BIBIN CHIDAMBARANATHAN: KINEMATICS OF MACHINERY

KINEMATICS OF MACHINERY

1. 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 Grubler’s 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:

· 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.

2. Kutzbach criterion, Grashoff's law

Kutzbach criterion:

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

Can we intuitively derive Kutzbach’s modification of Grubler’s 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

Let’s 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

3. 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) Watt’s 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. Watt’s 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 (1^stinversion)

2) Oscillating cylinder engine mechanism (2^ndinversion)

3) Crank and slotted lever mechanism (2^ndinversion)

4) Whitworth quick return motion mechanism (3^rdinversion)

5) Rotary engine mechanism (3^rdinversion)

6) Bull engine mechanism (4^thinversion)

7) Hand Pump (4^thinversion)

· 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 α and the return stroke is executed when the crank rotates through angle β or 360 – α. Therefore, when the crank rotates uniformly, we get,

Time to cutting = α = α

Time of return β 360 – α

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 α or 360 -2θ. During the return stroke, the angle covered is 2θ or β.

Therefore, Time to cutting = 360 -2θ = 180 – θ

Time of return 2θθ = α = α . β 360 – α

· 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) Oldham’s 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 θ = x. PR

and Sin θ = y. QR

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

(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. Oldham’s 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 Oldham’s 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=xω where, ω = angular velocity of each shaft in rad/sec v = linear velocity in m/sec

4. 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.

5. 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 α or 360 -2θ. During the return stroke, the angle covered is 2θ or β.

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 α in Counter clockwise direction during working stroke, the link CD rotates through 180. We can observe that / α >/ β. Hence time of working stroke is α /β times more or the return stroke is α /β 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.

Time to cutting = α = α

Time of return β 360 – α

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

6. 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 α =P Cos α 2

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

5. Hooke’s joint:

Hooke’s 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 Hooke’s 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 α. When the vehicles steer to the right as shown in the figure, the short link BL is turned so as to increase α, where as the link LK causes the other short link AK to turn so as to reduce α. The fundamental equation for correct steering is, CotΦ–Cosθ = b / l

In the above arrangement it is clear that the angle Φ through which AK turns is less than the angle θ 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 θ, the corresponding value of Φ and (Cot Φ – Cos θ) 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 α. 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 servant’s 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.

7. 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. Grasshopper’s mechanism, c. Robert’s mechanism,

d. Tchebicheff’s 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=(AF–FP)(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. Robert’s 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 Robert’s 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.

8. 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 = wr, where w = angular velocity of the link and r = distance from pivot.

Acceleration Components

Normal Acceleration: Aⁿ = w²r. Points toward the center of rotation
Tangential Acceleration: A^t = ar. In a direction perpendicular to the link
Coriolis Acceleration: A^c = 2w(dr/dt). In a direction perpendicular to the link
Sliding Acceleration: A^s = 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 (w). 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 + A^t + A^c + A^s

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.

9. 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 = wr, 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, V_A, 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: V_B = V_A + V_B/A where V_B = absolute velocity of point B, V_A = absolute velocity of point A, and V_B/A is the velocity of point B relative to point A.

10. Velocity Analysis of Four Bar Mechanisms:

Problems solving in Four Bar Mechanisms and additional links.

11. Velocity Analysis of Slider Crank Mechanisms:

Problems solving in Slider Crank Mechanisms and additional links.

12. Acceleration Analysis of Four Bar Mechanisms:

Problems solving in Four Bar Mechanisms and additional links.

13. Acceleration Analysis of Slider Crank Mechanisms:

Problems solving in Slider Crank Mechanisms and additional links.

14. Kinematic analysis by Complex Algebra methods:

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

15. Vector Approach:

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

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

Computer programming for simple mechanisms

17. Coincident points, Coriolis Acceleration:

Coriolis Acceleration: A^c = 2w(dr/dt). In a direction perpendicular to the link.

A slider attached to ground experiences only sliding acceleration.

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.

18. 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.

19. 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)

20. 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.

21. 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.

22. High speed cams:

High speed cams

23. Circular arc and Tangent cams:

Circular arc
Tangent cam

24. Standard cam motion:

Simple Harmonic Motion
Uniform velocity motion
Uniform acceleration and retardation motion
Cycloidal motion

25. Pressure angle and undercutting:

Pressure angle

· Undercutting

Unit IV 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.

