1. Analytical and Applied Kinematics
Vito Moreno
(cell)
Office EB1, hours Thursdays 10:00 to 5:00

2. Kinematics Introduction
This course introduces a unified and analytical approach to two (2) and
three (3) dimensional kinematics and planar and spatial geometry and
constraint motion.
Applications to: mechanisms, robotics, biomechanics…
Some topics covered:
Coordinate transformation operators
Displacement operators
Motion invariants
Velocity and acceleration operators
Link and joint constraints
Analytical methods of mechanism synthesis and analysis
Geometric error modeling
Computational methods in kinematics and geometry

4. Kinematics Introduction
Reference texts: Kinematics and Mechanisms Design, Suh,C.H. and Radcliffe, C.W., John Wiley and Sons, Mechanism Design, Erdman, A.G., Sandor, G.N., Kota, S., Prentice Hall, 4th ed Mechanism and Dynamics of Machinery, Mabie, H.H., Reinholtz, C.F., John Wiley and Sons, 4th ed Theoretical Kinematics, Bottema, O., Roth, B., Dover Publications, Introduction to Theoretical Kinematics, McCarthy, J.M. The MIT Press, 1990.

5. Kinematics Introduction
Basic definitions: Kinematics is part of Solid Mechanics Statics – study of forces and moments apart from motion Kinetics – study of the action of forces and moments on the motion of bodies Dynamics Kinematics – Study of the relative motion apart from forces

6. Kinematics Introduction
Mechanism – combination of several rigid bodies which are connected is such a way that relative motion between them is allowed. Function of a mechanism – to transmit or transform motion from one rigid body to another (source to output). Types of mechanisms- Gear systems Cam systems Planar and spatial linkages

7. Kinematics Introduction
Gears
Cams
Planar and spatial linkages

8. Kinematics Introduction
Machine – a mechanism or combination of mechanisms for
the purpose of transferring force or motion.
Motion
Plane (2D) motion – translation, rotation
Spatial (3D) Motion
Helical – pitch rotation and translation
Spherical – all points at a fixed distance from a given point
Cylindrical – free rotation and translation along an axis

9. Kinematics Introduction
The link – a solid (rigid) body which is connected to n other links
Linkage – links connected by joints
Erdman and Sandor, Table 1.1 Planar Link Types

10. Kinematics Introduction
Joints
Kinematic Pair (joint) = connection between two links which allows
certain relative motion
Lower pair – relative motion described by single (1) coordinate
e.g. revolute, prismatic, rolling pairs
Higher pair – more than one degree of freedom
roll/slip, spherical ball and socket
Kinematic Chain – a set of links connected by joints

11. Kinematics Introduction
Suh & Radcliffe
Fig 1.1 Kinematic Pairs
Pg 4

12. Kinematics Introduction
Degree of Freedom –no of independent parameters
(input coordinates) to completely
define the position of a rigid body
2D – 3 dof, 3D – 6 dof Y
Unconstrained rigid link
Three independent variables X

13. Kinematics Introduction
Before joining, multiple links will have 3n DOF Y X

14. Kinematics Introduction
Connections between links result in loss of DOF Y
ground X
Pin joints loose 2 DOF, have only 1 DOF called f1 joint
Degree of Constraint = number of freedoms a free body looses after
it is connected to a fixed link
DOC+DOF=3

15. Erdman and Sandor, Table 1.2 Dimensional Joints

16. Erdman and Sandor, Table 1.2 Dimensional Joints

17. Kinematics Introduction
Four Bar Linkage Notation
Mobility analysis by Gruebler’s equation
Coupler
link 3 4 2
Input
link
Follower
link
fixed 1
fixed 1
Four Bar Linkage is a Single DOF system -1 input coordinate required to define
position of all members

18. Kinematics Introduction
Sliding connection reduces DOF also
Coupler
link 2 3
Input
link Y
fixed
Output
link 1 4 X
Slider-Crank Linkage Notation 1
fixed

19. Kinematics Introduction
F1=15 (12+3)
F=3(12-1)-2(15)=+3
At Q, 3 links,2 joints

20. Kinematics Introduction
Velocity equivalent linkage
(Kinematic diagram)
More complicated linkage 8
(roll/slide)
DOF= 3(n-1)-2f1-1f2
n=7
f1=7
f2=1
F=3(7-1)-2(7)-1(1)=+3
DOF= 3(n-1)-2f1-1f2
n=10
f1=12
f2=0
F=3(10-1)-2(12)-0(1)=+3

21. Kinematics Introduction
Exceptions to Grueblers equation
n=5
f1=6
DOF 3(5-1)-2(6)=0
But motion is allowed
3rd link is redundant
Mfg errors could cause binding
E&S Fig 1.26
Overconstrained linkage
n=3
f1=3
DOF 3(3-1)-2(3)=0
But motion is allowed
Sum of radii = dist between pivot points
E&S Fig 1.27

22. Kinematics Introduction
Exceptions to Grueblers equation
Passive or redundant DOF
Rotation of 4 does not affect arm 3
n=4
f1=3, f2=1roll/slide
DOF 3(4-1)-2(3)-1(1)=+2
Welded roller to arm(3)
n=3, f1=2, f2=1
DOF=3(3-1)-2(2)-1(1)=+1
Slipping prevented between roller and cam
n=4, f1=4, f2=0
DOF=3(4-1)-2(4)=+1
E&S Fig 1.26
Overconstrained linkage
E&S Fig 1.28

23. Kinematics Introduction
6 Bar Linkages
Tracer Points
(binary links)
Ternary links
Watt Linkage
Stephenson Linkage
E&S Fig 1.13 a-d

24. Kinematics Introduction

25. Kinematics Introduction Changing the fixed Link - Inversion
Basic Slider -Crank
Whitworth Quick Return Link
input
input
output
output
input
Oscillating Cylinder Pump
Pump Mechanism
output
input
output

26. Kinematics Introduction – Force and Transmission of Motion
n-n transmission of force
and motion

27. Kinematics Introduction – Force and Transmission of Motion
For maximum mechanical advantage

28. Kinematics Introduction – Homework #1
Determine the DOF for the
Mechanisms shown 2 1 3

29. Kinematics Introduction – Homework #14 5

30. Kinematics Introduction – Homework #16 7 8 9

31. 10

32. Kinematics Introduction – Homework #111
Honda