. Note: Successfully constrained motion Consider a foot step bearing. In this case, the shaft may rotate in bearing or it can move upwards. If the load is applied on the shaft to prevent upward movement, then the shaft will only rotate. Now this motion is considered as successfully constrained motion. The relation between number of pairs (p) and number of links l is given below. l 2p 4... (a) The relation between number of links l and number of joints (j) is also given below. j 3 2 l 2...(b) The above equations (a) and (b) are used only for lower pair. To make these formulae to valid for higher pair, we have to consider each higher pair as equivalent to two lower pairs with an additional link. Determine, whether the following are kinematic chain (or) not? 1. rrangement of three links Here no.of pair = 3 No.of links l = 3 pply equation, l 2p link 1 Fig Since LHS RHS, it is not a kinematic chain. So no relative motion is possible. It is known as structure. (or locked chain). 2. rrangement of four bar chain Here No.of links l 4; No.of pair p 4 link 3 link 2

21 Basics of Mechanisms 1.11 pply the equation (a) l 2p link 3 link link 2 Since LHS = RHS, it satisfies the equation (a). lso, pply the equation (b) link 1 Fig j 3 2 l Since LHS = RHS, it satisfies the equation (b). Since it satisfies the both equations (a) and (b), it is kinematic chain. 1.6 INTRODUCTION TO MECHNISM If one of the links of a kinematic chain is fixed, then the chain is known as mechanism. If a mechanism uses four links, then it is known as +y -z c.w c.c.w c.w c.c.w - x +x c.c.w c.w +z -y c.w = clockwise c.c.w = counter clockwise Fig. 1.14

22 Mechanics of Machinery Kerala simple mechanism and if the mechanism uses more than four links, then it is known as compound mechanism. If the mechanism is used to transmit power (or) to do work, then it is known as machine. To obtain a different mechanism, a different link can be fixed. The main function of mechanism is to transmit (or) to modify motion. nd the main function of machine is to obtain mechanical advantage. 1.7 NUMBER OF DEGREES OF FREEDOM (OR) MOBILITY Number of degrees of freedom is also known as movability (or) mobility. If a body is in space, it has 6 degrees of freedom. 1. The body may move in x (or) -x directions. 2. It may move in y (or) -y directions. 3. It may move in z (or) -z directions. 4. lso the body may rotate clockwise (or) anticlockwise about x axis. 5. It may rotate clockwise (or) anticlockwise about y axis. 6. It may rotate clockwise (or) anticlockwise about z axis. So totally it has 6 degrees of freedom. If the body is resting on a floor, then it may not move in -y direction. nd also, it may not rotate about x axis and z axis. In this case, the body D Link 3 C Link 4 D Link 3 L ink 4 Lin k 2 E Link 5 Link 2 C Link 1 B Link 1 B (a) Four bar chain Fig (b) Five bar chain

23 Basics of Mechanisms 1.13 has only 3 degrees of freedom. Similarly, we can arrest the motions (linear motion and rotary motion) and reduce the number of degrees of freedom. If four bar chain is considered, it has only one degree of freedom. ie The link 4 can only rotate. So only is sufficient to define the relative position of all other links. But in case of five bar chain, 1 and 2 are required to define the relative position of all other links. So the number of degrees of freedom for 5 bar chain is KUTZBCH CRITERION (OR) GRUBLER S EQUTION Kutzbach criterion is used to determine the number of degrees of freedom (or) movability (n) of a mechanism. This number of degrees of freedom can be determined directly from the number of links, number of joints and type of pair. Number of degrees of freedom n 3 l 1 2j h Kutzbach Criterion where l number of links j number of binary joints or lower pairs. h number of higher pairs If there is no higher pair,then h = 0 For three bar mechanism, l 3; j 3; h 0 So, n 3 l 1 2j h Number of degrees of freedom for three bar mechanism n 0 3 C 1 2 B Three bar mechanism Fig. 1.16

