assignment 1 introduction of machines and mechanisms o… · assignment – 1 introduction of...

KINEMATICS OF MACHINES (2131906)

B.E. Semester – III Department of Mechanical Engineering Darshan Institute of Engineering and Technology, Rajkot 1

ASSIGNMENT – 1 INTRODUCTION OF MACHINES AND

MECHANISMS Theory

(1) Distinguish between:

(i) Mechanism and machine; (ii) Analysis and synthesis of mechanisms;

(iii) Kinematics and dynamics.

(2) Define: kinematic link, kinematic pair, and kinematic chain.

(3) What are rigid and resistant bodies? Elaborate.

(4) State and explain types of constrained motion.

(5) Define kinematic pair. How it is classified?

(6) Differentiate giving examples:

(i) Lower and higher pairs; (ii) Closed and unclosed pairs; (iii) Turning and rolling

pairs.

(7) What do you mean by degree of freedom of a kinematic pair? How are pairs

classified? Give examples.

(8) What is degree of freedom of a mechanism? How is it determined?

(9) How are the degree of freedom and the number of joints in a linkage can be found

when the number of links and the number of loops in a kinematic chain are

known?

(10) Define Grashof’s law. State how is it helpful in classifying the four-link mechanisms

into different types?

(11) Describe various inversions of a slider-crank mechanism giving examples.

(12) What are quick-return mechanisms? Where are they used? Discuss the functioning

of any one of them.

(13) Enumerate the inversions of a double slider-crank chain. Give examples.

(14) Describe briefly the functions of elliptical trammel and scotch yoke.

(15) In what way Oldham’s coupling is useful in connecting two parallel shafts when

the distance between their axes is small?



EXAMPLES (1) For the kinematic linkages shown in Figure 1.1, find the number of binary links

( ), ternary links ( ), other links ( ), total links N, loops L, joints or pairs ( ),

and degree of freedom (F).

Figure 1.1

[(a)

(b)

(c) ]

(2) Show that the linkages shown in Figure.1.2 are structures. Suggest some changes to

make them mechanisms having one degree of freedom. Number of links should be

changed by more than 1.

Figure 1.2



ASSIGNMENT – 2 SYNTHESIS OF MECHANISM Theory

(1) Define the following term related to synthesis:

a. Type synthesis

b. Number synthesis

c. Dimensional synthesis

d. Function generation

e. Path generation

f. Body guidance

(2) Explain two position synthesis of slider crank mechanism.

(3) Explain two position synthesis of crank and rocker mechanism.

(4) Explain three position synthesis.

(5) Explain Freudenstein”s equation.

(6) Explain Bloch method of synthesis.

Examples

(1) Four bar crank–rocker quick return linkage for specified time ratio = 1:1.25 with

45 output rocker motion. Design the synthesis.

(2) Design a four bar link mechanism if the motions of the input and output links are

governed by a function . Assume to vary from 30 to 120

and from 60 to 130 . The length of the fixed link is 30 mm. Use chebychev

spacing of accuracy points.

(3) Synthesis a function generator to solve the equation

over the range

using three precision positions.

(4) Using Bloch method , synthesis and also draw a four bar linkages to meet the

following specifications of angular position, velocities and accelerations at one of

its positions are



ASSIGNMENT – 3 VELOCITY AND ACCELERATION ANALYSIS

Theory

(1) Describe the procedure to construct diagram of a four-link mechanism.

(2) What is velocity of rubbing? How is it found?

(3) What is instantaneous centre of rotation? How do you known the number of

instantaneous centres in a mechanism?

(4) State and prove the Kennedy’s theorem as applicable to instantaneous centres of

rotation of three bodies. How is it helpful in locating various instantaneous centres

of a mechanism?

(5) State and explain angular velocity ratio theorem as applicable to mechanisms.

(6) What are centripetal and tangential components of acceleration? When do they

occur? How are they determined?

(7) Describe the procedure to draw velocity and acceleration diagrams of a four-link

mechanism. In what way the angular accelerations of the output link and the

coupler are found?

(8) What is an acceleration image? How are they helpful in determining the

accelerations of offset points on a link?

(9) What is Coriolis acceleration component? In which cases does it occur? How is it

determined?

(10) Explain the procedure to construct Klein’s construction to determine the velocity

and acceleration of a slider-crank mechanism.

Examples (1) In a four-link mechanism, the crank AB rotates at 36 rad/s. The lengths of the links

are: AB = 200 mm, BC = 400 mm, CD = 450 mm and AD = 600 mm. AD is the fixed

link. Ate the instant when AB is at right angle to AD, determine the velocity of

(i) the mid-point of link BC,

(ii) a point on link CD, 100 mm from the pin connecting the links CD and AD.

