外文翻译中英文-平面机械程序库 AMESim仿真软件
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外文翻译中英文-平面机械程序库
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A planar mechanical library in theAMESim simulation software. Part I:Formulation of dynamics equationsWilfrid Marquis-Favre*, Eric Bideaux, Serge ScavardaLaboratoire dAutomatique Industrielle, Institut National des Sciences Applique es de Lyon,Ba t. St Exupe ry, 25, avenue Jean Capelle, F-69621 Villeurbanne Cedex, FranceReceived 25 March 2003; received in revised form 17 December 2004; accepted 8 February 2005Available online 17 March 2005AbstractThis paper presents the mathematical developments of a planar mechanical library imple-mented in the AMESim simulation tool. Body and joint components are the basic componentsof this library. Due to the library philosophy requirements, the mathematical models of thecomponents have required a generic vector calculus based formulation of the constraint equa-tions. This formulation uses a set of dependent generalized coordinates. The dynamics equa-tions are obtained from the application of Jourdain?s principle combined with the Lagrangemultiplier method. The body component mathematical models consist of differential equationsin terms of the dependent generalized coordinates. The joint component mathematicalmodels are based on the Baumgarte stabilization schemes applied to the geometrical, kine-matic and acceleration constraint equations. The Lagrange multipliers are the implicit solutionof these Baumgarte stabilization schemes. The first main contribution of this paper is theexpression of geometrical constraints in terms of vectors and their exploitation in this form.The second important contribution is the adaptation of existing formulations to the AMESimphilosophy.? 2005 Elsevier B.V. All rights reserved.1569-190X/$ - see front matter ? 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.simpat.2005.02.006*Corresponding author. Tel.: +33 4 72 43 85 58.E-mail address: wilfrid.marquis-favreinsa-lyon.fr (W. Marquis-Favre)./locate/simpatSimulation Modelling Practice and Theory 14 (2006) 2546Keywords: AMESim; Planar mechanics; Dynamics equations; Constraint equations; Lagrange multi-pliers; Baumgarte stabilization1. IntroductionThis paper, organized in two parts, presents a new library for the simulation toolAMESim 2. The first part is dedicated to the theoretical developments of thelibrary. The second part shows the composition of the library as it was primarilyimplemented in AMESim and illustrates it with an application example of aseven-body mechanism. This library proposes components belonging to the planarmechanical domain. The objective with this library was not to compete with multi-body system software tools that are better adapted to this domain. The objective wasmore to enlarge the range of industrial applications capable of being treated byAMESim. From a theoretical point of view the challenge of implementing thislibrary was to fit existing mechanical formulations to the inherent requirements ofAMESim philosophy. The solution has been found by adapting the dynamic equa-tions expressed from Jourdain?s principle and the Lagrange multiplier methodtogether with Baumgarte?s stabilization. Also a generic feature of the formulationhas been researched over the library components (bodies and joints) and one keycontribution of this paper is concerned with this generic feature. Basically the formu-lation consists of expressing the geometric constraints associated with joints in termsof vectors and carrying out the developments of this form. The result is the set up,for kinematic and acceleration constraints, of a unique expression that fits every jointpresented in the library.The generic feature of the formulation proposed thus enables the derivation ofjoint contraints to be systematized. One can then imagine a new joint with its corre-sponding vector constraint and derive straightforwardly the corresponding mathe-matical model by applying the proposed formulation. Also, in the context ofpredefined component models, the given formulation clearly shows the frontiers ofthe different mathematical models in terms of inputs and outputs. Therefore it alsohelps to define in which models output equations must be implemented. Also, theformulation proposed intrinsically enables closed loop structures to be dealt with.AMESim (for Advanced Modeling Environment for performing Simulations) isorganized in component libraries. The components, represented by symbolicallytechnologically suggested icons, can be interconnected exactly like the system understudy. AMESim was first applied to electrohydraulic engineering systems with simpleone-dimensional mechanical systems (like inertia, springs, and dampers in transla-tion or in rotation). It recently opened its libraries to a variety of other componentdomains. One can now carry out modeling, analysis and simulation for systems con-sisting of pneumatic, powertrain, hydraulic resistance, thermal, electromagnetic andcooling components for instance. The restriction to only one-dimensional motion forthe mechanical components has motivated the development of a two-dimensionalmechanical library.26W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 2546Section 2 presents an overview of some multibody codes and object-orientedtools, as