Modal Analysis of Exhaust System

Transcription

1 Modal Aalysis of Exhaust System Marcus Myré Jörge Olsso Departmet of Mechaical Egieerig Uiversity of Karlsroa/Roeby Karlsroa, Swede 1999 Thesis submitted for completio of Master of Sciece i Mechaical Egieerig with emphasis o Structural Mechaics at the Departmet of Mechaical Egieerig, Uiversity of Karlsroa/Roeby, Karlsroa, Swede. Abstract: A modal aalysis is performed o a exhaust system for cars. Natural frequecies ad mode shapes are correlated betwee experimetal ad fiite elemet models. MAC-values are calculated. Theories ad guidelies for modal aalyses of exhaust systems are discussed. Keywords: Modal Aalysis, Modal Correlatio, Exhaust System, Structural Dyamics, Fiite Elemet Method.

2 Acowledgemets This wor was carried out at the Departmet of Mechaical Egieerig, Uiversity of Karlsroa/Roeby, Karlsroa, Swede, uder the supervisio of Dr. Göra Broma. The wor was iitiated i March 1999 as a co-operatio betwee AP Torsmasier Techical Ceter AB ad the Departmet of Mechaical Egieerig at the Uiversity of Karlsroa/Roeby. We wish to express our gratitude to Dr. Göra Broma ad Ph.D. studet Aders Jösso for their scietific guidace ad support throughout the wor. At AP Torsmasier we wat to tha M.Sc. Kristia Althii, Research & Developmet Maager, M.Sc. Per Folesso, Predictio Egieer, ad Aders Svesso, Laboratory Maager for all the support ad for maig this wor possible. Fially, we would lie to tha our colleagues i the Master of Sciece programme ad all the other members of the Departmet of Mechaical Egieerig for valuable discussios ad support. Karlsroa, October 1999 Marcus Myré Jörge Olsso

3 Cotets 1 Notatio 5 2 Itroductio 7 3 Exhaust System Desig 8 4 Theories Structural Dyamics Sigle-Degree-Of-Freedom System (SDOF) Multiple-Degree-of-Freedom System (MDOF) Experimetal Modal Aalysis SDOF Polyomial Polyreferece Frequecy Polyreferece Mode Idicator Fuctio (MIF) Correlatio Graphical Comparisos of Natural Frequecies Graphical Compariso of Mode Shapes Numerical Correlatio Methods 18 5 Fiite Elemet Models FE-model of Rear Part Model I Model II FE-model of Frot Part FE-model of Assembled Exhaust System 24 6 Modal Measuremets Measuremets o Rear Part Hammer Excitatio Shaer Excitatio Measuremets o Frot Part Measuremets o Assembled Exhaust System Ifluece of Measuremet Coditios Excitatio Amplitude Boudary Coditios Temperature 34 7 FE- ad Test-model Correlatio 36

4 7.1 Rear Part Frot Part Assembled Exhaust System 39 8 Coclusios 41 Refereces 43

5 1 Notatio c F K M m s t X X-Y-Z x x Viscous dampig Force Stiffess matrix Stiffess Mass matrix Mass Displacemet Time Maximum displacemet vector Co-ordiate system Displacemet vector Displacemet Modal Displacemet vector Dampig ratio Mode Shape vector Frequecy Idices b d i j P X Number Damped Number Number Number Theoretically Experimetally 5

6 Abbreviatios EMA FEM FRF MAC MDOF MSF Experimetal Modal Aalysis Fiite Elemet Method Frequecy Respose Fuctio Modal Assurace Criterio Multi Degree of Freedom Modal Scale Factor 6

