New entity in statics – “Face Force”

Developing new engineering concept through graph

representations

Kinematical Linkage

Consider a kinematical linkage and its graph representation

Kinematical Linkage Static Structure

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

In classical machine theory there is no known variable corresponding to the absolute velocity

of a joint in the linkage

?

Kinematical Linkage Static Structure

?

In search for this variable in statical structureswe shall make a detour to …

the works of Maxwell

Maxwell Diagrampoints in the diagram are associated with the

faces of the structure

II

VI

IIIV

I

IV

O

Maxwell Diagramlines in the diagram are associated with the

rods of the structure

II

VI

IIIV

I

IV

O

DualityIn our approach we proved that the diagram can be

considered as velocity diagram of a linkage

II

IIIIV

O

V

IVI

O1O2O3O4

DualityIn our approach we proved that the diagram can be

considered as velocity diagram of a linkage

DualityIn our approach we proved that the diagram can be

considered as velocity diagram of a linkage

Therefore Maxwell diagram describes both the forces in the structure and the velocities

in the mechanism

II

VI

IIIV

I

IV

O

The length of the segment in the diagram corresponds to the force in the rod of the

structure

II

VI

IIIV

I

IV

O

Or to the relative linear velocity of the link in the dual mechanism

II

VI

IIIV

I

IV

O

The coordinates of the points in the diagram correspond to the absolute velocities of

mechanism joints

II

VI

IIIV

I

IV

O

But there is no known variable to which it corresponds in the structure.

II

VI

IIIV

I

IV

O

Face force – a multidimensional generalization of mesh currents

It was proved that face forces manifest some properties of electric potentials.

pathrodFF ii

rodface i