JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 1, pp. 139-162, Warsaw 2006 GEARS AND GRAPHS Józef Wojnarowski Jerzy Kopeć Stanisław Zawiślak Faculty of Mechanical Engineering and Computer Science, University of Bielsko-Biala, Poland e-mail: szawislak@ath.bielsko.pl The paper presents, among others, a survey of works connected with the problem of themodeling of gears bymeans of versatile graph theory models. This approach to the problems of gearmodeling allows compu- ter based analysis and synthesis. Some recent papers claim that graph representations and derived methods belong to the branch of artificial intelligence due to the possibility of obtaining automatically versatile re- sults, e.g. different constructional design solutions of mechanisms. Some examples are enclosed to explain which class of tasks is solved bymeans of the graphtheoryapproach, i.e.modelingbymeansof bondgraphsand linear graphs. The following problems have been considered: derivation of systems of equations describing the behaviour of gear subsystems and detection of a redundant wheel. The survey is based on over 60 papers published mainly within last 10 years, some of them in world-wide high level scientific magazines. Key words: modeling, analysis, design automation, artificial intelligence 1. Introduction The graph theory is a branch of mathematics which has versatile appli- cations to many fields of engineering, which was summarized in well-known monographs (Deo, 1974;Wojnarowski, 1977a,Murota, 1987). Since the edition of these books, many new results have been achieved, especially in the field of the modeling of mechanisms. The aim of this paper is to present a selected range of applications of gra- phs for themodeling of gears. Recently, this area of application of graphs has been extensively developed all over the world. The term ”graphs”, used in the 140 J.Wojnarowski et al. title of the paper, covers a wide range of objects, i.e. graphs and hypergraphs considered by the graph theory (i.e. linear graphs, digraphs,weighted graphs), as well as bond graphs. The term ”gears” is connected with different types of gears: car gear boxes, planetary gears, mixed geared trains as well as parts of machines enclosing geared pairs or geared mechanisms (e.g. geared robot wrists). In fact, as has been just mentioned, bond graphs have strong theore- tical connections with graphs in the sense of a pure graph theory, but for their application purpose there is not any need to analyze them. Nevertheless, this direction of investigations is developed as well (Ort and Martens, 1974). The methods described underneath are based not only on graphs themselves, but also on different algebraic objects assigned to the graphs: matrices, structural numbers, vector spaces and systems of equations. In general, the idea of application of graphs to the modeling of technical objects consists in: automation of analysis, generation of all possible construc- tional forms or solutions, synthesis of particular mechanical systems, finding particular parts inside technical systems (i.e. a redundant wheel), optimiza- tion of particular engineering problems, e.g. the shortest path, the maximum flow or transmission system containing a gear. This approach is recognized as a subarea of the artificial intelligence which should be added to its standard elements like pattern recognition, machine translation or expert systems. This idea arose from the proposal of Shai and Preiss (1999) and it is supported by the present paper. In the opinion of the authors of the present work, the di- rection of investigation concerning the usage of artificial intelligence methods is worth carrying on and propagating Shai’s statement that methods based upon graphs are just such ones. Another scientist who independently started works in this field is Schmidt et al. (2000) who called the methodology developed by her team as ”graph grammars”. It consists in algorithmization of engineering tasks in a special meta-language (quasi-language). A simple graph G (Bondy andMurty, 1977; Deo, 1974) is a pair (V,E), where V – a set of vertices, E – a set of edges and E is a subset of the cartesian product of the set V . A weighted graph is a triple (V,E,W), where W is a weight function having the domain E and a particular set of values, e.g. R – set of real numbers. The weight can be a real number, e.g. a distance in [km] whenwemodel a road network, but it