Duality between Statical and Kinematical Engineering Systems

Transcription

1 Pape 00, Civil-Comp Ltd., Stiling, Scotland Poceedings of the Sixth Intenational Confeence on Computational Stuctues Technology, B.H.V. Topping and Z. Bittna (Editos), Civil-Comp Pess, Stiling, Scotland. Duality between Statical and Kinematical Engineeing Systems O. Shai Depatment of Mechanics Mateials and Systems Tel-Aviv Univesity, Isael Abstact The wok pesented in the pape is one of the outcomes of a geneal eseach called Multidisciplinay Combinatoial Appoach (MCA) aimed to assist in obtaining a geneal pespective ove engineeing systems. The appoach is focused on developing geneal discete mathematical models, called Combinatoial Repesentations and associating them with diffeent engineeing systems. Once the engineeing system is associated with a specific combinatoial epesentation, analysis and othe foms of engineeing easoning can be computeized and conducted solely upon the combinatoial epesentation. The wok intoduced in this pape enables to tansfe knowledge fom vaious domains of engineeing to stuctual mechanics and vice vesa. The pape is concentated on tansfeing knowledge by means of a new duality elation between statical and kinematical systems that has been established in 00. Keywods: gaph theoy, combinatoial epesentations, duality, beams, tusses, gea systems, mechanisms. Intoduction The duality elations between statical and kinematical systems ae established in the pape though systematic mathematical pocesses based on elations between genealized discete mathematical models, called Combinatoial Repesentations. The fist duality pesented in the pape is between deteminate tusses and mechanisms []. It is shown that a combinatoial epesentation of a mechanism is a potential gaph epesentation (PGR), wheeas the epesentation of a tuss is the flow gaph epesentation (FGR). It is then poved that the potential and flow gaph epesentations ae mutually dual, thus making the tusses and mechanisms dual as well, as is depicted in the example of Figue a. Anothe duality

2 elation that has been ecently established is the duality between statical column systems and mechanisms, and its special case - the duality between statical beams and gea systems (Figue. b). Establishing the duality between statics and kinematics povides a channel fo knowledge and infomation tansfe, yielding immediate pactical and theoetical implications, some of which ae descibed in this pape. One of the significant theoetical implications is poving the coespondence between theoems and methods in the two fields. In the pape this issue is demonstated by showing that Maxwell-Cemona diagam fo foces in tusses and velocity polygon of mechanisms ae actually dual methods. 0 0 ' 9 9 = P V ' P Figue : Example of dual systems fom static and kinematical domains. Tuss and its dual mechanism, Beam and its dual planetay gea system. Futhe application of the duality elation is demonstated by developing ules fo validity checking. It is shown that though the duality elation, methods fo checking the mobility of mechanisms become applicable to checking the stability of tusses and vice vesa. One of the most pomising pactical applications of the appoach is opening a new avenue of eseach fo facilitating the pocess of design of engineeing systems. Employing the dualism, civil enginee faces an oppotunity to seach fo a solution to his poblem also among known mechanisms. Once a mechanism pefoming the desied task is found, the designe is just left to tansfom it to the coesponding dual static system.

3 Combinatoial Repesentations and thei Duality The esults epoted in this pape ae based on fou Combinatoial Repesentations - Potential Gaph Repesentations (PGR), Potential Line Gaph Repesentations (PLGR), Flow Gaph Repesentations (FGR) and Flow Line Gaph Repesentations (PLGR). Tables and summaize the popeties of these epesentations, accompanied with thei elations to elevant engineeing fields. Futhe infomation appeas in detail in ([], [] and []). Tables and employ some basic tems of gaph theoy, thus following ae few necessay definitions: Netwok gaph is a diected gaph G = <V, E>, whee V is the vetex set and E is the edge set. The vetex upon which the aow is diected is called the head vetex and the othe is called the tail vetex. Each edge e is assigned a vecto called flow F (e) and each vetex v is assigned a vecto called potential π (v). Subtaction of the potential of the tail vetex fom the potential of the head vetex fo a specific edge is called the potential diffeence - (e) of that edge. Combinatoial Repesentations (CR) Main popety Repesented engineeing system Engineeing Edge intepetation of the gaph elements Vetex Flow Gaph Repesentation - G F (FGR) Flow law - The vecto sum of the flows in evey cutset of G F is equal to zeo. Q G ) F(G ) = 0 ( F F Deteminate tuss. Tuss element: od, eaction, extenal foce in the tuss. The flow in the edge is intepeted as the foce in the coesponding tuss element. Pinned joint of the tuss Potential Gaph Repesentation- G (PGR) Potential law The vecto sum of the potential diffeences in evey cicuit of G is equal to zeo. B ( G ) (G ) = 0 Mechanism. Link in the mechanism. The potential diffeence is intepeted as the elative velocity between the end joints of the link. Joint in the mechanism. The potential is intepeted as the linea velocity of the joint. Table. Flow and Potential Gaph Repesentations and thei usage.

