2.1 Degree of Freedom (DOF) or
Mobility
MENG 371 Notes, Chapter 2
Dr. Keith Hekman
September 9, 2003
2.2 Types of Motion
• Pure Rotation – the body possesses on point
(center of rotation) that has no motion with respect
to the “stationary” frame of reference. All other
points move in circular arcs
• Pure Translation – all points on the body describe
parallel (curvilinear or rectilinear) paths.
• Complex motion – a simultaneous combination of
rotation and translation
• DOF – number of independent parameters
(measurements) that are needed to uniquely define
its position in space at any instant of time.
• Rigid body in a plane has 3 DOF. (x,y,θ)
• Rigid body in space has 6 DOF (3 translation, 3
rotation)
2.3 Links, joints, and kinematic chains
• Links – building blocks
• Node – attachment points
– Binary link – two nodes
– Ternary link – three nodes
– Quaternary link – four nodes
• Joint – connection between two or more links (at
their nodes) which allows motion
– Classified by type of contact, number of DOF, type of
physical closure, or number of links joined
Kinematic chains, mechanisms,
machines, link classification
Joint Classification
• Type of contact - line, point,
surface
• Number of DOF – full
joint=1DOF, half joint=2DOF
(removes half as many DOF)
• Form closed (closed by
geometry) or Force closed
(needs an external force to keep
it closed)
• Joint order = number of links-1
•
•
•
•
–
–
–
–
2.4 Determining Degree of Freedom
• For simple mechanisms calculating DOF is simple
Open Mechanism
DOF=3
Kinematic chains –links joined together for motion
Mechanisms – grounded kinematic chain
Machines – mechanism designed to do work
Link classification
Closed Mechanism
DOF=1
Ground – fixed w.r.t. reference frame
Crank – pivoted to ground, makes complete revolution
Rocker – pivoted to ground, has oscillatory motion
Coupler - link has complex motion, not attached to
ground
2.4 Determining Degree of Freedom
• Gruebler’s equation for planar mechanisms
M=3L-2J-3G
• Where
–
–
–
–
M=degree of freedom or mobility
L=number of links
J=number of joints (half joints count as 0.5)
G=number of grounded links =1
M=3(L-1)-2J
2.4 Determining Degree of Freedom
2.4 Determining DOF (Examples)
M=3(L-1)-2J1-J2
• Kutzbach’s equation for planar mechanisms
M=3(L-1)-2J1-J2
• Where
–
–
–
–
M=degree of freedom or mobility
L=number of links
J1=number of full joints
J2=number of half joints
Open Mechanism
Closed Mechanism
• For Spatial Mechanisms
M=6(L-1) -5J1-4J2-3J3-2J4-J5
2.4 Determining DOF (Examples)
2.4 Determining DOF (Examples)
M=3(L-1)-2J1-J2
M=3(L-1)-2J1-J2
2.5 Mechanisms and Structures
• Number Synthesis – determination of the number
and order of links and joints necessary to produce
motion of a particular DOF
• Book gives details
• Mechanism – DOF>0
• Structure – DOF=0
• Preloaded Structure –
DOF<0, may require
force to assemble
2.7 Paradoxes
• Greubler criterion does not
include geometry, so it can
give wrong prediction
• Usually when things are
the same
Gears
2.6 Number Synthesis
E-quintet
Total Links
Binary
Ternary
Quaternary
Pentagonal
Hexagonal
4
4
0
0
0
0
6
4
2
0
0
0
6
5
0
1
0
0
8
7
0
0
0
1
8
4
4
0
0
0
8
5
2
1
0
0
8
6
0
2
0
0
8
6
1
0
1
0
2.8 Isomers
• Greek for having equal parts
• Refers to valid ways to assemble different types of
links
• Only one valid fourbar isomer
• Two valid sixbar isomers
• Third one fails DOF test, as the DOF is not
distributed over the linkage.
Fourbar Isomer
• Only way to construct a fourbar isomer is to have
one binary link next to another binary link.
Stephenson’s Sixbar Isomer
• One way to construct a sixbar isomer is to have the
two ternary links separated.
Watt’s Sixbar Isomer
• One way to construct a sixbar isomer is to have the
two ternary links attached.
