A Vibro-Acoustic Study of Vehicle Suspension Systems: Experimental and Mathematical Component Approaches

Transcription

1 A Vibro-Acoustic Study of Vehicle Suspension Systems: Experimental and Mathematical Component Approaches Eskil Lindberg Doctoral Thesis Stockholm 2013 Material and Structural Acoustics Group The Marcus Wallenberg Laboratory for Sound and Vibration Research Department of Aeronautical and Vehicle Engineering Postal address Visiting address Contact Royal Institute of Technology Teknikringen 8 Tel: MWL/AVE Stockholm eskill@kth.se SE Stockholm Sweden

2 Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framläggs till offentlig granskning för avläggande av teknologie doktorexamen onsdag den 22 maj 2013, 13:15 i sal F3, Lindstedtsvägen 26, KTH, Stockholm. TRITA-AVE-2013:17 ISSN ISBN c Eskil Lindberg, 2013

3 A Vibro-Acoustic Study of Vehicle Suspension Systems: Experimental and Mathematical Component Approaches Eskil Lindberg Material and Structural Acoustics Group The Marcus Wallenberg Laboratory for Sound and Vibration Research Department of Aeronautical and Vehicle Engineering Royal Institute of Technology Abstract The objective of the present work is to study the vehicle suspension as a vibro-acoustic system of high complexity, consisting of many sub-systems with fundamentally different acoustical properties. In a parallel numerical and experimental modelling effort, important contributions to the understanding of its behaviour have been achieved. These findings are based on a balance between component investigations and global modelling of the complete system; they have been formulated for the transmission of both tyre-road excitation and friction-induced vibrations in the brake system. Initially an experimental study was conducted on a full vehicle test rig studying the broadband interior brake noise problem of, here named, roughness noise. The purpose of the study was twofold: first, to determine if the transmission from the source to the interior of the vehicle was structure-borne; second, to study the complexity of the suspension as a vibro-acoustic system. Parameters affecting the vibro-acoustic source were varied to gain understanding of the source mechanisms. This experimental study laid the foundation of the first part of this thesis (paper A) and provided the directions for the second part, the development of a mathematical modelling approach (paper B and C). In these two papers, methods for analysing the complex vibro-acoustic transfer of structure-borne sound in a vehicle suspension system were developed. The last part was then focussed on the wheel rim influence on the vibro-acoustic behaviour (paper D) of the suspension system. As a whole, the work clearly demonstrates that it is possible to conduct component studies of subsystems in the vehicle suspension system; and from these component studies it is possible draw conclusions that very well may avoid severe degradations in the interior noise of future vehicle generations. Keywords: Acoustics; Vehicle suspension; Disc brake; Wheel rim Roughness noise; Interior tyre-road noise Component mode synthesis; Undeformed coupling interface;

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5 Dissertation The work presented in this doctoral thesis was carried out at the Department of Aeronautical and Vehicle Engineering, the Royal Institute of Technology (KTH) in Stockholm, Sweden. This thesis consists of two parts. The first part gives an overview of the research with a summary of the performed work. The second part collects the following scientific articles: Paper A. E. Lindberg, N.-E. Hörlin and P. Göransson. An Experimental Study of Interior Vehicle Roughness Noise from Disc Brake Systems. Applied Acoustics; (3), pp Paper B. E. Lindberg, N.-E. Hörlin and P. Göransson. Component Mode Synthesis Using Undeformed Interface Coupling Modes to Connect Soft and Stiff Substructures. Shock and Vibration; (1), pp Paper C. E. Lindberg, M. Östberg, N.-E. Hörlin and P. Göransson. A Vibro- Acoustic Reduced Order Model Using Undeformed Coupling Interface Substructuring With Application to Rubber Bushing Isolation in Vehicle Suspension Systems. Submitted to Applied Acoustics Paper D. E. Lindberg. Tyre-Road Noise Experimental Component Investigation of the Structural Dynamic Behaviour of the Rim. Report, ISBN v

6 The following papers are not included in this thesis due its content being out of the scope of this thesis. * E. Lindberg, P. B. U. Andersson. Experimental Investigation of Sound Power Radiation From Partly Open Enclosure With Numerous Interior Objects. 19th International Congress on Acoustics, 2007, Madrid. * P. B. U. Andersson and E. Lindberg. Boundary element method for intensity potential approach: Predicting the radiated sound power from partially enclosed noise sources. Acta Acustica united with Acustica; (4), pp Some of the results have been presented in conferences. * E. Lindberg, N.-E. Hörlin, P. Göransson. Experimental Study of Wire Brush Brake Noise on a Personal Car. SAE Brake Colloquium and Exhibition 2009, Tampa, FL. (Oral only) * E. Lindberg. Tyre-Road Noise Experimental Component Investigation of the Structural Dynamic Behaviour of the Rim. AIA-DAGA Conference on Acoustics 2013, Merano. vi

7 Acknowledgements First of all I would like to thank my supervisors Nils-Erik Hörlin and Peter Göransson for your support and guidance. Nisse, you have challenged me and have given me new view on things, thank you! Peter, thank you for being there when I needed you!! Martin Östberg thank you for the good cooperation with paper C. A big thanks goes to Kent Lindgren for his devotion and expertise in the lab! The industrial partners of this work are acknowledged: SAAB automobile, Opel and Daimler AG. People that deserve extra credit are: Anders Sköld, Maurice Claessens, Daniel Sachse, Ralf Lehmann, Otto Gartmeier and Eric Bauer. To my fellow PhD-students, thank you all for your support and friendship. Special thanks goes to Eleonora Nordgren and Mathias Barbagallo! Finally to my family, mamma, pappa and Ida, thank you! vii

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9 Contents I Overview and Summary 1 1 Introduction Societal motivation of thesis Industrial motivation of this thesis Background Overview Outline Brake noise and friction-induced sound and vibrations Background Friction-induced noise Brake noise classification Roughness noise theory Interior brake roughness noise Experimental setup Results and discussion Brake pressure System loading Vehicle speed Conclusions and findings Findings Conclusions Component mode synthesis approach Background Introduction ix

10 4.2 Theory General problem Change of basis Local modes Results and discussion Test structure Vibro-acoustic response Evaluation of the approach Conclusions UCI as a framework Background Local substruture models and global assembly Global model description Results total transmitted power Transmission path Discussion Conclusions Wheel rim influence on interior noise Background Approach Interior noise and rim parmeter correlations Discussion Conclusion Outlook 53 II Appended Papers 61 x

11 Part I Overview and Summary 1

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13 CHAPTER 1 Introduction This chapter summarises the overall objectives for the research that is discussed in this thesis. After a brief introduction of the scope of the work the aims are briefly addressed ending with the outline of the thesis. 1.1 Societal motivation of thesis This is an applied study where structure-borne noise in vehicle suspension systems is investigated. The societal motivation was to build competence and knowledge that could be used in the development of vehicles with reduced negative impact on the environment. Two environmental concerns, i.e. reducing noise pollution and energy consumption, prevail. The functional aspect tying these problems together is the structural mass. Both structure-borne sound and energy consumption can be argued to be correlated to the structural mass of the vehicle. Generally speaking, high structural mass leads to both less structure-borne noise and higher energy/fuel consumption of the vehicle and vice versa. Hence, in vehicles there must be a trade-off between energy consumption and structural borne noise. Fortunately, problems of structure-borne noise are not only governed by the structural mass: In addition, damping and isolation treatments can be designed together with optimisation of the stiffness and the mass properties to open up profound possibilities. However, the minimisation of the structural mass for a given noise problem requires a deep understanding of the vibroacoustic system and the dominant source mechanisms, is required. This is where this thesis intends to contribute to advance in the current state-of-the-art. 3

14 4 1 Introduction 1.2 Industrial motivation of this thesis From an industrial/vehicle manufacturers, point of view the objective is to produce vehicles which are attractive for customers, in other words meeting their demands; e.g. on the interior acoustic environment, and on the fuel consumptions of the vehicle. In this sense, the commercial and societal goals coincide well, thus underlining the relevance of the topic as such. For the present research, the industrial motivation as set by SAAB Automobile AB, the initial industrial partner in the project, was to gain knowledge and understanding of brake noise in general. This was in anticipation of an increasing importance for this noise source as the weight of the cars of tomorrow could be be expected to be substantially reduced compared to the present fleet. The initial focus of the research was set on a particular source of interior noise, i.e. a brake noise problem usually referred to as moan noise, but gradually changed into a more general study of the complete vehicle suspension system and of a broadband noise phenomenon related to brake pad-disc interaction in general. This was driven by the first results, which opened up for a more general investigation of the suspension system as such, reducing the emphasis on the actual source mechanism in favour of a study of the broadband transfer of vibro-acoustic power through the suspension as a whole. 1.3 Background Research is the search for new and novel knowledge, where the scientific study is the active, systematic and methodical process of accumulating this knowledge. The methods used in the scientific study can be many e.g. cartography, case study, classification, experience, experiment, interview, mathematical model, simulation, statistical analysis and ethnography. The findings that are discussed in this work are mainly based on two of these methods i.e., experiments and mathematical models. They are communicated as this thesis which is a collation of papers that is summarised in a general introduction. The introduction is written with the purpose of giving a shorter and less technical summary of the papers, and of providing a motivation to the thesis. This thesis consists of four appended papers A to D; in paper A and D experimental studies were conducted and in paper B and C a mathematical approach was developed. Often a distinction between basic- and applied- research is made. Basic research is

