ta202a - manufacturing processes ii mechanismshome.iitk.ac.in/~mlaw/ta202a/lecture3-4.pdf ·...

IIT KanpurMachine ToolDynamics Laboratory

TA202A - Manufacturing Processes II

Mechanisms

Lecture 3 - 4

Mohit Law

Mechanical Engineering

[email protected]

mailto:[email protected]


Design of mechanical systems

The design of any mechanical system needs proper

understanding of:

i. The geometrical aspects of motion (kinematics), and

ii. The various forces involved in motion (kinetics and

dynamics, i.e., mechanics)

2


Mechanism and machines

• Mechanism: is a combination of rigid or restraining bodies so

shaped that they can move upon each other with a definite

relative motion

• Machine: is a mechanism or a collection of mechanisms which

performs useful mechanical work

• Every machine is a mechanism, but not vice versa

3


Structure of this lecture

• What is a mechanism?

– Building blocks of mechanisms

• Mechanisms of interest to us

– Cams

– Belt-pulley, chain-sprocket

– Gears

• Spur, Bevel, Worm, Trains

– Quick return

– Indexing

• How different components of mechanisms are made

4


Building blocks of mechanisms – kinematic pairs

5

Source: http://planning.cs.uiuc.edu/node109.html

Degree of freedom

of a kinematic pair is

given by the number

of independent

coordinates required

to specify the

relative movement

http://planning.cs.uiuc.edu/node109.html


Mechanisms you are familiar with

6

https://www.youtube.com/watch?v=nb1pPOSAgnQ

https://www.youtube.com/watch?v=nb1pPOSAgnQ


Desired motion

7

What kind of mechanism will provide me this motion?

Source: Shigley and Uicker, Theory of Machines and Mechanisms


Cam

8

• Follower motion, 𝑦 is the ordinate, and the cam

motion , 𝜃 is on the abscissa.

• In general: 𝑦 = 𝑦 𝜃

• 1st derivative is the measure of steepness: 𝑦′ 𝜃 =𝑑𝑦

𝑑𝜃

• 2nd derivative is a measure of the radius of curvature:

𝑦′′ 𝜃 =𝑑2𝑦

𝑑𝜃2


Types of Cams

9

Convex cam Concave cam

Cylindrical cam Spherical cam Flat/wedge cam

Radial cam

https://www.youtube.com/watch?v=GYVgGSQjX2U

https://www.youtube.com/watch?v=GYVgGSQjX2U


Cam shafts

10

https://www.youtube.com/watch?v=NZXVTkPPpVQ

https://www.youtube.com/watch?v=NZXVTkPPpVQ


How are Cams designed?

11

Simple harmonic motion Cycloidal motion 8th order polynomial

• Usually, the total lift, dwell, and return cycles depend on application, and are

predetermined.

• However, there are many possible choices of follower motion that can achieve the

desired lift/return and dwell

• The key step is cam design is hence the right choice for these motions

• We would preferably like smooth velocity, acceleration and jerk profiles

• Matching derivatives of displacement diagrams with the desired motion profile



Displacement diagrams and their importance

12


Consider a cam with a follower that rises with a parabolic motion from a dwell to another dwell such that the total lift is 𝐿 and the total rotation angle is 𝛽

𝑦 = 𝐴𝜃2 + 𝐵𝜃 + 𝐶

For the first half of the motion, consider the following equation of the parabola

First three derivatives of this w.r.t 𝜃:

𝑦′ = 2𝐴𝜃 + 𝐵; 𝑦′′ = 2𝐴; 𝑦′′′ = 0

To match the position and slope with those of the preceding dwell properly, at:

(1)

(2)

𝜃 = 0 → 𝑦 0 = 𝑦′ 0 = 0, which from (1) and (2) gives → 𝐵 = 𝐶 = 0

𝜃 =𝛽

2→ 𝑦 =

𝐿

2; which

from (1) gives → 𝐴 =2𝐿

𝛽2



13


𝑦 = 𝐴𝜃2 + 𝐵𝜃 + 𝐶 𝑦′ = 2𝐴𝜃 + 𝐵; 𝑦′′ = 2𝐴; 𝑦′′′ = 0(1) (2)