well as the environmental requirements of AMESim. These requirementshave some implications on how the 2D library is built. Section 3 details the theoret-ical developments that enabled the mathematical models of the library componentsto be set up. Section 4 concludes this first part.2. Constraints of AMESim library philosophyAfter a brief overview of multibody code principles and some object-orientedtools, a presentation of AMESim requirements is given.Concerning multibody codes a state of the art is given by 23. Details are notreproduced here and readers are referred to this book for a more profound presen-tation. Although more than a decade has passed and certain tools are no longerdeveloped and others have changed, this state of the art book gives a good idea ofthe main principles that can be used as a basis for multibody codes. Also this over-view enables the library proposed to be positioned with respect to these codes. Thereare different approaches for writing dynamic equations. The approaches most repre-sented in multibody codes are, the NewtonEuler equations applied to each body,the NewtonEuler equations applied to sets of bodies, Lagrange?s equations andKane?s equations 13,14. The variables, in whose terms the dynamic equations arewritten, are either absolute coordinates or relative coordinates. Also supplementarymethods are used for reducing the index of the DifferentialAlgebraic Equations.The principal ones are the coordinate partitioning method, the projection matrixmethod, the Baumgarte stabilization and the penalty formulation 9. The first twomethods aim at working with a set of independent generalized coordinates whilethe Baumgarte stabilization enables the constraints, together with the differentialequations, to be handled and the penalty formulation increases the differential sys-tem order by introducing extra dynamics into the model.In the domain of the object-oriented tools to which AMESim may be attached,certain enable multibody systems to be treated with a different approach to the mod-elling. For instance Dymola 21 is, like AMESim, based on well-identified techno-logical components in a pluridisciplinary context but it sets up the mathematicalmodel in a different way. Basically each component model consists of equationsnot oriented in terms of variable assignments nor organized a priori. Then, at thecomponent connection stage, all the mathematical models are gathered in an implicitform and the compilation carries out the variable assignments and the organizationof the equations in a consistent manner. Thus the order of the whole model is glob-ally reduced and a number of constraints are a priori symbolically eliminated. Like-wise, tools based on bond graph (e.g. 20Sim 1 or MS1 18) can deal with multibodysystems in a pluridisciplinary context (e.g. 4,7). The essential feature of bond graphlanguage is its ability to describe the energy topology of a model at an acausal level.This enables all the model variables to be globally assigned and all the equations tobe globally organized. This also eliminates superfluous dependencies of the multi-body models.W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 254627It is now important to show the key features of AMESim to justify how the planarmechanical library has been implemented. Its feature oriented towards engineeringsystems and its user friendliness make AMESim work with well-identified technolog-ical components, symbolically manipulated by means of icons. These icons are inter-connected, one to the other and identically to the engineering system architectureunder study. Fig. 1 shows an example of a door locking system using a permanentmagnet modelled in AMESim. The icons displayed here belong to the magnetic,mechanical and signal libraries. This simple example shows the coupling betweenmechanical and magnetic domains where one circuit, fed by a permanent magnet(right-hand side magnetic circuit), is forced to move with respect to another passivecircuit (left-hand side circuit). The main components consist of a permanent magnet(rectangle with a compass needle inside), three magnetic circuit parts characterizedby a certain reluctance (rectangles with ?square? ports with a diagonal cross inside),two variable air-gaps (vertical twin rectangles), two mechanical nodes (both sides ofthe air-gap components), a signal generator with a signal-to-displacement converter(in the centre of the right-hand side circuit), and a component for the set of the mag-netic medium characteristics (BH diagram in a circle). Each component can beassociated with one model from a set of component compatible mathematicalmodels. As soon as the model has been chosen the component conserves thismathematical model.Contrary to acausal tools, AMESim works with component models that haveequations both a priori oriented in terms of variable assignments and organized. Thisfeature requires implementing new models in a predefined calculus scheme. Also themathematical formulation of a component model has to be organized in order to fitinto other potential component connections. So each component associated with amathematical model has a predetermined set of input and output variables. It canthus be considered as a causal model. The connection of the components enablesthe exchange of those variables on the way out a component for those variables thatare calculated by its mathematical model (outputs) and, the exchange of those vari-ables on the way in a component for those variables that are calculated by a con-nected component mathematical model (inputs). This causal feature of AMESimphilosophy is the main constraint when implementing new components. This differsFig. 