15 1.4 Overview 5 the scientific study, which is said to be driven by the curiosity of the researcher with the primary goal of extending the human knowledge and theoretical understanding of our existence. Applied research on the other hand is goal oriented, the goals can be commercial or social, set by e.g. companies, states or unions. In the onset of this thesis work the goals were formulated by SAAB Automobile AB together with Vinnova and KTH. Even though the scientific work in this thesis would not have started if it was not for the industrial need, the actual end result is not different from any basic research work. In the end, the most significant impact of the knowledge gained may very well be the modelling of the vehicle suspension as a vibro-acoustic system as such. This stands in contrast to a detailed investigation of a specific phenomenon, for instance friction induced vibrations. 1.4 Overview The common denominator for the parts of the current work is the vehicle suspension system itself. It was concluded from an early stage of the research performed in this project (paper A), that the vehicle suspension as a vibro-acoustic system is too complicated to be fully understood from purely global analyses, such as the full vehicle experimental studies in paper A. The questions generated by that part of the work could be summarised as: i) what is governing the friction source, which parameters are affecting the generation of vibrations in the contact? ii) how is the brake system assembly (calliper, pads, disc etc.) affecting the final interior noise? iii) how do vibrations generated in the disc-pad interaction transfer from the brake system in to the vehicle? The vehicle industry has seen a tremendous development since the early introduction of the mass-produced cars. It is a well-known fact that vehicles make noise, however the attitudes related to it have continuously shifted over the years. There are several reasons for this: some are attributed to the overall performance of the vehicles themselves, such as higher speeds (noisier) and more powerful engines (noisier); others are attributed to advances in technologies related to improved noise encapsulations and vehicle cabin isolation of air-borne noise. However, with increasing environmental concerns related to transport, there is a growing trend towards lightweight body struc-

16 6 1 Introduction tures, potentially leading to lower fuel consumption and lower CO 2 emissions. This has two sides, as seen from the perspectives of the present research. On one hand, the contradictory relation between structural mass and noise gives reasons for some concern; this concern may partially be addressed with encapsulation and isolation advances for air-borne noise and, in addition, with the results discussed and obtained in this thesis. With the gain in the understanding of the interior noise associated with the transfer of vibro-acoustic energy through the suspension system, the potential for improvements is substantial. In retrospect, it is clear that in order to understand how the components interact and influence the vibro-acoustics, a mathematical modelling approach had to be developed. What was not as obvious, was which type of modelling would in the end facilitate the analysis of complex geometries, such as the vehicle suspension system. The finite element method (FEM) is well-known for its flexibility and versatility, however, with increasing geometrical complexity the method tends to involve increasing model sizes and corresponding computational challenges. Furthermore, to investigate the vehicle suspension system, materials with very different stiffness and non proportional frequency dependence (i.e. rubber) have to be considered, such as rubber bushings and steel arms: this is a problem. Finally the large number of components in the vehicle suspension system as a whole, forms a computational and modelling problem in itself. To summarise the research had to adress a modelling problem with three main issues: i) system complexity (the number of components), ii) geometrical complexity and iii) material complexity. The method in this thesis addresses all these modelling issues (however it is not said that it is the only way to come to a solution). In simple words it could be explained in the following. In paper B a substructuring approach is suggested. Substructuring is simply a method where the local, in this case, dynamic behaviour of a component is separately described together with coupling conditions for the specific coupling interfaces. The local component may be described with a reduced set of equations in the global formulation. The idea is that the global modelling size may be reduced with this approach, hence making it possible to solve with the computer resources of today. In paper B the local components were reduced using normal mode superposition, overcoming some of the geometrical complexity, where a reduced set of these modes are used in the global formulation. In its classical version, component mode synthesis (CMS) does not reduce the the modelling size of the coupling interfaces, a limitation which could be a potential problem in the current context. In paper B this problem is

17 1.5 Outline 7 suggested to be overcome by using to an advantage the material complexity mentioned above. Typically, the rubber steel mismatch in structural stiffness is large for rubber bushings connected to steel arms, and in paper B it is shown that if a stiffness ratio between interacting bodies for a given geometry is large then it is enough to model the coupling interfaces as undeformable (for a given acceptable error). This reduces each coupling interface to only 6 degrees of freedom (DOF) from whatever number of DOFs (often in the order of several thousands). The suggested method of undeformed coupling interfaces (UCI) was further explored in paper C where the simplicity of the coupling conditions was further utilised. Here, the UCI-method was suggested as a fundament of a reduced order model (ROM) of a subsystem of a vehicle suspension system consisting of one link arm connected to a vehicle body through two rubber bushings. There is one component of the vehicle suspension system which frequently is neglected, namely the wheel rim. As a further step in demonstrating the importance of adequate tools for dealing with structural-borne noise in the suspension system, an experimental component investigation of the relative influence of this particular part was performed (paper D). In this investigation it was shown that the interior noise in a vehicle using two different sets of rims could have in the order of 5 db difference for the same nominal driving conditions. In this work it was shown that this difference could, contrary to common knowledge, be associated with the stiffness of the rim. In summary, the subject of structure-borne noise in vehicle suspension system will continue to be a challenge for researchers and vehicle industry for many years to come. With the results discussed in this thesis, a point of view on how this challenge may be tackled is given, that possibly could alleviate some of these problems. 1.5 Outline In the next chapters a summary of the research discussed in papers A to D is found. Chapter 2 gives an overview of the related research in the field of frictioninduced noise and vibrations, and disc brake noise in general. Chapter 3 presents an overview of the experimental study on the disc brake roughness noise.

18 8 1 Introduction Chapter 4 presents an overview of the mathematical modelling approach of component mode synthesis using undeformed coupling interfaces. Chapter 5 presents an overview of the reduced order modeling approach suggested for vibro-acoustic modelling of the vehicle suspension system. Chapter 6 presents an overview of a component investigation of the wheel rim. Chapter 7 gives an outlook for future research. Appended papers A to D give the complete results of the thesis.

19 CHAPTER 2 Brake noise and friction-induced sound and vibrations The first part of this thesis consists of an experimental investigation of the interior broadband vehicle brake noise phenomenon referred to as wire brush noise, roughness noise or rubbing noise. This chapter aims at giving a background into the theory of the related friction-induced noise and vibration problem. It serves as a starting point for building the understanding of the suspension system influence to the problem. 2.1 Background The subject of friction-induced sound and vibrations is truly multidisciplinary and have many applications e.g. music acoustics (bow instruments), seismology and railway noise (curve squeal). The common factor of all of these is that an unstable friction force creates a dynamic excitation of the interacting bodies. A friction force is a nonconservative entity and is governed by the tangential stresses created by the relative motion of two bodies Shpenkov (1995). Friction forces may, from a micro-structural viewpoint, be explained as adhesive junctions formed by asperities in contact, and the shear force needed to cause breakaway Sheng (2008). When asperities break loose, they release stored elastic energy, resulting in a vibro-acoustic response. In addition, the ploughing effects of abrasion wear Wriggers (2006) can also be a mechanism in the vibro-acoustic source. The exact type of excitation is highly dependent on the properties of the contact, such as the surface roughness, the sliding speed, and the contact pressure Persson (2000). Various types of noise phenomena with different 9

20 10 2 Brake noise and friction-induced sound and vibrations spectral frequency contents are associated with friction, such as tonal and broadband noise, either due to feedback of structural resonances or surfaces roughness. In 1979, Yokoi and Nakai (1979) made a classification of friction-induced noise to be either rubbing or squealing. According to this work, rubbing is a broadband phenomenon where a large frequency range is excited, whereas squealing is a high pitch tonal noise where only a narrow frequency band is excited. 2.2 Friction-induced noise In the literature, a vast amount of knowledge can be found on the acoustic behaviour of some of the important local contact parameters, primarily from different experimental studies. For instance, Yokoi and Nakai (1979, 1980, 1981a,b, 1982) have shown experimentally in a series of papers that noise levels (sound recorded close to the contact and vibrations of one of the contact bodies) have strong correlation with both surface roughness and sliding speed. Furthermore, these results have been confirmed by others and further studied on similar effects for various types of materials and setups, Othman and Elkholy (1990); Othman et al. (1990); Stoimenov et al. (2007); Ben Abdelounis et al. (2010); Zahouani et al. (2009); Jibiki et al. (2001). However, most publications concerning friction-induced noise focus understandably on the squealing noise, perhaps due to its perceived annoyance. This type of noise is mostly associated with a stick-slip phenomenon, a concept sometimes used loosely but in general stems from a variation in the friction force. It can originate from the difference of dynamic and static friction coefficient or varying normal force caused by for example resonances, and can occur on many different length scales. Therefore the source mechanism may be seen as a local or global phenomenon of the surface in contact. For further discussion of stick-slip phenomena see for instance Akay (2002); Sheng (2008); Persson (2000); Chen et al. (2005). 2.3 Brake noise classification Despite the multitude of brake noise phenomena, it may be argued that all of them can be classified as belonging either to squealing and/or roughness friction-induced noise. However, it is also common to classify brake noise according to where in the frequency spectra the noise phenomenon can be expected. For instance, Akay (2002) uses this kind of classification which serves as a good tool to find a common name of the phenomenon and thus, the appropriate literature on the subject. In fact, there are

