KOM Slides_02D_Velocity and 03C_Acceleration Analysis.pdf

1. Configuration or Space Diagram Velocity Diagram
Velocity Calculation
Velocity Calculation for Point “ C ”
Ratio Method:

2. Configuration or Space Diagram
Velocity Diagram
Problem No. 1 Scale Factor : 1: 2
a, d
b
c
𝜐𝑏𝑎 𝜐𝑐𝑏
𝜐𝑐𝑑
Velocity Diagram
Scale Factor : 1 cm = 10 cm/s

3. Links Velocity
Symbol
Velocity
(cm/s)
Angular
Velocity
Symbol
Angular
Velocity
(rad/s)
AB 𝜐𝑏𝑎
50.24 ω𝑏𝑎
12.56
BC 𝜐𝑐𝑏
15.38
CD 𝜐𝑐𝑑
38.23 ω𝑐𝑑
4.77
TABLE
Configuration or Space Diagram
Velocity Diagram
Problem No. 1

4. Problem No. 2
Configuration or Space Diagram
E
Velocity Diagram
Scale Factor : 1cm = 100 cm/s
a, e
b
c
𝜐𝑏𝑎 𝜐𝑐𝑏
𝜐𝑐𝑒
𝜐𝑑𝑎
d
𝜐𝑑𝑏
𝜐𝑑𝑐
Velocity Diagram

5. Problem No. 2
Configuration or Space Diagram
Links Velocity
Symbol
Velocity
(cm/s)
Angular
Velocity
Symbol
Angular
Velocity
(rad/s)
BA 𝜐𝑏𝑎
471.30 ω𝑏𝑎
31.42
CB 𝜐𝑐𝑏
340 ω𝑐𝑏
5.67
DB 𝜐𝑑𝑏
170
CE 𝜐𝑐𝑒
400
Point D 𝜐𝑑𝑎 or 𝜐𝑑𝑒
410
E
Scale Factor : 1cm = 100 cm/s
a, e
b
c
𝜐𝑏𝑎 𝜐𝑐𝑏
𝜐𝑐𝑒
𝜐𝑑𝑎
d
𝜐𝑑𝑏
𝜐𝑑𝑐
Velocity Diagram
TABLE

6. Problem No. 3
Configuration or Space Diagram
Scale Factor : 1:2
O,02, O1
𝜐𝑎𝑜
Velocity Diagram
𝑂1
𝑂2
Scale Factor : 1cm = 100 mm/s
a
d
𝜐𝑑𝑎
𝜐𝑑02 b
c
𝜐𝑐𝑏
𝜐𝑐𝑜1
𝜐𝑏𝑎
𝜐𝑑𝑏
Links Velocity
Symbol
Velocity
(mm/s)
Angular
Velocity
Symbol
Angular
Velocity
(rad/s)
OA 𝜐𝑎𝑜
867.64 ω𝑎𝑜
18.84
DA 𝜐𝑑𝑎
280 ω𝑑𝑎
1.84
BA 𝜐𝑏𝑎
180 ω𝑏𝑎
1.97
CB 𝜐𝑐𝑏
810 ω𝑐𝑏
8.90
Velocity
at P
𝜐𝑑𝑜2
960
Velocity
at C
𝜐𝑐𝑜1
170
V = 867.079 mm/sec
o
a
b
d
c
172.682
965.953
a
Velocity Diagram

7. Problem No. 3
Scale Factor : 1cm = 100 cm/s
𝑂1
𝑂2

8. Problem No. 4
Configuration or Space Diagram
Scale Factor : 1 : 2
b
𝜐𝑎𝑜2
=
Velocity Diagram
ω𝑎𝑜2
× AO2
Scale Factor : 1 cm = 1m/sec
O2,
a
O4,
𝜐𝑏𝑎
𝜐𝑏𝑜4
p
𝜐𝑝𝑎
𝜐𝑝
𝜐𝑎𝑜2
Links Velocity
Symbol
Velocity
(mm/s)
Angular
Velocity
Symbol
Angular
Velocity
(rad/s)
AO2 𝜐𝑎𝑜2
4500 ω𝑎𝑜2
45.00
BA 𝜐𝑏𝑎
4490 ω𝑏𝑎
44.90
BO4 𝜐𝑏𝑜4
6510 ω𝑏𝑜4
73.97
PA 𝜐𝑝𝑎
4490
Velocity at
P
𝜐𝑝𝑜2
6200

