# Dynamic Characteristics of Automobile Exhaust System Components

Dynamic Characteristics of Automobile Exhaust System Components Thomas Englund Karlskrona, 2003 Department of Mechanical Engineering Blekinge Institute of Technology SE-371 79 Karlskrona, Sweden © Thomas Englund Blekinge Institute of Technology Licentiate Dissertation Series No. 2003:05 ISSN 1650-2140 ISBN 91-7295-027-7 Published 2003 Printed by Kaserntryckeriet AB Karlskrona 2003 Sweden Acknowledgements This work was carried out at the Department of Mechanical Engineering, Blekinge Institute of Technology, Karlskrona, Sweden, under the supervision of Professor Göran Broman and Professor Kjell Ahlin. I wish to express my appreciation to my supervisors for their support and guidance throughout this work. My thanks go also to my colleagues and friends at the Department. I especially want to thank M.Sc. Johan Wall for many interesting discussions and a fruitful cooperation. I am also grateful to Associate Professor Mikael Jonsson at the Division of Computer Aided Design, Department of Mechanical Engineering, Luleå University of Technology, Luleå, Sweden, for his help during this work. Last but not least I want to thank the staff at Faurecia Exhaust Systems AB, especially M.Sc. Kristian Althini and M.Sc. Håkan Svensson, for valuable support and discussions. I gratefully acknowledge the financial support from the Swedish Foundation for Knowledge and Competence Development, Faurecia Exhaust Systems AB and the Faculty Board of Blekinge Institute of Technology. Thomas Englund 3 Abstract Demands on emission control, and low vibration and noise levels have made the design of automobile exhaust systems a much more complex task over the last few decades. This, combined with increasing competition in the automobile industry, has rendered physical prototype testing impractical as the main support for design decisions. The aim of this thesis is to provide a deeper understanding of the dynamic characteristics of automobile exhaust system components to form a basis for improved design and the development of computationally inexpensive theoretical component models. Modelling, simulation and experimental investigation of a typical exhaust system are performed to gain such an understanding and evaluate ideas of component modelling. Modern cars often have a gas-tight bellows-type flexible joint between the manifold and the catalytic converter. This joint is given special attention since it is the most complex component from a dynamics point of view and because it is important for reducing transmission of engine movements to the exhaust system. The joint is non-linear if the bellows consists of multiple plies or if it includes an inside liner. The first non-linearity is shown to be weak and may therefore be neglected. The non-linearity due to friction in the liner is, however, highly significant and gives the joint complex dynamic characteristics. This is important to know of and consider in exhaust system design and proves the necessity of including a model of the liner in the theoretical joint model when this type of liner is present in the real joint to be simulated. It is known from practice and introductory investigations that also the whole system sometimes shows complex dynamic behaviour. This can be understood from the non-linear characteristics of the flexible joint shown in this work. An approach to the modelling of the combined bellows and liner joint is suggested and experimentally verified. It is shown that the exhaust system is essentially linear downstream of this joint. Highly simplified finite element models of the components within this part are suggested. These models incorporate 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 4 and the experimental results is very good, which verifies the usability of these component models. A major conclusion is that in coming studies of how engine vibrations affect the exhaust system it may be considered as a linear system if the flexible joint consists of a bellows. If the joint also includes a liner, the system may be considered as a linear sub-system that is excited via a non-linear joint. Keywords: Correlation, Exhaust system, Experimental investigation, Finite element model, Flexible joint, Modal analysis, Non-linear, Simplified modelling, Structural dynamics, Updating. 5 Thesis Disposition This thesis comprises an introductory part and the following appended papers: Paper A Englund, T., Wall, J., Ahlin, K. and Broman, G., ‘Significance of nonlinearity and component-internal vibrations in an exhaust system’, Proceedings of the 2nd WSEAS International Conference on Simulation, Modelling and Optimization, Greece, 2002. Paper B Englund, T., Wall, J., Ahlin, K. and Broman, G., ‘Automated updating of simplified component models for exhaust system dynamics simulations’, Proceedings of the 2nd WSEAS International Conference on Simulation, Modelling and Optimization, Greece, 2002. Paper C Wall, J., Englund, T., Ahlin, K. and Broman, G., ‘Modelling of multi-ply bellows flexible joints of variable mean radius’, Proceedings of the NAFEMS World Congress 2003, USA, 2003. Paper D Englund, T., Wall, J., Ahlin, K. and Broman, G., ‘Dynamic characteristics of a combined bellows and liner flexible joint’. Submitted for publication. In the introductory part chapter one gives the general context of the work, chapters two to four provide a brief introduction to the field of structural dynamics and chapters five to seven present and discuss the specific research problem of this thesis. 6 The Author’s Contribution to the Papers The appended papers are prepared in collaboration with co-authors. The present author’s contributions are as follows: Paper A Responsible for planning and writing the paper. Carried out approximately half of the theoretical modelling, simulations and experimental investigations. Paper B Took part in the planning and writing of the paper. Carried out approximately half of the simulations and development of the updating routine. 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. Responsible for the theoretical modelling and simulations. Took part in the experimental investigations. 7 Table of Contents 1 Introduction 2 Vibration Analysis 2.1 Introduction to Vibration 2.2 Linear and Non-linear Vibrations 2.3 Solution Methods 3 Experimental Investigations 3.1 Introduction to Experimental Modal Analysis 3.2 Data Acquisition 3.3 Modal Parameter Extraction 4 Correlation and Updating 5 Research Problem 5.1 Background 5.2 Exhaust System Components 5.3 Aim and Scope 6 Summary of Papers 6.1 Paper A 6.2 Paper B 6.3 Paper C 6.4 Paper D 7 Conclusions and Future Research 8 References 9 11 11 11 14 17 17 17 20 21 25 25 26 29 31 31 31 32 32 33 35 Appended Papers Paper A Paper B Paper C Paper D 39 55 71 91 8 1 Introduction Increased demands on improvements in product quality in a wide sense together with demands on reduced development costs and time-to-market [1, 2] have made physical prototype testing impractical as the main support for design decisions. The trend is towards virtual prototyping [3-7] to save time and other resources in the development process itself and in order to find more optimal solutions to market demands. Theoretical modelling and simulation make it possible to investigate many different design solutions and gain a better fundamental understanding of the influence of various design parameters on product characteristics. The importance of this is indicated in figure 1 [8, 9]. Figure 1. The design process paradox. During the development process more and more details concerning the product design need to be fixed. At the same time, knowledge about the design problem increases. This is the classic dilemma or paradox of product development, and there is, of course, a strong desire to raise the knowledge curve as much as possible as early as possible while the freedom of design is still high in order to avoid costly design changes late in the process. Theoretical modelling and simulation facilitate this. 9 The improved fundamental understanding usually obtained through theoretical modelling and simulation also facilitates re-use of knowledge in future development projects, which further promotes overall efficiency of the product developing company. Many products are, however, complex; and it should be pointed out that to be able to trust the theoretical models and the simulation procedures, experimental investigations are necessary for verification [2, 10]. This can often be done by using sub-systems or analogy with earlier products, instead of a full physical prototype of the present product. The aim should be to use an optimum of physical testing through intelligent coordination with modelling and simulation and not to exclude physical testing entirely. Investigations regarding the dynamic characteristics of products have become increasingly important and comprehensive [8]. There is an increased general awareness of dynamics problems, and companies are forced by legislation and customer’s demands to lower vibration and noise levels in their products. A parallel explanation for the increased activity within this field is the immense development of computer capacity and software, which has made far more comprehensive investigations possible. This thesis is a part of a co-operation project between the Department of Mechanical Engineering at Blekinge Institute of Technology, Karlskrona, Sweden and Faurecia Exhaust Systems AB, Torsås, Sweden. The overall aim of the project is to find a procedure for effective modelling and simulation of the dynamics of customer-proposed automobile exhaust system designs at an early stage in the product development process, to support the dialogue with customers and for overall optimisation. To be suited for this it is important that the theoretical system model is as computationally inexpensive as possible while yet being accurate enough for the characteristics it is supposed to describe. Further background information and a specification of the aim and scope of this thesis are described in chapter five. 10 2 Vibration Analysis 2.1 Introduction to Vibration Vibration is a common feature of everyday life though most people probably do not reflect much on it. Some vibrations are useful and desirable, as, for example, in music instruments, loudspeakers and machines sorting mixtures of stone and sand. Some vibrations are undesirable or even harmful, as for example in turbine blades, bridges and exhaust systems. Perhaps the most serious vibrations are those arising from earthquakes. The study of vibrations considers oscillatory motions of a dynamic system. A dynamic system can be defined as “A combination of matter which possesses mass and whose parts are capable of relative motion” [11]. This means that all structures, which have mass (inertia) and elasticity (stiffness), are capable of vibrating. The discipline which deals with this is often called structural dynamics. Vibrations arise as a result of dynamic loading, which sometimes gives a resonant response. This happens when the structure vibrates with such a frequency that stiffness and inertia forces are cancelled out. The frequencies at which this happens are often called resonance frequencies or (undamped) natural frequencies [12] and are associated with certain vibration forms that are called mode shapes. If an undamped system is excited with an external force, which frequency equals a resonance frequency of the system, the response approaches infinity. However, damping, which dissipates vibration energy, is always present in real structures and limits the resonance amplitude. Since the amount of damping is often low in typical structures, the vibration amplitude may, however, be very large, and the structure may collapse at loads considerably below the static collapse load. A well-known historical example of structural collapse caused by resonance is the Tacoma Narrows bridge, which collapsed due to wind-induced vibrations only a few months after that it had been opened for traffic [13]. 2.2 Linear and Non-linear Vibrations A system is defined as linear if it fulfils the principle of superposition; otherwise the system is non-linear [14, 15]. 11 An important property of a linear system is that when it is excited with a sinusoidal force the steady-state response becomes sinusoidal with the same frequency as the excitation frequency. The amplitude and the phase of the response are functions of the excitation frequency. The steady-state response of a non-linear system excited with a sinusoidal force generally includes additional frequency components, see figure 2. Energy is then transferred between frequencies [16, 17]. Figure 2. Possible output for a linear and non-linear system when excited with a sinusoidal force. A real system is always more or less non-linear, so a linear model is always an approximation. It is often, however, a good approximation. Examples of nonlinearity which may be present in mechanical systems are friction, progressive stiffness and gap [15, 18, 19]. To illustrate how non-linearity can affect the dynamics of a system, a simple example from [19] is considered. A mass, M = 945 Kg, is connected to ground by a linear spring, with stiffness of K = 24 MN/m, and a linear viscous damper with a damping coefficient of C = 7700 Ns/m. In addition, an elastic ideal-plastic friction element connects the mass to the ground, see figure 3. The behaviour of the friction element is also shown in figure 3. When the deformation gives a force magnitude in this element that is below Ff = 246 N it acts like a linear spring with stiffness of Kf = 24 MN/m. For further 12 deformation, the force remains at Ff. When the motion changes direction the element again acts as a linear spring until -Ff is reached. Figure 3. A system which includes a non-linear elastic ideal-plastic friction element. The system is excited with a sinusoidal force, and the steady-state response is calculated. The steady-state response amplitude at the excitation frequency over excitation force amplitude, A, is plotted against excitation frequency, see figure 4. This is done for three different excitation amplitude levels: 0.2·Ff, Ff and 5·Ff. As shown, the excitation level significantly affects the dynamics of the system. For a linear system the response would have been independent of the excitation level. In summary, analysis and understanding of non-linear systems are often much more difficult, time-consuming and complex than for linear systems. 13 −7 Displacement / Excitation force (m / N) 7 x 10 A = 5⋅Ff 6 5 4 3 A = 0.2⋅Ff 2 A = Ff 1 0 100 150 200 Frequency (Hz) 250 300 Figure 4. Response for different excitation levels. 2.3 Solution Methods Real structures have a continuous distribution of mass and stiffness and thus have an infinite number of so-called degrees of freedom (DOFs). DOFs can be described as the number of independent variables necessary to define the configuration of the studied system [11]. In structural dynamics, the range of problems for which a closed form solution can be found is very small. Instead, numerical methods which introduce an approximation by restricting the number of DOFs are often necessary. A popular method in structural dynamics is the finite element method (FEM). Since the middle of the twentieth century this method has developed into a powerful tool which may be applied to almost all practical problems of mathematical physics. A few examples of the many textbooks about the method are [20-22]. Essentially, the method transforms ordinary or partial differential equations into a finite system of algebraic equations, or partial differential equations into a finite system of ordinary differential equations. The solution of such a system of equations is a discrete approximation of the solution of the original differential equation(s). 