7 2 Itroductio The eed for aalyses of the dyamic behaviour of exhaust systems i the automotive idustry has icreased, primarily due to demads o a drastic shorteig of the time for product developmet. Higher combustio temperatures have also complicated the desig due to weaeig of the material with icreased temperature. Aother importat reaso is the aim at a lower fuel cosumptio, which ca be met by for example a decreased mass of the car. This, i tur, has resulted i more ad more lea structures ad a icreased use of ew materials such as alumiium ad high stregth steel. These materials have low dampig ad high vibratio trasmissio. Also cosiderig icreased demads o comfort, it is easily realised that desigig exhaust systems has become much more difficult. Some papers dealig with the dyamic behaviour of exhaust systems have bee foud [1-3]. How to describe some of the compoets of exhaust systems has bee studied before at the Departmet of Mechaical Egieerig, Uiversity of Karlsroa/Roeby [4-7]. The aim of this wor is to perform a modal aalysis of a Volvo S/V 70 exhaust system, fabricated by AP Torsmasier AB. Iformatio from the literature ad experiece from studyig this specific exhaust system will be used to gai coclusios that may be geerally useful whe performig modal aalyses withi the product developmet process. 7

8 3 Exhaust System Desig The modal aalysis is performed o a Volvo S/V 70 exhaust system (see figure 3.1). This represets a commo type of exhaust system desig of today s cars. It cosists of two mai parts, oe frot ad oe rear part. Figure 3.1. Volvo S/V 70 exhaust system. The frot part cosists of a welded exhaust maifold (ot show i the figure), which is the exhaust systems coectio to the egie. After the maifold is the flexible bellows, which allows for thermal expasio ad reduces trasmissio of axial ad bedig motios from the maifold to the rest of the exhaust system. The egie i Volvo S/V 70 is, as i most moder cars, mouted i the trasverse directio. Therefore, the rollig of the egie results i rather large movemets at the ed of the maifold. If the coectio to the rest of the exhaust system at this locatio were stiff, it would cause very large movemets ad bedig stresses i the exhaust system. The frot part also cotais two catalytic coverter mooliths. 8

9 The catalytic coverter uit is placed as close to the egie as possible to obtai a good cleaig effect of the exhaust fumes as quic as possible after the egie has bee started. The cleaig process starts whe the catalytic coverter reaches a certai temperature called the light-off temperature. The frot part is show i figure 3.2. Figure 3.2. Frot part of exhaust system (without exhaust maifold). The rear part cosists of three pipes welded together with two mufflers, oe big muffler i the middle ad oe smaller i the rear ed of the system. The rear part is show i figure 3.3. The mufflers cosist of a sheet metal shell, which has pipes with holes i the wall goig through chopped E-Glass (Glass wool). Chopped E-Glass gives a good soud silecig effect. I the large muffler the pipe is maig a S -tur ad i the small muffler the pipe goes straight trough. The coectios betwee the pipes ad mufflers are welded. 9

10 Figure 3.3. Rear part of exhaust system. 10

11 4 Theories 4.1 Structural Dyamics To itroduce some basic cocepts ad quatities of structural dyamics a short discussio is give below. For a more complete discussio the reader is referred to for example Craig [8] or Ima [9] Sigle-Degree-Of-Freedom System (SDOF) The reaso why a structure is movig is because a force (or forces) is affectig or has affected the structure. Figure 4.1 shows a sigle degree of freedom (SDOF) system. c m F(t) x(t) Figure 4.1. Sigle-degree-of-freedom system cosistig of mass, sprig ad viscous dampig. The equatio of motio for this system is m x cx x F(t) (4.1) were m is the mass of the body, c is the dampig costat of the viscous damper, is the stiffess of the sprig, x is the displacemet of the mass (elogatio of the sprig), ad F is the exteral force o the mass. Without exteral force ad dampig the system has a atural frequecy of 11

12 (4.2) m Icludig the effect of dampig the resoace frequecy is 2 d 1 (4.3) where c (4.4) 2m is called the dampig ratio. Dampig has usually mior effect o the resoace frequecies i a exhaust system because of the low material dampig Multiple-Degree-of-Freedom System (MDOF) Whe describig vibratios of a more complex structure a multiple degree of freedom approach is eeded. The structure is ofte modelled as a discrete system of a fiite umber of masses. Figure 4.2 shows a udamped system m1 m2 m -1 m x1 x2 x -1 x Figure 4.2. Udamped -degree-of-freedom system. The equatios of motio for each mass become 12