9. Instantaneous Centre Method

10. Instantaneous Centre Method

11. Instantaneous Centre Method
Types of Instantaneous Centres
1. Fixed Instantaneous Centres
2. Permanent Instantaneous Centres
3. Neither Fixed nor Permanent Instantaneous Centres

12. Instantaneous Centre Method
Location of Instantaneous Centres

13. Instantaneous Centre Method
Problem No. 1
𝜐𝑏𝑎 = ω𝑏𝑎 × BA
1
2
3
4 1 2
3
4
𝐼13
𝜐𝑏𝑎 𝑜𝑟 𝜐𝑏 = ω𝑏𝑎 × BA
𝜐𝑏
𝐵𝐼13
=
𝜐𝑐
𝐶𝐼13
ω𝑐𝑏
=

14. Instantaneous Centre Method
Problem No. 1
𝜐𝑏𝑎 = ω𝑏𝑎 × BA
3. Fig. shows a pin Joined four bar linkage having the following dimensions : Fixed link AD = 4 m ;
Driving link AB = 1.5 m, Driven link CD = 2.5 m ; Connecting link BC = 3 m Angle BAD = 60°. Link
AB revolves at 25 rpm. Determine: (i) Angular velocity of link CD and (ii) Angular velocity of link
BC.
1
2
3
4

15. Instantaneous Centre Method
Problem No. 1
𝜐𝑏𝑎 𝑜𝑟 𝜐𝑏 = ω𝑏𝑎 × BA
𝜐𝑏
𝐵𝐼13
=
𝜐𝑐
𝐶𝐼13
ω𝑐𝑏
=

16. Instantaneous Centre Method
Problem No. 2
𝐼13
1 2
3
4
𝐼24
𝜐𝑎
=
𝜐𝑏
𝐵𝐼13
ω𝑏𝑎
=
𝐴𝐼13

17. Instantaneous Centre Method
Problem No. 3
𝜐𝑎
=
𝜐𝑏
𝐵𝐼13
𝐴𝐼13
1 2 3 4 5 6
12 23 34 45 56
13 24 35 46
14 25 36
15 26
16
1
𝜐𝑎𝑜 𝑜𝑟 𝜐𝑎 = ω𝑎𝑜 × AO
𝐼34
𝐼45
𝐼14
𝐼56 𝐼16 at infinity
𝐼23
2
3
4
5
6
1
6
1
𝐼15
𝜐𝑏
=
𝜐𝑑
𝐷𝐼14
𝐵𝐼14
𝜐𝑑
=
𝜐𝑒
𝐸𝐼15
𝐷𝐼15

18. Instantaneous Centre Method
Problem No. 4
1
2
3
4
5
6
1 2 3 4 5 6
12 23 34 45 56
13 24 35 46
14 25 36
15 26
16
1
2
3
4
5
6
1
1
𝐼12
𝐼23
𝐼34
𝐼14
𝐼35
𝐼16 𝐼56
𝐼13
𝐼15
𝜐𝑝𝑜1 𝑜𝑟 𝜐𝑝 = ω𝑝𝑜1 × PO1
𝜐𝑝
=
𝜐𝑞
𝑄𝐼13
𝑃𝐼13
𝜐𝑞
=
𝜐𝑠
𝑆𝐼15
𝑄𝐼15
𝜐𝑠
=
𝜐𝑇
𝑇𝐼16
S𝐼16
𝐼24