14 Applying FEM for a structural dynamics problem the resulting system of ordinary differential equations typically is [ M ]{u&&} + [C ]{u&} + [ K ]{u} = {F (t )} (1) where [M] is a mass matrix, [C] is a viscous damping matrix, [K] is a stiffness matrix, {F} is a load vector, and { u }, { u& } and { u&& } are vectors of displacement, velocity and acceleration of the DOFs, respectively. Dots indicate time derivative, and t is time. When equation (1) has been solved for { u }, other quantities such as strains, stresses and reaction forces can be obtained from underlying equations. In contrast to the mass and stiffness matrices the damping matrix cannot in general be calculated theoretically. It is instead often introduced to approximate the overall energy dissipation, obtained from measurements or experience. Often so-called proportional damping is used [23]. There are different methods for solving equation (1). If the system is linear, modal superposition [12, 20, 24] is often used since it is then generally the most computationally effective method. Initially an eigenvalue problem then needs to be solved for the most important mode shapes and corresponding natural frequencies; these give an indication in themselves of the dynamic characteristics of the structure. Many eigensolvers exist, for example, the Lanczos method [25], and the subspace method [26]. The response of the system is then expressed in terms of a linear combination of the mode shapes. If proportional damping is assumed this makes it possible to produce a system of uncoupled differential equations by introducing so-called modal coordinates. When applying modal superposition, so-called modal damping is often used, which means that a damping ratio is specified for each uncoupled differential equation (mode). When using modal superposition the modes that do not significantly affect the results are often excluded. Which modes that are important to include are determined by, for example, the frequency content of the excitation [20]. The number of uncoupled equations is equal to the number of modes included in the analysis, so by excluding insignificant modes the number of equations can be reduced significantly. This reduction and decoupling are the reasons for the effectiveness of the method of modal superposition. For non-proportional damping special considerations are necessary to decouple the differential equations [23, 27]. 15 The uncoupled equations can be solved in different ways. For example, by using numerical methods based on difference approximations of the time derivatives, by using the Duhamel integral [12], or by using methods for digital filter design [28]. Alternatively, if the excitation is periodic and only the steady-state response is of interest, it is generally much more effective to solve the problem in the frequency domain. This makes it possible to find the steady-state response without first solving for the initial transient. If the system is non-linear, modal superposition is not valid and direct integration [12, 20, 24] is then often used. This means that equation (1) is integrated numerically in its original form by methods based on difference approximations of the time derivatives, and that a general damping matrix can be used without special considerations. Direct integration is usually more computationally expensive than the method of modal superposition. However, if only short duration events are to be studied, the computational cost of solving the eigenvalue problem may not be compensated for by the effectiveness resulting from modal superposition. Direct integration is therefore sometimes used for linear systems too. There are several methods based on difference approximations of the time derivatives, for example, the Newmark method [29], the central difference method [20] and the Runge-Kutta methods [30]. When the system is non-linear, it is often necessary to use an iterative procedure to solve the system of equations at each time step, for example, the Newton-Raphson iteration scheme or some of its variants [20]. This adds, of course, to the computational cost. There are a number of commercial software packages which support the above tasks, or parts of them, for example, ABAQUS [31], ANSYS [32], I-deas [33] and Nastran [34]. 16 3 Experimental Investigations 3.1 Introduction to Experimental Modal Analysis In theoretical modelling and simulation approximations are made at two main levels. Firstly, an idealised mathematical model is constructed by various assumptions and simplifications of reality. Secondly, approximate methods are mostly used to solve the idealised model. Furthermore, there are parameters such as damping, stiffness and friction properties that are hard or impossible to determine theoretically. Thus, it is necessary to have close interaction between modelling, simulation and experimental investigation for verification and understanding. Experimental modal analysis (EMA) [16, 23, 27, 35] is a well-established method in structural dynamics since the 1970s. It relies on many different knowledge domains such as transducer technology, signal processing, dynamics and numerical analysis. It is necessary to have enough knowledge in each of these fields to perform an EMA of good quality. An EMA consists of two main steps. In the first, experimental data are acquired and in the second step, often called modal parameter extraction, the modal parameters, that is, natural frequencies, ωr, mode shapes, {ψ}r, and damping ratios, ξr, where r represents a specific mode, are determined on the basis of the acquired data. These parameters define the dynamics of a linear system and can be used to correlate and in some cases update the theoretical model. 3.2 Data Acquisition When acquiring data during an EMA, the structure is typically excited with an impulse hammer or a shaker. Both approaches have their advantages and disadvantages. Shaker measurements are often more time-consuming but are also often more accurate [16, 23, 27, 36]. Transducers are essential components since it is very important to measure accurately both the input to the structure and the response. The excitation force is typically measured using a piezoelectric force transducer and the response is typically measured with piezoelectric accelerometers. The structure is often excited via a so-called stinger, which is a thin rod that is 17 used to reduce the influence of the attachment of the shaker on the structure. The stinger also acts as a mechanical fuse. The frequency content of the time signals are often calculated by using the fast Fourier transform (FFT) algorithm to obtain frequency response functions (FRFs). An FRF, Hpq, is defined as the displacement amplitude in point p, Xp, over the force amplitude in point q, Fq; see equation (2). H pq (ω ) = X p (ω ) (2) Fq (ω ) Note that both X and F are dependent on the frequency. Furthermore, they are complex in order to accommodate both amplitude and phase information. Alternative forms of FRFs, where the velocity or acceleration amplitude replaces the displacement amplitude in equation (2), are also frequently used. A typical FRF is shown in figure 5. Since the FRFs are complex, both the magnitude and the phase are plotted to give the complete information. There are a number of alternative ways of illustrating FRFs; see, for example, [27]. For non-linear structures, sometimes so-called first order FRFs, are used to describe the dynamic behaviour of the system. A first order FRF is defined as the spectral ratio of the response to the excitation force at the excitation frequency [15]. A sinusoidal excitation force (from a shaker) is generally preferred when performing measurements on non-linear structures so that the excitation force can be accurately controlled, and the non-linearity accurately quantified [15, 16, 27]. 18 Phase (rad) 2 0 −2 −3 10 −4 Magnitude (m / N) 10 −5 10 −6 10 −7 10 −8 10 20 40 60 80 100 120 Frequency (Hz) 140 160 180 200 Figure 5. A typical FRF (obtained from measurements on a modified Volvo S/V 70 exhaust system). Before the final data are acquired many aspects must be considered and checked, for example, regarding suspension of the test object, selection of excitation and response points and assessment of data quality [16, 27, 36]. If a shaker is used, a broad range of excitation signals exists and it must be decided which one is the most appropriate for the present case. Coordination with modelling and simulation is often useful during these considerations, for example, in the selection of excitation and measurement points. None of these tasks is trivial. They often take up most of the time spent on an EMA, and are of vital importance for its quality. 19 3.3 Modal Parameter Extraction An FRF, Hpq, for a linear system with viscous damping can be described by equation (3) or (4) [23]. ∗ ⎛ Apqr ⎞ Apqr ⎜ ⎟ + H pq (ω ) = ⎜ jω − λ∗r ⎟⎠ r =1 ⎝ jω − λ r N ∑ H pq (ω ) = b0 + b1 ( jω ) + b2 ( jω ) 2 + K + bn ( jω ) n a 0 + a1 ( jω ) + a 2 ( jω ) 2 + K + a 2N ( jω ) 2N (3) (4) where r represents a specific mode, n represents the number of zeros of the FRF, N represents the number of modes included in the analysis, ω represents frequency and * represents the complex conjugate. By taking the inverse fast Fourier transform (IFFT) of an FRF the corresponding impulse response function (IRF), h, is obtained. An IRF can be described by equation (5) [23]. hpq (t ) = ∑ (A N pqr r =1 ∗ ∗ ⋅ e λ r t + Apqr ⋅ e λr t ) (5) where t represents time. Equations (3)-(5) are central in modal parameter extraction. Several modal parameter extraction methods exist [16, 23, 27]. Some methods work in the frequency domain and relate to equation (3) or (4), and some work in the time domain and relate to equation (5). The common aim of the methods is to estimate the residues, A, and the poles, λ, or the coefficients a0, a1,…, a2N, b0, b1,…, bn so that the equations match the experimental data as closely as possible. A least square matching is often used. If equation (4) is used, the residues and poles can be determined from the coefficients of this equation. The modal parameters (ωr, {ψ}r, ξr) can easily be calculated when the residues and poles are known. It is important that the system studied is linear, or at least approximately linear, for the modal parameter extraction methods to produce meaningful results, since equations (3)–(5) are only valid for linear systems. 20 4 Correlation and Updating When correlating theoretical and experimental results the theoretical model is often found to be less accurate than desired. To develop a theoretical model that reflects reality in a satisfactory way, results from an EMA can be used as references to update the theoretical model [16, 27, 37]. Before the updating can be performed it is necessary to correlate theoretical and experimental results in a straightforward and objective way in order to discern possible differences. Several methods are available for this purpose [16, 27]. One such is simple tabulation of experimental and theoretical natural frequencies. A more informative method is to plot experimental and theoretical natural frequencies against each other. From this plot it is possible not only to see the amount of correlation but also to draw conclusions about the nature of discrepancies. A typical plot is shown in figure 6. The diagonal line represents perfect matching between experimental and theoretical natural frequencies. Theoretical natural frequency (Hz) 160 140 120 100 80 60 40 20 0 0 20 40 60 80 100 120 140 Experimental natural frequency (Hz) 160 Figure 6. A typical frequency plot (obtained from measurements on a modified Volvo S/V 70 exhaust system). 21 For the comparison to make sense it is very important that correlated mode pairs (CMPs) are used; in other words, the theoretical mode shapes must be compared with their experimental counterparts. A simple way to compare mode shapes is to animate them and compare them visually. Another common, and more objective, method is to determine the modal assurance criterion (MAC) matrix, which is a tool to numerically quantify the degree of conformance between two sets of mode shapes. A MAC-value of unity indicates perfect correlation, and a MAC-value of zero indicates no correlation. An example of a MAC-matrix is shown in figure 7. Figure 7. A typical MAC-matrix (obtained from measurements on a modified Volvo S/V 70 exhaust system). The MAC-matrix can also be used to check that sufficient measurement points are used during the EMA. This special form of the MAC-matrix is called AutoMAC. The experimental mode shapes are then correlated against themselves, which means that the diagonal values become unity. The criterion used to judge if the measurement points are sufficient is to check that the offdiagonal entities of the AutoMAC-matrix are small enough. So-called spatial 22 aliasing is then avoided; in other words, it is possible to distinguish between the different mode shapes by using the chosen measurement points. Another tool that may be used when correlating experimental and theoretical results is the coordinate MAC (CoMAC); see figure 8. This tool provides a numerical quantification of the correlation presented as a function of the individual DOFs. The lower the CoMAC-values the greater the differences between experimental and theoretical results for the corresponding DOFs. The regions of low CoMAC-values are not, however, necessarily where the theoretical model needs to be updated, since errors in one part of a model may have a large influence on other parts of the model. 1 0.9 0.8 0.7 CoMAC 0.6 0.5 0.4 0.3 0.2 0.1 0 10 20 30 40 50 DOF Figure 8. A typical CoMAC (obtained from measurements on a modified Volvo S/V 70 exhaust system). Yet another way of comparing experimental and theoretical results is to overlay FRFs. More sophisticated methods for comparison of FRFs, like, for example, the frequency response assurance criterion (FRAC), also exist. The aim of updating is to adjust the theoretical model to minimise discrepancy between experimental and theoretical results. It is important to include all the 23 parameters which are significant for the discrepancy. If not, the model may be adjusted incorrectly and will be a compromise of unknown quality [38]. The measure of discrepancy to be minimised as a function of the chosen model parameters is often called the objective function, and is based on the methods for correlating the experimental and theoretical results described above. An example of an objective function may be some weighted average of the discrepancies between experimental and theoretical natural frequencies over the included modes. When using this objective function it is, however, important to use constraints to assure that CMPs are compared. Finding a combination of parameters that minimises an objective function is a typical optimisation problem and many methods to solve such problems exist [16, 27]. 24 5 Research Problem 5.1 Background Original purposes of an automobile exhaust system were to lead exhaust gases from the engine to the rear end of the car to avoid toxic substances from entering the passenger cabin, and reduce the tremendous noise that would be present if the exhaust gases had left the engine directly. This is still relevant but a modern exhaust system has additional functionality, and is a rather complex product. A brief background is given below and a description of the components of a typical modern exhaust system is given in the next section. In many industrialised countries increasing demands on reduced emissions of toxic and environmentally harmful substances led to the introduction of the catalytic converter as a standard exhaust system component in the 1980s. These demands also led to successively higher combustion temperatures and more sophisticated combustion control systems. The exhaust system is today an important and integral part of combustion and emission control. In the 1980s it also became increasingly common with transverse engine orientation. This gives different engine movements relative to the exhaust system compared to longitudinal engine orientation and requires a highly flexible joint close to the engine to reduce transmission of these movements to the exhaust system. Due to its location in relation to the catalytic converter and the combustion control system sensors it is crucial for combustion control that this joint is gas-tight. To meet the above requirements a steel bellowstype joint was introduced. The increasing demands on comfort have led to the introduction of additional, larger and more sophisticated sound-silencing mufflers. The addition of more and more components generally increases flow resistance, which conflicts with the desire to minimise the reduction of engine power output. It also increases the weight of the car. Decreased engine efficiency and increased weight result in increased fuel consumption and emissions. In summary, demands on emission control, and low vibration and noise levels have made the design of automobile exhaust systems considerably more complex. Great changes have taken place over the last decades. The bellowstype joint in particular has caused car and component manufacturers severe dynamics problems because of lagging ability of predicting its characteristics 25 and mutual interaction with the rest of the exhaust system as a function of its design parameters. There is a great need for better understanding of the dynamic characteristics of the new components and their influence on the system dynamics [39]. 