Dynamics Study of an Automobile Exhaust System

8/13/2019 Dynamics Study of an Automobile Exhaust System

1/98


2/98

Johan Wall

Blekinge Institute of TechnologyLicentiate Dissertation Series No. 2003:08

ISSN 1650-2140ISBN 91-7295-030-7

Published 2003Printed by Kaserntryckeriet ABKarlskrona 2003Sweden


3/98

3

Acknowledgements

This work was carried out at the Department of Mechanical Engineering,Blekinge Institute of Technology, Karlskrona, Sweden, under the supervisionof Professor Gran Broman and Professor Kjell Ahlin, to whom I will beforever grateful for their guidance and support.

I would like to thank my friends and colleagues at the Department for creatinga pleasant and productive working environment. Special thanks go to M.Sc.Thomas Englund, my closest co-worker, for interesting discussions andfruitful collaboration. I would also like to thank the staff at Faurecia Exhaust

Systems AB for valuable support. Finally, I wish to express my gratitude toAssociate Professor Mikael Jonsson at the Division of Computer AidedDesign, Department of Mechanical Engineering, Lule University ofTechnology, Lule, Sweden, for his involvement in this work.

I gratefully acknowledge financial support from the Swedish Foundation forKnowledge and Competence Development, Faurecia Exhaust Systems AB,and the Faculty Board of Blekinge Institute of Technology.

Karlskrona, November 2003

Johan Wall


4/98

4

Abstract

Low vibration levels are a critical objective in automobile exhaust systemdesign. It is therefore important for design engineers to be able to predict,describe and assess the dynamics of various system design proposals during

product development.

The aim of this thesis is to provide a deeper understanding of the dynamics ofautomobile exhaust systems to form a basis for improved design and thedevelopment of a computationally inexpensive theoretical system model.Modelling, simulation and experimental investigation of a typical exhaust

system are performed to gain such an understanding and to evaluatemodelling ideas. Special attention is given to the influence of the bellows-typeflexible joint on the dynamics of the exhaust system.

The investigations show that the exhaust system is essentially lineardownstream of the flexible joint. Highly simplified finite element models ofthe major components within this part are suggested. These modelsincorporate adjustable flexibility in their connection to the exhaust pipes and a

procedure is developed for automatic updating of these parameters to obtain better correlation with experimental results. The agreement between the

simulation results of the updated models and the experimental results is verygood, which confirms the usability of these models.

Furthermore, the investigations show the great reduction of vibrationtransmission to the exhaust system that the bellows-type joint, either with orwithout an inside liner, gives in comparison with a stiff joint. For thecombined bellows and liner joint, vibration transmission is, however, higherthan for the bellows alone. Inclusion of the liner also makes the exhaustsystem behaviour significantly non-linear and complex, whereas the system

behaviour proves to be essentially linear when the joint has no liner. This

shows the importance of including a model of the liner in the theoreticalsystem model when the liner is present in the real system. The choice ofwhether or not to include a liner in the real system is obviously a complexissue. The advantages of reduced bellows temperature and improved flowconditions should be weighed against the disadvantages found in this work.

By combining the above findings it is concluded that in coming studies ofhow engine vibrations affect the exhaust system, the latter may be consideredas a linear system if the flexible joint consists of a bellows. If the joint also


5/98

5

includes a liner, the system may be considered as a linear subsystem that isexcited via a non-linear joint.

Keywords: Exhaust system, Experimental investigations, Finite elementmethod, Flexible joint, Modal analysis, Model updating, Simplified modelling,Structural dynamics.


6/98

6

Thesis

Disposition

This thesis comprises an introductory part and the following appended papers:

Paper A

Englund, T., Wall, J., Ahlin, K. & Broman, G., (2002), Significance of non-linearity and component-internal vibrations in an exhaust system,

Proceedings of the 2nd WSEAS International Conference on Simulation, Modelling and Optimization , Skiathos Island, Greece.

Paper B

Englund, T., Wall, J., Ahlin, K. & Broman, G., (2002), Automated updatingof simplified component models for exhaust system dynamics simulations,


Paper C

Wall, J., Englund, T., Ahlin, K. & Broman, G., (2003), Modelling of multi- ply bellows flexible joints of variable mean radius, Proceedings of the NAFEMS World Congress 2003 , Orlando, U.S.A.

Paper D

Wall, J., Englund, T., Ahlin, K. & Broman, G., Influence of a bellows-typeflexible joint on exhaust system dynamics. Submitted for publication.


7/98

7

The Authors Contribution to the Papers

The papers appended to this thesis are the result of joint efforts. The presentauthors contributions are as follows:

Paper A

Took part in the planning and writing of the paper. Carried out approximatelyhalf of the theoretical modelling, simulations and experimental investigations.

Paper B

Responsible for planning and writing the paper. Carried out approximatelyhalf of the development of the updating routine and the theoreticalsimulations.

Paper C

Took part in the planning and writing of the paper. Carried out approximately

half of the theoretical modelling, simulations and experimental investigations.

Paper D

Responsible for planning and writing the paper. Carried out approximatelyhalf of the experimental investigations.


8/98

8

Table of Contents

1 Introduction 9 1.1 Automobile Exhaust System Design 9 1.2 Dynamics Analysis 11

1.2.1 Theoretical Modelling and Simulation 13 1.2.2 Experimental Investigation 14 1.2.3 Coordination 15

1.3 Product Development and Virtual Prototypes 18 2 Aim and Scope 21 3 Summary of Papers 22

3.1 Paper A 22 3.2 Paper B 22 3.3 Paper C 23 3.4 Paper D 23

4 Conclusions and Future Research 24 5 References 26

Appended Papers

Paper A 31Paper B 47Paper C 63Paper D 83


9/98

9

1 Introduction

1.1 Automobile Exhaust System Design

An automobile exhaust system has several functions. Originally, it was usedfor silencing the noise caused by high pressure exhaust gases leaving theengine and for transporting these hot and toxic gases away from the driverscompartment. Nowadays, it is also an important and integral part ofcombustion and emission control. For this to work properly there must be noleakage upstream of the catalytic converter. The durability of that part of thesystem is therefore crucial. Customer demands for comfort and long productlife guarantee also for the exhaust system as a whole are additional reasons forthe increasing importance for design engineers to be able to predict, describeand assess the dynamics of various system design proposals during productdevelopment. The above considerations converge into the critical objective ofobtaining low vibration levels in the exhaust system.

A modern exhaust system generally consists of a manifold, a flexible joint, acatalytic converter, mufflers, and pipes. A typical system is shown in figure 1(manifold not included).

Figure 1. Typical exhaust system design (Modified Volvo S/V 70 system).

The flexible joint is included primarily to reduce transmission of enginemovements to the exhaust system. As it is usually located between themanifold and the catalytic converter it needs to be gas-tight. It must alsowithstand high temperatures, and should combine high flexibility with high


10/98

10

strength and durability. A steel bellows-type joint is therefore commonlyused. It generally consists of a multi-ply bellows combined with an insideliner and an outside braid. The liner is included to reduce the temperature ofthe bellows and improve flow conditions. The braid is included formechanical protection and to limit the possible extension of the joint. The

parts are connected with end caps. A model of the joint is shown in figure 2.

Figure 2. Basic design of the flexible joint.

Experience shows that this joint sometimes causes complex dynamic behaviour of the exhaust system and this has caused car and componentmanufacturers severe problems (Althini [1]).

The catalytic converter is included to convert toxic and environmentallyharmful gases into less harmful gases. The mufflers are included to reduce thenoise level.

A few publications concerning dynamics analysis of exhaust systems have been found in the literature. Verboven et al. [2] discuss experimentalinvestigations of exhaust system dynamics in general while focusing on acomparison of an ordinary experimental modal analysis with a running modeanalysis. Deweer et al. [3] perform an experimental modal analysis andemphasise the usefulness of theoretical models during the pre-test phase tomake better experiments. Belingardi and Leonti [4] deal with problems relatedto building finite element models with which to study the dynamic behaviour


11/98

11

of exhaust systems. Piombo et al. [5] point out the direct coupling between thedynamic behaviour of the exhaust system and its fatigue resistance. Ling et al.[6] perform finite element simulations of exhaust system dynamics to studyvibration transmission to the car body and show the potential benefits ofincorporating simulations into the exhaust system design process.

The above references do not include studies of the influence of the bellows-type flexible joint on the dynamics of exhaust systems; neither do they includediscussions of non-linear dynamics analyses.

The flexible joint has been studied by, for example, Broman et al. [7],Cunningham et al. [8], and Englund et al. [9] in relation to the automobile

exhaust system application. In addition to these references, there arenumerous publications regarding bellows flexible joints in general. For areview, see [7].

