Rajalakshmi Engineering College, Thandalam

ERODE SENGUNTHAR ENGINEERING COLLEGE.

Department of Mechanical Engineering

Branch : MECHANICAL ENGINEERING

Sem/Year : III sem. / II year

Subject Name: KINEMATICS OF MACHINERY

Objective:

To study the mechanism, machine and the geometric aspect of motion.

TEXT BOOKS:

1. Rattan S.S., Theory of Machines, Tata McGraw-Hill Publishing Company Ltd., New Delhi, 1998.

2. Shigley J.E. and Uicker J.J., Theory of Machines and Mechanisms, McGraw-Hill, Inc. 1995.

3. Khurmi R.S. and Gupta J.K., Theory of Machines, Eurasia Publishing House, New Delhi,2006

REFERENCES:

1. Thomas Bevan, Theory of Machines, CBS Publishers and Distributors, 1984.

2. Ghosh A. and Mallick A.K., Theory of Mechanisms and Machines, Affiliated East-West Pvt. Ltd., New Delhi, 1988.

3. Rao J.S. and Dukkipati R.V., Mechanism and Machine Theory, Wiley-Eastern Ltd., New Delhi, 1992.

NOTES OF LESSON

Unit I Basics of Mechanisms

Introduction:

Definitions : Link or Element, Pairing of Elements with degrees of freedom, Grublers criterion (without derivation), Kinematic chain, Mechanism, Mobility of Mechanism, Inversions, Machine.

Kinematic Chains and Inversions :

Kinematic chain with three lower pairs, Four bar chain, Single slider crank chain and Double slider crank chain and

their inversions.

Mechanisms:

i) Quick return motion mechanisms Drag link mechanism, Whitworth mechanism and Crank and slotted lever mechanism

ii) Straight line motion mechanisms Peaceliers mechanism and Roberts mechanism.

iii) Intermittent motion mechanisms Geneva mechanism and Ratchet & Pawl mechanism.

iv) Toggle mechanism, Pantograph, Hookes joint and Ackerman Steering gear mechanism.

Terminology and Definitions-Degree of Freedom, Mobility

Kinematics: The study of motion (position, velocity, acceleration). A major goal of understanding kinematics is to develop the ability to design a system that will satisfy specified motion requirements. This will be the emphasis of this class.

Kinetics: The effect of forces on moving bodies. Good kinematic design should produce good kinetics.

Mechanism: A system design to transmit motion. (low forces)

Machine: A system designed to transmit motion and energy. (forces involved)

Basic Mechanisms: Includes geared systems, cam-follower systems and linkages (rigid links connected by sliding or rotating joints). A mechanism has multiple moving parts (for example, a simple hinged door does not qualify as a mechanism).

Examples of mechanisms: Tin snips, vise grips, car suspension, backhoe, piston engine, folding chair, windshield wiper drive system, etc.

Key concepts:

Degrees of freedom: The number of inputs required to completely control a system. Examples: A simple rotating link. A two link system. A four-bar linkage. A five-bar linkage.

Types of motion: Mechanisms may produce motions that are pure rotation, pure translation, or a combination of the two. We reduce the degrees of freedom of a mechanism by restraining the ability of the mechanism to move in translation (x-y directions for a 2D mechanism) or in rotation (about the z-axis for a 2-D mechanism).

Link: A rigid body with two or more nodes (joints) that are used to connect to other rigid bodies. (WM examples: binary link, ternary link (3 joints), quaternary link (4 joints))

Joint: A connection between two links that allows motion between the links. The motion allowed may be rotational (revolute joint), translational (sliding or prismatic joint), or a combination of the two (roll-slide joint).

Kinematic chain: An assembly of links and joints used to coordinate an output motion with an input motion.

Link or element:

A mechanism is made of a number of resistant bodies out of which some may have motions relative to the others. A resistant body or a group of resistant bodies with rigid connections preventing their relative movement is known as a link.

A link may also be defined as a member or a combination of members of a mechanism, connecting other members and having motion relative to them, thus a link may consist of one or more resistant bodies. A link is also known as Kinematic link or an element.

Links can be classified into 1) Binary, 2) Ternary, 3) Quarternary, etc.

Kinematic Pair:

A Kinematic Pair or simply a pair is a joint of two links having relative motion between them

Example:

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.

Types of Kinematic Pairs:

Kinematic pairs can be classified according to

i) Nature of contact.

ii) Nature of mechanical constraint.

iii) Nature of relative motion.

i) Kinematic pairs according to nature of contact :

a) Lower Pair: A pair of links having surface or area contact between the members is known as a lower pair. The contact surfaces of the two links are similar.

Examples: Nut turning on a screw, shaft rotating in a bearing, all pairs of a slider-crank mechanism, universal joint.

b) Higher Pair: When a pair has a point or line contact between the links, it is known as a higher pair. The contact surfaces of the two links are dissimilar.

Examples: Wheel rolling on a surface cam and follower pair, tooth gears, ball and roller bearings, etc.

ii) Kinematic pairs according to nature of mechanical constraint.

a) Closed pair: When the elements of a pair are held together mechanically, it is known as a closed pair. The contact between the two can only be broken only by the destruction of at least one of the members. All the lower pairs and some of the higher pairs are closed pairs.

b) Unclosed pair: When two links of a pair are in contact either due to force of gravity or some spring action, they constitute an unclosed pair. In this the links are not held together mechanically. Ex.: Cam and follower pair.

iii) Kinematic pairs according to nature of relative motion.

a) Sliding pair: If two links have a sliding motion relative to each other, they form a sliding pair. A rectangular rod in a rectangular hole in a prism is an example of a sliding pair.

b) Turning Pair: When on link has a turning or revolving motion relative to the other, they constitute a turning pair or revolving pair.

c) Rolling pair: When the links of a pair have a rolling motion relative to each other, they form a rolling pair. A rolling wheel on a flat surface, ball ad roller bearings, etc. are some of the examples for a Rolling pair.

d) Screw pair (Helical Pair): if two mating links have a turning as well as sliding motion between them, they form a screw pair. This is achieved by cutting matching threads on the two links.

The lead screw and the nut of a lathe is a screw Pair

e) Spherical pair: When one link in the form of a sphere turns inside a fixed link, it is a spherical pair. The ball and socket joint is a spherical pair.

Degrees of Freedom:

An unconstrained rigid body moving in space can describe the following independent motions.

1. Translational Motions along any three mutually perpendicular axes x, y and z,

2. Rotational motions along 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).

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 no. of restraints.

To find the number of degrees of freedom for a plane mechanism we have an equation known as Grublers equation and is given by

F = 3 ( n 1 ) 2 j1 j2

F = Mobility or number of degrees of freedom

n = Number of links including frame.

j1 = Joints with single (one) degree of freedom.

J2 = Joints with two degrees of freedom.

If F > 0, results in a mechanism with F degrees of freedom.

F = 0, results in a statically determinate structure.

F < 0, results in a statically indeterminate structure.

Kinematic Chain:

A Kinematic chain is an assembly of links in which the relative motions of the links is possible and the motion of each relative to the others is definite (fig. a, b, and c.)

In case, the motion of a link results in indefinite motions of other links, it is a non-kinematic chain. However, some authors prefer to call all chains having relative motions of the links as kinematic chains.

Linkage, Mechanism and structure:

A linkage is obtained if one of the links of kinematic chain is fixed to the ground. If motion of each link results in definite motion of the others, the linkage is known as mechanism. If one of the links of a redundant chain is fixed, it is known as a structure.

To obtain constrained or definite motions of some of the links of a linkage, it is necessary to know how many inputs are needed. In some mechanisms, only one input is necessary that determines the motion of other links and are said to have one degree of freedom. In other mechanisms, two inputs may be necessary to get a constrained motion of the other links and are said to have two degrees of freedom and so on.

The degree of freedom of a structure is zero or less. A structure with negative degrees of freedom is known as a Superstructure.

Motion and its types:

If the motion between a pair of links is limited to a definite direction, then it is completely constrained motion. E.g.: Motion of a shaft

If the motion in a definite direction is not brought about by itself but by some other means, then it is incompletely constrained motion.

The three main types of constrained motion in kinematic pair are,

1.Completely constrained motion : If the motion between a pair of links is limited to a definite direction, then it is completely constrained motion. E.g.: Motion of a shaft or rod with collars at each end in a hole as shown in fig.

2. Incompletely Constrained motion : If the motion between a pair of links is not confined to a definite direction, then it is incompletely constrained motion. E.g.: A spherical ball or circular shaft in a circular hole may either rotate or slide in the hole as shown in fig.

3. Successfully constrained motion or Partially constrained motion: If the motion in a definite direction is not brought about by itself but by some other means, then it is known as successfully constrained motion. E.g.: Foot step Bearing.

Machine:

It is a combination of resistant bodies with successfully constrained motion which is used to transmit or transform motion to do some useful work. E.g.: Lathe, Shaper, Steam Engine, etc.