13 13 0 ) ( x x x m 0 0 ) ( 0 ) ( x x x m x x x x m x x x x m (4.5) This is a coupled system of secod order ordiary differetial equatios. Usig matrix otatio gives 0 Kx x M (4.6) where m m m m M, x x x x x, (4.7) K Assumig a harmoic solutio of the form t i Xe x (4.8) where X is the vector of amplitudes, gives the eigevalue problem 0 M X K ) ( 2 (4.9)

14 This is a system of liear algebraic equatios. Notrivial solutios exist if 2 det( K M) 0 (4.10) Equatio (4.10) is called the characteristic equatio. The roots of this equatio are called eigevalues, which give the atural frequecies of the system. Substitutig these eigevalues bac ito equatio (4.9) the mode shape correspodig to each atural frequecy ca be obtaied. Whe studyig a cotiuous body or a structure the stiffess matrix K ad the mass matrix M of the discrete MDOF-model is ofte obtaied by the fiite elemet method. 4.2 Experimetal Modal Aalysis Experimetal Modal Aalysis (EMA) is the iterpretatio of test data collected uder a test o a vibratig structure. The purpose of a EMA is to create a model of the vibratio properties ad the behaviour of the test structure. The properties of iterest are resoace frequecies, dampig ratios ad mode shapes of the structure. The modal properties are estimated from the frequecy respose fuctios (FRFs) obtaied from the test data. I the FRF a pea of the magitude mars every resoace frequecy. Each resoace frequecy ca be associated with a certai mode shape that represets the deflectio shape of the structure. There are several methods available for estimatio of the mode shapes, both sigle ad multiple degree of freedom methods. The estimatio techiques, also called curve-fittig methods, are used to geerate a aalytical fuctio that approximates the measured FRFs. The methods used i this wor are the SDOF polyomial, the polyreferece ad the frequecy polyreferece methods, which are briefly explaied below. For a more complete discussio the reader is referred to for example Maia et al [10] or Ewis [11]. 14

15 4.2.1 SDOF Polyomial SDOF polyomial is sigle degree of freedom estimatio techique. This method treats each mode separately, as if that mode was the oly oe preset i the structure. This sigle-mode assumptio is ot quite true because the eighbourig modes do cotribute to the total respose at the resoace of the aalysed mode. The advatage of this techique is that it is easy to lear ad produces solutios quicly. However, it is ot recommeded if there are several peas i the FRF close to each other i frequecy. Furthermore, if the mode shape accuracy is of high priority this techique is ot recommeded Polyreferece Polyreferece is a multiple degree of freedom estimatio techique that uses a curve-fittig algorithm i the time domai. Values for atural frequecies, dampig ratios, ad residue reflect the simultaeous ifluece of multiple modes across multiple referece locatios. The major advatage of this techique is its ability to use all the available FRFs whe calculatig the modal parameters. The demad o the FRFs is that they use the same frequecy iterval ad also the same amout of spectral lies Frequecy Polyreferece Frequecy polyreferece is also a multiple degree of freedom estimatio techique. It is similar to the polyreferece techique but uses the frequecy domai istead of the time domai Mode Idicator Fuctio (MIF) The mode idicator fuctio provides a visual idicatio of the modal desity. A MIF-diagram ca be created from several FRFs ad idicates where there is a major mode of the structure. A valley i the diagram represets the mode. If some mode i the structure is poorly excited it ca be hard to see it i the FRFs but it is easy to locate it i the MIF-diagram. There are some methods available i I-DEAS [12]. They are the Normal 15

16 MIF, the Power Spectrum MIF, ad the Multivariate MIF. I this wor the Multivariate MIF is used because this method ca hadle several referece co-ordiates simultaeously. 4.3 Correlatio Correlatio is a process where data from the experimet are compared with theoretical results. There are several methods available, which are more or less complicated. Two graphical methods are Graphical compariso of atural frequecies ad Graphical compariso of mode shapes. They are easy to use, but they are very time cosumig for models with may odes. Two umerical methods for compariso of mode shapes are the Modal scale factor ad the Modal assurace criterio. The methods are briefly explaied below. For a more complete discussio the reader is referred to for example Maia et al [10] or Ewis [11] Graphical Comparisos of Natural Frequecies Whe comparig the umerically calculated frequecies with the experimetally determied frequecies it is suitable to use a simple plottig techique. This is doe by plottig the experimetal value agaist the predicted oe for each of the modes icluded i the compariso as show schematically i figure 4.3. Measured frequecy * * * * * * * Predicted frequecy Figure 4.3. Graphical comparisos of atural frequecies. 16

