Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 2, pp. 415-433, Warsaw 2010 LINEAR-GRAPH AND CONTOUR-GRAPH-BASED MODELS OF PLANETARY GEARS Józef Drewniak Stanisław Zawiślak University of Bielsko-Biala, Department of Mechanical Engineering Fundamentals, Bielsko-Biała, Poland; e-mail: jdrewniak@ath.bielsko.pl; szawislak@ath.bielsko.pl Analysis and synthesis of mechanisms are basic engineering tasks. They can suffer fromerrors due to versatile reasons.The graph-basedmethods of analysis and synthesis of planetary gears can be alternative methods for accomplishing of the aforementioned tasks, which additionally allow for checking of their correctness. In the paper, two graph-basedmethods of analysis of planetarygears arediscussed.Anexemplaryplanetarygear is analysed bymeans of the graph-basedmethods aswell as the traditio- nalWillismethod. Force and torque analyseswereperformedaswell.An algorithmic approach–which implies from the graphmodels – allows for checking of versatile variants of designs in an easy and schematic way, which can lead to optimisation of the design within a conceptual phase of the design procedure. Keywords: contour equationmethod,Hsu’s graph, f-cycle equation, gear ratio variants 1. Introduction Analysis and synthesis of mechanisms can be performed by means of versa- tile methods. These tasks can suffer from human errors. So, it is reasonable to have some alternative methods which allow for comparison of results and for detection of almost unavoidable mistakes. The graph-based methods de- liver such alternative approaches for modelling of a wide class of mechanical systems. Two of them are considered in the present paper. Therewere also someother attempts tomodelplanetarygears viadiagrams e.g. Wolf’s pictograms (Müller andWilk, 1996) or PKP-schemata and nume- rical codes (Ivančenko et al., 1974), but these methods did not further evolve due to lack of generalization and lack of connections with other branches of 416 J. Drewniak, S. Zawiślak mathematics.Also themethodbased on signal flowgraph theory formodelling of gears (Bonnell and Hess, 1968; Wojnarowski and Lidwin, 1975; Uematsu, 1997) has not been too frequently used. On the contrary, the graph basedme- thods have been independently, intensively developed for several recent years all over the world, see e.g. (Uyguroğlu and Demirel, 2005; Wojnarowski et al., 2006). Besides the papers, there are some monographs where the graph methods were described in details and illustrated by means of representative examples. The linear graph-based methodology of modelling of mechanisms was extensively described in Tsai (2001). The word ”linear” will be usually omitted in the rest of the present paper for simplicity. The contour based approach was discussed in Marghitu and Crocker (2001) as well as Marghitu (2005). The so-called contour graph is a symmetrical digraph without loops. The paperWojnarowski et al. (2006) includes a review of the graph-basedme- thods for gears and relevant references (58 items) almost totally different than these cited here, including topics connected with bond-graphs. Especially, the lastmentioned work encloses a short list of Prof.Wojnarowski’s achievements dedicated to this topic. He and Prof. Arczewski pioneered the graph-based modelling of mechanical systems in the 70s of the previous century in Poland. The present paper focuses on the graph-based analysis of planetary gears (Za- wiślak, 2007, 2008) but the graph relatedmodelling belongs to a wider family of the algebraic structurebased approaches (Shai, 2001)which allow especially for classification of mechanisms (Davies, 1968), conceptual design (Zawiślak, 2006), enumeration of structures (Tuttle et al., 1989) and synthesis ofmechani- cal systems (Schmidt et al., 2000) aswell as determination of angular velocities of elements of multi-body systems (Arczewski and Dul, 1995). Powerfulness of the graph-based modelling of mechanical systems consists in the fact that a graph is inseparably connected with