24 Mechanics of Machinery Kerala For four bar mechanism, l 4, j 4; h 0 Then n 3 l 1 2j h Number of degrees of freedom for 4 bar mechanism For five bar mechanism, l 5; j 5; h 0 n 1 D C B Four bar mechanism Fig D 4 3 The n 3 l 1 2j h Number of degrees of freedom n 2 Consider this three link mechanism Here number of links l = 3 ; binary joints j = 2 higher pair h = 1 We know, Kutzbach criterion, n 3 l 1 2j h 3 E 5 2 B 1 Five bar mechanism Fig C Consider the four link mechanism Here, no. of links 4 1 Three link mechanism Mechanism with a higher pair Fig Binary joints 3

25 Basics of Mechanisms 1.15 higher pair h 1 n 3 l 1 2j h wheel Grubler s Criterion for Plane Mechanism The Grubler s criterion applies to mechanisms with only single degree of freedom joints, where the overall movability of the mechanism is equal to one. By substituting n 1 and h 0 in Kutzbach equation, n 3 l 1 2j h we get 3l 2j 4 0 This equation is known as the Grubler s criterion for plane mechanism with constrained motion. plane mechanism with a movability of one and only single degree of freedom joints cannot have odd number of joints. The example of this mechanism is Four-bar mechanism Slider-crank mechanism. where link l 4 and joint j THREE IMPORTNT KINEMTIC CHINS: Mechanism with a higher pair Fig (with four lower pairs) Various mechanisms. 1. Four bar chain (or) quadric cyclic chain (4 turning pairs) 2. Single slider crank chain (Three turning pairs and one sliding pair) 3. Double slider crank chain (Two turning pairs and two sliding pairs) 1 Four link mechanism

26 Mechanics of Machinery Kerala 1.10 FOUR BR CHIN: Connecting rod (or) coupler C D 3 Rocker (lever) (follower) 4 crank (Driver) 2 1 fixed link(frame) B Fig Four bar chain The simplest and the basic kinematic chain is a four bar chain. It consists of four links link (1): Fixed link (frame) link (2): Rocker (lever (or) follower) link (3): Connecting rod (coupler) link (4): Crank (driver) Grashof s law for a four bar mechanism Grashof s law states that 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. D L in k 3 C Link 4 L in k 2 Link 1 B Fig. 1.22

27 Basics of Mechanisms 1.17 In this four bar mechanism, link 4 is shortest length. link 3 is longest length ccording to Grashof s law, Length link 4 link 3 length link 1 link 2 In four bar chain, when the crank (link 4 ) is the driver, the mechanism is transforming rotary motion into oscillating motion Kinematic inversions If link 1 is fixed and other links are in relative motion, then it is known as one mechanism. If link 2 is fixed and other links are in relative motion, then it is known as another mechanism. If different links are fixed to get different mechanisms, then the different mechanisms are known as kinematic inversions. By fixing, in turn, different links in a kinematic chain, kinematic inversion is obtained Inversions of four bar chain Important inversions of four bar chain are 1. Beam engine (crank and lever mechanism) 2. Coupling rod of a locomotive (Double crank mechanism) 3. Watt s indicator mechanism (Double lever mechanism) 1. Beam engine: (Crank and lever mechanism) It consists of four links. This mechanism converts the rotary motion into reciprocating motion. The crank (link 2) rotates about. This motion is transmitted to oscillating lever (link 4) through connecting rod (link 3). The oscillating lever oscillates about D and it makes the piston to reciprocate inside the cylinder.

28 Mechanics of Machinery Kerala O scillating lever (link 4) E Piston rod D C cylinder (link1) Link 3 (connecting rod) Frame(Link 1) B Fig Beam engine crank (Link 2) 2. Coupling rod of a locomotive (Double crank mechanism) Link 4 D Wheels Link 3 C Link 2 Link 1 B Fig Coupling rod of a locomotive This chain consists of four links as shown in Fig This mechanism is used to transmit rotary motion from one wheel to the other wheel. In train, the power from leading wheel is transmitted to trailing wheel. Here D and BC are two cranks having equal length. They are connected to two different wheels. The link CD (link 3) is the coupling rod coupling both wheels. The fixed link (1) maintains the constant centre to centre distance between the two wheels.