19. Instantaneous Centre Method
Arnold Kennedy Theorem

20. Relative Acceleration Method
Tangential Component
Radial Component
a’
m
b’

21. Relative Acceleration Method

22. Relative Acceleration Method

23. Relative Acceleration Method
For a Slider : When the slider (point) moves along a straight line, then its radial acceleration will be
zero

24. Relative Acceleration Method
Problem No. 1
ω
α
Space Diagram
𝑝1
′
x
Scale : 1cm = 10 m/s2
𝑓𝑎𝑝1
𝑟
a’
𝑓𝑎𝑝1
𝑡
𝑓𝑎𝑝1
y 𝑓𝑏𝑎
𝑟
𝑓𝑏𝑎
𝑡
𝑝2
′
z
𝑓𝑏𝑝2
𝑟
𝑓𝑏𝑝2
𝑡
b’
𝑓𝑏𝑝2
𝑓𝑏𝑎

25. Relative Acceleration Method
Problem No. 1
Links Velocity
Symbol
Velocity
(m/s)
Angular
Velocity
Symbol
Angular
Velocity
(rad/s)
Acc.
Symbol
Radial
Comp.
Acc.
Radial
Comp.
(m/s2)
Acc.
Symbol
Tangential
Comp.
Acc.
Tangential
Comp.
(m/s2)
Angular
Acc.
Symbol and
Value (rad/s2)
TOTAL
Acceleration
Symbol & Value
(m/s2)
AP1 𝜐𝑎𝑝1
03 ω𝑎𝑝1
10 𝑓𝑎𝑝1
𝑟 30 𝑓𝑎𝑝1
𝑡 09 α𝑎𝑝1
= 30 𝑓𝑎𝑝1
= 31.6
BA 𝜐𝑏𝑎
2.1 ω𝑏𝑎
5.86 𝑓𝑏𝑎
𝑟
11.67
𝑓𝑏𝑎
𝑡 13.6 α𝑏𝑎 = 37.78 𝑓𝑏𝑎 = 18
BP2 𝜐𝑏𝑝2
2.25 ω𝑏𝑝2
6.13 𝑓𝑏𝑝2
𝑟 14.06 𝑓𝑏𝑝2
𝑡 28 α𝑏𝑝2
= 77.78 𝑓𝑏𝑝2
= 31.33
TABLE

26. Relative Acceleration Method
Problem No. 2
Space Diagram
𝑜2
′
x
y
z
𝑜4
′
a’
b’
p’
𝑓𝑎𝑜2
𝑟
𝑓𝑏𝑎
𝑟
𝑓𝑏𝑜4
𝑟
Scale : 1cm = 50 m/s2
𝑓𝑎𝑜2
𝑡
𝑓𝑎𝑜2
𝑓𝑏𝑜4
𝑡
𝑓𝑏𝑎
𝑡
𝑓𝑝𝑎
𝑓𝑏𝑜4
𝑓𝑝𝑜2
𝑓𝑏𝑎

27. Relative Acceleration Method
Problem No. 2
Links Velocity
Symbol
Velocity
(m/s)
Angular
Velocity
Symbol
Angular
Velocity
(rad/s)
Acc.
Symbol
Radial
Comp.
Acc.
Radial
Comp.
(m/s2)
Acc.
Symbol
Tangential
Comp.
Acc.
Tangential
Comp.
(m/s2)
Angular
Acc.
Symbol and
Value (rad/s2)
TOTAL
Acceleration
Symbol & Value
(m/s2)
AO2 𝜐𝑎𝑜2
4.5 ω𝑎𝑜2
45 𝑓𝑎𝑜2
𝑟 202.5 𝑓𝑎𝑜2
𝑡 30 α𝑎𝑜2
= 300 𝑓𝑎𝑜2
= 204.71
BA 𝜐𝑏𝑎
4.49 ω𝑏𝑎
44.49 𝑓𝑏𝑎
𝑟 201.6 𝑓𝑏𝑎
𝑡 610 α𝑏𝑎 = 6100 𝑓𝑏𝑎 = 635
BO4 𝜐𝑏𝑜4
6.513 ω𝑏𝑜4
73.79 𝑓𝑏𝑜4
𝑟 482.03 𝑓𝑏𝑜4
𝑡 680 α𝑏𝑜4
=
7727.27
𝑓𝑏𝑜4
= 835
PA 𝜐𝑝𝑎
4.49 ω𝑝𝑎
44.49 𝑓𝑝𝑎
𝑟 201.6 𝑓𝑝𝑎 = 640
Velocity
at P
𝜐𝑝𝑜2
6.20 𝑓𝑝𝑜2
= 490
TABLE

