Systematic conceptual engineering design using graph representations Research

Systematic conceptual engineering design using graph representations.

Research Objectives Development of Systematic design methods to facilitate conceptual engineering design using discrete mathematical models called combinatorial representations that are based on graph theory as a medium for knowledge transfer. • Design through Common Graph Representation. • Design through Dual Graph Representation. • Identification and usage of special properties obtained by graphs.

Problem solving with Graph Representations Different problems from different domains Not Really! All can be represented by a common bipartite graph Chessboard problem Tensegrity Satellite communications

Common Graph Representation Special Properties Tensegrity solved Chessboard problem solved Satellite problem solved Chessboard problem Tensegrity Satellite communications Solving one of the problems in its domain solves the analogous problems using the graph to transfer the solution. Special properties of the graph are reflected in the domains represented.

Design using Common Graph Representations It was found that the same type of graph representations, say G can be associated with more than one engineering domain, say D 1 and D 2. In this case, G can be used to transfer solution from D 1 to D 2 and vice-versa. Step 1: Defining engineering problem in original domain. • Function Definition – What it does. • Use of “Black Box” Function Definition (Pahl and Wallace, 1996) Alternating angular velocity drive V t Design Problem Rectified angular velocity output V t Original Design engineering Problem domain

Design using Common Graph Representations Step 2: Transforming problem to Graph Representation level. • Use of “common language” to describe system function. • Flow or Potential variables to describe system. Alternating Potential = input t Alternating angular velocity drive V t Design Problem Rectified angular velocity output Original V t Rectified Potential = outp CGR Design Common Graph Problem Representation engineering domain t

Design using Common Graph Representations Step 3: Locate a solution in another engineering domain. • Engineering domain must share common representation. • Flow or Potential variables translated to corresponding terminology of secondary engineering domain. Secondary engineering domain – Electrical engineering Electric circuit is found that rectifies an alternating Alternating Rectified voltage source: The Full Wave Potential = input Potential rectifier = output t Alternating voltage source V t Design Problem t CGR Common Graph Representation Original Rectified voltage engineering output domain Design Problem V t Secondary engineering domain

Design using Common Graph Representations Step 4: Transfer solution from engineering domain to Graph Representation level. • Each structure element in the engineering level is translated into it’s equivalent element representation in the graph through deterministic steps. • Graph topology insures proper representation of properties and system behavior. A 2 CGR Common Graph Representation 1 0 C 3 B 4 Original engineering domain Secondary engineering domain

Design using Common Graph Representations Step 5: Building new design at the engineering level using the graph solution. • Each element in the graph representation is represented at the engineering CGRdeterministic steps. level as an equivalent element through Common Graph Representation • Graph topology again insures that proper representation of properties and system behavior is transferred to engineering solution. This structural procedure on the graph representation ensures: • Each edge corresponds to an element in the mechanical system. Original Secondary engineering • Each vertex corresponds to a point in theengineering mechanical domain system where velocity is measured.

Design using Common Graph Representations Step 5: Building new design at the engineering level using the graph solution. C A AA 22 A 1 1 C A C CCC 00 3 4 B CGR Common Graph Representation Original engineering domain Secondary engineering domain

Design using Common Graph Representations Step 5: Building new design at the engineering level using the graph solution. • C elements both possess the same potential. A 2 1 CC 0 C 3 3 A C B C 44 B BB CGR Common Graph Representation Original engineering domain Secondary engineering domain

Design using Common Graph Representations Step 5: Building new design at the engineering level using the graph solution. A 2 A C B 1 C 0 3 A C 4 B B CGR Common Graph Representation Original engineering domain Secondary engineering domain

Linear to Angular Design Mechanical Design process can be made simpler by first designing linear systems and then converting to angular systems. • Potential ( ) can be represented as tangential velocity with edges possessing angular velocity. A A • Flow (F) can be represented as force acting around an 2 axis (Moment). 1 1 C 0 0 A 3 C B 4 B CGR Common Graph Representation Original engineering domain Secondary engineering domain

