The document discusses the velocity and acceleration analysis of a slider crank mechanism using Klein's construction method. It provides specific calculations for linear and angular velocities and accelerations at a crank angle of 55°, considering both uniform and non-uniform crank speeds. The analysis includes step-by-step procedures for constructing configuration and acceleration diagrams, as well as simulations using GeoGebra software.

1. Theory of Machines-I
Velocity and Acceleration Analysis of Mechanisms
Kleins Construction Method(Graphical Method)
Slider Crank Mechanism
Prof. K N Wakchaure
Department of Mechanical Engineering
Sanjivani College of Engineering, Kopargaon

2. The crank of a slider crank mechanism rotates clockwise at a constant speed of 300 r.p.m. The crank is 150
mm and the connecting rod is 600 mm long.
Determine: 1. Linear velocity and acceleration of the Slider P and
2. Angular velocity and angular acceleration of the connecting rod, at a crank angle of 55° from inner dead
centre position.
Given Data:
Crank Length r= 150 mm
Connecting Rod Length, l=600mm
Crank Angle θ=55°
Crank speed 𝑵 𝒄/𝒐=300 rpm
Link True Length Scaled Length (1:5)
OC 150mm 30mm
CP 600mm 120mm
55°
O
C
P
IDC
Configuration Diagram (Scale 1:5)
O
Find:
1. 𝑽 𝑷
2. 𝒂 𝑷
3. 𝝎 𝒑/𝒄
4. 𝜶 𝒑/𝒄
Given:
𝜔 𝒄/𝑜 =
2 ∗ 𝜋 ∗ 𝑁𝒄/𝑜
60
= 31.42 𝑟𝑎𝑑/𝑠

3. Step 1. Draw a configuration Diagram with suitable Scale.
Kleins Construction
Scale: 1:5
For uniform velocity of Crank

4. Step 2. Draw a line perpendicular to OP through point O.
Kleins Construction
For uniform velocity of Crank

5. Step 3. Extend CP (Connecting Rod) until it intersects previous line
Kleins Construction
For uniform velocity of Crank

6. Step 4. Mark Point M at the intersection of two lines.
Kleins Construction
For uniform velocity of Crank

7. Step 5. Join OM and CM by solid lines.
Kleins Construction
For uniform velocity of Crank

8. Link Velocity Vector Length Formula Velocity
OC 𝑉𝑐/𝑜 𝑜𝑐 30 mm 𝑂𝐶 ∗ 𝜔 ∗ 𝑆𝐹 4713 𝒎𝒎
𝒔
CP 𝑉𝑝/𝑐 𝑐𝑝 17.58mm CM∗ 𝜔 ∗ 𝑆𝐹 2761.81 𝒎𝒎
𝒔
Slider P 𝑉𝑝/𝑎 𝑎𝑝 28.17mm 𝑂𝑃 ∗ 𝜔 ∗ 𝑆𝐹 4426.13 𝒎𝒎
𝒔
Step 6. ∆OCM is the velocity diagram of Given Mechanism
Kleins Construction
Component Significance
OC Velocity of Link OC
CM Velocity of Link CP
OM Velocity of Slider P
For uniform velocity of Crank

9. Step 7. Find Angular Velocity of Connecting Rod CP
Kleins Construction
Oa M
C
Link Velocity Length Formula Velocity
OC 𝑉𝑐/𝑜 30 mm 𝑂𝐶 ∗ 𝜔 ∗ 𝑆𝐹 4713 𝒎𝒎
𝒔
CP 𝑉𝑝/𝑐 17.58mm 𝑂𝑃 ∗ 𝜔 ∗ 𝑆𝐹 2761.81 𝒎𝒎
𝒔
Slider P 𝑉𝑝/𝑎 28.17mm 𝑂𝑃 ∗ 𝜔 ∗ 𝑆𝐹 4426.13 𝒎𝒎
𝒔
𝑉𝑐/𝑜
𝑉𝑝/𝑐
𝑉𝑝/𝑎
𝑉𝑝/𝑐= PC*ω 𝑝/𝑐 = 2761.81 𝒎𝒎
𝒔 ω 𝑝/𝑐 = 2761.81/PC
ω 𝑝/𝑐 = 2761.81/600 ω 𝑝/𝑐 = 4.60 rad/sec
For uniform velocity of Crank