28. Relative Acceleration Method
Problem No. 3
Space Diagram
Scale : 1cm = 100 mm Length
𝜐𝑎𝑜
𝜐𝑐𝑎
𝜐𝑑𝑐
𝜐𝑐𝑞
𝜐𝑑𝑞 𝜐𝑏𝑑
𝜐𝑏𝑔
o, q, g
a
b
c
d
Scale : 1cm = 0.25 m/s

29. Relative Acceleration Method
Problem No. 3
Space Diagram
Scale : 1cm = 100 mm Length
Acceleration Diagram Scale : 1cm = 2 m/s2
o’
a’
𝑓𝑎𝑜
𝑟
= 𝑓𝑎𝑜
x
𝑓𝑐𝑎
𝑟
𝑓𝑐𝑎
𝑡
q’
𝑓𝑐𝑞
𝑟
y
c’
𝑓𝑐𝑞
𝑓𝑐𝑎
𝑓𝑑𝑞
𝑟
z
𝑓𝑑𝑞
𝑡
d’
𝑓𝑑𝑐
𝑟
m
𝑓𝑑𝑐
𝑡
𝑓𝑑𝑐
𝑓𝑑𝑞
𝑓𝑏𝑑
𝑟
n
𝑓𝑏𝑑
𝑡
g’
b’
𝑓𝑏𝑔
𝑡 = 𝑓𝑏𝑔
𝑓𝑏𝑑
𝑓𝑐𝑞
𝑡

30. Problem No. 3
Links Velocity
Symbol
Velocity
(m/s)
Angular
Velocity
Symbol
Angular
Velocity
(rad/s)
Acc.
Symbol
Radial
Comp.
Acc.
Radial
Comp.
(m/s2)
Acc.
Symbol
Tangential
Comp.
Acc.
Tangential
Comp.
(m/s2)
Angular
Acc.
Symbol and
Value (rad/s2)
TOTAL
Acceleration
Symbol & Value
(m/s2)
AO 𝜐𝑎𝑜
0.94 ω𝑎𝑜
6.28 𝑓𝑎𝑜
𝑟 5.89 𝑓𝑎𝑜
𝑡 0 α𝑎𝑜= 0 𝑓𝑎𝑜 = 5.89
CA 𝜐𝑐𝑎
1.00 ω𝑐𝑎
1.67 𝑓𝑐𝑎
𝑟 1.67 𝑓𝑐𝑎
𝑡 15 α𝑐𝑎 = 25 𝑓𝑐𝑎 = 15.1
CQ 𝜐𝑐𝑞
1.14 ω𝑐𝑞
7.86 𝑓𝑐𝑞
𝑟 8.96 𝑓𝑐𝑞
𝑡 3.6 α𝑐𝑞 = 24.83 𝑓𝑐𝑞 = 9.65
DC 𝜐𝑑𝑐
1.08 ω𝑑𝑐
8.64 𝑓𝑑𝑐
𝑟 9.33 𝑓𝑑𝑐
𝑡 5 α𝑑𝑐 = 40 𝑓𝑑𝑐 =10.58
DQ 𝜐𝑑𝑞
1.14 ω𝑑𝑞
7.86 𝑓𝑑𝑞
𝑟 8.96 𝑓𝑑𝑞
𝑡 3.6 α𝑑𝑞 = 24.83 𝑓𝑑𝑞 = 9.65
BD 𝜐𝑏𝑑
0.45 ω𝑏𝑑
0.90 𝑓𝑏𝑑
𝑟 0.405 𝑓𝑏𝑑
𝑡 7.21 α𝑏𝑑 = 14.44 𝑓𝑏𝑑 = 7.22
BG 𝜐𝑏𝑔 0.88 *** 𝑓𝑏𝑔
𝑟 0 𝑓𝑏𝑔
𝑡 8.4 α𝑏𝑞 = 21 𝑓𝑏𝑔 = 8.4
TABLE