29 Basics of Mechanisms Watt s Indicator mechanism (Double lever mechanism) It is used to indicate the intensity of the steam (or) gas pressure inside the cylinder. It consists of four links. C Link2 C D Link1 B B F F D Link4 E E Link3 Indicator plunger Indicator cylinder Fig Watt s indicator mechanism 1. Fixed link (link 1) at 2. Link C (link 2) 3. Link CE (link 3) 4. Link BFD (The link BF and FD have no relative motion with each other. So both are considered as one link). The displacement of the link BFD is directly proportional to the gas pressure inside the cylinder. This displacement is shown by the tracing point E at the end of link CE. The continuous (solid) lines show the initial position of the mechanism and the dotted lines show the extreme position when the pressure of gas is sufficiently more SINGLE SLIDER CRNK CHIN This mechanism is used to convert the rotary motion into reciprocating motion and vice versa. It is a modification of basic four bar chain. It consists of one sliding pair and three turning pairs.

30 Mechanics of Machinery Kerala Guides Connecting rod (Link 3) Crank (Link 2) Frame (Link 1) Frame (Link 1) (Link 4) Piston Fig Single slider crank chain Link 1 and 2 - Turning pair link 2 and 3 - Turning pair link 3 and 4 - Turning pair link 4 and 1 - Sliding pair Link 1 : Frame (Fixed link) Link 2 : Crank Link 3 : Connecting rod Link 4 : Cross head (or) slider (or) piston Inversions of single slider crank chain By fixing, in turn, different links in a kinematic chain, an inversion is obtained. Important inversions: 1. Pendulum pump (or) Bull engine 2. Oscillating cylinder engine 3. Rotary internal combustion engine (or) Gnome engine. 4. Crank and slotted lever quick return motion mechanism. 5. Whitworth quick return motion mechanism.

31 Basics of Mechanisms Pendulum pump (or) Bull engine Here, the link 4 (cylinder) is fixed, when the crank (link 2) rotates, the connecting rod oscillates about a pin pivoted to the fixed link 4 at and the piston attached to the piston rod reciprocates. This type of mechanism is used in duplex pump to supply feed water to the boilers. Cylinder (Link 4) Connecting rod (Link 3) Piston rod (Link 1) Crank (Link 2) Cylinder (Link 4) Fig Pendulum pump 2. Oscillating cylinder engine This mechanism is used to convert rotary motion into oscillatory motion. Here the connecting rod (link 3) is fixed. When the crank rotates, Piston rod (Link 1) C ra nk (Link 2) cy linder (Link 4) Connecting rod (Link 3) Fig Oscillating cylinder engine

32 Mechanics of Machinery Kerala the piston attached to piston rod reciprocates and the cylinder oscillates about a pin pivoted to the fixed link at. 3. Rotary internal combustion engine (or) Gnome engine It consists of seven cylinders and all revolve about fixed centre D. In this mechanism, the crank (link 2) is fixed. When the connecting rod (link 4) rotates, the piston (link 3) reciprocates inside the cylinders (link 1). Connecting rod (Link 4) Piston (Link 3) Fixed crank (Link 2) Cylinder (Link 1) D Fig Rotary internal combustion engine 4. Crank and slotted lever quick return motion mechanism This mechanism is used in shaping machines and slotting machines. Here, the link (3) (This link (3) corresponds to connecting rod in reciprocating engine mechanism) is fixed. The crank CB (link 2) rotates with uniform speed. slider (link 1) is attached to the crank pin B. It slides along slotted bar P and so the slotter bar P oscillates about the pivoted point. The link (or) a connecting rod connects the P with ram which carries the tool and reciprocates. The cutting stroke occurs when the crank rotates from position CB 1 to CB 2 (ie through an angle ). The return stroke occurs when the crank rotates from position CB 2 to CB 1 (ie through an angle ).