21 2.3 Brake noise classification 11 many different names of brake noises, sometimes without a clear and unique correspondence. Most attention concerning brake noise has been focused on the high-pitch tonal-noise phenomenon, such as, the disc brake squeal, (DBS) phenomenon (out of thousands see Chen et al. (2005); Papinniemi et al. (2002); Kinkaid et al. (2003); Hoffmann and Gaul (2008)). Another, scarcely studied, tonal brake noise problem is the moan phenomenon (to the knowledge of the author the core of the literature is Nack and Joshi (1995); Gugino et al. (2000); Wang et al. (2003); Kim et al. (2005); Kim and Park (2006); Hoffmann and Gaul (2008)). Common for these tonal brake noise problems are that they are considered to have a strong dynamic coupling to the supporting structure. This coupling is known to be a non-linear feedback between structural dynamics and unsteady surface contact, Sheng (2008); Chen et al. (2005), where structural resonances create an unstable friction force that feeds energy back into the resonance. For instance, DBS occurs when a circumferential mode of the disc is triggered, and a dynamic force is created at the same frequency as the mode, resulting in an unstable system. In addition, there are many other brake noise phenomena which may not arise from an unstable non-linear feedback phenomenon, which has to some extent been studied in literature, for example (hot and cold) judder and roughness noise (wire brush). Judder occurs when the disc exhibits non-uniform behaviour in the circumferential direction, e.g. disc thickness variations Sheng (2008); Hoffmann and Gaul (2008). However, the roughness brake noise problem seems not to have been studied much in the literature. The source may be described as a broadband rubbing/roughness/scratch noise. Its broadband nature, compared to the tonal nature of for instance DBS, suggests a fundamental difference of the generating mechanisms. Perhaps the most prominent is that the roughness phenomenon does not exhibit a strong link between structural resonances and the source mechanism itself. A supporting illustration comes from flow acoustics, where steady mean flow whistle noise, such as the tones from a flute, are by necessity considered as non-linear feedback mechanisms. In contrast, noise sources of broadband character, are often considered to have a weak link between system resonances and sound generating mechanism, e.g. the broadband part of a turbulent sound generated by a fan Carley and Fitzpatrick (2000); Powell (1961) Roughness noise theory The surface roughness and the sliding speed are two very important parameters when characterising roughness noise. How the surface roughness of the interacting bodies affects noise generation has been studied for quite some time. For instance, Yokoi and

22 12 2 Brake noise and friction-induced sound and vibrations Nakai (1982) used a so called pin-on-rim setup where they pressed a steel rod onto an unlubricated rotating disc. The surface roughness of the disc was varied between measurements, and they concluded that there is a correlation between increasing surface roughness and an increase in sound pressure and vibration levels in the system. They found that the noise could be predicted with the simple formula, Eq. (2.1): Lp (db) = 20 log 10 ( H H ref ) m, (2.1) where H is a statistical value of the surface roughness, and m = 0.8 for the overall value of the sound pressure level, (SPL) and m = 1.2 for the peak SPL for the rod resonance frequencies. They showed that there is a strong correlation between increased sliding speed and the sound and vibration levels observed in the system. The relative sound pressure level change due to increasing speed could then be approximated by the relation: Lp (db) = 20 log 10 ( V V ref ) n, (2.2) where V is the sliding speed and n is a value that can range between 0.6 and 1.1. The correlation between surface roughness and noise has also been confirmed Othman and Elkholy (1990) and Ben Abdelounis et al. (2010). Furthermore, Othman and Elkholy (1990) also stated that the correlation is independent of the contact sample size and material. As a matter of fact, also the correlation between speed and noise stated by Yokoi and Nakai (1982) has been confirmed by Smyth and Rice (2009); Ben Abdelounis et al. (2010). Moreover, Smyth and Rice (2009) showed that the sliding speed had no effect on the frequency content of the roughness noise, and Ben Abdelounis et al. (2010) showed that the noise dependency on the surface roughness and the sliding speed could be separated. They found that the problem may be modelled using the same variables as Yokoi and Nakai (1982), but using 0.8 m 1.16 and 0.7 n 0.96, as Lp (db) = 20 log 10 (( V V ref ) n ( H H ref ) m ). (2.3) In addition, Othman et al. (1990) have stated that The magnitude SPL is sensitive to variation in contact load; increasing the contact load tends to increase the SPL and vice versa. In that paper a spring stylus was run over a rough surface and the roughness was estimated from the sound generated.

23 CHAPTER 3 Interior brake roughness noise This chapter presents an experimental study of interior brake roughness noise. The experiments were conducted in a laboratory environment, the test object was a small passenger car with disc brakes. The study showed that the problem of interior roughness noise can be viewed as a structural-borne noise problem well correlated to both the vehicle speed and the brake pressure. 3.1 Experimental setup The experiments discussed here were performed under laboratory conditions, where a small passenger car was put on rollers (see Fig. 3.1). The test object was selected for its size and weight, based on the hypothesis that roughness noise is more prominent for a lightweight vehicle. The main part of this study consisted of several noise and vibration recordings, for different vehicle speeds and brake forces. In each measurement the brake force and vehicle speed were kept constant through the recording, and all noise and vibration signals were acquired simultaneously. The sound pressure was recorded both inside the passenger cabin and close to the brake system. The accelerations were recorded at numerous locations in the brake system. For the sake of conciseness the results herein are from measurements conducted while driving only the left front wheel of vehicle. Detailed explanation of the setup and the measurements can be found in paper A, together with a discussion of sources of errors in the measurements. 13

24 14 3 Interior brake roughness noise Figure 3.1: Photograph of the full vehicle experimental rig. 3.2 Results and discussion Brake pressure It is clear from the observations made in the experimental study that the interior roughness noise may be considered as a broadband phenomenon (see paper A). The measurements show that increased brake pressure leads to a broadband increase of the interior SPL. This may also be observed concerning the vibration levels of the brake pad (see paper A). In Fig. 3.2 the normalised total (0.1-1 khz) levels of the acceleration of the calliper (X-,Y- and Z-directions) and interior SPL are plotted as a function of brake pressure, the corresponding vehicle speed is 2.9 km/h and all curves are normalised to the acceleration level of the calliper in Y-direction for a brake pressure of 1/2 bar and speed of 2.9 km/h. Two visualisation lines are added, the dashed line shows the corresponding 3 db per doubling of brake pressure slopes i.e. linearly proportional brake pressure (P), and the dashdotted line shows the slope that is proportional to the brake pressure

25 3.2 Results and discussion 15 Normalised Level 0.1 1kHz [db] Visualisation line P Visualisation line P 2/3 Interior SPL Calliper x Calliper y Calliper z Brake pressure (log scale) [bar] 10 Figure 3.2: Total normalised Levels between 0.1-1kHz, Function of brake pressure. Interior SPL, calliper acceleration levels, circles interior SPL, calliper acceleration in Z-,Y-,X-direction, crosses, stars and triangles respectively, with the corresponding vehicle speed of 2.9 km/h, normalised with the acceleration level of the calliper Y- direction for a brake pressure of 1/2 bar and speed of 2.9 km/h. Two visualisation line are added, the dashed line show a 3 db per doubling of brake pressure slope i.e. linearly proportional to the brake pressure (P), the dashdotted line slope is proportional to the brake pressure as P 2/3. as P 2/3. Note that in Fig. 3.2 the brake pressure is plotted in a logarithmic scale. As discussed in the previous chapter, increasing the brake pressure while keeping the sliding speed constant will result in an increase of the interior noise in the vehicle and the vibrations of the brake system. Based on the assumption that the vibro-acoustic frictional source is proportional to the stored elastic energy released in the breakaway when the asperities break loose, it could be argued that the source level should be linearly proportional to the contact pressure and hence increased by 3 db per doubling of the contact pressure. Indeed, as may be seen in Fig. 3.2, the total (0.1-1 khz) acceleration level of the calliper in the vertical (Z-) direction follows this quite well. On the other hand, the slopes of the interior SPL and the calliper acceleration in the X- and Y-directions appear to have slopes that are proportional to the brake pressure as P 2/3. Keep in mind that the con-

26 16 3 Interior brake roughness noise tact plane of disk and pad is the XZ-plane and the sliding direction is in the Z-direction. In an attempt to describe the vibro-acoustic source, one could argue that the vibrations levels generated in the contact zone should be proportional to the stiffness (or to the resistance in motion). The broadband character of this source may then be explained as local stiffness variations in the contact zone, where the overall level is considered to be proportional to the DC component of the stiffness. Assuming that the DC component of the stiffness in the normal direction could be represented by Hertz contact theory and the tangential resistance to motion by an elasto-plastic analogy of the Coulomb s friction law, then, for Hertz contact theory the normal elastic contact stiffness is proportional to the contact force (Johnson (1985)) as K n Fn 2/3, where K n is the normal contact stiffness and F n is the normal force. From the elasto-plastic analogy of the Coulomb s friction law one could then argue that the friction force can be viewed as a plastic stiffness as K plastic t F n µ where µ is the friction coefficient. From this the following vibration level relation could then be formed, La (db) = 10 log 10 ( Pα ), (3.1) P α ref where P ref is an arbitrary reference contact pressure, and α = 2/3 for the levels in the normal direction to the contact zone and α = 1 in the tangential (or sliding direction). If the contact between pads and disc could be described by Hertz theory and Coulomb s friction law then it is interesting to see that there seems to be a link between the normal contact stiffness (Hertz theory) and the interior SPL. It should also be noted that the interior SPL has a slope very similar to both the calliper accelerations in X- and Y- direction. Possible explanations for these behaviours, see Fig. 3.2, might be: i) most of the vibro-acoustic energy is realised when asperity break loose in the sliding direction in the frictional contact, and due to the orientation (sliding direction coincides fairly well with the vertical direction) of the brake system the vertical direction is mostly excited, ii) the vibration levels of X- and Y-directions are not uniquely dependent on the normal contact stiffness, instead they might be dependent on the plastic frictional contact stiffness, iii) the interior SPL is more related to calliper vibration levels in the X- and Y-directions than in the Z-direction. Possibly the suspension with shock absorber isolates and absorbs these vibrations more efficiently. Based on the type of brake-discpad assembly, on the suspension system (transfer path) and on the hypothesis from Eq. (3.1), a relation for the interior SPL may be written as,

