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Theory of machines

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THEORY OF MACHINES FOR VTU, AMIE, DME STUDENTS..
The study of a mechanism involves its analysis as well as synthesis.
Analysis is the study of motions and forces concerning different parts
of an existing mechanism. Whereas Synthesis involves the design of its
different parts.
Mechanics: It is that branch of scientific analysis which deals with
motion, time and force.
Kinematics is the study of motion, without considering the forces
which produce that motion. Kinematics of machines deals with the
study of the relative motion of machine parts. It involves the study of
position, displacement, velocity and acceleration of machine parts.
Dynamics of machines involves the study of forces acting on the
machine parts and the motions resulting from these forces.
Plane motion: A body has plane motion, if all its points move in
planes which are parallel to some reference plane. A body with plane
motion will have only three degrees of freedom. i.e., linear along two
axes parallel to the reference plane and rotational/angular about the
axis perpendicular to the reference plane. (eg. linear along X and Z
and rotational about Y.)The reference plane is called plane of motion.
Plane motion can be of three types. 1) Translation 2) rotation and 3)
combination of translation and rotation.
Translation: A body has translation if it moves so that all straight
lines in the body move to parallel positions. Rectilinear translation is a
motion wherein all points of the body move in straight lie paths.
Eg. The slider in slider crank mechanism has rectilinear translation.

Published in: Engineering
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Theory of machines

  1. 1. Shashidhar_gs@yahoo.co.in ANALYSIS AND SYNTHESIS OF MECHANISMS AND MACHINES BY: G.S SHASHIDHARA.,DME, BE, AMIE(I)
  2. 2. Shashidhar_gs@yahoo.co.in The study of a mechanism involves its analysis as well as synthesis. Analysis is the study of motions and forces concerning different parts of an existing mechanism. Whereas Synthesis involves the design of its different parts. Mechanics: It is that branch of scientific analysis which deals with motion, time and force. Kinematics is the study of motion, without considering the forces which produce that motion. Kinematics of machines deals with the study of the relative motion of machine parts. It involves the study of position, displacement, velocity and acceleration of machine parts. Dynamics of machines involves the study of forces acting on the machine parts and the motions resulting from these forces. Plane motion: A body has plane motion, if all its points move in planes which are parallel to some reference plane. A body with plane motion will have only three degrees of freedom. i.e., linear along two axes parallel to the reference plane and rotational/angular about the axis perpendicular to the reference plane. (eg. linear along X and Z and rotational about Y.)The reference plane is called plane of motion. Plane motion can be of three types. 1) Translation 2) rotation and 3) combination of translation and rotation. Translation: A body has translation if it moves so that all straight lines in the body move to parallel positions. Rectilinear translation is a motion wherein all points of the body move in straight lie paths. Eg. The slider in slider crank mechanism has rectilinear translation. (link 4 in fig.1.1)
  3. 3. Shashidhar_gs@yahoo.co.in Translation, in which points in a body move along curved paths, is called curvilinear translation. The tie rod connecting the wheels of a steam locomotive has curvilinear translation. (link 3 in fig.1.2) Rotation: In rotation, all points in a body remain at fixed distances from a line which is perpendicular to the plane of rotation. This line is the axis of rotation and points in the body describe circular paths about it. (Eg. link 2 in Fig.1.1 and links 2 & 4 in Fig.1.2) Translation and rotation: It is the combination of both translation and rotation which is exhibited by many machine parts. (Eg. link 3 in Fig.1.1) Link or element: It is the name given to any body which has motion relative to another. All materials have some elasticity. A rigid link is one, whose deformations are so small that they can be neglected in determining the motion parameters of the link. A link or element need not to be rigid body, but it must be a 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 link should have the following two characteristics: 1. It should have relative motion.
  4. 4. Shashidhar_gs@yahoo.co.in 2. It must be resistant body. TYPES OF LINKS: In order to transmit motion, driver and the follower may be connected by the following three types of links: (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 is partly deformed in a manner not to affect the transmission of motion. Eg: belts, ropes, chains and wires are flexible links and transmit tensile forces only. (3) Fluid link: A fluid link is one which is formed by having a fluid in a receptacle and the motion is transmitted through the fluid by pressure or compression only. Eg: hydraulic presses, jacks and brakes. Fig.1.3
  5. 5. Shashidhar_gs@yahoo.co.in Binary link: Link which is connected to other links at two points. (Fig.1.3 a) Ternary link: Link which is connected to other links at three points. (Fig.1.3 b) Quaternary link: Link which is connected to other links at four points. (Fig1.3 c) Pairing elements: the geometrical forms by which two members of a mechanism are joined together, so that the relative motion between these two is consistent are known as pairing elements and the pair so formed is called kinematic pair. Each individual link of a mechanism forms a pairing element. Fig.1.4 Kinematic pair Fig.1.5 Degrees of freedom (DOF): It is the number of independent coordinates required to describe the position of a body in space. A free body in space (fig 1.5) can have six degrees of freedom. I.e., linear positions along x, y and z axes and rotational/angular positions with respect to x, y and z axes. In a kinematic pair, depending on the constraints imposed on the motion, the links may loose some of the six degrees of freedom. 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 may consist of several parts, which are rigidly fastened together, so that they do not move relative to one another.