10. Step 8. locate Midpoint G of Link CP by bisection method
Kleins Construction
For uniform velocity of Crank

11. Step 9. Draw a circle consisting C is a center and CM is a radius
Kleins Construction
For uniform velocity of Crank

12. Step 10. Draw another circle having G is center and GC as A radius
Kleins Construction
For uniform velocity of Crank

13. Step 11. These two circles intersects at two points and mark theme as K and L
Kleins Construction
For uniform velocity of Crank

14. Step 12. Join points K and L and draw a faint line between these two points
Kleins Construction
For uniform velocity of Crank

15. Step 13. Line joining K and L intersects Connecting Rod CP at point Q and OP at point N.
Kleins Construction
For uniform velocity of Crank

16. Kleins Construction
Step 14. Draw line CQ, QN and ON.
For uniform velocity of Crank

17. Kleins Construction
Step 14. Quadrilateral OCQN represents Acceleration Diagram of given Mechanism.
For uniform velocity of Crank

18. Kleins Construction
Elements Significance
OC Centripetal Acceleration of Crank
CQ Centripetal Acceleration of Connecting Rod
QN Tangential Acceleration of Connecting Rod
CN Resultant Acceleration of Connecting Rod
ON Acceleration of Slider/ Piston P
Step 14. Quadrilateral OCQN represents Acceleration Diagram of given Mechanism.
For uniform velocity of Crank

19. Kleins Construction
Step 15. Acceleration Analysis.
Link Acceleration Vector Length Magnitude Acceleration
OC 𝑎 𝑐/𝑜
𝑐 OC 30 mm 𝑂𝐶 ∗ 𝜔2
∗ 𝑆𝐹 148082.46 𝒎𝒎
𝒔 𝟐
𝑎 𝑐/𝑜
𝑡 -- -- --
CP 𝑎 𝑝/𝑐
𝑐 CQ 2.57 mm 𝑂𝑄 ∗ 𝜔2
∗ 𝑆𝐹 12685.73 𝒎𝒎
𝒔 𝟐
𝑎 𝑝/𝑐
𝑡 QN 24.68 mm 𝑄𝑁 ∗ 𝜔2
∗ 𝑆𝐹 121861.99 𝒎𝒎
𝒔 𝟐
Slider P 𝑎 𝑝/𝑎 ON 14.7 mm 𝑂𝑁 ∗ 𝜔2
∗ 𝑆𝐹 77560.4 𝒎𝒎
𝒔 𝟐

20. Kleins Construction
Step 16. Rotate quadrilateral about point O in the direction of rotation of crank through 180° to
get actual acceleration diagram.
N
O
C
Q
For uniform velocity of Crank

21. Kleins Construction
For uniform velocity of Crank
Step 16. Simulation using GeoGebra Classic Software

22. Kleins Construction
For uniform velocity of Crank

23. Kleins Construction
For Non uniform velocity of Crank
The crank of a slider crank mechanism rotates clockwise at speed of 300 r.p.m with increasing the rate of
500rad/𝒔𝒆𝒄 𝟐. The crank is 150 mm and the connecting rod is 600 mm long.
Determine: 1. Linear velocity and acceleration of the Slider A and
2. Angular velocity and angular acceleration of the connecting rod, at a crank angle of 55° from inner dead
centre position.

24. The crank of a slider crank mechanism rotates clockwise at speed of 300 r.p.m with increasing the rate of
500rad/𝒔𝒆𝒄 𝟐
. The crank is 150 mm and the connecting rod is 600 mm long.
Determine: 1. Linear velocity and acceleration of the Slider A and
2. Angular velocity and angular acceleration of the connecting rod, at a crank angle of 55° from inner dead
centre position.
Given Data:
Crank Length r= 150 mm
Connecting Rod Length, l=600mm
Crank Angle θ=55°
Crank speed 𝑵 𝒄/𝒐=300 rpm
Crank anular accel.α=500rad/𝒔𝒆𝒄 𝟐
Link True Length Scaled Length (1:5)
OC 150mm 30mm
CP 600mm 120mm
55°
O
C
P
IDC
Configuration Diagram (Scale 1:5)
O
Find:
1. 𝑽 𝒂
2. 𝒂 𝒂
3. 𝝎 𝒑/𝒄
4. 𝜶 𝒑/𝒄