1. Example of an AMESim model representation.28W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 2546from other object-oriented tools, based on acausal component models or acausalphenomenon models, like Dymola, or tools with a bond graph input (e.g. 20-Simor MS1).Fig. 2 gives an example of two components in the mechanical (a mass in transla-tion) and the hydraulic (a two way hydraulic pump) domains respectively. The con-necting ports of the components show the variables exchanged by them andespecially the outputs (?exiting? arrows) passed to the connected components andthe inputs (?entering? arrows) received from the connected components. These con-necting ports are intimately associated with power ports since two of the variablesexchanged at these ports are power variables.Fig. 2 examples illustrate two key features of a library oriented simulation tool.The first one is the domain port concept. It shows how AMESim can deal with plu-ridisciplinary systems. The second feature is the connecting port constraints. Sinceone component mathematical model requires given inputs to then calculate its stateand its outputs, not all combinations of the component connections are allowed. Forinstance the Fig. 2 examples cannot be connected one to the other by any port. How-ever a mass component may be connected to a spring component or a dampercomponent.Another key feature of a library oriented simulation software tool is the modular-ity concept. This often results in symmetrical components with respect to their con-necting port. This symmetry property, though not generalized to all components inAMESim, has been adopted for the planar mechanical library. The reason will ap-pear obvious when components of this library are presented.In the context of planar mechanisms and rigid bodies the library is not restrictedto any mechanical domain application. The library also accepts closed loop struc-tures. Although relative coordinates are generally more efficient for dynamic equa-tion formulation, AMESim philosophy requires the use of absolute coordinates.The absolute coordinates of the mass center have been chosen for each body. Nev-ertheless the planar feature of the library does not require any specific variables forthe body orientation. Thus the absolute angular position has been chosen for eachbody as well. Once again, due to AMESim philosophy, the equations of the compo-nents cannot be globally reorganized when the components are connected. This for-bids the use of the coordinate partitioning method or the projection method todecrease the index of the DifferentialAlgebraic Equation systems. For this reasonthe Baumgarte stabilization has also been used in the library.Fig. 2. Example of two AMESim components.W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 2546293. Theoretical developments of the library componentsAs has been explained in the previous section the library must be organized inwell-identified technological components. It has been decided to base the planarmechanical library on a body component and on joint components. The body com-ponent is associated with a supposed rigid material item of a mechanism. Its behav-ior is essentially governed by its kinetic state. The joint components are associatedwith the abstract items that represent the attachment of bodies in a mechanism. Theyare supposed to be ideal and their mathematical model is based on the constraintsthat they impose on the connected bodies.3.1. Body component mathematical modelThe mathematical model of the body component is based on Jourdain?s Principleformulation (e.g. 5,23)1:A? P?1where A* is the virtual power developed by the acceleration quantities and P* thevirtual power developed by the actions on the body.In the library philosophy there is no a priori privileged candidate for the roleof the generalized coordinates. For a planar motion, the generalized coordinates,which have been chosen, are the absolute mass center coordinates projected ontothe absolute frame of reference xGi;yGi and the absolute angular position hi(Fig.3). This choice enables the more general case of a body motion to be dealt with.The body motion restriction will be determined by the joint constraints, as shownlater.With this choice of generalized coordinates xGi;yGi;hi Eq. (1) members may nowbe writtenA? Ax_ x?Gi Ay_ y?Gi Ah_h?iP? Qx_ x?Gi Qy? mig_ y?Gi Qh_h?i2with mithe body mass and g the gravity acceleration. We consider here y0as theascendant vertical axis. The star superscript indicates virtual quantities. The coeffi-cients of the virtual velocities in A* are derived from the kinetic coenergy (e.g. 6)of a body by the equation:AqdoTdto_ q?oToqwith q a generalized coordinate31A nomenclature is given in Appendix A.30W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 2546Applied to Fig. 3 body in planar motion these quantities are written simply:Ax mi xGiAy mi yGiwith Iithe body moment of inertia around Gi; z0Ah