  6. 6. Shashidhar_gs@yahoo.co.in Mechanism: A mechanism is a combination of rigid or resistant bodies so formed and connected that they move upon each other with definite relative motion. such as the crank- connecting rod mechanism of the I.C. engines, steering mechanisms of automobiles……. etc. Machine: machine is a device which receives energy and transforms it into some useful work. A machine consists of a number of parts or bodies with successfully constrained motion which is used to transmit or transform motion to do some useful work. A machine is a mechanism or collection of mechanisms which transmit force from the source of power to the resistance to be overcome. 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. Eg: railway bridge, a roof truss, machine frames etc. Difference between a Machine and Structure: MACHINE STRUCTURE 1. The parts of a machine moves relative to one another. 1. The members of a structure do not move relative to one another. 2. A machine transforms the available energy into some useful work. 2. In a structure no energy is transformed into useful work. 3. The link of a machine may transmit both power and motion. Eg: Lathe, shaper, steam engine etc. 3. The members of a structure transmit forces only. Eg: Railway bridges, roof trusses, machine frame. Difference between a Machine and Mechanism MACHINE MECHANISM 1. Machine modifies mechanical work. 1. Mechanism transmits and modifies motion.
  7. 7. Shashidhar_gs@yahoo.co.in 2. A machine is a practical development of any mechanism. 2. A mechanism is a part of a machine. 3. A machine may have number of mechanisms for transmitting mechanical work or power. Eg: Lathe, Shaper, Steam engine etc. 3. A mechanism is the skeleton outline of the machine to produce motion between various links. Eg: Clock work, type-writer, an indicator to draw P.V diagrams of an engine etc. Rigid Body: is that body whose changes in shape are negligible compared with its overall dimensions or with the changes in position of the body as a whole, such as rigid link, rigid disc…..etc. Kinematic pair: When two elements or links are connected together in such a way that their relative motion is constrained, form a kinematic pair. Therefore, in order to compel a body to move in a definite path, it must be paired with another. If the constraint is not complete (not definite path) the pair is termed as incomplete or unsuccessful. Kinematic Pair 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 (i.e. in a definite direction), the pair is known as kinematic pair. Types of kinematic pairs: (i) Based on nature of contact between elements: (a) Lower pair. If the joint by which two members are connected has surface contact, the pair is known as lower pair. Eg. pin joints, shaft rotating in bush, slider in slider crank mechanism.
  8. 8. Shashidhar_gs@yahoo.co.in Fig.1.6 Lower pairs (b) Higher pair. If the contact between the pairing elements takes place at a point or along a line, such as in a ball bearing or between two gear teeth in contact, it is known as a higher pair. Fig.1.7 Higher pairs
  9. 9. Shashidhar_gs@yahoo.co.in (ii) Based on relative motion between pairing elements: (a) Sliding pair.Sliding pair is constituted by two elements so connected that one is constrained to have a sliding motion relative to the other. DOF = 1 Eg: rectangular rod in a rectangular hole. (b) Turning pair (revolute pair). When connections of the two elements are such that only a constrained motion of rotation of one element with respect to the other is possible, the pair constitutes a turning pair. DOF = 1 Eg: circular shaft revolving inside a bearing. (c) Cylindrical pair. If the relative motion between the pairing elements is the combination of turning and sliding, then it is called as cylindrical pair. DOF = 2 Fig.1.8 Sliding pair Fig.1.9 Turning pair Fig.1.10 Cylindrical pair (d) Rolling pair. When the pairing elements have rolling contact, the pair formed is called rolling pair. Eg. Bearings, Belt and pulley. DOF = 1 Eg: rolling wheel on a flat surface, ball and roller bearings etc.
