Relations Between Engineering Fields

Duality between engineering systems Since graph representations are mathematical entities, mathematical relations can be established between them, such as duality relations: (giG1)=Tdual(gj G2) G1 G2 Tdual gj gi

Duality between engineering systems Duality between graph representations yield the duality relations between the represented engineering systems. G1 G2 Db Da D T’ gj gi sj T si

Dual Representation Design Technique

Duality between Linkages and Structures Duality between Stewart platforms and Robots Duality between Planetary and Beam Systems

Duality relation between linkages and structures

Consider a kinematical linkage and its graph representation

Consider a kinematical linkage and its graph representation

Now, consider a static structure and its graph representation Kinematical Linkage Static Structure

There exist a mathematical relation between the representations of the two systems Kinematical Linkage Static Structure

Therefore there is the duality relation between the represented engineering systems Kinematical Linkage Static Structure

The relative velocity of each link of the linkage is equal to the internal force in the corresponding rod of the structure Kinematical Linkage Static Structure

The equilibrium of forces in the structure is thus equivalent to compatibility of relative velocities in the linkage Kinematical Linkage Static Structure

Checking system stability through the duality Stable ???? 8 5 9 2 4 7 10 11 1 12 6 3 11 7 3 4 12 2 1 5 8 9 10 6 Due to links 1 and 9 being located on the same line Definitely locked !!!!! 8 12 ’ 2 1 10 6 7 3 5 9 R’ 4 12 ’ 9 10 R’ 11 6 7 8 2 3 5 1 4

Duality relation between Stewart platform and serial robot

Consider a Stewart platform system. 1 2 3 4 5 6 P

BUILDING THE GRAPH REPRESNTATION OF THE STEWART PLATFORM 3 3 2 2 4 4 1 1 5 5 6 6

BUILDING THE DUAL GRAPH REPRESNTATION 1 2 3 4 5 6 P 1 P 2 3 4 6 5 P 3 2 4 1 5 6

BUILDING THE DUAL SPATIAL LINKAGE (SERIAL ROBOT) 1 2 3 4 5 6 P 1 P 2 3 4 5 6 1 2 3 4 5 6 P

BUILDING THE DUAL SPATIAL LINKAGE (SERIAL ROBOT) 1 2 3 4 5 6 P 1 P 2 3 4 5 6 1 2 3 4 5 6 P

VERIFYING THE DUALITY RELATION BETWEEN THE TWO SYSTEMS The correspondence in geometric positions of the kinematical pairs of the linkage and the legs of the Stewart platform. P

VERIFYING THE DUALITY RELATION BETWEEN THE TWO SYSTEMS Each force in the Stewart platform leg is directed in the same direction as the relative angular velocity of the corresponding kinematical pair in the dual serial robot. P

Duality relation between planetary and beam systems

Consider a simple two-gear system. B 1 2

BUILDING THE GRAPH REPRESNTATION OF THE GEAR SYSTEM A C B 1 2 2 C rC B 1 rB A rA

VERIFYING THE ISOMORPHISM OF THE REPRESENTATION A C B 1 2 2 C w1/0 rC B rA x w1/0 1 rB A rA

VERIFYING THE ISOMORPHISM OF THE REPRESENTATION A C B 1 2 2 C w1/0+w2/1 rC B rA x w1/0+rB x w2/1 1 rB A rA

VERIFYING THE ISOMORPHISM OF THE REPRESENTATION A C B 1 2 2 C w1/0+w2/1+w0/2=0 rC B rA x w1/0+rB x w2/1+rC x w0/2=0 1 rB A rA

VERIFYING THE ISOMORPHISM OF THE REPRESENTATION A C B 1 2 2 C w1/0+w2/1+w0/2=0 rC B rA x w1/0+rB x w2/1+rC x w0/2=0 1 rB A rA

VERIFYING THE ISOMORPHISM OF THE REPRESENTATION A C B 1 2 2 C w1/0+w2/1+w0/2=0 rC B rA x w1/0+rB x w2/1+rC x w0/2=0 1 rB A rA

DEDUCING BEHAVIORAL EQUATIONS FROM THE GRAPH KNOWLEDGE A C B 1 2 2 C DA+w2/1+w0/2=0 rC B rA x DA+rB x w2/1+rC x w0/2=0 1 rB A rA

DEDUCING BEHAVIORAL EQUATIONS FROM THE GRAPH KNOWLEDGE A C B 1 2 2 C DA+DB+w0/2=0 rC B rA x DA+rB x DB+rC x w0/2=0 1 rB A rA

DEDUCING BEHAVIORAL EQUATIONS FROM THE GRAPH KNOWLEDGE A C B 1 2 2 C DA+DB+DC=0 rC B rA x DA+rB x DB+rC x DC=0 1 rB A rA

DEDUCING BEHAVIORAL EQUATIONS FROM THE GRAPH KNOWLEDGE A A C B I B 1 2 2 C DA+DB+DC=0 rC B rA x DA+rB x DB+rC x DC=0 1 rB A rA

BUILDING THE DUAL GRAPH REPRESENTATION C C I A A A C B B B 1 2 2 C DA+DB+DC=0 rC B rA x DA+rB x DB+rC x DC=0 1 rB A rA

DEDUCING BEHAVIORAL EQUATIONS FROM THE DUAL REPRESENTATION A A C I B B 1 2 2 C FA+DB+DC=0 rC B rA x FA+rB x DB+rC x DC=0 1 rB A rA

DEDUCING BEHAVIORAL EQUATIONS FROM THE DUAL REPRESENTATION A A C I B B 1 2 2 C FA+FB+DC=0 rC B rA x FA+rB x FB+rC x DC=0 1 rB A rA

DEDUCING BEHAVIORAL EQUATIONS FROM THE DUAL REPRESENTATION A A C I B B 1 2 2 C FA+FB+FC=0 rC B rA x FA+rB x FB+rC x FC=0 1 rB A rA

DEDUCING BEHAVIORAL EQUATIONS FROM THE DUAL REPRESENTATION A A C I B B 1 2 2 C FA+FB+FC=0 rC B rA x FA+rB x FB+rC x FC=0 1 rB A rA

BUILDING THE DUAL BEAM SYSTEM C I A A C B B 1 2 2 C C rC FA+FB+FC=0 rC B B rA x FA+rB x FB+rC x FC=0 rB 1 rB A A rA rA

VERIFYING THE DUALITY RELATION BETWEEN THE TWO SYSTEMS C I A A C B B 1 2 2 C C rC PA+FB+FC=0 rC B B rA x PA+rB x FB+rC x FC=0 rB 1 rB A A rA rA

VERIFYING THE DUALITY RELATION BETWEEN THE TWO SYSTEMS C I A A C B B 1 2 2 C C rC PA+RB+FC=0 rC B B rA x PA+rB x RB+rC x FC=0 rB 1 rB A A rA rA

VERIFYING THE DUALITY RELATION BETWEEN THE TWO SYSTEMS C I A A C B B 1 2 2 C C rC PA+RB+RC=0 rC B B rA x PA+rB x RB+rC x RC=0 rB 1 rB A A rA rA

VERIFYING THE DUALITY RELATION BETWEEN THE TWO SYSTEMS C I A A C B B 1 2 2 C C rC PA+RB+RC=0 rC B B rA x PA+rB x RB+rC x RC=0 rB 1 rB A A rA rA