Iihi4Qx, Qy, and Qhare the generalized forces including the constraint actions resultingfrom the fact that xGi, yGi, and hiare not necessarily independent after the connectionof a body component to a joint component. From Eq. (1) and by taking a compat-ible virtual transformation with the joints as they exist at time t, we can now writethe three identities that constitute the formulation basis for the body components.These three identities aremi xGi Qxmi yGi Qy? migIihi Qh5This formulation requires that the expression of the three generalized forces Qx, Qy,and Qhbe further developed in order to fit any potential connected joint component.First let us inspect the case of a body with only one connecting port at a point M.Let us also consider simply a given action on the body characterized by a wrenchabout point M (e.g. 17):fWg :F Fx x0 Fy y0forceMM Cz z0torque about point M(6The virtual power developed by this action isP?F ?V0?M MM ?X0?i7whereV0?M is the virtual absolute velocity of point M andX0?iis the virtual abso-lute angular velocity of the body. The velocity transport (e.g. 11) enables Eq. (7) tobe written as:Fig. 3. Schema of a body in planar motion.W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 254631P?F ?V0?Gi F ?X0?i? GiM? ?!?MM ?X0?iF ? _ x?Gi x0 _ y?Gi y0 MM GiM? ?!?F?_h?i z08From Eq. (8) we can clearly identify the generalized forces used in the dynamic for-mulation of a body component:QxF ? x0 FxQyF ? y0 FyQhMM GiM? ?!?F? z0 CzF ? z0? GiM? ?!9Since GiM? ?!is a characteristic vector of the body, the variables Fx, Fy, and Cz, char-acterizing the given force at point M, are the only variables passed to the body at theconnecting port. The variables Qx, Qy, and Qhare calculated in the body componentmodel on Eq. (9) basis.It is shown in the next section that the equation formulation for the generalizedforces (Eq. (9) applies for any type of joint component connected to a body compo-nent. The expressions of Fx, Fy, and Czvary with the type of the connected joint butare calculated in the joint component.3.2. Joint component mathematical modelFirst a general formulation is given for the joint component mathematical model.It is then illustrated in the example of a translational joint.Let us consider this time two bodies connected by a joint. By the only fact thatboth bodies are connected (a joint component between two body components) theirgeneralized coordinates (xGi, yGiand hifor body i and xGj, yGjand hjfor body j) areno longer independent. In the library philosophy the constraint equations are ex-pressed in the joint component, which in turn furnishes the constraint actions tothe body components. These constraint actions correspond to the variables Fx, Fy,and Czpreviously presented and passed to each body component. The generalexpressions of these variables are now determined.The joints considered in the planar mechanical library generate only geometricalconstraints. These constraints may be expressed in a general way in an implicit formby Eq. (10) (e.g. 15).gkq1;.;qn 0for k 1 to m10with n the number of generalized coordinates involved in the constraints and m theconstraint number.It is supposed here that the constraints are scleronomic 16, which means thattime does not explicitly appear in the constraint equations. At the kinematic levelthese equations become32W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 2546_ gkq1;.;qn; _ q1;.; _ qn ogkoqi_ qi 0for k 1 to m and with the Einstein implicit summation conventionon repeated subscript i11In the case of bodies connected by a joint the generalized coordinates areqTxGiyGihixGjyGjhj?12The library philosophy imposes working with a set of dependent coordinates. ALagrange multiplier is then associated with each constraint equation of the joint be-tween both bodies. Let kk(k = 1 to m) be these Lagrange multipliers. We now devel-op the expressions of the resulting constraint terms Fxi, Fyi, and Czion the body i sideand Fxj, Fyj, and Czj, on the body j side in terms of the Lagrange multipliers kk(k = 1to m) and the joint geometry.Let us first consider the Fig. 4 schematic representation of two bodies in a generalplanar motion (note that in the context of a planar motion, zi zj z0). The joint ischaracterized by the geometric axes M; xil and N; xjl respectively on bodies i and j(no peculiarity is shown at this point in order to keep the generality of the develop-ment). These two axes are defined by their relative positions in the correspondingbody frames (relative coordinates (xM,yM) and (xN,yN) with respect to body frames,respectively, for points M and N, and angular relative positions hiland hjlfor the vec-tor axes xiland xjl).The geometrical joints considered between bodies i and j may also be expressed bythe following equations (this conjecture is at least verified for joints considered in thelibrary)2:fkOM? ?!;ON?!; xil; yil; xjl; yjl? 0for k 1 to m13Next the kinematic equations corresponding to these geometrical constraints areobtained by differentiation with respect to time and in the same reference frame. Letthe absolute frame of reference be the frame of differentiation. The kinematic con-straints are (see the appendix for the calculus details and the notations used)fk;OM!? _ xGi x0 _ yGi y0?GiM? ?!?fk;OM!?fk; xil? xil?fk; yil? yil?_hi z0fk;ON!?