CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering

Theory of Machines (ME 203)

INDEX Sr.

Date of

No.

performance

1

2

3 4 5 6 7

No. of

Title

Pages

Study of Fundamentals of Mechanism Study Of Inversions of Mechanism Study Straight Line Motion Mechanisms And Pantograph Study of Steering Mechanisms Velocity And Acceleration Diagram Tutorial on Gear Construction of Involute Gear Tooth Profile

8

Study of Gear Train

9

Study of Friction

10

Study of Cam And Cam Profile

Page 1

Marks

Date of

Sign of

Assessment Faculty

CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering

Theory of Machines (ME 203)

Date:

Experiment No 1 STUDY OF FUNDAMENTALS OF MECHANISM AIM:

To study of fundamentals of Mechanism

OBJECTIVES: 1.

To study fundamentals of mechanisms.

INTRODUCTION: The theory of machine and mechanisms is an applied science which is used to understand the relationships between the geometry & motions of the parts of the machine or mechanism & the forces which produce these motions. The theory of machine may be sub-divided into the following four branches. 1. Kinematics: It is that branch of theory of machines, which deals with the relative motion between the various parts of the machines. 2. Dynamics: It is that branch of theory of machines, which deals with the forces and their effects, while acting upon the machine parts in motion. 3. Kinetics: It is that branch of theory of machines, which deals with the inertia forces which arise from the combined effect of the mass and motion of the machine parts. 4. Statics: It is that branch of theory of machines, which deals with the forces and their effects, while the machine parts are at rest. The mass of the parts is assumed to be negligible.. MECHANISM & MACHINE: Each part of a machine, which moves relative to some other part is known as a kinematic link or link. The word link is used to designate a machine part or a component of mechanism. A link is a member or a combination of members of a mechanism connecting other members and having motion relative to them. When a mechanism is required to transmit the power or to do some particular type of work, it then becomes a machine. Therefore a machine is a mechanism, which apart from imparting definite motions to the parts, also transmits and modifies the available mechanical energy into some kind of desired work.

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering

KINEMATIC CHAIN: The two links or elements of a machine when in contact with each other are said to form a pair. If the relative motion between them is completely or successfully constrained the pair is known as kinematic pair. When the kinematic pairs are coupled in such a way that the last link is joined to the first link to transmit definite motion, it is called a kinematic chain. The following three types of kinematic chains are important from the subject point of view. 1) Four bar chain mechanism. 2) Single slider crank chain. 3) Double slider crank chain. FOUR BAR CHAIN: The simplest and the basic kinematic chain is a four bar chain or quadric cycle chain. It is much-preferred mechanical device for mechanization and control of motion due to its simplicity and versatility. Figure:

As shown in the above figure a four bar chain consists of four links connected in form of quadrilateral by four pin joints. In a four bar chain, one of the links, in particular the shortest link will make a complete revolution relative to the other three links. Such a link is called the “crank” or “driver”. Here, link A0A is the crank. Link BB0 makes partial rotation or oscillates and is known as “lever” or “rocker” and AB, which connects the crank, and lever is called “connecting rod” or “coupler”. The fixed link A0B0 is known as the “frame” of the mechanism. SINGLE SLIDER CRANK CHAIN: A single slider crank chain is a modification of the basic four bar chain. It consists of one sliding pair and three turning pairs. This type of mechanism converts rotary motion into reciprocating motion and vice versa. However, when it is used in an automobile by adding valve mechanism etc. it becomes a machine, which converts the available energy (force on the piston) into the desired energy (torque of the crank shaft). The torque is used to move a vehicle. Reciprocating pumps, reciprocating compressors and steam engines are other examples of machines derived from the slider crank mechanism.

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering In a single slider crank mechanism, as shown in figure below, between cylinder & piston there is a sliding pair and other are turning pairs. Figure: The link 1 corresponds to the frame of the engine, which is fixed. The link 2 corresponds to the crank, link 3 corresponds to the connecting rod and link 4 corresponds to cross head. As the crank rotates, the cross head reciprocates in the guides and thus the piston reciprocates in tee cylinder. DOUBLE SLIDER CRANK CHAIN: A double slider crank chain is a modification of the basic four bar chain. It consists of two sliding pair and two turning pairs. For example Elliptical trammel as shown in fig below link 1 & 3 are the slider which forms sliding pairs with fixed link (link 4) and link 2 is connecting rod having turning pair with sliders ( link 1 and link 2).

REVIEW QUESTIONS: 1. Define Mechanism 2. Differentiate between Machine ,Mechanism and Structure 3. Define kinematic pair and give its classification 4. Explain Grashoff’s Law. 5. Compare four bar chain mechanism and slider crank mechanism. 6. Explain Kuzback’s criterion and Gurebler’s criterion.