other algebraic structures like e.g. sub- graphs (e.g. trees, cycles, cliques, paths), dual graph, matrices, vector spaces of cuts and cycles, structural numbers and matroids as well as algorithms connected with these structures. By interpreting themechanical knowledge in terms of graphs, we solve mechanical problems via graph models (Shai and Preiss, 1999). The graphs applied by Shai had other rules of assignment than these discussed in the present paper. Versatile linear graph models were considered by Tsai (2001) but Hsu’s graphmodels of gears (Hsu, 2002) are utilized in the following considerations. In the case of analysis of car gearboxes, the graph-based methodology is also efficient using additionally transformations of a basic graph according to a particular gear drive (Zawiślak et al., 2008; Zawiślak, 2008). An introductory comparison of the graph and contour methods for analysis of gears was done by the authors in Drewniak and Zawiślak (2010). Linear-graph and contour-fraph-based models... 417 The application of the considered methods for analysis of an exemplary planetary gear is given underneath. This is a coupled gear (in German: Kop- pelgetriebe). The considered gear has an internal closed loopwhat causes that there is also a circle (loop) made of the stripped edges in the graph represen- tation of the gear. Such structures were claimed (Tsai, 2001) as impossible to analyse via the graph method but the presented considerations, which lead to compatible results for all three discussed methods, denied this statement. The considered gear is suitable for designing a gearbox as an introductory layout where several inputs and outputs are available. If clutches and brakes are added, then some elements can be fixed and a respectable angular velocity is equal to zero. Force and torque analyses were here performed in a traditional way. But this stage of analysis can also be done via the graph-based approach, which is especially useful for mechanisms (Marghitu, 2005). Dynamics of gears can be studied via the graph approach uponAndrews’smethodology (Andrews et al., 1997). The general idea of the graph-basedmodelling of mechanical systems con- sists in the following steps: • discretisation of amechanical system. Thismeans that appropriate sim- plifications have to bemade. Some aspects are omitted. Some structural elements are considered as essential and they are interpreted as graph vertices. Some connections or relationships between these elements are abstracted. They are represented via graph edges, usually some system of weights can be assumed to the edges or vertices and edges, • assignment of the graph to the mechanism (especially planetary gear) based upon special rules. There are several different rules depending on the object ofmodellingandproblems solved via the graphbasedmethod, • derivation of special subgraphs, e.g. f-cycles or contours. These subgra- phs canbe singled outbasedupon the graph-theoretical algorithmswhat causes that the approach is simple and algorithmic, • listing the codes of these graph elements. The encoding rules are clearly defined what allows for avoidance of mistakes, • generation of a system of equations in an algorithmic way using the codes. These codes allow for management (assignment) of the indica- tors of variables existed in the considered equations in a straightforward manner, • solution of the obtained system in a chosen algebraic way to obtain ne- eded angular velocities, ratios, forces, accelerations etc. Different sets of 418 J. Drewniak, S. Zawiślak unknowns are considered but if these unknowns are not essential for the solutions then they are excluded from the considerations via appropriate algebraic transformations. The similar routine can be formulated for the reverse order of activities, i.e. synthesis: going from the graph generation towards creation of a gear functional structure (Tsai, 2001). The considered planetary gear is presented in Fig.1. Fig. 1. Functional scheme of an exemplary planetary