17 The plotted poits should lie o or close to a straight lie of uity slope. If they lie close to a lie with a differet slope it is probably because of a icorrect material property i the theoretical model. If the poits lie scattered widely aroud the straight lie there is a serious failure i the model that represet the test structure ad a re-evaluatio of the model is ecessary. A case of particular iterest is whe the poits deviate slightly from the ideal lie i a systematic, rather tha radom, fashio. This situatio suggests that there is a specific characteristic resposible for the deviatio ad that this caot simply be attributed to experimetal errors Graphical Compariso of Mode Shapes A coveiet approach to compare the experimetal mode shapes with the theoretical mode shapes is by a plot similar to the oe used for the atural frequecies. I this method the ormalised displacemet amplitude of each (or some) ode i the mode shape vector is plotted, experimetal versus theoretical, as show schematically i figure Experimetal displacemet * * * * -1 * * Predicted 1 displacemet * -1 Figure 4.4. Graphical compariso of mode shape. 17

18 The idividual poits i this plot relate to specific co-ordiates o the model. A basic demad is therefore that the locatios of the poits i the experimetal model correspod with the odes i the theoretical model. The poits are expected to lie close to a straight lie passig through the origi. If both sets of mode shape data cosist of mass-ormalised eigevectors, the straight lie should have a uity slope. The patter of ay deviatio from this requiremet ca idicate quite clearly the cause of the discrepacy. If the poits lie close to a straight lie of a slope differet from 1, either oe of the mode shapes may ot be mass ormalised or some other form of scalig error exists i the data. If the poits are widely scattered about a lie, there are probably cosiderable iaccuracies i oe or both of the sets of data. If the scatter is excessive, the two eigevectors that are beig compared do ot relate to the same mode Numerical Correlatio Methods There also exist a few umerical methods for the correlatio procedure. The umerical methods compute some simple statistical properties for a pair of modes. Oe method is the Modal Scale Factor (MSF) that represet the slope of the best straight lie through the poits (see figure 4.4). This parameter gives o idicatio of the quality of the fit of poits to the straight lie, simply its slope. Aother method, which is used i this wor, is the Modal Assurace Criterio (MAC). The MAC calculates the least square deviatio of the poits from the straight lie correlatio accordig to the formula: MAC ( P, X ) b1 X b b1 ( ) ( ) ( ) ( ) X X b b P b1 2 b ( ) ( ) P b P b (4.11) were X is the experimetally measured mode shape ad P is the theoretically predicted mode shape. Idex b deotes the ode umber. 18

19 5 Fiite Elemet Models I order to predict the atural frequecies ad mode shapes of the exhaust system, differet Fiite Elemet (FE) models are made. As for the measuremets, the exhaust system is divided ito two parts (three models) i order to simplify the correlatio. The models are the rear part, the frot part ad fially the assembled system. The Laczos method is used to solve the modal calculatios i I-DEAS. Free-free boudary coditios are used for all models i this wor. This is the easiest to realise ad is also geerally recommeded for the first models i the product developmet process. At a early developmet stage the boudary coditios that the exhaust system will have uder operatio are ot always ow. By usig free-free boudary coditios it is still possible to develop FE-models of the exhaust system, or parts of it, that correspod well with measuremets. For example, stiffess values of coectios etc. ca be foud. As the product developmet process proceeds the FE-models ca the be aalysed with other boudary coditios. 5.1 FE-model of Rear Part Model I A simple way of modellig the rear part of the exhaust system is to assume that the mufflers are so much stiffer tha the pipes that all bedig ad twistig deflectios are due to the pipes. With this assumptio the mufflers ca be replaced by rigid elemets ad lumped mass elemets simulatig the mufflers mass ad momet of iertia (which were experimetally determied). The model cosists of 17 beam elemets, four rigid elemets ad two lumped mass elemets. Model I is show i figure 5.1 ad the results are give i table 5.1. X-Y-Z is the calculatio co-ordiate system i I-DEAS with X alog the mai directio of the pipes. 19