𝑦 = 2𝐿𝜃

𝛽

2

; 𝑦′ =4𝐿𝜃

𝛽2 ; 𝑦′′ =4𝐿

𝛽2 ; 𝑦′′′= 0

The maximum slope occurs at the inflection point, where 𝜃 =𝛽

2, substituting this into

𝑦′from (3) → 𝑦′𝑚𝑎𝑥 =2𝐿

𝛽

(3)

𝜃 = 0 → 𝑦 0 = 𝑦′ 0 = 0 → 𝐵 = 𝐶 = 0

𝜃 =𝛽

2→ 𝑦 =

𝐿

2→ 𝐴 =

2𝐿

𝛽2

For the first half of the parabola



14


𝑦 = 𝐴𝜃2 + 𝐵𝜃 + 𝐶 𝑦′ = 2𝐴𝜃 + 𝐵; 𝑦′′ = 2𝐴; 𝑦′′′ = 0(1) (2)

Since the slope must match that of the first parabola at 𝜃 =𝛽

2→ 𝑦′𝑚𝑎𝑥 =

2𝐿

𝛽

(4)

For the second half of the parabola: 𝜃 = 𝛽 → 𝑦 = 𝐿; 𝑦′ = 0

Hence: 𝑦 = 𝐿 = 𝐴𝛽2 + 𝐵𝛽 + 𝐶; and 𝑦′ = 0 = 2𝐴𝛽 + 𝐵

(5)

From (4) and (5) →2𝐿

𝛽= 2𝐴

𝛽

2+ 𝐵 (6)

Solving (4) and (6) simultaneously → 𝐴 = −2𝐿

𝛽2 ; 𝐵 =4𝐿

𝛽; 𝐶 = −𝐿 (7)

Substituting (7) into (1) and (2) for the second half of the motion, we get:

𝑦 = 𝐿 1 − 2 1 −𝜃

𝛽

2

𝑦′ =4𝐿

𝛽1 −

𝜃

𝛽𝑦′′ = −

4𝐿

𝛽2𝑦′′′ = 0 (8)


Displacement diagram and derivatives for parabolic motion

15


For the second half of the motion(kinematic derivatives)

𝑦 = 𝐿 1 − 2 1 −𝜃

𝛽

2; 𝑦′ =

4𝐿

𝛽1 −

𝜃

𝛽; 𝑦′′ = −

4𝐿

𝛽2 ; 𝑦′′′ = 0

For the first half of the motion (kinematic derivatives)

𝑦 = 2𝐿𝜃

𝛽

2

; 𝑦′ =4𝐿

𝛽2; 𝑦′′ =

4𝐿

𝛽2; 𝑦′′′= 0

𝑦 = 𝑦 𝜃 ; 𝜃 = 𝜃(𝑡)

Time derivatives:

ሶ𝑦 =𝑑𝑦

𝑑𝑡=

𝑑𝑦

𝑑𝜃

𝑑𝜃

𝑑𝑡= 𝑦′𝜔;

ሷ𝑦 = 𝑦′′𝜔2; ഺ𝑦 = 𝑦′′′𝜔3

Cams are driven by constant-speed shafts → 𝜃 = 𝜔t

Acceleration profile is discontinuous, hence this parabolic profile is not a good choice for the cam


How are Cams designed?

16

Simple harmonic motion Cycloidal motion 8th order polynomial

• Usually, the total lift, dwell, and return cycles depend on application, and are

predetermined.

• However, there are many possible choices of follower motion that can achieve the

desired lift/return and dwell

• The key step is cam design is hence the right choice for these motions

• We would preferably like smooth velocity, acceleration and jerk profiles

• Matching derivatives of displacement diagrams with the desired motion profile



Exam question from 2019 - II

17

Question 4 [4]

Consider that in the design of cams, travel of the follower can be described by either of these

equations:

Option 1: 𝑦 = 2𝐿 𝜃

𝛽

2for when 0 <

𝜃

𝛽<

1

2; and 𝑦 = 𝐿 1 − 2 1 −

𝜃

𝛽

2 for when

1

2<

𝜃

𝛽< 1

Option 2: 𝑦 =𝐿

2 1 − cos

𝜋𝜃

𝛽

wherein 𝑦 is the rise motion of the follower for some input rotational motion 𝜃 of the cam, such that

𝑦 = 𝑦(𝜃); and 𝐿 is the lift of the follower.

