OPTIMIZATION OF AN ENGINE MOUNTING SYSTEM FOR VIBRO-ACOUSTIC COMFORT IMPROVEMENT

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1 OPTIMIZATION OF AN ENGINE MOUNTING SYSTEM FOR VIBRO-ACOUSTIC COMFORT IMPROVEMENT Marco La Civita ad Aldo Sestieri Departmet of Mechaics ad Aeroautics Uiversity of Rome "La Sapieza" 184 Rome, ITALY ABSTRACT I this paper a optimizatio procedure for egie moutig systems is developed to provides good comfort coditios. The algorithm follows basic assumptios adopted by a importat producer compay, but a more geeral formulatio is developed that provides very iterestig results. Differet solutio procedures are used showig that the approach based o Geetic Algorithms is geerally more efficiet 1 INTRODUCTION Improvemet of vibro-acoustic comfort i automobiles is becomig more ad more demadig i NVH desig so that big efforts are accomplished to obtai ay possible reductio of oise ad vibratio. A major source of vibro-acoustic discomfort, especially at low frequecies, is the egie moutig system that geerates both disturbig vibratios ad upleasat boom oise i the vehicle's compartmet. For ay type of automobile a differet threshold level is established that should ot be overcome. Of course the problem is quite complex because soud ad vibratio are both frequecy depedet. However whilst vibratio displacemets usually decrease with frequecy, the acoustic problem is more itricate because it ivolves the quality of oise that would establish which spectrum shape is more or less fastidious for the huma ears. Sice this subject is o stadard ad much work is ecessary to obtai a sufficiet kowledge o it, usually oe oly tries to miimise the db oise level i the whole frequecy bad of iterest. I this paper a optimizatio of suitable desig variables of the egie supports is proposed. Because of the oliear character of the problem ad the may desig variables ivolved, a appropriate model of the system is required as well as the defiitio of a suitable objective fuctio accoutig for the comfort stategy adopted, that is a choice of the automobile's producer differet strategies drive differet solutios. Most of the available works o the subject try to reduce the egie forces by shiftig appropriately the atural frequecies of the system! the approach is ot simple ad a reasoable result relies o a suitable defiitio of the desig goal. Heyes et al.l cosider directly the forces trasmitted from the egie to the vehicle body ad defie a acceptable egie motio. Automobiles' producers use their ow approachs that are ot usually available i the literature. We follow here the mai lies of a procedure developed by FIAT. 2 ANALYSIS OF THE SYSTEM The cosidered system ca be divided ito the three classical subsystems source, trasmissio paths ad receiver. The source geeratig both the vibratios ad the oise i the iteral compartmet (structure bore soud) is the egie system ad particularly the ubalaced forces arisig i the kiematic motio of the pisto-coectig rod mechaism. Typically, i a four-cylider, four-stroke egie, with a 18 simmetric shaft, as the oe cosired i the preset case, the first order iertia forces ad torques are balaced, while the secod order disturbaces are ot. Therefore they are resposible of the vibratio pheomeo, together with the torque oscillatios due the gas actio o the pisto. The trasmissio paths from the source to the receiver are the mouts betwee the egie ad the chassis. Sice the mouts should provide a somehow hard groud of the egie block to the chassis structure, while reducig the forces trasmitted from the egie to the chassis, they are made of elastomers presetig a appropriate force-strai characteristic. Because of these cotrastig requiremets, the choice of the mouts is ot a easy task to avoid iterferece betwee the egie ad the chassis durig limit ruig coditios such as bumps ad sudde brakes, a rigid mout would be required, while a low stiffess would be ecessary to isolate the egie from the chassis. To obtai this goal a oliear hardeig stiffess ca be used, with a first low stiffess liear part ad a icreasig oliear behaviour to boud the maximum dis- 1998