20 Figure 5.1. Model I, Rear part. Table 5.1. Estimated modes from the first FE-model. Mode Freq. [Hz] Descriptio Y-Bedig Z-Bedig Torsio Y-Bedig Z-Bedig Y-Bedig Y-Bedig Model II The FE-model of the rear part is adjusted to correspod better with measuremets. Two combied modificatios are used. The stiffess of the mufflers are modified by Key-I sectios, ad the coectio betwee the exhaust pipes ad mufflers are weaeed by usig weaer beam elemets for the very first part of the pipes comig out from the mufflers. This is to 20

21 simulate that the welded coectio betwee the muffler ad the pipe is ot completely stiff. Model II is show i figure 5.2 ad the results are give i table 5.2. Figure 5.2. Model II, Rear part. Table 5.2. Estimated modes from the secod FE-model. Mode Freq. [Hz] Descriptio Z-Bedig Y-Bedig Z-Bedig Torsio Y-Bedig Y-Bedig Z-bedig 3 21

22 5.2 FE-model of Frot Part The frot part of the exhaust system is more difficult to model tha the rear part. This is maily because it cotais a flexible uit, cosistig of a bellows with a braid ad a ier-lier (meshrig), see figure 5.3. Figure 5.3. Flexible uit. A method of modellig the bellows by beam elemets has bee described by Broma et al [6]. However, at preset there are o existig methods to describe the bellows i combiatio with the braid ad the ier-lier i a simple way i a FE-model. A test to just iclude the bellows itself i the model (eglectig the braid ad the ier-lier) gave poor agreemet with experimetal results. The secod mode i the FE-model is related to the flexible uit. This mode does ot exist i the measuremets. A possible reaso for this is that the FEmodel is etirely liear while the real system icludes oliearities, for example the flexible uit. The bellows is pre-compressed by the ier-lier (meshrig). This gives the characteristics show i figure 5.4. Whe the compressio forces are smaller tha the pre-compressio, the stiffess of the uit is high compared to a ucompressed bellows. For pullig loads 22

23 the stiffess is also very high because of the braid i combiatio with the high radial stiffess of the bellows. Compressio of Bellows Extra legth whe Bellows is ucompressed 0 Force Figure 5.4. Characteristics of flexible uit i the exhaust system. Furthermore, probable oliear behaviour of the ier-lier itself also implies that a oliear structural dyamics aalysis would be ecessary. This is however beyod the scope of this wor. For the reasos above the flexible uit is modelled by a beam elemet that is stiffer tha the ucompressed bellows. Also the frot part is at first modelled roughly to estimate locatios of the accelerometers for a good measuremet. Whe the modal parameters are determied the FE-model is improved. To speed up the improvemet wor, beam properties for the catalytic- ad bellows sectios are defied by the Key-I method i I-DEAS. Advatages with this method are that it is very easy to chage the properties ad that fictive properties ca be used. The fial FE-model is show i figure 5.5 ad the results are give i table

24 Figure 5.5. FE-model of frot part. Table 5.3. Estimated modes of frot FE-model. Mode Freq. [Hz] Descriptio Z-Bedig Bellows mode Z-Bedig Y-Bedig Z-bedig FE-model of Assembled Exhaust System The FE-model of the assembled exhaust system is show i figure 5.6 ad the results of the modal calculatios are give i table

25 Figure 5.6. FE-model of assembled exhaust system. Table 5.4. Estimated modes of assembled FE-model. Mode Freq. [Hz] Descriptio Z-Bedig Y-Bedig Z-Bedig Y-Bedig Y-Bedig Z-Bedig Y-Bedig Torsio Z-Bedig Y-Bedig Z-Bedig 5 25