4 Combinatoial Repesentations (CR) Main popety Repesented engineeing system Engineeing Edge intepetation of the gaph elements Vetex Flow Line Gaph Repesentation G LF (FLGR) Teminal equation - Flows in all edges of G LF have two othogonal components: otational and linea, elated as follows: FRi = i FLi, whee i is some constant vecto associated with the edge. Flow law - The vecto sum of the flows in evey cutset of G F is equal to zeo. Q G ) F(G ) = 0 ( LF LF Pilla stuctue A pilla, eaction o extenal foce applied to the hoizontal plate. The linea flow in the edge is intepeted as the foce in the coesponding element. The otational flow is the moment exeted by the element upon the plates to which it is connected. The constant vecto i thus defines the location of pilla in elation to the plate. A plate suppoted by the pillas Potential Line Gaph Repesentation- G L (PLGR) Teminal equation Potential Diffeences in all edges of G L have two othogonal components: otational and linea, elated as follows: Li = i Ri, whee i is some constant vecto associated with the edge. Potential law The vecto sum of the potential diffeences in evey cicuit of G is equal to zeo. B ( G L ) (G L ) = 0 Mechanism Kinematical couple of mechanism links. The otational potential diffeence is intepeted as the elative angula velocity between the end joints of the link. The linea potential diffeence is the toso of the kinematical pai. The constant vecto i is thus equal to the adius vecto fom the coesponding junction to some common efeence point. Link in the mechanism. The potential is intepeted as the angula velocity of the link. Table. Flow and Potential Line Gaph Repesentations and thei usage. Sepaation of the CR to two tables is not abitay. It can be seen fom these Tables that thee is a cetain coespondence between the popeties descibed in thei left and ight sides. Indeed it can be poved that the coesponding CR ae actually dual: Duality between flow and potential gaphs []. Given a flow gaph G F execute the following * * steps: build its dual gaph G F, equate the potential diffeences in the edges of G F to the flows in the coesponding edges of G F. It then follows fom the popeties of dual gaphs [] that these potential diffeences satisfy the potential law in G * F. Thus, G * F can be consideed as a valid potential gaph G. Finally, it can be postulated that fo each flow gaph G F thee exists a dual potential gaph G and vice vesa.

5 Fom the duality between the flow and potential gaph epesentations one can deduce the duality elation between tusses and mechanisms, as is outlined in the diagam in Figue. Flow Gaph Repesentation FGR Q( G) = B(G*) mutual dualism Potential Gaph Repesentation PGR Repesents Repesents Deteminate tusses mutual dualism Mechanisms Figue : Diagam explaining the mutual dualism between tusses and mechanisms. Simila pocedue can be pefomed to pove the duality between Potential and Flow Line Gaph Repesentations []. Duality between statical and kinematical engineeing systems. Fom the duality between PGR and FGR, one can conclude that the engineeing systems they epesent, namely mechanisms and tusses ae also dual. This duality elation is defined as follows: - thee is a link in the mechanism fo each od, mobile suppot eaction o extenal foce in the tuss. - the vecto of the elative velocity of each mechanism link is equal to the vecto of the foce acting in the coesponding element in the tuss.

6 P E V E / O B A C E P = V E /O A B C O D D R V V E E / O/O E B O A D (c) C R Figue : Duality between mechanisms and tusses - Mechanism, Dual Tuss, (c) Potential Gaph Repesentation of the Mechanism, Flow Gaph Repesentation of the tuss. P E B A D R C O Figue shows an example of a mechanism and its coesponding dual tuss, obtained though building the coesponding gaph epesentations. In the same way as it was done fo mechanisms and tusses, one may conclude the duality between mechanisms and pilla stuctues fom the duality between PLGR and FLGR. This duality implies that fo each mechanism thee is a coesponding dual pilla stuctue and vice vesa. Table, summaized the elations between the two engineeing systems. Popeties of a pilla stuctue Plate / set of foces acting on the plate. Foce i. Plane paallel to the plates of the system. = adius vecto between two adjacent i, j joints upon which intenal/extenal foces ae acting. - adius vecto to the point of action of i foce i. Popeties of the dual mechanism Face / set of links limiting the face. Relative angula velocity of the kinematical pai i *. Plane of the mechanism links. Li,j vecto paallel to a link in the mechanism whose end kinematic joints ae i and j. i * - adius vecto of the kinematical pai i *. Table. Coespondence between the popeties of pilla stuctue and its dual mechanism.