33 Basics of Mechanisms 1.23 Connecting rod Ram Cutting stroke Return stroke Too l Line of stroke R 1 P R R 2 Q P 1 P 2 Slider (Link 1) B C B 1 (90 o - 2 ) B 2 Fixed (Link 3) Slotted bar (Link 4) Fig Crank and slotted lever quick return motion mechanism Note that is greater than. ie. The angle made by cutting stroke is greater than the angle made by return stroke. Since the crank rotates with uniform angular speed, the return stroke occurs quickly. That is why, it is called quick return motion mechanism. 5. Whitworth quick return motion mechanism This mechanism is also mostly used in shaping machines and slotting machines. Here the link 2 is fixed. (This link 2 corresponds to crank in a reciprocating steam engine). The driving crank C (link 3) rotates with uniform speed. The slider (link 4) is attached to the crank pin. It slides along the slotted bar P (link 1). So the slotted bar P oscillates about a

34 Mechanics of Machinery Kerala connecting rod Line of stroke Slotted bar (Link 1) Slider (Link 4) Fig P 1 P 2 D pivoted point D. The connecting rod PR connects the slotted bar and ram which carries the tool and reciprocates. When the driving crank C moves from C 1 to C 2 (clockwise) through an angle, the tool is moving forward stroke (cutting stroke). When the driving crank C moves from C 2 to C 1 through an angle (clockwise), the tool is moving back (return stroke). Since is greater than and the driving crank C rotates with uniform angular velocity, the return stroke is quicker than the cutting stroke. Thus it executes the quick return motion DOUBLE SLIDER CRNK CHIN It consists of two turning pairs and two sliding pairs Inversions of Double slider crank chain Three important inversions 1. Elliptical trammel 2. Scotch yoke mechanism 3. Oldham s coupling C P 1 R 1 Ram R Driving crank (Link 3) cutting stroke Return stroke R 2 Fixed (Link 2) Whitworth quick return motion mechanism Tool

35 Basics of Mechanisms Elliptical trammel Elliptical trammel is an instrument used for drawing ellipses. Refer the fig. Sider (Link 3) Connecting Bar (Link 2) P Slotted plate (Link 4) Fixed B Sider (Link 1) Fig Elliptical trammels This inversion is obtained by fixing the link 4 (slotted plate). This slotted plate has two straight grooves at right angle to each other. (Just like + sign). The slider (link 1) and slotted plate (link 4) form one sliding pair. The slider (link 3) and slotted plate (link 4) form another sliding pair. The connecting bar (link 2) and slider (link 1) form one turning pair; The connecting bar (link 2) and slider (link 3) form another turning pair. When the sliders slide along their respective grooves, any point on connecting bar (say point P) traces out an ellipse. Here P is the half of major axis of the ellipse and BP is the half of minor axis of the ellipse.

36 Mechanics of Machinery Kerala 2. Scotch yoke mechanism Scotch yoke mechanism is used to convert rotary motion into reciprocating motion. The inversion is obtained by fixing any one of the sliders link 1 or link 3. Link 3 Link 1 Link 2 Link 4 B 1 Fig Scotch Yoke Mechanism In this mechanism, link 1 is fixed. In such arrangement, the whole frame ie link 4 will reciprocate. When the crank (link 2) rotates about B as centre, the whole frame (link 4) reciprocates. The fixed link 1 guides the frame. 3. Oldham s coupling This coupling is used for connecting two parallel shafts whose axes are at a small distance apart. In this mechanism, when one shaft rotates, the other shaft also rotates at the same speed. The link 1 (flange) is rigidly fastened to the end of the driving shaft by forging. shaft. The link 3 (another flange) is rigidly fastened to the end of the driven The supporting frame (link 2) is fixed. The link 1 and link 3 have diametrical slots cut in their inner faces.