_ xGj x0 _ yGj y0?GjN? ?!?fk;ON!?fk; xjl? xjl?fk; yjl? yjl?_hj z0 0for k 1 to m142Even if Eqs. (10) and (13) correspond to the same constraint equations, there is a fundamentaldistinction between their expressions. In fact gkmay be defined as a linear form on Rnwhile fkmay bedefined as a linear form on the Cartesian product of two dimensional vector spaces E6. Having dispelledthis ambiguity the distinction in the constraint notation is no longer applied in the rest of the paper.W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 254633It is worthwhile noting that all vectors in Eq. (14) must be expressed in the absoluteframe of reference.The constraint terms contributing to the generalized forces are obtained from thegeneral expression given byQqXmk1ofkoqkkXmk1o_fko_ qkk15Using the generalized coordinates defined for Fig. 4 bodies in planar motion andEq. (14) for the expression of the kinematic constraint equations, the constraintterms contributing to the generalized forces arefor body i:Qxi kkfk;OM!? x0Qyi kkfk;OM!? y0Qhi ?kkfk; xil? xilfk; yil? yil? z0kkfk;OM!? z0?GiM? ?!?8:for body j:Qxj kkfk;ON!? x0Qyj kkfk;ON!? y0Qhj ?kkfk; xjl? xjlfk; yjl? yjl? z0kkfk;ON!? z0?GjN? ?!?8:16with the Einstein implicit summation convention on the repeated subscript k.By analogy to the case of a given action (Eq. (9) the following variables can nowbe clearly expressed:Fig. 4. Geometrical characteristics of two bodies in planar motion.34W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 2546for body i:Fxi kkfk;OM!? x0Fyi kkfk;OM!? y0Czi ?kkfk; xil? xilfk; yil? yil? z08:for body j:Fxj kkfk;ON!? x0Fyj kkfk;ON!? y0Czj ?kkfk; xjl? xjlfk; yjl? yjl? z08:17These variables are calculated in the joint component and passed respectively tobody i and body j components that calculate the terms given by relations (18) tocomplete the generalized forces Qhiand Qhj.Fi? z0? GiM? ?!?withFi Fxi x0 Fyi y0for body iFj? z0? GjN? ?!?withFj Fxj x0 Fyj y0for body j18In turn each body component integrates its dynamic model and furnishes, to thejoint component, the absolute position of point M for body i and point N for body jand the absolute angular position of axis xilfor body i and axis xjlfor body j (Eqs.(19) and (20). The absolute position of a point M (respectively, N) is obtained withthe absolute position of the body mass center Gi(respectively, Gj) and the relativeposition vector of this point M (respectively, N) projected onto the absolute frameof reference. The absolute angular position of the axis xil(respectively, xjl) is simplythe sum of the body absolute angular position and the relative angular position ofthis axis xil(respectively, xjl) in the body frame. The relative position vectors ofpoints M and N and the relative angular positions of axes xiland xjlare parametersin the body componentsx0M xGi xMcoshi? yMsinhiy0M yGi xMsinhi yMcoshih0M hi hil8:19x0N xGj xNcoshj? yNsinhjy0N yGj xNsinhj yNcoshjh0N hj hjl8:20In the joint component the Lagrange multipliers kkremain to be calculated. Thisis addressed later in this section. Before this we will illustrate the above mathematicalformulation onto the example of a translational joint.W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 2546353.3. Illustration of the joint dynamics formulation on the translational joint exampleA translational joint between points M and N of two bodies may be schematicallyrepresented by Fig. 5.The geometrical constraints imposed by a translational joint may be expressed byf1OM? ?!;ON?!; yil? MN? ?!? yil ON?!? yil? OM? ?!? yil 0f2OM? ?!;ON?!; yjl? MN? ?!? yjl ON?!? yjl? OM? ?!? yjl 021An alternative to this formulation is, for instance, MN? ?!? yil 0 and xjl? yil 0 butEqs. (21) enable the symmetry property of the translational joint component to beconserved in its model formulation. Then Eq. (22) give the variables calculated inthe translational joint component and passed to the body components (see Eq. (17).for body i:Fxi k1? yil ? x0 k2? yjl? x0 k1sinh0M k2sinh0NFyi k1? yil ? y0 k2? yjl? y0 ?k1cosh0M? k2cosh0NCzi ?k1ON?!? OM? ?!? yilhi? z0 ?k1x0N? x0M?cosh0M y0N? y0M?sinh0M?8:for body j:Fxj k1 yil? x0 k2 yjl? x0 ?k1sinh0M? k2sinh0NFyj k1 yil? y0 k2 yjl? y0 k1cosh0M k2cosh0NCzj ?k2ON?!? OM? ?!? yjlhi? z0 ?k2x0N? x0M?cosh0N y0N? y0M?sinh0N?8:22Fig. 5. Schematic representation of a translational joint between two bodies.36W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 2546Furthermore the generalized forces due to the constraint terms in the body compo-nent mathematical models are calculated with Eqs. (16).Until this point bodies with only one connecting point have been considered. For-mulation development is strictly identical, as in those seen above, for bodies withseveral connecting points. Previous calculation is only lengthened proportionallyto the number of connecting points. The generalized forces due to different con-straints in this case are simply summed up for the number of connecting ports.Now the calculation of the Lagrange multipliers is addressed.3.4. Calculation of the Lagrange multipliersThis formulation, in dependent coordinates, led to the use of the Lagrange mul-tiplier method. The Lagrange multipliers are implicit solutions of the constraintequations embedded in the joint components and thus must be calculated in thesecomponents. AMESim enables implicit equations to be programmed, the