7 Figue shows an example of a mechanism and its dual pilla stuctue. Inteesting esult emeges, when the duality elation is applied to a one-dimensional case of engineeing systems. The one-dimensional plate acted upon by a numbe of vetical foces is actually a beam acted upon extenal foces and eactions. The one dimensional mechanism on the othe hand has all links paallel to one anothe, combining it to the fact that it also satisfies toso cicuit ule, leads us to conclude that it is actually a gea system. Consequently, one-dimensional deteminate beams ae dual to the gea mechanisms, as is shown in the Figue above. C B A B A O C 0 (c) Figue : Pilla stuctue, Flow Gaph Repesentation of the Pilla stuctue, Potential Gaph Repesentation of the Mechanism, (c) Dual Mechanism.

8 Pactical aspects of the duality elations.. The mutual dualism between Maxwell-Cemona and image velocity diagams Image velocity diagam is a known gaphical method fo velocity analysis of mechanisms []. Once the image velocity diagam of a mechanism is constucted, the elative velocities of evey mechanism link can be measued diectly fom it. Knowing that the elative velocities in the mechanism links ae equal to the intenal foces in the ods of the dual tuss, it can be concluded that the image velocity can be used to pefom the static analysis of a tuss. Futhemoe, one can build the diagam diectly fom the tuss, without even consideing its dual mechanism as is explained in the cuent subsection. This pocess becomes clea when summaizing all the popeties of the image velocity diagam while simultaneously ewiting them in the teminology of stuctual analysis, as is done in Table. Fom Table, it follows that the image velocity method completely coincides with the known Maxwell-Cemona diagam algoithm fo static analysis of deteminate stuctues []. Consequently, Maxwell-Cemona and Image velocity methods ae mutually dual methods. Image velocity popeties In mechanism teminology Each point coesponds to a joint in a mechanism. The diving link is epesented by a line in its velocity diection and length popotional to its velocity value. The line connects the points coesponding to the end joints of the link. The elative velocity of a link is epesented by a line paallel to the link elative velocity, which connects the points coesponding to the end joints of the link. In teminology of the dual tuss Each point coesponds to a non-bisected aea in the tuss closed by tuss elements. The extenal foce is epesented by a line in its diection with length popotional to its magnitude. The line connects the points coesponding to the aeas sepaated by the extenal foce. The axial foce in a od is epesented by a line paallel to the od, connecting the points coesponding to the aeas sepaated by it. Table. Popeties of the image velocity diagam in the teminology of both mechanisms and the dual tusses. Figue pesents a fou ba chain, its dual tuss and the image velocity diagam. One can veify that this diagam also pesents the static analysis diagam of the dual tuss, namely its Maxwell-Cemona diagam. 8

9 II' IV' II V O I' III' O P ' I III O IV I' = P' V I P V III' V V IV' V II' V V O F III F F IV II F ' F F 0 (c) Figue : The coespondence between image velocity and Maxwell-Cemona diagams. The fou ba chain. Its dual tuss. (c) The image velocity Maxwell-Cemona diagam.. Checking stability though dual mechanisms Sometimes the poblem of checking stability of statical system equies its thoough numeical investigation. Employing the duality elations may in some cases assist in avoiding such a pocedue. Conside, fo example, two deteminate tusses of Figue a and b. Only the fist one of these two is stable. Reaching this conclusion without pefoming calculations, is not easy even fo expets in mechanical engineeing. On the othe hand, consideing the mechanisms dual to these tusses makes the task easie. Figue c shows the mechanism dual to tuss of Figue a, wheeas Figue d shows the mechanism dual to tuss of Figue b. In the mechanism of Figue d, links and 9 ae co-linea, in contast to the mechanism of Figue c. Theefoe, it is easy to deive that the dual mechanism of the fist tuss on Figue d is locked while the dual mechanism of the second tuss is not. This makes it possible to postulate that the tuss of Figue a is not stable, wheeas the tuss of Figue b is stable. This example stengthens the claim that thee ae popeties which ae had to detect in the pimal epesentation, wheeas they ae tanspaent in the dual. 9