can be also a vector of two or more components. This will be illustrated in the following sections. A hypergraph H is a pair (V,HE) where HE is a set of hyperedges i.e. the family of subsets of V .Abondgraphconstitutes the idea of representing relationships between different physical parameters of the system in form of a pictogram or an icon coupled with adequate algebraic formulas Gears and graphs 141 and joining rules. The linear graphs and bond graphs are matched with other algebraic objects to create powerful models of engineering systems based on the rules of assignment ”system-graph” and derived intelligent methodology concerningdifferent problems listed underneath.Thegraphs canbe considered as models of such different technical objects like e.g.: (a) vibrating systems (Wojnarowski, 1977a), (b) trusses (Shai and Preiss, 1999a,b), (c) mechanisms (ChenandYao, 2000) (d) gears (Wojnarowski, 1977b;Choi andBryant, 2002), (e) robot wrists (Lin, 1990), (f) networks of particular media (e.g. water, gas, railway, electric current, etc.) (Deo, 1974). The problems considered bymeans of these modeling are as follows: • analysis of vibrating systems (determination of eigen-requencies) • finding the redundant gear wheel, velocity calculations, dynamical ana- lysis • enumeration of all possible constructional forms of a gear • optimization of a particular problem (layout, ratios, weight of the gear- box) • synthesis of a system with particular properties, e.g. degree of freedom, transmission function. The question arises how it can be possible to model such different objects and consider such different problems by means of the same class of objects? It is especially characteristic for Polish scientists because interest in this field of knowledge is relatively narrow, just opposite to the situation in western countries, America and The Far East. The answer to such a formulated qu- estion consists in modeling process, i.e. what is assigned to a vertex, what is recognized as an edge or hyper-edge, what is a set of weights or how is a bond graph built as well as how to assign more advanced algebraic structures as matrices andmatrix-based systems of equations or generalized networks. Mo- reover, themystery is veiled in thesematrix formulas and systems of quations which can be automatically created in dedicated computer programs where a graph (representing a mechanical system) is the initial data. These standard packages are widely used, e.g. MODELICA, 20-Sim, SIMULINKor others. In general,methods derived from themodeling of graphs are algorithmizable and programmable, which is an essential benefit arising from the effort of learning and applying these graphs. As has been previously said, different mechanical systems can bemodeled bymeans of graphs, but in this paper we focus our attention on some class of gears. 142 J.Wojnarowski et al. In the paper by Summers et al. (2001), the authors distinguished two pur- poses of modeling a system: (a) representing the structure of the system and (b) representing the behavior of the system. It would be best to have one model for these two purposes but, till now, not enough attempts have been made in this context. Both attitudes can be performed by means of graphs: the structure bymeans of simple graphs and behavior bymeans of bond gra- phs. A more detailed explanation of what is understood under these terms is presented afterwords in the present paper. Besides the explanation of the essence of the modeling process, the goal of this paper is also to provide a wide review of recent papers on these topics published in well known scientific magazines and on a query throughout the internet and private communications obtained via snail-mail from authors or even via personal meetings during conferences. It gives an impression of the state-of-art of this field nowadays. Usually, authors discuss usage of linear gra- phs and bondgraphs separately, this paper tries to set both these approaches together. The paper encloses also some exemplary considerations as well as a list of approx. 60 references. The reader – whomight be then eager tomake a detailed study of the described papers – can easily find vast areas for further investigations, which is also the aim of this paper. 