gear: 1,2, . . . ,6 sun wheels, wheels with internal toothings and planets; h,H – arms; A,B,. . . ,F – characteristic points; ω,M – angular velocities and torques A special planetary gear is considered. It encloses an internal closed loop formed by wheels and planets. The mobility of the structure is equal to 2. It means that two elements have to be driven. Some considerations concerning possible inputs and outputs are presented in Section 4 (Tables 1 and 2). The mobility W (DOF, degree of freedom) for the consideredmechanism, i.e. pla- netarygear, canbecalculateduponthe following formula (Grübler-Kutzbach’s equation) W =3n−2c5− c4 =18−12−4=2 (1.1) where: n – number of links (movable elements), c5 – number of full joints (one degree of freedom), e.g. rotational type; c4 – number of half joints, e.g. meshing type; moreover c5 and c4 are equal to the total number of edges and diagonals of a polygon (c5 = 6) and the number of stripped edges (c4 = 4), respectively. We have also n= 6, i.e. number of graph vertices – in the case of linear graph representation of a planetary gear (see Fig.2). Linear-graph and contour-fraph-based models... 419 Nevertheless, it is possible to analyse this case. However, there are some restrictions on the number of teeth and dimensions to assure the possibility of assemblage of the gear and proper gearing in two parallel toothings. In Tsai (2001), there are several tableswhere themechanical properties ofmechanisms are expressed via characteristics of their assigned graphs. For the considered gear, the following data for teeth numbers and the module are assumed: z1 = 15; z2 = 24; z3 = 63(−63); z4 = 18; z5 = 21 and z6 = 60(−60), m = 2mm, where negative values of the teeth numbers are considered for the internal gearing in the case of the Willis method, and one common module for all meshings has been assumed. The original approach discussed in the present paper consists in: • usage of Tsai’s derivation of the f-cycle equationsmethod for Hsu’s type graphs, which originally were usedmainly for synthesis purposes, • analysis of the closed-loop gear via the graph-based approach, which has also other novelty aspects. Generally, the contour method has been sel- domutilized inmechanics till now, and for open layouts only.Besides the mentionedbooks, there are fewpublishedpapersdealingwith this appro- ach for gears. Moreover, the examples of gears given in Prof.Marghitu’s books are relatively simple. However, really complicated structures we- re analysed for versatile mechanisms without geared subsystems. In the present paper, a contour graph has been generalized for the analysed compound gear (closed-power-loop), i.e. double described vertices we- re introduced and additional arcs were considered to assure orientation of every contour, which means that the orientation of contours can be chosen arbitrarily, • comparisonof thedetailed rulesof assignment ’mechanical system-graph’ for two consideredgraphapproaches is given in thepresentpaper–which were only roughly listed together in Drewniak and Zawiślak (2010), • analysis of forces, torques and, especially, efficiency of the gear is made, which makes the presented considerations complex and comprehensive. Usage of a free body diagram makes the analysis very effective. 2. Graph based model of a planetary gear Graph-basedmodels ofmechanisms, especially gears, have been developed for manyyears.Despite the fact that several different graphmodels ofmechanisms were discussed in Tsai (2001) – the Hsu graphs have been chosen, and they 420 J. Drewniak, S. Zawiślak are utilized as models of planetary gears in the present paper. It seems that this model is the most adequate. The rough idea of modelling is as follows: only some general properties or aspects of a mechanism are taken into account, e.g. analysis of kinematics. Therefore, the main rotating elements of a mechanical system are represen- ted by graph vertices and the relations among them are modelled via edges. The applied rules of