27 3.2 Results and discussion 17 Lp (db) = 10 log 10 ( P2/3 + γp ), (3.2) P 2/3 ref + γp ref where γ = H t /H n is a constant that gives a measure of the relative influence of each excitation direction to the interior noise H t = P sound /Ft exci and H n = P sound /Fn exci. This constant γ could be experimentally determined. From the shaker measurements discussed in paper A the constant was determined as, γ = 1000 H t ( f ) 2 d f H n ( f ) 2 d f (3.3) where f is the frequency in Hz and H n and H t is the measured transfer functions of sound pressure over excitation force in Y- and Z-direction respectively. In the current investigation γ was found to be In Fig. 3.3 the interior SPL is plotted together with three visualisation lines from Eq. (3.1) with α = 2/3, α = 1 and Eq. (3.2) using the experimental γ = These data are recorded for the brake pressure range of bar since this was the range in which the best agreement for the SPL could be found in Fig A clear correlation for the interior SPL may be observed in Fig. 3.3 with the combined model. It should be noted that the extreme values, a zero γ or an infinite γ, correspond to a model proportional to only P 2/3 or P respectively. In the current investigation the experimental γ was found to be small and hence the interior SPL had a slope close to the curve for P 2/3. However, having said that, for a different design of the brake and suspensions system a different γ would most likely have been found. Thus, from the combined Hertz contact theory and a Coulomb s stiffness analogy model of the interior SPL, it is suggested that the orientation of the brake and suspension system is an important design parameter for the reduction of interior roughness noise System loading Yet another operational parameter having a significant influence on both the interior noise and acceleration levels of the brake system, is the applied brake pressure. This is quite obvious from the clear trend that can be seen with increased brake pressure and a corresponding increased acoustical response (see Fig. 9 paper A). This dependence between interior noise and the contact pressure has to the knowledge of the author not been extensively investigated before. Thus, the strong link between the brake pressure and the interior noise is a new and original result arising out of this work.

28 18 3 Interior brake roughness noise Normalised Level 0.1 1kHz [db] Visualisation line P Interior SPL Visualisation line P 2/3 +P 0.1 Visualisation line P 2/ Brake pressure [bar] Figure 3.3: Total normalised Levels between 0.1-1kHz, Function of brake pressure. Interior SPL, with the corresponding vehicle speed of 2.9 km/h, normalised SPL for a brake pressure of 2.5 bar and speed of 2.9 km/h. Three visualisation line are added, the dashed line show a 3 db per doubling of brake pressure slope i.e. linearly proportional to the brake pressure (P), the dashdotted line slope is proportional to the brake pressure as P 2/3 and the solid line show the combined model in Eq. (3.2). Observing the shifting in the frequency of the peaks in Fig. 9(b), paper A, together with the amplification in the different frequency bands, see Fig. 9(a), paper A, indicates that there might also be a process where the vibro-acoustic system is affected by the brake pressure. Hence, not only the source mechanisms are affected by the contact pressure but also the actual transmission paths between the source and the interior SPL. One possible reason for this behaviour shown in Fig. 9, paper A, could be the coupling conditions in the contact zone itself, that is, the increased brake pressure leads to stronger coupling and hence the vibro-acoustic system is changed. Another hypothesis might be that an increased brake pressure will also lead to an increased brake force, and this force must be carried by the calliper. This may pre-load the bushings and result in geometrical non-linearities when different connectors change relative position, thus also changing the vibro-acoustic system. In Fig. 3.4 the acceleration levels of the outer brake pad are shown, with Fig. 3.4(a)

29 3.2 Results and discussion 19 showing a zoom of the three highest peaks that could be seen in Fig. 9(b), paper A, for the three different non-zero brake pressures. In Fig. 3.4(b) the acceleration levels are drawn for three different situations, the thick black and the dashed lines are for the two measurements with a shaker excitation. The thick black line is for the case of a 110 kg external load and was used to simulate the brake force at high brake pressure. The dashed line is for the case when no external load was used, for the same amplitude of electric signal to the shaker. In both cases the brake pressure was kept at 5 bar. Since an adaptor was necessary to enable shaker excitation, the thin solid line in Fig. 3.4(b) is included to show that the system still is fairly intact, despite the external load. The thin solid line shows the results from similar measurements as in Fig. 3.4(a), except that the adaptor was used, when 5 bar brake pressure was used for a vehicle speed of 1.9 km/h. For further information on the adaptor setup see paper A. Interestingly enough, when comparing the curves in Fig. 3.4, there is a shift upwards of the two highest (in frequency) peaks with the load (Fig. 3.4(b)) and this shift seems to be of the same order of magnitude as may be observed in Fig. 3.4(a). Hence, the hypothesis that the static brake force loading may change the vibro-acoustic system properties appears to be valid. The static loading influence on the interior noise is highly dependent on the vibroacoustic transfer path properties, thus governing how much noise is transferred into the vehicle compartment for a given excitation. The effect of the static loading from the brake force is modelled using measured transfer functions (see Eq. (3.4)). These were measured (calliper acceleration to interior sound pressure) with and without the 110 kg load and an estimate of the interior noise was made from the acceleration of the calliper. Three different transfer functions were measured separately with a shaker, exciting the calliper in the three coordinate directions and simultaneously measuring the acceleration, in the direction of the excitation force, in the excitation point and the interior sound pressure. The interior noise was estimated by using acceleration (all three coordinate directions) measured on the calliper for the case of 5 bar brake pressure and 1.3 km/h vehicle speed. The total level was estimated by treating each coordinate directions as being uncorrelated. In Fig. 3.5 the total level of the transfer function estimation with and without the external load is shown together with the directly measured interior SPL. The plot is zoomed in at the frequency region between Hz since it is the region where the strongest amplification of interior SPL can be observed for increased brake pressure (see Fig. 9(a) paper A). The interior noise

30 20 3 Interior brake roughness noise [bar] 3.3 [bar] 1.8 [bar] Level [db re. 1 μm/s 2 ] Frequency [Hz] (a) Acceleration levels of the outer pad in the disc rotational axis direction, when applying external pressure to the brake liquid. dashed line 1.8 bar, thin line 3.3 bar and thick line 5 bar. Vehicle speed 1.3 km/h. No adaptor, (Zoomin from Fig. 9(b) paper A, 5 Hz resolution Shaker, 0 [kg] Shaker, 110 [kg] Roller, 0 [kg] Level [db re. 1 μm/s 2 ] Frequency [Hz] (b) Acceleration levels of the outer pad in the disc rotational axis direction. Dashed line shaker excitation of brake calliper in disc rotational axis direction no external load, no rotation, 5 bar pressure. Thick line shaker excitation of brake calliper in the disc rotational axis direction a 110 kg external load, no rotation, 5 bar pressure. Thin line roller excitation, 5 bar. All using an adaptor, 5 Hz resolution Figure 3.4: Graph comparing the frequency shifts, effect of an external force and the brake force, for the pad acceleration in normal direction to disc.

31 3.2 Results and discussion 21 was then estimated using p est 2 = a rol 2 H sha 2 p, a x + a rol y 2 H sha 2 p, a y + a rol z 2 H sha 2 p, a z x (3.4) where p est is the estimated interior sound pressure and a rol is the acceleration measured in X, Y or Z direction when braking with the roller. H sha p, a is a transfer function between interior sound pressure and driving point acceleration (X, Y or Z direction) either with static loading or with out loading, measured with a shaker. The estimated interior noise for the two cases is compared with the measured interior noise from the same measurement as when a rol was recorded in (see Fig. 3.5). There are of course limitations with this procedure of estimating the interior noise and the assumptions made. The first assumption implies that the brake system (calliper, disc and pad assemble) could be treated as a rigid body suggesting that the movement of the system could be described using six degrees of freedom (DOF), the three spatial directions and the three rotations around the corresponding axis. These DOFs could ideally be represented using six independent shaker measurements of the brake system. Here, only three independent shaker measurements were used. Secondly, assuming that the different shaker measurements could be conducted separately, also implies that all measured directions could be treated as being uncorrelated. These assumptions may appear very crude, since there will of course be correlation between the different DOFs, and a three DOF representation of the motion of the brake system may be a crude approximation. But, interestingly enough the results presented in Fig. 3.5 gives a good representation of the interior noise despite these simplifications. Furthermore, it appears that the model using the loaded transfer functions have a better representation of the directly measured results. Hence, it could be argued that the system loading effect discussed above has an influence on the transfer path problem. Moreover, it has also been demonstrated that there is a clear link between vibrations in the calliper and the noise inside the vehicle Vehicle speed To investigate the influence of the vehicle speed, Fig. 3.6 shows the sound pressure, inside the vehicle (a), and acceleration levels, of the brake pad (b), respectively when 5 bar pressure is applied to the brake liquid for the three speeds 1.3, 1.9 and 2.9 km/h. The plotted pad vibration was measured in the Y-direction (inward disc rotational axis direction). What can be noted in Fig. 3.6(a) is that changing the vehicle speed essentially corresponds to a broadband increase of the overall SPL inside the plotted

32 22 3 Interior brake roughness noise Measurement Estimation without load Estimation with load SPL [db re. 20 μpa] Frequency [Hz] Figure 3.5: Measured and estimated SPL for 5 bar brake pressure and 1.3 km/h using the adaptor. Thick line, direct measurement. Thin line, estimation using unload transfer function. Dashed line, estimation using load transfer function. frequency range. Consequently, no frequency band seems to be affected more than another, as was the case in Fig. 3.4 for different brake pressures. It is futhermore clear from Fig. 3.6(b) that the peaks in the frequency response do not shift with increasing speed. Roughness noise is probably present in all vehicles with solid material friction brakes. However, it is mostly masked by other noise sources. To the knowledge of the author most of the cases where wire brush (roughness) noise is reported as a problem are for low vehicle speed when background levels are lower, so the effects of tyre-road and other sources are of course important in how the noise event is perceived. In this investigation the masking effects were minimised to allow for a better picture of the generation of the excitation itself. One of the goals of the present work was to see how the interior roughness brake noise in the vehicle correlates to the sliding speed. Furthermore, it gave a possibility to study how results from experimental studies of simplified setups such as the pinon-rim setup by Yokoi and Nakai (1982), may be related to the problem of an entire brake system including the vibro-acoustic transfer path problem. In the literature it may be found that statistical values describing the surface roughness are important pa-

33 3.2 Results and discussion [km/h] 1.9 [km/h] 1.3 [km/h] SPL [db re. 20 μpa] Frequency [Hz] (a) Sound pressure levels Level [db re. 1 μm/s 2 ] [km/h] 1.9 [km/h] 1.3 [km/h] Frequency [Hz] (b) Acceleration levels of brake pad,y-direction Figure 3.6: Levels for different speeds. Thick black line 1.3 km/h, dashed line 1.9 km/h and thick gray line 2.9 km/h. Brake pressure in all cases 5 bar.