26. Spur gear Terminology and definitions:

Spur Gears:
External
Internal
Definitions

27. Fundamental Law of toothed gearing and Involute gearing:

Law of gearing
Involutometry and Characteristics of involute action
Path of Contact and Arc of Contact
Contact Ratio
Comparison of involute and cycloidal teeth

28. Inter changeable gears, gear tooth action, Terminology:

Inter changeable gears
Gear tooth action
Terminology

29. Interference and undercutting:

Interference in involute gears
Methods of avoiding interference
Back lash

30. Non standard gear teeth: Helical, Bevel, Worm, Rack and Pinion gears (Basics only)

Helical
Bevel
Worm
Rack and Pinion gears

RACK AND PINION WORM GEAR

31. Gear trains:

Gear Train Basics
The velocity ratio, m_V, of a gear train relates the output velocity to the input velocity.
For example, a gear train ratio of 5:1 means that the output gear velocity is 5 times the input gear velocity.

32. Parallel axis gear trains:

Simple Gear Trains – A simple gear train is a collection of meshing gears where each gear is on its own axis. The train ratio for a simple gear train is the ratio of the number of teeth on the input gear to the number of teeth on the output gear. A simple gear train will typically have 2 or 3 gears and a gear ratio of 10:1 or less. If the train has 3 gears, the intermediate gear has no numerical effect on the train ratio except to change the direction of the output gear.
Compound Gear Trains – A compound gear train is a train where at least one shaft carries more than one gear. The train ratio is given by the ratio m_V = (product of number of teeth on driver gears)/(product of number of teeth on driven gears). A common approach to the design of compound gear trains is to first determine the number of gear reduction steps needed (each step is typically smaller than 10:1 for size purposes). Once this is done, determine the desired ratio for each step, select a pinion size, and then calculate the gear size.
Reverted Gear Trains – A reverted gear train is a special case of a compound gear train. A reverted gear train has the input and output shafts in –line with one another. Assuming no idler gears are used, a reverted gear train can be realized only if the number of teeth on the input side of the train adds up to the same as the number of teeth on the output side of the train.

33. Epicyclic gear trains:

· If the axis of the shafts over which the gears are mounted are moving relative to a fixed axis , the gear train is called the epicyclic gear train.

· Problems in epicyclic gear trains.

34. Differentials:

Used in the rear axle of an automobile.
To enable the rear wheels to revolve at different speeds when negotiating a curve.
To enable the rear wheels to revolve at the same speeds when going straight.

Unit V Friction

35. Surface contacts:

Basic laws of friction
Pivot and collar, introduction and types.
Problem on flat pivot, Problems on conical pivot.

36. Sliding and Rolling friction:

Sliding contact bearings
Rolling contact bearings
Problems in bearings

37. Friction drives:

Friction drives
Positive drives and Slip drives
Speed ratio

38. Friction in screw threads:

· Friction in screw and nut

· Friction in screw jack

· Problems in screw jack

39. Friction clutches:

Single plate clutches and Multi-plate clutches
Uniform wear theory and Uniform pressure theory
Problems in clutches

40. Belt and rope drives:

Belt drives, Open belt drives and Crossed belt drives
Length of the belt and Angle of lap
Power transmitted by a belt drive
Problems in belt drives

41. Friction aspects in Brakes:

Brakes, Types
Mechanical brakes, band brakes
Braking torque calculations
Self locking brakes
Problems in brakes

42. Friction in vehicle propulsion and braking:

Vehicle dynamics
Vehicle propulsions
Braking aspects in vehicles

Mechanisms

A mechanism is a combination of rigid or restraining bodies so shaped and connected that they move upon each other with a definite relative motion. A simple example of this is the slider crank mechanism used in an internal combustion or reciprocating air compressor.

Machine

A machine is a mechanism or a collection of mechanisms which transmits force from the source of power to the resistance to be overcome,and thus perform a mechanical work.

Plane and Spatial Mechanisms

If all the points of a mechanism move in parallel planes, then it is defined as a plane mechanism.
If all the points do not move in parallel planes then it is called spatial mechanism.