Sketch, neatly the two displacement diagrams for both options, i.e. plot 𝑦 on the ordinate and 𝜃/𝛽 on

the abscissa. Take the start value for 𝜃/𝛽 as ‘0’, and the final value as ‘1’. On these displacement

diagrams, overlay the velocity (𝑦′ ), acceleration (𝑦′′′ ), and jerk (𝑦′′′ ), and remember that 𝜃 = 𝜔𝑡.

Comment on which of the two options is preferred to help describe the motion of the follower, and

to design the cam. Also discuss why. For a full grade, sketches should be neat and reasoned analysis

must be presented.


How are cams manufactured?

18

On CNC machines


Belt + pulley drive

19

https://www.youtube.com/watch?v=hkIyGrLDy3A

https://www.youtube.com/watch?v=hkIyGrLDy3A


Chain + sprocket drives

20

https://www.youtube.com/watch?v=tXVE5O_jJi8

https://www.youtube.com/watch?v=tXVE5O_jJi8


Gears

21

Gears are a positive drive, i.e., no slipping, i.e. for gear with equal number of teeth, angular velocity of gear 1 is the same as angular velocity of gear 2

https://www.youtube.com/watch?v=P4rNX0gCm3E

https://www.youtube.com/watch?v=P4rNX0gCm3E


Spur gear nomenclature

22


𝑃 =𝑁

𝑑

𝑚 =𝑑

𝑁

Diametrical pitch:

Module of the gear:

𝑁 – number of teeth; 𝑑 – pitch circle diameter

𝑝 =𝜋𝑑

𝑁= 𝜋𝑚Circular pitch:


The involute gear

23

https://en.wikipedia.org/wiki/Involute_gear

https://en.wikipedia.org/wiki/Involute_gear


Law of gearing

• the angular velocity ratio of all gears of a

meshed gear system must remain constant,

• also the common normal at the point of contact

must pass through the pitch point.

• Let 𝑣1 and 𝑣2 be the velocities of the point Q on

the wheels 1 and 2 respectively. If the teeth are

to remain in contact, then the components of

these velocities along the common normal MN

must be equal, i.e.,

24

https://www.ques10.com/p/24423/state-explain-law-of-gearing/

𝑣1 cos𝛼 = 𝑣2 cos 𝛽 → 𝜔1 × 𝑂1𝑄 cos𝛼 = 𝜔2 × 𝑂2𝑄 cos𝛽 →𝜔1

𝜔2=

𝑂2𝑁

𝑂1𝑀=

𝑂2𝑃

𝑂1𝑃

https://www.ques10.com/p/24423/state-explain-law-of-gearing/


Spur gears

25

https://www.youtube.com/watch?v=49IOAHJ-V4I

Spur gears are used to transmit rotary motion between parallel shafts

𝑣 = 𝑟𝐴𝜔𝐴 = 𝑟𝐵𝜔𝐵 →𝜔𝐴

𝜔𝐵=

𝑟𝐵

𝑟𝐴=

𝑁𝐵

𝑁𝐴



Spur gear train

26

𝜔6 =𝑁2

𝑁3

𝑁4

𝑁5

𝑁5

𝑁6𝜔2

What is the speed of the driven gear?



Gear trains

27

What is the speed of the driven gear, gear # 8?



Rack and pinion (spur gears)

28




Bevel gears

29


Axis of the shafts intersect



Bevel gears

30


Where are bevel gears used?

31

https://www.youtube.com/watch?v=eef7MutOVME

https://www.youtube.com/watch?v=eef7MutOVME


Worm gears

32


Normally used with nonintersecting shafts which are usually at a shaft angle of 90°



Everyday example of use of worm gears

33

Source: www.mardustrial.com

http://www.mardustrial.com/


Worm gears

34

• One rotation of the worm moves one tooth on the worm wheel

(gear)

• Very high gear ratios

• For example, if you have

– 8 teeth, ratio is 1/8

– 24 teeth, ratio is 1/24

– …

• These are one directional, and hence ‘self-locking’


Gear trains

35

Given the speed of gear 2, what will be the speed of gear 7?