34 24 3 Interior brake roughness noise rameters that govern the frictional noise source mechanism, but measurement results in literature have also suggested that surface roughness parameters and sliding speed parameters may be independently studied, Ben Abdelounis et al. (2010). In other words, sliding speed as a noise generating parameter may be studied without knowledge of the parameters describing the surface roughness. In addition, the size of the contact pairs should not affect the behaviour. It may be seen in Fig. 3.6 that there is indeed a broadband increase of both interior noise levels and brake system vibrations. As stated by Smyth and Rice (2009), the noise dependence on sliding speed should not affect the frequency content, but only increasing the overall level of the noise, which is verified for the current problem see Fig It may also be argued that sliding speed dependence of the noise problem is simpler to model than the brake pressure effect, as there seems to be little system altering effect associated with change of the speed. In Fig. 3.7 the interior total (0.1-1 khz) normalised SPL is shown as a function of the vehicle speed for the brake pressure 1.3 and 5 bar respectively, the normalisation chosen as the total SPL from the lowest speed using the same brake individual brake pressure. The limits from the two equations Eq. (2.2) (Yokoi and Nakai (1982)) and Eq. (2.3) (Ben Abdelounis et al. (2010)) are also included in the graph. In fact, it can be seen that even though these formulas were derived from simplified measurements and the results from this study is for a much more complex setup, the results correlate surprisingly well. 3.3 Conclusions and findings The main results from measurements performed on a full vehicle laboratory test rig are presented. The test rig was designed by the author for the purpose of full vehicle in-situ measurements of brake noise. Evidently it is possible to reproduce the effects of friction-induced noise in a vehicle using this test rig Findings The vibro-acoustic excitation of the roughness noise vibrations may be divided into two components. That is, i) vibro-acoustic excitation in the sliding direction, in these measurements a linear proportionality to the brake-pressure was found, ii) the vibroacoustic excitation in normal direction of the contact plane, in these measurements a non-linear proportionality to the brake-pressure was found (the brake pressure to the power of 2/3 is proportional to the acceleration levels). The proportionalities in both

35 3.3 Conclusions and findings 25 Normalised SPL 0.1 1kHz [db] Ben Abdelounis et al. Yokoi and Nakai 1.3 [bar] 5.0 [bar] Roller speed [km/h] Figure 3.7: Total normalised SPL between khz as a function of roller speed. Crosses, interior noise for a brake pressure of 1.3 bar. Circles, interior noise for a brake pressure of 5 bar. Dashed line, limits from Eq. (2.3). Dotted line, limits from Eq. (2.2). Normalisation total SPL of individual signal and 1.3 km/h. cases can be argued to be linked to the stiffness, where the resisting force (stiffness) in the sliding direction is directly proportional to the brake pressure which is consistent with Coulomb s friction law. The contact stiffness in the normal direction is proportional to the contact pressure to the power of 2/3, according to the Hertz contact theory. Combined models of the Hertz contact theory and the Coulomb s friction law may then be used to predict the relative total interior SPL change due to a change in brake pressure Conclusions It is concluded that the vehicle phenomenon of interior disc brake roughness noise (wire brush) is a purely structure-borne noise problem, hence not air-borne (see paper A). Moreover, the noise phenomenon is dependent on the brake pressure and the vehicle speed. Both interior SPL and brake system vibrations increase with increasing brake pressure and sliding speed. Furthermore, increased brake pressure may lead to system altering effects, and this loading of the system has to be included in order to accurately model the problem.

36

37 CHAPTER 4 Component mode synthesis approach This chapter introduces a component mode synthesis approach where the undeformable attribute of an interface between a soft and a stiff connector is directly enabled in the formulation of the coupling condition. It is based on the assumption of undeformed coupling interfaces and on the classic Craig-Bampton method. It is shown that the computational cost can be greatly reduced using the undeformed coupling interfaces approach compared both to a direct finite element solution as well as to the classic Craig-Bampton method. For a system built of components of the same properties as the rubber bushing/ linking arm assembly, the accuracy is shown to be very good from an engineering perspective (less than 1% error). 4.1 Background From the experimental study (in previous chapter) it was concluded that disc brake roughness noise may be viewed as structure-borne noise phenomenon. Furthermore, system altering effects of the brake force were observed for the transfer path system. These two findings lead to the conclusion that a deeper knowledge of the vibro-acoustic system characteristics of the vehicle suspension was needed. This part of the thesis introduces an approach to model multifaceted systems using a finite element approach; the number of degrees of freedom is reduced to a small number of local normal modes and a set of six undeformed interface coupling modes per coupling interfaces. An example of such a multifaced systems is the suspension system of a vehicle, with this approach problems associated with the rubber bushings and complex geometries are handled. This is shown to lead to computationally fast and accurate modelling of such a system. 27

38 28 4 Component mode synthesis approach Introduction Herein, a reduction technique for modelling of suspension systems is suggested, having the potential of overcoming the problem of an unnecessary large number of DOFs in the mathematical description of the vibro-acoustic field. Reduction methods in structural dynamics (vibro-acoustics) often use the concept of modes: a set of normal modes (eigenmodes) are generated from an eigenvalue problem. The theory of modal superposition states that a reduced set eigenmodes can be used to span an approximate solution of the original problem. The choice of the set of included modes in the approximated solution may come from a physical motivation. The normal choice is only to include modes with eigenfrequencies below a particular upper frequency limit. The theory of how reduced subsystems can be coupled together is commonly referred to as component mode synthesis (CMS). CMS can be described as a method where the local behaviour of individual substructures is described by a set of reduced local eigenmodes. Force and displacement continuity between the substructures are enforced by a set of coupling functions (constraint modes). There exists several different versions of the CMS method, which are distinguished by the use of local eigenmodes: fixed interface, Craig and Bampton (1968); free interfaces, (Rixen (2004); hybrid, MacNeal (1971)). Common to all is that continuity between substructures is ensured by a static condensation. The static solution of the inner DOFs is computed by successively prescribing either a unit displacement or unit force (fixed and free interfaces respectively) at the coupling interface one DOF at a time. This procedure generates a solution vector for each interface DOF and these vectors may be seen as coupling modes. Together with the local eigenmodes, the solution of the fully coupled problem may now be spanned. For a deeper description of the different formulations, the reader is referred to the review paper of De Klerk et al. (2008). A limitation of classic CMS methods such as the Craig-Bampton (C&B) method is that the reduction order is limited by the size of the coupling interface, hence this method is only appropriate for systems with small coupling interfaces, Junge et al. (2009); Tran (2009); Balmes (1996). Furthermore, non-conforming meshes between different components may appear e.g. because different components are developed by different groups of engineers, Farhat and Geradin (1992). The problem with nonconforming mesh interfaces is usually dealt with by introducing additional constraints in the form of so called Lagrange multipliers (see for instance rix (1998); Farhat and Geradin (1992)).

39 4.2 Theory 29 Research on the interface reduction is a research area which attracted considerable interest, for instance, Tran (2001, 2009); Balmes (1996); Herrmann et al. (2010). Most of the methods use static condensations of the inner modes on the global problem. Thus, the new system only comprises interface DOFs, that may be used to generate yet another set of basis function by solving the new eigenvalue problem. However, a physically motivated truncation criterion based on the eigenfrequency cannot be used to choose a reduced set of interface DOFs eigenmodes basis function. Instead, the method is limited to criteria as e.g. evaluation of the strain energy of each mode, ger (2000) or singular value decomposition Balmes (2005); Herrmann et al. (2010). In this thesis a physically motivated technique of interface reduction is presented. The technique is proposed for coupling between soft and stiff parts, such as the connection to rubber bushings in the vehicle suspension system. The physical reasoning is based on the assumption that an interface between a soft and a stiff part will have an approximately undeformed interface shape. Hence, the displacement of the interface can be described by the six rigid body motions (three orthogonal spatial directions and the three rotations around these axes), van der Valk (2010). This method also allows for a substantial reduction of the original problem and completely eliminates the problem of non-conforming meshes: only a conforming coordinate system is needed when generating the undeformed interface displacement functions. 4.2 Theory General problem In Fig. 4.1 the system used in this section is defined to give an basic understanding of the theory behind the modelling approach. The vibro-acoustic displacement field u(x) is defined in the domain Ω which is entirely enclosed by the boundary Ω. The boundary is subdivided into a coupling interface boundary Σ and the remaining boundary Γ. Λ is the union of boundary Γ and the domain Ω. The continuum mechanical displacement field u(x) may be written in a FE discretized representation. The displacement DOF vector may also be subdivided into U Σ and