Marks obtained:

Signature of faculty:

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Date:

CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering Theory of Machines (ME 203)

Date:

Experiment No 2 STUDY OF INVERSIONS OF MECHANISM AIM: To study of inversions of mechanism OBJECTIVES: To study inversions of 4-bar and slider crank mechanism. To study applications of inversion of mechanism. INVERSION OF MECHANISM: When one of the links is fixed in a kinematic chain, it is called a mechanism. So we can obtain as many mechanisms as the number of links in a kinematic chain by fixing, in turn, different links in a kinematic chain. The method of obtaining different mechanisms by fixing different links in a kinematic chain is known as inversion of the mechanism. 1 Inversion of Four Bar Chain.  Beam engine.  Coupling rod of locomotives.  Watts’s indicator mechanism. 2. Inversion of single slider cranks Chain.  Pendulum pumps or bull engine.  Oscillating cylinder.  Gnome engine.  Crank and slotted quick return motion mechanism.  With-worth quick return motion mechanism. 3. Inversion of Double slider cranks Chain.  Elliptical trammels.  Scotch-yoke mechanism  Oldham’s coupling.

Four Bar Chain

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering

. Single slider crank chain

Double slider cranks Chain. QUESTIONS: 1. Enlist and explain the various inversions of 4-bar chain. 2. Explain the inversion of slider crank mechanism when link 2 is fixed (other than that used in a machine tool). 3. Explain Crank and slotted lever quick return motion mechanism and Whit worth quick return motion mechanism with neat sketches. 4. Compare the inversions of slider crank mechanism having link 2 and link 3 fixed respectively and used in a machine tool. 5. Enumerate the inversion of a double slider crank chain. Give examples.

Marks obtained:

Signature of faculty:

Page 6

Date:

CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering Theory of Machines (ME 203)

Date:

Experiment No 3 STUDY STRAIGHT LINE MOTION MECHANISMS AND PANTOGRAPH AIM: To study straight-line motion mechanisms and pantograph. OBJECTIVES: 1. 2.

To study Peaucellier’s, Hart’s, Scott Russell’s and Watt’s mechanisms. To study Pantograph.

INTRODUCTION: Straight-line mechanisms: One of the most common forms of the constraint mechanisms is that it permits only relative motion of oscillatory nature along a straight line. The mechanisms used for this purpose are called straight-line mechanisms. They are of the following two types: 1) In which only turning pairs are used. 2) In which one sliding pair is used. These two types of mechanisms may produce exact straight line motion or approximate straight motion. Exact straight line motion mechanisms made up of turning pairs: There are mainly two types of mechanisms having turning pairs which produces exact straightline motion. They are: 1) Peaucellier’s mechanism A Peaucellier’s mechanism consists of eight links (fig. 1) such that,

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering 2) Hart’s mechanism A Hart mechanism consists of six links such that, AB = CD, AD = BC and OE = OQ. OE is the fixed link and OQ the rotating link. The links are arranged in such a way that ABDC is a trapezium (AC || BD). Pins E and Q on the links AB and AD respectively, and the point P on the link CB are located in such a way that, AE = AQ = CP AB AD CB As OQ rotates about O, P moves in a line perpendicular to EO.

Exact straight line motion mechanisms made up of turning sliding pair: 1) Scott Russell’s mechanism A Scott Russell’s mechanism consists of three movable links (fig. 3) OQ, PS and slider S that moves along OS. The links are connected in such a way that, OA = QP = QS. P moves in a straight line perpendicular to OS as the slider S moves along OS. Approximate straight-line motion mechanisms: The approximate straight line motion mechanisms are the modifications of the four bar chain mechanisms. Following mechanisms producing approximate straight line motion are important from the subject point of view. 1) Watt’s mechanism It consists of four links OQ, OA, QB and AB. OQ is a fixed link. Links OA and QB can oscillate about center O and Q respectively. If P is point on the link AB such that, PA = QB PB OA Then for small oscillation of OA and QB, P will trace an approximate straight line. 2) Modified Scott Russell’s mechanism. In this mechanism the links are proportioned in a way that QS is the mean proportional between OQ and QP. P will approximately traverse a straight line perpendicular to OS. 3) Grasshopper mechanism

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering This mechanism is a derivation of the modified Scott Russell’s mechanism in which the sliding pair at S is replaced by a turning pair. This is achieved by replacing the slider with a link AS perpendicular to OS in the mean position P will move in an approximate straight line if QS is the mean proportional between OQ and QP. Pantograph: A pantograph is an instrument used to reproduce to an enlarged or a reduced scale and as exactly as possible the path described by a given point. It is mostly used for the reproduction of plane areas figures such as maps, plans etc. on enlarged or reduced scales. It is sometimes used as an indicator rig in order to reproduce to a small scale the displacement of the crosshead and therefore of the piston of a reciprocating steam engine. It is also used to guide cutting tools.

