Duality between Engineering Systems Since graph representations are

Duality between Engineering Systems • Since graph representations are mathematical entities, mathematical relations can be established between them, such as duality relations: (gi G 1)=Tdual(gj G 2) • Duality between graph representations yields the duality relations between the represented engineering systems. (si D 1)=T(gi G 1) (sj D 2)=T’(gi G 1) G 1 D 2 gi si (si D 1)=Tdual(sj D 2) T D G 2 gj T’ D 1 sj

Potential Graph Representation is a directed graph where: • Each vertex is associated a vector variable called ‘potential’ • Each edge is associated a vector called ‘potential difference’ equal to the vector difference between the potentials of the corresponding vertices. • The potential law: the vector sum of the potential differences of the edges forming a circuit in the potential graph is equal to zero.

Flow Graph Representation is a directed graph where: • Each edge is associated a vector called ‘flow’. • The sum of flows of the edges forming a cutset in the flow graph is equal to zero.

Duality between Potential and Flow Graph Representations. • For each potential graph gi we consider a flow graph g*i satisfying: – gi is dual to g*i. – the orientation of the potential difference of some edge of gi is set to be the same as that of the flow in its dual edge * in g*i. Potential Graph Flow Graph

Duality between Potential and Flow Graph Representations. • Since gi and g*i are dual: • The potential law for gi: • The flow law for g*i: • Thus potential differences of gi and the flows of g*i satisfy the same set of linear equations. • Then the potential difference of edge of gi is equal to the flow of its dual edge * in g*i.

Representing Mechanisms with the Potential Graph Representation Rules for representing a mechanism by the potential graph representation: • Each pinned joint of the mechanism is represented as a vertex in the graph. • Each link is represented by an edge whose end vertices represent the corresponding pinned joints. • The potential of the vertex is equal to the velocity of motion of the corresponding joint. • The potential difference of the edge is equal to the relative linear velocity of the corresponding link. • The potential law maps the physical law of structural integrity of the mechanism

Representing Structure with the Flow Graph Representation Rules for representing a structure by the flow graph representation: • Each joint of the structure is represented as a vertex in the graph. • Each rod is represented by an edge whose end vertices represent the corresponding joints. • The flow of the edge is equal to the force in the corresponding rod. • The flow law maps the physical law of static equilibrium of forces.

Due to the duality relation between the representations of the two systems… Kinematical Linkage Static Structure

…there is a duality relation between the engineering systems themselves 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