40 30 4 Component mode synthesis approach Figure 4.1: The vibro-acoustic displacement u(x) and the body force g(x) in the domain Ω where Σ = Ω \ Γ, and Λ = Ω Γ. x = [x 1 ; x 2 ; x 3 ] is the position in space U Λ, where U Σ corresponds to the nodal displacement vector belonging to the coupling boundary Σ and U Λ is the nodal displacement vector belonging to Λ. The corresponding matrix representation of the problem may be written as [ ( ) ( )] ( ) ( ) KΣΣ K ΣΛ ω 2 MΣΣ M ΣΛ UΣ FΣ =, (4.1) U Λ F Λ K T ΣΛ K ΛΛ M T ΣΛ M ΛΛ where K and M are the stiffness and mass matrices respectively, U and F are the displacement and force vectors respectively Change of basis This section explains how to generate the (nodal) displacement vector representation of the basis functions used in the change basis CMS approach. The Classical C&B substructuring method uses one set of basis function to span the local DOF of each substructure, and another set of functions for the coupling of the structures. The two sets of functions will be herein referred to as local modes and coupling modes Local modes The local modes are generated by a constrained eigenvalue problem. On the coupling interface Σ a homogeneous (zero) Dirichlet condition is imposed as U Σ = 0. Together

41 4.2 Theory 31 with a zero force conditions on Γ. Hence, the eigenvalue problem may be written as ( KΛΛ ω 2 M ΛΛ ) UΛ = 0 (4.2) The solution of Eq. (4.2) gives the eigenvalues ω 2 n and eigenvectors φ n. The total displacement vectors of the constrained eigenvalue problem, including the constrained coupling interface DOFs as zeroes, is written as in Eq. (4.3) where each column corresponds to a displacement eigenvector: [ Φ = ] 0 = φ Λ,(1) φ Λ,(n) [ 0 Φ Λ ] (4.3) This is the first set of basis functions used in the projection from local DOFs to generalised DOFs. Coupling modes To allow for a kinematic coupling to an adjacent substructure, basis functions which are non-zero on Σ are needed. These are constructed by static solutions of M coup different, linearly independent boundary value problems having different non-zero prescribed displacement conditions on Σ. These boundary constraints are imposed via the FE displacement vector U Σ = ψ Σ,(m) where F Σ = 0 and F Λ = 0. Denoting the corresponding solution of U Λ of the remaining displacement DOFs by U Λ = ψ Λ,(m), the mth solution is obtained by solving the static problem ( ) ( ) ( KΣΣ K ΣΛ ψσ,(m) 0 = (4.4) ψ Λ,(m) 0) hence, K T ΣΛ K ΛΛ ψ Λ,(m) = K 1 ΛΛ KT ΣΛ ψ Σ,(m). (4.5) The M coup solutions of Eq. (4.5) may be organised as shown in Eq. (4.6) where each column corresponds to the displacement vector (or mode ) for a given imposed displacement vector ψ Σ,(m) : [ ] [ ] ΨΣ ψσ,(1) ψ Ψ = = Σ,(m) (4.6) Ψ Λ ψ Λ,(1) ψ Λ,(m) This is the second set of basic function used in the projection from local DOFs to generalised DOFs.

42 32 4 Component mode synthesis approach Modal projection If the two projection basis functions of the local and the coupling modes are assembled the total projection basis Θ reads Θ = [Ψ, Φ] = The projection of each component may then be written as [ ] ΨΣ 0. (4.7) Ψ Λ Φ Λ S = Θ T K Θ, W = Θ T M Θ, (4.8) and G = Θ T F, U modal = ΘQ, (4.9) with the following transformed system of generalised coordinates [ ( ) SΣΣ 0 0 S ΛΛ ω 2 ( WΣΣ W T ΣΛ W ΣΛ W ΛΛ )] ( ) QΣ = Q Λ ( ) GΣ. (4.10) G Λ If the eigenvectors φ Λ in the projection basis are mass normalised then W ΛΛ is an diagonal matrix with only ones in the diagonal, S ΛΛ is a diagonal matrix with the eigenvalues ω 2 n in the diagonal. The mass and the stiffness submatrices of the interface DOFs, W ΣΣ and S ΣΣ are full matrices, the coupling (interface or inner DOFs) mass submatrix (W ΣΛ ) is also full. Hence, also the global mass matrix W is full. However, the fast solution properties of the diagonal matrix W ΛΛ can still be utilised by condensing the solution to the interface DOFs. Classic Craig-Bampton In Eq. (4.10) the general form of the classic C&B method is formulated. For the special case when Ψ Σ is chosen as the identity matrix I then Eq. (4.10) corresponds exactly to the classic C&B method, which is based on the calculation of the inner response (on Λ) for the case of successive unit displacement of each DOF (on Σ), while keeping all other DOFs on the interface fixed. This procedure of repeating the calculation of Eq. (4.5) as many times as there are DOFs on Σ, is equivalent to replacing Ψ Σ with the unit matrix. Hence, no reduction of interface (Σ) DOFs is made, since multiplying by a square unit matrix will not change the dimension of the resulting matrix.

43 4.3 Results and discussion 33 Rigid interfaces six DOFs So far only a general form of the classic C&B CMS has been presented. In the presented theory, the possibility of using any function that can be described on Σ has been shown for the calculation of the static coupling modes ψ. As stated before in the classic formulation of the C&B CMS, an unreduced interface formulation is used. Here, a different set of displacement conditions are constructed to generate a different set of static coupling modes using Eq. (4.5). This approach allows for a truncation of the coupling interface. At this point the UCI approach is introduced. The deformations of an interface between two connected bodies could be argued to be governed (among other parameters) by the relative stiffness of the bodies. If there is a large relative difference of the stiffness, then the coupling interface may translate and rotate, but keeping its original shape almost undeformed. The stiffer body interfaces behave almost as a free boundary and the softer body interfaces behave similar to a spatially prescribed translation and rotation, that is, identical to the stiffer body translation and rotation. These assumptions of undeformed coupling interfaces allow for a restriction of coupling modes to represent six different rigid motions of the interface, e.g. three translations and three rotations around a common rotation point. In all other aspects these coupling modes correspond to the classic C&B coupling modes. The main advantage of this approach is that the number of coupling modes is reduced from the number of FE DOFs associated to the coupling boundary Σ, to six. Another important feature is that mesh compatibility is not required, only the translation directions, rotation axes and rotation point have to be compatible. Formally, not even the geometry has to be compatible. For a deeper description of the CMS UCI approach see paper B. 4.3 Results and discussion In order to evaluate the usefulness of the proposed approach, a test structure is implemented. The question is whether the approach with the UCI may be used for vibroacoustic modelling of built up substructures of fundamentally different stiffness.

44 34 4 Component mode synthesis approach Figure 4.2: The test structure used in this thesis consisting of seven substructures. Boundary conditions are indicated by R (Rigid or Fixed) or F (unit surface force condition in all the spatial directions) Test structure The test structure herein used is shown in Fig. 4.2, the properties of which are chosen to resemble the vibro-acoustic problem of a triangular linking arm in a vehicle suspension system, the linking arm often being connected to the stiff vehicle body in three positions via rubber bushings. It consists of a cross (substructure 3 in Fig. 4.2) in order to resemble the linking arm. The rubber bushings are included as blocks (substructures 2, 4 and 6 in Fig. 4.2) and three connecting parts are included (substructures 1, 5 and 7 in Fig. 4.2) to mimic the stiffness of the vehicle body. A more detailed description of the test object can be found in the appended paper B Vibro-acoustic response From Figs. 5 and 6 in paper B, three different frequency regions may be recognised. These are defined as low (0-100 Hz), mid ( Hz) and high ( Hz), mainly for evaluation purposes. In the low frequency range there are rigid body resonances behaviours of substructure 3. At mid frequencies the behaviour is mostly governed by the mass law, and at high frequencies there are three elastic modes of the cross that governs the response. In Fig. 4.3 the solution (spatial root mean squared value (RMS) of the displacement magnitude of substructure 3) is shown for different Young s modulus of the soft connectors. When the relative Young s modulus moves closer to unity, then the first rigid body resonances moves up in frequency

45 4.3 Results and discussion 35 Displacement [db re. 1 pm] E rel = 10 6 E rel = 10 4 E rel = 10 2 E rel = Frequency [Hz] Figure 4.3: The vibro-acoustic response, integral of the total displacement of substructure 3 for four different Young s module of the soft connectors, presented as the relative Young s modulus. (see Fig. 4.3) Evaluation of the approach Relative Young s modulus In Fig. 4.4 the overall relative difference, in the defined frequency ranges (low, mid and high), is shown for a changing stiffness of the soft connectors. In other words, all material properties are kept constant except the Young s modulus of the substructures 2, 4 and 6. What may be seen in Fig. 4.4 is that all the results for C&B and UCI have results that may be considered as accurate results from an engineering point of view, with a deviation less than 1%, for the relative Young s modulus E rel less than 10 3 (relative Young s modulus is defined as, E rel = E soft /E stiff ). As expected, the displacements at higher frequencies are the hardest to predict with the UCI approach. This is probably due to the fact that at high frequencies, the size of the interfaces become significant in comparison to the wave length. In this test case, a fairly large interface was used. In real vehicle suspension systems, the interfaces between structures and bushings are usually smaller. Note that the test case evaluated here should be viewed as a proof of concept, verifying the possibility of modelling built up structures with softer and stiffer parts, such as the vehicle suspensions. So far, results have only been shown including

46 36 4 Component mode synthesis approach all modes for the C&B and the UCI approach, which highlight the consequences of the UCI approach compared to the C&B method. It should be noted that the C&B method without any inner reduction should give exactly the same result as the direct FE since only a projection is made Relative difference [ ] C&B Low C&B Mid C&B High UCI Low UCI Mid UCI High Relative Young s modulus [ ] Figure 4.4: The total relative difference between the modal solutions and the direct solution for the three frequency bands, using all inner modes. Function of the stiffness of the soft contactors (function of relative stiffness). C&B method (dashed lines) UCI approach (solid lines). 4.4 Conclusions A component mode synthesis approach that uses undeformed coupling interfaces is proposed. The approach enables a significant reduction of the original problems, where classic CMSs are limited to reduction of DOFs not associated with the coupling interfaces. The approach also overcomes any problem of non-conforming mesh of different components. It is demonstrated that specific systems may be modelled using the suggested approach, giving results that from an engineering point of view are quite acceptable. For the approach itself to be valid, the stiffness of the connecting bodies must be fundamentally different, such as, for the rubber bushing connected to a steel linking arm in a vehicle suspension system.