The following sections of this chapter provide a brief introduction to the fieldof structural dynamics and the general context of the present study. Thespecific aim and scope of this thesis is presented and discussed in chapter two.

1.2 Dynamics Analysis

The foundation for the modern study of classical dynamics was provided bySir Isaac Newton (1642-1727). His famous principles form the basis ofnumerous derived formulas used in dynamics studies. The field considered inthe present study is that of structural dynamics. Some examples of the manytext books in structural dynamics are those by Craig [10], Tedesco et al. [11],and Inman [12].

Many mechanical structures are subjected to dynamic (time varying) loadingin some form. This may lead to undesirable vibration levels. The resulting

problems may range from mild discomfort to structural breakdown. Todaythere is an increased general awareness of dynamics problems, and companiesare forced by legislation and customers demands to lower vibration and noiselevels in their products. At the same time, there is usually a desire to reducethe weight of the products. Lightweight structures could, however, sometimes

be more vulnerable to vibration. This emphasises the need for structuraldynamics analyses during product development. A parallel explanation for theincreased activity within this field is the immense development of computer


12/98

12

capacity and software, which has made far more comprehensive investigations possible.

Resonance is an important phenomenon in structural dynamics. This occurswhen a structure is excited at a frequency that coincides with one of its naturalfrequencies. The natural frequencies of a structure are the frequencies which astructure will select if allowed to vibrate freely without any excitation(Dynamic Testing Agency [13]). Each natural frequency is related to aspecific deflection form, or a so-called mode shape. The response amplitudeincreases dramatically at resonance and is limited only by the damping

present in the structure. Continued vibration at resonance may lead tostructural failure. Avoiding resonance is therefore usually necessary.

To be able to trust theoretical models and simulation procedures as support fordesign decisions and optimisation, they should be experimentally validated.This can be facilitated using a coordinated approach, see figure 3.

Figure 3. Coordinated approach to structural dynamics analysis.

In short, theoretical models for description of interesting productcharacteristics are developed. These models are implemented in computersoftware and used for simulations. Adjustments are made between themodelling and simulation parts until the simulation yields reasonable results.The simulation results are then compared to experimental results obtained byusing subsystems or an analogy with previous products. The coordination alsomeans that theoretical models and simulations are used to design goodexperiments. It is still, however, not always possible to predict all potential

problems beforehand; performing experimental investigations on several


13/98

13

different occasions might therefore be needed to produce quality measurementdata. The process of modelling, simulation and experimental investigation isiterated until good agreement between theoretical and experimental results isachieved. Simulation of the theoretical models can then be used as aneffective tool for the optimisation of the complete product. Shouldoptimisation imply design changes that significantly change the relevance ofthe assumptions of the theoretical models, the whole procedure is repeated.More detailed descriptions of the product are successively created, ifnecessary, as the development proceeds. This approach makes better decisions

possible earlier on in the development process, resulting in a reduction of theresources used, reduced time-to-market, and improved quality. When acompletely new product is developed, many complete iterations are usually

needed. When a new variant of a product is developed much prior work can be re-used.

Theoretical modelling, simulation and experimental investigation, and thecoordination between these parts, are described in more detail below inrelation to structural dynamics. Optimisation/Design is not explicitlyconsidered in this thesis.

1.2.1 Theoretical Modelling and Simulation

A model is a simplified description of a real system, often expressed inmathematical terms. It may be used to simulate certain aspects of the

behaviour of a system in order to better understand it.

One way of classifying theoretical models in structural dynamics is accordingto the number of independent response variables needed to uniquely define itslocation in space as it moves [13]. This is called the number of degrees offreedom (DOFs), N . Real structures are continuous ( N = ) and it is thereforenot usually possible to obtain closed-form solutions unless rather large

simplifications are made. Numerical methods are mostly necessary, giving anapproximate solution to the problem.

A commonly used numerical method in structural dynamics is the finiteelement method (FEM). The development of this method took off in themiddle of the twentieth century, accompanied by the growing possibility touse computers for the required computations. In the beginning it was

primarily used in the aerospace industry but has now developed into a powerful general tool which may be applied to almost all problems ofmathematical physics. The method produces approximate solutions to


14/98

14

differential equation problems. The division of the solution domain into manygeometrically simple subregions (finite elements) makes it possible to dealwith arbitrarily shaped solution domains.

Examples of the many text books describing this method are those by Bathe[14], Huebner et al. [15], and Zienkiewicz and Taylor [16]. Commercial finiteelement (FE) software packages are widely available. Examples used in thisthesis are ABAQUS [17] and I-deas [18].

In structural dynamics analyses the natural frequencies and correspondingmode shapes are obtained by solving an eigenvalue problem, which is anidealisation of the structure into a system without any energy dissipation

(damping) and in free vibration. The FEM is often used to obtain the massmatrix and the stiffness matrix of the eigenvalue problem for the structure athand. How to solve eigenvalue problems is discussed by, among others, [14].

1.2.2 Experimental Investigation

There is no way of knowing how accurate simulation results are unless theyare compared to some reference. When the objective is to describe thedynamic behaviour of a mechanical system, experimental modal analysis

(EMA) is useful for producing reference results. This is because the naturalfrequencies and mode shapes, which are a major outcome of an EMA, can becompared to the theoretical natural frequencies and mode shapes withoutmaking any assumptions regarding the nature and amount of damping in thestructure studied.

Experimental results are generally considered more realistic than simulationresults. This is because they are derived from a physical structure withoutmaking any assumptions about it. There are, however, a number ofuncertainties in performing an EMA. These can be categorised into three main

groups: errors during data acquisition, errors during signal processing, anderrors during data analysis. The analyst must use his/her knowledge andexperience to avoid such errors.

Depending on the objective of the analysis, different levels of knowledge and planning are needed to perform the EMA. An example of a simple modal testis the measurement of natural frequencies and damping ratios. A morecomplicated modal test may involve the measurement of all modal

parameters, including mode shapes, to be compared with the results of a FEanalysis [13]. Among other things, test planning normally involves decisions


15/98


16/98


17/98

17

A way of comparing theoretical and experimental natural frequenciesgraphically is shown in figure 5. This natural frequency comparison plotmakes it possible not only to see the degree of correlation but also the natureof the discrepancies. It is essential that CMPs are compared for the plot to bemeaningful.

0 50 100 150 2000

50

100

150

200

Experimental natural frequency (Hz)

T h

e o r e t

i c a

l

n a t u r a

l

f r e q u e

n c y

(

H z

)

Figure 5. Typical natural frequency comparison plot.

For more about correlation methods see [19] and [20].

It is widely recognised that FE analysis in structural dynamics is a complextask and that theoretical results do not always correlate well with experimentaldata (Maguire [23], Ewins and Imregun [24]). This implies that the FE modelsoften contain errors and need to be corrected.

A theoretical model may contain different types of error. Configuration errorsoccur as a result of the fact that the model is an approximation of what theanalyst believes the system to be. Omission of important physicalrelationships, mismodelling of joints and other boundary conditions, and


18/98

18

linearity assumption of not truly linear structures are examples of possibleconfiguration errors. Discretisation errors occur as a result of the fact that anumerical method usually produces approximate results in discrete points ofthe solution domain. The refinement of the FE mesh is, for example, related tothe accuracy of the result. Parameter errors occur because the physical

properties of the structure studied are seldom known exactly. Incorrectmaterial properties are one example of a parameter error.

Updating is the process of tuning the parameters in a theoretical model so thatsimulation results correlate better with experimental data. Since updatingalgorithms can only handle parameter errors ([19], Chen [25]), other types oferror, if present, must be corrected before model updating is undertaken. The

assessment of whether or not the model includes such errors (of significancefor the characteristics it is supposed to describe) is often referred to as modelverification. This is fairly straightforward for discretisation errors; checking amodel for configuration errors, on the other hand, is not an easy task. If themodel is not verified, valuable resources might be wasted on updating thatwould not produce a meaningful outcome anyway. The verification process isdescribed by, among others, [25].

There are many rather subjective choices left for the analyst when doingmodel updating. A benchmark study has, however, shown that an acceptable

result can be achieved with many different updating algorithms and residuals(Link and Friswell [26]). This indicates that the choice of algorithms andresiduals is not as critical as that of what parameters to include in the updating

procedure. If the chosen parameters are not the erroneous ones a solution thatmeets the update criterion, as stated by the analyst, may still be found. Theupdated parameters would, however, have lost their physical meaning(Avitabile [27]). An introduction to model updating is given by [19]. Thesubject is also thoroughly treated by Friswell and Mottershead [28].