(6.55 m/s; 1.45m/s)



Fig. 3.1

(2) For the mechanism shown in Fig. 3.1, determine the velocities of the points C, E

and F and the angular velocities of the links BC, CDE and EF.

(0.83 m/s; 0.99 m/s; 5.4 rad/s ccw; 8.3 rad/s ccw; 6.33 rad/s ccw)

(3) A toggle mechanism is shown in Fig. 3.2 along with the dimensions of the links in

mm. Find the velocities of the points B and C and the angular velocities of links AB,

BQ and BC. The crank rotates at 50 rpm in the clockwise direction.

(0.13 m/s; 0.106 m/s; 0.74 rad/s ccw; 1.3 rad/s ccw; 1.33 rad/s cw)

(4) Draw the velocity polygon for the mechanism shown in Fig. 3.3. Find the angular

velocity of link AB. (1.75 rad/s

cw)

(Hint: assume the length of vector vef and complete the velocity polygon. Determine

the velocity scale).

(5) For the mechanism shown in Fig. 3.4, determine the angular velocity of link AB.

(60.3 rad/s ccw)

Figure 3.2



Figure 3.3

Figure 3.4

(6) In the mechanism shown in Fig. 3.5, O and A are fixed CD = 200 mm, OA = 60 mm,

AC = 50 mm and OB (crank) = 150 mm. = 90º. Determine the velocity of

slider D for counter-clockwise rotation of OB at 80 rpm. (0.32

m/s)

Figure 3.5

(7) A crank and rocker mechanism ABCD has the following dimensions:

AB = 0.75 m, BC = 1.25 m, CD = 1 m, AD = 1.5 m

BE = 437.5 mm, CE = 87.5 mm and CF = 500 mm



E and F are two points on the coupler link BC. AD is the fixed link. BEC is read

clockwise and F lies on BC produced. Crank AB has an angular velocity of 20.94

rad/s counter-clockwise and a deceleration of 280 rad/s2 at the instant DAB =

60º. Find

(i) The instantaneous acceleration of C, E and F

(ii) The instantaneous angular velocities and accelerations of links BC and CD.

[(i) 166 m/s2, 330 m/s2, 161 m/s2 (ii) wbc = 5.92 rad/s cw, wcd = 11.5 rad/s ccw, bc

= 229 rad/s2 ccw, cd = 100 rad/s2 ccw]

(8) Figure 3.6 shows a mechanism in which O and Q are the fixed centres. Determine

the acceleration of the slider S and the angular acceleration of the link BQ for the

given configuration.

[14.5 m/s2 towards left; 114 rad/s2 cw]

Figure 3.6

(9) In a simple steam engine, the lengths of the crank and the connecting rod are 100

mm and 400 mm respectively. The weight of the connecting rod is 50 kg and its

centre of mass is 220 mm from the cross-head centre. The radius of gyration about

the centre of mass is 120 mm. If the engine speed is 300 rpm, determine for the

position when the crank has turned 45º from the inner-dead centre,

(i) The velocity and acceleration of the centre of mass of the connecting rod.

(ii) The angular velocity and acceleration of the rod.

(iii) The kinetic energy of the rod.

[(i) 2.7 m/s, 80 m/s2 (ii) 5.7 rad/s, 173 rad/s2 (iii) 194 N.m]

(10) Figure 3.7 shows a toggle mechanism in which the crank OA rotates at 120 rpm.

Find the velocity and the acceleration of the slider at D.

[170mm/s; 830 mm/s2]



Figure 3.7



ASSIGNMENT – 4 SPECIAL MECHANISMS Theory

(1) Explain Hart straight line motion mechanism with the help of neat sketch and

prove that tracing point describes a straight line path.

(2) Derive the equation for finding out the ratio of angular velocities of two shafts of

Hooke’s joint.

(3) Explain Peaucellier mechanism with neat sketch.

(4) What is condition for correct steering? Sketch and explain Devis steering gear

mechanism. Also explain Ackermann steering gear mechanism.

Examples (1) A Hook’s joint is used to connect two shafts inclined at an angle 20 degree. Find

maximum and minimum velocity of driven shaft and the angles of which the driven

shaft will have the same speed as that of the driving shaft which rotates at 240

rpm.