26 6 Modal Measuremets A measuremet model cosists of a umber of odes that represet poits o the real structure where measuremets are performed. The resolutio of the mode shapes is better if may poits are used. O the other had the measuremets are the more time cosumig. May poits must be used if mode shapes of high order are of iterest. The placig of the trasducers (measurig poits) is also importat. It is easier to detect the motio of the structure if the trasducers are placed where the amplitudes of the motio are high. O the other had the trasducers will the have higher ifluece o the behaviour of the structure. The mass of the trasducers ca have cosiderable ifluece for light-weighted structures. If the trasducers are mouted at poits havig oe or small amplitudes the motio of the structure is hard to measure. To fid good places for the trasducers it is a good idea to simulate the structure by a first simple FE-model. Iterative improvemets of both the FE-model ad the measurig model are ofte ecessary. The measuremets o the exhaust system are performed for the three cases described before. 6.1 Measuremets o Rear Part I this case it is ow from the FE-calculatios that there are oly few atural frequecies i the frequecy rage of iterest. The measuremet model cosists of 19 odes accordig to figure 6.1. The FRFs were tae i the two directios perpedicular to the pipes. It is difficult to measure alog the pipes, especially with hammer excitatio. Node 52 has a local co-ordiate system to mae it possible to excite perpedicular to the pipe. All other poits satisfy the excitatio directio i the global co-ordiate system. 26

27 Figure 6.1. Measuremet model of rear part. Free-free boudary coditios were approximated by hagig the part i rubber bads with low modulus of elasticity. The measuremet set-up is show i figure 6.2. Figure 6.2. Measuremet set-up (with shaer excitatio). 27

28 6.1.1 Hammer Excitatio Hammer excitatio was performed i all odes i Y ad Z directios. A plastic top o the excitatio hammer was used to provide the structure with as much excitatio eergy as possible i the frequecy rage of iterest. To use polyreferece methods more tha oe referece is eeded. The accelerometers were fasteed i ode 53 ad 59 for measurig i Y ad Z directios. This choice was based o the mode shapes from the first FEmodel. Results from the measuremets are preseted i table 6.1. Mode Table 6.1. Modal parameters from hammer ad shaer excitatio (polyreferece). Hammer Freq. [Hz] Dampig [%] Shaer Freq. [Hz] Dampig [%] Descriptio Y-Bedig Z-Bedig Z-Bedig Torsio Y-Bedig Y-Bedig Z-Bedig ? ? ? 28

29 6.1.2 Shaer Excitatio Shaer excitatio was carried out i ode 50 i the Y ad Z directios ad i ode 48 i the Y directio. Accelerometers were mouted i all odes i the Y ad Z directios. Results from the measuremets are preseted i table 6.1. For mode 1 ad 2 it is hard to achieve good coherece because they are coupled ad their frequecies are close to each other. To achieve better coherece ad clear peas i the FRFs a sie sweep was used i the iterval Hz. Modes 8-10 were hard to idetify because of too few measuremet poits, which gave bad resolutio for the higher modes. The differece i dampig values betwee hammer ad shaer excitatio is probably related maily to the resolutio i spectral lies. I the shaer measuremet the rigid body modes are high i compariso with the frequecies of iterest. This results i bad resolutio for the wated peas ad maes the FRF jagged. The ris of ot discoverig the highest value of the peas is high, which clearly affects the dampig values. I the hammer measuremet the rigid body modes are ot a problem. Other factors that may affect the dampig values are the measuremet setup ad the fact that a mass is added (the mass of the force trasducer) whe the shaer is used. If the dampig factor is of great iterest it is recommeded to use a sie sweep i both directios aroud the atural frequecy with small frequecy steps. 6.2 Measuremets o Frot Part The measuremet model cosists of 17 odes accordig to figure 6.3. Oly hammer excitatio was doe o this part, sice the results from the rear part showed oly small differeces betwee hammer ad the shaer excitatio, ad shaer measuremets are very time cosumig. Effective shaer measuremets require a measurig system with may chaels. 29