Invalid Sixbar Isomer
• This is an invalid isomer as the DOF is not
distributed through the mechanism
This is a structure
Effective link
2.9 Linkage Transformation
• A slider can be replaced by a link of infinite length
Geneva Mechanism
2.10 Intermittent Motion
• Series of Motions and Dwells
• Dwell – no output motion with input motion
• Examples: Geneva Mechanism, Linear Geneva
Mechanism, Ratchet and Pawl
Linear Geneva Mechanism
Ratchet and Pawl
2.11 Inversion
• Created by
grounding a
different link in a
kinematic chain
• Different behavior
for different
inversions
3 Stephenson 6-bar inversions
2 Watt’s 6-bar inversions
2.12 Grashof Condition
• Fourbar linkage is simplest linkage with 1DOF
• Grashof condition predicts behavior of linkage
based only on links length
– S=length of shortest link
– L=length of longest link
– P,Q=length of other two links
For case of S+L<P+Q
• Ground link adjacent to shortest => crank-rocker
• Ground shortest link => double crank
• Ground link opposite shortest link – Grashof double
rocker with shortest link capable of making a
complete rotation
• If S+L P+Q the linkage is Grashof with at least one
link capable of making a complete rotation
• Otherwise the linkage is non-Grashof with no link
capable of making a complete rotation relative to
ground
For the case of S+L>P+Q
• All inversions will be double rockers
For the case of S+L=P+Q
• Book says all inversions will be double cranks or
crank rockers (true if S=P,L=Q)
• Indeterminate point when links are aligned
(change points)
Parallelogram form
Deltoid form
Anti parallelogram
form
Barker’s Complete Classification
Type
s+l vs
p+q
Inversion
1
<
L1=s=ground
I-1
2
<
L2=s=input
I-2
3
<
L3=s=coupler
I-3
4
<
L4=s=output
5
>
L1=l=ground
6
>
L2=l= input
II-2
Class 2 rocker-rocker-rocker
RRR2
Triple-rocker
7
>
L3=l= coupler
II-3
Class 3 rocker-rocker-rocker
RRR3
Triple-rocker
8
>
L4=l= output
II-4
Class 4 rocker-rocker-rocker
RRR4
Triple-rocker
9
=
L1=s=ground
III-1
Change point crank-crank-crank
SCCC
SC double-crank
10
=
L2=s=input
III-2
Change point crank-rocker-rocker
SCRR
SC crank-rocker
11
=
L3=s=coupler
III-3
Change point rocker-crank-rocker
SRCR
SC double-rocker
12
=
L4=s=output
III-4
Change point rocker-rocker-crank
SRRC
SC rocker-crank
Class
Barker’s Designation
Code
Also Known as
Grashof crank-crank-crank
GCCC
double-crank
Grashof crank-rocker-rocker
GCRR
crank-rocker
Grashof rocker-crank-rocker
GRCR
double-rocker
I-4
Grashof rocker-rocker-crank
GRRC
rocker-crank
II-1
Class 1 rocker-rocker-rocker
RRR1
Triple-rocker
13
=
Two equal pairs
III-5
Double change point
S2X
Parallelogram or
deltoid
14
=
L1=L2=L3=L4
III-6
Triple change point
S3X
Square
2.14 Springs as links
• Springs remove a degree of freedom (1 more
equation)
• Examples: desk arm lamp, garage door
2.13 Linkages of more than 4 bars
5-bar 2DOF
Geared 5-bar 1DOF
•Provides for more complex motion
•Watt’s sixbar – 2 fourbar linkages in series
•Stephenson’s sixbar – 2 fourbar linkages in parallel
2.15 Compliant Mechanisms
• Compliant “link” capable of significant deflection
acts like a joint
• Also called a “living hinge”
• Advantage: simplicity, no assembly, little friction
2.16 Micro Electro-Mechanical
Systems (MEMS)
• Micromachines range in size from few micrometers
to a few millimeters
• Shape is made on large scale, then photographically
reduced on wafer and etched.
• Can make
compliant
mechanisms
in MEMS
2.17 Practical Considerations
2.18 Motors and Drivers
• Read on your own