2 placemet ad avoid shock at the ed of the stroke. The receiver is simply the ma, perceivig the oise by his ears ad the vibratios through his eds. 3 OPTIMIZATION PROCEDURE To icrease the iteral comfort a optimizatio procedure must be used, i.e. a objective fuctio must be specified together with appropriate costraits. Some used typologies of objective fuctios are described below. The objective fuctio ca accout appropriately for the trasmitted forces from the egie to a rigid structure through the mouts. Typica_l]y, i such a cases, oly few frequecies of the trasmitted forces are cosidered[ This approach does ot accout for the elastic behaviour of the vehicle system; moreover, although it miimizes the forces trasmitted through the mouts, it does ot ecessarily provide the optimal respose i the poits of iterest. The objective fuctio pealizes the presece of atural frequecies of the egie-mouts system (a 6 dots system) withi the frequecy bad of the forcig terms[ 6 1. The limitatio is the same as the oe cosidered above, with the icoveiece of a o direct correlatio with the forces miimizatio through the mouts. The objective fuctio accouts oly for the miimizatio of the acoustic pressure i the vehicle compartmet[ 7 1. The objective fuctio accouts oly for the miimizatio of vibratios[ 8 1. It is worthwile to poit out that this last choice strikes with the former oe (acoustic comfort) i terms of the moutig stiffess, if separate optimizatios are performed. The approach used at CRF (Cetro Ricerche Fiat) aalyses both the vibratio ad acoustic effects, accoutig for several load coditios ad a high umber of costraits. However they strictly divide the oliear characteristic of the mouts ito two liear stiffesses; moreover the solutio is searched by a extesive parametric aalysis kow as DOE (Desig of Experimets)[ 9 1. I this paper we used the strategy proposed by FIAT ad the moutig scheme adopted by them for the refereced vehicle. It cosists of 3 mouts, two of which, characterized by three differet stiffesses alog three axes, are put as close as possible to the axis alog which the torques are applied ad devolve to aother mout, characterized by a sigle axial stiffess (rod-like mout), the task of avoidig egie rotatio. We followed the CRF approach with respect to the objective fuctio ad costraits, but formulated a more complete mathematical model ad used directly a oliear optimizatio algorithm. 4 MATHEMATICAL FORMULATION Two differet models were cosidered to defie the objective fuctio for the oise ad vibratios, respectively. The logical scheme to determie the variables of iterest is the followig at the gear correspodig to the regime coditio of the vehicle, oe loads the egie with a give percetage of the maximum available torque which is cosidered costat i time (static aalysis). This allows to determie the workig poits of the mouts o their characteristic curves. The stiffesses are cosequetly liearized by the value of the taget at the workig poit. By usig these stiffesses ad the dyamic excitatios due to iteral ubalace of the egie, a procedure is developed to compute the forces trasmitted from the egie to the chassis through the mouts. Whe the stiffess values are multiplied by the acoustic trasfer fuctios (ATF's) of the fitted vehicle, the soud pressure i the poits of iterest is determied. This output provides the first part of the objective fuctio. For the foot-board vibratios, a differet procedure is followed. I this case it is ecessary to reduce the acceleratio due to the egie shake. The egie shake is felt by the driver as a fastidious vibratio of the foot-board ad it is due to a dyamic iteractio betwee the vehicle suspesios ad the egie moutig, ragig from 6-8 Hz util Hz it appears as a aomalous isolated peak i the trasfer fuctio modulus betwee the groud displacemet ad the vertical displacemet of the chassis. I order to determie the appropriate trasfer fuctio providig the secod part of the objective fuctio, the system egie block-mouts-chassis- suspesios-wheels is described by a 13 dots system (figure 1 ) 6 dots for the egie (m i the figures) correspodig to 3 traslatios ad 3 rotatios, 1 for the vertical motio of each wheel (r i the figure), 3 for the chassis correspodig to vertical traslatio, pitch ad roll. Four differet referece systems are used. They are z X Figure 1 Model for the egie shake 1999