  10. 10. Shashidhar_gs@yahoo.co.in Fig.1.11 (a) Ball bearing Fig.1.11(b) Belt and pulley (e) Spherical pair. A spherical pair will have surface contact and three degrees of freedom. Eg. Ball and socket joint. DOF = 3 Eg: ball and socket joint. (f) Helical pair or screw pair. When the nature of contact between the elements of a pair is such that one element can turn about the other by screw threads, it is known as screw pair. Eg. Nut and bolt. DOF = 1 Eg: lead screw and the nut of a lathe. Fig.1.12 Ball and socket joint Fig.1.13 Screw pair (iii) Based on the nature of mechanical constraint. (a) Closed pair. Elements of pairs held together mechanically due to their geometry constitute a closed pair. They are also called form-closed or self-closed pair. All the lower pairs and some of the higher pairs are closed pairs. A cam and follower pair(higher pair) and screw pair (lower pair) belong to the closed pair. (b) Unclosed or force closed pair. Elements of pairs held together by the action of external forces constitute unclosed or force closed pair .Eg. Cam and follower.
  11. 11. Shashidhar_gs@yahoo.co.in Fig.1.14 Closed pair Fig. 1.15 Force closed pair (cam & follower) Constrained motion: In a kinematic pair, if one element has got only one definite motion relative to the other, then the motion is called constrained motion. (a) Completely constrained motion. If the constrained motion is achieved by the pairing elements themselves, then it is called 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 called completely constrained motion. Eg: piston and cylinder (in a steam engine) form a pair and 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. Fig.1.16 Completely constrained motion (b) Incompletely constrained motion. When relative motion between pairing elements takes place in more than one direction, it is
  12. 12. Shashidhar_gs@yahoo.co.in called incompletely constrained motion. The change in the direction of impressed force may alter the direction of relative motion between the pair. Eg. Shaft in a circular hole. (it may either rotate or slide in a hole. These both motions have no relationship with the other). Fig.1.18 Incompletely constrained motion (c) 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 called successfully constrained motion. If constrained motion is not achieved by the pairing elements themselves, but by some other means, then, it is called successfully constrained motion. Eg. Foot step bearing, where shaft is constrained from moving upwards, by its self weight. Fig.1.17 Foot strep bearing Kinematic chain: A kinematic chain is a group of links either joined together or arranged in a manner that permits them to move relative
  13. 13. Shashidhar_gs@yahoo.co.i to one another. If the lin motion is possible, it results o.in links are connected in such a lts in a locked chain or structure. Fig.1.19 19 Locked chain or structure Mechanism: A mechani means that the motion of a definite and predictab Usually one of the links of t nism is a constrained kinemati any one link in the kinematic c able motion relative to each o the kinematic chain is fixed in a Fig.1.20 Slider er crank and four bar mechanism If, for a particular position of the other links of the ch unconstrained kinematic c ion of a link of the chain, the pos chain can not be predicted, then c chain and it is not mechanism. Fig.1.21 Un Unconstrained kinematic chain way that no atic chain. This c chain will give of the others. n mechanism. sms. ositions of each en it is called as
  14. 14. Shashidhar_gs@yahoo.co.in Machine: A machine is a mechanism or collection of mechanisms, which transmit force from the source of power to the resistance to be overcome. Though all machines are mechanisms, all mechanisms are not machines. Many instruments are mechanisms but are not machines, because they do no useful work nor do they transform energy. Eg. Mechanical clock, drafter. Fig.1.21 Drafter Planar mechanisms: When all the links of a mechanism have plane motion, it is called as a planar mechanism. All the links in a planar mechanism move in planes parallel to the reference plane. Degrees of freedom/mobility of a mechanism: It is the number of inputs (number of independent coordinates) required to describe the configuration or position of all the links of the mechanism, with respect to the fixed link at any given instant. An unconstrained rigid body moving in space can describe following independent motions: 1. Translational motions along any three mutually perpendicular axes x, y and z. 2. Rotational motions about these axes Thus a rigid body possesses six degrees of freedom. The connection of a link with another imposes certain constraints on their relative motion. The number of restraints can never be zero (joint is disconnected) or six (joint becomes solid).
