graphsLinear graph models                                                                                                                                                                                                                            intro Introduction to linear graphs                                                                             p.1
                                         graphs.intro Introduction to linear graphs
                                         1 Engineersoftenusegraphicaltechniquesto
                                                                                                                                                                                                                              linear graphs
                                         aid in analysis and design. We will use linear
                                                                                                                                                                                                                              topology
                                         graphstorepresentthetopologyorstructure
                                         of a system modeled as interconnected lumped
                                         elements.
                                         2 Thisrepresentstoustheessentialstructure
                                         of the system in a minimalist form. In this
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                                         Figure intro.1: a modern New York subway map in the style of Vignelli (Jake Berman).
              graphsLinear graph models                                                         intro Introduction to linear graphs   p.2
              way,itis like Massimo Vignelli’s famous
              1972 NewYorksubwaysystem“map,”which
              inspired widespread adoption of his style (see
                                1
                                                                           1. Vignelli was a brilliant Minimalist designer of many prodcucts,
              Figure intro.1). Besides minimalism, the key
                                                                           fromdishestoclothing, but he was most known for his graphic design.
              idea in Vignelli subway maps is that the details
                                                                           Great places to start studying Vignelli are the documentary Design is
                                                                           One(2012)andTheVignelliCanon.
              of the tunnels’ paths are irrelevant and, in fact,
              distracting to the person attempting to get from
              onestation to another.
              3 Inasimilarway,alineargraphrepresents
              the system in a minimalist style, with only two
              types of objects:
                                                                           edges
                 1. Asetofedges,eachofwhichrepresents
                     anenergyportassociatedwithasystem
                     element. Each edge is drawn as an
                     oriented line segment “            ”.
                                                                           nodes
                 2. Asetofnodes,eachofwhichrepresents
                     a point of interconnction among system
                     elements. Each node is drawn as a dot
                     “ ”.
                                                                                            edge
              4 Alledgesbeginandendatnodes. Thenodes                       V           V             V
                                                                             ref        1             2
              represent locations in the system where distinct                                F
                                                                                     node          node
              across-variable values may be measured.
              For example, wires that connect elements                     Figure intro.2: an edge with nodes. The across
                                                                           variable is V = V1 −V2.
              are actually nodes at which voltage may be
              measured. Putting an edge together with nodes,
              wehaveFigureintro.2.
              5 Itisimportanttonotethatlineargraphs
                                                                           nonlinearity
              canrepresent nonlinear system elements—the
              nameisareferencetothelinesused.
              6 Itiscommontochooseanodeofthegraph
                                                                           reference node
              as the reference node, to which all across-
              variables are referenced. Due to its similarity to
                                                                           ground
              the electronic ground, we often use these terms
              interchangeably.
              7 Figureintro.3showshowalineargraph
              canbeconstructedforasimpleRC-circuit.
              Notethatthewiresbecomenodes,the
              elements becomeedges,andthereference
              noderepresentsthecircuit ground. In a similar
              manner,wewillconstructlineargraphsof
              graphsLinear graph models                                                          sign Introduction to linear graphs   p.3
              circuits, mechanical translational systems, and
              mechanical rotational systems.
                                                           R                                     R
                                                  +
                                          V       −                       C
                                           s
                                                                                                           C
                                                                                      V
                                                                                        s
              Figure intro.3: an example of a linear graph representation of an RC-circuit.