Kinematic chain with three lower pairs

It is impossible to have a kinematic chain consisting of three turning pairs only. But it is possible to have a chain which consists of three sliding pairs or which consists of a turning, sliding and a screw pair.

The figure shows a kinematic chain with three sliding pairs. It consists of a frame B, wedge C and a sliding rod A. So the three sliding pairs are, one between the wedge C and the frame B, second between wedge C and sliding rod A and the frame B.

This figure shows the mechanism of a fly press. The element B forms a sliding with A and turning pair with screw rod C which in turn forms a screw pair with A. When link A is fixed, the required fly press mechanism is obtained.

Kutzbach criterion, Grashoff's law

Kutzbach criterion:

Fundamental Equation for 2-D Mechanisms: M = 3(L 1) 2J1 J2

Can we intuitively derive Kutzbachs modification of Grublers equation? Consider a rigid link constrained to move in a plane. How many degrees of freedom does the link have? (3: translation in x and y directions, rotation about z-axis)

If you pin one end of the link to the plane, how many degrees of freedom does it now have?

Add a second link to the picture so that you have one link pinned to the plane and one free to move in the plane. How many degrees of freedom exist between the two links? (4 is the correct answer)

Pin the second link to the free end of the first link. How many degrees of freedom do you now have?

How many degrees of freedom do you have each time you introduce a moving link? How many degrees of freedom do you take away when you add a simple joint? How many degrees of freedom would you take away by adding a half joint? Do the different terms in equation make sense in light of this knowledge?

Grashoff's law:

Grashoff 4-bar linkage: A linkage that contains one or more links capable of undergoing a full rotation. A linkage is Grashoff if: S + L < P + Q (where: S = shortest link length, L = longest, P, Q = intermediate length links). Both joints of the shortest link are capable of 360 degrees of rotation in a Grashoff linkages. This gives us 4 possible linkages: crank-rocker (input rotates 360), rocker-crank-rocker (coupler rotates 360), rocker-crank (follower); double crank (all links rotate 360). Note that these mechanisms are simply the possible inversions (section 2.11, Figure 2-16) of a Grashoff mechanism.

Non Grashoff 4 bar: No link can rotate 360 if: S + L > P + Q

Lets examine why the Grashoff condition works:

Consider a linkage with the shortest and longest sides joined together. Examine the linkage when the shortest side is parallel to the longest side (2 positions possible, folded over on the long side and extended away from the long side). How long do P and Q have to be to allow the linkage to achieve these positions?

Consider a linkage where the long and short sides are not joined. Can you figure out the required lengths for P and Q in this type of mechanism

2. Kinematic Inversions of 4-bar chain and slider crank chains:

Types of Kinematic Chain: 1) Four bar chain 2) Single slider chain 3) Double Slider chain

Four bar Chain:

The chain has four links and it looks like a cycle frame and hence it is also called quadric cycle chain. It is shown in the figure. In this type of chain all four pairs will be turning pairs.

Inversions:

By fixing each link at a time we get as many mechanisms as the number of links, then each mechanism is called Inversion of the original Kinematic Chain.

Inversions of four bar chain mechanism:

There are three inversions: 1) Beam Engine or Crank and lever mechanism. 2) Coupling rod of locomotive or double crank mechanism. 3) Watts straight line mechanism or double lever mechanism.

Beam Engine:

When the crank AB rotates about A, the link CE pivoted at D makes vertical reciprocating motion at end E. This is used to convert rotary motion to reciprocating motion and vice versa. It is also known as Crank and lever mechanism. This mechanism is shown in the figure below.

2. Coupling rod of locomotive: In this mechanism the length of link AD = length of link C. Also length of link AB = length of link CD. 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 mechanism. This is shown in the figure below.

3. Watts straight line mechanism or Double lever mechanism: In this mechanism, the links AB & DE act as levers at the ends A & E of these levers are fixed. The AB & DE are parallel in the mean position of the mechanism and coupling rod BD is perpendicular to the levers AB & DE. On any small displacement of the mechanism the tracing point C traces the shape of number 8, a portion of which will be approximately straight. Hence this is also an example for the approximate straight line mechanism. This mechanism is shown below.

2. Slider crank Chain:

It is a four bar chain having one sliding pair and three turning pairs. It is shown in the figure below the purpose of this mechanism is to convert rotary motion to reciprocating motion and vice versa.

Inversions of a Slider crank chain:

There are four inversions in a single slider chain mechanism. They are:

1) Reciprocating engine mechanism (1st inversion)

2) Oscillating cylinder engine mechanism (2nd inversion)

3) Crank and slotted lever mechanism (2nd inversion)

4) Whitworth quick return motion mechanism (3rd inversion)

5) Rotary engine mechanism (3rd inversion)

6) Bull engine mechanism (4th inversion)

7) Hand Pump (4th inversion)

1. Reciprocating engine mechanism :

In the first inversion, the link 1 i.e., the cylinder and the frame is kept fixed. The fig below shows a reciprocating engine.

A slotted link 1 is fixed. When the crank 2 rotates about O, the sliding piston 4 reciprocates in the slotted link 1. This mechanism is used in steam engine, pumps, compressors, I.C. engines, etc.

2. Crank and slotted lever mechanism:

It is an application of second inversion. The crank and slotted lever mechanism is shown in figure below.

In this mechanism link 3 is fixed. The slider (link 1) reciprocates in oscillating slotted lever (link 4) and crank (link 2) rotates. Link 5 connects link 4 to the ram (link 6). The ram with the cutting tool reciprocates perpendicular to the fixed link 3. The ram with the tool reverses its direction of motion when link 2 is perpendicular to link 4. Thus the cutting stroke is executed during the rotation of the crank through angle and the return stroke is executed when the crank rotates through angle or 360 . Therefore, when the crank rotates uniformly, we get,

Time to cutting = =

Time of return 360

This mechanism is used in shaping machines, slotting machines and in rotary engines.

3. Whitworth quick return motion mechanism:

Third inversion is obtained by fixing the crank i.e. link 2. Whitworth quick return mechanism is an application of third inversion. This mechanism is shown in the figure below. The crank OC is fixed and OQ rotates about O. The slider slides in the slotted link and generates a circle of radius CP. Link 5 connects the extension OQ provided on the opposite side of the link 1 to the ram (link 6). The rotary motion of P is taken to the ram R which reciprocates. The quick return motion mechanism is used in shapers and slotting machines. The angle covered during cutting stroke from P1 to P2 in counter clockwise direction is or 360 -2. During the return stroke, the angle covered is 2 or .

Therefore,

Time to cutting = 360 -2 = 180

Time of return 2 = = . 360

4. Rotary engine mechanism or Gnome Engine:

Rotary engine mechanism or gnome engine is another application of third inversion. It is a rotary cylinder V type internal combustion engine used as an aero engine. But now Gnome engine has been replaced by Gas turbines. The Gnome engine has generally seven cylinders in one plane. The crank OA is fixed and all the connecting rods from the pistons are connected to A. In this mechanism when the pistons reciprocate in the cylinders, the whole assembly of cylinders, pistons and connecting rods rotate about the axis O, where the entire mechanical power developed, is obtained in the form of rotation of the crank shaft. This mechanism is shown in the figure below.

Double Slider Crank Chain:

A four bar chain having two turning and two sliding pairs such that two pairs of the same kind are adjacent is known as double slider crank chain.

Inversions of Double slider Crank chain:

It consists of two sliding pairs and two turning pairs. They are three important inversions of double slider crank chain. 1) Elliptical trammel. 2) Scotch yoke mechanism. 3) Oldhams Coupling.

1. Elliptical Trammel:

This is an instrument for drawing ellipses. Here the slotted link is fixed. The sliding block P and Q in vertical and horizontal slots respectively. The end R generates an ellipse with the displacement of sliders P and Q.

The co-ordinates of the point R are x and y. From the fig. cos = x. PR

and Sin = y. QR

Squaring and adding (i) and (ii) we get x2 + y2 = cos2 + sin2

(PR)2 (QR)2

x2 + y2 = 1

(PR)2 (QR)2

The equation is that of an ellipse, Hence the instrument traces an ellipse. Path traced by mid-point of PQ is a circle. In this case, PR = PQ and so x2+y2 =1 (PR)2 (QR)2

It is an equation of circle with PR = QR = radius of a circle.

2. Scotch yoke mechanism: This mechanism, the slider P is fixed. When PQ rotates above P, the slider Q reciprocates in the vertical slot. The mechanism is used to convert rotary to reciprocating mechanism.

3. Oldhams coupling: The third inversion of obtained by fixing the link connecting the 2 blocks P & Q. If one block is turning through an angle, the frame and the other block will also turn through the same angle. It is shown in the figure below.

