Generalized analysis method of engine suspensions based on bond graph modeling and feedback control theory

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1 Generalized analysis method of ine susensions based on bond grah modeling and feedback ntrol theory P.Y. RICHARD *, C. RAMU-ERMENT **, J. BUION *, X. MOREAU **, M. LE FOL *** *Equie Automatique des ystèmes Continus et Hybrides uélec - Camus de Rennes Avenue de la Boulaie, BP 8, F-55 Cesson-évigné Cedex FRANCE **Laboratoire d Automatique et de Productique EP 6 CNR Université de Bordeaux - ENERB 5, Cours de la Libération, F-45 Talence Cedex ***CF Gomma Barre-Thomas A 94, Route de Lorient, F-5 Rennes Abstract - The issue of vibration ntrol in automotive industry has often been illustrated using the simle aradigm of a srung mass. ome revious work introduced a method based on the well-known rinciles of feedback ntrol in order to formalize as well as to analyze this roblem. It states that a srung mass essentially behaves like a closed loo system in which the mass stands for the rocess, while its susension lays the role of a regulator. The great simlicity of this method makes it very attractive. Yet, its rimary formulation only alies to susensions without inertia. Using bond grah modeling, this aer shows that the aroach can be made more general and extended to the analysis of any susension device, at the rice of a slight adatation. The case of a hydraulic ine mount is then resented as an alication examle. The generalized method roosed makes it easy to otimize such a system as for the stability robustness of the ntrol it realizes. Key-words - Vibrations, ine mount, bond-grah, feedback ntrol.. Introduction Vibration ntrol has recently emerged as one of the most imortant issues of research in automotive industry. Among the vibration sources that affect the dynamics of a vehicle, we essentially find the road (through its unevenness) and the ine. Both of these sources must be isolated from the vehicle me in order to ensure its mfort [], []. As a nsequence, two distinct devices have to be develoed the main vehicle susension, which is dedicated to isolating the me from the wheel train, and the ine one. In this aer, we shall articularly be interested in the latter. However, the fundamental rinciles remain the same in either case. The aer is organized as follows in section, we focus on a generic high level scheme of a susension system with its environment, restricted to one dimension. tarting from this first model, we briefly recall in section the analysis method reorted in [4], which is based on a articular closed-loo reresentation of the whole system (srung mass lus susension). Then, using a bond grah model instead, we derive in section 4 a new closed-loo reresentation, which is a more general one. Thus we rove that the original aroach is only suitable for susensions without inertial effects. In a last section, we study the alication examle of a vehicle ine srung by means of a hydraulic mount. Taking benefit from the generalized reresentation obtained, we show that the whole system analysis can be erformed ractically the same way as in the original aroach. We secially emhasize the stability robustness of the imlicit ntrol achieved by the mount.

2 . Basic model of a susension system Much work has already been done in order to model susension systems. Considering just one dimension of motion, the simlest model available is a srung mass, as deicted in Fig.. Needless to say that this model is a very general one, which uld as well reresent both kinds of susensions referred to above. However, the notations we adot rresond to the case of an ine susension F M F Z F Z exc mass dislacement force exerted by thesusension uon theme me dislacement adot it as a starting oint, since it is widely sread as well in the literature as in the industrial ractice. everal quantities may be used in order to characterize the susension stiffness. First, equation () defines what will further be called the theoretical stiffness. Cth F Z Z () An exerimental stiffness should also be introduced, as given by equation (). It rresonds to the usually externalexciting force measured transfer function when using a servohydraulic testing machine, namely the one between a srung mass sinusoidal dislacement injected on the ine side and force exerted by thesusension uon themass the resulting force exerted uon the me when the latter is ket unmoved. F C ex () Z Z Engine Z Z M F exc Frame F usension F Fig. srung mass basic scheme The inuts of this system are resectively the force F exc and the dislacement Z.. Feedback ntrol theory based rimary formulation The central oint of the method roosed in [4] nsists in nverting the mechanical scheme of Fig. into a block diagram with such a structure that the rinciles of feedback ntrol theory beme alicable for analysis. For that urose, it is necessary to define the transfer functions of both entities in the system. From an external oint of view, the susension is usually nsidered as a stiff system. As will be seen later on, such an aroach is a quite reducing one, which may neglect some imortant hysical effects of real devices, esecially when frequency increases. We shall however If we admit that the susension does not include any inertia, then it generates a unique effort, which exerts as well on the mass as on the me, and whose magnitude only deends on its deflection. In such nditions, it is clear that the susension behaves like a singleinut/single-outut system. Under the assumtion of full linearity, the theoretical stiffness C th defines its inutoutut transfer function. Moreover, the exerimental stiffness C ex is equal to C th. Hence the whole system can be modeled by a closed-loo block diagram analog to that of a ntrol loo, acrding to Fig.. The srung mass lays the role of the ntrolled rocess, while the susension lays that of the ntroller. F exc F ( ) C th M Z Fig. srung mass basic closed-loo model Z Thus it aears that the action of the susension uon the srung mass can be interreted as a feedback ntrol. The erformances of this ntrol are straightforwardly deduced from the roerties of the associated oen-loo transfer function, the exression of which is given by equation (). OL th () M C