QUESTIONS: 1.

Give one presentation on any one of the straight line motion mechanism.

2.

Draw a neat sketch of Hart’s mechanism and prove that it produces an exact straight line motion.

3.

How is Hart’s mechanism different from Peaucellier’s mechanism?

4.

Describe Watt’s mechanism for straight line motion and derive the condition under which the straight line is traced.

5.

Explain Scott Russell’s mechanism with neat sketch.

6.

Explain the working of a pantograph and state its uses.

Marks obtained:

Signature of faculty: Page 9

Date:

CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering Theory of Machines (ME 203)

Date:

Experiment No 4 STUDY OF STEERING MECHANISM

AIM: To study Steering Mechanisms. OBJECTIVES: 1. 2.

To study Davis steering mechanism. To study Ackerman steering mechanism.

INTRODUCTION: The steering gear mechanism is used for changing the direction of two or more of the wheel axles with reference to the chassis, so as to move the automobile in any desired path. Usually the two back wheels have a common axis, which is fixed in direction with reference to the chasis and the steering is done by the means of the front wheels. In automobiles, the front wheels are placed over the front axles, which are pivoted at the points fixed to the chassis. The back wheels are placed over the back axles at the two ends of the differential tube. When the vehicle takes a turn, the front wheels along with the respective axles turn about the pivoted joints. The back wheels remain straight and do not turn. Therefore, the steering is done by the means of front wheels only. In order to avoid skidding (i.e. slipping of the wheels sideways), the two front wheels must turn about the same instantaneous center I which lies on the axis of the back wheels. If the instantaneous center of the two front wheels do not coincide with the instantaneous center of the back wheels, the skidding of the front or the back wheels will definitely take place, which will cause more wear and tear of the tires. To satisfy this condition, the axis of the wheel on the inside of the curve must be turned through a larger angle than the axis of the wheel on the outside of the curve. The condition for the correct steering mechanism is given by the equation cotΦ - cot = c/b Where, c= distance between the pivots of the front axle. b= wheel base.

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering There are mainly two steering mechanisms which are important for the subject point of view. 1. Davis steering gear. 2. Ackerman steering gear. QUESTIONS: 1. Explain with neat sketch steering gear mechanism and derive condition that must be satisfied for the correct steering. 2. Explain the Davis steering gear mechanism with neat sketch and prove that the gear is theoretically correct. 3. Explain Ackerman steering gear mechanism with neat sketch. 4. What is the difference between Davis gear and Ackerman steering gear? Which is better in your opinion?

Marks obtained:

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Page 11

Date:

CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering Theory of Machines (ME 203)

Date:

Experiment No 5 STUDY VELOCITY AND ACCELERATION DIAGRAM AIM: To Study Velocity and Acceleration Analysis Method. OBJECTIVES: 1. To study the velocity and acceleration of links in a mechanism. 2. To study the various methods of analysis.

INTRODUCTION:When the body is displaced along a straight path, then it is called linear displacement. As shown in figure 10.1 , the body A is displaced linearly from position 1 to position 2 through a distance ‘S’ The SI units of linear displacement is mm or m.the rate of change of linear displacement of body with respect to time is called linear velocity It is denoted by ‘V’. V=

dS dt

Where, S = linear displacement of body (mm OR m). t = time of displacement (sec). Position 1

Position 2

S Figure 10.1 When the body is displaced along a circular path, then it is called Angular displacement. As shown in figure 10.2 , Link OA which rotates about center ‘O’. inclined at angle with X Axis. Let link OA move by angle in time dt. The rate of change of angular diaplacement of body with respect to time is called as angular velocity. It is denoted by ‘ω’. Page 12

CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering

A d 

O

Figure 10.2

d dt Where, θ = angular displacement of body (mm OR m). t = time of displacement (sec).

ω=

REVIEW QUESTIONS: 1. Explain the Freudenstein’s Theorem. 2. Explain Klein’s Construction method. EXAMPLES:Exp 1. The dimension of a four bar chain mechanism are as shown in Figure 1. Where the link AD is fixed and the crank AB rotates at a uniform speed of 240 r.p.m. in clockwise direction. When the crank is at 600 with the horizontal, find the angular velocity of links BC and CD and also the absolute velocity of points E of the structure BEC. Exp 2. The dimension and configuration of the four bar mechanism, shown in Figure 2 are as follows: AB = 300 mm; BC = 360 mm; CD = 360 mm, and AD = 600 mm.

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering The angle AP1P2 = 60°. The crank P1A has an angular velocity 10 rad/s and an angular acceleration of 30rad/s2, both clockwise. Determine the angular velocities and angular accelerations of P2B, and ABand the velocity and acceleration of the joint B. Figure 1.