An application of the third inversion of the double slider crank mechanism is Oldhams coupling shown in the figure. This coupling is used for connecting two parallel shafts when the distance between the shafts is small. The two shafts to be connected have flanges at their ends, secured by forging. Slots are cut in the flanges. These flanges form 1 and 3. An intermediate disc having tongues at right angles and opposite sides is fitted in between the flanges. The intermediate piece forms the link 4 which slides or reciprocates in flanges 1 & 3. The link two is fixed as shown. When flange 1 turns, the intermediate disc 4 must turn through the same angle and whatever angle 4 turns, the flange 3 must turn through the same angle. Hence 1, 4 & 3 must have the same angular velocity at every instant. If the distance between the axis of the shaft is x, it will be the diameter if the circle traced by the centre of the intermediate piece. The maximum sliding speed of each tongue along its slot is given by

v=x where, = angular velocity of each shaft in rad/sec v = linear velocity in m/sec

3. Mechanical Advantage, Transmission angle:

The mechanical advantage (MA) is defined as the ratio of output torque to the input torque. (or) ratio of load to output.

Transmission angle.

The extreme values of the transmission angle occur when the crank lies along the line of frame.

4. Description of common mechanisms-Single, Double and offset slider mechanisms - Quick return mechanisms:

Quick Return Motion Mechanisms:

Many a times mechanisms are designed to perform repetitive operations. During these operations for a certain period the mechanisms will be under load known as working stroke and the remaining period is known as the return stroke, the mechanism returns to repeat the operation without load. The ratio of time of working stroke to that of the return stroke is known a time ratio. Quick return mechanisms are used in machine tools to give a slow cutting stroke and a quick return stroke. The various quick return mechanisms commonly used are i) Whitworth ii) Drag link. iii) Crank and slotted lever mechanism

1. Whitworth quick return mechanism:

Whitworth quick return mechanism is an application of third inversion of the single slider crank chain. This mechanism is shown in the figure below. The crank OC is fixed and OQ rotates about O. The slider slides in the slotted link and generates a circle of radius CP. Link 5 connects the extension OQ provided on the opposite side of the link 1 to the ram (link 6). The rotary motion of P is taken to the ram R which reciprocates. The quick return motion mechanism is used in shapers and slotting machines.

The angle covered during cutting stroke from P1 to P2 in counter clockwise direction is or 360 -2. During the return stroke, the angle covered is 2 or .

2. Drag link mechanism :

This is four bar mechanism with double crank in which the shortest link is fixed. If the crank AB rotates at a uniform speed, the crank CD rotate at a non-uniform speed. This rotation of link CD is transformed to quick return reciprocatory motion of the ram E by the link CE as shown in figure. When the crank AB rotates through an angle in Counter clockwise direction during working stroke, the link CD rotates through 180. We can observe that / >/ . Hence time of working stroke is / times more or the return stroke is / times quicker. Shortest link is always stationary link. Sum of the shortest and the longest links of the four links 1, 2, 3 and 4 are less than the sum of the other two. It is the necessary condition for the drag link quick return mechanism.

3. Crank and slotted lever mechanism:

It is an application of second inversion. The crank and slotted lever mechanism is shown in figure below.

In this mechanism link 3 is fixed. The slider (link 1) reciprocates in oscillating slotted lever (link 4) and crank (link 2) rotates. Link 5 connects link 4 to the ram (link 6). The ram with the cutting tool reciprocates perpendicular to the fixed link 3. The ram with the tool reverses its direction of motion when link 2 is perpendicular to link 4. Thus the cutting stroke is executed during the rotation of the crank through angle and the return stroke is executed when the crank rotates through angle or 360 . Therefore, when the crank rotates uniformly, we get,

Time to cutting = =

Time of return 360

This mechanism is used in shaping machines, slotting machines and in rotary engines.

5. Ratchets and escapements - Indexing Mechanisms - Rocking Mechanisms:

Intermittent motion mechanism:

1. Ratchet and Pawl mechanism: This mechanism is used in producing intermittent rotary motion member. A ratchet and Pawl mechanism consists of a ratchet wheel 2 and a pawl 3 as shown in the figure. When the lever 4 carrying pawl is raised, the ratchet wheel rotates in the counter clock wise direction (driven by pawl). As the pawl lever is lowered the pawl slides over the ratchet teeth. One more pawl 5 is used to prevent the ratchet from reversing. Ratchets are used in feed mechanisms, lifting jacks, clocks, watches and counting devices.

2. Geneva mechanism: Geneva mechanism is an intermittent motion mechanism. It consists of a driving wheel D carrying a pin P which engages in a slot of follower F as shown in figure. During one quarter revolution of the driving plate, the Pin and follower remain in contact and hence the follower is turned by one quarter of a turn. During the remaining time of one revolution of the driver, the follower remains in rest locked in position by the circular arc.

3. Pantograph: Pantograph is used to copy the curves in reduced or enlarged scales. Hence this mechanism finds its use in copying devices such as engraving or profiling machines.

This is a simple figure of a Pantograph. The links are pin jointed at A, B, C and D. AB is parallel to DC and AD is parallel to BC. Link BA is extended to fixed pin O. Q is a point on the link AD. If the motion of Q is to be enlarged then the link BC is extended to P such that O, Q and P are in a straight line. Then it can be shown that the points P and Q always move parallel and similar to each other over any path straight or curved. Their motions will be proportional to their distance from the fixed point. Let ABCD be the initial position. Suppose if point Q moves to Q1 , then all the links and the joints will move to the new positions (such as A moves to A1 , B moves to Q1, C moves to Q1 , D moves to D1 and P to P1 ) and the new configuration of the mechanism is shown by dotted lines. The movement of Q (Q Q1) will be enlarged to PP1 in a definite ratio.

4. Toggle Mechanism:

In slider crank mechanism as the crank approaches one of its dead centre position, the slider approaches zero. The ratio of the crank movement to the slider movement approaching infinity is proportional to the mechanical advantage. This is the principle used in toggle mechanism. A toggle mechanism is used when large forces act through a short distance is required. The figure below shows a toggle mechanism. Links CD and CE are of same length. Resolving the forces at C vertically F Sin =P Cos 2

Therefore, F = P . (because Sin /Cos = Tan ) 2 tan Thus for the given value of P, as the links CD and CE approaches collinear position (O), the force F rises rapidly.

5. Hookes joint:

Hookes joint used to connect two parallel intersecting shafts as shown in figure. This can also be used for shaft with angular misalignment where flexible coupling does not serve the purpose. Hence Hookes joint is a means of connecting two rotating shafts whose axes lie in the same plane and their directions making a small angle with each other. It is commonly known as Universal joint. In Europe it is called as Cardan joint.

5. Ackermann steering gear mechanism:

This mechanism is made of only turning pairs and is made of only turning pairs wear and tear of the parts is less and cheaper in manufacturing. The cross link KL connects two short axles AC and BD of the front wheels through the short links AK and BL which forms bell crank levers CAK and DBL respectively as shown in fig, the longer links AB and KL are parallel and the shorter links AK and BL are inclined at an angle . When the vehicles steer to the right as shown in the figure, the short link BL is turned so as to increase , where as the link LK causes the other short link AK to turn so as to reduce . The fundamental equation for correct steering is, CotCos = b / l

In the above arrangement it is clear that the angle through which AK turns is less than the angle through which the BL turns and therefore the left front axle turns through a smaller angle than the right front axle. For different angle of turn , the corresponding value of and (Cot Cos ) are noted. This is done by actually drawing the mechanism to a scale or by calculations. Therefore for different value of the corresponding value of and are tabulated. Approximate value of b/l for correct steering should be between 0.4 and 0.5. In an Ackermann steering gear mechanism, the instantaneous centre I does not lie on the axis of the rear axle but on a line parallel to the rear axle axis at an approximate distance of 0.3l above it.

Three correct steering positions will be:

1) When moving straight.2) When moving one correct angle to the right corresponding to the link ratio AK/AB and angle . 3) Similar position when moving to the left. In all other positions pure rolling is not obtainable.

Some Of The Mechanisms Which Are Used In Day To Day Life.

BELL CRANK:

GENEVA STOP:

BELL CRANK: The bell crank was originally used in large house to operate the servants bell, hence the name. The bell crank is used to convert the direction of reciprocating movement. By varying the angle of the crank piece it can be used to change the angle of movement from 1 degree to 180 degrees.

GENEVA STOP: The Geneva stop is named after the Geneva cross, a similar shape to the main part of the mechanism. The Geneva stop is used to provide intermittent motion, the orange wheel turns continuously, the dark blue pin then turns the blue cross quarter of a turn for each revolution of the drive wheel. The crescent shaped cut out in dark orange section lets the points of the cross past, then locks the wheel in place when it is stationary. The Geneva stop mechanism is used commonly in film cameras.

ELLIPTICAL TRAMMEL

PISTON ARRANGEMENT

ELLIPTICAL TRAMMEL: This fascinating mechanism converts rotary motion to reciprocating motion in two axis. Notice that the handle traces out an ellipse rather than a circle. A similar mechanism is used in ellipse drawing tools.

PISTON ARRANGEMENT: This mechanism is used to convert between rotary motion and reciprocating motion, it works either way. Notice how the speed of the piston changes. The piston starts from one end, and increases its speed. It reaches maximum speed in the middle of its travel then gradually slows down until it reaches the end of its travel.