  15. 15. Shashidhar_gs@yahoo.co.in Degrees of freedom of a pair is defined as the number of independent relative motions, both translational and rotational. A pair can have degrees of freedom = 6 – number of restraints. Grubler’s equation: Number of degrees of freedom of a mechanism is given by F = 3(n-1)-2l-h. Where, F = Degrees of freedom n = Number of links = n2 + n3 +……+nj, where, n2 = number of binary links, n3 = number of ternary links…etc. l = Number of lower pairs, which is obtained by counting the number of joints. If more than two links are joined together at any point, then, one additional lower pair is to be considered for every additional link. h = Number of higher pairs Examples of determination of degrees of freedom of planar mechanisms: (i)
  16. 16. Shashidhar_gs@yahoo.co.in F = 3(n-1)-2l-h Here, n2 = 4, n = 4, l = 4 and h = 0. F = 3(4-1)-2(4) = 1 I.e., one input to any one link will result in definite motion of all the links. (ii) F = 3(n-1)-2l-h Here, n2 = 5, n = 5, l = 5 and h = 0. F = 3(5-1)-2(5) = 2 I.e., two inputs to any two links are required to yield definite motions in all the links. (iii) F = 3(n-1)-2l-h
  17. 17. Shashidhar_gs@yahoo.co.in Here, n2 = 4, n3 =2, n = 6, l = 7 and h = 0. F = 3(6-1)-2(7) = 1 I.e., one input to any one link will result in definite motion of all the links. (iv) F = 3(n-1)-2l-h Here, n2 = 5, n3 =1, n = 6, l = 7 (at the intersection of 2, 3 and 4, two lower pairs are to be considered) and h = 0. F = 3(6-1)-2(7) = 1 (v) F = 3(n-1)-2l-h Here, n = 11, l = 15 (two lower pairs at the intersection of 3, 4, 6; 2, 4, 5; 5, 7, 8; 8, 10, 11) and h = 0. F = 3(11-1)-2(15) = 0 (vi) Determine the mobility of the following mechanisms.
  18. 18. Shashidhar_gs@yahoo.co.in (a) F = 3(n-1)-2l-h Here, n = 4, l = 5 and h = 0. F = 3(4-1)-2(5) = -1 I.e., it is a structure (b) F = 3(n-1)-2l-h Here, n = 3, l = 2 and h = 1. F = 3(3-1)-2(2)-1 = 1 (c) F = 3(n-1)-2l-h Here, n = 3, l = 2 and h = 1. F = 3(3-1)-2(2)-1 = 1 Inversions of mechanism: A mechanism is one in which one of the links of a kinematic chain is fixed. Different mechanisms can be obtained by fixing different links of the same kinematic chain. These are called as inversions of the mechanism. By changing the fixed link, the number of mechanisms which can be obtained is equal to the number of links. Excepting the original mechanism, all other mechanisms will be known as inversions of original mechanism. The inversion of a mechanism does not change the motion of its links relative to each other.
  19. 19. Shashidhar_gs@yahoo.co.in Four bar chain: Fig 1.22 Four bar chain One of the most useful and most common mechanisms is the four-bar linkage. In this mechanism, the link which can make complete rotation is known as crank (link 2). The link which oscillates is known as rocker or lever (link 4). And the link connecting these two is known as coupler (link 3). Link 1 is the frame. Inversions of four bar chain:
  20. 20. Shashidhar_gs@yahoo.co.in Fig.1.23 Inversions of four bar chain. Crank-rocker mechanism: In this mechanism, either link 1 or link 3 is fixed. Link 2 (crank) rotates completely and link 4 (rocker) oscillates. It is similar to (a) or (b) of fig.1.23. Fig.1.24
  21. 21. Shashidhar_gs@yahoo.co.in Drag link mechanism. Here link 2 is fixed and both links 1 and 4 make complete rotation but with different velocities. This is similar to 1.23(c). Fig.1.25 Double crank mechanism (Coupling rod of Locomotive) . This is one type of drag link mechanism, where, links 1& 3 are equal and parallel and links 2 & 4 are equal and parallel. When AB rotates about A, the crank DC rotates about D. This mechanism is used for coupling locomotive wheels. Since links AB and CD work as cranks, this mechanism is also known as double crank or crank-crank or drag-crank mechanism. Fig.1.26
  22. 22. Shashidhar_gs@yahoo.co.i Double rocker mechani makes complete rotation, o.in nism. In this mechanism, link 4 is n, whereas links 3 & 4 oscillate (Fi Slider crank chain: This one sliding pair and three is called crank. Link 3 ha motion and is called conn and is called slider. Link 1 convert rotary motion to r fixed. Link 2 Fig.1.23d) his is a kinematic chain having fou ee turning pairs. Link 2 has rotary has got combined rotary and necting rod. Link 4 has reciproc Inversions of slider cran mechanism is obtained by (a) crank fixed (b) con Rotary engine (Gnome mechanism. (crank fixe It is a rotary cylinder V-ty back, rotary internal com now-a-days gas turbines four links. It has ary motion and d reciprocating rocating motion nism is used to is frame (fixed). This mechani o reciprocating and vice versa. Fig1.27 ank chain: Inversions of slider cr fixing links 2, 3 and 4. crank onnecting rod fixed ( Fig.1.28 c) slider fixed e Engine) – I inversion of slid ixed) type internal combustion engin mbustion engines were used in es are used in its place. It con slider crank ine. Sometimes aviation. But onsists of seven
  23. 23. Shashidhar_gs@yahoo.co.in cylinders in one plane and all revolves about fixed centre A as shown in Fig. 1.29, while the crank OA (link 2) is fixed. In this mechanism, when the connecting rod (link 4) from pistons are connected to A rotates, the piston (link 3) reciprocates inside the cylinders forming link 1. Fig. 1.29 Whitworth quick return motion mechanism–I inversion of slider crank mechanism. This is first inversion of slider mechanism, where, crank 1 is fixed. Input is given to link 2, which moves at constant speed. Point C of the mechanism is connected to the tool post D of the machine. During cutting stroke, tool post moves from D to D. The corresponding positions of C are C and C as shown in the fig. 1.30. For the point C to move from C to C, point B moves from B to B, in anti-clockwise direction. I.E., cutting stroke takes place when input link moves through angle BOB in anticlockwise direction and return stroke takes place when input link moves through angle BOB in anti-clockwise direction.