30 Figure 6.3. Measuremet model of frot part. The measuremets were performed i almost the same way as for the rear part. Nodes 1 to 7 have local co-ordiate systems to mae it possible to excite perpedicular to the pipe. All other poits satisfy the excitatio directio i the global co-ordiate system. Results from the measuremets are preseted i table 6.2. Table 6.2. Modal parameters from hammer excitatio (polyreferece). Mode Freq. [Hz] Dampig [%] Descriptio Z-Bedig Z-Bedig Z-Bedig Z-Bedig Y-Bedig Z-Bedig 3 30

31 6.3 Measuremets o Assembled Exhaust System To detect ad idetify all modes i the frequecy rage of iterest may odes were used. To create this measuremet model the models for the rear ad frot parts were joied, with the exceptio that a few close odes were excluded. The measuremet model of the etire exhaust system is show i figure 6.4. The measuremet set-up is show i figure 6.5. Hammer excitatio was performed accordig to earlier descriptios. Results from the measuremets are preseted i table 6.3. Figure 6.4. Measuremet model of assembled exhaust system. 31

32 Figure 6.5. Measuremet set-up of assembled exhaust system. Table 6.3. Modal parameters from hammer excitatio (polyreferece). Mode Freq. [Hz] Dampig [%] Descriptio Z-Bedig Y-Bedig Z-Bedig Y-Bedig Y-Bedig Z-Bedig Y-Bedig Torsio Z-Bedig Y-Bedig Z-Bedig 5 32

33 6.4 Ifluece of Measuremet Coditios To evaluate how differet parameters affect the measuremet results some tests were made o the frot part of the exhaust system. The set-up is show i figure 6.6. Figure 6.6. Set-up for ivestigatio of differet measuremet coditios Excitatio Amplitude Noliearities may give differet FRFs for differet amplitudes of the excitatio sigal. Figure 6.7 shows how the FRFs chage with differet excitatio force amplitude. With icreased amplitude the pre-compressio of the bellows may be exceeded, resultig i decreased stiffess. This is the reaso why the amplitude of the FRFs decrease with higher excitatio force amplitude. The lower stiffess of the bellow also lowers the resoace frequecy. 33

34 Figure 6.7. FRFs for differet excitatio force amplitudes Boudary Coditios Differet ways of realisig the free-free boudary coditios may give differet FRFs. Three choices of hagig placemets of the rubber bads were tested. Results are give i figure 6.8. Resoace frequecies remaied the same but the amplitudes chaged differetly for differet modes Temperature No tests were doe for the temperature ifluece. However, cosiderig that the material modulus of elasticity is depedet o temperature, show i figure 6.9, it is obvious that the dyamic behaviour is depedet o temperature. 34

35 Figure 6.8. FRFs for differet hagig arragemets. Figure 6.9. Modulus of elasticity versus temperature, from [13]. 35

36 7 FE- ad Test-model Correlatio MAC values are calculated to get a idicatio of how well the measured ad the theoretical modes correspod. The correlatio is performed i I- DEAS Test ad the MAC matrixes are show i figure 7.1 to 7.3. Table 7.1 to 7.3 iclude values from the modal pairig. 7.1 Rear Part Figure 7.1. MAC matrix for rear part. 36

37 Table 7.1. Mode pairig for rear part. Measured FEM Measured FEM Diff. MAC mode mode Freq. [Hz] Freq. [Hz] [%]

38 7.2 Frot Part Figure 7.2. MAC Matrix for frot part. Table 7.2. Mode pairig for frot part. Measured FEM Measured FEM Diff. MAC mode mode Freq. [Hz] Freq. [Hz] [%]

39 7.3 Assembled Exhaust System Figure 7.3. MAC matrix for assembled exhaust system. 39

40 Table 7.3. Mode pairig for assembled exhaust system. Measured FEM Measured FEM Diff. MAC mode mode Freq. [Hz] Freq. [Hz] [%]