RACK AND PINION

RATCHET

RACK AND PINION: The rack and pinion is used to convert between rotary and linear motion. The rack is the flat, toothed part, the pinion is the gear. Rack and pinion can convert from rotary to linear of from linear to rotary. The diameter of the gear determines the speed that the rack moves as the pinion turns. Rack and pinions are commonly used in the steering system of cars to convert the rotary motion of the steering wheel to the side to side motion in the wheels. Rack and pinion gears give a positive motion especially compared to the friction drive of a wheel in tarmac. In the rack and pinion railway a central rack between the two rails engages with a pinion on the engine allowing the train to be pulled up very steep slopes.

RATCHET: The ratchet can be used to move a toothed wheel one tooth at a time. The part used to move the ratchet is known as the pawl. The ratchet can be used as a way of gearing down motion. By its nature motion created by a ratchet is intermittent. By using two pawls simultaneously this intermittent effect can be almost, but not quite, removed. Ratchets are also used to ensure that motion only occurs in only one direction, useful for winding gear which must not be allowed to drop. Ratchets are also used in the freewheel mechanism of a bicycle.

WORM GEAR

WATCH ESCAPEMENT.

WORM GEAR: A worm is used to reduce speed. For each complete turn of the worm shaft the gear shaft advances only one tooth of the gear. In this case, with a twelve tooth gear, the speed is reduced by a factor of twelve. Also, the axis of rotation is turned by 90 degrees. Unlike ordinary gears, the motion is not reversible, a worm can drive a gear to reduce speed but a gear cannot drive a worm to increase it. As the speed is reduced the power to the drive increases correspondingly. Worm gears are a compact, efficient means of substantially decreasing speed and increasing power. Ideal for use with small electric motors.

WATCH ESCAPEMENT: The watch escapement is the centre of the time piece. It is the escapement which divides the time into equal segments. The balance wheel, the gold wheel, oscillates backwards and forwards on a hairspring (not shown) as the balance wheel moves the lever is moved allowing the escape wheel (green) to rotate by one tooth. The power comes through the escape wheel which gives a small 'kick' to the palettes (purple) at each tick.

GEARS

CAM FOLLOWER.

GEARS: Gears are used to change speed in rotational movement. In the example above the blue gear has eleven teeth and the orange gear has twenty five. To turn the orange gear one full turn the blue gear must turn 25/11 or 2.2727r turns. Notice that as the blue gear turns clockwise the orange gear turns anti-clockwise. In the above example the number of teeth on the orange gear is not divisible by the number of teeth on the blue gear. This is deliberate. If the orange gear had thirty three teeth then every three turns of the blue gear the same teeth would mesh together which could cause excessive wear. By using none divisible numbers the same teeth mesh only every seventeen turns of the blue gear.

CAMS: Cams are used to convert rotary motion into reciprocating motion. The motion created can be simple and regular or complex and irregular. As the cam turns, driven by the circular motion, the cam follower traces the surface of the cam transmitting its motion to the required mechanism. Cam follower design is important in the way the profile of the cam is followed. A fine pointed follower will more accurately trace the outline of the cam. This more accurate movement is at the expense of the strength of the cam follower.

STEAM ENGINE.

Steam engines were the backbone of the industrial revolution. In this common design high pressure steam is pumped alternately into one side of the piston, then the other forcing it back and forth. The reciprocating motion of the piston is converted to useful rotary motion using a crank.

As the large wheel (the fly wheel) turns a small crank or cam is used to move the small red control valve back and forth controlling where the steam flows. In this animation the oval crank has been made transparent so that you can see how the control valve crank is attached.

6. Straight line generators, Design of Crank-rocker Mechanisms:

Straight Line Motion Mechanisms:

The easiest way to generate a straight line motion is by using a sliding pair but in precision machines sliding pairs are not preferred because of wear and tear. Hence in such cases different methods are used to generate straight line motion mechanisms:

1. Exact straight line motion mechanism.

a. Peaucellier mechanism, b. Hart mechanism, c. Scott Russell mechanism

2. Approximate straight line motion mechanisms

a. Watt mechanism, b. Grasshoppers mechanism, c. Roberts mechanism,

d. Tchebicheffs mechanism

a. Peaucillier mechanism :

The pin Q is constrained to move long the circumference of a circle by means of the link OQ. The link OQ and the fixed link are equal in length. The pins P and Q are on opposite corners of a four bar chain which has all four links QC, CP, PB and BQ of equal length to the fixed pin A. i.e., link AB = link AC. The product AQ x AP remain constant as the link OQ rotates may be proved as follows: Join BC to bisect PQ at F; then, from the right angled triangles AFB, BFP, we have AB=AF+FB and BP=BF+FP. Subtracting, AB-BP= AF-FP=(AFFP)(AF+FP) = AQ x AP .

Since AB and BP are links of a constant length, the product AQ x AP is constant. Therefore the point P traces out a straight path normal to AR.

b. Roberts mechanism:

This is also a four bar chain. The link PQ and RS are of equal length and the tracing pint O is rigidly attached to the link QR on a line which bisects QR at right angles. The best position for O may be found by making use of the instantaneous centre of QR. The path of O is clearly approximately horizontal in the Roberts mechanism.

a. Peaucillier mechanism

b. Hart mechanism

Unit II Kinematics

Velocity and Acceleration analysis of mechanisms (Graphical Methods):

Velocity and acceleration analysis by vector polygons: Relative velocity and accelerations of particles in a common link, relative velocity and accelerations of coincident particles on separate link, Coriolis component of acceleration.

Velocity and acceleration analysis by complex numbers: Analysis of single slider crank mechanism and four bar mechanism by loop closure equations and complex numbers.

7. Displacement, velocity and acceleration analysis in simple mechanisms:

Important Concepts in Velocity Analysis

1. The absolute velocity of any point on a mechanism is the velocity of that point with reference to ground.

2. Relative velocity describes how one point on a mechanism moves relative to another point on the mechanism.

3. The velocity of a point on a moving link relative to the pivot of the link is given by the equation: V = r, where = angular velocity of the link and r = distance from pivot.

Acceleration Components

Normal Acceleration: An = 2r. Points toward the center of rotation

Tangential Acceleration: At = r. In a direction perpendicular to the link

Coriolis Acceleration: Ac = 2(dr/dt). In a direction perpendicular to the link

Sliding Acceleration: As = d2r/dt2. In the direction of sliding.

A rotating link will produce normal and tangential acceleration components at any point a distance, r, from the rotational pivot of the link. The total acceleration of that point is the vector sum of the components.

A slider attached to ground experiences only sliding acceleration.

A slider attached to a rotating link (such that the slider is moving in or out along the link as the link rotates) experiences all 4 components of acceleration. Perhaps the most confusing of these is the coriolis acceleration, though the concept of coriolis acceleration is fairly simple. Imagine yourself standing at the center of a merry-go-round as it spins at a constant speed (). You begin to walk toward the outer edge of the merry-go-round at a constant speed (dr/dt). Even though you are walking at a constant speed and the merry-go-round is spinning at a constant speed, your total velocity is increasing because you are moving away from the center of rotation (i.e. the edge of the merry-go-round is moving faster than the center). This is the coriolis acceleration. In what direction did your speed increase? This is the direction of the coriolis acceleration.

The total acceleration of a point is the vector sum of all applicable acceleration components:

A = An + At + Ac + As

These vectors and the above equation can be broken into x and y components by applying sines and cosines to the vector diagrams to determine the x and y components of each vector. In this way, the x and y components of the total acceleration can be found.

8. Graphical Method, Velocity and Acceleration polygons :

Graphical velocity analysis:

It is a very short step (using basic trigonometry with sines and cosines) to convert the graphical results into numerical results. The basic steps are these:

1. Set up a velocity reference plane with a point of zero velocity designated.

2. Use the equation, V = r, to calculate any known linkage velocities.

3.Plot your known linkage velocities on the velocity plot. A linkage that is rotating about ground gives an absolute velocity. This is a vector that originates at the zero velocity point and runs perpendicular to the link to show the direction of motion. The vector, VA, gives the velocity of point A.

4.Plot all other velocity vector directions. A point on a grounded link (such as point B) will produce an absolute velocity vector passing through the zero velocity point and perpendicular to the link. A point on a floating link (such as B relative to point A) will produce a relative velocity vector. This vector will be perpendicular to the link AB and pass through the reference point (A) on the velocity diagram.

5. One should be able to form a closed triangle (for a 4-bar) that shows the vector equation: VB = VA + VB/A where VB = absolute velocity of point B, VA = absolute velocity of point A, and VB/A is the velocity of point B relative to point A.

9. Velocity Analysis of Four Bar Mechanisms:

Problems solving in Four Bar Mechanisms and additional links.

10. Velocity Analysis of Slider Crank Mechanisms:

Problems solving in Slider Crank Mechanisms and additional links.

11. Acceleration Analysis of Four Bar Mechanisms:

Problems solving in Four Bar Mechanisms and additional links.