(2) A universal joint is used to connect two shafts which are inclined at 20 degree and

the speed of the driving shaft is 1500 rpm.Find extreme angular velocity of the

driven shaft and its maximum acceleration



ASSIGNMENT – 5 GEARS Theory

(1) Draw the gear teeth profile and define different terms related to gear.

(2) Explain law of Gearing.

(3) Derive equation for Length of path of contact, Arc of contact and Contact ratio.

(4) What is interference? How it should be avoided? Derive the equation for minimum

numbers of teeth on pinion/wheel to avoid interference.

(5) Derive equation for efficiency of spiral gears.

Examples (1) Each of two gears in a mesh has 48 teeth and a module of 8 mm. the teeth are of

200 involute profiles. The arc of contact is 2.25 times the circular pitch. Determine

the addendum.

(2) Two involute gears in mesh have 200 pressure angle. The gear ratio is 3 and the

number of teeth on the pinion is 24. The teeth have a module of 6 mm. the pitch

line velocity is 1.5 m/s and the addendum equal to one module. Determine the

angle of action of the pinion (The angle turned by the pinion when one pair of

teeth is in the mesh) and the maximum velocity of sliding.

(3) Two involute gears in a mesh have a module of 8 mm and a pressure angle of a

200. The larger gear has 57 while the pinion has 23 teeth . If the addenda on a

pinion and gear wheel are equal to one module, find the

i. Contact ratio

ii. Angle of action of the pinion and the gear wheel

iii. Ratio of the sliding to rolling velocity at the beginning contact, pitch point,

end of contact.

(4) Two 200 involute spur gears mesh externally and give a velocity ratio of 3. The

module is 3 mm and the addendum is equal to 1.1 module. If the pinion rotates at

120 rpm. Determine the

i. Minimum number of teeth and each wheel to avoid interference

ii. Contact ratio.

(5) Two 20º involute spur gears have a module of 10 mm. The addendum is equal to

one module. The larger gear has 40 teeth while the pinion has 20 teeth. Will the

gear interference with the pinion?



(6) Two 20º involute spur gears have a module of 10 mm. The addendum is one

module. The larger gear has 50 teeth and the pinion has 13 teeth. Does

interference occur? If it occurs, to what value should the pressure angle be

changed to eliminate interference?

(7) The following data relate to two meshing involute gears:

No of teeth on the gear wheel 60

Pressure angle 20º

Gear ratio 1.5

Speed of the gear wheel 100rpm

Module 8 mm

The addendum on each wheel is such that the path of approach and the path of

recess on each side are 40% of the maximum possible length each. Determine the

addendum for the pinion and the gear and the length of the arc of contact.

(8) A pinion of 20º involute teeth rotating at 275 rpm meshes with a gear and

provides a gear ratio of 1.8. the number of teeth on the pinion is 20 and the module

is 8mm. if the interference is just avoided, determine

i. The addendum on the wheel and the pinion

ii. The path of contact

iii. The maximum velocity of sliding on both sides of the pitch point.

(9) A pinion of 32 involute teeth and 4 mm module drives a rack. The pressure angle is

20°. The addendum of both pinion and rack is the same. Determine the maximum

permissible value of the addendum to avoid interference. Also, find the number of

pairs of teeth in contact.

(10) The centre distance between two meshing spiral gears is 260 mm and the angle

between the shafts is 65°. The normal circular pitch is 14 mm and the gear ratio is

2.5. The driven gear has a helix angle 35°. Find the

i. No of teeth on each wheel

ii. Exact centre distance

iii. Efficiency assuming the friction angle to be 5.5°

(11) A drive is made up of two spiral gear wheels of the same hand. same diameter and

of normal pitch of 14 mm. the centre distance between the axes of the shafts is

approximately 150 mm. the speed ratio is 1.6 and the angle between the shafts is

75º. Assuming a friction angle of 6 º ,determine

i. Spiral angle of each wheel iii. Efficiency of the drive

ii. No of teeth on each wheel iv. Max. efficiency



ASSIGNMENT – 6 GEAR TRAINS

Examples

(1) In an epicyclic gear train, an arm carries two gears A and B having 36 and 45 teeth

respectively. If the arm rotates at 150 r.p.m. in the anticlockwise direction about the

centre of the gear A which is fixed. Determine the speed of gear B. If the gear A

instead of being fixed makes 300 r.p.m. in the clockwise direction, what will be the

speed of gear B?

(2) An epicyclic gear consists of three gears A, B and C as shown in figure. The gear A has

72 internal teeth and gear C has 32 external teeth. The gear B meshes with both A and

C and is carried on an arm EF which rotates about the centre of A at 18 r.p.m. If the

gear A is fixed, determine the speed of gears B and C.