  24. 24. Shashidhar_gs@yahoo.co.in Fig.1.30 Crank and slotted lever quick return motion mechanism – II inversion of slider crank mechanism (connecting rod fixed). This mechanism is mostly used in shaping machines, slotting machines and in rotary internal combustion engines. In this mechanism, the link AC (i.e. link 3) forming the turning pair is fixed, as shown in Fig. 1.31. The link 3 corresponds to the connecting rod of a reciprocating steam engine. 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 . The line of stroke of the ram (i.e. ) is perpendicular to AC produced.
  25. 25. Shashidhar_gs@yahoo.co.i o.in Oscillating cylinder eng mechanism (connecting Pendulum pump or bul mechanism (slider fixed In this mechanism, the in link 4 (i.e. sliding pair), a Fig.1.31 engine–II inversion of slider cr ing rod fixed). Fig.1.32 r crank ull engine–III inversion of sli xed). inversion is obtained by fixing th , as shown in Fig. 1.33. In this ca slider crank the cylinder or case, when the
  26. 26. Shashidhar_gs@yahoo.co.in 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. 1.33. Fig.1.33 Double slider crank chain: It is a kinematic chain consisting of two turning pairs and two sliding pairs. Scotch –Yoke mechanism. This mechanism is used for converting rotary motion into a reciprocating motion. Turning pairs – 12, 23; Sliding pairs – 34, 41.
  27. 27. Shashidhar_gs@yahoo.co.in Fig.1.34 Hand pump: Inversions of double slider crank mechanism: Elliptical trammel. This is a device which is used for generating an elliptical profile.
  28. 28. Shashidhar_gs@yahoo.co.i o.in In fig. 1.35, if AC = p and BC
  29. 29. Rearranging , of an ellipse. The path trac and minor axis equal to 2p Oldham coupling. Thi mechanism, An oldham's shafts whose axes are at a in such a way that if one the same speed. Fig.1.35 = q, then, x = q.cosθ and y = p Ɵ Ɵ = 1 This is th p.sinθ. the equation ith major axis raced by point C is an ellipse, with and 2q respectively. his is an inversion of double 's coupling is used for connecting le slider crank ing two parallel afts are coupled t also rotates at small distance apart. The shaft e shaft rotates, the other shaft a
  30. 30. Shashidhar_gs@yahoo.co.in Fig.1.36 Gears and gear trains: Introduction: The slip and creep in the belt or rope drives is a common phenomenon, in the transmission of motion or power between two shafts. The effect of slip is to reduce the velocity ratio of the drive. In precision machine, in which a definite velocity ratio is importance (as in watch mechanism, special purpose machines..etc), the only positive drive is by means of gears or toothed wheels. Friction Wheels: Kinematically, the motion and power transmitted by gears is equivalent to that transmitted by friction wheels or discs in contact with sufficient friction between them. In order to understand motion transmitted by two toothed wheels, let us consider the two discs placed together as shown in the figure 4.1. A gear is a toothed wheel with the teeth cut on the periphery of a cylinder or a cone, or sometimes on elliptical discs. Gears are mounted on the axles or shafts, and keyed to them. Two gears mounted one on each of the driving and driven shafts are arranged so that the teeth of one will mesh with the teeth of the other. When one of the discs is rotated, the other disc will be rotate as long as the tangential force exerted by the driving disc does not exceed the maximum frictional resistance between the two discs. But when the tangential force exceeds the frictional resistance, slipping will take place between the two discs. Thus the friction drive is not positive a drive, beyond certain limit. Gears are machine elements that transmit motion by means of successively engaging teeth. The gear teeth act like small levers. Gears are highly efficient (nearly 95%) due to primarily rolling contact between the teeth, thus the motion transmitted is considered as positive. Gears essentially allow positive engagement between teeth so high forces can be transmitted while still undergoing essentially rolling contact. Gears do not depend on friction and do best when friction is minimized. Gear Classification:
  31. 31. Shashidhar_gs@yahoo.co.in