assignment start from the idea: the singled outmain gear elements like e.g. gear wheels, planets and arms are considered as vertices. Especially, all elements rotating around themain physical axis of a planetary gear are represented by the vertices of a polygon, moreover the mutual rela- tions among them: ’rotation around the same axis’ are coded via the polygon edges and its diagonals. The latter are not drawn for simplicity of the picture. However, they are used for determination of some f-cycles. Fig. 2. Modified Hsu’s linear graph of the planetary gear (see: variant – Table 2, row 2) Therelation for gearwheels: ’tobe ingear’ is representedbya stripped line. Therelation ’tobeanarmforagearedwheel (planetwheel)’ is representedbya continuous line.An exemplary graphof theplanetary gear (Fig.1) is presented in Fig.2. Some vertices are described in an unusualmanner, e.g. 3/6 and 1/4. It is to be underlined that wheels 3 and 6 as well as 1 and 4 are connected by a stiff axis and they create a united element with two toothings. The set of f-cycles is as follows: (1,2)h; (2,3)h; (4,5)H and (5,6)H (Tsai, 2001), where only adequate descriptions of geared wheels are used, i.e. 1 instead of 1/4. Precisely, these notions are the codes of the f-cycles. Namely, first cycle (1,2)h consists of threeedges {(1,2),(2,h),(1,h)}. As it canbeseen: edge (1,h) isnot drawn explicitly and it is hidden in the shaded polygon. Edge (2,h) represents a pair ’planet and arm’. Edge (1,2) represents gearing of twowheels. The rule for a code of the f-cycle is that we write a description of the stripped edge in brackets and the description of the arm outside the brackets. Moreover, the Linear-graph and contour-fraph-based models... 421 names of vertices are arranged in increasing order starting fromnumbers. The adequate detailed cycles can be written for the remaining three codes. Every f-cycle generates one equation. The following system of equations can be derived based upon thementio- ned f-cycles according to the formulated rules ω1−ωh =−N21(ω2−ωh) ω2−ωh =+N32(ω3−ωh) ω4−ωH =−N54(ω5−ωH) ω5−ωH =+N65(ω6−ωH) (2.1) where: ωi, i = 1,2, . . . ,6; ωh and ωH are angular velocities of respective gear elements; Nij – ratios, Nij = zi/zj, the signs ’–’ and ’+’ depend on the external and internal gearing, respectively, zi, zj – teeth numbers of thewheel and pinion, respectively. Due to the layout of the considered planetary gear, the following equalities can be considered ω3 =ω6 ω1 =ω4 (2.2) Solving the system of equations, the following solutions are derived ω3 = (N54N65+1)ωH −ω1 N54N65 ωh = ω1+N21N32ω3 1+N21N32 (2.3) Taking into account the input angular velocities ω1 =157s −1 ωH =87.5s −1 (2.4) the numerical values of the output angular velocities are obtained ω3 =ω6 = 87.5 ( 1+ 21 18 60 21 ) −157 60 18 =66.65s−1 (2.5) ωh = 157+66.6524 15 63 24 1+ 63 15 =84.025s−1 The achieved results will be compared with other methods of gear analysis. 3. Contour based model of the planetary gear The contour method of modelling of a mechanical system consists in creation of a special graph enclosing the contours, i.e. closed circles built of arrows 422 J. Drewniak, S. Zawiślak connecting the vertices representing the elements of the system. The conto- ur graph of the gear presented in Fig.1 is shown in Fig.3. The rule is that the contour starts in a supporting system (body of a gear) and passes via elements onwhich angular (or linear)movement is passed by.We end the con- tour after returning to the support. All closed independent loops generated in this way have to be drawn. Then the list of codes of the derived contours is created. Next, in turn, the contour equations can bewritten in an algorithmic manner manipulating the indices in the following way: all relative velocities for every consecutive pair of element codes have to be inserted into the deri- ved equations. There are relative angular velocities of the mechanical system elements inside the obtained system of equations. It is an unavoidable stage