dynamicsequations, however, globally obtained on a complete system, are generally a Differ-entialAlgebraic Equation system of index 3 10,22 with the geometrical constraintsprogrammed in this way. This DifferentialAlgebraic Equation index causes seriousand even critical problems for numerical resolution. Implementation of kinematicconstraint equations or even the second derivative with respect to time (accelerationorder) of the geometrical constraint equations would decrease the index. Howeverthis also causes a loss of information at the geometrical or even kinematic leveland the numerical solution has a high chance of diverging from the exact solution.In order to circumvent this problem, the Baumgarte stabilization 3 is used to cal-culate the Lagrange multipliers in an implicit manner. The Baumgarte stabilization isbased on a control scheme 20 of the constraint errors to make them convergetowards zero. Its calculus scheme is a PID control on the kinematic constraint equa-tions which then also use the constraint equations at the geometrical and accelera-tion orders (e.g. 8). In this manner all the information about the constraints iskept and the judicious choice of the PID schema gains enables the constraint errorsto converge towards zero.If Eq. (13) represents the geometrical constraint equations then the Baumgartestabilization calculus scheme is given by:fk 2a_fk b2fk 0for k 1 to m23where a/b and b may, respectively, be interpreted as a damping coefficient and a nat-ural pulsation of the solution for fk. The same numerical coefficients have been arbi-trarily chosen for all the constraint equations.The use of this stabilization scheme obliges the body component to supply an-other piece of information other than the information concerning the point M posi-tion and on the angular position of the joint axis. In fact the kinematic and theacceleration constraint equations are obtained from the differentiation, with respectto time, in the absolute frame of reference of the geometrical constraint equation(13). The first differentiation gives (see the appendix for calculation details andnotations)W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 254637fk;OM!? _ x0M x0 _ y0M y0?fk;ON!? _ x0N x0 _ y0N y0?_hifk; xil? yil?fk; yil? xil?_hjfk; xjl? yjl?fk; yjl? xjl? 0for k 1 to m24Eq. (24) shows that body components must also supply the absolute velocities ofpoints M and point N and the absolute angular velocities of both bodies (whichare equal to the absolute angular velocities of the joint axes). These quantities forpoint M (respectively, for point N by substituting i by j and M by N) are given by_ x0M _ xGi?_hixMsinhi yMcoshi_ y0M _ yGi_hixMcoshi? yMsinhi(25The second differentiation, with respect to time, in the absolute frame of reference ofthe geometrical constraint equations gives (see the appendix for calculation detailsand notations):fk;OM!? x0M x0 y0M y0?fk;ON!? x0N x0 y0N y0?fk;OM!2_ x0M2 _ y0M2?fk; ON!2_ x0N2 _ y0N2?2fk;OM!;ON!_ x0M_ x0N _ y0M_ y0N?hifk; xil? yil?fk; yil? xil?hjfk; xjl? yjl?fk; yjl? xjl?_h2ifk; xil? xilfk; yil? yil?_h2jfk; xjl? xjlfk; yjl? yjl?2_hi_hjfk; xil; xjlfk; yil; yjl?cos h0M?h0N?h? fk; xil; yjl?fk; yil; xjl?sin h0M?h0N?i?2_ x0M_hifk;OM!; xilsinh0Mfk;OM!; yilcosh0M?_hjfk;OM!; xjlsinh0Nfk;OM!; yjlcosh0N?2_ y0M_hifk;OM!; xilcosh0M?fk;OM!; yilsinh0M?_hjfk;OM!; xjlcosh0N?fk;OM!; yjlsinh0N?2_ x0N_hifk;ON!; xilsinh0Mfk;ON!; yilcosh0M?_hjfk;ON!; xjlsinh0Nfk;ON!; yjlcosh0N?2_ y0N_hifk;ON!; xilcosh0M?fk;ON!; yilsinh0M?_hjfk;ON!; xjlcosh0N?fk;ON!; yjlsinh0N?0 for k 1 to m26Eq. (26) shows in turn that the absolute accelerations of points M and N and theabsolute angular accelerations of both bodies (equal to the absolute angular acceler-ations of the joint axes) are needed for a joint component model where Baumgartestabilization is implemented. The acceleration of point M (respectively, for point Nby substituting i by j and M by N) is given by x0M xGi?hixMsinhi yMcoshi ?_h2ixMcoshi? yMsinhi y0M yGihixMcoshi? yMsinhi ?_h2ixMsinhi yMcoshi(27The peculiarity of each joint generally eliminates a number of terms in these equa-tions. Nevertheless Eqs. (24) and (26) enable generic kinematic and acceleration con-straint equations to be expressed in terms of the variables supplied to a jointcomponent.38W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 2546The kinematic and acceleration constraint equations are now illustrated on thetranslational joint example. The geometrical constraint equations in this case aregiven by Eqs. (21). In terms of the variables passed at the connecting port of the jointcomponent these equations become:? x0N? x0M?sinh0M y0N? y0M?cosh0M 0? x0N? x0M?sinh0N y0N? y0M?cosh0N 0(28The corresponding kinematic constraint equations are? yil?V0M yil?V0N ?OM? ?! ON?!?X0i? yil? 0? yjl?V0M yjl?V0N ?OM? ?! ON?!?X0j? yjl? 08:( )? _ x0N? _ x0M?sinh0M _ y0N? _ y0M?cosh0M?_hix0N? x0M?cosh0M? y0N? y0M?sinh0M? 0? _ x0N? _ x0M?sinh0N _ y0N? _ y0M?cosh0N?_hjx0N? x0M?cosh0N? y0N? y0M?sinh0N? 08:29Finally the acceleration constraint equations are? yil?J0M yil?J0N c0i? yil? MN? ?!? 2X0i? ? yilhi?V0M2X0i? yilhi?V0N 0? yjl?J0M yjl?J0N c0j? yjl? MN? ?!? 