12. Acceleration Analysis of Slider Crank Mechanisms:

Problems solving in Slider Crank Mechanisms and additional links.

13. Kinematic analysis by Complex Algebra methods:

Analysis of single slider crank mechanism and four bar mechanism by loop closure equations and complex numbers.

14. Vector Approach:

Relative velocity and accelerations of particles in a common link, relative velocity and accelerations of coincident particles on separate link

15. Computer applications in the kinematic analysis of simple mechanisms:

Computer programming for simple mechanisms

16. Coincident points, Coriolis Acceleration:

Coriolis Acceleration: Ac = 2(dr/dt). In a direction perpendicular to the link.

A slider attached to ground experiences only sliding acceleration.

A slider attached to a rotating link (such that the slider is moving in or out along the link as the link rotates) experiences all 4 components of acceleration. Perhaps the most confusing of these is the coriolis acceleration, though the concept of coriolis acceleration is fairly simple. Imagine yourself standing at the center of a merry-go-round as it spins at a constant speed (). You begin to walk toward the outer edge of the merry-go-round at a constant speed (dr/dt). Even though you are walking at a constant speed and the merry-go-round is spinning at a constant speed, your total velocity is increasing because you are moving away from the center of rotation (i.e. the edge of the merry-go-round is moving faster than the center). This is the coriolis acceleration. In what direction did your speed increase? This is the direction of the coriolis acceleration.

Example:1

Unit III Kinematics of CAM

Cams:

Type of cams, Type of followers, Displacement, Velocity and acceleration time curves for cam profiles, Disc cam with reciprocating follower having knife edge, roller follower, Follower motions including SHM, Uniform velocity, Uniform acceleration and retardation and Cycloidal motion.

Cams are used to convert rotary motion into reciprocating motion. The motion created can be simple and regular or complex and irregular. As the cam turns, driven by the circular motion, the cam follower traces the surface of the cam transmitting its motion to the required mechanism. Cam follower design is important in the way the profile of the cam is followed. A fine pointed follower will more accurately trace the outline of the cam. This more accurate movement is at the expense of the strength of the cam follower.

17. Classifications - Displacement diagrams

Cam Terminology:

Physical components: Cam, follower, spring

Types of cam systems: Oscilllating (rotating), translating

Types of joint closure: Force closed, form closed

Types of followers: Flat-faced, roller, mushroom

Types of cams: radial, axial, plate (a special class of radial cams).

Types of motion constraints: Critical extreme position the positions of the follower that are of primary concern are the extreme positions, with considerable freedom as to design the cam to move the follower between these positions. This is the motion constraint type that we will focus upon. Critical path motion The path by which the follower satisfies a given motion is of interest in addition to the extreme positions. This is a more difficult (and less common) design problem.

Types of motion: rise, fall, dwell

Geometric and Kinematic parameters: follower displacement, velocity, acceleration, and jerk; base circle; prime circle; follower radius; eccentricity; pressure angle; radius of curvature.

18. Parabolic, Simple harmonic and Cycloidal motions:

Describing the motion: A cam is designed by considering the desired motion of the follower. This motion is specified through the use of SVAJ diagrams (diagrams that describe the desired displacement-velocity-acceleration and jerk of the follower motion)

19. Layout of plate cam profiles:

Drawing the displacement diagrams for the different kinds of the motions and the plate cam profiles for these different motions and different followers.

SHM, Uniform velocity, Uniform acceleration and retardation and Cycloidal motions

Knife-edge, Roller, Flat-faced and Mushroom followers.

20. Derivatives of Follower motion:

Velocity and acceleration of the followers for various types of motions.

Calculation of Velocity and acceleration of the followers for various types of motions.

21. High speed cams:

High speed cams

22. Circular arc and Tangent cams:

Circular arc

Tangent cam

23. Standard cam motion:

Simple Harmonic Motion

Uniform velocity motion

Uniform acceleration and retardation motion

Cycloidal motion

24. Pressure angle and undercutting:

Pressure angle

Undercutting

UNIT IV

GEARS

SPUR GEARS

Gears are used to transmit power between shafts rotating usually at different speeds.

Some of the many types of gears are illustrated below :

TERMINOLOGY OF SPUR GEAR

Fig. 1: Terminology of spur gear

A pair of meshing gears is a power transformer, a coupler or interface which marries the

speed and torque characteristics of a power source and a power sink (load).

A single pair may be inadequate for certain sources and loads, in which case more complex combinations such as the above gearbox, known as gear trains, are

necessary.

In the vast majority of applications such a device acts as a speed reducer in which the power source drives the device through the high speed low torque input shaft, while power is fed from the device to the load through the low speed high torque output shaft.

Speed reducers are much more common than speed -up drives not so much because they reduce speed, but rather because they amplify torque.

Thus gears are used to accelerate a car from rest, not to provide the initial low speeds

(which could be accomplished by easing up on the accelerator pedal) but to increase

the torque at the wheels which is necessary to accelerate the vehicle.

These notes will consider the following aspects of spur gearing :-

Overall kinetics of a gear pair (for cases only of steady speeds and loads)

Tooth geometry requirements for a constant velocity ratio (eg. size and conjugate action)

Detailed geometry of the involute tooth and meshing gears

The consequences of power transfer on the fatigue life of the components The essentials of gear design.

Some of the main features of spur gear teeth are illustrated. The teeth extend from the root, or dedendum cylinder (or colloquially, "circle" ) to the tip, or addendum circle,both these circles can be measured.

The useful portion of the tooth is the flank (or face), it is this surface which contacts the mating gear.

The fillet in the root region is kinematically irrelevant since there is no contact there,but it is important insofar as fatigue is concerned.

CONDITION FOR CONSTANT VELOCITY RATIO OF TOOTHED WHEELS

(LAW OF GEARING)

Consider the portions of the two teeth, one on the wheel 1 (or pinion) and the other on the wheel 2, as shown by thick line curves in Fig. 2.

Let the two teeth come in contact at point Q, and the wheels rotate in the directions as shown in the figure. Let TT be the common tangentand MN be the common normal to the curves at the point of contact Q.

From the centres O1and O2 draw O1M and O2N perpendicular to MN. A little

consideration will show that the point Q moves in the direction QC, when considered as a point on wheel 1, and in the direction QD when considered as a point on wheel

Let v1 and v2 be the velocities of the point Q od the wheels I and 2 respectively. If the teeth are to remain in contact, then the components of these ye locities akng the

common normal MN must be equal.

From above, we see that the angular velocity ratio is inversely proportional to the ratio of the distances of the point P from the centres O1 and O2, or the common normal to the two surfaces at the point of contact Q intersects the line of centres at point P which divides the centre distance inversely as the ratio of angular velocities

Fig. 2. Law of gearing.

Therefore in order to have a constant angular velocity ratio for all positions of the

wheels, the point P must be the fixed point (called pitch point) for the two wheels.

In other words, the common normal at the point of contact between a pair of teeth must always pass through the pitch point. This is the fundaniental condition which must be satisfied while designing the profiles for the teeth of gear wheels. It is also known as law of gearing.

INTERFERENCE IN INVOLUTE GEARS

Fig. 3 shows a pinion with centre O1 in mesh with wheel or gear with centre O2.

MN is the common tangent to the base circles and KL is the path of contact between the two mating teeth.

Fig.3.

Fig.3. Interference in involute gears.

A little consideration will show, that if the radius of the addendum circle of pinion is increased to O1N the point of contact L will move from L to N.

When this radius is further increased, the point of contact L will be on the inside of base circle of wheel and not on the involute profile of tooth on wheel. The tip of tooth on the pinion will then undercut the tooth on the wheel at the root and remove part of the involute profile of tooth on the wheel.

This effect is known as !nterference, and occurs when the teeth are being cut. In brief, the phenomenon when the tip of tooth undercuts the root on its mating gear is known as interference.

Similarly, if the radius of the addendum circle of the wht l increases beyond O2M then the tip of tooth on wheel will cause interference with the tooth on pinion.

The points M and N are called interference points. Obviously, interference may be avoided if the path of contact does not extend beyond interference points. The limiting value of the radius of the addendum circle of the pinion is O1N and of the wheel is O2M.

From the above discussion, we conclude that the interference may only be avoided, if the point of contact between the two teeth is always on the involute profiles of both the teeth. In other words, interference may only be prevented (f the addendu addendum circles of the two mating gears cut the common tangent to the base circles between the points of tangency.

Maximum length of path of contact,

MN = MP + PN = r sin +Rsin =(r +R) sin

OVERALL KINETICS OF A GEAR PAIR

Fig : 4 Overall kinetics of a gear pair

Fig : 4 Overall kinetics of a gear pair

Analysis of gears follows along familiar lines in that we consider kinetics of the overall assembly first, before examining internal details such as individual gear teeth.

The free body of a typical single stage gearbox is shown.

The power source applies the torque T1 to the input shaft, driving it at speed 1 in the sense of the torque (clockwise here).