3 a fixed vehicle coordiate system 'Rv with the x-axis parallel to the ceterlie chassis axis ad directed opposite to the movig directio, the y-axis parallel to the forward wheel axis ad directed from the left to the right wheel, ad the z-axis perpedicular to x ad y; a fixed referece frame 'R, parallel to 'Rv, with origi located at the ceter of mass of the egie; a egie referece frame 'Re with z-axis parallel to the crakshaft, directed toward the gearbox, x-axis coicidet with the cylider 1 axis ad y axis perpedicular to x ad z such to form a right frame; a mout frame 'Rm, with z-axis parallel to the mout axis, ad the other two orthogoal to z. 4.1 Model for the oise Statics The workig poit of the mouts is give by the displacemet ad rotatio of the egie block uder the actio of its weight, the elastic forces of the mouts ad the torque i the ruig load coditio. Of course the torque is depedet o the agular velocity of the egie, but, for a sake of simplicity, its average value is cosidered. For the equilibrium of forces ad torques i 'R oe has 2.." i'; - mg = Ic; +cqs = Cqs is the quasi-static torque previously discussed. The ukow is the vector iic 1 whose compoets are the traslatios ad rotatios of the ceter of mass i the R frame. Therefore oe must express the terms i the equilibrium equatios i fuctio of iic. For the i-th mout, the elastic force due to a displacemet u is 2 r = s(ui)" + Aiui (2) where 3 ('. ~;= ~ ~I ) y e~ c., Ai=.X' 'y.x' ) Usually the power i eq. (2) assumes oly odd ad positive values, such that the characteristic curve of the mouts 1 We use the tilde to represet a vector havig both traslatioal ad rotatioal compoets, ad the bar to represet a vector with either traslatios or rotatios 2 By power of a vector we mea the power of all its compoets. 3 The prime (') deote variables i the mout referece... (1) (3) (4) resembles physically a hard stiffess typically the values 5 or 3 better describe a more or less hard sprig. Sice the equatios are solved i the 'R frame, it is ecessary to perform a coordiate trasformatio from 'Rm, ito that is obtaied by multiplyig the compoets i 'Rm, by a suitable trasformatio matrix E>;, thus obtaiig u; = E>,u (5) They ca be substituted ito eq. (2) to yield the elastic force f; of the mout i terms of the displacemet u; coectig mout ad egie i the geeral referece To express u; i terms of the ukow iic a oliear relatioship ivolvig trigoometric fuctios would be ecessary. Assumig however that both traslatios ad rotatios are small, oe simply obtais (7) where aj. is the compoet of the i-th mout positio vector alog the j-axis. By substitutig eq. (7) ito (6) oe obtais f; i terms of iic To solve the set of equatios (1) it is still ecessary to determie the torques c; provided by the mouts. By usig the small displacemets assumptio, oe ca write [ f;]- _- (6) (8) c; = f; = A;f; (9) ad thus c; ca be determied. Oce solved the equilibrium set of equatios, iic is obtaied by equatio (7) together with the mouts workig poits uder the give load coditio. The value of the stiffess K; used i the dyamic computatios is obtaied by the taget to the characteristic curve of the i-th mout i the workig poit so far determied Therefore, from equatio (6) K- df; I '- du;.. work,g p1t (1) (11) where "diag" deotes a operator trasformig the give vector ito a diagoal matrix. 2

4 Dyamics First let us trasform K; (eq. (11 )) ito Ki i the mout referece Rm, usig 8;. It is coveiet to add to this stiffess aother term of viscous dampig for the mout i. Deote with K~; such ew quatity (12) (3 is a appropriate coefficiet accoutig for the dyamic stiffeig of the mout at low frequecies (e.g., at about 1Hz, (3 ca rage from 1.1 to ). By usig agai the assumptio of a small egie motio, a dyamic liearised equatio ca be writte, i.e. Miic(t)=fd(t)+ Lf;(t) (13) where fd are the excitig forces resposible of the egie vibratios ad due to the secod order ubalaced disturbaces. M is the iertia matrix give by m m 1=1 m M= (14) fxx fxy fxz fxy lyy fyz fxz fyz fzz By Fourier trasformig equatio (13) oe obtais -w 2 Muc(w) = fd(w) + Lf;(w) (15) The sum i equatio (13) ca be expressed i ters of iic as see i the previous sectio. First a coordiate trasformatio is performed from K~; i Rm, to Kd, i. The oe writes ad, by meas of equatio (7), oe has A; Kd; lli = A; Kd; A; T Uc (16) By substitutig this relatioship ito equatio (15), the ukow iic is obtaied. The forces trasmitted to the vehicle are the elastic forces through the mouts, so that oe has (17) This equatio provides the iputs to the ATF's so that it is possible to determie the oise pressure p(w) i ay poit of the compartmet by p(w) = 2sum[ATF(w)fv 1 (w)] (18) i=i where "sum" deotes the sum of the whole vector compoets i the argumet. 4.2 Model for vibratios For the egie block ad the mouts the model is the same as the oe used for the aalysis of oise. The chassis is cosidered as a rigid body due to the low frequecy rage of iterest. While for the egie 6 dofs must be cosidered, for the chassis oly 3 are ecessary (vertical traslatio, pitch ad roll) to represet correctly the egie shakel 9 1. Similarly a simple model ca be used for the wheel-suspesio system, with a sigle dof it is represeted by a mass ad a sprig accoutig for the wheel mass ad the tire's stiffess i parallel with a sprig ad a damper accoutig for the suspesio system (figure 2). The equatio for this subsystem is w 2 mrz2 + ks(z3- z2) + JWCs(Z2- Z3)- kr(z2- zi) = (19) where z1 is the road iput ad z3 is the displacemet of the coectio poit with the chassis (figure 2). FRF Computatio z3 Figure 2 Wheel-suspesio subsystem The wheel-footboard FRF ca be obtaied by couplig the equatios of all the the subsystems ivolved. Cosiderig agai small displacemets ad rotatios, the dyamic equatio of the chassis ca be writte as ad, Fourier trasformig (2) 4 -w 2 Mviiev(w) = Lft,(w)+ Lfs 1 (w) (21) i=j where lie. 4 is the vector of displacemet ad rotatios of the chassis ceter of mass, i.e. ad J=l mv Mv= fxx ) (!yy 4 Te arrow superscript deotes a vector avig oly some dofs of the origial defied vector. 21

5 The sums o the right had side of equatio (21) are the sums of the trasmitted forces to the chassis from the mouts fm, ( w) ad from the 4 suspesios f. 1 ( w). For the mouts it is possible to develop a model aalogous to the model of oise, although i this case oly some dofs are cosidered ad the chassis is dyamically coupled to the egie-mouts subsystem. Therefore, i the elastic forces the differece betwee the egie-mouts coectio poits ad the chassismouts coectio poits must be cosidered. Similarly to the expressio used i equatio (7), the displacemets of the egie-mout coectio poits ad the chassis-mouts coectio poits are represeted, respectively, by where -bz, by, b~, ) -bx, (22) As for the A'[ matrix elemets, the elemets of B'[ represet the compoets of the i-th mout positio vector alog the j axis i the referece frame v. The elastic forces i terms of the ceter of mass displacemets of the chassis ad egie are give respectively by ft, = Kct;(u;m - u;j = Kct;(AT iicm - B'[ iicu) (23) Therefore equatio (21) ca be writte as (24) For the elastic forces at the chassis-suspesio coectio poits oe has (figure 2) fs, = k,, (z2,, - Z3,,) + JWCs; (z2,, - Z3,,) (25) J,, is a force with a sigle compoet alog z because this is the oly degree of freedom of the wheel-suspesio subsystem. To determie the force appearig i equatio (21) J., must be mupltiplied by the vector -d, )T 'x thus obtaiig = Ld;J,, (26) j=l j=l For the egie the dyamic equatio is similar to the oe cosidered i the model of oise, i.e. (27) The differeces cocer the absece of iteral excitatios that are ot of iterest i the wheel-footboard FRF computatio ad the expressio of the elastic forces of the mouts, because they ivolve also the chassis motio. Oce solved the system ivolvig the coupled subsystems, the respose at the driver's foot-board ca be determied for a give road iput z1r at ay wheel. Sice the comfort specificatios are give o the FRF modulus, the road iput is put costat ad equal to oe for all the wheels[ 9 J. 5 OPTIMIZATION Goal of ay optimizatio problem is the choice of a set of desig variables able to maximize or miimize some features of the system. As for the