5.2 Exhaust System Components A typical exhaust system, belonging to a Volvo S/V 70, is shown in figure 9. It comprises primarily a manifold, a bellows-type flexible joint, a catalytic converter, mufflers and pipes. This particular exhaust system consists of a front assembly and a rear assembly connected with a sleeve joint. Figure 9. A Volvo S/V 70 exhaust system. 26 The manifold collects exhaust gases from the engine cylinders into a single pipe. It has smooth curves to give a flow that reduces the engine power output as little as possible and is as uniform as possible when the exhaust gases enter the catalytic converter. A uniform flow is beneficial for the cleaning efficiency. Cast-iron and fabricated (welded) manifolds are available on the market. Fabricated manifolds have become successively more popular. Some of the reasons for this are that they generally improve engine performance and have a lower weight and thermal inertia compared to cast-iron manifolds. The lower weight decreases fuel consumption and the lower thermal inertia decreases the time for the catalytic converter to reach full activity when a cold engine is started. On the other hand, cast-iron manifolds are usually less expensive. The flexible joint generally consists of a gas-tight flexible bellows, an inside liner and an outside braid. These parts are shown in figure 10. Figure 10. The bellows, the liner and the braid. The bellows is often multi-plied since this results in a lower stiffness than a single-ply bellows for a given strength. Low stiffness is beneficial for decoupling the engine from the exhaust system. The liner is used for reducing the temperature of the bellows and improving flow conditions. The braid is used for mechanical protection and to limit the extension of the joint. The 27 bellows, the liner and the braid are rigidly connected at the ends with endcaps. See figure 11 for a schematic assembly of the joint. More information on this joint can be found in [5] and [40], which also contains further references considering bellows joints. Figure 11. Schematic assembly of the flexible joint. It is known from practice and introductory investigations that this joint has a strong influence on the dynamics of the exhaust system [41]. The purpose of a catalytic converter is to convert harmful exhaust gases into less harmful gases before they are expelled to the environment [42]. Certain materials, known as catalytic materials, cause such chemical reactions without being affected themselves. Most modern cars have a so-called three-way catalytic converter, which refers to the three emission types it reduces, namely carbon monoxide, hydrocarbons and nitrogen oxides. The catalytic converter includes ceramic or metallic monoliths coated with the catalytic materials. The exhaust gases pass through the monoliths that are designed to expose as large an area as possible to obtain a good cleaning effect with a minimum of catalytic materials, which are very expensive. It is necessary to control the fuel-to-air ratio on a continuous basis for a catalytic converter to work properly. Mufflers are designed to reduce noise arising during combustion. There are basically two principles governing this process: reflection and absorption [42]. 28 In reflection mufflers, the exhaust gases are led through chambers of different lengths. The lengths are designed to cancel out the sound waves in the frequency interval where the engine makes the most noise. Absorption mufflers consist of one chamber. Perforated pipes, in which the exhaust gases pass, are led through the muffler, which is filled with a sound-silencing material. The exhaust system investigated in the present study has two absorption mufflers: a large intermediate muffler and a small rear muffler, see figure 9. The above parts are connected with pipes. The lengths and the cross-sections of the pipes influence the engine performance as well as the dynamic and acoustic behaviour of the exhaust system. In addition to being connected to the engine, the exhaust system is usually attached to the chassis of the car by rubber hangers. These provide flexibility and reduce vibration transmission to the passenger cabin. For the present exhaust system two hanger attachments are placed at the intermediate muffler and a third is placed just downstream the rear muffler, see figure 9. Modern exhaust systems also often include heat radiation shields at critical locations, see figure 9. 5.3 Aim and Scope The aim of this thesis is to provide a deeper understanding of the dynamic characteristics of automobile exhaust system components to form a basis for improved design and the development of computationally inexpensive theoretical component models in accordance with the overall project aim stated in chapter one. Modelling, simulation and experimental investigation of a typical exhaust system are performed to gain such an understanding and evaluate ideas of component modelling. The flexible joint is given special attention since it is the most complex component from a dynamics point of view, and since it is important for reducing transmission of engine movements to the exhaust system. Such movements can cause cabin noise and structural durability problems [3, 7, 43]. The braid of the flexible joint is not included in this study. By studying the flexible joint separately, before it is included in a complete system analysis, the knowledge obtained can be used also in a wider context. 29 This type of joint is, for example, also used in piping systems at power stations and in marine applications. The excitation from an automobile engine is usually in the frequency interval of 30-200 Hz [3, 7]. Excitations at lower frequencies may arise as a result of road irregularities, acceleration, breaking, and gear shifting. Thus, the frequency interval of interest is 0-200 Hz. The manifold is not included in this study. A related thesis is that by Wall [44], which focuses on how the dynamics of the assembled exhaust system is affected by the flexible joint. 30 6 Summary of Papers 6.1 Paper A This paper considers the dynamic characteristics of the exhaust system shown in figure 9. The flexible joint is not included in the study. A theoretical and an experimental modal analysis are performed. It is shown that the non-linearity of the exhaust system downstream of the flexible joint is negligible. Furthermore, it is shown that shell vibrations of the mufflers and the catalytic converter as well as ovalling of the pipes are negligible in the frequency interval of interest. This means that the pipes can be modelled by using beam elements, and the catalytic converter and the mufflers can be modelled by using lumped mass and mass moment of inertia elements. Additional short beam elements are used with these component models to account for flexibility at the connections as described in paper B. 6.2 Paper B In this paper experimental natural frequencies and mode shapes obtained from paper A are used as a reference to update the theoretical model. The sum of the differences between theoretically and experimentally obtained natural frequencies is chosen as the objective function to be minimised. Constraints are used on the correlation between theoretically and experimentally obtained mode shapes, considering the modal assurance criterion matrix, to ensure that correlated mode pairs are compared. The stiffness properties of the short beam elements used to model flexibility at the connections between the mufflers/catalytic converter and the pipes are used as the parameters to be adjusted during the updating. The theoretical model is built and solved in the finite element software ABAQUS. The updating is performed by using the sequential quadratic programming algorithm in the Optimization Toolbox in MATLAB. To obtain an automated updating procedure, the two software packages need to interact with each other. This is established by an in-house MATLAB script. The agreement between the updated theoretical and experimental results is very good, which verifies the usability of these component models. 31 6.3 Paper C This paper considers the flexible joint. The braid and the liner are not included in the study. A straightforward way of modelling the bellows is to use shell finite elements. Due to the convoluted geometry of the bellows this would, however, lead to a computationally expensive model. Instead it is modelled with beam elements using an equivalent pipe analogy. This has proved successful in previous research on single-ply bellows with a constant mean radius. The bellows studied in this paper is double-plied and has a variable mean radius. Adjustments are suggested by which the previous research can be extended to model this kind of bellows too. Experimental investigations of the axial and bending load cases are performed for verification. The correlation between theoretical and experimental results is very good. The experimental investigations reveal, however, that the bellows is slightly nonlinear, but this non-linearity is weak and may be neglected in the present application. However, a hypothetical qualitative explanation for the nonlinearity is provided. 6.4 Paper D In this paper the bellows of paper C is combined with an inside liner. The braid is not included. An approach for modelling the combined bellows and liner joint is developed and is experimentally verified for axial and bending load cases. The correlation between experimental and theoretical results is good. It is shown that the dynamic characteristics of the joint are strongly dependent on the relation between the excitation force level and the friction limit of the liner. Peak responses are, for example, significantly reduced when the excitation level approximately corresponds to the friction limit. This is due to friction-based damping. The liner thus makes the dynamics of the joint significantly non-linear and complex, and it is therefore important to consider these effects in joint design. 32 7 Conclusions and Future Research The present thesis addresses the dynamic characteristics of automobile exhaust system components. Modelling, simulation and experimental investigation of a typical exhaust system show that the major part of the system is essentially linear. Highly simplified finite element models of the components within this part are suggested. These models incorporate 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 very good, which verifies the usability of these component models. The flexible joint, usually located between the manifold and the catalytic converter, proves to be complex. It is non-linear if the bellows consists of multiple plies, or if it includes an inside liner. The first non-linearity is shown to be weak and may therefore be neglected. The non-linearity due to friction in the liner is, however, highly significant and gives the joint complex dynamic characteristics. This is important to know of and consider in exhaust system design and proves the necessity of including a model of the liner in the theoretical joint model when this type of liner is present in the real joint to be simulated. It is known from practice and introductory investigations that also the whole system sometimes shows complex dynamic behaviour. This can be understood from the non-linear characteristics of the flexible joint shown in this work. An approach to the modelling of the combined bellows and liner joint is suggested and experimentally verified. Future research should investigate the potential of making this model less computationally expensive. Future research on the joint should also investigate how the braid affects its dynamic characteristics. A major conclusion is that in coming studies of how engine vibrations affect the exhaust system it may be considered as a linear system if the flexible joint consists of a bellows. If the joint also includes a liner, the system may be considered as a linear sub-system that is excited via a non-linear joint. How to simulate such a system in a computationally effective way forms an 33 interesting question for future work. This may also be expanded into the more general question of how to model and simulate a general system that has linear relations between most of its degrees of freedom but that includes small but significant non-linear parts and is excited at some arbitrary point(s). To obtain more realistic excitation levels in the vibration analyses of exhaust systems, which is shown to be important since the flexible joint is non-linear, a theoretical model for simulation of the engine dynamics should be developed and included in future studies. Also the rubber hangers used to attach the exhaust system to the chassis should be investigated and included. Interesting questions for future work may also be how the high temperatures and the flow of the exhaust gases affect the dynamics of the exhaust system. 34 8 References 1. Katzenbach, A. and Haasis, S., ‘The international engineering process of the future’, AutoTechnology, vol. 1, April, 2001. 2. Winterkorn, M., ‘Challenges facing automotive product development’, AutoTechnology, vol. 1, February, 2001. 3. Belingardi, G. and Leonti, S., ‘Modal analysis in the design of an automotive exhaust pipe’, Int. J. of Vehicle Design, vol. 8, no. 4/5/6, 1987. 4. Carfagni, M. and Caprioli, D., ‘Virtual scooter prototype in the design of comfort’, Proceedings of the International Modal Analysis Conference – IMAC, USA, 2003. 5. Cunningham, J., Sampers, W. and van Schalkwijk, R., ‘Design of flexible tubes for automotive exhaust systems’, ABAQUS Users’ Conference, Netherlands, 2001. 6. Sellgren, U., ‘Simulation-driven design – motives, means, and opportunities’, Doctoral thesis, Department of Machine Design, Royal Institute of Technology, Sweden, 1999. 7. Verboven, P., Valgaeren, R., van Overmeire, M. and Guillaume, P., ‘Some comments on modal analysis applied to an automotive exhaust system’, Proceedings of the International Modal Analysis Conference – IMAC, USA, 1998. 8. Genta, G., ‘Vibrations of structures and machines’, (second edition), Springer-Verlag, USA, 1995. 9. Ullman, D. G., ‘The mechanical design process’, (second edition), McGraw-Hill, USA, 1997. 10. Hibbit, D., ‘Future trends and challenges in software development’, Proceeding of the NAFEMS World Congress 2003, USA, 2003. 11. Tse, F. S., Morse, I. E. and Hinkle, R. T., ‘Mechanical vibrations theory and applications’, (second edition), Allyn and Bacon, USA, 1978. 35 12. Craig, R. R., ‘Structural dynamics’, John Wiley & Sons, USA, 1981. 13. http://www.lib.washington.edu/specialcoll/tnb, October 2003. 14. Oppenheim, A. V. and Schafer, R. W., ‘Discrete-time signal processing’, (second edition), Prentice-Hall, USA, 1999. 15. Worden, K. and Tomlinson, G. R., ‘Nonlinearity in structural dynamics’, IOP Publishing, UK, 2001. 16. Maia, N. M. M. and Silva, J. M. M., (eds.), ‘Theoretical and experimental modal analysis’, Research Studies Press, UK, 1997. 17. Moon, F. C., ‘Chaotic and fractal dynamics: an introduction for applied scientists and engineers’, John Wiley & Sons, USA, 1992. 18. Ferreira, J. V., ‘Dynamic response analysis of structures with nonlinear components’, Doctoral thesis, Department of Mechanical Engineering, Imperial College of Science, Technology and Medicine, UK, 1998. 19. Guillen, J., ‘Studies of the dynamics of dry-friction-damped blade assemblies’, Doctoral thesis, Department of Mechanical Engineering, University of Michigan, USA, 1999. 20. Bathe, K. J., ‘Finite element procedures’, Prentice-Hall, USA, 1996. 21. Huebner, K. H., Thornton, E. A. and Byrom, T. G., ‘The finite element method for engineers’, (third edition), John Wiley & Sons, USA, 1995. 