31. Klein’s Construction (only for Slider Crank Mechanism)
Problem No. 4
Space Diagram
N

32. Klein’s Construction (only for Slider Crank Mechanism)
Problem No. 4
Space Diagram Scale : 1cm = 1m/s
Acceleration Diagram Using Relative Acc. Method for comparing with
Klein’s method
𝜐𝑏𝑛
𝜐𝑎𝑜 𝜐𝑏𝑎
o, n b
a
N
Velocity diagram Using Relative Velocity Method for comparing with
Klein’s method
o’, n’
a’
𝑓𝑎𝑜
𝑟
=𝑓𝑎𝑜
𝑓𝑏𝑎
𝑟
x
𝑓𝑏𝑎
𝑡
b’
𝑓𝑏𝑛
𝑡
=𝑓𝑏𝑛
𝑓𝑏𝑎
Scale : 1cm = 50 m/s2
Scale : 1cm = 100 length

33. Klein’s Construction (only for Slider Crank Mechanism)
Problem No. 4
Velocity Diagram
(Region OAC)
C
N
X
M
P
F
H
Acceleration Diagram
(Region OAHFO)
𝜐𝑎𝑜 𝑜𝑟 𝜐𝑎 = ω𝑎𝑜 × AO x scale factor
𝜐𝑏𝑎 = ω𝑎𝑜 × AC x scale factor
𝜐𝑏𝑛 = ω𝑎𝑜 × OC x scale factor
𝑓𝑏𝑎
=
𝑓𝑎𝑜
𝑟
ω𝑎𝑜
2 × AO x Scale factor
=
𝑓𝑏𝑎
𝑟
ω𝑎𝑜
2 × AH x Scale factor
=
𝑓𝑏𝑎
𝑡
ω𝑎𝑜
2 × HF x Scale factor
=
𝑓𝑏𝑛
𝑡
ω𝑎𝑜
2 × OF x Scale factor
= ω𝑎𝑜
2 × AF x Scale factor

34. Problem No. 4
Links Velocity
Symbol
Velocity
(m/s)
Angular
Velocity
Symbol
Angular
Velocity
(rad/s)
Acc.
Symbol
Radial
Comp.
Acc.
Radial
Comp.
(m/s2)
Acc.
Symbol
Tangential
Comp.
Acc.
Tangential
Comp.
(m/s2)
Angular
Acc.
Symbol and
Value (rad/s2)
TOTAL
Acceleration
Symbol & Value
(m/s2)
AO 𝜐𝑎𝑜
4.189 ω𝑎𝑜
41.89 𝑓𝑎𝑜
𝑟 175.75 𝑓𝑎𝑜
𝑡 0 α𝑎𝑜= 0 𝑓𝑎𝑜 =175.75
BA 𝜐𝑏𝑎
2.80 ω𝑏𝑎
5.61 𝑓𝑏𝑎
𝑟 15.68 𝑓𝑏𝑎
𝑡 95 α𝑏𝑎 = 190 𝑓𝑏𝑎 = 96.28
BN 𝜐𝑏𝑛
3.10 **** **** 𝑓𝑏𝑛
𝑟 0 𝑓𝑏𝑛
𝑡 165 ∗∗∗∗ 𝑓𝑏𝑛 = 165
TABLE

35. Klein’s Construction (only for Slider Crank Mechanism)
Problem No. 4
Y
Y1

36. Klein’s Construction (only for Slider Crank Mechanism)
Problem No. 4