10 (c) R R Figue : Example of stable and non-stable tusses and thei dual mechanisms. A stable tuss A non-stable tuss (c, d) Coesponding dual mechanisms.. Employing duality fo designing engineeing systems The duality connection between mechanisms and stuctues can be applied fo synthesis of new engineeing systems. The main idea behind this appoach lies in the fact that if a mechanism possesses some special engineeing popeties, then its dual tuss o pilla stuctue possess the exact same popeties. In the following example the idea is employed to solve a static design poblem. Suppose one needs to design a static system, such that when a small foce is applied to one of its joints, a much geate foce is poduced in one of its ods. Such a static system can be obtained immediately by using the duality between tusses and mechanisms. This is done by fist finding a known mechanism having simila velocity chaacteistics, namely, a mechanism that fo a small elative velocity in its diving link poduces in its othe link a much geate elative velocity. One of many known mechanisms satisfying this equiement is pesented on Figue a. The velocity of link of this mechanism is consideably lage than that of the link. The tuss dual to this mechanism is pesented in Figue d. Accoding to the duality popety, the tuss possesses the same foce chaacteistics as the velocity 0

11 chaacteistics of the mechanism, i.e. a small extenal foce P causes a much geate foce in od. (c) Figue : Tuss design. Oiginal mechanism, Coesponding potential gaph epesentation. (c) Dual flow gaph epesentation and the esulting tuss. Futhe eseach. The geneal vecto fomulation of the gaph epesentations involved should not limit us to two-dimensional systems. The vaiables of FLGR and PLGR both may acquie abitay spatial oientations without contadicting the embedded ules and popeties of these epesentations. Conside fo example a spatial stuctue pesented in Figue 8:

12 P 0 C D E P A B C D E F B A F I P 0 F B C ω A F 0 D E A B C D E (c) Figue 8. Stewat platfom and its dual. Stewat platfom, Coesponding FLGR, (c) Dual PLGR and the coesponding dual spatial mechanism. The stuctue shown in Figue. 8 pesents a statical aspect of the known Stewat platfom. The PLGR dual to the gaph appeaing in Figue 8b appeas in Figue 8c. The gaph is built of seially connected edges and thus coesponds to a mechanism with seially connected links, also known as a seial manipulato mechanism. By means of this elation one can now convet all the easoning pocesses upon the Stewat platfom to pocesses upon the dual mechanism, thus gaining new abilities that wee not peviously available. This esult esonates with a eseach done in obotics fo establishing duality between seial and paallel manipulatos [,8] that was mainly based on the scew theoy.

13 Conclusions The pape has intoduced some of the theoetical and pactical contibutions of two duality elations: between mechanisms and deteminate tusses, and between mechanisms and pilla stuctues. It should be noted that although the highlight of the pape was the contibution to the field of civil engineeing, the geneal appoach intoduced is applicable to many othe engineeing fields as well. Up until today the appoach has been employed in statics, kinematics, electonics and hydaulics. In additional to theoetical contibution of the appoach it may have implications in design, analysis, optimization, checking validity and othe aspects of engineeing eseach. Refeences [] O.Shai, The Duality Relation between Mechanisms and Tusses, Mechanism and Machine Theoy, (), -9, Mach, 00. [] O.Shai, Utilization of the Dualism between Deteminate Tusses and Mechanisms, accepted fo publication, Mechanism and Machine Theoy, Febuay, 00. [] O.Shai, "Duality Relation between Static Pillas and Kinematical Systems though Gaph Repesentations", submitted to Jounal of Mechanical Design, 00. [] M.N.Swamy, K.Thulasiaman, Gaphs: Netwoks and Algoithms, Wiley, New Yok, 98. [] R.L.Noton, Design of Machiney, McGaw-Hill, New Yok,99. [] S.P.Timoshenko, D.H.Young, Theoy of Stuctues, McGaw-Hill, 99. [] K.J. Waldon, K.H. Hunt, Seial-Paallel Dualities in Actively Coodinated Mechanisms, Int. J. Robotics Reseach 0(), -80, Octobe 99. [8] F.Gosselin, J.Lallemand, A New Insight into the Duality between Seial and Paallel Non-Redundant and Redundant Manipulatos, Robotica, 9, - 0, 00.