22. Zienkiewicz, O. C. and Taylor, R. L., ‘The finite element method’, (fifth edition), Butterworth-Heinemann, UK, 2000. 23. Allemang, R., ‘Vibrations: analytical and experimental modal analysis’, Report UC-SDRL-CN-20-263-662, University of Cincinnati, USA, 1994. 24. Hitchings, D., (ed.), ‘A finite element dynamics primer’, NAFEMS, Scotland, 1992. 25. Lanczos, C., ‘An iteration method for the solution of the eigenvalue problem of linear differential and integral operators’, Journal of Research of the National Bureau of Standards, vol. 45, 1950. 36 26. Bathe, K. J., ‘Solution methods of large generalized eigenvalue problems in structural engineering’, Report UC SESM 71-20, Civil Engineering Department, University of California, USA, 1971. 27. Ewins, D. J., ‘Modal testing: theory practise and application’, (second edition), Research Studies Press, UK, 2000. 28. Brandt, A. and Ahlin, K., ‘A digital filter method for forced response computation’, Proceedings of the International Modal Analysis Conference – IMAC, USA, 2003. 29. Newmark, N. M., ‘A method of computation for structural dynamics’, ASCE Journal of Engineering Mechanics Division, vol. 85, 1959. 30, Chapra, S. C. and Canale, R. P., ‘Numerical methods for engineers’, (third edition), McGraw-Hill, USA, 1998. 31. ABAQUS, HKS, Inc., http://www.abaqus.com. 32. ANSYS, ANSYS, Inc., http://www.ansys.com. 33. I-deas, EDS PLM Solutions, Inc., http://www.eds.com. 34. MSC.Nastran, MSC.Software, Inc., http://www.mscsoftware.com. 35. Avitabile, P., ‘Experimental modal analysis, a simple non-mathematical presentation’, Sound and Vibration, January, 2001. 36. Ahlin, K. and Brandt, A., ‘Experimental modal analysis in practice’, Saven EduTech AB, Sweden, 2001. 37. Deweer, J., van Langenhove, T. and Grinker, S., ‘Identification of the best modal parameters and strategies for FE model updating’, SAE Noise & Vibration Conference & Exposition, USA, 2001. 38. Avitabile, P., ‘Model updating – endless possibilities’, Sound and Vibration, September, 2000. 39. Ahltini, K., Research and Development Manager, Faurecia Exhaust Systems AB, Sweden, personal communication, October 2003. 37 40. Broman, G., Jönsson, A. and Hermann, M., ‘Determining dynamic characteristics of bellows by manipulated beam finite elements of commercial software’, Int. J. of Pressure Vessels and Piping, vol. 77, Issue 8, 2000. 41. Wall, J., Englund, T., Ahlin, K. and Broman, G., ‘Influence of a bellowstype flexible joint on exhaust system dynamics’. Submitted for publication. 42. Bosch, R., ‘Automotive handbook’, (fifth edition), Bentley Publishers, USA, 2001. 43. Ling, S. F., Pan, T. C., Lim, G. H. and Tseng, C. H., ‘Vibration isolation of exhaust pipe under vehicle chassis’, Int. J. of Vehicle Design, vol. 15, no. 1/2, 1994. 44. Wall, J., ‘Dynamics study of an automobile exhaust system’, Licentiate thesis, Department of Mechanical Engineering, Blekinge Institute of Technology, Sweden, 2003. 38 Paper A Significance of Non-linearity and Component-internal Vibrations in an Exhaust System 39 Paper A is published as: Englund, T., Wall, J., Ahlin, K. and Broman, G., ‘Significance of nonlinearity and component-internal vibrations in an exhaust system’, Proceedings of the 2nd WSEAS International Conference on Simulation, Modelling and Optimization, Greece, 2002. 40 Significance of Non-linearity and Component-internal Vibrations in an Exhaust System Thomas L Englund, Johan E Wall, Kjell A Ahlin, Göran I Broman Abstract To facilitate overall lay-out optimisation inexpensive dynamics simulation of automobile exhaust systems is desired. Identification of possible non-linearity as 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 of the engine and to reduce the transmission of vibrations to the rest of the exhaust system. This joint is significantly non-linear due to internal friction, which makes some kind of non-linear analysis necessary for the complete exhaust system. To investigate the significance of non-linearity and internal vibrations of other components a theoretical and experimental modal analysis of 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 also shown that shell vibrations of the catalyst and mufflers as well as ovalling of the pipes are negligible in the frequency interval of interest. The results implies, for further dynamics studies, that the complete system could be idealised 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 other components within the linear sub-system could also be modelled in a simplified way. Such simplified component models are suggested. The agreement between theoretical and experimental results is very good, which indicates the validity of the simplified modelling. Keywords: Correlation, Dynamics, Exhaust system, Linear sub-system, Modal analysis, Non-linear joint. 41 1. Introduction There is a trend to use more computer simulations in the design of products. This is mainly due to demands on shortened time to market, higher product performance and greater product complexity. To be useful in the design process it is important that the simulation models are kept as simple as possible while still being accurate enough for the characteristics they are supposed to describe. To reveal weaknesses in the simulation models experimental investigation is often necessary. The simulation models can then be updated to better correlate with experimental results. This study is a part of a co-operation project between the Department of Mechanical Engineering at the Blekinge Institute of Technology, Karlskrona, Sweden and Faurecia Exhaust Systems AB, Torsås, Sweden. The overall aim of the project is to find a procedure for effectively modelling and simulating the dynamics of customer-proposed exhaust system lay-outs at an early stage in the product development process, to support the dialogue with the costumer and for overall lay-out optimisation. Demands on, for example, higher combustion temperatures, reduced emissions, reduced weight, increased riding comfort and improved structural durability have made the design of exhaust systems more delicate over the years and more systematic methods have become necessary. Examples of linear studies of exhaust systems are the works by Belingardi and Leonti [1] and Ling et al. [2], who focus on simulation models, and the work by Verboven et al. [3] who focus on experimental analysis. An introductory study of the present exhaust system is that of Myrén and Olsson [4]. Most modern cars have the engine mounted in the transverse direction. A flexible joint between the manifold and the rest of the exhaust system is therefore included to allow for the motion of the engine and to reduce the transmission of vibrations to the rest of the exhaust system. Recent suggestions of a stiffer attachment of the exhaust system to the chassis, as discussed by for example DeGaspari [5], with the purpose of reducing weight, makes this joint even more important. The commonly used type of joint is significantly non-linear due to internal friction, which makes some kind of non-linear dynamics analysis necessary for the complete system. Thus a more comprehensive approach seems necessary. This paper represents an early step and focuses on the part of the exhaust system that is downstream the flexible joint. The purpose is to verify the assumption that this part is 42 essentially linear so that, in the further studies, the complete system could be idealised into a linear sub-system that is excited via the non-linear flexible joint. The purpose is also to find a computationally effective and experimentally 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 the system 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 a sleeve joint. Both are welded structures of stainless steel. The front part is attached to the manifold by a connection flange. The engine and manifold are not 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 information on this type of joint is given by, for example, Cunningham et al. [6] and Broman et al. [7]. The front assembly, see figure 2, consists of this joint, the catalyst and pipes. 43 Figure 2. Front assembly. The catalyst includes a honeycomb ceramic and the outside shell structure is rather complicated. Thus, detailed modelling would be computationally expensive. The rear assembly, see figure 3, consists of pipes, an intermediate muffler and a rear muffler. Figure 3. Rear assembly. Perforated pipes pass through the mufflers. The mufflers are filled with sound silencing material. Their outside shell structure is also rather complicated. Besides the connection to the manifold the exhaust system is attached to the chassis 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. The frequency interval of interest for the modal analysis is obtained by considering that a four-stroke engine with four cylinders gives its main excitation at a frequency of twice the rotational frequency. Usually the 44 rotational speed is below 6000 rpm. Excitation at low frequencies may arise due 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 present analysis. It is assured that the flexible joint does not have any internal deformations. 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 outside shell structure of the mufflers and the catalyst are modelled with linear quadrilateral shell elements using the CAD-geometry. The mass of the internal material is distributed evenly to the shell elements. The pipes are modelled using parabolic beam elements. The flexible joint is modelled by stiff beam elements with a fictive density to reflect its mass and mass moment of inertia. Lumped mass elements are used to model the connection flange, attachments for the hangers, nipples and the heat shield. Connection between the beam elements representing the pipes and the shell elements representing the 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 of nodes are approximately 2200. This initial model is used as a basis for determining suitable transducer locations for the experimental modal analysis of 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 experimental modal analysis (EMA) the exhaust system is suspended, at the hanger attachments and at the connection flange, using soft adjustable rubber bands as shown in figure 4. 45 Figure 4. The measurement set-up. From the initial FE-analysis it is known that the motion is mainly in the plane (y-z) perpendicular to the length-direction (x) of the system. To be able to excite the system in both the y- and z- directions in one set-up the shaker is inclined. After consulting the FE-model several possible excitation points are tested. The final excitation point is taken just upstream the intermediate muffler, as seen in figure 4. The shaker is connected to the exhaust system via a stinger and a force transducer. A burst random signal is used to excite the exhaust system to avoid possible leakage problems. An HP VXI measuring system with 16 available channels is used. Five triaxial accelerometers could therefore be used in each measuring round. The accelerometers are attached on top of the exhaust system. To minimise the influence of the extra mass loading the accelerometers are evenly spread over the exhaust system in each measuring round. 46 Again considering the results from the initial FE-model it is concluded that 25 evenly distributed measuring points should be sufficient to represent the mode shapes in the frequency interval of interest. Using the AutoMAC, see figure 5, the chosen measurement points are checked to avoid spatial aliasing. The small off-diagonal terms in the AutoMAC indicate that the chosen measurement points sufficiently well describe the modes in the frequency interval of interest. 1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 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 response function (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 may arise that the static preload could have an undesired influence when the system hang horizontally. To ensure that this is not the case the exhaust system is also hanged vertically and some arbitrary FRFs are measured. The 47 difference in natural frequencies is negligible between the two set-ups and it is therefore concluded that the initial set-up is satisfactory. I-DEAS Test [9] is used to acquire the FRFs. The FRFs are exported to MATLAB [10] where they are analysed using the experimental structural dynamics toolbox developed by Saven Edutech AB [11]. The poles are extracted in the time domain using a global least square complex exponential method. The residues are found using a least squares frequency domain method. To improve the quality of the extracted modal parameters only data in the y- and z-directions are used. To get as good a fit as possible the curve fitting procedure is conducted in two steps; first in the interval 5-90 Hz and then in the interval 90-150 Hz. Above 150 Hz no significant modes are found, as seen in a typical FRF shown in figure 6. Velocity / Reaction force (m / Ns) Frequency response function −2 10 −3 10 −4 10 −5 10 50 100 150 Frequency (Hz) 200 250 Figure 6. Typical FRF. 5. Simplification and Correlation Determining the natural frequencies of the mufflers and the catalyst experimentally it is seen that no significant local modes are present in the frequency interval of interest. This was also found by Verboven et al. [3]. 48 Therefore the modelling of the mufflers and the catalyst, which are responsible for most of the model size in the initial FE-model, can be significantly simplified. The mufflers and the catalyst are modelled by lumped mass and mass moment of inertia elements. The properties of these elements are obtained from the original FE-model. If more suitable in a general case these properties can also be obtained directly from the CAD-model or experimentally. 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. No significant 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 and mufflers/catalyst, short beam elements with individual properties are used. These elements are located at the true connection locations, that is, with reference to the real system. Thus, they are placed between the rigid elements that are connected to the lumped mass and mass moment of inertia elements and the beam elements representing the pipes. These individual beam properties are updated so that the difference between theoretical and experimental results is minimised. The updating procedure uses 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 of over 90 % compared to the initial FE-model is obtained. Simplifications of this type are important if direct time integration becomes necessary for the non-linear dynamics analysis of the complete system. It is also important when a large number of simulations are necessary for overall exhaust system lay-out optimisation. The FE modes are calculated 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 converted into real-valued modes. 49 6. Results and Correlation To correlate the mode shapes from the updated FE-model and the experimental 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, which indicates 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 is shown in figure 8. The 45-degree line represents perfect matching. The crosses indicate the frequency match for each correlated mode pair. 50 Theoretical natural frequency (Hz) 150 100 50 0 0 50 100 Experimental natural frequency (Hz) 150 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 is normal for this type of modelling and measurement process [14]. The results are summarised in table 1. 51 Table 1. Results. Mode 1 2 3 4 5 6 7 8 9 10 a Experimental Frequency Damping (Hz) (%) 10.9 0.32 12.9 0.52 34.9 0.49 36.4 0.30 59.1 0.69 67.1 1.5 80.8 0.79 101 1.6 127 0.91 139 2.3 Theoretical Frequency (Hz) 10.9 12.8 35.8 36.9 57.3 69.7 83.7 101 126 135 Correlationa (%) MAC 0.24 -1.0 2.6 1.3 -3.0 3.9 3.6 0.30 -0.60 -2.9 0.95 0.93 0.88 0.85 0.93 0.85 0.91 0.96 0.64 0.60 The correlations are calculated before rounding off. The damping values are given as the fraction of critical damping and the correlation value is the relative difference between experimental and theoretical natural frequencies. Above 150 Hz no significant modes are found. 