  32. 32. Shashidhar_gs@yahoo.co.i 1. External gears: 2. Internal gears Gears may be classified acc revolution. The axes may b 1. Gears for connecting par 2. Gears for connecting inte 3. Gears for neither paralle o.in according to the relative position y be arallel shafts, intersecting shafts, llel nor intersecting shafts. Gears for connecting pa parallel shafts: 1. Spur gears: Spur gears have straight teeth, and a many spur gears are u reductions. Each time a ge the teeth collide, and this stress on the gear teeth. T most of the gears in your c n of the axes of ars are the most common type o d are mounted on parallel shaft used at once to create very gear tooth engages a tooth on th his impact makes a noise. It also . To reduce the noise and stress car are helical. e of gears. They fts. Sometimes, ery large gear the other gear, lso increases the ess in the gears,
  33. 33. Shashidhar_gs@yahoo.co.in PHOTO OF SPUR GEARS Spur gears are the most commonly used gear type. They are characterized by teeth, which are perpendicular to the face of the gear. Spur gears are most commonly available, and are generally the least expensive. They are used in small watches, precision measuring instruments, machine tools to gear boxes fitted in motor cars and aero engines, etc. · Limitations: Spur gears generally cannot be used when a direction change between the two shafts is required. · Advantages: Spur gears are easy to find, inexpensive, and efficient. 2. Parallel helical gears: The teeth on helical gears are cut at an angle to the face of the gear. Helical gears are used to connect parallel shafts as well as non-parallel, non-intersecting shafts. When two teeth on a helical gear system engage, the contact starts at one end of the tooth and gradually spreads as the gears rotate, until the two teeth are in full engagement.
  34. 34. Shashidhar_gs@yahoo.co.in
  35. 35. Shashidhar_gs@yahoo.co.in This gradual engagement makes helical gears operate much more smoothly and quietly than spur gears. For this reason, helical gears are used in almost all car transmission. Because of the angle of the teeth on helical gears, they create a thrust load on the gear when they mesh. Devices that use helical gears have bearings that can support this thrust load. One interesting thing about helical gears is that if the angles of the gear teeth are correct, they can be mounted on perpendicular shafts, adjusting the rotation angle by 90 degrees. Helical gears to have the following differences from spur gears of the same size: 1. Tooth strength is greater because the teeth are longer, 2. Greater surface contact on the teeth allows a helical gear to carry more load than a spur gear 3. The longer surface of contact reduces the efficiency of a helical gear relative to a spur gear Rack and pinion (The rack is like a gear whose axis is at infinity.): Racks are straight gears that are used to convert rotational motion to translational motion by means of a gear mesh. (They are in theory a gear with an infinite pitch diameter).
  36. 36. Shashidhar_gs@yahoo.co.in PHOTO OF RACK AND PINION Small gears are called pinions and racks are a series of teeth on a rectangular bar. They may be considered as spur gears of infinite radii. In theory, the torque and angular velocity of the pinion gear are related to the Force and the velocity of the rack by the radius of the pinion gear, as is shown. Perhaps the most well-known application of a rack is the rack and pinion steering system used on many cars in the past, lathe, drilling, planer, and steep rail tracks etc., are fitted with rack and pinion to convert rotary motion to straight line motion. Gears for connecting intersecting shafts: Bevel gears are useful when the direction of a shaft's rotation needs to be changed. They are usually mounted on shafts that are 90 degrees apart, but can be designed to work at other angles as well.
  37. 37. Shashidhar_gs@yahoo.co.i o.in PHO OTO OF BEVEL GEARS ars can be straight, spiral or hyp lly have the same problem as th engages; it impacts the corresp The teeth on bevel gears bevel gear teeth actually gear teeth, as each tooth all at once. Just like with spur gears, gear teeth. These spiral tee starts at one end of the whole tooth. ypoid. Straight straight spur esponding tooth s, the solution to this problem is teeth engage just like helical teeth he gear and progressively sprea On straight and spiral bev each other, but they must can engage with the axes i This feature is used in m differential and the input input pinion to be mounte evel gears, the shafts must be per st also be in the same plane. The es in different planes. many car differentials. The ring ut pinion gear are both hypoid. T ted lower than the axis of the rin to curve the eth: the contact eads across the erpendicular to he hypoid gear, ing gear of the . This allows the ring gear.