47 CHAPTER 5 UCI as a framework This chapter outlines the reduced order model (ROM) built using the UCI as proposed in this thesis. The method may be used to reduce the order of a global problem: this is done by subdividing the global system to substructures interacting through a suitable number of UCIs. The local dynamic behaviour of each substructure may then be modelled with a, for that particular problem, best suited description. The feasibility of the method is demonstrated by a sensitivity analysis of the vibro-acoustic power isolation in a vehicle suspension system, comprised of a link arm connected to a vehicle car body through two rubber bushings. 5.1 Background As shown in the previous chapter, the use of an UCI-approach may be a powerful tool in the establishment of a reduced order model (ROM). In this chapter a more general approach to the use of an UCI modelling is formulated. The potential in this method is demonstrated in a parametric analysis on subsystem level, allowing for an analysis of the flow of vibro-acoustic energy through a hypothetical suspension system into a vehicle car body. A hypothetical vehicle suspension system, with a link arm connected to a car body via rubber bushings, is modelled using the UCI-approach. The link arm is modelled by the C&B-UCI-approach, where the inner degrees of freedom (DOF)s are tranformed by a normal mode superposition, using a reduced set of the calculated inner modes. The rubber in the bushings is described by a realistic and frequency dependent visco-elastic 37

48 38 5 UCI as a framework material model. The model has non-proportional damping, which effectively prohibits the direct use of a modal approach. In order to reduce the size of the bushing models, the rotational symmetry of the bushings is used in a reduction of the spatial dimension of the resulting problem, allowing for a computationally efficient two-dimensional finite element modelling Östberg et al. (2010). Furthermore, the UCI enables a straightforward connection between the suspension system and an interface dynamic stiffness matrix of the car body. In this case this dynamic stiffness matrix components were calculated and supplied by a car manufacturer from a full body-in-white FEM. In order to assess the importance of the orientation of the rubber bushings, in the actual installation, and to illustrate the potential of optimising the vibro-acoustic properties of the vehicle with respect to these, a parameter study is undertaken where the mounting orientations of the bushings are varied. The results show that the global vibro-acoustic properties can vary by several orders of magnitude. 5.2 Local substruture models and global assembly F i 1 i 1 Λ i i 2 U i 2 U i 1 F i 2 Figure 5.1: Principal sketch of a substructure Λ i with two interfaces Σ i 1 and Σi 2. The proposed substructuring methodology relies on the assumption of undeformed coupling interfaces, allowing for a dynamic description of each interface in terms of six generalised forces and displacements (of which three are rotational forces {moments} and three rotational displacements {angles}). The two interfaces 1 and 2 of

49 5.2 Local substruture models and global assembly 39 the substructure i, (Fig. 5.1) are described in terms of dynamic stiffness matrices, relating the generalised forces for interface m, F i m = [F i x m, Fi y m, Fi z m, Mi x m, Mi y m, Mi z m ]T, to the generalised displacements, U i m = [U i x m, Ui y m, Ui z m, Φi x m, Φi y m, Φi z m ]T, (T denotes transpose) by [ F i 1 F i 2 ] = [ D i 11 D i 12 D i 21 D i 22 ] [ U i 1 U i 2 ] + [ T i 1 T i 2 ]. (5.1) The first three components of the generalised force vector are the corresponding Cartesian components and the other three are the moment vector Cartesian components. Similarly, the generalised displacement vector has three displacement Cartesian components and three rotational Cartesian components, while T i m accounts for the local force contributions internal to substructures. Furthermore, the interface dynamic stiffness matrix, [ ] D D i i 11 D i 12 D i 21 D i (5.2) 22 is defined, where D i 12 = (Di 21 )T. An assembly of the dynamic stiffness components is readily performed, details may be found in paper C, resulting in the global dynamic stiffness matrix for the idealised vehicle suspension system Global model description A schematic representation of the global assembly of the substrucures can be seen in Fig. 5.2 where; the car geometry is represented by a block, the bushings are represented by the 3D equivalents of the bushing models, and the link is represented by the actual geometry used in the substructure model described below. Bushings 1 and 2 are connected to the car at interfaces 1 and 2 respectively. The axial direction of bushing 1 is aligned with the z-axis, while bushing 2 is aligned with the x-axis. In order to demonstrate the UCI modelling, three different types of sub-models are used to construct the global model. The link arm is modelled using the CMS-UCI explained in the previous chapter. The two rubber bushings are modelled using a UCI-ROM derived from a 2D axisymmetric model using a frequency dependent visco-elastic material model. Finally the car body model is a frequency dependent UCI-ROM from a full car body finite element model. A more thorough explanation of the different models may be found in paper C. One key advantage of any ROM in general and the proposed method in particular

50 40 5 UCI as a framework Figure 5.2: Principal sketch of the assembled system, where the light green Λ 1,(Arm) part represents the link arm, the dark green parts Λ 2,(Bush1) and Λ 3,(Bush2) the rubber bushings and the brown part the car body Λ 4,(Car). The red surface on the far left of the link arm indicates the external force location. is that parameter studies are easily conducted on a substructure level. As an illustration of the potential in the usage of the UCI-ROM, a study is undertaken in which the orientations of the rubber bushings, which are used to connect the link arm to the car body, are varied Results total transmitted power The vibro-acoustic power transmitted into the generic car structure is evaluated. The purpose is to investigate the sensitivity of the vibration isolation provided by the suspension system and its dependence on the rubber bushing configuration. Thus, in the following all discussion referring to transmitted power means the net flow of energy into the car substructure. As excitation, distributed unit traction forces are applied uniformly at one surface (surface indicated with red in Fig. 5.2). The power (P i n) is calculated from the interface displacements (U i n) and traction forces (F i n) vectors for each substructure and coupling interface n = 1, 2 using where denote the complex conjugate and R the real part. P i n = 1 2 R( Fi n iωu i n ), (5.3)

51 5.2 Local substruture models and global assembly Ω x1 Ω y1 Ω y2 Ω y Ω x1 Ω y1 Ω y2 Ω y Powe r int o c ar [W] Powe r int o c ar [W] Fr e que nc y [Hz ] (a) Force excitation in x-direction Fr e que nc y [Hz ] (b) Force excitation in y-direction Figure 5.3: Total power into the car substructure. Four different bushing rotations are shown; range deg for each 3 deg. Rotations around x (blue) and y (green) respectively of bushing 1, rotations around y (grey) and z (red) respectively of bushing 2. Black dashed line shows the original orientation while the solid black line is included to show the individual frequency response (of one of the responses with high power). In Figs. 5.3 the total vibro-acoustic power transmitted into the car structure, and its dependence on the bushing (tilting) orientation angles, are illustrated. The bushings are rotated around their local x0 and y0 axes separately (hence not rotating around the axial direction of the bushing z0 see Fig. 3 paper C). This corresponds to rotations with an angle Ω1x around the global x-axis and angle Ωy1 around the global y-axis, respectively, for bushing 1. For bushing 2, Ωy2 and Ωz2 rotations around the global y- and z-axis, respectively, are used. The excitation is applied in either the global x-direction (Fig. 5.3(a)) or in the y-direction (Fig. 5.3(b)). It is apparent that the effect of rotating the bushings is most significant for frequencies above 200 Hz; for rotations around Ω1x and Ωz2 for which the variations of the power relative to the original configurations are of orders of magnitudes. The solid black lines in Fig. 5.3 is included to visualise the individual frequency response of one of the configurations giving a high response. To further emphasize the potential of reorienting the bushings and to illustrate the robustness of the achieved results, the most influential rotation angles for each load

52 42 5 UCI as a framework Ωx 1 of max Ωx 1 of min 45 Ratio Ωz 2 of max Ωz 2 of min 30 Ratio 25 Ω 1 x [deg] Ratio max min [-] Ω 2 z [deg] Ratio max min [-] Frequency [Hz] (a) Force excitation in x-direction, angular parmeter Ω 1 x Frequency [Hz] (b) Force excitation in y-direction, parmeter Ωz 2 angular Figure 5.4: Ratio of the angular configuration giving the maximum of transmitted power, for each frequency, and the one giving the minimum (red curve NB y-axis right side), and at what angles the maxima and minima are found (blue and black curve respectively NB! y-axis left side). case (Ω 1 x for x-excitation and Ω 2 z for y-excitation) are studied more closely in Fig Here the angles for which a minimum and maximum of transmitted power occurs is shown, this for each frequency (1 Hz frequency resolution) together with the power ratio between these extremal angles (right y-axis red line NB two y-axis in the graph) Transmission path Considering the complexity of the vehicle suspension system, it is evident that a deeper knowledge of the system at hand would be needed to establish efficient design strategies. If, for example, an optimisation with respect to isolation is required, the most important factors determining the total energy flow would represent an interesting evaluation criteria. Analysing the different energy transmission paths, the results in Fig. 5.5 are obtained for the two load cases separately. Here the percentage of the total power (summed over the frequency band Hz) going through each bushing is shown, each subsubfigure is plotted as a function of angular orientation of the bushings with two vertical lines added indicating the angle at which the maximum and minimum of total transmitted power occur.

