Proceedings of the ASME 2011 International Design Engineering Technical Conferences &
Computers and Information in Engineering
IDETC/CIE 2011
August 28-31, 2011, Washington, DC, USA
The correction to Grubler’s criterion for calculating the
Degrees of Freedom of Mechanisms
Shai offer
School of Mechanical Engineering
Faculty of Engineering
Tel-Aviv University
Tel-Aviv ,Israel
The outline of the talk
- The extended Grubler’s equation
- Self-stress (SS)
- Self-stress in Assur Graph singularity 
finite motion
- Number of self-stresses can be found:
1.Assembly method.
2. Force equilibrium method.
Extended Grubler’s Equation
Extended Grubler = Grubler + SStopology + SSgeometry
Instantaneous
Continuous
Self-stress – An assignment of scalars to the links
satisfying the force equilibrium for each joint.
Extended Grubler  the correct instantaneous mobility but
NOT the global mobility
Examples:
C
6
7
4
2
A
B
C
1
5
A
2
4
3
D
8
2
A
B
6
5
C
6
5
1
)a(
4
3
1
B
)b(
)c(
Grubler =0
Grubler =0
Grubler =0
Instantaneous geometrical SS =1
Topological SS =1
Continuous geometrical SS =1
Extended Grubler =1
Extended Grubler =1
Extended Grubler =1
Infinitesimal motion
Finite motion
Finite motion
Mobility at the singular position of an Assur Graph –
Tetrad
1
A
8
Extended Grubler = 0
4
6
D
7
3
B
2
C
5
1
A
2
B 3
(0,II)
4
8
7
(1  7)  (3  5)  (2  6) = 0
=(0, II)
D
6
C
5
Grubler = 0
Instantaneous Geometrical SS = 1
Characterization done by
equimomental lines and face force
Extended Grubler = 0 + 1 = 1
Infinitesimal motion
Continuous singularity of the Tetrad- by using sliders
A
1
Grubler =0
Continuous geometrical SS=1
C
5
2
Extended Grubler = 0 + 1 = 1
8
6
4
B
3
D
7
A
1
C
5
2
8
6
4
B
3
D
7
𝒀𝟐𝑨 + 𝑿𝟐𝑫 = 𝑳𝟐𝟐 − 𝑿𝟐𝑩 + 𝑳𝟐𝟔 − 𝑳𝟐𝟒 − 𝑿𝟐𝑩
= 𝒄𝒐𝒏𝒔𝒕
How can we find the
number of
Self-Stresses?
1. Assembly method.
2. Force equilibrium method.
Assembly Method
When you have to insert a link between two joints
so that their location may not be changed (stationary points)
 there is a self-stress.
Rigid body between two predetermined Stationary points = SS
L
Link
L
Singularity
of a dyad
L
L=∞
Singularity
of a Triad
Continuous
Singularity of a
Triad
Self Stress in Tetrad due to Singularity  Infinitesimal and
finite motions
Infinitesimal
motion
Infinitesimal
motion
L
Replacing with sliders
 finite motion
Replication of the singular Tetrad  finite floating mechanism (four selfstresses)
Replication of the singular double triad  finite floating mechanism
Infinitesimal motion
L
Infinitesimal motion
Replacing with sliders  finite
motion
Replication of the singular double Triad finite floating mechanism
Force equilibrium method
M≠=0
FF =0
M=≠0
F =0
Two self-stresses  Two infinitesimal motions
Second internal
force
First internal
Second self-of
Equilibrium
First
self-stress
force Forces.
stress(red)
(blue)
Grubler = 0
Two Self-Stresses
Extended Grubler = 0 + 2 = 2, two independent
infinitesimal motions
Future Work/Research
1. The proofs rely on the properties of rigidity matrix.
2. In 3d the correction of Grubler’s equation is also
Self-Stresses.
3. The relation between instantaneous mobility and
global mobility through the existent Self-Stresses.
4. Computer program that will “invent” floating
mechanisms of replicating Assur Graphs at their
singular positions, (also in 3d).
Thank you!!
For more information you are invited to write to:
shai@eng.tau.ac.il