of the method. However, the unwanted quantities can be eliminated and the system can be solved in a step-by-step manner. A detailed explanation of the methodology can be studied upon (Marghitu andCrocker, 2001). Underneath, only the needed details tailored to planetary gear modelling will be given. Fig. 3. Contour graph of the planetary gear The following contours can be distinguished for the contour graph (Fig.3) of the considered planetary gear: (I) 0→ 1→ 2→h→ 0 (II) 0→h→ 2→ 3→ 0 (III) 0→ 4→ 5→H → 0 (IV) 0→ 4→ 5→ 6→ 0. Contour (I) generates the following pairs of indicators {(1,0);(2,1);(h,2);(0,h)}. They are used in the first two equations (1) of system (3.1). The contour can be interpreted as follows: we start it from 0, i.e. the support for the axis of wheel 1, which is geared with planet wheel 2, planet 2 rotates around its own axis mounted to h and rotates together with Linear-graph and contour-fraph-based models... 423 the arm h around the axis depicted by 0. This is the end of the contour: 0 → 1 → 2 → h → 0. Analogous explanations can be given for the three remaining contours. The contour graph is presented in Fig.3 where the special drawing rules byMarghitu and Crocker (2001) were applied. It encloses four contours I-IV. Theorientations of the contours are shownvia crooked arrows insideparticular contours I-IV. Based upon the distinguished contours, the following system of equations can be written (1) ω10+ω21+ωh2+ω0h =0 rA×ω21+rB ×ωh2 =0 (2) ωh0+ω2h+ω32+ω03 =0 rB ×ω2h+rC ×ω32 =0 (3) ω40+ω54+ωH5+ω0H =0 rF ×ω54+rE ×ωH5 =0 (4) ω40+ω54+ω65+ω06 =0 rF ×ω54+rD×ω65 =0 (3.1) where rk are the position vectors of the points k = A,B,. . . ,F shown in Fig.1; ωij – means vectors of relative angular velocity of the i-th element in relation to the element j. Every contour generates two equations connected with angular velocities and some other for forces and torques. The latter are not used in the present paper.Thefirst equation for everypair is a sumofangularvelocities designated by a contour, the second one is a sum of cross products of the respective arm multiplied by the angular velocity. In the second case, some summands are omittedwhen the arm length is equal to zero. The following relations are used for simplification of the system: — connected with the rule for exchanging of an order of the indicators ωi0 =−ω0i i=1,2, . . . ,6 ω0h =−ωh0 ω0H =−ωH0 (3.2) — connected with the rule of transformation of the relative velocities into general ones, i.e. rotating around themain axis denoted by 0 ωi0 =ωi i=1,2, . . . ,6 ωh0 =ωh ωH0 =ωH (3.3) — connected with geometrical relations of the considered planetary gear rA = r1 rF = r4 rB = r1+r2 rE = r4+r5 rC = r1+2r2 rD = r4+2r5 (3.4) 424 J. Drewniak, S. Zawiślak For example: rA is equal to the pitch radius of geared wheel 1, so it is equal to r1. An analogous explanation can be given for remaining equalities (3.4) for arms rB to rF . The transformed system can be written in the following form −rAω1−rAω21−rAωh2−rAω0h =0 rAω21+rBωh2 =0 −rCωh0−rCω2h−rCω32−rCω03 =0 rBω2h+rCω32 =0 −rEω40−rEω54−rEωH5−rEω0H =0 rFω54+rEωH5 =0 −rDω40−rDω54−rDω56−rDω06 =0 rFω54+rDω56 =0 (3.5) To solve system (3.1), the following actions have been undertaken: we omit the arrows, i.e. we turn the vector equations into scalar ones. It can be do- ne because the velocities act as vectors along the same direction. Moreover, the senses (i.e. orientations along the given direction) will be established via the solution of the system. The vector multiplication (cross product) can be simplified to the scalar one because the angles between the arms and veloci- ties in the case of gears with cylindrical wheels are always equal to 90◦. For other mechanisms enclosing cranks, pistons, cylinders, sliders, followers, etc., a detailed analysis of angles has to be done (Marghitu and Crocker, 2001) to simplify the system of equations correctly. Additionally, the first equation in (3.5) was multiplied by −rA, the third by −rC, etc., the resultant equations