25. Step 1-12 remains same for Non uniform speed.
Procedure for Velocity diagram is same as of Uniform speed.
Kleins Construction
For Non uniform velocity of Crank

26. Step 13. Draw a line Perpendicular to OC through Point O
Kleins Construction
For Non uniform velocity of Crank

27. Step 14. Calculate Distance OO1
Kleins Construction
OO1=
𝑂𝐶∗𝛼
𝑆𝐹∗𝜔2
=
150∗500
5∗31.422
=15.19mm
For Non uniform velocity of Crank

28. Step 15. Draw a horizontal Line (along line of action of Piston ) through point O1.
For Non uniform velocity of Crank
Kleins Construction

29. Step 16. Draw a horizontal Line through point O1 which intersects line KL at point N1.
For Non uniform velocity of Crank
Kleins Construction

30. Step 17. New polygon O1CQN1 is the acceleration diagram of slider crank mechanisms where crank rotates with
varying speed.
For Non uniform velocity of Crank
Kleins Construction

31. Kleins Construction
Step 18. Acceleration Analysis
Elements Significance
OC Centripetal Acceleration of Crank
OO1 Tangential Acceleration of Crank
O1C Resultant Acceleration of Crank
CQ Centripetal Acceleration of Connecting Rod
QN1 Tangential Acceleration of Connecting Rod
CN1 Resultant Acceleration of Connecting Rod
O1N1 Acceleration of Slider/ Piston P
For Non uniform velocity of Crank

32. Kleins Construction
Step 19. Acceleration Analysis
Link Acceleration Vector Length Magnitude Acceleration
OC
𝐚 𝐜/𝐨
𝐜
OC 30 mm 𝐎𝐂 ∗ 𝛚 𝟐
∗ 𝐒𝐅 148082.46 𝐦𝐦
𝐬 𝟐
𝐚 𝐜/𝐨
𝐭
OO1 𝐎𝐂∗𝛂
𝐒𝐅∗𝛚 𝟐=15.19mm 𝐎𝐎𝟏 ∗ 𝛚 𝟐
∗ 𝐒𝐅 75000 𝐦𝐦
𝐬 𝟐
CP
𝐚 𝐩/𝐜
𝐜
CQ 2.57 mm 𝐎𝐐 ∗ 𝛚 𝟐
∗ 𝐒𝐅 12685.73 𝐦𝐦
𝐬 𝟐
𝐚 𝐩/𝐜
𝐭
QN1 15.63 mm 𝐐𝐍𝟏 ∗ 𝛚 𝟐
∗ 𝐒𝐅 77160.83 𝐦𝐦
𝐬 𝟐
Slider P 𝐚 𝐩/𝐚 O1N1 28.97 mm 𝐎𝐍𝟏 ∗ 𝛚 𝟐
∗ 𝐒𝐅 143027.91 𝐦𝐦
𝐬 𝟐
For Non uniform velocity of Crank

33. Kleins Construction
Step 20. Acceleration Analysis
For Non uniform velocity of Crank
Angular Acceleration
of Connecting Rod
𝛼 𝑝/𝑐
𝑎 𝑝
𝑐
𝑡
= . 𝛼 𝑝
𝑐
∗ 𝑃𝐶=77160.83 𝒎𝒎
𝒔 𝟐
𝑃𝐶 = 600𝑚𝑚
𝛼 𝑝/𝑐 = 128.6 𝑟𝑎𝑑/𝑠𝑒𝑐2
Acceleration of Piston
𝑎 𝑝/𝑎
𝑎 𝑝/𝑎=143027.91 𝒎𝒎
𝒔 𝟐
𝑎 𝑝/𝑎=143.027 𝒎
𝒔 𝟐

34. Kleins Construction
Step 21. Simulation using GeoGebra Classic Software
For Non uniform velocity of Crank

35. • Velocity and acceleration Analysis of Slider crank Mechanism
• By considering crank rotation with
1. Uniform speed
2. Varying Speed
• Simulation with GeoGebra Software.
Summery

36. •THANKYOU…
Prof. K NWakchaureTheory of Machines-I