3 4. Generalized formulation When no restrictive assumtion is made about the susension, the whole mass lus susension system can be modeled by means of a high level bond grah, where the susension itself is reresented as a -ort field, acrding to Fig.. e F exc I M Z & F F & f Z Fig. srung mass high level bond grah In is turn, this bond grah can be transformed into a closed-loo block diagram, as deicted in Fig. 4. stands for the susension transfer function between its, inuts Z & ( ), Z & ( ) and its oututs ( ) F F. It is a -dimension square matrix, the four elements of which can be exressed as the singleinut/single-outut transfer functions (4). F F F F Their symbolic mutation is ossible from a hysical level bond grah model of the susension. Then, making them aear searately, one can develo the closed-loo block diagram of Fig. 4. Furthermore, if one chooses as new inuts of the susension sub model the srung mass and the me resective dislacements instead of the rresonding velocities, one gets the final block diagram of Fig. 5. F exc M F F (4) Fig. 4 srung mass generalized closed-loo model M Z F F exc F Z Fig. 5 srung mass generalized and develoed closed-loo model ince no articular assumtion has been made about the susension in the resent aroach, the resulting model is erfectly general. It obviously aears to be rather different from that of Fig.. In fact, both models only beme equivalent rovided the following structural roerty The former model must therefore be nsidered as a very articular case of the latter. Yet the imortant roerty highlighted by the rimary formulation, which states that a susension naturally erforms a regulation of its deflection near the static equilibrium osition, is still verified in the general case. But it should be noticed that the characteristics of the feedback ntrol achieved by the susension, esecially its stability robustness degree, only deend on. Indeed, Fig. 5 shows that the rresonding oen-loo transfer function is given by equation (5).