are gathered in system (3.5). Then the unwanted relative velocities can be eliminated. Assuming the cylindrical gear wheels, we have the relationships ri = di 2 = zimi 2 (3.6) where: mi is the module of the i-th wheel, zi – teeth number of the i-th wheel, i=1,2, . . . ,6. Due to the fact that it is a universal design havingmulti-inputs andmulti- outputs, all modules were assumed as equal. It causes that the module is not present in the formulas for theoutputangular velocities. In the case of different modules, these formulas would be a little more complicated but the system is also solvable. The solution to system of equations (3.5) is as follows ω6 =ω3 = ωH(2r4+2r5)−r4ω1 2r5+r4 =66.65s−1 (3.7) ωh = r1ω1+(r1+2r2)ω3 2(r1+r2) = 84.025s−1 We obtained the same numerical results (3.7) as in the case of the linear graph-basedmodelling – see formulas (2.5). It confirms the correctness of the performedmodelling and analysis. Linear-graph and contour-fraph-based models... 425 4. Analysis of an exemplary planetary gear by means of the Willis method and data comparisons The correctness of kinematical analysis of the considered gear will be addi- tionally checked via the classical Willis method. We also analyse several va- riants of constructional data. The second set of the test data is as follows: z1 = 18; z2 = 21; z3 = 60(−60); z4 = 15; z5 = 24 and z6 = 63(−63); mi =m=2[mm], where the negative values of teeth numbers are considered for internal gearing in the case of theWillis method. The given quantities are: ω1 = 157s −1; ωh = 30s −1; the unknowns are ωh and ω3 = ω6. The same systems of equations derived based upon the two considered graph methods solved for the new unknowns give similar general solutions. The ratio of an arbitrary planetary gear, including differential ones and those having mobility W ­ 2, can be calculated by means of the Willis for- mula. In the case of the considered gear, we have to consider the partial ratios ih13 and i H 46, where the upper indices h or H determine the virtually fixed link of the gear, after applying to all the gear links additional velocities −ωh or −ωH, respectively. Then the relative velocity of the arm h or H is equal to zero. Therefore, the ratio ih13 of the geared wheels, considering the order from 1 to 3, in the case of ”theoretically fixed” arm h and z3 < 0 can be expressed as ω1−ωh ω3−ωh = z3 z1 (4.1) Similarly, the ratio iH46 of the gearedwheels, considering the order from 4 to 6, in the case of ”theoretically fixed” arm H and z6 < 0 can be written as ω4−ωH ω6−ωH = z6 z4 (4.2) Taking into account the additional kinematic conditions: ωI = ω1 = ω4 and ω3 = ω6, we can calculate the unknown angular velocities and the needed kinematic ratio iIH ω6 =ω3 =ωh [z1(ωI−ωh) ωhz3 +1 ] =−8.10s−1 (4.3) ωH = z4ωI−z6ω3 z4−z6 =23.65s−1 iIH = ( ωI ωH ) ωh=30s −1 ∼=6.638 The same values of these quantities were obtained for both graph-based me- thods. It confirms equivalence of them and the possibility of mutual checking of their correctness. 426 J. Drewniak, S. Zawiślak In Fig.4, free body diagrams for forces and torques acting on the geared wheels and arms are presented. Some considerations on these quantities are given in the next Section. Analysing different inputs and outputs, different ratios can be achieved. The results for the first set of data are presented in Table 1. Further analysis can be done for other values of the input velocities or numbers of teeth. The numerical values of respective velocities and ratios in the second set of data are given in Table 2. Table 1.Gear ratios for the first set of data; outputs: ωh and ω3 =ω6 No. Input angular Output angular Ratio velocity [s−1] velocity [s−1] 1 ω1 =157 ω6 =66.65 (i16)ωH =2.36 ωH – known (ωH =87.5) 2 ω1 =157 ωh =84.03 (i1h)ωH =1.87 ωH – known (ωH =87.5) 3 ωH =87.5 ω6 =66.65 (iH6)ω1 =1.31 ω1 – known (ω1 =157) 4 ωH =87.5 ωh =84.03 (iHh)ω1 =1.04 ω1 – known (ω1 =157) Table 2.Gear ratios for the second set of data; outputs: ωH and ω3 =ω6 No. Input angular Output angular Ratio velocity [s−1] velocity [s−1] 1 ω1 =157 ω6 =−8.10 (i16)ωh =−19.38 ωh – known (ωh =30) 2 ω1 =157 ωH =23.65 (i1H)ωh =6.638 ωh – known (ωh =30) 3 ωh =30 ω6 =−8.10 (ih6)ω1 =−3.70 ω1 – known (ω1 =157) 4 ωh =30 ωH =23.65 (ihH)ω1 =1.26 ω1 – known (ω1 =157) For the third data set, we assume that the output arm h is fixed via a brake (ωh = 0s −1) and the other data from the second case remain. It is a particular version of the considered gear, which can also have a particular Linear-graph and contour-fraph-based models... 