variables, the objective fuctio may be subjected to some costrait. The possibility of obtaiig a efficiet ad accurate solutio, especially for oliear problems, depeds o the problem's dimesio, i.e. umber of variables, type of objective fuctio ad costraits. It is worthwile to poit out that the goal of a optimizatio problem ca ot be easily icluded i a sigle objective fuctio therefore it ca be difficult to establish a correct strategy for ay problem. Differet solutio procedures are proposed i the literature to solve oliear optimizatio problems. Iitially the optimizatio of the preseted problem was obtaied by a MATLAB software based a a multi-objective approach that uses the Goal Attaimet Method proposed i[loj. Although this procedure provided a better solutio tha the stadard oe obtaied by CRF, it was observed that it was strogly depedet o the startig coditio, i.e. it is usually sub-optimal. To overcome this trouble a code based o the Geetic Algorithms[1 1 J was developed that led to a more robust approach. The advatages of this approach are summarized i the followig poits simple implemetatio o each computer; complete compatibility with developed softwares for the computatio of the elemets of the objective fuctios (acoustics ad vibratios); possibility of dealig with discrete variables, that ca be used to choose amog differet cofiguratios of the egie mouts withi the optimizatio process; lower sesitivity of the optimizatio solutio from the startig poit. 5.1 Formulatio of the problem The first eccesary step is a appropriate choice of the egie mouts cofiguratio (umber of mouts, mouts locatio, mouts stiffess etc.). Simple cosideratios put a limitatio to the miimum stiffess that ca be used for the mouts. The choice of the objective fuctio is very troublesome, i that it coditios the correct achievmet of the goal. Differet choices lead to quite differet solutios, so that a good sesitivity is required to the desiger. Fially, to achieve a solutio havig a physical meaig, it is ecessary to impose suitable costraits techical costraits o the mouts realizatio, geometrical costraits o the desig of the egie va, costraits imposed by uavail- 22

6 ~.. =. ability of ATF's i some poits of the chassis, etc. As previously stated, the adopted cofiguratio is typical of the producer's desig phylosophy. Followig the producer's suggestios, we defied 29 optimizatio variables, after havig assumed that the oliear characteristic of the mouts would be of order 5, i.e. the elastic forces would be of the type fe 1 = a; x + b;x 5 The variables are 3 coordiates for each mout locatio (totally 9), 2 values a; ad b; for each directio of two out of three mouts (12) plus a pair of values for the sigle stiffess of the road-like mout (2), 3 agles characterizig the directio of two out of the three mouts (6). With referece to the techical costraits, a static stiffess k ~ 5 kg/mm is ecessary for life ad fatigue requiremets of the mouts[ 9 1; a suitable ratio betwee radial ad axial stiffess is suggested, i.e. krad ~ kax/4, to make simpler the costructio of the mouts. The geometrical costraits are usually imposed to avoid iterferece betwee the egie block ad some parts of the chassis durig particular limit maouvres of the vehicle. It was suggested that the maximurr displacemets of the mouts-chassis coectio poits mus rage withi ±15 mm i the 3 directios. The limit maouvre! cosidered are those more critical for the displacemets, i.e. upwards ad dowwards bumps, first gear acceleratio, back ward acceleratio, right ad left curves i limit coditios o grip. Such dyamic loads are cosidered as static to semplif) the displacemets computatio ad because oe is maill iterested to their maximum value istead of time histories The displacemet values ca be obtaied by solvig a syste aalogous to the equilibrium equatios (1 ), after the itroduc tio of the terms f ma ad Cma, idicatig forces ad torque! for the chose maouvre. { i)'; + fma - mg = l=l lc; +cma = With referece to the ATF's, it is worthwile to poit out tha, they are determied experimetally, so that they are ot available at ay poit of the egie box. Therefore it is ecessary to look for a optimum locatio of the mouts i the eighborhood of the iitial mouts locatio such that the ATF's ca be cosidered costats. This boud is assumed withi ±5 mm alog the 3 directios. All these costraits are itroduced ito the optimizatio problem as equatios or iequatios together with the objective fuctio. 