37 Basics of Mechanisms 1.27 Flange (Link 1) Interm ediate piece (Link 4) Ton gue Driving shaft C D Flange (Link 3) Ton gue Supporting fram e (Link 2) Fig Oldham s coupling B Driven shaft Interm ediate piece (Link) In between to these two flanges, an intermediate piece (link 4) is sliding. It has two tonges-two diametrical projections on both faces - at right angles to each other. The link 4 slides on the slots of the two links 1 and 3. The rotary motion is given to link 1. So the intermediate piece also rotates with the same speed. Hence the link 3 also rotates with the same speed, i.e., links 1, 3 and 4 have the same angular velocity at every instant. The distance between the axes of the shafts is constant and therefore the centre of the intermediate piece will follow the path of a circle with radius equal to the distance between the axes of the two shafts. Therefore, maximum sliding speed of each tongue of the intermediate piece in the slot will be given by the peripheral velocity of the centre of the disc along its circular path. Peripheral velocity of the disc ngular velocity of the shaft distance between the axes of the shafts V r

38 Mechanics of Machinery Kerala 1.13 MECHNICL DVNTGE It is defined as the ratio of driven link torque to driver link torque. In a four bar mechanism as shown in Fig the link PQ is driving link and RS is driven link. R Driving link Q Driven link R Q P S Fig Four bar Mechanism The mechanical advantage of four bar mechanism varies depending on the position of the links. So both angles (,) changes continuously. When the is too small let the mechanical advantage becomes infinite. T R Driven torque T Q Driving torque M.. Ideal T R T Q Q R M. ctual T R T Q considering friction 1.14 TRNSMISSION NGLE It s defined as the angle between the driven link RS and link QR. If the angle is too small, then mechanical advantage approaches zero.

39 Basics of Mechanisms DESCRIPTION OF COMMON MECHNISM Offset Slider crank mechanism as a quick return Mechanism. Offset slider mechanism This mechanism is used to convert the rotary motion into reciprocating motion and vice versa. It is a modification of basic four bar chain. It consists of one sliding pair and three turning pairs. Guides Connecting rod (Link 3) Crank (Link 2) e = offset Frame (Link 1) (Link 4) Piston Fig Off-set slider crank chain Link 1 and 2 - Turning pair link 2 and 3 - Turning pair link 3 and 4 - Turning pair link 4 and 1 - Sliding pair Link 1 : Frame (Fixed link) Link 2 : Crank Link 3 : Connecting rod Link 4 : Cross head (or) slider. In some specific applications the common line of slider and crank centre is offset by e distance as shown in Fig 1.36 which is known as offset slider crank mechanism. The same mechanism can be used as a quick return mechanism because of the slider has different velocities during forward and return stroke.

40 Mechanics of Machinery Kerala F O O= r = Crank Radius B= =Connecting rod e R B F B R M 120 mm Fig Let O r Crank radius B l Length of the connecting rod e off set distance The farthest position of slider is obtained when the crank O R and connecting rod R B R of slider are at a distance r l from crank centre O. OB R O R R B R r l similarly the closest position of the slider is obtained, when...(i) OB F B F F O F l r from B F OM and B R OM cos MOB F OM e OB F l r Hence Q cos 1 cos MOB R OM e OB R r l e r l cos 1 e l r...(ii)

41 Basics of Mechanisms 1.31 Thus return stroke angle forward stroke angle Q R 180 Q F 180 The ratio of forward to return stroke timing is given as Time of forward stroke Time of return stroke angle of forward stroke angle of return stroke Q F Q R pplication: 1. It s used in automatic packing machines. 2. In assembly line it is used to push parts to work area. 3. It is also used in punching / rivetting press Pantograph pantograph is an instrument having path four-bar links with lower pairs used to re-produce a path exactly similar to one traced out by a point on the linkage. The path may be on an enlarged (or) reduced scale as required. B C O D B E C D Fig E