2X0j? ? yjl?hi?V0M2X0j? yjlhi?V0N 08:( )? x0N? x0M?sinh0M y0N? y0M?cosh0M?hix0N? x0M?cosh0M? y0N? y0M?sinh0M? 2_hi? _ x0N? _ x0M?cosh0M? _ y0N? _ y0M?sinh0M? 0? x0N? x0M?sinh0N y0N? y0M?cosh0N?hjx0N? x0M?cosh0N? y0N? y0M?sinh0N? 2_hj? _ x0N? _ x0M?cosh0N? _ y0N? _ y0M?sinh0N? 08:30It is worthwhile noting in Eq. (30) that the squared terms of the absolute angularvelocities of bodies vanish by using the geometrical constraint equations. Some termswill often vanish, particularly for the joints presented in the library.4. ConclusionThe purpose of this paper is not to develop a new mechanical theory or a new for-mulation for writing dynamics equations of planar multibody mechanical systemsW. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 254639(one may find valuable information on this topic in 19 or 12). Instead the issue wasmore to fit existing formulations to the AMESim requirements in order to build anew library in the domain of planar mechanical systems. The tool requirements wereprincipally modularity and communicability between components which must havepredefined mathematical models, in particular between body and joint components.This has led to a formulation in dependant coordinates, so the Lagrange multipliermethod was used. The generalized coordinates are the absolute Cartesian coordi-nates of the body mass centers and the absolute angular positions of the bodies.The body component mathematical model consists of the Lagrange equations writ-ten in terms of the previously given generalized coordinates. The joint componentsconsist of joint constraint equations from which the Lagrange multipliers, as implicitsolutions, are numerically obtained. Baumgarte stabilization schemas have beenimplemented onto the constraint equations in order to decrease the DifferentialAlgebraic Equation index without degrading the numerical solution.The main originality of the mathematical model implementation, caused by Ame-sim requirements, consists of geometrical constraint equations explicitly expressed interms of vectors and exploited in this form (Eq. (13) instead of expressed in terms ofscalars (Eq. (10). This was required to show the generic formulation of constraintactions and the variables received by body components (Eq. (9), whatever the con-nected joint component. In fact Eq. (10) would not have enabled terms (18) to bedeveloped, these terms are to be treated in the body component.The generic formulation proposed in this paper (in particular for joint con-straints), though dedicated to causal component mathematical models, may beused for systematically deriving the equations associated with joint constraints.One can even imagine a symbolical routine that generates these equations with thevector constraint as the only input. Concerning the use of Baumgarte?s stabilizationwith Lagrange?s equations and the Lagrange multiplier method, the formulation ismore classical. Here it shows how to split the different mathematical submodelsaccording to the modular and component approach of an object-oriented tool.Also it is quite well adapted to tools that do not globally re-oriente in terms of var-iable assignment and re-organize the mathematical model. Moreover, in the contextof object-oriented tools, one advantage of the formulation presented is that the gen-erated DifferentialAlgebraic Equation system is generally of index one even forclosed loop structures. In fact, by using Baumgarte?s stabilization, it was not neededto reformulate a posteriori the model that otherwise would have been at least ofindex three. The drawback to this approach is that the size of the state model is in-creased and it has, in addition, certain number of algebraic equations and implicitvariables.One logical perspective to the formulation proposed is its extrapolation to threedimensional mechanisms. There is no conceptual difficulty for using the same schemeby combining Lagrange?s equations, the Lagrange multiplier method and Baumg-arte?s stabilization as well as splitting the models into the same components, namelybodies and joints. Basically the main difficulty is intrinsic to the three dimensionalmechanism domain. This renders the generic formulation proposed in this papermore tedious. The main reason resides in the necessity of taking the body orientation40W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 2546into account. For planar mechanisms only the rotation around one absolute refer-ence fixed axis was needed. This has greatly simplified some mathematical develop-ments in the formulation. With three dimensional mechanisms it is necessary tochoose a set of generalized coordinates to represent the absolute orientations ofbodies. Euler?s angles or Euler?s parameters are examples of such generalized coor-dinates. With three of Euler?s angles the main drawback is the possible occurrenceof singularities where there is either non unique solutions or discontinuities. It is thennecessary to switch to another set of Euler?s angles. Otherwise Euler?s parameters arewell designed for these cases but there are four of them and so this requires an extraalgebraic relation to be taken into account in the formulation 24.Part II of this paper details the library composition and illustrates it with theexample of a seven-body mechanism 23. It also shows how actuating items areincluded in the joint components.Appendix A. Nomenclature a ?bscalar product between two vectors a ?bcross product between two vectors_ a dadt; a d2adt2first and second time differentiationsoaobpartial derivativea*virtual quantities avectoracolumn vectoraTmatrix transposeA*virtual power developed by acceleration quantitiesAacceleration quantitiesTkinetic coenergyP*virtual power developed by mechanical actionsQgeneralized forcesO; x0; y0; z0absolute frame of referenceGi; xi; yi; zibody frameMi; xil; yil; ziljoint privileged framexGi;yGiabsolute coordinates of the body mass centerhiabsolute angular body positionxMi;yMirelative coordinates of point Mihilrelative angular position of joint privileged framex0Mi;y0Miabsolute coordinates of point Mih0Mabsolute angular position of joint privileged framemi, Iibody mass, central moment of inertia about z0ggravity accelerationfWgwrenchFforce(Fx,Fy)force absolute coordinatesMMtorque about point MW. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 254641Appendix B. Notation conventions on partial derivatives with respect to a vectorWe introduce here a conventional notation for sake of conciseness. The vectornotedf aofo arepresents the partial derivative in the absolute frame of reference off with respect to the vector a. We suppose here that the constraint equations onlyconsist of sums of scalar products of the form a a ?b. In this casef a ab. The vectorcomponents off aare the partial derivatives of f with respect to the vector compo-nents of a. With the scalar product properties, we may manipulate the partial deriv-atives with respect to a vector as we do with partial derivatives with respect toscalars. Furthermore the partial derivative off awith respect to a second vector isa scalar noted f a;bof aob. In the same context, for constraint equations consisting onlyof sums of scalar products f a;b fb; a a. The notation introduced here differs fromthat in 19 for instance, in the sense thatf ais not treated as a row vector but as a fullvector. Finally we denote f a2 f a; a.Appendix C. Development of the kinematic and acceleration constraint equationsThe kinematic constraint equations are obtained by differentiation with respect totime in the absolute frame of reference of Eq. (13) (see appendix on notationconventions):dfkOM? ?!;ON?!; xil; yil; xjl; yjl?dt 0for k 1 to m( )ofkoOM? ?!?dOM? ?!dtofkoON?!?dON?!dtofko xil?d xildtofko yil?d yildtofko xjl?d xjldtofko yjl?d yjldt 0for k 1 to m( )fk;OM!?V0M fk;ON!?V0N fk; xil?X0i? xil?fk; yil?X0i? yil?fk; xjl?X0j? xjl?fk; yjl?X0j? yjl? 0for k 1 to m31Cztorque component on z0V0Mabsolute velocity of point MJ0Mabsolute acceleration of point MX0iabsolute angular velocity of body i c0iabsolute angular acceleration of body iqgeneralized coordinatesgk, fkgeometrical constraint expressionskkLagrange multipliersa, bparameters of the Baumgarte stabilization42W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 2546These equations are used for the joint component generic model (Eq. (24). The con-straint terms used in the body component are obtained by expressing Eq. (31) interms of the generalized velocities. Using the velocity transport from points M(respectively, N) to points Gi(respectively, Gj) achieves this.fk;OM!?V0Gi X0i? GiM? ?!?fk;ON!?V0Gj?X0j? GjN? ?!?fk; xil?X0i? xil?fk; yil?X0i? yil?fk; xjl?X0j? xjl?fk; yjl?X0j? yjl? 0for k 1 to m( )fk;OM!?V0Gi fk;ON!?V0Gj?GiM? ?!?fk;OM!?fk; xil? xil?fk; yil? yil?X0iGjN? ?!?fk;ON!?fk; xjl? xjl?fk; yjl? yjl?X0j 0for k 1 to mwhich leads to Eq. (14).Starting again from Eq. (31), their differentiation with respect to time in theabsolute frame of reference givesfk;OM!?J0Mfk;ON?!?J0Nofk;OM!oOMdOM? ?!dt?V0Mofk;ON!oON?!dON?!dt?V0Nofk;OM!oON?!dON?!dt?V0Mofk;ON!oOM? ?!dOM? ?!dt?V0Nofk;OM!o xild xildtofk;OM!o yild yildtofk;OM!o xjld xjldtofk;OM!o yjld yjldt01A?V0Mofk;ON!o xild xildtofk;ON!o yild yildtofk;ON!o xjld xjldtofk;ON!o yjld yjldt01A?V0Nfk; xil?dX0i? xil?dtfk; yil?dX0i? yil?dtfk; xjl?dX0j? xjl?dtfk; yjl?dX0j? yjl?dtofk; xiloOM? ?!dOM? ?!dtofk; xiloON?!dON?!dtofk; xilo xjld xjldtofk; xilo yjld yjldt !?X0i? xil?ofk; yiloOM? ?!dOM? ?!dtofk; yiloON?!dON?!dtofk; yilo xjld xjldtofk; yilo yjld yjldt !?X0i? yil?ofk; xjloOM? ?!dOM? ?!dtofk; xjloON?!dON?!dtofk; xjlo xild xildtofk; xjlo yild yildt !?X0j? xjl?W. Marquis-Favre et al. / Simulation Modelling Practice and Theory 14 (2006) 254643ofk; yjloOM? ?!dOM? ?!dtofk; yjloON?!dON?!dtofk; yjlo xild xildtofk; yjlo yild yildt !?X0j? yjl?0for k1tomUsing the hypothesis of constraint equations only constituted by a sum of vectorscalar products, we may also state that these scalar products never imply axes of thesame joint privileged frame (e.g. xil? xilor xil? yiland the same with substitution of iby j). This explains that terms such asofk; xilo xil,ofk; xilo yil,ofk; yilo xilandofk; yilo yil(and the same withsubstitution of i by j) do not appear in the previous equations. Using again the nota-tion convention of the partial derivatives with respect to a vector we obtainfk;OM!?J0M fk;OM!?J0N fk;OM!2V0Mhi2 fk;OM!2V0Nhi2 2fk;OM!;ON!V0M ?V0N 2 fk;OM!; xilX0i? xil?fk;OM!; yilX0i? yil? fk;OM!; xjlX0j? xjl?fk;OM!; yjlX0j? yjl?V0M 2 fk;OM!; xilX0i? xil?fk;ON!; yilX0i? yil? fk;ON!; xjlX0j? xjl?fk;ON!; yjlX0j? yjl?V0N fk; xil c0i? xil?X0i?X0i? xil?hifk; yil c0i? yil?X0i?X0i? yil?hifk; xjl c0j? xjl?X0j?X0j? xjl?hifk; yjl c0j? yjl?X0j?X0j? yjl?hi fk; xil; xjlX0j? xjl? fk; xil; yjlX0j? yjl?hi?X0i? xil? fk; yil; xjlX
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