E

44 mm

40 mm C

B 60 mm

80 mm

72 mm

60

A

D 100 mm

Figure 2.

C

360 mm B 360 mm 300 mm

A

60

D 600 mm

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering Exp 3. The dimensions of the various links of a mechanism, as shown in Fig. 3, are as follows : AB = 30 mm ; BC = 80 mm ; CD = 45 mm ; and CE = 120 mm.

Fig-3 The crank AB rotates uniformly in the clockwise direction at 120 r.p.m. Draw the velocity diagram for the given configuration of the mechanism and determine the velocity of the slider E and angular velocities of the links BC, CD and CE. Also draw a diagram showing the extreme top and bottom positions of the crank DC and the corresponding configurations of the mechanism.Find the length of each of the strokes. Exp 4. In a mechanism as shown in Fig. 4, the link AB rotates with a uniform angular velocity of 30 rad/s. The lengths of various links are : AB = 100 mm ; BC = 300 mm ; BD = 150 mm ; DE = 250 mm ; EF = 200 mm ; DG = 165 mm.Determine the velocity and acceleration of G for the given configuration.

Fig-4 Exp 5.The crank and connecting rod of a reciprocating engine are 150 mm and 600 mm respectively. The crank makes an angle of 60° with the inner dead centre and revolves at a uniform speed of 300 r.p.m. Find, by Klein’s construction, 1. Velocity and acceleration of the piston, 2. Velocity and acceleration of the mid-point D of the connecting rod, and 3. Angular velocity and angular acceleration of the connecting rod.

Marks obtained:

Signature of faculty: Page 15

Date:

CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering Theory of Machines (ME 203)

Date:

Experiment No. 6 Introduction of GEAR AIM: To study Different gears. OBJECTIVES: 1. To study the different types of Gear. 1. To understand law of gearing. INTRODUCTION: The slipping of a belt or rope is a common phenomenon, in the transmission of motion or power between two shafts. The effect of slipping is to reduce the velocity ratio of the system. In precision machines, in which a definite velocity ratio is of importance (as in watch mechanism), the only positive drive is by means of gears or toothed wheels. A gear drive is also provided, when the distance between the driver and the follower is very small. Different types of gear: 1.Spur gear 2.Helical gear 3.Warm gear 4.Bevel gear Terminology of gear as shown in fig-8.1

Fig 8.1.Terminology of Gear

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering Law of Gearing: Constant angular velocity ratio for all positions of the wheels, the point 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 fundamental condition which must be satisfied while designing the profiles for the teeth of gear wheels. It is also known as law of gearing.

Fig 8.2 law of Gearing QUESTIONS: 1. Explain the terms : (i) Module, (ii) Pressure angle, and (iii) Addendum. 2. Explain Comparison Between Involute and Cycloidal Gears. 3. Expalin Length of path of contact and length of arc of contact in detail. 4. Derive an expression for the minimum number of teeth required on the pinion in order to avoid interference in involute gear teeth when it meshes with wheel. 5. Derive an expression for minimum number of teeth required on a pinion to avoid interference when it gears with a rack. 6 . Two spur gears of 24 teeth and 36 teeth of 8 mm module and 20° pressure angle are in mesh. Addendum of each gear is 7.5 mm. The teeth are of involute form. Determine : 1. the angle through which the pinion turns while any pair of teeth are in contact, and 2. the velocity of sliding between the teeth when the contact on the pinion is at a radius of 102 mm. The speed of the pinion is 450 r.p.m.

Marks obtained:

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Date:

CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering Theory of Machines (ME 203)

Date:

Experiment No 7 CONSTRUCTION OF INVOLUTE GEAR TOOTH PROFILE AIM: To determine the involute tooth profile of given module, pressure angle and PCD. OBJECTIVES: 2.

To understand the construction the construction method of involute gear profile.