42 Mechanics of Machinery Kerala It is made up of bars connected by turning pairs. The bars B and BC are extended to O and E respectively, such that O/OB D/BE Thus, for all relative positions of the bars, the triangles OD and OBE are similar and the points O, D and E are in one straight line. It may be noted that point E traces out the same path as described by point D. From similar triangles OD and OBE, we find that OD/OE D/BE Let point O be fixed and the points D and E move to some new positions D and E. Then OD/OE OD/OE Now we can know that the straight line DD is parallel to the straight line EE. Hence, if O is fixed to the frame of a machine by means of a turning pair and D is attached to a point in the machine which has rectilinear motion relative to the frame, then E will also trace out a straight line path. Similarly, if E is constrained to move in a straight line, then D will trace out a straight line parallel to the former. pantograph is mostly used for the reproduction of plane areas and figures such as maps, plans etc., on enlarged or reduced scales. It is, sometimes, used as an indicator rig in order to reproduce to a small scale the displacement of the crosshead and therefore of the piston of a reciprocating steam engine. It is also used to guide cutting tools. modified form of pantograph is used to collect power at the top of an electric locomotive Straight line Generators 1. Paucellier Mechanism Paucellier mechanism consists of eight links as shown in Fig OQ QC ; OB O ; BP P C CB OQ is fixed link and QC is rotating link. It can be proved that as the link QC moves around Q, P moves in a straight line perpendicular to QO.

43 Basics of Mechanisms 1.33 E P C O B Q R N Fig Paucellier Mechanism. Hence, OC OP constant. It can be proved as follows. From the triangle OE and PE O 2 OE 2 E 2...(1) Subtracting equation (2) from (1) P 2 EP 2 E 2...(2) O 2 P 2 OE 2 EP 2 OE EP OE EP OP OC Since O and P are constant length, therefore, the product of OP OC remains constant. Hence the Point P traces a straight line path, perpendicular to the diameter OR.

44 Mechanics of Machinery Kerala 2. Watt s Mechanism It s one of the simplest straight line generator mechanism. It consists of a four bar chain PQRS with fixed at P and S. In the mean position. The crank PQ and the followed link RS are parallel and coupler link QR is perpendicular to the link PQ and RS. R S O P Q Fig Links PQ and RS can oscillate about centers P and S respectively. It s seen that if O is a point on the line QR such that OQ OR RS PQ Then, for small oscillations of PQ and RS, O will trace an approximately straight line. Thus coupler link QR into two parts, which are inversely proportional to the lengths of the link PQ and RS. lso refer Watt s Indicator Mechanism in Section Steering mechanisms Steering mechanism in automobile is used to control the direction of a vehicle s motion. The perfect steering is achieved when all the four wheels are rolling perfectly under all conditions of running. While taking turns the condition of perfect rolling is satisfied if the axes of the front wheel when produced meet the rear wheel axis at one point. Fig. (1.41). Then this point (I) is the instantaneous centre of the vehicle.

45 W he el B a se b Basics of Mechanisms 1.35 Y C I Fig Steering Mechanism Perfect steering angle From the figure it is clear that the inside wheel is required to turn through a greater angle than the outer wheel. Larger the steering angle, smaller the turning circle; and by referring Fig for correct steering, cot Y C b Wheel Track Y b C b cot C b cot cot C b...(1) where angle of outside lock angle of inside lock. Equation 1 represents the basic condition for the steering mechanism for perfect rolling of all wheels.

46 Mechanics of Machinery Kerala ckermann steering mechanism ckermann steering mechanism is based upon the four bar chain mechanism with two longer links having equal lengths and two shorter links having equal lengths. If the vehicle has to travel in a straight line, the longer links should be parallel to each other and the shorter links are having an angle inclined with the longitudinal axis. (Fig. 1.42(a)) C K L B D B K + L E b a (a) For Straight Drive F Fig Rear xis (b) For Right Drive J So the turning of the vehicle is done by varying the angle of. If increases, then the vehicle turns right (Fig. 1.42(b)) and if decreases then it turns left. (Fig.1.41). Refer Fig Values of the angle for the given value of can be calculated from the ratio between the small and large linkages, such as D and angle. B