Once the simulation model has been validated it can be used to support design

decisions in the product development process.

1.3 Product Development and Virtual Prototypes

The competitive advantage of short lead time in product development (PD) isobvious. On some markets changes in consumer preferences are so rapid thata product may be obsolete even before it is launched. The average product


19/98

19

lifetime has been compressed significantly over the last decade and the needfor an efficient PD process is therefore evident.

Design changes generally become more difficult and costly as a PD project progresses. On the other hand, knowledge of the design problem increases, asthe project progresses, as discussed by, for example, Ullman [29]. This design

process paradox is schematically illustrated in figure 6. A major step towardsa more effective PD process is therefore to make the learning curve as steep as

possible at the beginning of a project. This facilitates investigation of design proposals when making changes is still an economically viable option.

Figure 6. The design process paradox.

Virtual prototyping is a way to accelerate the learning process in a PD project.In this thesis a virtual prototype is referred to as models and simulation

procedures used to predict, describe and assess product characteristics. Themain advantage of virtual prototypes is that a large number of virtual testscan be performed instead of a few costly tests on physical prototypes. Once asimulation model has been developed, the additional cost of further analysis isusually very low compared to building new physical prototypes. Furthermore,


20/98

20

virtual prototypes can be utilised at an early stage in the PD process, incontrast to full physical prototype testing that naturally needs to be conductedat a late stage in the process. An overview of virtual prototyping is given byWang [30].

The use of virtual prototypes may not only save time and money, it couldeven improve the depth and quality of the analysis (Thomke [31]). Adrawback is that the virtual prototypes may be of lower fidelity than full

physical prototypes, that is, they cannot identify all the design problems onemight find using a physical prototype. In spite of this, the overall learningspeed is still higher when using virtual prototypes as each design iteration is

performed much faster (Thomke & Fujimoto [32]). As discussed earlier,

simulation models should, however, always be validated if they are to beuseful in the PD process.

The use of computer simulations as a design tool is increasing. This is particularly evident among automobile manufactures. To be able to compete,manufactures rely heavily on virtual prototypes. Examples of case studiesrelated to the automobile industry are those of [31] studying BMW, [32]studying Toyota, and Hudi et al. [33] studying Audi. All these studiesdemonstrate the positive impact of virtual prototyping on PD performance.


21/98

21

2 Aim and Scope

This thesis is part of a co-operation project between the Department ofMechanical Engineering at Blekinge Institute of Technology, Karlskrona,Sweden and Faurecia Exhaust Systems AB, Torss, Sweden. The overall aimof the project is to find a procedure for effective modelling and simulation ofthe dynamics of customer-proposed automobile exhaust system designs at anearly stage in the product development process, to support the dialogue withcustomers and for overall optimisation. To be suited for this it is importantthat the theoretical system model is as computationally inexpensive as

possible while yet being accurate enough for the characteristics it is supposed

to describe.

The aim of this thesis is to provide a deeper understanding of the dynamics ofautomobile exhaust systems to form a basis for improved design and thedevelopment of a computationally inexpensive theoretical system model.Modelling, simulation and experimental investigation of a typical exhaustsystem are performed to gain such an understanding and to evaluatemodelling ideas. Special attention is given to the influence of the bellows-typeflexible joint on the dynamics of the exhaust system.

The excitation from an automobile engine is usually in the frequency intervalof 30-200 Hz ([2], [4]). Excitations at lower frequencies may arise as a resultof road irregularities, acceleration, breaking, and gear changing. Thus, thefrequency interval of interest is set at 0-200 Hz.

The exhaust system is studied at room temperature. This is, of course, asimplification. An exhaust system is exposed to temperatures that clearlyaffect some of its mechanical properties. The real hangers are replaced byelastic rubber bands, which may also influence the dynamic response of thesystem. The manifold and the braid of the flexible joint are not included inthis study.

The implications of the above simplifications are not addressed in this thesis.The idea is that these aspects could be added, in a later stage of the project, tothe base of understanding gained through the present work.

A related thesis is that by Englund [34], which focuses on the dynamiccharacteristics of the major components in an exhaust system.


22/98

22

3 Summary of Papers

3.1 Paper A

In this paper the dynamic characteristics of a typical exhaust system areinvestigated. The flexible joint is not included in the study. It is shown thatshell vibrations of the catalytic converter and the mufflers as well as ovallingof the pipes are negligible in the frequency interval of interest. This impliesthat the pipes can be modelled by using beam elements, and that the catalyticconverter and the mufflers can be modelled by using lumped mass and massmoment of inertia elements. Additional short beam elements are used withthese models to account for the flexibility at the connections, as described in

paper B. It is also shown that non-linearity in the part of the systemdownstream of the flexible joint is negligible. This implies that this part can

be considered as a linear subsystem in dynamics studies of the completeexhaust system.

3.2 Paper B

In this paper an automated updating procedure is developed. A simulationmodel of the part of the exhaust system that is downstream of the flexible

joint is built in the commercial finite element software ABAQUS using thesimplified component models suggested in paper A. The sum of thedifferences between theoretically and experimentally obtained naturalfrequencies is chosen as the objective function to be minimised. Constraintsare used on the correlation between theoretically and experimentally obtainedmode shapes, considering the modal assurance criterion matrix, to ensure thatcorrelated mode pairs are compared. The properties of the short beamelements used to model flexibility at the connections between themufflers/catalytic converter and the pipes are used as the parameters to beadjusted during updating. Updating is performed by using the sequentialquadratic programming algorithm in the Optimization Toolbox in MATLAB.To obtain an automated procedure, communication is established between thetwo software packages by an in-house MATLAB script. The correlation

between results from the updated theoretical model and the experimentalresults is very good, indicating the usability of these component models andthat the updating procedure works well.


23/98

23

3.3 Paper C

In this paper a computationally effective simulation model of the bellowsflexible joint is presented. A straightforward way of modelling the bellows isto use shell finite elements. Due to the convoluted geometry of the bellowsthis would, however, require a high number of elements. The bellows modelwould thus constitute a large part of the model of the complete exhaustsystem. For more effective dynamics simulations, a beam finite elementrepresentation of the bellows has been presented in prior work. This papersuggests adjustments by which this procedure can be extended to model alsomulti-ply bellows of variable mean radius. Experimental investigations of theaxial and bending load cases are performed. The correlation betweentheoretical and experimental results is very good. The experimentalinvestigations reveal, however, that the bellows is slightly non-linear, but thisnon-linearity is weak and may be neglected in the present application.

Nonetheless, a hypothetical qualitative explanation for the non-linearity is provided.

3.4 Paper D

In this paper the influence of the flexible joint on the dynamics of the exhaustsystem is investigated. The ability of different joint configurations to reducevibration transmission from the engine to the exhaust system is studied.Measurements show the great reduction of vibration transmission to theexhaust system that the bellows-type joint, with and without an inside liner,gives in comparison with a stiff joint. However, for the combined bellows andliner joint, vibration transmission is higher than for the bellows alone. Thismakes the choice of including a liner in the real application a complex issue.It is also shown how the combined bellows and liner joint makes the exhaustsystem dynamics significantly non-linear and complex, leading to the

conclusion that a model of the liner needs to be included in the theoreticalsystem model when the liner is present in the real system. Furthermore, thenon-linearity of the double-plied bellows reported in paper C is seen to have aminor influence. This confirms the validity of using the linear beam model ofthe bellows when studying exhaust system dynamics.


24/98

24

4 Conclusions and Future Research

Dynamics of an automobile exhaust system is the subject of this work.

The investigations show that the exhaust system is essentially lineardownstream of the flexible joint. Highly simplified finite element models ofthe major components within this part are suggested. These modelsincorporate adjustable flexibility in their connection to the exhaust pipes, anda procedure is developed for automatic updating of these parameters to obtain

better correlation with experimental results. The agreement between thesimulation results of the updated models and the experimental results is very

good, which confirms the usability of these models.

Furthermore, the investigations show the great reduction of vibrationtransmission to the exhaust system that a bellows-type joint, with and withoutan inside liner, gives in comparison with a stiff joint. However, for thecombined bellows and liner joint, vibration transmission is higher than for the

bellows alone. The inclusion of the liner also makes the exhaust system behaviour significantly non-linear and complex, whereas the system behaviour proves to be essentially linear when the joint has no liner. Thisdemonstrates the importance of including a model of the liner in the

theoretical system model when the liner is present in the real system. It is alsoclear that the non-linearity due to the multiple plies of the bellows does notsignificantly affect the dynamics of the complete system. Using the linear

beam model of the bellows itself is thus sufficient in this application. Thechoice of including a liner in the real system is clearly a complex issue.Advantages of reduced bellows temperature and improved flow conditionsshould be weighed against the disadvantages found in this work.