Kinematic Pairs

A mechanism has been defined as a combination so connected that each moves with respect to each other.A clue to the behaviour lies in in the nature of connections,known as kinetic pairs.
The degree of freedom of a kinetic pair is given by the number independent coordinates required to completely specify the relative movement.

Lower Pairs

A pair is said to be a lower pair when the connection between two elementsis through the area of contact.Its 6 types are :

Higher Pairs

A higher pair is defined as one in which the connection between two elements has only a point or line of contact. A cylinder and a hole of equal radius and with axis parallel make contact along a surface. Two cylinders with unequal radius and with axis parallel make contact along a line. A point contact takes place when spheres rest on plane or curved surfaces (ball bearings) or between teeth of a skew-helical gears. in roller bearings, between teeth of most of the gears and in cam-follower motion. The degree of freedom of a kinetic pair is given by the number independent coordinates required to completely specify the relative movement.

Wrapping Pairs

Wrapping Pairs comprise belts, chains, and other such devices.

To define a mechanism we define the basic elements as follows :

Link

A material body which is common to two or more kinematic pairs is called a link.

Kinematic Chain

A kinematic chain is a series of links connected by kinematic pairs. The chain is said to be closed chain if every u link is connected to atleast two other links, otherwise it is calledan open chain. A link which is connected to only one other link is known as singular link.If it is connected to two other links, it is called binary link.If it is connected to three other links, it is called ternary link, and so on. A chain which consists of only binary links is called simple chain. A type of kinematic chain is one with constrained motion, which means that a definite motion of any link produces unique motion of all other links. Thus motion of any point on one link defines the relative position of any point on any other link.So it has one degree of freedom.

The process of fixing different links of a kinematic chain one at a time to produce distinct mechanisms is called kinematic inversion.Here the relative motions of the links of the mechanisms remain unchanged.
First, let us consider the simplest kinematic chain,i.e., achain consisting of four binary links and four revolute pairs. The four different mechanisms can be obtained by four different inversions of the chain.

Slider Crank mechanism

It has four binary links, three revolute pairs, one prismatic pair.By fixing links 1, 2, 3 in turn we get various inversions.

Double Slider Crank mechanism

It has four binary links, two revolute pairs, two sliding pairs.Its various types are :

Scotch Yoke mechanism:

Here the constant rotation of the crank produces harmonic translation of the yoke.Its four binary links are :

Fixed Link
Crank
Sliding Block
Yoke

The four kinematic pairs are :

revolute pair (between 1 & 2)
revolute pair (between 2 & 3)
prismatic pair (between 3 & 4)
prismatic pair (between 4 & 1)

Oldhams Coupling:

It is used for transmitting anbgular velocity between two parallel but eccentric shafts

Elliptical Trammel:

Here link 4 is fixed. Any point on the link 2 describes an ellipse as it moves.The mid-point of the link 2 will obiviously describe a circle.

Let n be the no. of links in a mechanism out of which, one is fixed, and let j be the no. of simple hinges(ie, those connect two links.) Now, as the (n-1) links move in a plane, in the absence of any connections, each has 3 degree of freedom; 2 coordinates are required to specify the location of any reference point on the link and 1 to specify the orientation of the link. Once we connect the linmks there cannot be anyrelative translation betweenthem and only one coordinate is necessary to specify their relative orientation.Thus, 2 degrees of freedom (translation) are lost, and only one degree of freedom (rotational) is left. So, no. of degrees of freedom is:
F=3(n-1)-2j
Most mechanisms are constrained, ie F=1. Therefore the above relation becomes,
2j-3n+4=0
,this is called Grubler's Criterion.
Failure of Grubler's criterion
A higher pair has 2 degrees of freedom .Following the same argument as before, The degrees of freedom of a mechanism having higher pairs can be written as,
F=3(n-1)-2j-h
Often some mechanisms have a redundant degree of freedom. If a link can move without causing any movement in the rest of the mechanism, then the link is said to have a redundant degree of freedom.
Example of redundant degree of freedom

The objective of kinematic analysis is to determine the kinematic quantities such as displacements, velocities, and accelerations of the elements of a mechanism when the input motion is given. It establishes the relationship between the motions of various components of the linkage.

