1. The correction to Grubler’s criterion for calculating the Degrees of Freedom of Mechanisms 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 Shai offer School of Mechanical Engineering Faculty of Engineering Tel-Aviv University Tel-Aviv,Israel

2. The outline of the talk - The extended Grubler’s equation - Self-stress (SS) -S-Self-stress in Assur Graph singularity  finite motion -N-Number of self-stresses can be found: 1.Assembly method. 2. Force equilibrium method.

3. Extended Grubler = Grubler + SS topology + SS geometry Continuous Instantaneous Self-stress – An assignment of scalars to the links satisfying the force equilibrium for each joint. Extended Grubler’s Equation Extended Grubler  the correct instantaneous mobility but NOT the global mobility

4. Examples: Grubler =0 Instantaneous geometrical SS =1 Extended Grubler =1 Grubler =0 Topological SS =1 Extended Grubler =1 Grubler =0 Continuous geometrical SS =1 Extended Grubler =1 Finite motion Infinitesimal motion

5. 1 2 3 4 5 6 7 8 (1  7)  (3  5)  (2  6) = 0 =(0, II) 1 2 3 4 5 6 7 8 (0,II) AB C D A B C D Extended Grubler = 0 Grubler = 0 Instantaneous Geometrical SS = 1 Extended Grubler = 0 + 1 = 1 Infinitesimal motion Mobility at the singular position of an Assur Graph – Tetrad Characterization done by equimomental lines and face force

6. 2 1 3 4 5 6 8 A B C 7 D 2 1 3 4 5 6 8 A B C 7 D Grubler =0 Continuous geometrical SS=1 Extended Grubler = 0 + 1 = 1 Finite motion Continuous singularity of the Tetrad- by using sliders

7. How can we find the number of Self-Stresses? 1. Assembly method. 2. Force equilibrium method.

8. 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.

9. Rigid body between two predetermined Stationary points = SS L L L L=∞ Link Singularity of a dyad Singularity of a Triad Continuous Singularity of a Triad

10. L L Self Stress in Tetrad due to Singularity  Infinitesimal and finite motions Infinitesimal motion Replacing with sliders  finite motion

11. L Replication of the singular Tetrad  finite floating mechanism (four self- stresses)

12. L Infinitesimal motion Replacing with sliders  finite motion Replication of the singular double triad  finite floating mechanism

13. Replication of the singular double Triad  finite floating mechanism

14. MM F F F MM F Force equilibrium method

15. First internal force Equilibrium of Forces. First self-stress (blue) Second internal force Second self- stress(red) Two self-stresses  Two infinitesimal motions Grubler = 0 Two Self-Stresses Extended Grubler = 0 + 2 = 2, two independent infinitesimal motions

16. 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).

17. Thank you!! For more information you are invited to write to: shai@eng.tau.ac.il