7. Conclusions A dynamics study of an exhaust system that consists of a non-linear flexible joint and a main part including pipes, mufflers and a catalyst is presented. The good agreement between the theoretical and experimental modal analysis, as well as the satisfactory results of the linearity check, implies, for further dynamics studies, that the complete system could be idealised into a linear sub-system that is excited via the non-linear flexible joint. It is also shown that shell vibrations of the catalyst and mufflers as well as ovalling of the pipes are negligible in the frequency interval of interest. This implies that the pipes could be modelled by beam elements and that the other components within the linear sub-system could be modelled by lumped mass and mass moment of inertia elements. The mass and inertia properties can be obtained either from a CAD-model or experimentally. Short beam elements with individual properties can be used successfully to model the flexibility of 52 the connections between the mufflers/catalyst and the pipes. Automated updating of these individual properties is recommended since doing it manually is time consuming and difficult. The agreement between results from the updated FE-model and the experimental investigations is very good. This implies that such simplified modelling is a valid approach and it may turn out important in coming nonlinear analyses, since such analyses are often computationally expensive. 8. Acknowledgements The support from Faurecia Exhaust Systems AB is gratefully acknowledged, especially from Håkan Svensson. The authors also gratefully acknowledge the financial support from the Swedish Foundation for Knowledge and Competence Development. 9. References 1. Belingardi, G. and Leonti, S., ‘Modal analysis in the design of an automotive exhaust pipe’, Int. J. of Vehicle Design, vol. 8, no. 4/5/6, 1987. 2. Ling, S.-F., Pan, T.-C., Lim, G.-H. and Tseng, C.-H., ‘Vibration isolation of exhaust pipe under vehicle chassis’, Int. J. of Vehicle Design, vol. 15, no. 1/2, 1994. 3. Verboven, P., Valgaeren, R., van Overmeire, M. and Guillaume, P., ‘Some comments on modal analysis applied to an automotive exhaust system’, Proceedings of the international Modal Analysis Conference – IMAC, Santa Barbara, USA, 1998. 4. Myrén, M. and Olsson, J., ‘Modal analysis of exhaust system’, Master Thesis, Department of Mechanical Engineering, University of Karlskrona/Ronneby, Karlskrona, Sweden, 1999. 5. DeGaspari, J., ‘Lightweight exhaust’, Mechanical Engineering, May 2000. 53 6. Cunningham, J., Sampers, W. and van Schalkwijk, R., ‘Design of flexible tubes for automotive exhaust systems’, ABAQUS Users’ Conference, 2001. 7. Broman, G., Jönsson, A. and Hermann, M., ‘Determining dynamic characteristics of bellows by manipulated beam finite elements of commercial software’, Int. J. of Pressure Vessels and Piping, vol. 77, Issue 8, 2000. 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., ‘Automated updating of simplified component models for exhaust system dynamics simulations’, 2nd WSEAS International Conference on Simulation, Modelling and Optimization, Skiathos Island, Greece, 2002. 14. Ewins, D. J., ‘Model validation: Correlation for updating’, Sādhanā, vol 25, part 3, 2000. 54 Paper B Automated Updating of Simplified Component Models for Exhaust System Dynamics Simulations 55 Paper B is published as: Englund, T., Wall, J., Ahlin, K. and Broman, G., ‘Automated updating of simplified component models for exhaust system dynamics simulations’, Proceedings of the 2nd WSEAS International Conference on Simulation, Modelling and Optimization, Greece, 2002. 56 Automated Updating of Simplified Component Models for Exhaust System Dynamics Simulations Thomas L Englund, Johan E Wall, Kjell A Ahlin, Göran I Broman Abstract To facilitate overall lay-out optimisation simplified component models for dynamics simulations of automobile exhaust systems are desired. Such optimisation could otherwise be computationally expensive, especially when non-linear analyses are necessary. Suggestions of simplified models of the mufflers and the catalyst are given. To account for the flexibility at the connections between those components and the pipes short beam elements with individual properties are introduced at these locations. An automated updating procedure is developed to determine the properties of these beam elements. Results from an experimental modal analysis are used as the reference. The theoretical model of the exhaust system is built in the finite element software ABAQUS. The updating procedure uses the sequential quadratic programming algorithm included in the Optimization Toolbox of the software MATLAB to minimise the sum of the differences between experimentally and theoretically obtained natural frequencies. Constraints are used on the correlation between the experimentally and theoretically obtained mode shapes by considering the MAC-matrix. Communication between the two software packages is established by an in-house MATLAB script. The correlation between results from the updated theoretical model and the experimental results is very good, which indicates that the updating procedure works well. Keywords: Correlation, Optimisation, Updating. Dynamic, Exhaust 57 system, Modal analysis, 1. Introduction Demands on shortened time to market, higher product performance and greater product complexity in combination with the fast development of computers 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 to study certain product characteristics and for overall introductory systems optimisation. The models and simulations should reflect the interesting characteristics of the real system accurately enough to support relevant design decisions. To gain confidence of this some kind of experimental verification is often 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 theoretical models. Procedures for more automated updating have therefore attained interest within the analysis community. See for example the works by van Langenhove et al. [1] and Deweer et al. [2] regarding updating of dynamic systems. Avitabile [3] discusses different updating criteria and points out the importance of the choice of parameters in the updating procedure. Chen and Ewins [4] describe the effect of discretisation errors when updating finite element models. This study is a part of a co-operation project between the Department of Mechanical Engineering at the Blekinge Institute of Technology, Karlskrona, Sweden and Faurecia Exhaust Systems AB, Torsås, Sweden. The overall aim of the project is to find a procedure for effectively modelling and simulating the dynamics of customer-proposed exhaust system lay-outs at an early stage in the product development process, to support the dialogue with the costumer and for overall lay-out optimisation. An accompanying paper is that of Englund et al. [5], which focuses on simplified and experimentally verified modelling of a typical automobile exhaust system. The updating of the simplified models of the components within that system is performed according to the procedure described in the present paper. The MATLAB Optimization 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 by an in-house MATLAB script to obtain automated updating. 58 2. Exhaust System Design The studied automobile exhaust system is shown in figure 1. The mass of the system 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 a sleeve joint. Both are welded structures of stainless steel. The front part is attached to the manifold by a connection flange. The engine and manifold are not 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 type of 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 rear assembly consists of pipes, an intermediate muffler and a rear muffler. Perforated pipes pass through the mufflers. The mufflers are filled with sound silencing material. Besides the connection to the manifold the exhaust system is attached to the chassis 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 pipes are modelled using quadratic beam elements and the mufflers and the catalyst are modelled using lumped mass and mass moment of inertia elements. Such 59 simplified modelling is valid in the frequency interval of interest [5]. These elements are connected to the beam elements representing the pipes by rigid elements. The properties of the lumped mass and mass moment of inertia elements are obtained from a finite element (FE) model where these parts are modelled with shell finite elements [5]. If more suitable in a general case these properties can also be obtained directly from the CAD-model or experimentally. To simulate the flexibility of the connections between the pipes and mufflers/catalyst, short beam elements with individual properties are used. These elements are located at the true connection locations, that is, with reference to the real system. Thus, they are placed between the rigid elements that are connected to the lumped mass and mass moment of inertia elements and the beam elements representing the pipes. Lumped mass elements are used to model the connection flange attached to the flexible joint, attachments for the hangers and the heat shield. Free-free boundary conditions are used and the natural frequencies and mode shapes are solved for by the Lanczos method. More information about the theoretical model 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 are calculated 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 converted into real-valued modes. The experimental modal analysis (EMA) is performed using free-free boundary conditions. To exclude the influence of the non-linearity of the flexible joint it is assured that it does not have any internal deformations. Thus it 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 significant modes occur above 150 Hz for this particular exhaust system [5]. 60 4. Updating The experimentally obtained natural frequencies and mode shapes are used to update the theoretical model. If, in a general case, a full physical prototype does not exist results from a detailed finite element model can be used as the reference. The selection of parameters to be included in the updating procedure is important. This is true whether the updating is based on frequency differences, mode shape differences or frequency responses [3]. Except for the connections between the mufflers/catalyst and the pipes the theoretical model of 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 the following three stiffness related properties; the two area moments of inertias and the polar area moment of inertia of the short beam elements representing the connections. All connections have individual properties. Altogether this gives 18 independent parameters to consider when updating the theoretical model. Figure 2. Front assembly. 61 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 natural frequencies are calculated. It can then roughly be concluded which parameters that are important to consider when updating the theoretical model of the exhaust system. Ten parameters are found to be significantly more important than the others. Using this approach the possibility to detect interaction between parameters is lost. Considering also these effects can be very time consuming. The procedure used in this work is a compromise between accuracy and time consumption. The aim is not necessarily to find the global optimum, but rather a solution that is good enough. Since many parameters are still involved an automated updating procedure using the Optimization Toolbox in MATLAB is developed. A constrained optimisation is performed using a sequential quadratic programming (SQP) algorithm [6]. The optimisation algorithm is supplied with start-values, bounds, constraints and optimisation criterion. The optimisation criterion chosen, which is to be minimised, is the sum of the differences in natural frequency within each correlated mode pair. Constraints are used on the correlation between theoretical and experimental mode shapes using the diagonal values of the MAC-matrix. The modal assurance criterion (MAC) is a technique to quantify the correlation between two sets of mode shapes. This constraint is important since it forces the algorithm to use correlated mode pairs when calculating the optimisation criterion. Good agreement is sought for both natural frequencies and mode shapes. Using constraints and bounds limits the search space, which usually reduces the number of function evaluations, that is, the problem converges faster [6]. Since natural frequencies and mode shapes must be solved for many times during the updating procedure ABAQUS and MATLAB interact with each 62 other. An in-house MATLAB script, taking advantage of MATLAB’s ability of reading and writing ASCII-files, is used to transfer data between the two different software packages. The optimisation procedure is schematically shown in figure 4. Yes Optimum? Terminate Solution No The MATLAB Toolbox calculates a new parameter combination. ABAQUS solves for natural frequencies and mode shapes. The MATLAB Toolbox calculates optimisation criterion and constraints. Figure 4. Automated updating procedure. Setting appropriate tolerances for the search algorithm in the Optimization Toolbox 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 of function evaluations without finding a much better solution. On the other hand setting them to loosely the search algorithm might not find the correct optimum. To be able to set the tolerances for the optimisation algorithm in a straightforward way all the ten parameters are scaled to be between zero and unity. 63 An important aspect to consider is that SQP is a gradient-based optimisation routine. This means that it only finds local optima, that is, different optima can 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 some kind of derivate-free optimisation method. In this work the start-values for the short beam element properties are taken from the beam elements representing the pipes at the connections in the theoretical model. If the start-values are good, that is, are near an optimum, the search algorithm finds this optimum faster. 5. Results and Discussion A comparison between the results from an initial theoretical model, that is, a model without the short beam elements accounting for the flexibility at the connections between the mufflers/catalyst and the pipes, and the experimental results shows that this model is far too stiff. Some of the theoretical natural frequencies are more than fifty per cent higher than the corresponding natural frequencies obtained experimentally. In a first step to achieve a theoretical model that correlates better with the experimental results Young’s modulus, of the fictive material of the short beam elements representing the connections, is updated. The same value of this parameter is used for all connections. The comparison between results from this roughly updated model, and the experimental results are summarised 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 mode switch between these modes, see figure 5. Furthermore, it can be seen in the figure that some of the off-diagonal values are high. This also indicates bad correlation. 64 Table 1. Results after the first update. Mode 1 2 3 4 5 6 7 8 9 10 a Experimental Frequency (Hz) 10.9 12.9 34.9 36.4 59.1 67.1 80.8 101 127 139 Theoretical Frequency (Hz) 10.6 13.1 35.9 42.1 50.0 74.6 82.9 86.2 116.5 141.2 Correlationa (%) MAC -2.4 1.2 2.8 16 -15 11 2.6 -14 -7.9 1.5 0.95 0.93 0.58 0.67 0.84 The correlations are calculated before rounding off. Figure 5. The MAC-matrix after the first update. 