4 OL (5) M M If we mare this exression to the one we had obtained in the ideal case of a susension without inertia, we can nclude that the transfer function which characterizes the susension as a ntroller for the rocess nstituted by the srung mass has the following exression C s (6) This transfer function is homogeneous to a stiffness. That is why we choose to call it the seudo-stiffness. It C s is must be underlined that in the general case, ( ) different from the exerimental stiffness, the exression of which is defined by equation (7). C ex (7) The incidence between both revious entities imlies the structural roerty, which is strictly fulfilled in the case of a susension with no internal mass. From an analytical oint of view, C s also differs from the theoretical stiffness, since the symbolic exression of the latter is C th Z Z Z Z (8) Besides, one can observe that C th does not reresent a true transfer function, since it deends on the magnitudes of Z and Z, unless either the new structural roerty be satisfied (which is once again true for a susension with no mass), either the bounded ndition Z & bât be imosed. Then, and only then, C th identifies itself with its exerimental unterart C th (9) 5. Alication examle Acrding to the notations already used, we now exlicitly nsider the srung mass as that of a vehicle ine. Moreover, we suose its susension to be realized by means of a hydraulic mount as deicted in Fig. 6. This device is a rubber mount that includes an inertia track without deuler. everal models have already been roosed for such a mount []. The one we use here is a hysical level bond grah. It has been extracted from a more mlex reresentation reorted in a revious ntribution [5]. We recall it in Fig. 7, with all its bonds being numbered for ractical reason. Fig. 6 hydraulic ine mount scheme Once causality has been assigned using CAP, a formal analysis of this bond grah under the assumtion of full linearity analytically yields the four inut-outut transfer functions (). e f e f7 e7 f e7 f7 () In ractice, one first has to derive the junction structure equations, which is done in a systematical fashion rovided the causality. Then nstitutive relationshis of the bond grah terminal elements have to be added. Each of them should be written in a symbolic form, using Lalace variable instead of time variable t, as a causally oriented relationshi between ower variables. At this stage, some redundances must be cancelled. Thus one gets a mathematical model exressed by a minimal set of algebraic equations. Alying Mason algorithm, one can reduce those equations between the different inuts and oututs, in order to exress,,, as rational functions of argument, with efficients being themselves functions of the model hysical arameters. The result is given by equations () to (4). The denominator is of urse the same for all these transfer functions. It can be seen that and are rigorously equal. Besides, the analysis of their mmon exression shows that they

5 structurally only differ from the oosite of through one efficient of the numerator that of the term with degree in. In return, must be clearly distinguished from. Only the efficients of terms with degrees and in their resective numerators are identical. C R bc k c F 4 f z& F 7 f z& 6 C C I I 6 4 TF TF R R TF TF TF C C I I I I Fig. 7 hydraulic ine mount hysical level bond grah ( C C ) ( bc ( C C ) R C C ) ( C ( I I ) R bc ) ( I I )) bc C C ( I I ) ( C C R C C C C ( I I ) ) c ( C C ) ( bc ( C C ) R C C ) k c C () k C ( C ( I I ) R bc ) (( ) I ( ) I )) bc C C ( I I ) C C R C C C C ( I I ) ( ) () () c ( C C ) ( bc ( C C ) R C C ) ( C ( C ( I I ) R bc ) ( ) I ( ) I ) C ( I I ) 4 C C ( bc ( I I ) R ( I I ) C C ( ) I I ( C C R C C C C ( I I ) ) k (4) It is interesting to examine the nsequences of the revious symbolic observations from a numerical oint of view. For this urose, we choose a tyical set of values for the mount arameters, which rresond to an existing device 4 4 I,8 kg/m,9 m 4 4 I,98 kg/m,9 m I 7,9 kg/m,75 m

6 5 kc,5 N/m C,55 m /Pa C, m / Pa 8 R,6 Pa s/m bc 46, N s/m It rovides the transfer function magnitude and hase lots of Figs. 8 and 9 (where linear scales have been chosen). As highlighted by these lots, the mount hysical arameters values are such that and remain ractically oosite to at any frequency in the useful domain. In return, distinctly deviates from as soon as the frequency exceeds Hz. o it is necessary to kee the me motionless to ensure the double roerty C C Cth ex s As a nclusion, the rimary formulation of section is not alicable to the case of the hydraulic ine mount, unless the vehicle me does not move (which is of urse unrealistic). Now, ming back to the generalized formulation develoed above, we shall examine, as an analysis examle, the mechanical feedback ntrol imlicitly achieved by the mount from a stability robustness oint of view. tability robustness is a roerty that characterizes the mount ability to ensure the ine vibrations isolation in site of ossible srung mass variations. This roerty directly deends on the oenloo transfer function OL C s (5) M M Using the analytical exression of, one is able to write the oen-loo transfer function in terms of the mount hysical arameters, as rovided by equation (6). Both the Bode diagram and the Black-Nichols chart of Fig. grahically describe its frequency roerties. These lots show that the nsidered mount is not otimally sized as for the stability robustness of the feedback ntrol it oerates. Indeed, the associated oen-loo transfer function does not exhibit a maximum hase at the frequency where its magnitude is unity. In order to ensure such a desired roerty, one theoretically just has to exress both frequencies where the magnitude is unity and the hase is maximum resectively as functions of the mount hysical arameters. Then, matching both resulting exressions, one gets a nstraint equation where a set of the mount arameters are involved (those which have an influence uon the resective ositions of both revious frequencies). In ractice, such a ste unfortunately exceeds the caabilities of the available symbolic mutation tools. o it cannot be erformed. However, without exlicitly knowing the resulting equation, some benefit might be taken from the simle fact the latter exists, in order to find out another way to otimize the mount. Thus, if it is ossible to write the revious equation as one of the arameters exlicit exression in terms of the others, then the otimality roerty can be ensured by just modifying the value of this articular arameter. Now such a ndition is quite a realistic one rovided that many degrees of freedom are a riori available (one equation for ossible arameters). x 4 N.s/m ntinuous line,, dotted line Hz Fig. 8 susension transfer function gains, Hz Fig. 9 susension transfer function hases