41 8 Coclusios I this wor a modal aalysis was performed o a Volvo S/V 70 exhaust system. Fiite elemet models were correlated with experimetal models. It was show that a rather simple fiite elemet model was sufficiet for describig the dyamic behaviour of the exhaust systems studied. Beam elemets, rigid elemets ad mass elemets were used. For the assembled system the agreemet betwee theoretical ad experimetal atural frequecies was withi approximately 10 % for the first eleve modes. This type of fiite elemet model is therefore probably sufficiet also for most other exhaust systems of similar type. For parts of the exhaust system it was more difficult to achieve good agreemet. Furthermore, it was foud that further studies o some compoets havig oliear characteristics are ecessary if these compoets are to be icluded i the aalysis i a more realistic way. For example if the flexible uit i the frot part of the exhaust system will be icluded. I this wor this uit was modelled by a beam elemet that is stiffer tha the ucompressed bellows. To achieve reliable models both theoretical ad experimetal modal aalyses should be performed, so that fiite elemet ad test models ca be compared ad successively improved. Free-free boudary coditios are recommeded for the first models. At a early stage of the product developmet process the boudary coditios that the exhaust system will have uder operatio are ot always ow. By usig free-free boudary coditios it is still possible to develop FE-models of the exhaust system, or parts of it, which correspod well with measuremets. For example, stiffess values of coectios etc. ca be foud. As the product developmet process proceeds the FE-models ca the be aalysed with other boudary coditios. For a good free-free hagig, may hagig poits ad rubber bads with low stiffess should be used. For the studied exhaust system some modes were strogly coupled. For such a system a sigle degree of freedom estimatio techique i the experimetal aalysis gives poor estimatio of the modal parameters. Therefore a multi degree of freedom techique was used. This is 41

42 recommeded wheever the mode shapes are of iterest. Sometimes it is also ecessary to use a sie sweep for a iterval of special iterest. 42

43 Refereces 1. Beligardi G. ad Leoti S.: Modal aalysis i the desig of a automotive exhaust pipe, It. J. Of Vehicle Desig, vol. 8, o. 4/5/6, UK Verbove P., Valgaere R., Va Overmeire M., ad Guillaume P.: Some commets o modal aalysis applied to a automotive exhaust sytem, Vrije Uiversiteit Brussel. (Year of publicatio uow.) 3. Shih-Fu Lig, Tso-Chei Pa, Geo-Hia Lim, ad Chig-Hua Tseg: Vibratio isolatio of exhaust pipe uder vehicle chassis, It. J. Of Vehicle Desig, vol. 15, os.1/2, UK Herma M. ad Jösso A.: Static Characteristics of Flexible Bellows, Master s Thesis, Departmet of Mechaical Egieerig, Uiversity of Karlsroa/Roeby, Karlsroa, Swede, Lidberg T. ad Mörtma S.: Dyamic Characteristics of Flexible Bellows, Master s Thesis, Departmet of Mechaical Egieerig, Uiversity of Karlsroa/Roeby, Karlsroa, Swede, Broma G., Herma M., ad Jösso A.: Modellig Flexible Bellows by Stadard Beam Fiite Elemets, Research Report 12/1999, Uiversity of Karlsroa/Roeby, Karlsroa, Swede, Berglud P. ad Pettersso J.: Dyamic Characteristics of Exhaust System Hagers, Master s Thesis, Departmet of Mechaical Egieerig, Uiversity of Karlsroa/Roeby, Karlsroa, Swede, Craig Roy. R., Jr.: Structural Dyamics, Joh Wiley & Sos Ic, Ima D. J.: Egieerig Vibratio, Pretice hall, Upper Saddle River NJ, Maia N. N. M., Silva J. M. M., He J., Lieve N. A. J., Li R. M., Sigle G. W., To W. M., Urgueria A. P. V.: Theoretical ad Experimetal Modal Aalysis, Research Studies Press Ltd, Somerset, Ewis D. J.: Modal Testig: Theory ad Practise, Joh Wiley & sos ic. Chichester,

44 12. I-DEAS Master Series 7, Structural Dyamics Research Corporatio (SDRC), 2000 Eastma Drive, Milford, Ohio 45150, USA. 13. Callister William D., Jr.: MATERIALS SCIENCE AND ENGINEERING A Itroductio, Joh Wiley & sos ic. Chichester,