6 RESULTS AND CONCLUSIONS As startig poit we assumed the stadard CRF solutio for the cosidered vehicle. The cotrol poit for the acoustic pressure was assumed o the right ear of the driver. With respect to the acoustic requiremet, the goal cocers a decrease of the soud pressure level below a give threshold. The objective fuctio is determied by computig the soud pressure level curve area above the threshold. With respect to the egie shake, the required goal is the reductio of the trasfer fuctio modulus below a give boud. I the plots preseted, the cotiuos lie is the CRF referece solutio, while the broke lie curves are the optimized results. For obvious reasos, all the graphs are suitably ormalized. Note that a.5 db differece i the soud pressure level diagram actually correspods to 5 db, while a.1 differece i the frequecy axis correspods to 12 Hz. I figure 3 the optimized characteristic curves of the three mouts are show the rod-like mout has a stiffess characteristic oly alog the axial directio, as established by desig. I figures 4 ad 5 the compariso is performed betwee CRF results ad oe of the best optimized results obtaied by the MATLAB algorithm. I figures 6 ad 7 the compariso cocers results obtaied by CRF ad by Geetic Algorithms. Egie side Gear side Rod-like mout z1e!ex1' EEx1o'. X ,.. e & ,_... [djx1'..... ~ > e... u... 2 Etdx1'.. ~ N e... LL ~2E]x.. X..... e t1. ~ ~ ~ 1 Gjx1' >-..., e.. LL Bx1' ~ N. ~.... u... ~.. X.... e. - If ~ E2 ~. >-. ll. ~ =. -1 ~a e ~ -1 Displacemet [mm] Displacemet [mm] Displacemet [mm] Figure 3 Optimized characteristics of the mouts Oe ca easily observe that the proposed model is quite satisfactory to describe both the acoustic ad vibratio problems related to the egie moutig system. The optimizatio procedures provided always better results tha the origial solutio withi the imposed costraits. Particularly, a importat reductio of the egie shake pheomeo was obtaied, satisfyig i this way requiremets typical of higher level vehicles; with respect to the oise a decrease of almost 5 db of the maximum level was achieved, thus obtaiig, i almost the whole frequecy badwidth, a lower SPL tha the stadard CRFvalues. Fially it is evidet that the Geetic Algorith approach usually provides more efficiet results tha those obtaied by the Goal Attaimet Method 23

7 &taodard 1-1 optimized H &tadard H opllmlzed[ O.J Frequecy [Hz) 6 o o OS 7 8 OQ Frequecy [Hz) Figure 4 Soud pressure level Figure 6 Soud pressure level Footboard respose FoolbOard respose 5 Figure 5 Egie shake Figure 7 Egie shake ACKNOWLEDGEMENT This work was partly developed for a mechaical egieerig degree thesis, i collaboratio with CRF. The cooperatio of CRF egieers is gratefully ackowledged. REFERENCES [1) Johso, S. R. ad Subhedar, J. W., Computer optimizatio of egie moutig systems, SAE Paper 79974, [2) Feg, Z. ad Hou, J., Optimizatio of egie moutig parameters with forbidde bads of multiple modal frequecies, Proc. IMAC 7, Las Vegas, [3) Syma, J. A., Heys, P. S. ad Vermeule, P. J., Vibratio isolatio of mouted egie through optimizatio, Mech. Mach. Theory, Jauary [5) Swaso, J. A. ad Hu, H. T., Otimizatio of aircraft egie suspesio systems, J. of Aircraft, Nov [6] Spiekerma, C. E., Radcliffe, C. J. ad Goodma, E. D., Optimal desig ad simulatio of vibratioal isolatio systems, J. of Mechaisms, Trasmissios, ad Automatio i Desig, Jue [7) Muller, M. ad Welti, U., The effect of egie mouts o the oise ad vibratio behavior of vehicles, SAE paper 9467, [8] Brett, J., Optimizatio of egie moutig systems to miimize vehicle vibratio, SAE paper , [9] CRF- Motiglio, M., Private commuicatios, [1) Differet authors, The Hadbook of Geetic Algorithms, Ed. L. Dares, NY, [11] Grace, A., Otimizatio Toolbox, Matworks Ic., 199. [4) Nel, C. B. ad Heyes, P. S., Experimetal verificatio of a optimizatio program for a frot wheel drive egie mout system, Proc.ISMA 21, Leuve,