2. Graphs as models of gears Methods of assignment of graphs to mechanical systems are different de- pending on a task which to be realized and a solution to be found. A compa- rison of these methods referring to gears was made by Prahasto (1992). The author had taken into account six methods proposed by Freudenstein, Ravi- sankar, Olson, Hsu,Wojnarowski and himself – basing on his supervisor’s (i.e. Prof. Andrews’) achievements. In this paper, we present methods based on graphs (structure analysis) and based on bondgraphs (dynamic analysis). It is worth noting that both presented approaches were generalized to a synthesis task aswell, but it is not the subject of the included examples. In thepaperbySummers et al. (2001), a definition of amodel is formulated as follows: ”Models are simplified abstract constructs used to predict beha- vior of a system and to get a quantitative understanding of their operation, improve their performance by variation of parameters, Gears and graphs 143 and find possible critical points before the actual system is built. Models can be catalogued as functional or structural”. It is almost a perfect definition. Aiming at an analysis of a structure, models should represent a systempreferably in variousways, which is achieved via graph-basedmodels. What does artificial intelligence of a graph-based system representation consist in? These models generalize structures of every system, they are as- signed or represent the flow of energy, power, rotational speeds, etc. Hence, the analysis can be performed automatically. Moreover, it is possible to so- lve the inverse problem, i.e. create a family of objects satisfying particular requirements (by some authors the term atlas of solutions” is used). Theassignment of a graph to amechanical system is amodelingprocedure. According to the definitionmentioned above, it is a simplified object, but its advantage is that some useful calculations can be made by means of a model which is adequate to the introductory mechanical system. In the following considerations we focus our attention on the modeling of gears. An exemplary gear is presented schematically in Fig.1a. In some consi- derations, we can abstract from themethod of supporting a system, therefore, in Fig.1b, the support is removed from the gear scheme. Fig. 1. Gear schemes; (a) general, (b) simplified Somemethodsof graphassignmentarepresented in the followingdiagrams: • corresponding to the bearing system (Fig.2a): G1 – displacement, Fig.2a (Hsu and Lin, 1994) 144 J.Wojnarowski et al. Fig. 2. Graphs assigned to a gear: (a) displacement, (b) displacement simplified, (c) conventional labelled, (d) canonical, (e) conventional, (f) geared kinematic chain Gears and graphs 145 G2 – canonical, Fig.2d (Chatterjee and Tsai, 1996) G3 – conventional, Fig.2e (Chen and Yao, 2000) G4 – geared kinematic chain, Fig.2f (Tsai, 2000) • corresponding to the system without constraints (Fig.2b): G5 – displacement, Fig.2b (Hsu and Lin, 1994) G6 – conventional labeled or coincident-joint graph, Fig.2c (Olson et al., 1991). The elements of the gear are represented by vertices of graphs G1-G3,G5, G6, except for case G4, where the elements are represented by polygons or edges. In the last mentioned case, the meshings (of engaged elements, geared pairs) aremarkedbyblackdots andturningpairsby single circles, respectively. The double concentric circles denote the common axis of several rotational elements fixed on common supports (bearings). The last rule is the same for all the graphs considered here. Relationships between gear elements are coded by means of graph edges. In the case of almost all the graphs, except for G4, the geared pairs (in mesh) are represented by dashed lines, and other relationships between the elements by continuous lines which are labeled or are notmarked byweights. If they are labeled, the assigned descriptionsmean the common axis, a−a, see Fig.1a. Furthermore, an edge can represent the so-called turning pair in the case when one of the elements is mounted on another one. Then the notation of the adequate rotational axis is considered as its label (marked on the general scheme), i.e. b or c, respectively. In the case of graphs G1 and G5, the coaxial elements are drawnasvertices of a grayfilled polygon. The idea of distinguishing the elements of a gear is realized in the case of graph G2 bymeans of the so-called levels: ground (support, bearings), first and second level. Graph G5 can be obtained from graph G6 by replacing the subgraph generated by vertices {1,3,4,6}, i.e. clique K4 and replacing it by a respectable polygon. Rules and derived methods of the analysis of gears are described in detail in the cited references. In the present paper, a methodology based on graph G5 is considered. The methodology of gear synthesis depicted in Fig.3 has been developed for different representations, but it is beyond the range of this paper and can be studied from the references listed at the end. The above mentioned graphs are used to analyse the layout of gears, ho- wever dynamics can be analyzed byPrahasto’s graphs (Prahasto, 1992),Woj- narowski’s graphs (Wojnarowski, 1977b) and bondgraphs. The usage of graphs and bondgraphs of selected gears will be presented in Section 4. 