7 OL 5 ( C C ) ( bc ( C C ) R C C ) ( C ( I I ) R bc ) ( I I )) bc C C ( I I ) M ( C C R C C C C ( I I ) ) k c C (6) Bode Diagrams nominal value of k c by a factor equal to,7 one gets a new otimized Bode diagram, and a new associated Black-Nichols chart, as deicted in Fig.. Phase (deg); Magnitude (db) Frequency (rad/sec) Nichols Charts Phase (deg); Magnitude (db) Bode Diagrams Frequency (rad/sec) Oen-Loo Gain (db) - 6 Nichols Charts Oen-Loo Phase (deg) Oen-Loo Gain (db) - Fig. actual system oen-loo As a nsequence, one can reasonably hoe it will be ossible to deal with the mount otimization roblem without having to solve the nstraint equation, by means of a simle PNL algorithm alied to an adequate arameter. Precisely, k c, which denotes the mount stiffness, seems to be a good candidate in this resect. From a hysical oint of view indeed, one can redict that this arameter will have more influence on the oen-loo transfer function magnitude than on its hase, since it is intuitively linked to the mount static stiffness. A sensitivity study nfirms this idea it shows that an increase of k c raises the frequency where the loo gain magnitude is unity but hardly modifies the one where its hase is maximum (the rresonding maximum, however, is reduced). Multilying the Conclusion Oen-Loo Phase (deg) Fig. imroved system oen-loo This aer focuses on a articular aroach to formalize the generic roblem of a srung mass, in order to facilitate its analysis. Based on a feedback ntrol interretation of the whole system, this aroach has first been reorted in revious ntributions. We have ointed out the insufficiency of its rimary formulation, which cannot take any class of susension devices into

8 acunt. Then, relying on bond grah modeling, we have roosed a new extended formulation, where the susension still lays the role of a ntroller with regard to the mass. Yet we have shown that in the most general case, the former monent should not be characterized any longer by its stiffness, as this entity is usually defined and/or measured, but rather by a so-called seudo stiffness, which is equal to the transfer function between the mass dislacement and the force it exerts uon the susension. Lastly, the alication of our generalized aroach to the examle of a vehicle ine susended by a hydraulic mount has allowed imroving the whole system robustness stability. References [] A. Genesseaux, A New Generation of Engine Mounts, AE Paer, N 9596,. 5-58, 995. [] D. Hrovat, urvey of Advanced usension Develoments Related Otimal Control Alications, Automatica, Vol., N, , 997. [] M. Le Fol, Concetion d une cale ilotée synchrone our la susension des groues moto-roulseurs à 4 cylindres en ligne, Journées Automatique et Automobile, Bordeaux, 999. [4] X. Moreau, Chater of the research roject final reort Non-integer differentiation in vibration isolation, GDR Automatique, Bordeaux, 999. [5] P.Y. Richard, J. Buisson, Y. Quenec'hdu, Modeling and simulation of ine susension units, ystems Analysis modelling imulation,. -8, 997.