When the kinematic dimensions and the configurations of the input link of a mechanism are prescribed, the configurations of all the other links are determined by displacement analysis.

Graphical Method
Analytical Method

In a graphical method of displacement analysis, the mechanism is drawn to a convenient scale and the desired unknown quantities are determined through suitable geometrical constructions and calculations.

The configurations of a rigid body in plane motion are completely defined by the locations of any two points on it.
Two intersecting circles have two points of intersection and one has to be careful, when necessary, to choose the correct point for the purpose in hand.
The use of tracing paper, as an overlay, is very convenient and very often provides an unambiguous and quick solution.
The graphical method fails if no closed loop with four links exists in the mechanism.

An analytical method of displacement analysis, is preferred whenever

high level of accuracy is required
a large number of configurations have to be solved
The graphical method fails.

In this method every link is represented by two dimensional are represented by two dimensional vectors expressed through complex notation. Considering each closed loop in the mechanism, a vector equation is established. Separating the real and imaginary parts , sufficient number of nonlinear algebraic equations are obtained to solve for the unknown quantities.

Let us consider a 4R linkage of given link lengths, viz., i=1, 2, 3, and 4. The configuration of the input link (2) is also prescribed by the angle θ₂, and we have to determine the configurations of the other two links, namely, the coupler and the follower, expressed by the angles θ₃ and θ₄.

Referring to Figure, all links are denoted as vectors, viz., l₁, l₂, l₃ and l₄. All angles are measured CCW from the X-axis which is along the fixed vector l₁, rendering ¸

₁=0. Considering the closed loop O₂O₄B_AO₂, we can write

Using complex exponential notation with ¸

₁=0 can be written as

Equating the real and imaginary parts of this equation separately to zero, we get

........(1a)

........(1b)

Thus, the two unknowns, namely, ¸

₃ and ¸
₄ can be solved from the two equations (1a) and (1b) as now explained.
Rearranging (1a) and (1b), we get

Squaring both sides of these two equations and adding, we obtain

Where

...............(2)

It may be noted that with the prescribed data, the coefficients a, b, and c of (2) are known. To solve for ¸

4, from (2) without ambiguity of quadrant, it is better to substitute

In (2) to yield

Thus, for a given position of the input link, two different values of ¸

4 are obtained as follows:

These two values correspond to the two different ways in which the 4R mechanism can be formed for any given value of ¸

2, as explained in Figure where the same problem has been solved by a graphical method.

To solve for the coupler orientation ¸

3, we can eliminate ¸

4 from (1a) and (1b) to get

Where

For a 4R linkage, the transmission angle ( μ ) is defined as the acute angle between the coupler (AB) and the follower ).₄B), as indicated in Fig. 2.11. If ( - ABO₄) is acute (Fig.2.11), then μ =- ABO₄. On the other hand, if - ABO₄ is obtuse, then μ =Π-- ABO₄. As explained in this figure, if μ = Π /2, then the entire coupler force is utilized to drive the follower. For good transmission quality, the minimum value of μ(μ_min)>300. For a crank-rocker mechanism, the minimum value of μ occurs when the crank becomes collinear with the frame, i.e.,

. If the swing angle (

) of the rocker is increased maintaining the same quick-return ratio, then the maximum possible value of μ _min decreases. If the forward and return strokes of the rocker take equal time, then (μ_min)max is restricted to

(see Problem 2.6). Therefore, such a crank rocker will have a poor transmission quality if

In the figure a rigid body 2 is shown to be in plane motion with respect to fixed link 1. The velocities of two points A and B of the rigid link are shown by V_A and V_B ,respectively. Two lines drawn through A and B in directions perpendicular to V_A and V_B meet at P. Let PA=r₁ and PB=r₂. The velocity of point B in the direction of AB is V_Bcos¸

, and that of point A in the same direction is V_Acos¸
. As the length of AB is fixed, the component of V_BA in the direction of AB is zero. Thus V_Bcos¸
=V_Acos¸
. From the triangle PAB, we have

Thus the velocities of the points A and B are proportional and perpendicular to PA and PB, respectively. So ,instantaneously the rigid body can be thought of as being momentarily on pure rotation about the point P.The velocity of any point C on the body at this instant is given by V_C=PC.V_B/r₂ in a direction perpendicular to PC .The point P is called instantaneous centre of velocity, and its instantaneous velocity is zero. Alternatively instantaneous centre of velocity of velocity can be described as a point which has no velocity with respect to the fixed link.