146 J.Wojnarowski et al. Fig. 3. Graph basedmethodology of the modeling of mechanical systems 3. Selected gear problems solved by means of graphs Problems which are considered in references have been mentioned briefly in Section 1. In this paper, some descriptions of contents of the references will be given. One of the highly exploited topics is the analysis of gears and geared mechanisms. This problem was considered in papers described Gears and graphs 147 hereafter. In Freudenstein and Yang (1972), a kinematic and static analysis of a spur-gear is presented. The authors present the rules on how to assign a graph to a gear and next how to perform the analysis on an exemplary Glover’s train. The graph creation rules are also described in Shai’s papers (Shai, 1999a,b). Another approach is presented in Lei and Lu [25], where the authors omitted thephase of graphrepresentation assigningamatrix to a gear. They derived a system of equations for kinematic and dynamic analysis of the gear. Topological analysis of gears can also be performed complying with the concept of kinematic fractionation (Liu and Chen, 2000) and kinematic units (Liu and Chen, 2001). Adifferent graph-basedmethodof gear analysiswaspresented in the thesis by Prahasto (1992). The author used the vector-network method propagated Andrews (Rogers and Andrews, 1975). The description of the procedure for analyzing planetary gear trains using the vector-network method is presented also in Prahasto and Andrews (1999). A special branch in the field of gear analysis is the enumeration of all possi- ble design solutions. The solution to this problem allows one, in consequence, to formulate synthesis process rules. The paper dedicated to this approach, see Castillo (2002), focused on the enumeration of 1-dof planetary gear train graphs based on functional constraints. Another problem is the optimization of gears or whole machine sub- systems including a gear. In Vetadzokowvska (1999), the author presents an algorithm for synthesis of an optimal 6-stage-gear for assumed transmission ratios for each stage. Graphs obtained by the author and diagrams of gears are included. Another approach to the optimization of gears is presented in Michelena and Papalambros (1995, 1997) – hypergraphs are used for the de- composition of an optimization task of gear trains. The next problem mentioned here is the synthesis of gears and geared mechanisms. Articulated gear mechanisms were synthesized in Chen and Liu (1999). The authors used hierarchical decomposition schemes. They created a database of so-called design primitives and obtained admissible mechani- sms. The design primitives have their graph representations. Furthermore, all possible graph representations can be efficiently enumerated.Two 3-dof exam- ples are presented for the sake of illustration. Anothermethod of enumeration of epicyclic gear trains is described in Shina and Krishnamurty (1993) whe- re the so-called standard code technique was used. An efficient methodology for structural synthesis of geared kinematic chains is presented in Hsu and Hsu (1997). The whole catalog of 1-dof geared kinematic chains (or rather their graph representations) is enclosed. Some fundamentals for thismethodo- 148 J.Wojnarowski et al. logy were presented by one of the authors in Hsu (1994), where he described mathematical gear models and a special structural code. The idea of the frac- tionation of gears (Liu and Chen, 2000) was used for topological synthesis of fractionated geared differential mechanisms (Chen and Yao, 2000). The con- siderations are based upon the 2-dof automotive gear differential. Theoretical grounds for the synthesis of gears are presented in Prasad Raju Pathapati and Rao (2002), where a new technique based on