For a single pair of gears the output shaft rotates at speed 2 in the opposite sense to the input shaft, and the torque T2 supplied by the gearbox drives the load in the sense

The reaction to this latter torque is shown on the free body of the gearbox apparently the output torque T2 must act on the gearbox in the same sense as that of the input torque T1.

The gears appear in more detail in Fig 5( i) below. O1 and O2 are th the centres of the pinion and wheel respectively. We may regard the gears as equivalent pitch

cylinders which roll together without slip - the requirements for preventing slip due to

the positive drive provided by the meshing teeth is examined below.

Unlike the addendum and dedendum cylinders, pitch cylinders cannot be measured directly; they are notional and must be inferred from other measurements.

Fig: 5 spur gear assembly

One essential for correct meshing of the gears is that the size of the teeth on the

pinion is the same as the size of teeth on the wheel.

One measure of size is the circular pitch, p, the distance between adjacent teeth

around the pitch circle Fig 5 ( ii); thus p = D/z where z is the number of teeth on a

gear of pitch diameter D.

The SI measure of size is the module, m = p/ - which should not be confused with the SI abbreviation for metre. So the geometry of pinion 1 and wheel 2 must be such that :

D1 / z1 = D2 / z2 = p / = m

That is the module must be common to both gears. For the rack illustrated above, both the diameter and tooth number tend to infinity, but their quotient remains the finite module.

The pitch circles contact one another at the pitch point, P Fig 5( iii), which is also

notional. Since the positive drive precludes slip between the pitch cylinders, the pinion's pitch line velocity, v, must be identical to the wheel's pitch line velocity

v = 1 R1 = 2 R2 ; where pitch circle radius R = D/2

Separate free bodies of pinion and wheel appear in Fig 5(iv).Ft is the tangential

component of action -reaction at the pitch point due to contact between the gears.

The corresponding radial component plays no part in power transfer and is therefore not shown on the bodies. Ideal gears only are considered initially, so the friction force due to sliding contact is omitted also.

The free bodies show that the magnitude of the shaft reactions must be Ft, and that for equilibrium :

Ft = T1 / R1 = T2 / R2 in the absence of friction.

The preceding concepts may be combined conveniently into :

1) T2 / T1 = D2 / D1 = z2 / z1 ; D = mz

That is, gears reduce speed and amplify torque in proportion to their teeth numbers. In practice, rotational speed is described by N (rev/min or Hz) rather than by (rad/s).

There exists a host of shapes which ensure conjugacy - indeed it is possible, within certain restrictions, to arbitrarily choose the shape of one body then determine the shape of the second necessary for conjugacy.

But by far the most common gear geometry which satisfies conjugacy is based on the involute, in which case both gears are similar in form, and the contact point's locus is a simple straight line - the line of action.

One method of generating an involute is shown in Fig 6. A. A generating cord, in which there is a knot C, is wrapped around a fixed cylinder - the base cylinder (idiomatically circle ) of radius Ro.

When the taut cord is subsequently unwound as shown in this animation, the knot traces out an involute whose polar coordinates may be expressed implicitly in terms of the variable generating angle , reckoned from the radius through the initial knot position, C'.

The coordinate origin is taken at the circle centre, O, with a fixed reference direction defined at some constant angle , also from the initial radius.

The tangent, TC, is normal to the involute at C, and since the tangent length TC is equal to the arc length TC', the polar coordinates of C ( r, ) are

2) r = Ro ( 1 + 2 ) = arctan

In order to see how the involute leads to gear teeth and conjugate action, we place aslightly different interpretation on the above model. The cord is wrapped around the base cylinder which in Fig 6.B is now free to rotate about its centre as the cord is pulled off in a fixed direction.

This fixed cord direction forms the line of action, tangent to the base cylinder at the fixed point T, and clearly satisfies conjugacy by cutting the fixed reference at the fixed pitch point P through which the pitch cylinder passes.

The line of action is inclined to the pitch point tangent at the pressure angle. The knot C always moves along the line of action, tracing out an involute with respect to the rotating cylinder.

The relation between the base and pitch circle radii is evidently :-

3) Ro = R cos

Extending this to two cylinders - representing meshing gears, 1 & 2 Fig C - the taut cord winds off one base cylinder and onto the other to form the line of action inclined at the pressure angle

The knot, C, on the mating involutes coincides with the contact point and moves along the line of action as the gears and base cylinders rotate. The pitch cylinders extend to the pitch point P situated at the intersection of the lines of action and of centres.

Evidently the distance between the cylinders does not affect the speed ratio since the base cylinder diameters are fixed.

The distance between knots - ie. between tooth flanks along the line of action is the base pitch, po, given by

po = Do / z = p cos = m cos . . . . . from ( 1 )

For continuous motion transfer, at least two pairs of teeth must be in contact as one of the pairs comes into or leaves mesh. The teeth in Fig.6.C are truncated in practice to permit rotation.

Involute generation by knotted cord is all very well conceptually, but hardly practicable as a basis for manufacturing.

Only one of the many methods of gear manufacture is considered here - the rack generation technique is fundamental to the understanding of gear behaviour.

CONJUGATE TOOTH ACTION

Fig : 7 Conjugate tooth action

We have seen that one essential for correctly meshing gears is that the size of the teeth ( the module ) must be the same for the two gears.

We now examine another requirement - the shape of teeth necessary for the speed ratio to remain constant during an increment of rotation; this behaviour of the contacting

surfaces (ie. the teeth flanks) is known as conjugate action.

Consider the two rigid bodies 1 and 2 which rotate about fixed centres, O, with angular velocities. The bodies touch at the contact point, C, through which the common tangent and normal are drawn.

The absolute velocity v of the contact point reckoned as a point on either body, is

perpendicular to the radius from that body's centre O to the contact point.

For the bodies to remain in contact, there must be no component of relative motion along the common normal, so that from the velocity triangles

v2 cos2 = v1 cos1 where v1 = O1C ; v2 = O2C

Note that the tangential components of velocity are generally different, so sliding must occur.

For the speed ratio to be constant therefore, from the above and similar triangles

O1C/v1 . O2C = O1C.cos /O2C.cos= O1C1 /O2C2 = O1 P / O2 P

ie. this ratio also must be constant.

This indicates that, since the centres are fixed, the point P is fixed too.

In general therefore, whatever the shapes of the bodies, the contact point C will move along some locus as rotation proceeds; but if the action is to be conjugate then the body geometry must be such that the common normal at the contact point passes always through one unique point lying on the line of centres - this point is the pitch point referred to above, and the pitch circles' radii are O1 P and O2 P.

There exists a host of shapes which ensure conjugacy - indeed it is possible, within certain restrictions, to arbitrarily choose the shape of one body then determine the shape of the second necessary for conjugacy.

But by far the most common gear geometry which satisfies conjugacy is based on the involute, in which case both gears are similar in form, and the contact point's locus is a simple straight line - the line of action.

One method of generating an involute is shown in Fig 8 .A. A generating cord, in which there is a knot C, is wrapped around a fixed cylinder - the base cylinder (idiomatically circle ) of radius.Ro.

When the taut cord is subsequently unwound as shown in this animation, the knot traces out an involute whose polar coordinates may be expressed implicitly in terms of the variable generating angle , reckoned from the radius through the initial knot position, C'.

The coordinate origin is taken at the circle centre, O, with a fixed reference direction defined at some constant angle , also from the initial radius.

The tangent, TC, is normal to the involute at C, and since the tangent length TC is equal to

the arc length TC', the polar coordinates of C ( r, ) are :-

6) r = Ro ( 1 + 2 ) ; = - + arc tan

In order to see how the involute leads to gear teeth and conjugate action, we place a slightly different interpretation on the above model.

The cord is wrapped around the base cylinder which in Fig.8.B is now free to rotate about its centre as the cord is pulled off in a fixed direction.

This fixed cord direction forms the line of action, tangent to the base cylinder at the fixed point T, and clearly satisfies conjugacy by cutting the fixed reference at the fixed pitch point P through which the pitch cylinder passes.

The line of action is inclined to the pitch point tangent at the pressure angle,. The knot C always moves along the line of action, tracing out an involute with respect to the rotating cylinder.

The relation between the base and pitch circle radii is evidently

Ro = R cos

Extending this to two cylinders - representing meshing gears, 1 & 2 Fig C - the taut cord winds off one base cylinder and onto the other to form the line of action inclined at the pressure angle.

The knot, C, on the mating involutes coincides with the contact point and moves along the line of action as the gears and base cylinders rotate.

The pitch cylinders extend to the pitch point P situated at the intersection of the lines of action and of centres. Evidently the distance between the cylinders does not affect the speed ratio since the base cylinder diameters are fixed.

A pinion tooth touches a wheel tooth at the contact point C (the knot) which moves up the line of action and along the teeth faces as rotation proceeds.

Since contact cannot occur outside the teeth, it takes place along the line of action only between the points Q2 and Q1 on the line of action and inside both addendum circles.

The line segment Q2Q1 is named the path of contact.