By combining the above findings it is concluded that in coming studies ofhow engine vibrations affect the exhaust system, the latter may be consideredas a linear system if the flexible joint consists of a bellows. If the joint alsoincludes a liner, the system may be considered as a linear subsystem that isexcited via a non-linear joint. How to simulate such a system in acomputationally effective way is an interesting question for future research.This may also be expanded into the more general question of how to modeland simulate a general system that has linear relations between most of itsdegrees of freedom but which includes small but significant non-linear partsand is excited at some arbitrary point(s).


25/98

25

To be able to include realistic excitation in the exhaust system dynamicssimulations, which has been demonstrated to be important since the flexible

joint is non-linear, a theoretical model for simulation of the engine dynamicsshould be developed and linked to the exhaust system model in future studies.Also, the rubber hangers used to attach the exhaust system to the chassisshould be investigated and modelled.

Interesting questions for future research may also include how the hightemperatures, the flow of the exhaust gases, and the braid of the flexible jointaffect the dynamics of the exhaust system.


26/98

26

5 References

1. Ahltini, K., (2003), Research and Development Manger, Faurecia ExhaustSystems AB, Personal communication.

2. Verboven, P., Valgaeren, R., Van Overmeire, M. & Guillaume, P.,(1998), Some comments on modal analysis applied to an automotiveexhaust system, Proceedings of the International Modal AnalysisConference IMAC , Santa Barbara, U.S.A.

3. Deweer, J., Van Langenhove, T. & Grinker, S., (2001), Identification of

the best modal parameters and strategies for FE model updating, Proceedings of the SAE Noise & Vibration Conference & Exposition ,Grand Traverse, U.S.A.

4. Belingardi, G. & Leonti, S., (1987), Modal analysis in the design of anautomotive exhaust pipe, International Journal of Vehicle Design ,8(4/5/6), pp. 475-484.

5. Piombo, B., Belingardi, G., Dardano, R. & Pavese, M., (1986),Automotive exhaust pipe: the modal analysis approach for design and

testing, Proceedings of the International Modal Analysis Conference IMAC , Los Angeles, U.S.A.

6. Ling, S.-F., Pan, T.-C., Lim, G.-H. & Tseng, C.-H., (1994), Vibrationisolation of exhaust pipe under vehicle chassis, International Journal ofVehicle Design , 15(1/2), pp. 131-142.

7. Broman, G., Jnsson, A. & Hermann, M., (2000), Determining dynamiccharacteristics of bellows by manipulated beam finite elements ofcommercial software, International Journal of Pressure Vessels and

Piping , 77(8), pp. 445-453.

8. Cunningham, J., Sampers, W. & van Schalkwijk, R., (2001), Design offlexible tubes for automotive exhaust systems, Proceedings of the 2001

ABAQUS Users Conference , Maastricht, The Netherlands.

9. Englund, T., Wall, J., Ahlin, K. & Broman, G., Dynamic characteristicsof a combined bellows and liner flexible joint. Submitted for publication.


27/98

27

10. Craig, R.R., (1981), Structural dynamics: an introduction to computermethods , John Wiley & Sons, New York, U.S.A.

11. Tedesco, J.W., McDougal, W.G. & Ross, C.A., (1999), Structuraldynamics: theory and applications , Addison Wesley, Menlo Park, U.S.A.

12. Inman, D.J., (1996), Engineering vibration , Prentice-Hall, New Jersey,U.S.A.

13. Dynamic Testing Agency, (1993), DTA handbook, Volume 3 ModalTesting , U.K.

14.

Bathe, K.-J., (1996), Finite element procedures , Prentice-Hall, NewJersey, U.S.A.

15. Huebner, K.H., Thornton, E.A. & Byrom, T.G., (1995), The finiteelement method for engineers , (third edition), John Wiley & Sons, NewYork, U.S.A.

16. Zienkiewicz, O.C. & Taylor, R.L., (2000), The finite element method ,(fifth edition), Butterworth-Heinemann, Oxford, U.K.

17. ABAQUS, ABAQUS, Inc., http://www.abaqus.com.

18. I-deas, Electronic Data Systems Corporation (EDS), http://www.eds.com.

19. Ewins, D.J., (2000), Modal testing: theory, practice and application ,(second edition), Research Studies Press, Baldock, U.K.

20. Maia, N.M.M. & Silva, J.M.M. (eds.), (1997), Theoretical andexperimental modal analysis , Research Studies Press, Baldock, U.K.

21. Allemang, R.J., (1994), Vibrations: analytical and experimental modalanalysis , UC-SDRL-CN-20-263-662, Structural Dynamics ResearchLaboratory, University of Cincinnati, Cincinnati, U.S.A.

22. Allemang, R.J. & Brown, D.L., (1982), A correlation coefficient formodal vector analysis, Proceedings of the International Modal AnalysisConference - IMAC , Orlando, U.S.A.


28/98

28

23. Maguire, J.R., (1996), A correlation benchmark for dynamic analysis, Proceedings of the DTA/NAFEMS/SECED Second InternationalConference: Structural dynamics modelling: Test, analysis andcorrelation , Cumbria, U.K.

24. Ewins, D.J. & Imregun, M., (1988), On the reliability of computationaldynamic response prediction capabilities (DYNAS), Journal of the

society of environmental engineers , 27, pp. 3-13.

25. Chen, G., (2001), FE model validation for structural dynamics ,Doctorial thesis, Dynamics Section, Department of MechanicalEngineering, Imperial College of Science, Technology and Medicine,

University of London, London, U.K.26. Link, M. & Friswell, M., (2003), Working group 1: generation of

validated structural dynamics models results of a benchmark studyutilising the GARTEUR SM-AG19 test-bed, Mechanical Systems andSignal Processing , 17(1), pp. 9-20.

27. Avitabile, P., (2000), Model updating endless possibilities, Sound andVibration , September 2000, pp. 20-28.

28. Friswell, M.I. & Mottershead, J.E., (1995), Finite element modelupdating in structural dynamics , Kluwer Academic Publishers,Dordrecht, The Netherlands.

29. Ullman, D.G., (1997), The mechanical design process , (second edition),McGraw-Hill, New York, U.S.A.

30. Wang, G.G., (2002), Definition and review of virtual prototyping, Journal of Computing and Information Science in Engineering , 2(3), pp.232-236.

31. Thomke, S.H., (1998), Simulation, learning and R & D performance:evidence from automotive development, Research Policy , 27(1), pp. 55-74.

32. Thomke, S. & Fujimoto, T., (2000), The effect of front-loading problem-solving on product development performance, Journal of Product Innovation Management , 17(2), pp. 128-142.


29/98

29

33. Hudi, J., Prokop, G., Pausch, M. & Kvasnicka, P., (2001), Integratedapplication of multibody simulation in the product-development process,

Proceedings of the ADAMS North American User Conference , Novi,U.S.A.

34. Englund, T., (2003), Dynamic characteristics of automobile exhaust system components , Licentiate thesis, Department of MechanicalEngineering, Blekinge Institute of Technology, Karlskrona, Sweden.


30/98

30


31/98

31

Paper A

Significance of Non-linearity andComponent-internal Vibrationsin an Exhaust System


32/98

32

Paper A is published as:

Englund, T., Wall, J., Ahlin, K. & Broman, G., (2002), Significance of non-linearity and component-internal vibrations in an exhaust system,

Proceedings of the 2nd WSEAS International Conference on Simulation, Modelling and Optimization, Skiathos Island, Greece.


33/98

33

Significance of Non-linearity and

Component-internal Vibrationsin an Exhaust System

Thomas L Englund, Johan E Wall, Kjell A Ahlin, Gran I Broman

AbstractTo facilitate overall lay-out optimisation inexpensive dynamics simulation ofautomobile exhaust systems is desired. Identification of possible non-linearityas well as finding simplified component models is then important. A flexible

joint is used between the manifold and the catalyst to allow for the motion ofthe engine and to reduce the transmission of vibrations to the rest of theexhaust system. This joint is significantly non-linear due to internal friction,which makes some kind of non-linear analysis necessary for the completeexhaust system. To investigate the significance of non-linearity and internal

vibrations of other components a theoretical and experimental modal analysisof the part of a typical exhaust system that is downstream the flexible joint is performed. It is shown that non-linearity in this part is negligible. It is alsoshown that shell vibrations of the catalyst and mufflers as well as ovalling ofthe pipes are negligible in the frequency interval of interest. The resultsimplies, for further dynamics studies, that the complete system could beidealised into a linear sub-system that is excited via the non-linear flexible

joint, that the pipes could be modelled with beam elements and that the othercomponents within the linear sub-system could also be modelled in asimplified way. Such simplified component models are suggested. The

agreement between theoretical and experimental results is very good, whichindicates the validity of the simplified modelling.