loops for the investigation of displacement isomorphism in planetary gear trains is described. The me- thodology is based on a graph representation, loop concept and the Hamming number concept. The 6-link PGT was considered. A detailed study of the synthesis of a bevel-gear-type robotic wrist mechanism bymeans of graphs is presented inLin (1990). The author created an atlas ofwristmechanisms for a particular task. Gears are also analyzed bymeans ofbondgraphs (Orlikowski, 2003; Choi and Bryant, 2002; Wojnarowski and Kopeć, 2003; Cichy, 2001). The last of the listed publications is a book (i.e. lecture notes), where methods of the transmission system modeling are described. Gears are just subsystems of such devices. In Choi and Bryant (2002), the bond graph method was co- upled with a finite element model of shafts of a gear, which is an unique approach. Orlikowski (2003) analyzes amore complex transmission system in- cluding geared pairs. In general, it can be stated that it is not easy to find a publication concerning strict application of bond graphs to the gear mode- ling. InWojnarowski andKopeć (2003), the authors intensively developed this topic. The considerations presented above can be summarized by the scheme presented in Fig.3. After a vast exploration of problems via generalization and abstraction, gear synthesis methods have been developed. The advantage of the synthesis methodology is comprehensively and fully proved in Castillo (2002), Lin (1990), Olson et al. (1987). 4. Exemplary problems In the present section, some selected methods are roughly described to give an impression what are the approaches to the gears modeling by means of graphs. The making use of these chosen methods is realised on different objects tounderline theversatile possibilities andthepowerfulnessof thegraph basedmethodology. Aswasmentioned previously, the essence of everymethod consists in the graph assigning rules. Three problems will be considered: Gears and graphs 149 • calculations of a gear ratio (see Section 4.1) • detection of redundant wheel (see Sectio 4.2) • determination of eigen-vibration of gears by means of bond graphs (see Section 4.3). 4.1. Representation of kinematic gear structure by means of graphs The goal of this section is to present amethod ofmodeling planetary gears bymeans of graphs and analyzing the existence of redundant gear wheels. Planetary gears have many advantages, e.g. they are compact, light and permit an essential reduction of rotational speed; therefore they are frequently used in vehicles, machining tools and heavy-duty machines. The graph the- ory is a useful tool for structural analysis and synthesis of kinematic chains containing gear wheels (Tsai, 2000). A redundant gear wheel in a planetary gear is such a wheel which can be removed without changing kinematic characteristics of the gear. The re- cognition of redundant gear wheels is important for making proper schemes of power transmission in automatic gear boxes. It has also a fundamental meaning for the automatic synthesis of gears. The theory has a further appli- cation which is not the subject of this paper, i.e. it allows one to enumerate gears having similar structure schemes, the so-called atlases of all possible so- lutions. This permit one to choose the most suitable solution and can help a designer in his conceptual work on a particular project, see e.g. Olson et al. (1991). In Hsu and Lin (1994), the method of assigning a graph to a gear has beenproposed.The graphwas called the displacement or kinematic graph of a gear.Detection of redundantgearwheels requires specialmatrix representation associated with this graph. Methods of assigning a kinematic graph to an exemplary gear, building a matrix (which is not used here) and searching for the redundant wheel are briefly presented below. Other graph representations of gears are also known. A layout of shafts and wheels of a planetary gear is usually presented by means of a scheme like shown in Fig.1. Due to the extent of the present work, only some elements of the graph-basedmethodology are enclosed. A full description can be found in the references listed at the end. The number of degrees of freedom can be calculated by means of Grubbler’s formula F =3(N −1)−2Po−Pz (4.1) Pz =N−F −1 150 J.Wojnarowski et al. where F – number of the degree of freedom N – number of elements Po – number of rotational pairs Pz – number of meshed