Fig : 9 The path of contact

The Figure shows clearly :

the contact point marching along the line of action

the path of contact bounded by the two addenda

the orthogonality between line of action and involute tooth flanks at the contact

point

how load is transferred from one pair of contacting teeth to the next as rotation

proceeds

relative sliding between the teeth - particularly noticable at the beginning and end

of contact

guaranteed tooth tip clearance due to the dedendum exceeding the addendum

a significant gap between the non-drive face of a pinion tooth and the adjacent

wheel tooth

The gap between the non-drive face of the pinion tooth and the adjacent wheel tooth is known as backlash. If the rotational sense of the pinion were to reverse, then a period of unrestrained pinion motion would take place until the backlash gap closed and contact with the wheel tooth re-established impulsively.

Shock in a torsionally vibrating drive is exacerbated by significant backlash, though a small amount of backlash is provided in all drives to prevent binding due to

manufacturing or mounting inaccuracies and to facilitate lubrication.

Backlash may be reduced by subtle alterations to tooth profile or by shortening the centre distance from the extended value, however we consider gears meshing only at the extended centre distance.

The average number of teeth in contact is an important parameter - if it is too low due to the use of inappropriate profile shifts or to an excessive centre distance for example, then manufacturing inaccuracies may lead to loss of kinematic continuity - that is to impact, vibration and noise.

The average number of teeth in contact is also a guide to load sharing between teeth; it is termed the contact ratio, given by

length of path of contact / distance between teeth along the line of action

= Q2PQ1 / base pitch, po and for extended centres with for the 20o system :

Gears having a contact ratio below about 1.2 are not normally recommended as the gears themselves, their shafts and bearings would all require especial care in design and manufacture to preserve conjugacy.

A pinion tooth touches a wheel tooth at the contact point C (the knot) which moves up the line of action and along the teeth faces as rotation proceeds.

Since contact cannot occur outside the teeth, it takes place along the line of action only between the points Q2 and Q1 on the line of action and inside both addendum circles. The line segment Q2Q1 is named the path of contact.

GEAR TRAIN

SIMPLE GEAR TRAIN:

The only way that the input and output shafts of a gear pair can be made to rotate in the

same sense is by interposition of an odd number of intermediate gears as shown in Fig , these do not affect the speed ratio between input and output shafts.

Such a gear train is called a simple train. If there is no power flow through the shaft of an intermediate gear then it is an idler gear.

COMPOUND GEAR TRAIN:

A gear train comprising two or more pairs is termed compound when the wheel of one stage is mounted on the same shaft as the pinion of the next stage.

A compound train as in the above gearbox is used when the desired speed ratio cannot be achieved economically by a single pair.

Applying ( 1) to each stage in turn, the overall speed ratio for a compound train is found to be the product of the speed ratios for the individual stages.

Selecting suitable integral tooth numbers to provide a specified speed ratio can beawkward if the speed tolerance is tight and the range of available tooth numbers is limited.

Unlike the above gearbox, the input and output shafts are coaxial in the train illustrated here; this is rather an unusual feature, but necessary in certain change speed boxes and the like.

In the next section we look at a particular gear train arrangement called an epicyclic

gear train, before focusing on details of gear tooth shape and manufacture.

EPICYCLIC GEAR TRAINS

An epicyclic train is often suitable when a large torque/speed ratio is required in a compact envelope. It is made up of a number of elements which are interconnected to form the train.

Each element consists of the three components illustrated below :

A central gear ( c) which rotates at angular velocity c about the fixed axis O-O

of the element, under the action of the torque Tc applied to the central gear's

integral shaft; this central gear may be either an external gear (also referred to as

a sun gear) Fig 13.a(1a), or an internal gear, Fig 13.a (1b).

An arm ( a) which rotates at angular velocity a about the same O-O axis under

the action of the torque, Ta - an axle A rigidly attached to the end of the arm

carries

A planet gear ( p) which rotates freely on the axle A at angular velocity p,

meshing with the central gear at the pitch point P - the torque Tp acts on the

planet gear itself, not on its axle.

The epicyclic gear photographed here without its arms consists of two elements. The central gear of one element is an exter+nal gear; the central gear of the other element is an internal gear.

The three identical planets of one element are compounded with ( joined to ) those of the second element.

We shall examine first the angular velocities and torques in a single three-component element as they relate to the tooth numbers of central and planet gears, zc and zp respectively.

The kinetic relations for a complete epicyclic train consisting of two or more elements may then be deduced easily by combining appropriately the relations for the individual elements.

All angular velocities, , are absolute and constant, and the torques, T, are external to the three-component element; for convenience all these variables are taken positive in one particular sense, say anticlockwise as here. Friction is presumed negligible, ie. the system is ideal.

There are two contacts between the components :

the planet engages with the central gear at the pitch point P where the action /

reaction due to tooth contact is the tangential force Ft, the radial component being

irrelevant;

the free rotary contact between planet gear and axle A requires a radial force

action / reaction; the magnitude of this force at A must also be Ft as sketched, for

equilibrium of the planet.

UNIT V

FRICTION

INTRODUCTION

When a body slides (rolls) or made to slide (roll) relative to a second body, with which it is in contact, there is a resistance to the relative motion. The resistance so

encountered is called friction.

The force resisting relative motion is called force of friction. Force of friction acts in a

direction opposite to that of relative motion and is tangential to the contacting surfaces

of the two bodies in contact.

At every joint in a machine, there is a loss of energy owing to friction. A proper

understanding about friction as a phenomenon enables us to reduce frictional forces.

In a number of applications, on the other hand, friction is considered to be quite

useful.

Friction drives like belt and rope drives, friction clutches, variable speed drives are

some of applications of this type.

TYPES OF FRICTION

Dry Friction

This type of friction exists between two bodies having relative motion and whose

contacting surfaces are dry and not separated by any lubricant.

It is further subdivided in two types as Sliding friction and rolling friction. Sliding friction is the friction in which the contacting surfaces have a sliding motion relative to each

other.

Rolling friction is the friction between two bodies in contact when they have a relative

motion of pure rolling.

Skin or Greasy Friction

When contact surfaces of two bodies, in relative motion, are separated by a film of

lubricant of small thickness, skin or greasy friction is said to exist between them.

This type of friction is also known as boundary friction.

Film or Viscous Friction

When contacting surfaces of two bodies, in relative motion, are completely separated

by a relatively thick film of fluid, viscous friction is said to exist between the two.

Limiting Friction

The maximum value of frictional force, which comes into play when one body slides or tends to slide over another body, is known as Limiting Friction.

Laws of Friction

1. Force of friction always acts in a direction in which the body tends to move.

2. Force of friction is directly proportional to the normal load between the

surfaces for a given pair of materials.

3. The force of friction depends upon the materials of the contacting surfaces.

4. The force of friction is independent of the area of contact surfaces for a given

normal load.

Coefficient of Friction

It is defined as the ratio between the limiting friction (F) and the normal reaction(R).

F / R

Figure. 1. Angle of friction ()

Let R is the resultant of normal reaction (RN) and the limiting friction (F). Then the angle

between R and RN is known as the angle of friction.

Tan = F / R. =

Angle of Repose

Consider that a body of weight (W) resting on an inclined plane. If the angle of

inclination of the plane to the horizontal is such that the body begins to move down

the plane, then the inclination of the plane is known as angle of repose.

The angle of repose is equal to angle of friction.

Figure.2. Inclined Plane

Body at Rest

Figure.3. Body at Rest

When a body is at rest on an inclined plane making an angle with the horizontal, the forces acting on the body are (Figure 3)

Let W = weight of body

RN, = Normal reaction

F' = force resisting the motion of body.

From equilibrium conditions, Wsin= F' a nd Wcos = RN.

If the angle of inclination of plane is increased, the body will just slide down the plane of its own when

W sin= F' = RN = W cos

Tan = = tan (or) =

This maximum value of angle of inclination of plane with the horizontal when the body

starts sliding of its own is known as the angle of repose or limiting angle of friction.

Motion up the Plane

Figure 4: Motion up the plane

Consider a body moving up an inclined plane under the action of a force F as shown

in Figure 4. Applying conditions of equilibrium and solving the equations obtained, we get the minimum force required to be applied, for equilibrium condition as Fmin = W sin (+)

Efficiency:

The efficiency of an inclined plane, when a body slides up the plane, is defined as the

ratio of the forces required to move the body without consideration and with consideration of force of friction. From the analysis the expression for the efficiency is found to be

Motion down the Plane

When the body moves down the plane, the force of friction F' (= Rn) acts in the

upwards direction and the reaction R, i.e. the combination of Rn and F' is inclined backwards.

Applying conditions of equilibrium we get the minimum force, required to be applied as

Fmin = W sin (-)

Efficiency:

Efficiency of the inclined plane when the body slides down the plane is defined as the

ratio of the forces required to move the body with and without the consideration of force of friction.

Square Threads

A square threaded screw used as a jack to raise a load W.

Faces of the square threads in the sectional vies are normal to the axis of the spindle.

Force F acting horizontally is the force at the screw thread required to slide the load

W up the inclined plane.