Keywords: Correlation, Dynamics, Exhaust system, Linear sub-system, Modalanalysis, Non-linear joint.


34/98


35/98

35

essentially linear so that, in the further studies, the complete system could beidealised into a linear sub-system that is excited via the non-linear flexible

joint. The purpose is also to find a computationally effective andexperimentally verified finite element (FE) model of this linear sub-system.This includes simplified modelling of the components.

2. Exhaust System Design and Excitation

The studied automobile exhaust system is shown in figure 1. The mass of thesystem is about 22 kg and it has a length of approximately 3.3 m.

Figure 1. The studied exhaust system.

The system consists of a front assembly and a rear assembly connected with asleeve joint. Both are welded structures of stainless steel. The front part isattached to the manifold by a connection flange. The engine and manifold arenot included in the study.

Between the manifold and the catalyst there is a flexible joint, consisting of a bellows expansion joint combined with an inside liner and an outside braid.This joint is significantly non-linear due to internal friction. More informationon this type of joint is given by, for example, Cunningham et al. [6] andBroman et al. [7].

The front assembly, see figure 2, consists of this joint, the catalyst and pipes.


36/98


37/98

37

rotational speed is below 6000 rpm. Excitation at low frequencies may arisedue to road irregularities, as discussed by, for example, Belangardi and Leonti[1] and Verboven et al. [3]. Thus, the interval is set to 0-200 Hz.

Free-free boundary conditions are generally desired to facilitate a comparison between the FE-results and the experimental results. This also makes it possible to easily exclude the influence of the non-linear joint in the presentanalysis. It is assured that the flexible joint does not have any internaldeformations. Thus it will move as a rigid body in the present analysis.

3. Initial Finite Element Model

An initial FE-model of the exhaust system is built in I-DEAS [8]. The outsideshell structure of the mufflers and the catalyst are modelled with linearquadrilateral shell elements using the CAD-geometry. The mass of theinternal material is distributed evenly to the shell elements. The pipes aremodelled using parabolic beam elements. The flexible joint is modelled bystiff beam elements with a fictive density to reflect its mass and mass momentof inertia. Lumped mass elements are used to model the connection flange,attachments for the hangers, nipples and the heat shield. Connection betweenthe beam elements representing the pipes and the shell elements representingthe mufflers/catalyst is obtained by rigid elements.

By comparing different mesh densities it is found that approximately 140 beam elements and 1900 shell elements are sufficient. The total number ofnodes are approximately 2200. This initial model is used as a basis fordetermining suitable transducer locations for the experimental modal analysisof the exhaust system.

The natural frequencies are solved for by the Lanczos method with free-free boundary conditions.

4. Experimental Modal Analysis

To sufficiently realise the free-free boundary conditions in the experimentalmodal analysis (EMA) the exhaust system is suspended, at the hangerattachments and at the connection flange, using soft adjustable rubber bandsas shown in figure 4.


38/98


39/98

39

Again considering the results from the initial FE-model it is concluded that 25evenly distributed measuring points should be sufficient to represent the modeshapes in the frequency interval of interest. Using the AutoMAC, see figure 5,the chosen measurement points are checked to avoid spatial aliasing. Thesmall off-diagonal terms in the AutoMAC indicate that the chosenmeasurement points sufficiently well describe the modes in the frequencyinterval of interest.

1 2

3 4

5 6

7 8

9 10

12

34

56

78

910

0

0.2

0.4

0.6

0.8

1

Figure 5. The AutoMAC-matrix.

The quality of the experimental set-up is further assured by a linearity check,a reciprocity check and by investigating the driving point frequency responsefunction (FRF). Also the coherence of some arbitrary FRFs is investigated.All the quality checks show satisfactory results.

Due to the long and slender geometry of the exhaust system concerns mayarise that the static preload could have an undesired influence when thesystem hang horizontally. To ensure that this is not the case the exhaustsystem is also hanged vertically and some arbitrary FRFs are measured. The


40/98

40

difference in natural frequencies is negligible between the two set-ups and it istherefore concluded that the initial set-up is satisfactory.

I-DEAS Test [9] is used to acquire the FRFs. The FRFs are exported toMATLAB [10] where they are analysed using the experimental structuraldynamics toolbox developed by Saven Edutech AB [11]. The poles areextracted in the time domain using a global least square complex exponentialmethod. The residues are found using a least squares frequency domainmethod. To improve the quality of the extracted modal parameters only datain the y- and z -directions are used. To get as good a fit as possible the curvefitting procedure is conducted in two steps; first in the interval 5-90 Hz andthen in the interval 90-150 Hz. Above 150 Hz no significant modes are found,

as seen in a typical FRF shown in figure 6.

50 100 150 200 250

10 5

10 4

10 3

10 2

Frequency response function

Frequency (Hz)

V e

l o c

i t y

/

R e a c t

i o n

f o r c e

( m

/

N s

)

Figure 6. Typical FRF.


41/98

41

5. Simplification and Correlation

Determining the natural frequencies of the mufflers and the catalystexperimentally it is seen that no significant local modes are present in thefrequency interval of interest. This was also found by Verboven et al. [3].Therefore the modelling of the mufflers and the catalyst, which areresponsible for most of the model size in the initial FE-model, can besignificantly simplified. The mufflers and the catalyst are modelled by lumpedmass and mass moment of inertia elements. The properties of these elementsare obtained from the original FE-model. If more suitable in a general casethese properties can also be obtained directly from the CAD-model orexperimentally. The lumped mass and mass moment of inertia elements are

connected to the beam elements representing the pipes by rigid elements.

The natural frequencies of the pipes are also investigated experimentally. Nosignificant ovalling modes are found in the frequency interval of interest,which confirms the validity of modelling the pipes by beam elements.

To simulate the flexibility of the connections between the pipes andmufflers/catalyst, short beam elements with individual properties are used.These elements are located at the true connection locations, that is, withreference to the real system. Thus, they are placed between the rigid elementsthat are connected to the lumped mass and mass moment of inertia elementsand the beam elements representing the pipes.

These individual beam properties are updated so that the difference betweentheoretical and experimental results is minimised. The updating procedureuses MATLAB [10] and ABAQUS [12] and is described in an accompanying

paper (Englund et al. [13]).

The updated FE-model has approximately 200 nodes. Thus a reduction ofover 90 % compared to the initial FE-model is obtained. Simplifications ofthis type are important if direct time integration becomes necessary for thenon-linear dynamics analysis of the complete system. It is also importantwhen a large number of simulations are necessary for overall exhaust systemlay-out optimisation.

The FE modes are calculated without consideration of damping and aretherefore real-valued. To be able to compare these modes with the modesobtained experimentally, which are complex due to damping, theexperimental modes are converted into real-valued modes.


42/98

42

6. Resul ts and Correlation

To correlate the mode shapes from the updated FE-model and theexperimental mode shapes a MAC-matrix is calculated, see figure 7.

Figure 7. The MAC-matrix.

Except for mode nine and ten the diagonal MAC-values are above 0.85, whichindicates good correlation. All the off-diagonal values in the MAC-matrix are

below 0.2. This indicates that the different mode shapes are non-correlated.

A comparison between theoretical and experimental natural frequencies isshown in figure 8. The 45-degree line represents perfect matching. Thecrosses indicate the frequency match for each correlated mode pair.


43/98

43

0 50 100 1500

50

100

150

Experimental natural frequency (Hz)

T h

e o r e t

i c a

l

n a t u r a

l

f r e q u e n c y

( H z

)

Figure 8. Theoretical and experimental natural frequencies.

The maximum difference in corresponding natural frequencies is below four per cent. The small and randomly distributed scatter of the plotted points isnormal for this type of modelling and measurement process [14].

The results are summarised in table 1.


44/98


45/98

45

the connections between the mufflers/catalyst and the pipes. Automatedupdating of these individual properties is recommended since doing itmanually is time consuming and difficult.

The agreement between results from the updated FE-model and theexperimental investigations is very good. This implies that such simplifiedmodelling is a valid approach and it may turn out important in coming non-linear analyses, since such analyses are often computationally expensive.