(3) In an epicyclic gear train, as shown in figure. The numbers of teeth on wheels A, B and

C are 48, 24 and 50 respectively. If the arm rotates at 400 r.p.m. clockwise,

Find: 1. Speed of wheel C when A is fixed 2. Speed of wheel A when C is fixed.



(4) An epicyclic gear train, as shown in figure has a sun wheel S of 30 teeth and two planet

wheels P-P of 50 teeth. The planet wheels mesh with the internal teeth of a fixed

annulus A. The driving shaft carrying the sun wheel transmits 4 kW at 300 r.p.m. The

driven shaft is connected to an arm which carries the planet wheels.

Determine the speed of the driven shaft and the torque transmitted, if the overall efficiency is 95%.

(5) An epicyclic gear train for an electric motor is shown in figure. The wheel S has 15

teeth and is fixed to the motor shaft rotating at 1450 r.p.m. The planet P has 45 teeth,

gears with fixed annulus A and rotates on a spindle carried by an arm which is fixed to

the output shaft. The planet P also gears with the sun wheel S. Find out the speed of

the output shaft.

If the motor is transmitting 2 H.P./1.5 kW, find the torque required to fix the annulus A.

(6) An epicyclic gear train, as shown in figure, is composed of a fixed annular wheel A

having 150 teeth. Meshing with A is a wheel B which drives wheel D through an idle



wheel C, D being concentric with A. The wheels B and C are carried on an arm which

revolves clockwise at 100 r.p.m. about the axis of A or D. If the wheels B and D have 25

teeth and 40 teeth respectively, find the number of teeth on C and the speed and

sense of rotation of C.

(7) An epicyclic train of gear is arranged as shown in following fig. How many revolution

does the arm , to which the pinions B and C are attached make :

1. When A makes one revolution clockwise and D makes half a revolution anticlockwise. 2. When A makes one revolution clockwise and D is stationary? The number of teeth on the gears A and D are 40 and 90 respectively.

(8) In an epicyclic gear train, the internal wheel A and B and compound wheels C and D

rotate independently about axis O. The wheels E and F rotate on pins fixed to the arm

G. E gears with A and C and F gears with B and D. All the wheels have the same module

and the number of teeth is: Tc= 28; Td = 26; Te= Tf= 18.

1. Sketch the arrangement 2. Find the number of teeth on A and B



3. If the arm G makes 100 rpm clockwise and wheel A is fixed, find the speed of B 4. If the arm G makes 100 rpm clockwise and wheel A makes 10 rpm. Counter clockwise, Find the speed of wheel B.

(9) Fig. shows Diagrammatically a compound epicyclic gear train. Wheel A, D and E are

free to rotate independently on spindle O, while B and C are compound and rotate

together on spindle P, on the end of arm OP. All the teeth on different wheel have the

same module. A has 12 teeth, B has 30 teeth and C has 14 teeth cut externally. Find

the number of teeth on wheel D and E which are cut internally.

(10) If the wheel A is driven clockwise at 1 rpm. While D is driven counter clockwise at 5

rpm. Determine the magnitude and direction of the angular velocities of arm OP and

wheel E.

(11) Fig. 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 the annular fixed wheel

E. Pinion C has 15 teeth and is integral with B( B, C being a compound gear wheel ).

Gear C meshes with annular wheel D, which is keyed to the machine shaft. The arm

rotates about the same shaft on which A is Fixed and carries the compound wheel B, C.

If the motor runs at 1000 rpm., find the speed of machine shaft. Find the torque

exerted on the machine shaft, if the motor develops a torque of 100 NM.



(12) An epicyclic gear train consists of a sun wheel S, a stationary internal gear E and three

identical Planet wheels P carried on a star-shaped planet carrier C. The sizes of

different toothed wheel are such that the planet carrier C rotates at 1/5th of the speed

of the sunwheel S. The minimum number of teeth on any wheel is 16. The driving

torque on the sun wheel is 100 Nm. Determine: 1. Number of teeth on different

wheels of the train, and 2. Torque necessary to keep the internal gear stationary.



ASSIGNMENT – 7 CAM AND FOLLOWERS

Theory

(1) What is a cam? What type of motion can be transmitted with a cam and follower

combination? What are its elements?

(2) How are the cams classified? Describe in detail.

(3) Discuss various types of cams.

(4) Compare the performance of Knife-edge, roller and mushroom followers.

(5) Define: base circle, pitch circle, trace point, pitch curve and pressure angle.

(6) What are the requirements of a high-speed cam?