65 0.91 0.72 0.70 In a final step the ten independent parameters are included in the automated updating procedure. The correlation between modes of this theoretical model and the experimental modes are calculated using the MAC-matrix, see table 2 and figure 6. The correlation is very good. All diagonal MAC values are above 0.85 except for mode nine and ten. Furthermore all the off-diagonal terms in the MACmatrix are below 0.2. As also seen all differences in natural frequencies are below four per cent. Table 2. Results after the final update. Mode 1 2 3 4 5 6 7 8 9 10 a Experimental Frequency (Hz) 10.9 12.9 34.9 36.4 59.1 67.1 80.8 101 127 139 Theoretical Frequency (Hz) 10.9 12.8 35.8 36.9 57.3 69.7 83.7 101 126 135 The correlations are calculated before rounding off. 66 Correlationa (%) MAC 0.24 -1.0 2.6 1.3 -3.0 3.9 3.6 0.30 -0.60 -2.9 0.95 0.93 0.88 0.85 0.93 0.85 0.91 0.96 0.64 0.60 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 the reference. If, in a general case, a full physical prototype does not exist results from a detailed finite element model can be used as the reference. The simplified component models can be used for, otherwise computationally expensive, overall lay-out optimisation and they can also be re-used when the same or similar components are to be included in other exhaust system assemblies. An automated updating procedure is developed. The sequential quadratic programming algorithm in MATLAB’s Optimization Toolbox is used to minimise the difference between theoretical and experimental natural frequencies. Constraints are used on the correlation between the theoretical and experimental mode shapes using the MAC-matrix. The natural 67 frequencies and mode shapes are solved for by ABAQUS. Communication between the two software packages is established by an in-house MATLAB script. The very good correlation between the updated theoretical model and the experimental results shows that the updating procedure works well. 7. Acknowledgements The support from Faurecia Exhaust Systems AB is gratefully acknowledged, especially from Håkan Svensson. The authors also gratefully acknowledge the financial support from the Swedish Foundation for Knowledge and Competence Development. 8. References 1. van Langenhove, T., Fredö, C. and Brunner, O., ‘FE model correlation & code shape updating using qualification test data. A case study on the olympus satellite’, Proceedings of NAFEMS World Congress, Como, Italy, 2001. 2. Deweer, J., van Langenhove, T. and Grinker, S., ‘Identification of the best modal parameters and strategies for FE model updating’, SAE Noise & Vibration Conference & Exposition, Grand Traverse, USA, 2001. 3. Avitabile, P., ‘Model updating – endless possibilities’, Sound and Vibration, September, 2000. . 4. Chen, G. and Ewins, D. J., ‘Perspective on modal updating performance’, Proceedings of the International Modal Analysis Conference – IMAC, San Antonio, USA, 2000. 5. Englund, T., Wall, J., Ahlin, K. and Broman, G., ‘Significance of nonlinearity and component-internal vibrations in an exhaust system’, WSEAS International Conference on Simulation, Modelling and Optimization, Skiathos Island, Greece, 2002. 6. MATLAB, The MathWorks, Inc., http://www.mathworks.com. 68 7. ABAQUS, HKS, http://www.abaqus.com. 8. Broman, G., Jönsson, A. and Hermann, M., ‘Determining dynamic characteristics of bellows by manipulated beam finite elements of commercial software’, Int. J. of Pressure Vessels and Piping, vol. 77, Issue 8, 2000. 9. Cunningham, J., Sampers, W. and van Schalkwijk, R., ‘Design of flexible tubes for automotive exhaust systems’, ABAQUS Users’ Conference, 2001. 69 70 Paper C Modelling of Multi-ply Bellows Flexible Joints of Variable Mean Radius 71 Paper C is published as: Wall, J., Englund, T., Ahlin, K. and Broman, G., ‘Modelling of multi-ply bellows flexible joints of variable mean radius’, Proceedings of the NAFEMS World Congress 2003, USA, 2003. 72 Modelling of Multi-ply Bellows Flexible Joints of Variable Mean Radius Johan E Wall, Thomas L Englund, Kjell A Ahlin, Göran I Broman Abstract Bellows flexible joints are included in automobile exhaust systems to allow for engine movements and thermal expansion and to reduce vibration transmission. Generally the joint consists of a flexible bellows, an inside liner and an outside braid. In this work the bellows is considered. A straightforward way to model the bellows is to use shell finite elements. Due to the convoluted geometry of the bellows that procedure requires however a high number 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 the commercial software I-DEAS and was verified against experimental results available in the literature for single-ply bellows of constant mean radius. This paper suggests adjustments by which this procedure can be extended to model also multi-ply bellows of variable mean radius. Experimental investigations of a double-ply bellows having decreasing mean radius towards its ends are included for verification. The agreement between theoretical and experimental results 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). The experimental investigation reveals an intriguing resonance frequency shift at small excitation force levels. Although considered to be of minor significance for the present application of the bellows, a hypothetic qualitative explanation to the observed phenomenon is given. Keywords: Beam model, Bellows, Dynamic, Experimental investigation, Flexible joint, Frequency shift, Multi-ply, Variable mean radius. 73 1. Notation A Area [m2] E Modulus of elasticity [Pa] G Shear modulus [Pa] h Height [m] I Area moment of inertia [m4] K Polar area moment of inertia [m4] L Length [m] R Radius [m] r Radius [m] t Thickness [m] ν Poisson’s ratio ρ Density [Kg/m3] Indices conv Convolution m Middle p Pipe 74 2. Introduction Bellows flexible joints are important components in automobile exhaust systems. A flexible connection between the manifold and the rest of the exhaust system is necessary to allow for deflections induced by engine movements and due to thermal expansion and to reduce vibration transmission. Recent suggestions of a stiffer attachment of the exhaust system to the chassis, as discussed by for example DeGaspari [1], with the purpose of reducing weight, makes this component even more important. Proper dimensioning of the flexible joint requires understanding of its dynamic characteristics and interaction with the rest of the exhaust system. This is studied in a co-operation project between the Department of Mechanical Engineering at Blekinge Institute of Technology, Karlskrona, Sweden and Faurecia Exhaust Systems AB, Torsås, Sweden. The overall aim of the project is to find a procedure for effectively modelling and simulating the dynamics of customer-proposed exhaust system lay-outs at an early stage in the product development process, to support the dialogue with the costumers and for overall lay-out optimisation. To be suited for that the simulation procedure cannot be too computationally expensive. This is especially important when the dynamics is non-linear, which will be considered in later studies. The models of the components of the exhaust system must therefore be as simple as possible while still giving a proper description of the dynamics of the system. The bellows flexible joint is the component within the exhaust system that is most difficult to describe inexpensively. Broman et al. [2] presented a method for determining the dynamic characteristics 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 the most straightforward way of modelling the bellows, the model size is reduced considerably by using this beam element model. Axial, bending and torsion degrees of freedom can be studied simultaneously and the modelling technique facilitates the interaction between the bellows and the rest of the exhaust 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 extended to model also a multi-ply bellows of variable mean radius. Experimental investigations are performed for verification. It is also tested if the procedure 75 can easily be implemented in other commercial software (in this case ABAQUS [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 gastight bellows combined with an inside liner and an outside braid. The liner was originally introduced for reduction of bellows temperature and for improved flow conditions. It also further reduces vibrations. The braid is used for mechanical protection and to limit the extension of the joint. The parts are connected with end-caps. The complete joint is significantly non-linear. In this paper the bellows is considered. More information on this type of joint is given by, for example, Cunningham et al. [5]. Gas-tight bellows Braid Liner End-cap Figure 1. Basic flexible joint design. The bellows of this paper is double-plied and it has smaller mean radius closer to the ends. For a given strength a multi-ply bellows has lower stiffness than a single-ply bellows. Low stiffness is desired to decouple the engine from the rest of the exhaust system. The frequency interval of interest for the analysis is obtained by considering that a four-stroke engine with four cylinders gives its main excitation at a frequency of twice the rotational frequency. Usually the rotational speed is 76 below 6000 rpm. Excitation at low frequencies may arise due to road irregularities, as discussed by, for example, Belangardi and Leonti [6] and Verboven 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 is adopted and extended in this paper. While the current bellows has a variable mean radius, with smaller radii closer to the ends, different equivalent pipes are used for different parts of the bellows. These equivalent pipes have different equivalent density, ρp, shear modulus, Gp, modulus of elasticity, Ep, area, Ap, area moment of inertia, Ip, and polar area moment of inertia, Kp. Three different equivalent pipes, with assumed constant mean radii, are used, see figure 2. 1 2 3 Figure 2. Three sections to be represented by different equivalent pipe models. Other differences are that the end caps of the bellows are included in the analysis and that the bellows has two plies instead of one ply. Furthermore the convolution profile is slightly different from the U-shaped profile considered in [2]. The bellows is made of stainless steel. The material properties are E = 193 GPa, ρ = 8000 kg/m3, and ν = 0.29. 77 The characteristic dimensions of the convolutions can be seen in figure 3. The convolution dimensions of the three different pipe sections are presented in table 1. Lconv r h r Rm Figure 3. Convolution dimensions. Table 1. Convolution dimensions (mm). Dimension Rm Lconv r h t Section 2 64.9 7.60 2.40 10.4 0.193 1 61.2 7.60 2.20 6.70 0.193 3 67.9 7.60 2.65 13.4 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 in a too heavy and stiff bellows. A different approach is therefore used. The mass of the bellows is measured. As the density and the remaining dimensions 78 are known, the material thickness, t, can be calculated. It is assumed that the thickness is constant all over the bellows and the same for each ply. The equivalent parameters, Ep, Gp, ρp, Ap, Ip, and Kp, are calculated in the way described in [2] for the three different equivalent pipes, see table 2. The axial and torsion stiffness are calculated using the linear static solver in I-DEAS considering one ply. These stiffness values are then multiplied by a factor two, to account for the two plies. This means that the plies are assumed to work independently of each other, in agreement with the EJMA-standard [8]. Table 2. Properties of the equivalent pipes. Property Ep [MPa] Gp [GPa] ρp [kg/m3] Ap [m2] Ip [m4] Kp [m4] Section 2 9.64 8.21 1.02⋅104 2.04⋅10-4 1.07⋅10-7 2.15⋅10-7 1 35.6 12.4 7120 1.92⋅10-4 9.00⋅10-8 1.80⋅10-7 3 7.03 6.28 1.29⋅104 2.13⋅10-4 1.23⋅10-7 2.46⋅10-7 As this is an early step in the analysis of the complete flexible joint, in which also the non-linear characteristics of the joint will be considered, an analysis procedure that with small adjustments can handle also this is sought. The analysis is therefore performed in ABAQUS instead of I-DEAS, because direct time integration might be needed when considering also the nonlinearity of the flexible joint. Both I-DEAS and ABAQUS have finite element formulations based on Timoshenko beam theory, which includes the influence of shear deformation and in dynamics the influence of rotary inertia. The latter must be considered for the pipe equivalents of the bellows but the influence of shear deformation on the bellows in bending is very small according to, for example, Morishita et al. [9] and Jakubauskas and Weaver [10], and should therefore be suppressed in the beam formulation. The model is solved in ABAQUS using the mode-based steady-state dynamics solver. This solver calculates the steady-state displacement amplitude as a function of frequency based on modal superposition. The first 79 step is to extract sufficiently many eigenmodes so that the dynamic response of the system is adequately modelled. To get a realistic result the modal damping of the system is specified. These damping ratios are obtained from experimental results. The damping ratios used in this work are given in table 3. The model is clamped at one end and is free at the other end. Two different load cases are considered. In the first case the bellows is excited with an axial harmonic force at the free end. In the second case a transverse harmonic force is used. The amplitude of the harmonic force is 0.1 N in both cases. The model is solved with a frequency step of 0.1 Hz. By comparing different mesh densities it is found that 20 linear beam elements are sufficient. The end-caps are modelled by lumped mass and mass moment of inertia elements. These elements are connected to the beam elements representing the bellows by rigid elements. During the experimental investigation a steel plate is welded on to the free end of the bellows. This plate is modelled with a lumped mass and mass moment of inertia element. The transducers are accounted for by lumped mass elements. 5. Experimental Investigations A measurement set-up that is useful also for the complete flexible joint in coming studies is desired. A sinusoidal excitation is used because it is well suited for non-linear analysis. The input signal level can be accurately controlled and the signal to noise ratio is good because all the input energy is concentrated at one frequency at the time. The main drawback is that it is time consuming because the measurements are performed frequency by frequency and that time is needed for the test specimen to reach steady state at each frequency. The signal analyser I-DEAS Test is used [11], which has a function called step sine closed loop control in the sine measurements module. Using this function the amplitude and/or the phase of the excitation force can be controlled. In the first case the bellows is excited in the axial direction and in the second case it is excited in the transverse direction. An HP VXI measuring system is used to acquire the experimental data. The experimental set-ups for the two different cases are described bellow. Due to the low stiffness of the bellows in 80 the transverse direction it is mounted so that its axial direction coincides with the gravitational force, to avoid undesired influence from gravitation on the dynamic behaviour. To secure that the experimental set-ups are satisfactory the influence of vibration feedback through the rigid table is investigated. No significant feedback is found. 5.1. Axial Measurements The bellows is rigidly mounted at one end in a chuck from a lathe, and is excited in the other end. The lathe chuck is mounted in a frame made of steel beams, which is assumed to be rigid. This frame is attached to a rigid table of considerable mass. The shaker is also mounted on this rigid table. The free end is connected to the shaker through a stinger and a force transducer. The measurement set-up can be seen in figure 4. Figure 4. Experimental set-up for the axial measurements. 81 To be able to excite the bellows in this set-up it has to be slightly modified. A metal plate is welded on to the free end. On this plate an accelerometer and a force transducer are mounted, see figure 5. Figure 5. Transducer placement in the axial set-up. The plate is rigid in the frequency interval of interest. The bellows is excited with a sinusoidal signal with an amplitude of 0.1 N. The frequency step is adapted so that a smaller increment is used near resonance frequencies, where the response changes rapidly. 5.2. Transverse Measurements The bellows is rigidly mounted at one end in a chuck from a lathe, and is excited in the other end. The lathe chuck is mounted on a rigid table of considerable mass. The shaker is also mounted on this table via a rigid angle bracket made of thick reinforced steel plates. The free end is connected to the shaker through a stinger and a force transducer. An accelerometer is placed on the opposite side of the bellows, see figure 6. The measurement set-up can be seen in figure 7. 82 Figure 6. Transducer placement in the transverse set-up. Figure 7. Experimental set-up for the transverse measurements. 