Face force – a multidimensional generalization of mesh currents

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

A New View on Engineering Systems through the Engineering Knowledge Genome: Structural Genes of Systems Topologies

General View on Systems Topologies Over Constrained Systems Under Constrained Systems Well constrained Systems Compositions of s-genes Gene 1 Gene 2 Gene 3

Definition of s-genes G is an s-gene IFF it is a well constrained graph and removing any set of vertices does not result in an s-gene. Examples s-genes in the following disciplines: Trusses Linkages Sketches These entities are genes because they are the building blocks of these and many other systems in different disciplines

Definition of s-genes in trusses A truss is an –gene IFF it is rigid determinate truss but does not contain any substructure that is a determinate truss A B D C P1 P2 P3 1 3 4 5 6 7 8 2 A B D P1 P2 P3 3 4 6 7 8 A B D C P1 P2 P3 1 3 4 5 6 8 2 s-gene mechanism mechanism

Removing vertex C result in Example of a system that is not an s-gene: A B D C P1 P2 P3 A B D P1 P2 P3 C Removing vertex C result in an s-gene Sub system that is an s-gene The first was not an s-gene

S-genes in well-constrained sketches B 1 2 3  L1 C A B C L1 1 3 2  L3 The sketch (geometric constraint) problem. The corresponding graph is a floating s-gene. L3 A B 1 2 3  L1 C The s-gene after grounding edge (2,C).

The process of constructing all the s-gene (in 2D, for now). (The process is complete and sound)

Ground edge is replaced by a triangle and two new ground edges. Basic s-gene – the dyad. A A1 A2 Ground edge is replaced by a triangle and two new ground edges. Fundamental Extension x y z t Dividing edge (x,y) to two by adding an inner vertex z and connecting z to an arbitrary vertex z, can be the ground. Regular extension

Example of creation s-genes B C E D O1 O2 O3 O4 F O5 (a) (b) (c) (b1) (b2) (c1) (c2) (A,O2) (B,O3) (A,B) (A,D) (A,E) The dyad.

The Map of all the genes in 2D

The Decomposition of Well-Constrained Systems into S-Genes B C D A A,B,C,D G E F E,F,G I H J O3 H,I,J K K O1 O2 The Decomposition Graph (obtained computationally by pebble game) Well Constrained Truss

Employing the Decomposition into S-Genes for Analysis 2* number of inner vertices = number of edges in the s-gene. I E F G H J K O3 O1 O2 1 2 3 4 7 5 6 8 2*4=8 J I K A B E C O1 O3 D F G H O2 The Determinate Truss I P6 H J K O3 O1 O2 PG P3 1 2 3 4 5 6 P8 P1 2*3=6 J K O3 O1 O2 1 2 3 4 5 6 2*3=6 P1 P6 P5 1 2 P6 2*1=2

Employing the s-genes for constructing well constrained sketches B 1 2 3  L1 C A B C L1 1 3 2  L3 The sketch problem The corresponding floating s-gene L3 A B 1 2 3  L1 C Grounding edge (2,C) L3 A B 1 2 3  L1 C Grounding edge (A,3) A,1,3,B 1 B C 2 The composition graph The composition graph

The process of constructing the geometric object by composing its s-genes 1 A 3 (b) (A,3)  L3 A B 1 2 3  L1 C A 3 (a) (A,3) A B 3 L3  C L1 1 (d) (A,3) 1 A (A,3) B 3 L3  (c) A B 3 L3  C L1 1 2 (e) 1 (A,3) B C 2

S-genes in mechanisms Strangely! S-genes in mechanisms are exactly the same as s-genes in trusses with the addition of a driving link

Deriving mechanisms’ S-genes from trusses’ s-genes B D C P1 P2 P3 1 3 4 5 6 7 8 2 (a) A Truss s-gene A B D C P1 P2 1 3 4 5 6 7 8 2 A B D C P2 P3 1 3 4 5 6 7 8 2 A B D C P1 P3 1 3 4 5 6 7 8 2 P3 P1 P2

A schematic graph of the mechanism A mechanism A schematic graph of the mechanism 1 2 3 4 5 6 7 8 9 10 11 A B C D E F J G H I A B C D E J G F H I 2 3 4 5 10 9 8 6 7 11

The decomposition of the schematic graph into s-genes B C D J 2 3 4 5 G H I 8 6 7 11 10 F 9 Tetrad Triad Diad

Decomposition of the mechanism Tetrad Triad Diad Velocities of inner joints are known A C 3 B A B C D J 2 3 4 5 E 4 1 E G H I 8 6 7 11 2 F 10 9 E 5 J D 10 F 6 G 9 7 8 H I 11

Employing s-genes for synthesis