427 practical meaning. In this case DOF of the planetary gear is equal to 1. Then the ratio i13 = i h 13 is determined bymeans of a formula adequate for the fixed axis and in the case when z3 < 0 ω1 ω3 = z3 z1 (4.4) The ratio iH46 can be determined similarly as in the general approach to the considered gear, see (4.3)3. The following values of angular velocities and the ratio iIH have been obtained ω3 = ω1z1 z3 =−47.10s−1 ωH = z4ω1−z6ω3 z4−z6 =−7.85s−1 (4.5) iIH = ( ωI ωH ) ωh=0 =−20 Also in this case, the same resultswere obtainedupon the graph-basedmodels. The designer can analyse several possibilities to change the input velocities or to fix some gear elements using brakes in the conceptual phase of gear design, e.g. within the three data sets analysed above. 5. Analysis of forces and torques Forces and torques acting in the gear can be analysed. The analysis is perfor- med in a step-by-step manner viewing the considerations from an input and an output. InFig.4, the free bodydiagram is presented.The fact that toothings 1 and 4 as well as 3 and 6 are fixed on the same elements (respectively) is taken into account especially as a condition of equilibrium of the corresponding forces F1,2 and F4,5. In Figs. 5, 6, 7, the third case is analysed. The scheme of the gear with an additional brake (placed at the input) is given in Fig.5. In Fig.6, analysis of velocities is shown. In Fig.7, the free body diagram for the considered gear in the third case is given. This type of diagram enables us to clearly analyse the forces and to spot the points where equilibrium of forces has to be achieved. Analysing the directions and senses of the angular velocities and torques, we can deduct which quantities are passive or active. Based uponFig.7, the forces and torques acting on the geared wheels and the arm H are analysed. The output torque of the gear is equal to MH =MIiIHηIH (5.1) where ηIH is the efficiency. 428 J. Drewniak, S. Zawiślak Fig. 4. Free body diagram of the considered gear with analysis of forces for the first and second analysed case Fig. 5. Scheme of the gear for the third case, i.e. with fixing of the arm h via a brake Therefore, it is necessary to calculate the efficiency of the gear ηIH in the case of the given efficiency of meshing of some pairs of the geared wheels, i.e. the external ones (1,2 and 4,5) as well as internal (2,3 and 5,6). The relative velocities of geared wheels 6 and 4 (in relation to the arm H) for a theoretically considered planetary gear are as follows (Abramov, 1976; Müller andWilk, 1996) ωH6 =ω6−ωH =−39.25s −1 < 0 ωH4 =ω4−ωH =164.85s −1 > 0 (5.2) Linear-graph and contour-fraph-based models... 429 Fig. 6. Analysis of velocities in the third case (with the arm h via a brake) Fig. 7. Free body diagram for the third case (arm h fixed via a brake) Because the senses of directions of the velocity ωH6 and the torque M6 are compatible (Fig.7), thengearedwheel 6 is an active gear in the series ofwheels 4-5-6, whereas wheel 4 is passive, thus M6|ω H 6 |η H 64 =M4|ω H 4 | (5.3) 430 J. Drewniak, S. Zawiślak From the condition of equilibrium for the arm H, we have MH =M4+M6 (5.4) The relative velocities of geared wheels 3 and 1 (in relation to the arm h) for a theoretically considered planetary gear (after introducing that the angular velocity ωh =0) are as follows ωh1 =ω1 =157s −1 > 0 ωh3 =ω3 =−47.10s −1 < 0 (5.5) hence wheel 1 is active and wheel 3 is passive M1|ω h 1 |η h 13 =M3|ω h 3 | (5.6) The equilibrium