The force F required to be applied is given by

Substituting tan= l / d and tan= and simplifying we get

A bar is, usually fixed to the screw head to use as a lever for the application of force.

Let f = force applied at the end of the bar of length L

Then

f L = F (d/2) = Fr or f = Fr/L = W r / L [tan (+)]

If the weight is lowered, the expressions for F and f are given by

Screw efficiency is defined as

Therefore the above equation can be obtained in terms of and as

The efficiency is maximum when d/d= 0 giving the necessary condition for maximum efficiency as

= 450 - /2

V-THREADS

In this case the faces are inclined to the axis of spindle. Figure shows a section of Vthread in which 2is the angle between the faces of the thread.

If RN is the normal reaction, then the axial component of Rn must be equal to W

i.e. W = Rn cos

This shows that the coefficient of friction (or tan) as used in relations for the square threads is to be replaced by or /cosor tan /costo adapt them to V-thread

PIVOTS AND COLLARS

When a rotating shaft is subjected to an axial load, the thrust (axial force) is taken

either by a pivot or a collar.

Examples are the shaft of a steam turbine, propeller shaft of a ship etc.

COLLAR BEARING

A collar bearing or simply a collar is provided at any position along the shaft and

bears the axial load on a mating surface.

The surface of the collar may be plane normal to the shaft or of conical shape

Figure 5: Collar Bearing

PIVOT BEARING

When the axial load is taken by the end of the shaft, which is inserted in a recess to

bear the thrust, Figure 6.

Figure 6: Pivot Bearing

It is called a pivot bearing or simply a pivot. It is also known as footstep bearing.

9

Friction torque of a collar bearing or pivot bearing is calculated on the basis of following two

assumptions:

Uniform Pressure theory

Uniform Wear theory

Each assumption leads to a different value of torque.

Uniform Pressure theory

In this case the intensity of pressure on the bearing surface is assumed to be constant

and the intensity of pressure is given by

Where Ro is the outer radius of the collar and Ri is the inner radius of the collar.

Uniform Wear theory

In this case wearing of the bearing surface is assumed to be uniform. Under this

assumption

Po ro = Pi ri = P. r = constant for uniform rate of wear

CONE CLUTCH

BELT AND ROPE DRIVE

INTRODUCTION

The velocity of the belt.

The tension under which the belt is placed on the pulleys.

The arc of contact between the belt and the smaller pulley

The conditions under which the belt is used.

It may be noted that

1. The shafts should be properly in line to insure uniform tension across the belt

section.

2. The pulleys should not be too close together, in order that the arc of contact on the

smaller pulley may be as large as possible.

3. The pulleys should not be so far apart as to cause the belt to weigh heavily on the

shafts, thus increasing the friction load on the bearings.

4. A long belt tends to swing from side to side, causing the belt to run out of the pulleys,

which in turn develops crooked spots in the belt

5. The tight side of the belt should be at the bottom, so that whatever sag is present on

the loose side will increase the arc of contact at the pulleys.

6. In order to obtain good results with flat belts, the maximum distance between the

shafts should not exceed 10 meters and the minimum should not be less than 3.5

times the diameter of the larger pulley.

Selection of a Belt Drive

Following are the various important factors upon which the selection of a belt drive depends:

1. Speed of the driving and driven shafts,

2. Speed reduction ratio,

3. Power to be transmitted,

4. Centre distance between the shafts,

5. Positive drive requirements,

6. Shafts layout,

7. Space available, and

8. Service conditions.

Types of Belt Drives

The belt drives are usually classified into the following three groups:

Light drives:

These are used to transmit small powers at belt speeds up to about 10 m is, as in

agricultural machines and small machine tools.

Medium drives:

These are used to transmit medium power at belt speeds over 10 m but up to 22 m as

in machine tools.

Heavy drives:

These are used to transmit large powers at belt speeds above 22 m/s, as in

compressors and generators.

Types of Belts

Fig.7.Types of belts.

Fig.7.Types of belts.

Though there are many types of belts used these days, yet the following are important

from the subject point of view:

Flat belt:

The flat belt, as shown in Fig. 7 (a), is mostly used in the factories and workshops,

where a moderate amount of power is to be transmitted, from one pulley to another when the two pulleys are not more than 8 meters apart.

V-belt:

The V-belt, as shown in Fig. 7(b), is mostly used in the factories and workshops,

where a moderate amount of power is to be transmitted, from one pulley to another, when the two pulleys are very near to each other.

Circular belt or rope:

The circular belt or rope, as shown in Fig. 7(c), is mostly used in the factories and

workshops, where a great amount of power is to be transmitted, from one pulley to

another, when the two pulleys are more than 8 meters apart.

If a huge amount of power is to be transmitted, then a single belt may not be

sufficient.

In such a case, wide pulleys (for V-belts or circular belts) with a number of grooves

are used.

Then a belt in each groove is provided to transmit the required amount of power from

one pulley to another.

ME 2203 - KINEMATICS OF MACHINERY

UNIT I BASICS OF MECHANISMS

1.Define Kinematic link.

It is a resistive body which go to make a part of a machine having relative motion between them.

2. Define Kinematic pair.

When two links are in contact with each other it is known as a pair. If the pair makes constrain motion it is known as kinematic pair.

3. Define Kinematic chain.

When a number of links connected in space make relative motion of any point on a link with respect to any other point on the other link follow a definite law it is known as kinematic chain.

4.Write the Grublers criterion for determining the degrees of freedom of a mechanism having plane motion.

n=3(l-1)-2j

h-Higher pair joint

l-Number of links

j-Lower pair joint

5.Define degree of freedom, what is meant by mobility. (Ap/May-2008)

The mobility of a mechanism is defined as the number of input parameters which must be independently controlled in order to bring the device into a particular position.

6.Write the Kutzbachs relation. (Ap/May-2008)

Kutzbachs criterion for determining the number of degrees of freedom or movability (n) of a plane mechanism is n=3(l-1)-2j-h

n-Degree of freedom.

l-Number of links.

h-Higher pair joint

j-Lower pair joint.

7.Define Grashoffs law and state its significance? (Ap/May-2008)

It states that in a planar four bar mechanism, the sum of shortest link length and longest link length is not greater than the sum of remaining two links length, if there is to be continuous relative motion between two members.

Significance:

Grashoffs law specifies the order in which the links are connected in a kinematic chain.

Grashoffs law specifies which link of the four-bar chain is fixed.(s+1)=(p+q) should be satisfied, if not, no link will make a complete revolution relative to another.

s= length of the shorter length

l= length of the longest link

p & q are the lengths of the other two links.

8.Define Inversion of mechanism.

The method of obtaining different mechanism by fixing different links in a kinematic chain is known as inversion of mechanism.

9.What is meant by Mechanical advantage of mechanism?

It is defined as the ratio of output torque to the input torque also defined as the ratio of load to effort.

M.A ideal = TB / TA

TB =driven (resisting torque)

TA =driving torque

10.Define Transmission angle.

The acute angle between follower and coupler is known as transmission angle.

11. Define Toggle position.

If the driver and coupler lie in the same straight line at this point mechanical advantage is maximum. Under this condition the mechanism is known as toggle position.

12.List out few types of rocking mechanism?

Pendulum motion is called rocking mechanism.

1. Quick return motion mechanism.

2. Crank and rocker mechanism.

3. Cam and follower mechanism.

13.Define pantograph?

It is device which is used to reproduce a displacement exactly in a enlarged scale. It is used in drawing offices, for duplicating the drawing maps, plans, etc. It works on the principle of 4 bar chain mechanism.

Eg. Oscillating-Oscillating converter mechanism

14.Name the application of crank and slotted lever quick return motion mechanism?

1. Shaping machines.

2. Slotting mechanism.

3. Rotary internal combustion engine.

15.Define structure?

It is an assemblage of a number of resistant bodies having no relative motion between them and meant for carrying loads having straining action.

16.What is simple mechanism?

A mechanism with four links is known as simple mechanism.

17.Define mechanism?

When one of the links of a kinematic chain is fixed, the chain is known as a mechanism.

18.Define equivalent mechanism; and spatial mechanism?

Equivalent mechanism: The mechanism, that obtained has the same number of the degree of freedom, as the original mechanism called equivalent mechanism.

Spatial mechanism: Spatial mechanism have special geometric characteristics in that all revolute axes are parallel and perpendicular to the plane of motion and all prism axes lie in the plane of motion.

19.Define double slider crank chain mechanism?

A kinematic chain which consists of two turning pair and two sliding pair is known as double slider crank mechanism.

20.Define single slider crank chain mechanism?

A single slider crank chain is a modification of the basic four bar chain. It consists of one sliding pair and three turning pair.

Applications:

Rotary or Gnome engines

Crank and slotted lever mechanism

Oscillating cylinder engine

Ball engine

Hand pump

21.Define Sliding pair.

In a sliding pair minimum number of degree of freedom is only one.

22.Define Turning pair.

In a turning pair also degree of freedom is one. When two l