pairs. Elements of the gear are labeled by numbers 0 – 6, furthermore positions of axes are marked by letters a, b and c. The kinematic chain, in which the support system is neglected, is presented in Fig.2b. In our case, the values of parameters are as follows: N = 7, Pz = 4 and therefore F = 2. One-dof planetary gear trains are considered in Olson et al. (1991), and the atlases of graphs assigned to 1-, 2- and 3-dof gears are listed in Tsai (2000). Based onHsu andLin (1994), the kinematic chain can be transformed into a displacement graph according to the rules roughly described and graphical- ly illustrated in Fig.2a. The displacement graph for a particular kinematic chain can be used for the determination of the equation of motion automati- cally (Freudenstein andYang, 1972). Themethodology is based upon thewell known idea of fundamental cycles of a graph. The sketch of the routine can be formulated as follows. A fundamental cycle (the so-called f-cycle), which corresponds to two en- gaged wheels i and j together with the coupled carrier k, consists of three vertices, one meshing edge i− j and two turning edges or/and edges of a po- lygon i−k and j−k. The vertex k, which is not incident to themeshing edge i− j, is a coupled transition vertex. For the sake of simplicity, the f-cycle is marked as (i,j)k. In this code, the first two numbers determine two meshing wheels (planet and geared wheel, respectively). The third one is the trans- mission/displacement vertex representing a carrier. The number of f-cycles is equal to the number of meshing edges in the kinematic chain. It can be se- en that there are four f-cycles: (5,3)6, (5,4)6, (2,1)3 and (2,4)3. The basic equation connected with the f-cycle (i,j)k describing relationships between angular velocities is as follows (Tsai, 2000) ωi−ωj =∓Nji(ωj −ωk) (4.2) where Nji = Zj/Zi and Zj, Zi – numbers of teeth on the geared wheels i and j, the sign (∓) depends on the type of engagement; it is positive if a positive rotation of the gear wheel i relative to the carrier k causes a positive rotation of the gear wheel j, otherwise it is negative. Gears and graphs 151 For the analyzed gear presented in Fig.1b, the following system of equ- ations can be written ω5−ω3 =+N35(ω3−ω6) ω5−ω4 =−N45(ω4−ω6) ω2−ω3 =+N12(ω1−ω3) ω2−ω3 =−N42(ω4−ω3) (4.3) Assuming the following numbers of teeth on consecutive geared wheels: Z1 =72,Z2 =16,Z3 =58,Z4 =44,Z′4 =32,Z5 =18, the transmission ratio R can be calculated. Additionally, we assume that element 6 is stationary (fixed, ω6 =0), input element is 1 and output 3. Therefore, the transmission ratio R is equal to R= ω1 ω3 (4.4) Solving the system of equations (4.3), a solution can be found in the following form R=1+ N42 N12 ( 1+ N35 N45 ) (4.5) The needed values are as follows: N42 = 2.75, N12 = 4.5, N35 = 3.22, N45 = 1.77. Finally, for the assumed data, the transmission ratio is equal to R≈ 2.72. The algorithm can be performed in the following steps: assignment of the graph to the system, automatic derivation of the kinematicmatrix, automatic generation of f-cycles, creation of equations (4.3), introduction of assumptions for the gear (e.g. the selected element is fixed). Further calculations can also be performed taking into account angular velocities of the carriers, etc. 4.2. Detection of the redundant wheel InHsuandLin (1994) analgorithmwasderived,whichenableddetection of the redundant gear wheel (in the case of assumed input and output elements). The algorithm consists in the analysis of base cycles in the displacement ki- nematic) graph. A sketch of the method is presented below in consecutive steps: (S1) creation of the displacement graph for the analyzed gear (S2) derivation of all f-cycles (S3) assumption of the input and output elements (S4) creation of sets of f-cycles 152 J.Wojnarowski et al. (S5) for every set consisting of 2 up to Pz −1 cycles, derivation of the asso- ciated set of vertices (S6) checking the criterion for the existence of redundantwheels: m=F+K where: m – number of unknowns, K – number of necessary equations needed for the determination of rotational speeds in the base cycles, F – number of degrees of freedom (dof), (S7) listing the sets of vertices forwhich the criterion is satisfied and checking which wheel is redundant in all the sets. Table 1.Analysis of existence of a redundant wheel K f-cycles Vertex set Condition m=F +K 2 (5,3)6; (5,4)6 {3,4,5,6} = (5,3)6; (2,1)3 {1,2,3,5,6} > (5,3)6; (2,4)6 {2,3,4,5,6} > (5,4)6; (2,1)3 {1,2,3,4,5,6} > (5,4)6; (2,4)3 {2,3,4,5,6} > (2,1)3; (2,4)3 {1,2,3,4} = The algorithm can be applied to the considered exemplary gear train (Fig.1b). The results of analysis for K = 2 are depicted in Table l. There are four f-cycles listed above, because there are Pz = 4 geared pairs. There- fore, two and three element subsets are considered in the routine. One can extract the following stages of analysis: 1. theoretical assumption of possible input and output elements in the con- sidered gear e.g. {1,3,4} 2. creaating of F +1 (F =dof) combinations of the mentioned elements, F =2 therefore F +1=3, and the set is {1,3,4} 3. making a 2-element set of cycles as listed in Table 1 4. checking condition (S6) and drawing conclusions. In our case: F =2,K =2, and m is equal to the cardinality of the vertex set. For the first row in Table 1 we have m = 4, therefore the condition is fulfilled. Analyzing possible inputs or outputs {1,3,4} for the set {1,2,3,4} (i.e. input 1, output 4), elements (5),6 are redundant. Analyzing possible inputs or outputs {3,4,6} for the set {3,4,5,6}, it follows that the elements (2),1 are redundant. The planetary gear wheel is highlighted by parentheses. Gears and graphs 153 The conclusion is that the gear has a redundant gear wheel depending on the choice of the input, output and ground. The above mentioned considera- tions indicate capabilities of the graph-based methodology for gear analysis. The most fruitful is the enumeration of all possible solutions – the so-called atlases of designs, which can be read in Tsai (2000) andwill be the subject of further investigations of the authors of this paper. 4.3. Analysis of gear dynamics with bondgraphs Achievement of design reliability and elimination of failures is easier since a variety of commercial CAD and structural analysis tools became available. However, on the preliminary stage of the design process (the conceptual one), a flexible, compact and ultimate method is needed. One of the valuable tools is the bondgraphmethod. Bondgraps describe the power flow in discrete or lump systems, and are cold power flow graphs. The bondgraphswere invented by Paynter, and deve- loped by Karnopp et al. (2000) and Thoma (1990). In this paper, torsional vibration is considered. As a unified pair of varia- bles, the angular velocity (flowvariable) and torsionalmoment (effort variable) are selected. The product of these variables gives the power P = ef =Mω (4.6) Physical properties of the modeled part, namely inertias and compliances as well as resistances, are regarded in the model. These properties are associa- tedwith standardbondgraph elements using commonprocedures (Karnopp et al., 2000; Thoma, 1990): inertias are described with I elements, compliances with C elements, energy dissipationwith R elements, dynamic and kinematic excitations with Se and Sf elements, respectively. Transformers TF descri- be the corresponding gear ratios. Table 2 contains collected set of standard bondgraph elements. The lines with half-arrows represent the power flow in the system. The goal is to write down equations from the graph (manually or automatical- ly) and skip differentiation. Comparing the linear graph theory, arcs are cold bonds and vertex elements, respectively. One can obtain the equation of mo- tion automatically by using a computer software. Figure 4shows theway from the existing element or its drawing (Fig.4a) to the dynamical model in the form of a bondgraph, the simplest model (see Fig.4b) with no compliance, then extended dynamical models with the compliance of the connecting shaft and friction losses regarded (Fig.4c), and finally with the wheel compliance (Fig.4d). 154 J.Wojnarowski et al. Table 2.Basic elements of bondgraphs Element Symbol Equation Effort source Se e= is known Flow source Sf f = is known Inertial element I :m f = 1 m ∫ e dt+f0 Compliance C : c e= 1 c ∫ f dt+e0 Resistance R : r e= rf 1-Nod 1 f = idem, ∑ ei =0 0-Nod 0 e= idem, ∑ fi =0 Transformer TF :n e1 =ne2, f2 =nf1 Gyrator GY : r e1 = rf2, e2 = rf1 Fig. 4. Development of a bondgraphmodel of a gear part: from a ”rough” graph to an enhanced model (causality analysis marks are not added to the graphs) Gears and graphs 155 The next stage is derivation of equations from the graph. The causality analysis procedure (see Karnopp et al., 2000) is required. It creates a proper order