INTRODUCTION: By using the principle of gear cutting by generation method, the involute tooth profiles of a gear can be drawn on paper. The speciallity of this method is that not only the face and flank comprising the tooth profile can be drawn, but the fillet curve portions of the teeth are also represented realistically. As indicated elsewhere in the book, this curve is a trochoid and not a circular arc. This aspect is not taken care of in the other methods described so far, which only good approximations are of tooth profiles. Using the generation method and depending upon the human and instrumental accuracy can draw a reasonably good replica of the tooth drawn, which can be used as a paper template for making a form tool or as a master profile to guide the tool in a coping machine. By this method, the trochoid is automatically generated along with sufficiently accurate tooth profiles, mainly in large gears with large modules (around 30 and above). With smaller modules, it becomes increasingly difficult to maintain the drafting accuracy and errors are likely to creep in. the generation method is described below step by step. PROCEDURE: 1. On a transparent tracing paper, draw the profile of the gear tooth cutting rack. This is the template of the tool for generating the tooth profiles. Fix it rigidly on the drawing board by adhesive tape or tack pins. The addendum of this cutting rack is 1.25 module with respect to its pitch line MM, which divides the rack into equal lengths of tooth thickness and tooth gap. This rack produces gears conforming to the basic rack as per IS: 2535, in which case the teeth have addendum equal to 1.25 in. 2. On a separate loose transparent tracing paper. Draw the pitch circle of the gear to be drawn on the paper. The profiles will be drawn. 3. The pitch line MM of the cutter rack is divided in short, equal parts, says 5 or 10 mm each, and numbered a, b, c as well as a’, b’, c’ etc. on the both sides of the vertical centerline in a similar manner. The pitch circle of the gear is divided at equal intervals and numbered 1, 2, 3 and 1’, 2’, 3’, etc. To facilitate correct alignment, draw thin vertical lines for the gear to be Page 18

CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering drawn. This is shown in fig 1. Obviously, smaller the intervals, more realistic and accurate will be ultimate result.

4. Next, place the loose tracing paper over the template in such a manner that the pitch line of the rack tool is tangent to the pitch circle of the gear at the pitch point P as shown in fig. 2. The vertical through P is the centerline of the pitch circle. 5. The rack that is under the transparent tracing paper having the gear pitch circle is visible through the paper. The profile of the rack is now traced on this paper by a sharp pencil.

6. Now, shift and rotate the paper, so that the points ‘a’ of rack and 1 of gear coincide. This means that the pitch line of the rack has now rolled on the pitch circle, so that the line is now tangent to the circle at point 1. In this condition the radial line thorough 1 will be exactly superimposed on the parallel line thorough ‘a’ of the rack. Draw the outline of the rack as before. Next shift and rotate the paper so that points b and 2 coincide in the above manner and draw the rack outline. Repeat the whole process for all the points on both sides of the centre line, pairing c – 3, h – 8, a’ – 1’… n' – 8’. Care should be taken to ensure that the pitch line of the rack is always tangent to the pitch circle at all successive points. 7. The desired profile is the envelope of the rack profiles thus drawn. Pass a smooth curve to delineate this envelope. It can be seen from fig 3, that the tooth profiles are gradually taking shape. It is particularly noticeable in case of the middle tooth in the figure. When the whole process is completed, the tooth profiles will look as shown in fig 4.

Marks obtained:

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Page 19

Date:

CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering

Theory of Machines (ME 203)

Date:

Experiment No 8 STUDY OF GEAR TRAIN AIM: To study Gear train. OBJECTIVES: 3. 4. 5.

To study the different types of Gear trains. To understand sun and planet type of epicycle gear train. To find speed of different gears in given gear train example.

INTRODUCTION: Gear train can be defined as a mechanism, which transmits power or motion from one shaft to another, with the help of gear wheels. The nature of the train used depends upon the velocity ratio required and relative position of the axes of shaft. Velocity Ratio (Speed Ratio): It is the ratio of the speed of the driver to the driven of follower and ratio of speed of any pair of gears in mesh is the inverse of their number of teeth. Speed Ratio = N1/N2 = T2/T1 Train Value: The value of a gear train is defined as the ratio of the angular speed of follower or driven gear to that of the driver. It will be the reciprocal of velocity ratio. Train Value= 2/1 = N2/N1. Where N1 = Speed of driver gear in r.p.m. N2 = Speed of driven gear in r.p.m. T1 = Number of teeth on driver. T2 = Number of teeth on driven. 1) Simple Gear Train(Figure 1) When there is only one gear on each shaft, it is known as simple gear train. When the number of intermediate gears are odd, the motion of the both the gears are in same direction. But if the number of gears are even, then motion of the driven will be in the opposite direction of the driver.

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering In the simple gear train , there is no effect of intermediate gears on train value of the gear train. This intermediate gears called ideal gears.

Figure 1 simple gear train 2) Compound Gear Train(Figure 2) In a compound gear train, each intermediate gear shaft carries two gears which are fastened together rigidly. Train Value = T6/T1 = (T1/T2) * (T3/T4) * (T5/T6) Compared to simple gear train, compound gear trains has the following advantages:  With the min. no. of teeth on pinion fixed, the size of the largest gear for the same velocity ratio is much less. This means that a compound gear train can provide a larger velocity ratio in limited space.  Faster pair of gears have to carry smaller tooth loads and hence these can be made of smaller pitch and narrow width, leading to further compactness.