  38. 38. Shashidhar_gs@yahoo.co.in Figure shows the input pinion engaging the ring gear of the differential. Since the driveshaft of the car is connected to the input pinion, this also lowers the driveshaft. This means that the driveshaft doesn't pass into the passenger compartment of the car as much, making more room for people and cargo. Neither parallel nor intersecting shafts: Helical gears may be used to mesh two shafts that are not parallel, although they are still primarily use in parallel shaft applications. A special application in which helical gears are used is a crossed gear mesh, in which the two shafts are perpendicular to each other. worm gear: Worm gearing is essentially a special form of helical gearing in which the teeth have inline contact and the axes of the driving and driven shafts are usually at right angles and do not intersect. A worm drive consists of a worm (essentially a screw) which may have one or more number of helical threads of trapezoidal shape cut on it and a worm wheel – a gear wheel with tooth profile consists of a small segment of a helix which engages with the worm.
  39. 39. Shashidhar_gs@yahoo.co.in PHOTO OF WORM AND WORM WHEEL Worm gears are used when large gear reductions are needed. It is common for worm gears to have reductions of 20:1, and even up to 300:1 or greater. Many worm gears have an interesting property that no other gear set has: the worm can easily turn the gear, but the gear cannot turn the worm. This is because the angle on the worm is so shallow that when the gear tries to spin it, the friction between the gear and the worm holds the worm in place. This feature is useful for machines such as conveyor systems, in which the locking feature can act as a brake for the conveyor when the motor is not turning. One other very interesting usage of worm gears is in the Torsion differential, which is used on some high-performance cars and trucks. They are used in machine tools like lathe, milling, drilling machines etc, to get large speed reduction. Advantages of Gear drives : 1. They are positive non-slip drives. 2. Most convenient for very small centre distances. 3. Unlike the belt and chain drives, by using different types of gears, it will be possible to transmit the power when the axes of the shafts are not only parallel, but even when non-parallel, intersecting, non-intersecting and co-planar or non-coplanar.
  40. 40. Shashidhar_gs@yahoo.co.in 4. In gear drives, unlike belt and chain, the velocity ratio will remain constant throughout. 5. They can be employed conveniently for low, medium and high power transmission. 6. Any velocity ratio as high as, even up to 300:1 can be obtained. 7. They have very high transmission efficiency. 8. Gears can be cast in a wide range of both metallic and non-metallic materials. 9. If required gears may be cast integral with the shafts. 10. Gears are employed for wide range of applications like in watches, precision measuring instruments, machine tools, gear boxes fitted in automobiles, aero engines, etc. Disadvantages of Gear drives : 1. They are not suitable for shafts of very large centre distances. 2. They always require some kind of lubrication. 3. At very high speeds noise and vibrations will be more. 4. They are not economical because of the increased cost of production of precision gears. 5. Use of large number of gear wheels in gear trains increases the weight of the machine. 4.3 Terminology for Spur Gears
  41. 41. Shashidhar_gs@yahoo.co.in Figure 4-4 Spur Gear
  42. 42. Shashidhar_gs@yahoo.co.in Terminology: Addendum: The radial distance between the Pitch Circle and the top of the teeth. Arc of Action: Is the arc of the Pitch Circle between the beginning and the end of the engagement of a given pair of teeth. Arc of Approach: Is the arc of the Pitch Circle between the first point of contact of the gear teeth and the Pitch Point. Arc of Recession: That arc of the Pitch Circle between the Pitch Point and the last point of contact of the gear teeth. Backlash: Play between mating teeth. Base Circle: The circle from which is generated the involute curve upon which the tooth profile is based. Center Distance: The distance between centers of two gears.
  43. 43. Shashidhar_gs@yahoo.co.in Chordal Addendum: The distance between a chord, passing through the points where the Pitch Circle crosses the tooth profile, and the tooth top. Chordal Thickness: The thickness of the tooth measured along a chord passing through the points where the Pitch Circle crosses the tooth profile. Circular Pitch: Millimeter of Pitch Circle circumference per tooth. Circular Thickness: The thickness of the tooth measured along an arc following the Pitch Circle Clearance: The distance between the top of a tooth and the bottom of the space into which it fits on the meshing gear. Contact Ratio: The ratio of the length of the Arc of Action to the Circular Pitch. Dedendum: The radial distance between the bottom of the tooth to pitch circle. Diametral Pitch: Teeth per mm of diameter. Face: The working surface of a gear tooth, located between the pitch diameter and the top of the tooth. Face Width: The width of the tooth measured parallel to the gear axis. Flank: The working surface of a gear tooth, located between the pitch diameter and the bottom of the teeth Gear: The larger of two meshed gears. If both gears are the same size, they are both called gears. Land: The top surface of the tooth. Line of Action: That line along which the point of contact between gear teeth travels, between the first point of contact and the last.