8. Acknowledgements

The support from Faurecia Exhaust Systems AB is gratefully acknowledged,especially from Hkan Svensson. The authors also gratefully acknowledge thefinancial support from the Swedish Foundation for Knowledge andCompetence Development.

9. References

1. Belingardi, G. and Leonti, S., (1987), Modal analysis in the design of anautomotive exhaust pipe, Int. J. of Vehicle Design , vol. 8, no. 4/5/6.

2. Ling, S.-F., Pan, T.-C., Lim, G.-H. and Tseng, C.-H., (1994), Vibrationisolation of exhaust pipe under vehicle chassis, Int. J. of Vehicle Design ,vol. 15, no. 1/2.

3. Verboven, P., Valgaeren, R., Van Overmeire, M. and Guillaume, P.,(1998), Some comments on modal analysis applied to an automotiveexhaust system, Proceedings of the international Modal AnalysisConference IMAC , Santa Barbara, USA.

4. Myrn, M. and Olsson, J., (1999), Modal analysis of exhaust system ,Master Thesis, Department of Mechanical Engineering, University ofKarlskrona/Ronneby, Karlskrona, Sweden,.

5. DeGaspari, J., (2000), Lightweight exhaust, Mechanical Engineering ,May 2000.


46/98

46

6. Cunningham, J., Sampers, W. and van Schalkwijk, R., (2001), Design offlexible tubes for automotive exhaust systems, ABAQUS UsersConference .

7. Broman, G., Jnsson, A. and Hermann, M., (2000), Determiningdynamic characteristics of bellows by manipulated beam finite elementsof commercial software, Int. J. of Pressure Vessels and Piping , vol. 77,Issue 8.

8. I-DEAS, EDS PLM Solutions, http://www.sdrc.com.

9. I-DEAS Test, MTS, http://www.mts.com.

10. MATLAB, The MathWorks, Inc., http://www.mathworks.com.

11. Experimental structural dynamics toolbox, Saven EduTech AB,http://www.saven.se.

12. ABAQUS, HKS, http://www.abaqus.com.

13. Englund, T., Wall, J., Ahlin, K. and Broman, G., (2000), Automatedupdating of simplified component models for exhaust system dynamics

simulations, 2nd WSEAS International Conference on Simulation, Modelling and Optimization , Skiathos Island, Greece.

14. Ewins, D.J., (2000), Model validation: Correlation for updating,S dhan , vol 25, part 3.


47/98

47

Paper B

Automated Updating ofSimplified Component Modelsfor Exhaust System Dynamics

Simulations


48/98

48

Paper B is published as:

Englund, T., Wall, J., Ahlin, K. & Broman, G., (2002),Automated updatingof simplified component models for exhaust system dynamics simulations,



49/98

49

Automated Updating of

Simplified Component Modelsfor Exhaust System Dynamics

Simulations

Thomas L Englund, Johan E Wall, Kjell A Ahlin, Gran I Broman

Abstract

To facilitate overall lay-out optimisation simplified component models fordynamics simulations of automobile exhaust systems are desired. Suchoptimisation could otherwise be computationally expensive, especially whennon-linear analyses are necessary. Suggestions of simplified models of themufflers and the catalyst are given. To account for the flexibility at theconnections between those components and the pipes short beam elements

with individual properties are introduced at these locations. An automatedupdating procedure is developed to determine the properties of these beamelements. Results from an experimental modal analysis are used as thereference. The theoretical model of the exhaust system is built in the finiteelement software ABAQUS. The updating procedure uses the sequentialquadratic programming algorithm included in the Optimization Toolbox ofthe software MATLAB to minimise the sum of the differences betweenexperimentally and theoretically obtained natural frequencies. Constraints areused on the correlation between the experimentally and theoretically obtainedmode shapes by considering the MAC-matrix. Communication between the

two software packages is established by an in-house MATLAB script. Thecorrelation between results from the updated theoretical model and theexperimental results is very good, which indicates that the updating procedureworks well.

Keywords: Correlation, Dynamic, Exhaust system, Modal analysis,Optimisation, Updating.


50/98

50

1. Introduction

Demands on shortened time to market, higher product performance andgreater product complexity in combination with the fast development ofcomputers have resulted in more simulations for prediction and evaluation of

product performance. Simplified modelling and inexpensive simulation procedures are often desired early in the product development process tostudy certain product characteristics and for overall introductory systemsoptimisation. The models and simulations should reflect the interestingcharacteristics of the real system accurately enough to support relevant designdecisions. To gain confidence of this some kind of experimental verification isoften necessary. If the correlation is not good enough the models need to be

updated. Doing this manually is usually a time consuming and difficult task,especially if there are many parameters to be updated in the theoreticalmodels.

Procedures for more automated updating have therefore attained interestwithin the analysis community. See for example the works by VanLangenhove et al. [1] and Deweer et al. [2] regarding updating of dynamicsystems. Avitabile [3] discusses different updating criteria and points out theimportance of the choice of parameters in the updating procedure. Chen andEwins [4] describe the effect of discretisation errors when updating finiteelement models.

This study is a part of a co-operation project between the Department ofMechanical Engineering at the Blekinge Institute of Technology, Karlskrona,Sweden and Faurecia Exhaust Systems AB, Torss, Sweden. The overall aimof the project is to find a procedure for effectively modelling and simulatingthe dynamics of customer-proposed exhaust system lay-outs at an early stagein the product development process, to support the dialogue with the costumerand for overall lay-out optimisation. An accompanying paper is that ofEnglund et al. [5], which focuses on simplified and experimentally verifiedmodelling of a typical automobile exhaust system. The updating of thesimplified models of the components within that system is performedaccording to the procedure described in the present paper. The MATLABOptimization Toolbox [6] is used for the updating procedure and ABAQUS[7] is used to solve for the natural frequencies and mode shapes.Communication between the two different software packages is established byan in-house MATLAB script to obtain automated updating.


51/98

51

2. Exhaust System Design

The studied automobile exhaust system is shown in figure 1. The mass of thesystem is about 22 kg and it has a length of approximately 3.3 m.

Figure 1. The studied exhaust system.

The system consists of a front assembly and a rear assembly connected with asleeve joint. Both are welded structures of stainless steel. The front part isattached to the manifold by a connection flange. The engine and manifold arenot included in the study.

Between the manifold and the catalyst there is a flexible joint. This joint is

significantly non-linear due to internal friction. More information on this typeof joint is given by, for example, Broman et al. [8] and Cunningham et al. [9].

The front assembly consists of this joint, the catalyst and pipes. The rearassembly consists of pipes, an intermediate muffler and a rear muffler.Perforated pipes pass through the mufflers. The mufflers are filled with soundsilencing material.

Besides the connection to the manifold the exhaust system is attached to thechassis of the car by rubber hangers. Two hanger attachments are placed at the

intermediate muffler and a third is placed just downstream the rear muffler,see figure 1.

3. Theoretical and Experimental Analysis

A theoretical model of the exhaust system is built in ABAQUS [7]. The pipesare modelled using quadratic beam elements and the mufflers and the catalystare modelled using lumped mass and mass moment of inertia elements. Such


52/98

52

simplified modelling is valid in the frequency interval of interest [5]. Theseelements are connected to the beam elements representing the pipes by rigidelements. The properties of the lumped mass and mass moment of inertiaelements are obtained from a finite element (FE) model where these parts aremodelled with shell finite elements [5]. If more suitable in a general casethese properties can also be obtained directly from the CAD-model orexperimentally.

To simulate the flexibility of the connections between the pipes andmufflers/catalyst, short beam elements with individual properties are used.These elements are located at the true connection locations, that is, withreference to the real system. Thus, they are placed between the rigid elements

that are connected to the lumped mass and mass moment of inertia elementsand the beam elements representing the pipes.

Lumped mass elements are used to model the connection flange attached tothe flexible joint, attachments for the hangers and the heat shield. Free-free

boundary conditions are used and the natural frequencies and mode shapes aresolved for by the Lanczos method. More information about the theoreticalmodel can be found in [5].

The results from the theoretical model are compared with natural frequencies

and mode shapes obtained experimentally. The theoretical mode shapes arecalculated without consideration of damping and are therefore real-valued. To be able to compare these modes with the modes obtained experimentally,which are complex due to damping, the experimental modes are convertedinto real-valued modes.

The experimental modal analysis (EMA) is performed using free-free boundary conditions. To exclude the influence of the non-linearity of theflexible joint it is assured that it does not have any internal deformations. Thusit will move as a rigid body in the present analysis. More about the EMA can

be found in [5].

The frequency interval of interest is 0-200 Hz but actually no significantmodes occur above 150 Hz for this particular exhaust system [5].