83 An excitation amplitude of 0.1 N is used and the frequency increment is adapted in the same way as in the axial case. 5.3. Linearity Check The validity of the linearity assumption is checked. The bellows is excited at different force levels, the response is measured and the frequency response function (FRF) is calculated. For a linear structure these FRFs are independent of excitation level. If non-linearity is present this is not true. Therefore the bellows is excited at different levels, in both the axial and transverse case. An example of such a test can be seen in figure 8, which shows FRFs corresponding to different excitation levels in the axial case. Depending on the excitation level a frequency shift of roughly 8 Hz at the third axial resonance frequency occurs. A discussion of this phenomenon is included in the next section. Frequency response function −4 Displacement / Reaction force (m / N) 10 0.1 N 2 N 5 N −5 10 −6 10 −7 10 175 180 185 Frequency (Hz) 190 Figure 8. FRFs for different excitation force levels. 84 195 6. Results and Discussion The resonance frequency shift observed in the linearity check is very interesting and somewhat intriguing. It appears at such small excitation levels and deflections that, for example, plastic material behaviour or geometric nonlinearity should not be the reason. A completely verified explanation cannot be given at the moment, but a hypothetic qualitative explanation is presented in the following. A similar experiment performed with a single-ply bellows does not show such shift. This indicates that the non-linearity could be related to the multiple plies. Although it is common to consider the plies to work independent of each other [8], the authors suspect that there are spots of contact between them. As the convolutions are exposed to bending, both in the axial and the transverse load case, relative (micro-)motion between the inner surface of the outer ply and the other surface of the inner ply, is induced, and thus there will be friction dissipation when the bellows vibrate. Friction does however usually not give a frequency shift - only a reduction of the vibration amplitude - but, if the friction limit, below which the two plies stick together, is slightly different between different locations along the bellows, the shift could be understood. Viewing the bellows as a chain of many “links”, among which most of them have a friction limit between the plies close to zero, but some of them have a higher friction limit, the chain will have a different equivalent stiffness for different excitation force levels. At low levels some of the “links” will have a much higher stiffness than the others, because the two plies are then stuck together and they act at these locations as one ply of double thickness (the bending stiffness is proportional to the thickness in cube). At increasing excitation force level, fever and fever “links” will have this higher stiffness, since the friction limit is exceeded in more and more “links”, and the equivalent stiffness of the chain then decreases. This goes on until the plies do not stick in any “links”. For further increasing excitation force levels, there will then be no further shift due to this friction-related non-linearity, only a reduction of vibration amplitude. For the present application of the bellows (exhaust system component) this non-linear phenomenon is probably of minor significance. The excitation levels should in practice mostly exceed the ones including this phenomenon. In any case, the frequency shift only seems to amount to a few percent, close to the theoretical linear resonance frequency. Furthermore, when the bellows is combined with the liner, this introduces a more significant non-linearity. 85 The issue is therefore not further scrutinized in this paper, and the measured resonance frequencies are accepted as “linear” in the comparison bellow. The FRF for the axial load case is shown in figure 9 for an excitation force level of 0.1 N. The FRF for the transverse load case is shown in figure 10 for an excitation force level of 0.1 N. The solid lines show the theoretical results and the dashed lines shows the experimental results. The results are summarised in table 3. The damping value is given as the fraction of critical damping and the correlation value is the relative difference between theoretical and experimental resonance frequencies. Frequency response function −2 Displacement / Reaction force (m / N) 10 −3 10 −4 10 −5 10 −6 10 −7 10 0 20 40 60 80 100 120 Frequency (Hz) 140 160 180 200 Figure 9. Theoretical and experimental FRFs in axial direction. 86 Frequency response function −1 Displacement / Reaction force (m / N) 10 −2 10 −3 10 −4 10 −5 10 −6 10 0 20 40 60 80 100 120 Frequency (Hz) 140 160 180 200 Figure 10. Theoretical and experimental FRFs in transverse direction. Table 3. Results. Mode 1 2 3 4 5 6 a b Experimental Frequency Damping (Hz) (%) 9.40 1.2 31.4 1.0 50.2 1.7 104 0.9 131 2.0 186 0.6 Theoretical Frequency (Hz) 9.10 32.9 50.3 104 127 182 Correlationa (%) Directionb -3.2 4.8 0.2 0.4 -2.6 -1.8 T A T A T A The correlations are calculated before rounding off. T=Transverse, A=Axial. The damping ratios are amplitude dependent due to the non-linear behaviour of the bellows. The damping ratios increase with increasing excitation force. 87 It should be mentioned that for mode five the damping ratio cannot be calculated by established methods from experimental data because this mode is not significant, see figure 10. This damping ratio is instead estimated on a trial and error basis so that the theoretical FRF resembles the experimental FRF near this mode. 7. Conclusions Simplified modelling of bellows flexible joints, used in for example automobile exhaust systems, is the subject of this study. Broman et al. [2] presented a method for determining the dynamic characteristics of single-ply bellows of constant mean radius by manipulated beam finite elements of a commercial software. This paper suggests how that procedure can be extended to model also a multi-ply bellows of variable mean radius. Experimental investigations are performed for verification. At small excitation force levels the response of the bellows is observed to be excitation dependent, that is, it is non-linear. The authors find this observation interesting and important to report, and a hypothetic qualitative explanation is discussed. For the present application of the bellows (exhaust system component) this non-linear phenomenon is however considered to be of minor significance, and its influence on the measured results seems to be small and is therefore neglected. This said, the correlation between theoretical and experimental results is very good, which verifies the extended modelling procedure. Furthermore, it is shown that the procedure can easily be implemented in another commercial software (in this case ABAQUS) than the one used in [2] (I-DEAS). 8. Acknowledgements The support from Faurecia Exhaust Systems AB is gratefully acknowledged, especially from Håkan Svensson. The authors also gratefully acknowledge the financial support from the Swedish Foundation for Knowledge and Competence Development. 88 9. References 1. DeGaspari, J., (2000), ‘Lightweight Exhaust’, Mechanical Engineering, May 2000. 2. Broman G., Jönsson A, and Hermann M., (2000), ‘Determining dynamic characteristics of bellows by manipulated beam finite elements of commercial software’, International Journal of Pressure Vessels and Piping, ISSN 0308-0161, vol. 77, Issue 8, 2000. 3. ABAQUS, HKS, http://www.abaqus.com. 4. I-DEAS, EDS PLM Solutions, http://www.sdrc.com. 5. Cunningham, J., Sampers, W. and van Schalkwijk, R., ‘Design of flexible tubes for automotive exhaust systems’, ABAQUS Users’ Conference, 2001 6. Belangardi, G. and Leonti, S., (1987), ‘Modal analysis in the design of an automotive exhaust pipe’, Int. J. of Vehicle Design, vol. 8, no. 4/5/6, 1987. 7. Verboven, P., Valgaeren, R., van Overmeire, M. and Guillaume, P., (1998), ‘Some comments on modal analysis applied to an automotive exhaust system’, Proceedings of the 16th International Modal Analysis Conference 1998, Santa Barbara, USA. 8. EJMA: Standards of the Expansion Joint Manufacturers Association, seventh edition, 1998. 9. Morishita, M., Ikahata, N. and Kitamura, S., (1989), ‘Dynamic analysis methods of bellows including fluid-structure interaction’, The 1989 ASME pressure vessels and piping conference, Hawaii. 10. Jakubauskas, V.F. and Weaver, D.S., (1998), ‘Transverse vibrations of bellows expansion joints – Part II: beam model development and experimental verification’, Journal of fluid and structures, 12, 457-473. 11. I-DEAS Test, MTS, http://www.mts.com. 89 90 Paper D Dynamic Characteristics of a Combined Bellows and Liner Flexible Joint 91 Paper D is submitted for journal publication as: Englund, T., Wall, J., Ahlin, K. and Broman, G., ‘Dynamic characteristics of a combined bellows and liner flexible joint’. 92 Dynamic Characteristics of a Combined Bellows and Liner Flexible Joint Thomas L Englund, Johan E Wall, Kjell A Ahlin and Göran I Broman Abstract A bellows combined with an inside liner and an outside braid is commonly used as a flexible joint in automobile exhaust systems to reduce transmission of engine movements to the exhaust system. It greatly influences the dynamics of the complete system. Understanding of its dynamic characteristics and a modelling method that facilitates systems simulation are therefore desired. This has been obtained in earlier works for the bellows itself. In this work an approach to the modelling of the combined bellows and liner joint is suggested and experimentally verified. Simulations and measurements show that the liner adds significant non-linearity and makes the characteristics of the joint complex. Results are presented for the axial and the bending load cases. In torsion influence of the liner is negligible. Peak responses are significantly reduced when the excitation level approximately corresponds to the friction limit of the liner. The complexity of the combined bellows and liner joint is important to know of and consider in exhaust system design and proves the necessity of including a model of the liner in the theoretical joint model when this type of liner is present in the real joint to be simulated. Keywords: Beam model, Dynamic, Experimental investigation, Flexible joint, Friction, Liner, Non-linear. 93 1. Notation A Area [m2] [C] Damping-matrix E Young’s modulus [Pa] F Force [N] [K] Stiffness-matrix k Stiffness [N/m] [M] Mass-matrix M Moment [Nm] L Length [m] R Radius [m] α Damping parameter β Damping parameter ε Strain ξ Damping ratio σ Stress [Pa] ω Angular frequency [rad/s] 94 Indices f Friction i Number l Liner y Yield 2. Introduction With the introduction of transverse engines and catalytic converters a highly flexible gas-tight joint became necessary in automobile exhaust systems to reduce transmission of engine movements to the exhaust system. A bellows combined with an inside liner and an outside braid is commonly used for this purpose between the manifold and the rest of the exhaust system. As shown by Wall et al. [1] this joint greatly influences the dynamics of the complete system. Understanding of its dynamic characteristics and a modelling method that facilitates simulations of the exhaust system at an early stage of the product development process are therefore desired. To be suited for this, the simulation procedure must be computationally inexpensive. The component models should therefore be as simple as possible while yet giving a proper description of their dynamic characteristics and mutual influence within the exhaust system. Such models have been suggested and experimentally verified by Englund et al. [2, 3] for the pipes, the mufflers and the catalytic converter. A method of modelling the bellows by manipulated beam finite elements, which considerably reduces the model size compared to a straightforward shell elements model, was presented by Broman et al. [4]. In this reference also a short historical background and further references on bellows joints can be found. This method was extended to apply also for multi-ply bellows of variable mean radius by Wall et al. [5]. This paper presents an approach to the modelling of the combined bellows and liner joint. Simulations are performed to gain understanding of its dynamic characteristics and experimental investigations are included for verification. 95 3. Basic design The design of a typical flexible joint is shown in figure 1. It consists of a gastight bellows combined with an inside liner and an outside braid. The gastightness is crucial for emission control since the joint is often upstream of the catalytic converter. The ends of the bellows, the liner and the braid are rigidly connected with the end-caps. General information on the design of this type of joint is also given by, for example, Broman et al. [4] and Cunningham et al. [6]. Figure 1. Basic design of the flexible joint. The bellows studied in this paper is double-plied and it has smaller mean radius closer to the ends. Low stiffness is beneficial to decouple the engine from the exhaust system and a double-plied bellows has lower stiffness than a single-ply bellows for a given strength. The liner is used for reduction of bellows temperature and for improved flow conditions. The liner consists of a strip of sheet metal that is winded cylindrically and folded. The cross-section of the winding is schematically shown in figure 1. When the coils move 96 relative to each other there is resistance due to friction, which depends on how hard the folding is and the coefficient of friction. The braid is used for mechanical protection and to limit the extension of the joint. It is not included in this study. 4. Modelling The bellows is modelled in ABAQUS [7] using a pipe analogy and a beam finite element formulation with certain parameters manipulated [4, 5]. Considering that the studied bellows has different mean radius three different equivalent pipes are used. By comparing different mesh densities it is found that 20 beam elements are sufficient. The liner is also modelled by beam elements with a pipe section. The wall thickness and the density of the material are set to give the correct mass of the liner. The liner and the bellows are only connected at the outer nodes of the elements at the ends. It is assumed that there is no contact elsewhere. The friction in the liner is modelled as a Coulomb type dry friction. When the force is below the friction limit, the liner behaves elastically (sticking). Above the friction limit the liner is slipping. The axial stiffness, k, and the axial friction limit, Ff, are determined experimentally (see section 6). The friction force is assumed to be symmetric and independent of frequency. To model the above in ABAQUS an ideal-plastic isotropic material is defined. The principal behaviour of this material is shown in figure 2, where E is Young’s modulus, σy is the yield stress and ε is the strain. The shear stress that may arise in the beam elements is not taken into account in the plasticity calculations. 97 σ σy E ε -σy Figure 2. Behaviour of the ideal-plastic material. The fictive Young’s modulus and the fictive yield stress are E= k⋅L Al (1) Ff Al (2) σy = where Al is the area of the pipe cross section representing the liner and L is its length. The plasticity calculations are performed numerically in so-called section points, see figure 3. When studying axial vibrations one section point is sufficient, but to get a good approximation of the bending resistance, Mf, for any axis of rotation more section points are necessary. 