  44. 44. Shashidhar_gs@yahoo.co.i Module: Millimeter of Pitc Pinion: The smaller of two Pitch Circle: The circle, from the center of the gea Diametral pitch: Teeth p Pitch Point: The point o gears, where the Line of Ce Pressure Angle: Angle perpendicular to the Line o Profile Shift: An increase a gear, introduced to low non-standard Center Dista Ratio: Ratio of the num of teeth on mating gears. Root Circle: The circle t passes through the bottom the tooth spaces. Root Diameter: diameter of the Root Circle Working Depth: The de to which a tooth extends the space between teeth the mating gear. 4.2 Gear-Tooth Action o.in itch Diameter to Teeth. meshed gears. le, the radius of which is equal ear to the pitch point. th per millimeter of pitch diamete t of tangency of the pitch circles Centers crosses the pitch circles. le between the Line of Actio e of Centers. ase in the Outer Diameter and Ro lower the practical tooth numbe stance. mbers e that tom of The rcle. depth ds into th on l to the distance eter. es of two meshing ction and a line Root Diameter of ber or acheive a
  45. 45. Shashidhar_gs@yahoo.co.in 4.2.1 Fundamental Law of Gear-Tooth Action Figure 5.2 shows two mating gear teeth, in which • Tooth profile 1 drives tooth profile 2 by acting at the instantaneous contact point K. • N1N2 is the common normal of the two profiles. • N1 is the foot of the perpendicular from O1 to N1N2 • N2 is the foot of the perpendicular from O2 to N1N2. Although the two profiles have different velocities V1 and V2 at point K, their velocities along N1N2 are equal in both magnitude and direction. Otherwise the two tooth profiles would separate from each other. Therefore, we have (4.1) 1 1 1 2 2 2 O N ω =O N ω or (4.2) O N = 2 2 1 1 ω 1 2 O N ω Figure 5-2 Two gearing tooth profiles We notice that the intersection of the tangency N1N2 and the line of center O1O2 is point P, and from the similar triangles, (4.3) 1 1 2 2 ΔO N P =ΔO N P Thus, the relationship between the angular velocities of the driving gear to the driven gear, or velocity ratio, of a pair of mating teeth is (4.4) O P = 2 1 ω 1 2 O P ω Point P is very important to the velocity ratio, and it is called the pitch point. Pitch point divides the line between the line of centers and its position decides the velocity ratio of the two teeth. The above expression is the fundamental law of gear-tooth action. From the equations 4.2 and 4.4, we can write, (4.5) O N 2 2 1 1 O P = 2 = 1 ω 1 2 O N O P ω which determines the ratio of the radii of the two base circles. The radii of the base circles is given by: cos cos (4.6) 1 1 1 2 2 2 O N =O P φ and O N =O P φ Also the centre distance between the base circles: φ
  46. 46. Shashidhar_gs@yahoo.co.in (4.7) O N O N O N O N 1 1 2 2 1 1 2 2 1 2 1 2 cos φ cos φ cos φ O O O P O P + = + = + = where φ is the pressure angle or the angle of obliquity. It is the angle which the common normal to the base circles make with the common tangent to the pitch circles. 4.2.2 Constant Velocity Ratio For a constant velocity ratio, the position of P should remain unchanged. In this case, the motion transmission between two gears is equivalent to the motion transmission between two imagined slip-less cylinders with radius R1 and R2 or diameter D1 and D2. We can get two circles whose centers are at O1 and O2, and through pitch point P. These two circles are termed pitch circles. The velocity ratio is equal to the inverse ratio of the diameters of pitch circles. This is the fundamental law of gear-tooth action. The fundamental law of gear-tooth action may now also be stated as follow (for gears with fixed center distance) A common normal (the line of action) to the tooth profiles at their point of contact must, in all positions of the contacting teeth, pass through a fixed point on the line-of-centers called the pitch point Any two curves or profiles engaging each other and satisfying the law of gearing are conjugate curves, and the relative rotation speed of the gears will be constant(constant velocity ratio).
  47. 47. Shashidhar_gs@yahoo.co.in FOR FULL EBOOK CONTACT: Shashidhar_gs@yahoo.co.in

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