Figure 2 compound gear train

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering 3) Reverted Gear Train: (figure 3) Reverted gear train is a special type of compound gear train in which the first and the last gear have the same axis. The distance between centers of shaft R1 + R2 = R3 + R4 If the teeth of all wheels have the same pitch the no. of teeth are proportional to the pitch radii. Then R1 = C * T1; R2 = C * T2; R3 = C * T3; R4 = C * T4 Where, C= Constant = m/2 M= pitch in module T1 + T2 = T3 + T4

Figure 3 Reverted gear Train 4) Epicyclic Gear Train: (Figure 4) In epicyclic gear train, the axes of the shafts, over which the gears are mounted, may move relative to a fixed axis. The train consists of a central gear & an epicycle gear, which produces Epicyclic motion by rolling around the periphery of the central gear. A arm contains the Bearings for the epicyclical gear to maintain the two gear wheels in mesh. It is also called the planetary or sun & planet gear train. In this nomenclature center gear is sun gear & the epicyclic gear will be the planet gear and arm is called the planet carrier. Some time redundant planet gears added for better force balance. The train may be simple or compound. The distinguish features of this train is that, there is relative motion between two or more of the areas of the wheels constituting the train. The epicyclic gear trains are useful for transmitting high velocity ratio with gear of moderate size in a comparatively lesser space. Unlike in simple or compound gear train, it is not possible to Page 22

CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering determine direction of rotation or relative magnitude of speed just from the look of gear train epicyclic gear train. His gear train used in the back gear of lathe, differential gear of the automobiles, hoists pulley blocks, wrist watches etc.

Figure 4 Epicyclic gear train

Velocity ratio of an epicyclic gear train: The velocity ratio of this type gear train can be found by following different methods. 1. 2. 3. 4.

Algebraic method Tabular method Graphical method vector method

Out of these methods here we are limiting over self for tabular method only, which is as under. Tabular Method: (Note: Before going in details draw the sketch of epicycle gear train with respective notations for ready reference.) 1. Let A be the arm, B is planet and C is sun. 2. Let the arm A fixed and wheel B is given one complete revolution in a clockwise direction. 3. As the gear B is assumed to make one revolution may be executing more than one revolution, values of all columns in the first raw of the table are multiplied by X and written in the second raw of the table. 4. Now the arm is not stationary. Therefore to account for rotation of arm, Y is added to the various quantities of second raw which will give the third raw of the table.

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering The method as outlined in steps 1,2,3 & 4 above can be written in tabulated form as back page. Opertaion Arm a fixed Multiply by x Add y Total Motion

Revolution of arm a 0 0 +y +y

Revolution of wheel b +1 X X+y X+y

Revolution of wheel b -Tb/Tc -x * Tb/Tc -x * Tb/Tc+ x Y - x * Tb/Tc

Total -Tb/Tc -x * Tb/Tc -x * Tb/Tc+ x Y - x * Tb/Td

Out of three quantities involved in last row, two rows of them are given. From these values X and Y can be determined. On substituting in third quantity, its magnitude can be estimated. Torque and tooth loads in epicyclic gear trains: The assumptions for this are as under: 1. The different gear moves with uniform speeds. (i.e. accelerations are not involved) 2. No internal friction losses. Assumption one reveals that, the algebraic sum of the torque applied externally must be equal to zero. T=0  Td + Tr + Th = 0 ----------(1) where, Td = Driving torque Tr = Resisting torque Th = Holding torque or Braking torque Assumption two reveals that the total energy must be equal to zero. Thus  T *  = 0 Or Td * d + Tr * r + Th * h = 0 --------(2) d, r, h = respective angular velicities. Out of above three, terms one becomes zero because of angular velocity of that term is zero (e.g. here holding, h = 0). So from above equation number (2) remaining values can be determined. QUESTIONS: 1.

List down the different gear trains and give the applications of each of them.

2.

What is an epicyclic gear train? How it is differ from a simple or compound gear train?

3.

Write merits and demerits of epicyclic gear trains.

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering

EXAMPLES:Exp 1. Figure 1 Shows an epicyclic gear train arrangement. Wheel E is a fixed wheel and wheels C & D are integrally cast, and mounted on one pin. If the arm A makes one rev/sec clockwise, determine the speed and the direction of rotation of wheels B & F

Figure 1 Exp 2. Figure 2 Shows an epicyclic gear train, pinion A has 15 teeth and is rigidly fixed to motor shaft. The wheel B has 20 teeth and gears with A, and also with annular fixed wheel ‘E’. Pinion C has 15teeth and is integral with ‘B’. Gear ‘C’ meshes with annular wheel D, which is keyed to the machine shaft. The arm rotates @’O’ and carries the compound wheels ‘B’ and ‘C’. If the motors runs at 1000rpm, find the speed of machine shaft.

Figure 2 EXP 3. The speed ratio of the reverted gear train shown in Fig is 12. The module of gears A and B is 3.125 mm and of gears C and D is 2.5mm. Calculate the suitable number of teeth for the gears. No gears is to have less than 24teeth.