(7) Deduce expressions for the velocity and acceleration of the follower when it moves

with simple harmonic motion.

Examples

(1) Draw the profile of a cam that gives a lift of 40 mm to a rod carrying a 20 mm

Diameter roller. The axis of the roller passes through the centre of the cam. The

least radius of the cam is 50 mm. The rod is to be lifted with simple harmonic

motion in a quarter revolutions and is to be dropped suddenly at half revolution.

Determine the maximum velocity and maximum acceleration during the lifting.

The cam rotates at 60 rpm (0.25 m/s; 3.155 m/s2)

(2) Layout the profile of a cam so that the follower

Is moved outwards through 30 mm during 180º of cam rotation with cycloidal

motion

Dwells for 20º of the cam rotation

Returns with uniform velocity during the remaining 160º of the cam rotation.

The base circle diameter of the cam is 28 mm and the roller diameter 8 mm. The

axis of the follower is offset by 6 mm to the left. What will be the maximum

velocity and acceleration of the follower during the outstroke if the cam rotates at

1500 rpm counter-clockwise? (3 m/s; 471.2 m/s2)

(3) Use the following data in drawing the profile of a cam in which knife-edged

follower is raised with uniform acceleration and deceleration and is lowered with



simple harmonic motion: Least radius of cam = 60 mm, Lift of follower = 45 mm,

Angle of ascent = 60º, Angle of dwell between ascent and descent = 40º, Angle of

descent = 75º. If the cam rotates at 180 rpm, determine the maximum velocity and

acceleration during ascent and descent. (Ascent: 1.62 m/s; 58.3 m/s2 Descent:

1.02 m/s; 46.05 m/s2)

(4) Draw the profile of a 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 the follower 30º and the distance between the cam centre and the

pivot of the follower 60 mm. The oscillating lever is 60 mm long with a roller of 8

mm diameter at the end. One oscillation of the follower is completed in one

revolution of the cam.

(5) Set out the profile of a cam to give the following motion to a flat mushroom contact

face follower:

- Follower to rise through 24 mm during 150º of cam rotation with SHM.

- Follower to dwell for 30º of the 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.

(6) A cam is required to give motion to a follower fitted with a roller 50 mm in

diameter. The lift of the follower is 30 mm and is performed

- with uniform acceleration for 12 mm, the cam turns through 45º.

- with uniform velocity for 12 mm, the cam turns through next 30º.

- with uniform deceleration for the remainder of the lift, the cam turns

through next 45º.

The follower falls through immediately with simple harmonic motion while the cam

turns through 120º. Then a period of dwell is followed for 120º of the cam angle.

Construct a lift and fall diagram on a cam angle base. Also draw the outline of the

cam. The least radius of the cam is 35 mm. The line of motion of the follower passes

through the centre of the cam axis.

(7) The following particulars relate to a symmetrical tangent cam having a roller

follower:



Minimum radius of the cam = 40 mm, Lift = 20 mm, Speed = 360 rpm, Roller

diameter = 44 mm, Angle of ascent = 60º

Calculate the acceleration of follower at the beginning of lift. Also find its values

when the roller just touches the nose and is at the apex of the circular nose. Sketch

the variation of displacement, velocity and acceleration during ascent.

(88.12 m/s2; 164 m/s2; and -92.6 m/s2; -111 m/s2)

(8) A flat-ended valve tapped is operated by a symmetrical cam with circular arcs for

frank and nose profiles. The total angle of action is 150º, base circle diameter 125

mm and the lift 25 mm. During the lift, the period of acceleration is half that of the

deceleration. Speed of cam shift is 1250 rpm. The straight line path of the tappet

passes through the cam axis. Find:

(i) Radii of the nose and the flank, and

(ii) Maximum acceleration and declaration during the lift.

(40.3 mm, 148 mm; 1465 m/s2; 808.8 m/s2)

(9) In a four-stroke petrol engine, the crank angle is 5º after t.d.c. when the suction

valve opens and 53º after b.d.c. when the suction valve closes. The lift is 8 mm, the

nose radius 3 mm and the least radius of the cam 18 mm. The shaft rotates at 800

rpm. The cam is of the circular type with a circular nose and flanks while the

follower is flat-faced. Determine the maximum velocity and the maximum

acceleration and retardation of the valve. When is the minimum force exerted by

the springs to overcome the inertia of moving parts weighting 250 g.

(1.3 m/s; 433.7 m/s; 161.4 m/s2;40.35 N)

assignment 1 introduction of machines and mechanisms o… · assignment – 1 introduction of...

Documents