53 5.2 Local substruture models and global assembly Ratio power [%] Ωx 1 [deg Bush 1 Max ] Ωy 1 [deg ] Bush 2 Min Ratio power [%] Ωx 1 [deg Bush 1 Max ] Ωy 1 [deg ] Bush 2 Min Ωy 2 [deg ] Ωz 2 [deg ] (a) Force excitation in x-direction Ωy 2 [deg ] Ωz 2 [deg ] (b) Force excitation in y-direction Figure 5.5: Portion of power transmitted through each bushing as a function of angular parameter change (total power in the Hz frequency range). Red lines: bushing 1, Blue lines: bushing 2. Vertical lines indicate the angle for the maximum (black dashed line) and minimum (green dashed-dotted line) of total energy flow into the generic car x y z ω x ω y ω z x y z ω x ω y ω z Power car [W] Power car [W] Frequency [Hz] (a) Bushing Frequency [Hz] (b) Bushing 2 Figure 5.6: Power transmitted into and out of the generic car for each DOF as a function of angular parameter change of Ω 2 z, for x-direction excitation. ω indicate rotational DOFs. NB logarithmic y-axis for both for negative and positive power, cut of absolute powers less the

54 44 5 UCI as a framework To further illustrate the complexity of the energy flow in the system, the energy flow in and out of the of the generic car structure of each individual DOFs is plotted (Fig. 5.6) for the angular parameter Ω 2 z, the results are for an excitation force in x-direction. The two subfigures Figs. 5.6(a) and 5.6(b) are respectively showing the power for bushings 1 and 2. The total power going into the car structure in this idealised test case must have positive power, since no power is injected into the car structure itself. Having said that, it may be observed that individual DOFs may have negative signs since the system is a closed circuit for the power transmission Discussion It is evident from Fig. 5.4 that; i) in the higher frequency range studied, the potential for optimisation is great, with a maximum power ratio of 40 and ii) the difference between max and min angles is quite stable (at least above 100 Hz) and large. This suggests that the results may be considered robust with respect to imperfection of the hardware implementation. In Fig. 5.5 it should be noted that i) the ratio between the powers passing through bushings 1 and 2 may be significantly altered by changing the angular orientations of the bushings, ii) for the two different excitation directions the resulting power ratios show totally different behaviour, hence a study of the isolation effects of the rubber bushings must include the appropriate excitation of the problem at hand iii) the power ratio between bushings 1 and 2 can vary much, even though the total power is still fairly constant (e.g. compare Figs. 5.3(b) and 5.5(b) for parameter Ω 1 y). It may be seen in Fig. 5.6 that, not only the total power transmitted into the car is changed when varying the angular parameter, but also the ratios of the different DOFs are highly affected by the change of angle. Rotating the bushings, with the corresponding change in their stiffness parameters (see Fig. 6 paper C), significantly alters the energy flow. This adds a further argument for the current UCI-ROM approach as the bushings cannot be modeled using a spring stiffness representation.

55 5.3 Conclusions Conclusions The substructuring methodology presented, where an undeformed coupling interface formalism is used to model a vehicle suspension system, has been found to be a powerful tool for investigating the complex mechanisms associated with structure-borne sound in vehicles. Using appropriate models for each substructure, a study of the influence of the rubber bushings used to connect the link arm to the car body is undertaken. It is found that, by rotating the bushings, the power transferred to the car body can be altered by orders of magnitude. On one hand, this opens up for radically decreasing the disturbances transmitted through the suspension system, but on the other hand as well identifying the possible sensitivity of the system to imperfections in the actual physical hardware installation, hence giving robust result. Finally, it should be stressed that the method discussed here is itself general, but to ease the understanding it is here illustrated for an idealised model.

56

57 CHAPTER 6 Wheel rim influence on interior noise This chapter discusses a study of the correlation of component properties to the interior noise. The component is the wheel rim, which is a crucial part in the transfer of acoustic power from tyre-road excitation to interior noise. For low frequencies this transfer can often be explained as a structure-borne noise problem. Often this component is ignored when its comes to is influence to the tyre-road noise; however, the results from this work show that the rim may have large influence on the interior noise. 6.1 Background So far in this thesis a component investigation of the brake system and the associated structural transfer of acoustic power have been discussed. Furthermore, a substructuring approach using undeformed coupling interfaces as a framework for connecting components in a vehicle suspension model has been introduced. There is one additional component involved in the acoustic power transfer in the vehicle suspension system that is of principal interest, the wheel rim. This study is of experimental nature where the system properties of the rim are correlated to the interior noise from the tyre-road interaction. When it comes to tyre-road noise, the interior noise field is not as well covered in the literature as the exterior noise. It may typically be subdivided into either structureborne or air-borne noise transmission Bekke et al. (2010), where low frequency problems are commonly considered as predominantly structure-borne, Molisani et al. (2003); Sakata et al. (1990) and vice versa. As an example, Sakata et al. (1990) showed it was possible to in a study correlating the interior noise and the measured spindle forces 47

58 48 6 Wheel rim influence on interior noise below 400 Hz. The rim itself as a subject of research in relation to interior and exterior tyre-road noise sees an increasing interest, partially fuelled by differences in interior noise levels that have been reported for different rims Sandberg and Ejsmont (2002); Curà and Curti (2004); Kindt et al. (2009); Kido and Ueyama (2005); Feng et al. (2009). Furthermore, the simplifying assumption of a rigid rim is not shared by all researchers in the field; for instance Yam et al. (2000) reports that the rim modes have an influence on the characteristics of tyre modes, and Hayashi (2007) has shown that for the tyre cavity mode the vibration levels in the suspension system may be altered significantly when changing numerically the out-of-plane torsional stiffness of the rim. Kindt et al. (2009) showed that, from mobility calculations of a tyre wheel assembly model, the results in the low frequency range differed substantially if a rigid wheel was used instead of a flexible one. In that paper it is also stated that different wheels can produce perceptibly different vehicle interior noise, in the frequency range Hz up to 5 db differences were found when comparing a steel and an aluminium wheel. This latter observation forms the starting point for the current study, where a statistical approach to correlate structure-borne interior tyre-road noise to rim system parameters is used. The basis for the study was road noise measurements combined with component vibration measurements and corresponding analysis, for different rims. 6.2 Approach The work discussed in this study takes as a starting point a measurement data base built during several years at a vehicle manufacturer. These measurement data were collected from a wide range of measurement types such as modal measurements performed on components and subsystems, road measurements and rolling road rig measurements. The result in this chapter is based on two of these, namely the road measurements and component mode measurements. The modal measurements have been post-processed in order to get the dynamic stiffness values of the rim, and these measured stiffness values have been used to validate calculated finite element stiffness results. An overview of the measurement and the calculations in this chapter can be found in paper D.

59 6.2 Approach 49 ID Rim 1 Rim 2 Rim 3 Rim 4 Rim 5 Spokes no spokes 7 spokes 7 spokes 7 2 spokes 10 spokes Material Steel Alu. Alu. Alu. Alu. Weight 8 kg 7.5 kg 10.3 kg 9.3 kg 7.4 kg Dimensions Static stiffness 1379 kn/m 3619 kn/m 6866 kn/m 5430 kn/m 5007 kn/m Table 6.1: Summary of rim properties, static stiffness is the 0 Hz component of the out-ofplane torsional stiffness of the disc of the rim as predefined Interior noise and rim parmeter correlations In this subsection results from rim parameter correlations are presented, where the correlations have been made for the rims listed in Table 6.1, where the most important parametric data for each rim are shown. These are in summary: rim identifier, number of spokes, material, mass, dimensions and static axial stiffness (calculated as discussed in paper D). Figure 6.1: Interior sound pressure levels (bottom lines, right y-axis) and calculated dynamic out-of-plane torsional stiffness of rim disc. SPL, four aluminium rims (Rim 2-5) different runs 60 km/h changing only the front rims. Stiffness calculations performed for the corresponding rims (Rim 2-5).

60 50 6 Wheel rim influence on interior noise Fig.6.1 shows the A-weighted interior SPL in the front seat of the vehicle for road measurements using the four different aluminium rims in Table 6.1 NB right y-axis. The calculated axial dynamic stiffnesses are also shown for the aluminium rims used, NB left y-axis. It may be seen that there is a clear tendency between a high SPL and a low rim stiffness and vice versa, for all investigated rims. This is particularly noticable in the frequency range Hz, with the exception for the peak at around 170 Hz where Rim 5 gives the highest SPL. To further illustrate the correlation between rim stiffness and interior noise, Fig.6.2 also includes results for a steel rim, confirming the previously observed link between low frequency interior noise and rim stiffness. To investigate in some more detail the Figure 6.2: Interior sound pressure levels (bottom lines, right y-axis) and calculated dynamic out-of-plane torsional stiffness of rim disc. SPL, two aluminium rims (Rim 3 & 5) and the steel rim (Rim 1) different runs 60 km/h changing the front rims. Stiffness calculations performed for the corresponding rims (Rim 1, 3 & 5). correlation of rim parameters to the interior SPL in Fig.6.3, the interior SPL is shown as a scatter plot for all the five studied rims as a function of static axial stiffness and mass, where the static stiffness was used since the dynamic stiffness does not vary much in the low frequency range (with an exception for the steel rim). Three predefined problematic frequency ranges were used where for each the maximum SPL in this particular frequency range is shown (the ranges are: , and Hz). To study the correlation, a logarithmic measure was used (db values),