53/98

53

4. Updating

The experimentally obtained natural frequencies and mode shapes are used toupdate the theoretical model. If, in a general case, a full physical prototypedoes not exist results from a detailed finite element model can be used as thereference.

The selection of parameters to be included in the updating procedure isimportant. This is true whether the updating is based on frequencydifferences, mode shape differences or frequency responses [3]. Except for theconnections between the mufflers/catalyst and the pipes the theoretical modelof the exhaust system is straightforward. Properties influencing the flexibility

(stiffness) of these connections are used when updating the theoretical model.There are six connections marked in figure 2 and 3. Each of them includes thefollowing three stiffness related properties; the two area moments of inertiasand the polar area moment of inertia of the short beam elements representingthe connections. All connections have individual properties. Altogether thisgives 18 independent parameters to consider when updating the theoreticalmodel.

Figure 2. Front assembly.


54/98

54

Figure 3. Rear assembly.

To sort out the important ones, a simple parameter study is performed. The parameters are modified by a factor ten, one at a time, and the naturalfrequencies are calculated. It can then roughly be concluded which parametersthat are important to consider when updating the theoretical model of theexhaust system. Ten parameters are found to be significantly more importantthan the others. Using this approach the possibility to detect interaction

between parameters is lost. Considering also these effects can be very timeconsuming. The procedure used in this work is a compromise betweenaccuracy and time consumption. The aim is not necessarily to find the globaloptimum, but rather a solution that is good enough. Since many parametersare still involved an automated updating procedure using the OptimizationToolbox in MATLAB is developed. A constrained optimisation is performedusing a sequential quadratic programming (SQP) algorithm [6]. Theoptimisation algorithm is supplied with start-values, bounds, constraints andoptimisation criterion. The optimisation criterion chosen, which is to beminimised, is the sum of the differences in natural frequency within eachcorrelated mode pair. Constraints are used on the correlation betweentheoretical and experimental mode shapes using the diagonal values of the

MAC-matrix. The modal assurance criterion (MAC) is a technique to quantifythe correlation between two sets of mode shapes. This constraint is importantsince it forces the algorithm to use correlated mode pairs when calculating theoptimisation criterion. Good agreement is sought for both natural frequenciesand mode shapes. Using constraints and bounds limits the search space, whichusually reduces the number of function evaluations, that is, the problemconverges faster [6].

Since natural frequencies and mode shapes must be solved for many timesduring the updating procedure ABAQUS and MATLAB interact with each


55/98

55

other. An in-house MATLAB script, taking advantage of MATLABs abilityof reading and writing ASCII-files, is used to transfer data between the twodifferent software packages. The optimisation procedure is schematicallyshown in figure 4.

Optimum?

The MATLABToolbox calculates a

new parametercombination.

ABAQUS solves fornatural frequenciesand mode shapes.

TerminateSolution

No

Yes

The MATLABToolbox calculates

optimisation criterionand constraints.

Figure 4. Automated updating procedure.Setting appropriate tolerances for the search algorithm in the OptimizationToolbox is not a trivial task. It usually has to be tuned for specific problems.Setting the tolerances to tight forces the algorithm to make a large number offunction evaluations without finding a much better solution. On the other handsetting them to loosely the search algorithm might not find the correctoptimum. To be able to set the tolerances for the optimisation algorithm in astraightforward way all the ten parameters are scaled to be between zero andunity.


56/98

56

An important aspect to consider is that SQP is a gradient-based optimisationroutine. This means that it only finds local optima, that is, different optimacan be found depending on the start-values. Some kind of multi-start

procedure can be used to reduce this problem. Another way is to use somekind of derivate-free optimisation method. In this work the start-values for theshort beam element properties are taken from the beam elements representingthe pipes at the connections in the theoretical model. If the start-values aregood, that is, are near an optimum, the search algorithm finds this optimumfaster.

5. Resul ts and Discussion

A comparison between the results from an initial theoretical model, that is, amodel without the short beam elements accounting for the flexibility at theconnections between the mufflers/catalyst and the pipes, and the experimentalresults shows that this model is far too stiff. Some of the theoretical naturalfrequencies are more than fifty per cent higher than the corresponding naturalfrequencies obtained experimentally.

In a first step to achieve a theoretical model that correlates better with theexperimental results Youngs modulus, of the fictive material of the short

beam elements representing the connections, is updated. The same value ofthis parameter is used for all connections. The comparison between resultsfrom this roughly updated model, and the experimental results aresummarised in table 1. The correlation is still not considered good enough.

As seen in table 1 mode six and seven is not correlating. This is due to a modeswitch between these modes, see figure 5. Furthermore, it can be seen in thefigure that some of the off-diagonal values are high. This also indicates badcorrelation.


57/98

57

Table 1. Results after the first update.

Experimental TheoreticalModeFrequency (Hz) Frequency (Hz)

Correlation a (%) MAC

1 10.9 10.6 -2.4 0.952 12.9 13.1 1.2 0.933 34.9 35.9 2.8 0.584 36.4 42.1 16 0.675 59.1 50.0 -15 0.846 67.1 74.6 117 80.8 82.9 2.68 101 86.2 -14 0.919 127 116.5 -7.9 0.72

10 139 141.2 1.5 0.70a The correlations are calculated before rounding off.

Figure 5. The MAC-matrix after the first update.


58/98

58

In a final step the ten independent parameters are included in the automatedupdating procedure. The correlation between modes of this theoretical modeland the experimental modes are calculated using the MAC-matrix, see table 2and figure 6.

The correlation is very good. All diagonal MAC values are above 0.85 exceptfor mode nine and ten. Furthermore all the off-diagonal terms in the MAC-matrix are below 0.2. As also seen all differences in natural frequencies are

below four per cent.

Table 2. Results after the final update.

Experimental TheoreticalModeFrequency (Hz) Frequency (Hz)

Correlation a (%) MAC

1 10.9 10.9 0.24 0.952 12.9 12.8 -1.0 0.933 34.9 35.8 2.6 0.884 36.4 36.9 1.3 0.855 59.1 57.3 -3.0 0.936 67.1 69.7 3.9 0.85

7 80.8 83.7 3.6 0.918 101 101 0.30 0.969 127 126 -0.60 0.64

10 139 135 -2.9 0.60a The correlations are calculated before rounding off.


59/98

59

Figure 6. The MAC-matrix after the final update.

6. Conclusions

Updating of simplified component models for simulation of the dynamic behaviour of an automobile exhaust system is the subject of this paper.Results obtained from an experimental modal analysis are used as thereference. If, in a general case, a full physical prototype does not exist resultsfrom a detailed finite element model can be used as the reference.

The simplified component models can be used for, otherwise computationallyexpensive, overall lay-out optimisation and they can also be re-used when thesame or similar components are to be included in other exhaust systemassemblies.

An automated updating procedure is developed. The sequential quadratic programming algorithm in MATLABs Optimization Toolbox is used tominimise the difference between theoretical and experimental naturalfrequencies. Constraints are used on the correlation between the theoretical


60/98

60

and experimental mode shapes using the MAC-matrix. The naturalfrequencies and mode shapes are solved for by ABAQUS. Communication

between the two software packages is established by an in-house MATLABscript.

The very good correlation between the updated theoretical model and theexperimental results shows that the updating procedure works well.

7. Acknowledgements

The support from Faurecia Exhaust Systems AB is gratefully acknowledged,

especially from Hkan Svensson. The authors also gratefully acknowledge thefinancial support from the Swedish Foundation for Knowledge andCompetence Development.

8. References

1. Van Langenhove, T., Fred, C. and Brunner, O., (2001), FE modelcorrelation & mode shape updating using qualification test data. A casestudy on the Olympus satellite, Proceedings of NAFEMS WorldCongress , Como, Italy.

2. Deweer, J., Van Langenhove, T. and Grinker, S., (2001), Identificationof the best modal parameters and strategies for FE model updating, SAE

Noise & Vibration Conference & Exposition , Grand Traverse, USA.

3. Avitabile, P., (2000), Model updating endless possibilities, Sound andVibration , September 2000.

.

4. Chen, G. and Ewins, D.J., (2000), Perspective on modal updating performance, Proceedings of the International Modal AnalysisConference IMAC , San Antonio, USA.

5. Englund, T., Wall, J., Ahlin, K. and Broman, G., (2002), Significance ofnon-linearity and component-internal vibrations in an exhaust system,WSEAS International Conference on Simulation, Modelling andOptimization , Skiathos Island, Greece.