condition for shaft I is expressed by the equation MI =−M1+M4 (5.7) Moreover, taking into account the equilibrium condition for wheels 3 and 6 (in the case when the total power is passed outside via the arm H), i.e. upon the assumption MII =M6−M3 =0 (5.8) we obtain the equality M3 =M6 (5.9) Summarizing the above considerations, it is possible to calculate the efficiency of the considered gear ηIH = MH MIiIH = 1+ ωH 4 |ωH 6 |ηH 64 ( ωH 4 ωh 3 |ωH 6 |ωh 1 ηH 64 ηh 13 −1 ) iIH ∼=0.79 (5.10) where ηh13 = η h 12η h 23 η H 64 = η H 65η H 54 (5.11) moreover, it was assumed that ηh12 = η H 54 = 0.99 (for the external meshing) and ηh23 = η H 65 =0.98 (for the internal meshing). 6. Final remarks Based upon the above-described considerations, the following conclusions can be drawn: two graph-basedmethods of modelling of mechanical systems have Linear-graph and contour-fraph-based models... 431 been used for analysis of an exemplary planetary gear. The obtained ratios (for two applied methods) as well as the results of Willis method are equal, which confirms the equivalence of these approaches. The usage of graph-based methods for analysis of the exemplary gear with a closed internal loop has been described and performed step by step giving a detailed explanation to all activities. Themethods are relatively simple, algorithmic and general. This confirms usefulness of these methods for checking of correctness of gear ana- lysis. Both the applied graph-based methods were slightly modified, i.e. Tsai’s andHsu’s approaches were joined as well as Marghitu’s graphwas tailored to a scheme having a closed loop. Additionally, it has been shown (by a counter example) that Tsai’s claim that the case when a graph has a circuit built of stripped lines is impossible for analysis is not a general rule. Moreover, the graph-based models give a powerful tool for modelling and representation of the knowledge about mechanical systems (Andrews et al., 1997; Shai and Preiss, 1999; Tsai, 2001; Zawiślak, 2007) what is needed for Artificial Intelligence based methods. The graph based models of gears and versatile mechanical systems are very effective in realisation of some other engineering tasks like e.g. synthesis and enumeration of design solutions,which is actually beyond the scope of the present paper but can be studied based upon the given references. Concerning the obtained results presented in Tables 1 and 2, it can be stated that the achieved ratios – shown in Table 1 – are approximately one, but in Table 2 the ratios above six and approximately nineteen are listed. 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ZawiślakS., 2008,Graphtheoreticbasedmodelsofplanetarygears, in:Theory of Machines and Mechanisms, J.Wojnarowski and I. Adamiec-Wójcik (Edit.), WydawnictwoAT-H, Bielsko-Biała, 2, 67-74 24. Zawiślak S., Szypuła Ł., Myśliwiec M., Jagosz A., 2008, Some ap- plications of graph transformations in modelling of mechanical systems, Pre- Proceedings Seventh InternationalWorkshop onGraph Transformation andVi- sual Modelling Techniques (GT-VMT 2008), Budapest, Hungary, 332-345 Grafowe i konturowe modele przekładni planetarnych Streszczenie Analiza i synteza mechanizmów są podstawowymi działaniami inżyniera. Nara- żone są one z różnych powodów na niezamierzone błędy. Metody analizy i syntezy przekładni planetarnych oparte na teorii grafówmogą byćmetodami alternatywnymi dla realizacji tych zadań, które pozwalają sprawdzić poprawność przeprowadzonych rozważań. W artykule rozważa się dwie metody grafowe modelowania przekładni. Przykładową przekładnię planetarną analizuje się tymimetodam, a wyniki porównu- je się z metodą Willis’a. Przeprowadzono także analizę sił, momentów i sprawności. Ujęcie algorytmiczne–którewynika zmodelowaniagrafami–pozwalana łatwe spraw- dzanie wielu wariantów rozwiązań w łatwym ujęciu schematycznym, co wpływa na optymalizację rozważanych rozwiązań konstrukcyjnychw fazie koncypowania. Manuscript received July 17, 2009; accepted for print October 19, 2009