If both links 1 and 2 are in motion, in a similar manner we can define relative instantaneous centre of velocity P₁₂ to be a point on 2 having zero velocity with respect to a coincident point on 1.Consequently the relative motion of link 2 with respect to 1 appears to be in pure rotation about P₁₂.

Thus if a mechanism has N links, the number of instantaneous centers possible are N (N-1)/2.

If two links have a hinged joint, the location of the hinge is the relative instantaneous centre because one link is in pure rotation with respect to the other about that hinge.
If relative motion between two links is pure sliding, the relative instantaneous centre lies at infinity on a line perpendicular to the direction of sliding.
If one link is rolling over another, the point of contact is the relative instantaneous centre.
If a link is sliding over a curved element, the centre of curvature is the relative instantaneous centre.
If the relative motion between two links is both rolling and sliding the relative instantaneous centre lies on the common normal to the surfaces of these links passing through the point of contact.

If three bodies are in relative motion with respect to each other, the three centers of velocity are collinear

Once the configuration of the mechanism is known, the velocities and acceleration are linear in the unknown quantities and hence are easy to solve. Consequently when the velocity and the acceleration analysis have to be carried out for a large number of configurations, the analytical method turns out to be more advantageous than the graphical method. Accuracy obtained in analytical method is also high.
Referring to the precious figure, assume that the configuration of the mechanism has already been determined, i.e., l₁,l₂, l₃,l₄, and θ₂ are prescribed and θ₃, θ₄are solved. The task is to determine the angular velocity and acceleration of the coupler and the follower if those of the crank are given. Towards this end differentiate with respect to time and obtain

………………………… (2a)

………………………… (2b)

We should note that the above equations are two simultaneous equations in the two unknown i.e.,

and

which can be solved to yield

The concept of velocity and acceleration images is used extensively in the kinematic analysis of mechanisms having ternary, quaternary, and higher-order links. If the velocities and accelerations of any two points on a link are known, then, with the help of images the velocity and acceleration of any other point on the link can be easily determined. An example is illustrated below:

Once the velocity analysis is complete, (2.34a) and (2.34b) again provide two linear equations in

and

which are obtained as
equation

A rigid link BCDE having four hinges is sown in figure. Let the angular velocity and acceleration of this be ω and α. The absolute velocity vectors of the E, B, C and D are shown in the figure as V_E, V_B, V_C, and V_D respectively. The velocity difference vectors are

And their magnitudes are, respectively,

So,

Hence the velocity diagram bcde is a scale drawing of the link BCDE. The figure bcde is called the velocity image of the link BCDE. The velocity image is rotated through 90o in the direction ω, as all the velocity difference vectors are perpendicular to the corresponding lined. The scale of the image is determined by and therefore the scale will be different for each link of a mechanism. The letters identifying the end points of the image are in the same sequence as that in the link diagram BCDE. The absolute velocity any point X on the link is obtained by joining the image of X(x) with the pole of the velocity diagram o.

Instantaneous Centre Method
Relative Velocity Method
First determine the number of instantaneous centers (N) by using the relation
Make a list of all the instantaneous centers in the mechanism.
Locate the fixed and permanent instantaneous centers by inspection.
Locate the remaining neither fixed nor permanent instantaneous centers by Kennedy’s theorem. This can be done by circle diagram. Mark points on a circle equal to the number of links in a mechanism.
Join the points by solid lines to show that these circles are already found. In the lines indicate the instantaneous centers corresponding to those particular two points.
In order to find the remaining instantaneous centers, join two such points that the line joining them forms two adjacent triangles in the circle diagram. The line which is responsible for completing two triangles should be a common side to the two triangles

The relative velocity method is based upon the velocity of the various points of the link.

Consider two points A and B on a link. Let the absolute velocity of the point A i.e. VA is known in magnitude and direction and the absolute velocity of the point B i.e. VB is known in direction only. Then the velocity of B may be determined by drawing the velocity diagram as shown.

