Mechanics and Machines Chapter 3 Courses

Kinematics Fundamental

1. 1 X.L.Wang
Topics to be Covered in the Course
ü1. Introduction to Mechanisms and Machines
ü2. Kinematics Fundamentals
3. Position and Displacement Analysis
4. Mechanism Design
5. Velocity Analysis
6. Acceleration Analysis
7. Cams: Design and Kinematic Analysis
8. Gears: Design and Kinematic Analysis
9. Dynamic Force Analysis

2. Chapter 3
Position and Displacement Analysis

3. 3 X.L.Wang
3. Position and Displacement Analysis
3.1 Introduction
3.2 Position analysis
3.3 Displacement analysis
3.4 Limiting position
3.5 Transmission angle

4. 4 X.L.Wang
1. Determine the position of all links in a mechanism as the
driver links are displaced.
2. Determine the limiting positions of a mechanism.
Objectives
§ Two types of methods
§ Graphical Analysis
§ Analytical Analysis

5. 5 X.L.Wang
nDefinition: to determine the position of all other links in a
mechanism as the driver link(s) is displaced.
- Configuration: the positions of all links
v Mobility=1
- Moving one link will
precisely position all
other links in the
mechanism.
What is position and displacement analysis?
n Given Point B’, find Point C’. New configuration is determined

6. 6 X.L.Wang
- The stresses in the materials must be kept well below
allowable levels.
Why position and displacement analysis?
n The design engineer must ensure that the proposed
mechanism will not fail under its operating conditions.
- To calculate the stresses, we need to know the static and
dynamic forces on the parts.
- To calculate the dynamic forces, we need to know the
accelerations.
???≤ ???
???=
???
???
F=??????

7. 7 X.L.Wang
Why position and displacement analysis?
- In order to calculate the accelerations, we must first find
the positions of all the links for each increment of input
motion.
- Then differentiate the position equations versus time to
find velocities, and then differentiate again to obtain the
expressions for acceleration.
???=
??????
??????
???=
??????
??????

8. 8 X.L.Wang
Displacement Analysis
1. Point B moves to B’
2. Point C must always lay on its constrained path
3. The length between B and C must stay constant
§ Graphical Analysis
C’
n Given Point B’, find Point C’.

9. 9 X.L.Wang
Displacement Analysis
C’’ can be ignored because the linkage must be disassembled and
reassembled to achieve this alternate configuration.
§ Graphical Analysis
C’’
C’

10. 10 X.L.Wang
Displacement Analysis
§ Graphical Analysis
-For crank-rockers mechanism, there are two regions of possible
motion which are termed assembly circuits.
- A mechanism is unable to move between assembly circuits
without being disassembled.
C’
C’’
By physically
disconnecting joint

11. 11 X.L.Wang
3.3 Displacement Analysis
§ Analytical Analysis
§ Law of sines
§ Law of cosines
§ Law of the sum of all interior angles in a general triangle

12. 12 X.L.Wang
3.3 Displacement Analysis
§ Analytical Analysis ( by laws of sines and cosines)
- Given the lengths (L1, L2 , L3 and L4) and a crank angle,
determine the interior joint angles.
§ Four-Bar Linkage

13. 13 X.L.Wang
3.3 Displacement Analysis
Given the lengths (L1, L2 and L3) and a crank angle, determine
the position of the slider and the interior joint angles.
§ Offset Slider-Crank
These equations can be used to determine the position of the links
in any mechanism configuration.
§ Analytical Analysis ( by laws of sines and cosines)

14. 14 X.L.Wang
Displacement Analysis
§ Analytical Analysis
Complex number notation of a position vector
( by position vector loop)
- Links are represented as position vectors

15. 15 X.L.Wang
Displacement Analysis
§ Analytical Analysis
Derivative
( by position vector loop)

16. 16 X.L.Wang
Displacement Analysis
§ Analytical Analysis
Position vector loop
n Given Point A, find Point B
( by position vector loop)
§ Four-Bar Linkage

17. 17 X.L.Wang
Displacement Analysis
§ Analytical Analysis
§ Four-Bar Linkage
( by position vector loop)

18. 18 X.L.Wang
Displacement Analysis
§ Analytical Analysis
§ Four-Bar Linkage
( by position vector loop)

19. 19 X.L.Wang
3.4 Limiting Position
Limiting position: the configuration of a mechanism that places
one of the follower links in an extreme location
position.

§ The crank and coupler becomes collinear.
§ Imbalance angle - the angle between the coupler
configuration at the two limiting positions

20. 20 X.L.Wang
A four-stroke engine
Ø Intake
Ø Compression
Ø Power
Ø Exhaust
3.4 Limiting Position
Limiting position: BDC ( bottom dead center)
TDC ( top dead center)
Piston stroke: is defined as its travel from TDC to BDC
or the reverse

21. 21 X.L.Wang
3.5 Transmission Angle

22. 22 X.L.Wang
3.5 Transmission Angle
§ Four-bar
§ Transmission angle - is measured between the coupler and
the output link.
§ Extreme values of the
transmission angle
§ The extreme transmission angle occurs when links 1 and 2
are colinear ( extended or overlapped).

23. 23 X.L.Wang
△ABD，△BCD









cos2
cos2
222
222
bccbl
addal
BD
BD
bc
addacb
2
cos2
cos
2222




min
o
,1cos0  ，
max
o
1cos180   ，－，
3.5 Transmission Angle
§ How to find the extreme values of γ

24. 24 X.L.Wang
3.5 Transmission Angle
§ Extreme values of the
transmission angle
§ Slider-Crank
§ Transmission angle - is measured between the coupler and
a line normal to the sliding direction.

25. 2023-3-20
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