for equations and permits automatic generation of solutions. Short per- pendicular lines andnumbersunder thebondsdenote the causality assignment order and equation form (details inKarnopp et al., 2000). The set of equations has a different form depending on the causality analysis. Figures 5a,b,c show possible causality forms and Fig.5d shows a case when a proper denoting graph (causality assignment) is impossible. Fig. 5. Bondgraphs for a simple model with different causalities The full set of equations has the form                          f1 = is known f5 = f3 e2 = is known e3 =−e4−e5 f3 = 1 n1 f1 e4 = J4 df4 dt e1 = 1 n1 e3 f2 =n2f5 f4 = f3 e5 =n2e2 (4.7) hence            f1 = is known e2 = is known e4 = J4 1 n1 df1 dt (4.8) 156 J.Wojnarowski et al. The arranged sets of equations for bondgraphs from Fig.5b and 5c have the form            e1 = is known f2 = is known e4 = J4 1 n2 df2 dt            e1 = is known e2 = is known df4 dt = J4(−n1e1−n2e2) (4.9) It is impossible to express a consistent system of equations for the graph from Fig.5d. This system of equations is contradictory                    f1 = is known f2 = is known e4 = J4 1 n1 df1 dt and e4 = J4 1 n2 df2 dt f4 = f3 = 1 n1 f1 and f4 = f5 = 1 n2 f2 (4.10) Figure 6 shows bondgraphs of compliant models after performing the cau- sality analysis. Fig. 6. A bondgraphmodel for compliant models with causality analysis marks The equations of dynamics for themodel shown inFig.4b andFig.6a have the form                              f1 = is known e2 = is known e4 = J41 1 n1 df1 dt e9 = 1 c4 ∫ ( − 1 n1 f1−f11 ) dt df11 dt = 1 J42 [e9−r42(f11−f15)−n2e2] (4.11) Gears and graphs 157 The subscripts correspond to the number of bonds. The equations of dy- namics of the model shown in Fig.4d and Fig.6b have the form                                                          f1 = is known e2 = is known e4 = 1 c41 ∫ ( − 1 n1 f1+f7 ) dt df7 dt = 1 J41 [−e4+e11−r41(f7−f10)] f10 = is known e11 = 1 c43 ∫ (−f7−f14) dt df14 dt = 1 J42 [e11−n2e2−r42(f14−f17)] f17 = is known f18 = c42n2 de2 dt (4.12) 5. Final remarks and conclusions Graphs can be recognized as versatile, concise and powerful models of gears.Theassignment of a graph toaparticular gear is only thefirst stepof the procedure. The assignment of further algebraic objects is an immanent phase of all described methods. Depending on the procedure of modeling, different tasks can be performed. One can mention here: analysis, synthesis, finding a particular part, optimization or generation of all possible design solutions. Only some of them is roughly described in this work. However, a wide range of publications on this topic is briefly presented. In all cases, the graph-based methodology allows automatization of a particular task and its realization by means of a computer routine. Graph-based methods can be coupled with FEMand evolutionary algorithms, whichmeans – together with all previously mentioned features – that the graph-basedmethodology can be recognized as an artificial intelligencemethod.Thesemethods are intensively developed just recently, and enjoy great interest confirmed by numerous investigations which are carried on. During a design andanalysis process, an engineer requires a tool for valida- tion of dynamic properties of a createdmachine.Abondgraphmodel is easy to 158 J.Wojnarowski et al. build andmodify, regarding or neglecting selected dynamical properties in the model, drawing or erasing adjacent bonds and elements. Some authors (Shai andPreiss, 1999a) claimed that the graphmethod can be recognized as a new branch of strong artificial intelligence (Kasperski, 2003). The paper supports this point of view due to the fact that the graph-basedmethods consist in the algorithmization of particular tasks. References 1. 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Załączono kilka przykładów dla wyjaśnienia, jakie klasy zadań można rozwiązywać za pomocą metod wywodzących się z teorii grafów; tj. modelowanie za pomocą grafów przepływu mocy i grafów liniowych. Omówiono następujące zagad- nienia: automatyczne tworzenie układów równań opisujących dynamikę podzespołów przekładni i wykrywanie kół nadmiarowych. Przegląd został oparty o prawie 60 prac, opublikowanych główniew okresie ostatnich 10. lat; część z nich pochodzi z uznanych czasopism specjalistycznych o ogólnoświatowym zasięgu. Manuscript received February 9, 2005; accepted for print September 28, 2005