Take some convenient point o, known as the pole.
Through o, draw oa parallel and equal to VA, to some convenient scale.
Through a, draw a line perpendicular to AB. This line will represent the velocity of B with respect to A, i.e.
Through o, draw a line parallel to VB intersecting the line of VBA at b.
Measure ob, which gives the required velocity of point B to the scale.

Consider two points A and B on the rigid link. The acceleration of the point A, i.e. aA is known in magnitude and direction and the direction of path of B is given. The acceleration of the point B is determined in magnitude and direction by drawing the acceleration diagram as discussed below:

From any point o , draw vector o a parallel to the direction of absolute at point A i.e. to some suitable scale as shown in figure.
We know that the acceleration of B with respect A i.e.aBA has the following two components:
Radial component of acceleration B with respect to A i.e.arBA and
Tangential component of acceleration of B with respect to A i.e.at BA
These two components are mutually perpendicular.
Draw vector a x parallel to the link AB such that
From point x, draw vector xb perpendicular to AB or vector a x and through o draw a line parallel to the path of the path of to represent the absolute acceleration of B i.e. aB. the vectors xb and a b intersect at b . Now the values of aB and atBA may be measured to the scale.
By joining the points a and b we may determine the total acceleration of B with respect A i.e. aB. The vector a b is known acceleration image of the link AB.

In a mechanism, a link is quite often guided along a prescribed path in another moving link. For the velocity and acceleration analyses of such a mechanism, the differences in the velocities and accelerations of two instantaneously coincident points belonging to the two links have to be determined. In this section, we shall derive the expressions for these quantities.

Figure shows a rotating rigid link (labeled 2) on which link 3 is moving along a straight line. The configurations at the instants t and (t+dt) are, respectively, shown by the symbols without and with a prime. Further, P₂ and P₃ represent two points on links 2 and 3, respectively, coincident at the instant t. The displacement of P₃ can be written as

Where P2P2 represents the displacement of P₂ and P₂ P₃ represents the displacement of P₃ with respect to link 2. Dividing both sides of the foregoing equation by ´

t and taking the limit

, we get

Where V_P3/2 is the velocity of P₃ as seen by an observer attached to link 2. The direction of V_P3/2 is tangential to the path of P₃ in link 2.

From figure we see that the block has moved through an additional transverse distance AP3’ because of rotation of link 2 and the radial motion of link 3 with respect to link 2.
When

From this equation we observe that the additional transverse distance is proportional to the square of the time elapsed. Therefore this displacement must be due to an additional acceleration of P3 in the transverse direction. If the magnitude of this additional is ac then

In vector notation, the above equation can be represented as

This extra transverse component of acceleration is known as the Coriollis component. The final expression will then be

It should be noted that the direction ac is obtained by rotating V_P3/2 through 90 degrees in the sense of Ƀ₂. For a straight path of P₃ on link 2, the direction of

is along the straight line.

When link 3 moves along a curvilinear path on the rotating link 2, the above equation can be written in terms of the components of

It may be noted that the magnitude of

is equal to

where C is the radius of curvature of the path of P₃ on link 2. the direction of V_P3/2 is obviously tangential to this path. For a straight line path of P on link 2, C becomes infinite and

Dimensional Synthesis of Mechanisms

Dimensional synthesis deals with the determination of kinematic dimensions (link lengths, offsets, etc.) of the mechanism to satisfy the required motion characteristics. Graphical as well as analytical methods are available for dimensional synthesis. The choice of the method depends largely on the type of problem to be solved. The problems can be classified as:

Motion generation
Path generation
Function generation
Dead-centre problems

Motion generation:

In motion generation, the required positions of the coupler are given and we have to find out mechanism.

Path generation:

In path generation a point on the link has to be guided through the required positions. Generally the point will be a point on the rigid body which is the coupler.

Function generation:

In function generation, the input and output angles are specified and we have to find the required mechanism. Two specified pairs of coordinated movements of the input and output links are to be generated.

BIBIN CHIDAMBARANATHAN

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Thursday, 13 September 2012

KINEMATICS OF MACHINERY

Unit II Kinematics

Unit III Kinematics of CAM

· Undercutting

Unit IV Gears

Gears are used to change speed in rotational movement.

Unit V Friction

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