98 Figure 3. Section points. If the liner is exposed to pure bending and uniform friction resistance is assumed along the circumference the exact bending resistance for any axis of rotation is Mf = 2 ⋅ Ff ⋅ Rl (3) π where Rl is the radius of the liner. Using a finite number of section points an approximate bending resistance is obtained. The maximum error in bending resistance is a function of the number of section points. Twenty points are used in this work. Axial and bending vibrations are studied. In torsion influence of the liner is negligible compared to the bellows and thus the prior model is sufficient [5]. In axial loading the liner slips at the coil that happens to have the weakest folding. This location is probably rather randomly distributed between different liners due to variations in fabrication and is thus hard to predict. In bending loading the liner slips at the coil that has the highest ratio of bending moment over friction bending resistance. When the joint is a part of the whole exhaust system in general motion, also this location is hard to predict. On the other hand, the exact slip location is then not so critical for the system and joint behaviour. Furthermore, numerical problems arise when connecting ideal-plastic elements in series. Therefore it is assumed that in the general case the liner slips at the mid-coil. This can be modelled by using a short ideal-plastic element in this mid-location, connected on each side to an elastic element to represent the rest of the liner. For the specific boundary conditions and bending loading used in the experimental set-up it is, however, most likely that the liner bends close to its 99 clamped end. Therefore the ideal-plastic element is located at this end in the simulations of this paper. The end-caps are considered as rigid pipes and are modelled with lumped mass and rotary inertia elements. Transducers and connecting devices used in the experiments are modelled with lumped mass elements. 5. Simulation The frequency interval of interest for the analysis is obtained by considering that a four-stroke engine with four cylinders gives its main excitation at a frequency of twice the rotational frequency. Usually the rotational speed is below 6000 rpm. Excitation at low frequencies may arise due to road irregularities, as discussed by, for example, Belangardi and Leonti [8] and Verboven et al. [9]. Thus, the interval is set to 0-200 Hz. Since the joint is strongly non-linear direct time integration, using the explicit solver of ABAQUS, is performed. In both the axial and bending cases the joint is clamped at one end and excited with a sinusoidal force at the other end. The response is taken in the excitation point. A frequency step of two Hz is used except at the first axial and bending peak responses, where a smaller step is used since the response here changes more rapidly. The model is simulated until steady state is reached for each excitation frequency. When the excitation force level is approximately the same as the friction limit, damping associated with the material is negligible compared to the friction based damping, but otherwise it affects the results. In ABAQUS it is possible to define so-called Rayleigh damping, according to [C ] = α [ M ] + β [ K ] (4) where [C] is the damping matrix, [M] is the mass matrix, [K] is the stiffness matrix, and α and β are damping parameters. For a linear system these parameters relates to the modal damping ratios, ξi, as (see, for example, Bathe [10]) ςi = β ⋅ωi α + 2ω i 2 (5) 100 where ωi is a certain natural frequency. In this work the values of α and β are calculated so that ξ1 = ξ3 = 2 % (ω1 equals the excitation frequency and ω3 = 3ω1 is the most dominant higher harmonic). This gives a system of two equations that is solved for each excitation frequency to obtain α and β. A damping ratio of two per cent gives a good overall agreement with experimental results when the system is excited with a force well below the friction limit. Since the calculation of the damping parameters, the increment in excitation frequency and storing of the simulated time response must be performed for each excitation frequency it would be very time-consuming to perform these tasks manually. Instead an automated procedure is used where MATLAB [11] and ABAQUS interact with each other. The damping parameters and the excitation frequency are determined in MATLAB and this information is used as input to ABAQUS where the calculations are performed. The calculated time response is then transferred back to MATLAB where it is stored. This procedure is automatically repeated for the excitation frequencies of interest. The interaction between the two software packages is performed by taking advantage of MATLAB’s ability of reading and writing ASCII-files. Since the system is non-linear the response generally includes higher harmonics. This is especially clear in the axial case when the excitation force level is approximately the same as the friction limit. To be able to analyse and compare simulated and experimental results in a straightforward way only the first harmonic is considered. The time responses are therefore transformed to the frequency domain, using the FFT algorithm in MATLAB, where the amplitude of the first harmonic is obtained. A flattop window is used to avoid possible leakage problems. The normalised response, defined as the amplitude of the first harmonic over the excitation force amplitude, is then calculated. Since the problem is non-linear the magnitude of the excitation force is significant. The behaviour of the joint is dramatically different depending on whether the excitation force is above or below the friction limit, so it is important that the force interval used in the simulation covers this shift. 6. Experimental investigation A sinusoidal excitation is used so that the input signal level can be accurately controlled. The signal to noise ratio is good because all the input energy is concentrated at one frequency at the time. Another advantage is that an 101 adaptive frequency increment can be used so that rapid changes in the response can be accurately captured. The main drawback is that it is time consuming because the measurements are performed frequency by frequency and that time is needed for the test specimen to reach steady state at each frequency. A Hewlett Packard VXI measuring system is used to acquire the experimental data and I-DEAS Test [12] is used as signal analyser. I-DEAS Test has a function called step sine closed loop control in the sine measurements module. The force level is iteratively adjusted until the target amplitude and/or phase is reached within a specified tolerance. The steady state acceleration amplitude of the fundamental frequency in the response is then saved and the procedure is repeated for the next excitation frequency. Retaining only the fundamental frequency of the response gives so-called first-order frequency response functions (FRFs). For a further discussion; see, for example, Maia and Silva [13]. The measurement set-up for axial vibrations is shown in figure 4. Figure 4. Experimental set-up for the axial measurements. 102 The joint is rigidly mounted at one end in a chuck from a lathe and is excited at the other end. The lathe chuck is mounted in a frame made of steel beams, which is assumed to be rigid. This frame is attached to a rigid table of considerable mass. The shaker is also mounted on this table. A metal plate is welded onto the free end of the joint and on this plate an accelerometer and a force transducer are mounted; see figure 5. The plate is rigid in the frequency interval of interest. The force transducer is connected to the shaker through a stinger. The measurement set-up for bending vibrations is shown in figure 6. Figure 5. Transducer placements in the axial set-up. Figure 6. Experimental set-up for the bending measurements. 103 Due to the low stiffness of the joint in the transverse direction it is mounted so that its axial direction coincides with the gravitational force, to avoid undesired influence from gravitation on the dynamic behaviour. The joint is rigidly mounted at one end in a chuck from a lathe and is excited at the other end. The lathe chuck is mounted on a rigid table of considerable mass. The shaker is also mounted on this table via a rigid angle bracket made of thick reinforced steel plates. The free end of the joint is connected to the shaker through a force transducer and a stinger and an accelerometer is placed on the opposite side; see figure 7. Figure 7. Transducer placements in the bending set-up. In both set-ups it is assured that vibration feedback through the rigid table is insignificant. The stiffness and the friction limit of the liner are determined experimentally by using principally the above described axial set-up. The axial stiffness, k, is obtained by exciting the liner with a force well below the friction limit. Plotting the FRF in a log-log Bode diagram format the stiffness is obtained from the low frequency asymptote (Ewins [14]). To determine the axial friction limit, Ff, the liner is excited with a sinusoidal force of low frequency and the force and displacements are registered. The amplitude of the excitation force is increased step by step until the friction limit is clearly reached. A plot is shown in figure 8. 104 25 20 15 Force (N) 10 5 0 −5 −10 −15 −20 −25 −4 −2 0 2 Displacement (m) 4 6 −3 x 10 Figure 8. The axial friction limit of the liner. 7. Results and discussion Theoretical and experimental results are compared in figure 9 for two different excitation force levels in the axial case, one below (10 N) and one above (25 N) the friction limit of the liner (17 N). Theoretical and experimental results are compared in figure 10 for two different excitation force levels in the bending case, one below (0.25 N) and one above (2.25 N) the friction bending resistance of the liner (corresponding to a transverse force of 1.0 N). Normalised responses (magnitude of first order FRFs) are plotted against excitation frequency for different excitation force levels. 105 Displacement / Reaction force (m / N) Theoretical, 25 N Theoretical, 10 N Experimental, 25 N Experimental, 10 N −4 10 −5 10 −6 10 0 50 100 Frequency (Hz) 150 200 Displacement / Reaction force (m / N) Figure 9. Theoretical and experimental axial results. Theoretical, 2.25 N Theoretical, 0.25 N Experimental, 2.25 N Experimental, 0.25 N −2 10 −3 10 −4 10 −5 10 −6 10 0 50 100 Frequency (Hz) 150 Figure 10. Theoretical and experimental bending results. 106 200 In the experimental investigation force drop out occurs at frequencies with large responses due to the weakness of the joint above the friction limit, making it impossible to excite the joint with a specified constant force amplitude. The excitation force here also includes higher harmonics. The magnitude of the experimental normalised response is therefore uncertain at these peaks. Otherwise the agreement between theoretical and experimental results is good, indicating that the simulation model can be used for further studies. Results for the simulation model excited with 10, 15, 20, 25 and 30 N in the axial direction, covering the friction limit, are shown in figure 11. −3 Displacement / Reaction force (m / N) 10 30 25 20 15 10 N N N N N −4 10 −5 10 −6 10 0 50 100 Frequency (Hz) 150 200 Figure 11. Simulated axial results for different force levels. The normalised response is small and the peak response frequencies are high at the lowest force level (10 N). This is because the liner is sticking, that is, no slipping occurs, and the stiffness of the liner is high compared to the bellows. At the intermediate force levels (15 and 20 N) both sticking and slipping takes place during different parts of the motion and large friction based damping result. The normalised response is higher and the peak response frequencies 107 are shifted downwards. Higher harmonics are significant in the response; see figure 12. Figure 12. Waterfall diagram when exciting the joint with 20 N in the axial case. At the higher force levels (25 and 30 N) the liner is mostly slipping and the influence of friction on the peak response frequencies becomes more and more negligible, so they approach those of the bellows itself [5] (with a small difference due to the mass of the liner). The normalised response is higher than for the low and intermediate excitation force levels but lower than the response of the bellows itself [5]. Results for the simulation model excited with 0.25, 0.75, 1.25, 1.75 and 2.25 N in the transverse direction, covering the friction bending resistance, are shown in figure 13. 108 Displacement / Reaction force (m / N) 2.25 1.75 1.25 0.75 0.25 −2 10 N N N N N −3 10 −4 10 −5 10 −6 10 0 50 100 Frequency (Hz) 150 200 Figure 13. Simulated bending results for different force levels. Comments regarding the normalised response are principally the same as for the axial case. The presence of higher harmonics at intermediate excitation levels is however less and at the higher excitation levels slipping occurs only for the lowest peak response. 8. Conclusions Dynamic characteristics of a combined bellows and liner flexible joint, commonly used in automobile exhaust systems, is the subject of this paper. An approach to the modelling of this joint is suggested and implemented in the commercial software ABAQUS. The simulation model is experimentally verified. Axial and bending vibrations are studied and it is shown that the liner adds significant non-linearity due to friction, which makes the characteristics of the joint complex. At excitation levels well below the friction limit of the liner the joint is very stiff (and essentially linear). This is not desired in applications since the purpose of the joint is to add flexibility. At intermediate excitation levels, close to the friction limit of the liner, stick-slip motion occurs, resulting in 109 high energy dissipation through friction based damping. At excitation levels well above the friction limit of the liner the behaviour of the joint approaches that of the bellows itself (and it becomes again more and more linear). Additional complications are that differences in the friction limit have been observed between joint specimens and that it also seems that the friction limit decreases after some time of use. The complexity of the combined bellows and liner joint is important to know of and consider in exhaust system design and proves the necessity of including a model of the liner in the theoretical joint model when this type of liner is present in the real joint to be simulated. 9. Acknowledgements The support from Faurecia Exhaust Systems AB is gratefully acknowledged, especially from Håkan Svensson. The authors also gratefully acknowledge the financial support from the Swedish Foundation for Knowledge and Competence Development. 10. References 1. Wall, J., Englund, T., Ahlin, K. and Broman, G., ‘Influence of a bellowstype flexible joint on exhaust system dynamics’. Submitted for publication. 2. Englund, T., Wall, J., Ahlin, K. and Broman, G., ‘Significance of nonlinearity and component-internal vibrations in an exhaust system’, Proceedings of the 2nd WSEAS International Conference on Simulation, Modelling and Optimization, Greece, 2002. 3. Englund, T., Wall, J., Ahlin, K. and Broman, G., ‘Automated updating of simplified component models for exhaust system dynamics simulations’, Proceedings of the 2nd WSEAS International Conference on Simulation, Modelling and Optimization, Greece, 2002. 4. Broman, G., Jönsson, A. and Hermann, M., ‘Determining dynamic characteristics of bellows by manipulated beam finite elements of commercial software’, International Journal of Pressure Vessels and Piping, ISSN 0308-0161, vol. 77, Issue 8, 2000. 110 5. Wall, J., Englund, T., Ahlin, K. and Broman, G., ‘Modelling of multi-ply bellows flexible joints of variable mean radius’, Proceedings of the NAFEMS World congress 2003, USA, 2003. 6. Cunningham, J., Sampers, W. and van Schalkwijk, R., ‘Design of flexible tubes for automotive exhaust systems’, ABAQUS Users’ Conference, Netherlands, 2001. 7. ABAQUS, HKS, http://www.abaqus.com. 8. Belingardi, G. and Leonti, S., ‘Modal analysis in the design of an automotive exhaust pipe’, Int. J. of Vehicle Design, vol. 8, no. 4/5/6, 1987. 9. Verboven, P., Valgaeren, R., van Overmeire, M. and Guillaume, P., ‘Some comments on modal analysis applied to an automotive exhaust system’, Proceedings of the 16th International Modal Analysis Conference – IMAC, USA, 1998. 10. Bathe, K. J., ‘Finite element procedures’, Prentice-Hall, UK, 1996. 11. MATLAB, The MathWorks, Inc., http://www.mathworks.com. 12. I-DEAS Test, MTS, http://www.mts.com. 13. Maia, N. M. M. and Silva, J. M. M., ‘Theoretical and experimental modal analysis’, Research Studies Press, UK, 1997. 14. Ewins, D. J., ‘Modal testing: theory practise and application’, Research Studies Press, UK, 2000. 111

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