Linear to Angular Design A A 2 A A 1 A 1 C C 0 0 C B B 1 3 C B 3 3 3 B 4 B B CGR Common Graph Representation Original engineering domain Secondary engineering domain

Linear to Angular Design • Edge 2 subject C elements bothtopossess the same • potential Linear element replaced by angular element 2 0 2 C A A C 22 CC A A 1 1 0 C B C C 3 B C C C 00 C 4 C 3 44 BB 0 CGR Common Graph Representation 4 Original engineering domain Secondary engineering domain

Looking at the complete mechanical rectifier where the driving input gear is subject to direction change: 2 C A 2 C C Rotates Anti-clock wise. 0 A 0 B C B 4 C C Rotates Clock wise. 4

Design using Common Graph Representations The same systematic process resulted in design through knowledge transfer of another available solution from the electronic engineering domain. Full Wave Rectifier Graph A Original engineering domain A 2 1 CGR Common Graph C 0 Representation 3 B Diode Bridge Graph B CGR Common Graph Representation 0 C 4 Secondary engineering domain Original engineering domain Secondary engineering domain

Design using special properties of Graph Representations Self Duality 1’ II 1 4 4’ II 3’ 3 3’ 1’ III 6 6’ IV I 2’ 2 5 4’ 5’ I 2’ III 6’ IV 5’

Design using special properties of Graph Representations Self Duality Every cutset a dual circleinand vice-versa Potentials in has Graph = Flows Dual Graph 1’ II 1 4 3’ I II 3 5 2 III 6 4’ I 2’ III 6’ IV IV Potential Law: Flow Law: 5’

Design using special properties of Graph Representations Self Duality Cutset does not have a=dual circle and vice-versa Potentials in Graph Flows in Dual Graph 1’ II 3’ I 4 3 5 4’ I 2’ III 2 1 Potential Law: IV Flow Law: Flow Law Broken = Illegal duality operation 5’

Design using special properties of Graph Representations Two Engineering systems in the Engineering Domain are transformed to graphs in the Graph Domain. The Graph Domain reveals properties that were not discovered at the Engineering level. These special properties may be transferred back to the Engineering Domain where they reflect the special properties in the Graph Domain. Gl Dj Special s 2 properties s 1 T g 2 Special g 1 properties

Special Properties of Dual Graphs 2 types of “rectifier” graphs Graph 1: Diode Bridge Graph 2: Full Wave rectifier A A II II IV I B C I C 0 III 0 B Dual to itself Potential Source can be automatically exchanged for Flow Source = A Not Dual to itself I III Potential Source cannot be automatically exchanged for Flow 0 ≠ Source II Resulting Graph is Illegal IV B C

Special Properties of Dual Graphs Graph 1: Diode Bridge Graph 2: Full Wave rectifier Not Dual to itself A A B C 0 Dual Statically Valid B A A B C C B 0 0 Dual Statically Non-Valid

Special Properties of Dual Graphs Graph 1: Diode Bridge Graph 2: Full Wave rectifier Not Dual to itself A A B C 0 Dual Statically Valid B A A B C C B B 0 0 Dual Statically Non-Valid

Design domain of concepts • Each element in the graph representation is represented at the engineering level as an equivalent element through deterministic steps. • A graph element can be represented by different structures possessing the same behavior. Graph element Equivalent Engineering structure X C Y Behavior

Design domain of concepts A 21 C 1 5 0 4 5 63 5 4 0 A 2 5 1 3 4 0 5 C 3 B C C 1 4 B B 2 C 0 D 3 B 0 A

Design domain of concepts 2 A 2 1 5 4 1 C 6 A A 0 D 3 5 C 3 B B B Mechanisms taken from : Mechanisms and Mechanical Devices Sourcebook By : Nicholas P. Chironis 4