6. MATLAB, The MathWorks, Inc., http://www.mathworks.com.


61/98

61

7. ABAQUS, HKS, http://www.abaqus.com.

8. Broman, G., Jnsson, A. and Hermann, M., (2000), Determiningdynamic characteristics of bellows by manipulated beam finite elementsof commercial software, Int. J. of Pressure Vessels and Piping , vol. 77,Issue 8.

9. Cunningham, J., Sampers, W. and van Schalkwijk, R., (2001), Design offlexible tubes for automotive exhaust systems, ABAQUS UsersConference .


62/98

62


63/98

63

Paper C

Modelling of Multi-ply BellowsFlexible Joints of VariableMean Radius


64/98

64

Paper C is published as:

Wall, J., Englund, T., Ahlin, K. & Broman, G., (2003), Modelling of multi- ply bellows flexible joints of variable mean radius, Proceedings of the NAFEMS World Congress 2003 , Orlando, U.S.A.


65/98

65

Modelling of Multi-ply Bellows

Flexible Joints of Variable MeanRadius

Johan E Wall, Thomas L Englund, Kjell A Ahlin, Gran I Broman

AbstractBellows flexible joints are included in automobile exhaust systems to allowfor engine movements and thermal expansion and to reduce vibrationtransmission. Generally the joint consists of a flexible bellows, an inside linerand an outside braid. In this work the bellows is considered. A straightforwardway to model the bellows is to use shell finite elements. Due to theconvoluted geometry of the bellows that procedure requires however a highnumber of elements, meaning that the bellows model would constitute a large

part of the model of the exhaust system. For more effective dynamics

simulations a beam finite element representation of the bellows has been presented in a prior work. This modelling procedure was implemented in thecommercial software I-DEAS and was verified against experimental resultsavailable in the literature for single-ply bellows of constant mean radius. This

paper suggests adjustments by which this procedure can be extended to modelalso multi-ply bellows of variable mean radius. Experimental investigations ofa double-ply bellows having decreasing mean radius towards its ends areincluded for verification. The agreement between theoretical and experimentalresults is very good, implying that the suggested extension of the modelling

procedure is valid. It is also shown that the procedure can easily be

implemented into other commercial software (in this case ABAQUS). Theexperimental investigation reveals an intriguing resonance frequency shift atsmall excitation force levels. Although considered to be of minor significancefor the present application of the bellows, a hypothetic qualitative explanationto the observed phenomenon is given.

Keywords: Beam model, Bellows, Dynamic, Experimental investigation, Flexible joint, Frequency shift, Multi-ply, Variable mean radius.


66/98

66

1. Notation

A Area [m2

]

E Modulus of elasticity [Pa]

G Shear modulus [Pa]

h Height [m]

I Area moment of inertia [m 4]

K Polar area moment of inertia [m4

]

L Length [m]

R Radius [m]

r Radius [m]

t Thickness [m]

Poissons ratio

Density [Kg/m 3]

Indices

conv Convolution

m Middle

p Pipe


67/98

67

2. Introduction

Bellows flexible joints are important components in automobile exhaustsystems. A flexible connection between the manifold and the rest of theexhaust system is necessary to allow for deflections induced by enginemovements and due to thermal expansion and to reduce vibrationtransmission. Recent suggestions of a stiffer attachment of the exhaust systemto the chassis, as discussed by for example DeGaspari [1], with the purpose ofreducing weight, makes this component even more important.

Proper dimensioning of the flexible joint requires understanding of itsdynamic characteristics and interaction with the rest of the exhaust system.

This is studied in a co-operation project between the Department ofMechanical Engineering at Blekinge Institute of Technology, Karlskrona,Sweden and Faurecia Exhaust Systems AB, Torss, Sweden. The overall aimof the project is to find a procedure for effectively modelling and simulatingthe dynamics of customer-proposed exhaust system lay-outs at an early stagein the product development process, to support the dialogue with thecostumers and for overall lay-out optimisation. To be suited for that thesimulation procedure cannot be too computationally expensive. This isespecially important when the dynamics is non-linear, which will beconsidered in later studies. The models of the components of the exhaustsystem must therefore be as simple as possible while still giving a properdescription of the dynamics of the system. The bellows flexible joint is thecomponent within the exhaust system that is most difficult to describeinexpensively.

Broman et al. [2] presented a method for determining the dynamiccharacteristics of single-ply bellows of constant mean radius by manipulated

beam finite elements of commercial software based on the assumption that the bellows is linear. Compared to a shell elements model, which would be themost straightforward way of modelling the bellows, the model size is reducedconsiderably by using this beam element model. Axial, bending and torsiondegrees of freedom can be studied simultaneously and the modellingtechnique facilitates the interaction between the bellows and the rest of theexhaust system, usually also modelled by finite elements. A short historical

background and further references on bellows studies can be found in [2].

In this paper it is investigated if the beam element procedure can be extendedto model also a multi-ply bellows of variable mean radius. Experimentalinvestigations are performed for verification. It is also tested if the procedure


68/98

68

can easily be implemented in other commercial software (in this caseABAQUS [3]) than the one used in [2] (I-DEAS [4]).

3. Basic Design of Flexible Joint and Excitation

The basic design of the flexible joint is shown in figure 1. It consists of a gas-tight bellows combined with an inside liner and an outside braid. The linerwas originally introduced for reduction of bellows temperature and forimproved flow conditions. It also further reduces vibrations. The braid is usedfor mechanical protection and to limit the extension of the joint. The parts areconnected with end-caps. The complete joint is significantly non-linear. In

this paper the bellows is considered. More information on this type of joint isgiven by, for example, Cunningham et al. [5].

End-cap

Braid Gas-tight bellows

Liner

Figure 1. Basic flexible joint design.

The bellows of this paper is double-plied and it has smaller mean radius closerto the ends. For a given strength a multi-ply bellows has lower stiffness than asingle-ply bellows. Low stiffness is desired to decouple the engine from therest of the exhaust system.

The frequency interval of interest for the analysis is obtained by consideringthat a four-stroke engine with four cylinders gives its main excitation at afrequency of twice the rotational frequency. Usually the rotational speed is


69/98

69

below 6000 rpm. Excitation at low frequencies may arise due to roadirregularities, as discussed by, for example, Belangardi and Leonti [6] andVerboven et al. [7]. Thus, the interval is set to 0-200 Hz.

4. Modelling of Bellows

Broman et al. [2] described how to model the dynamic characteristics of a bellows using a pipe analogy and by manipulating certain parameters of the beam finite element formulation in the software I-DEAS. This procedure isadopted and extended in this paper.

While the current bellows has a variable mean radius, with smaller radii closerto the ends, different equivalent pipes are used for different parts of the

bellows. These equivalent pipes have different equivalent density, p, shearmodulus, G p, modulus of elasticity, E p, area, A p, area moment of inertia, I p,and polar area moment of inertia, K p. Three different equivalent pipes, withassumed constant mean radii, are used, see figure 2.

1 2 3

Figure 2. Three sections to be represented by differentequivalent pipe models.

Other differences are that the end caps of the bellows are included in theanalysis and that the bellows has two plies instead of one ply. Furthermore theconvolution profile is slightly different from the U-shaped profile consideredin [2].

The bellows is made of stainless steel. The material properties are E = 193GPa, = 8000 kg/m 3, and = 0.29.


70/98

70

The characteristic dimensions of the convolutions can be seen in figure 3. Theconvolution dimensions of the three different pipe sections are presented intable 1.

h

R m

r

r

Lconv

Figure 3. Convolution dimensions.

Table 1. Convolution dimensions (mm).

SectionDimension1 2 3

Rm 61.2 64.9 67.9 Lconv 7.60 7.60 7.60

r 2.20 2.40 2.65h 6.70 10.4 13.4t 0.193 0.193 0.193

The thickness of each ply in the bellows is reduced during the forming process. The standard of the Expansion Joint Manufacturing Association,EJMA [8], suggests how to account for this for U-shaped bellows. For the

present convolution profile this correction is however insufficient, resulting ina too heavy and stiff bellows. A different approach is therefore used. Themass of the bellows is measured. As the density and the remaining dimensions


71/98


72/98

72

step is to extract sufficiently many eigenmodes so that the dynamic responseof the system is adequately modelled. To get a realistic result the modaldamping of the system is specified. These damping ratios are obtained fromexperimental results. The damping ratios used in this work are given in table3.

The model is clamped at one end and is free at the other end. Two differentload cases are considered. In the first case the bellows is excited with an axialharmonic force at the free end. I

Dynamics Study of an Automobile Exhaust System

Documents

Transcript of Dynamics Study of an Automobile Exhaust System