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering

Figure 3

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering Theory of Machines (ME 203) Date:

Experiment No 9 STUDY OF FRICTION AIM: To study of friction. OBJECTIVES:1. To study different types of friction. 2. To study different Laws of friction. INTRODUVTION:Friction is a force that resists the motion of an object that is in contact with another object or material. If the objects are not moving relative to each other, the friction force is called static. If they are moving, the friction is kinetic. There are three major types of friction: sliding, rolling and fluid friction. The cause of friction is a combination of molecular adhesion, surface roughness, and deformation effects. Sliding friction When two solid objects are in contact and a force is applied to slide one object against the other, sliding friction force resists the motion. If F is the force pushing on an object and Fr is the force of friction, the relationship between F and Fr will determine whether the object will slide or not move at all. Kinetic friction If force F is greater than friction Fr (written as F > Fr), then the object will slide or move. The friction is considered kinetic friction, which means moving friction. Fr = μN

Pushing force greater than friction force

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering Static friction If the pushing force F is less than the resistive force of friction Fr (written as F < Fr), there is no motion and the objects remain static with respect to each other. In this case, the friction is considered static friction, which means it is not moving. Static > Kinetic What is interesting is that the static friction that holds an object in place is greater than the kinetic friction that slows down a moving object. In other words, once you start an object moving, the friction decreases from the static friction holding the object in place. You have seen this in trying to slide a heavy box across the floor. It may be very difficult to move, but once it starts sliding, it is easier to push. Causes of sliding friction The causes of sliding friction are molecular attraction or adhesion between the materials, surface roughness of the materials, and deformation resistance in the case of soft materials. Rolling friction When a wheel of ball is in contact with a solid surface, and a force is applied to the wheel, static friction will prevent the wheel from sliding. Instead, the wheel will start to roll. Once the wheel is rolling, another type of friction takes over. Rolling friction is the resistive force that slows the wheel's motion on the other solid surface. It is different than static or kinetic friction. Much of rolling friction is caused by adhesion between the surfaces.

Rollers reduced friction Fluid friction When a solid object is in contact with a fluid, such as a liquid or gas, and a force is applied to either the object or to the fluid, there is a friction force that resists the motion. Examples where fluid friction occurs are water flowing through a hose, an airplane flying through the atmosphere and oil lubricating moving parts.

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering QUESTIONS: 1. Explain the following terms with neat sketch?    

Limiting friction Limiting angle of friction Angle of repose Co-efficient of friction

2. Derive the first principles and efforts required for the effort required to raise the load with a screw jack taking friction into consideration.

3. For screw jack having the nut fixed, derive the equation of efficiency,

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering Theory of Machines (ME 203) Date:

Experiment No 10 STUDY OF CAM AND CAM PROFILE

AIM: To study cam and cam profile. OBJECTIVES: 1. 2.

Study of different Cam mechanism. Problem solving on cam profile.

INTRODUCTION: A cam is rotating machine elements, which drives reciprocating or oscillating motion to another element known as follower. The cams are usually rotated at uniform speed by a shaft, but the follower motion is pre-determined and will be according to the shape of the cam. APPLICATIONS: Cams are widely used in automatic machines, internal combustion engines, and machine tools, printing control mechanisms. CLASSIFICATION OF CAM: Cam may be classified according to: - Shape. - Follower movement. - Manner of constraint of the follower. According to shape: 1. Wedge and Flat Cams. 2. Radial or Disc Cams. 3. Spiral Cams. 4. Cylindrical Cams. 5. Conjugate Cams. 6. Globodial Cams. 7. Spherical Cams. According to Follower movement: 1. Rise –Return-Rise (R-R-R) 2. Dwell-Rise-Return-Dwell (D-R-R-D) Page 30

CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering 3. Dwell-Rise-Dwell-Return-Dwell (D-R-D-R-D) According to manner of constraint of the follower: 1. Pre-loaded spring cam 2. Positive drive cam 3. Gravity cam From the above classification following two types are important: - Radial or Disc Cams - Cylindrical Cams CLASSIFICATION OF FOLLOWER: According to - the surface in contact.. - the motion of the follower. - The path of motion of the follower. According to the surface in contact:1. 2. 3. 4.

Knife edge follower. Roller follower. Flat face or Mushroom follower. Spherical faced follower.

According to motion of the follower:1. Reciprocating or Translating follower. 2. Oscillating or Rotating follower. According to the path of motion of the follower:1. Radial follower. 2. Off-set follower. TERMS USED IN RADIAL CAMS: 1) Base circle: It is smallest circle that can be drawn to the cam profile. 2) Trace point: It is reference point on the follower and is used to generate the Pitch Curve. In case of knife-edge follower, the knife edge represents the trace point and the pitch curve corresponds to the cam profile. In a roller follower, the Centre of the roller represents the trace point. 3) Pressure angle: It is the angle between the direction of the follower motion and a normal to the pitch curve. This angle is very important in designing a cam profile. If the pressure angle is too large, a reciprocating follower will jam in its bearings.