Gear trains

36

Given the speed of gear 2, what will be the speed of gear 9?

Note that there are spur gears, bevel gears, and a worm and worm wheel in

this gear train




37

Question 1 [1]

A gear train is shown on the right. It includes a

pulley drive, a set of bevel gears, two sets of spur

gears, and a worm and worm wheel pair. The

number of teeth (T) on each gear is as shown. If

the input speed of the driving pulley (# 2 in the

figure) is 30 RPM, what is the output speed of the

worm wheel? Also, if the required output torque

at the worm wheel is 10 Nm, what should be the

torque supplied by the motor driving the pulley #

2? Keep in mind that if the input gear rotates

faster than the output gear, then the gear train

amplifies the input torque.

Show all steps in your calculations for a full grade.

Question 2 [1]


How are gears made?

38

https://www.youtube.com/watch?v=8yNj_Ogu0-Ehttps://www.youtube.com/watch?v=0rnTh6c19HM

Gear hobbing Gear tooth form cutting

https://www.youtube.com/watch?v=8yNj_Ogu0-E

https://www.youtube.com/watch?v=0rnTh6c19HM


How you will make gears?

39


Steps in making a bevel gear

Step 1: Identify the raw material

Step 2: Mount the cylindrical work piece

in the chuck

Step 3: Measure the diameter and turn it

to size

Step 4: Drill a through hole

Step 5: Move to the milling machine

Step 8: Mount the cutter on to arbor

shaft

Step 6: Mount the gear blank in indexing head

Step 7: Adjust the angle of the indexing

head

Step 9: Cut the gear teeth one at a

time

… and, finally:


Steps in making a spur gear

Step 1: Identify the raw material

Step 2: Mount the cylinder in

the lathe

Step 3: Measure the diameter, and

turn it to size

Step 4: Drill a through hole

Step 5: Assemble the gear blank in

a mandrel

Step 6: Mount the mandrel in the

machine

Step 7: Turn the gear blank down

to size

Step 8: Move to the milling machine

Step 9: Mount the mandrel

and the cutter

Step 10: Mount the driving dog

for indexing

Step 11: Cut the gear teeth one

at a time

… and, finally:


Other interesting and relevant mechanisms

42

Quick return mechanism

Indexing mechanism


Quick return mechanism

43


Quick return mechanism – gear based

44

https://www.youtube.com/watch?v=zA0owiHQgpc

https://www.youtube.com/watch?v=zA0owiHQgpc


Geneva indexing mechanism

45

https://www.youtube.com/watch?v=dGxUl36IrB8

https://www.youtube.com/watch?v=dGxUl36IrB8


Geneva mechanism: how to size this?

46

http://benbrandt22.github.io/genevaGen/

http://benbrandt22.github.io/genevaGen/


Geneva mechanism: how to design this in CAD?

47

• https://www.instructables.com/id/Make-Geneva-Wheels-of-Any-Size-in-a-Easier-Way/

• https://newgottland.com/2012/01/08/make-geneva-wheels-of-any-size/

https://www.instructables.com/id/Make-Geneva-Wheels-of-Any-Size-in-a-Easier-Way/

https://newgottland.com/2012/01/08/make-geneva-wheels-of-any-size/


How are Geneva mechanisms made?

48

On CNC machines


One the most famous mechanisms

49

https://www.youtube.com/watch?v=ZO8QEG4x0wY

https://www.youtube.com/watch?v=ZO8QEG4x0wY



50

Question 2 [1]

Say you have a motor, and you want to use the motor to drive a mechanism that results in translational

motion. Sketch any two such mechanisms (other than what is already discussed in Question 4) that

can translate rotary motion to linear motion. Please make neat sketches and label all parts for a full

grade.


Structure of this lecture

• What is a mechanism?

– Building blocks of mechanisms

• Mechanisms of interest to us

– Cams

– Belt-pulley, chain-sprocket

– Gears

• Spur, Bevel, Worm, Trains

– Quick return

– Indexing

• How different components of mechanisms are made

51

ta202a - manufacturing processes ii mechanismshome.iitk.ac.in/~mlaw/ta202a/lecture3-4.pdf ·...

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