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering 4) Pitch point: It is a point on the pitch curve having the maximum pressure angle. 5) Pitch circle: It is a circle drawn from the centre on the cam through the pitch points. 6) Pitch curve: It is a curve generated by the trace point as the follower moves relative to the cam. For knife-edge follower, the pitch curve and the cam profile are same whereas for a roller follower, they are separated by the radius of the roller. 7) Prime circle: It is the smallest circle that can be drawn from the Center of the cam and tangent to the pitch curve. For a knife-edge and a flat face follower, the prime circle and the base circle are identical. For roller follower, the prime circle is larger than the base circle by the radius of the roller. QUESTIONS: 1.

A cam operating a knife-edged follower has the following data : (a) Follower moves outwards through 40 mm during 60° of cam rotation. (b) Follower dwells for the next 45°. (c) Follower returns to its original position during next 90°. (d) Follower dwells for the rest of the rotation. The displacement of the follower is to take place with simple harmonic motion during both the outward and return strokes. The least radius of the cam is 50 mm. Draw the profile of the cam when 1. the axis of the follower passes through the cam axis, and 2. the axis of the follower is offset 20 mm towards right from the cam axis. If the cam rotates at 300 r.p.m., determine maximum velocity and acceleration of the follower during the outward stroke and the return stroke.

2.

Design a cam to raise a valve with simple harmonic motion through 50 mm in 1/3 of a revolution, keep if fully raised through 1/12 revolution and to lower it with harmonic motion in 1/6 revolution. The valve remains closed during the rest of the revolution. The diameter of the roller is 20 mm and the minimum radius of the cam is 25 mm. The diameter of the camshaft is 25 mm. The axis of the valve rod passes through the axis of the camshaft. If the camshaft rotates at uniform speed of 100 r.p.m.; find the maximum velocity and acceleration of a valve during raising and lowering.

3.

A cam with 30 mm as minimum diameter is rotating clockwise at a uniform speed of 1200 r.p.m. and has to give the following motion to a roller follower 10 mm in diameter: (a) Follower to complete outward stroke of 25 mm during 120° of cam rotation with equal uniform acceleration and retardation ; (b) Follower to dwell for 60° of cam rotation ; (c) Follower to return to its initial position during 90° of cam rotation with equal uniform acceleration and retardation ; (d) Follower to dwell for the remaining 90° of cam rotation. Draw the cam profile if the axis of the roller follower passes through the axis of the cam. Determine the maximum velocity of the follower during the outstroke and return stroke and also the uniform acceleration of the follower on the out stroke and the return stoke.

4.

Draw the profile of a cam with oscillating roller follower for the following motion :

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CHAROTAR UNIVERSITY OF SCIENCE & TECHNOLOGY FACULTY OF TECHNOLOGY & ENGINEERING CHAMOS Matrusanstha Department of Mechanical Engineering (a) Follower to move outwards through an angular displacement of 20° during 120° of cam rotation. (b) Follower to dwell for 50° of cam rotation. (c) Follower to return to its initial position in 90° of cam rotation with uniform acceleration and retardation. (d) Follower to dwell for the remaining period of cam rotation. The distance between the pivot centre and the roller centre is 130 mm and the distance between the pivot centre and cam axis is 150 mm. The minimum radius of the cam is 80 mm and the diameter of the roller is 50 mm. 5.

6.

7.

Draw the profile of cam which is to give oscillatory motion to the follower with uniform angular velocity about its pivot. The base circle diameter is 50 mm, angle of oscillation of follower 30o and distance between the cam center and pivot of the follower 60 mm. The oscillating lever is 60 mm long with roller of 8 mm diameter at the end. One oscillation of the follower is completed in one revolution of the cam. Set out the profile of a cam to give the following motion to a flat face follower: - Follower to rise through 24 mm during 150o of cam rotation with SHM. - Follower to dwell for 30o of cam rotation. - Follower to return to the initial position during 90o of the cam rotation with SHM - Follower to dwell for the remaining 90o of cam rotation - Take minimum radius of the cam as 30 mm. Draw cam profile for given following motion of roller follower  0º - 60º Rise of 12 mm with uniform velocity  60º - 90º Dwell  90º - 180º Rise of 24 mm with harmonic motion  180º - 210º Dwell  at 210º Sudden fall of 12 mm  210º - 240º Dwell  240º - 360º Fall of 24 mm with uniform acceleration Base circle diameter is 40mm and follower diameter is 6 mm.

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