1. INTRODUCTION
The non-linear phenomena caused by "nite amplitude waves travelling in pipes and ducts are often neglected in predictive acoustic analyses because they signi"cantly increase the computationcost. Nevertheless,in several cases, they fundamentallymodify theresponse of the systemespecially when they are associatedwith resonancemodes. These e!ects concern not only the amplitude of the piping system response but also its frequency spectrum. Namely, the tone of the sound delivered at the tailpipe may be strongly a!ected by a non-linear behaviour. In this subject, the work of Chester [1] remains the most famous. He developed an analytical solution of the non-linear resonance of a viscous closed tube and showed that wave discontinuities may be generated. He also showed that these discontinuities are signi"cantly softened by viscous e!ects. The balance between the source strength and the viscous e!ects then determines if a non-linear model should be considered. Such phenomena are of great interest when dealing with a wind music instrument [2]. Typically, the high pitched component characterizing the tone of a brass instrument is shown to be caused by the non-linear propagation of sound waves [3].
Another "eld where the non-linear e!ects may strongly modify the acoustic response of a piping system is the automotive exhaust path. In fact, the improvement of engine e$ciencies is of an increasing interest to car manufacturers. One parameter that has an important e!ect on the engine performance is the static pressure at the exit of the engine exhaust valve. It is the well-known engine &&backpressure''. In the new generation of powerful engines, there is a tendency to lower that backpressure to enhance engine performance. This induces an increase of the acoustic pressure level at the exit of the manifold (180 dB, referenced to 20 lPa) due to the increased pressure di!erence downstream and upstream of the exhaust valve especially during rapid acceleration. The exhaust system manufacturers have the di$cult task of reducing that sound level to 100 dB at the tailpipe. Moreover, from a subjective point of view, the tone of a vehicle exhaust noise becomes a signi"cantfactor in the customer's judgement.Thus, as the exhaust noise becomesquieter in level, the psycho-acoustic criteria become increasingly signi"cant. That means that, in addition to reduction of its total level, the noise spectral contents in terms of low and high frequencies must be analyzed and optimized. Besides these requirements, the exhaust system designers need to work within the constraintthatthetotal volumeallocatedtotheexhaustlineisincreasinglysmall,especially in the small car, B car or town car segment. In this case the exhaust systems classically equipped with three boxes (catalytic converter, sub-mu%er and main mu%er) are replaced by exhaust lines equipped with only two boxes (catalytic converter and main mu%er). This implies the use of a long intermediate pipe and favours the formation of standing waves at lower frequencies. In normal conditions, these kinds of exhaust systems work perfectly and conform to design speci"cations. However, some experiments performed in exacting conditionssubjectivelyshowthattheseexhaustsystemsmay generatean undesirablesound that is characterized by its intermittence and by a metallic aspect particularly irritating to the human ear. This undesirable sound known as &&rasping noise'' contributes to the global noise emittedby the exhaust system and therefore may have to be taken into account in the pass-by noise computation. When rasping noise occurs it gives the impression of mechanical rattle, and so gives an impression of poor quality. A primary investigation of this phenomenon has shown that the rasping noise occurs when a powerful engine is started and quickly accelerated under cold conditions. It is greatlyreducedwhenthe engineandthe exhaustsystemarewarmedup.Raspingnoisedoes not normally occur with constant engine speed or slow run up. In previous work on this subject, the rasping noise is assumed to be caused by shock waves generated inside the exhaust pipe. Nirei et al. [4] postulated that a third of the rasping noise is emitted to the atmosphere as a shell radiation noise whereas the remaining part is emitted through the exhaust tailpipe ori"ce. However Okada et al. [5, 6] postulated that the total rasping noise is emitted through the exhaust outlet into the still atmosphere despite the fact that the associated metallic tone is perceived as radiation noise caused by vibration of the outer shell of the mu%er. Okada et al. also showed that the "nite amplitude compressive waves generated downstream of the exhaust valve and ampli"ed by resonance are steepened as they travel along the pipe. These non-linear e!ects hence de"ned give rise to shock waves after passing a distance called the transition distance which was analytically de"ned. In another case, Takayama et al. [7] visualized the #ow at the vicinity of a sudden area expansion within a simple exhaustpipe. Theydemonstrated by solving theunsteady two-dimensionalEuler equations that a planar shock wave occurs at the smallest section of the expansion area. It is di!racted at the corner of the area expansion and turns into cylindrical shock, which attenuates with propagation. Jonas et al. [8] studied the rasping noise as a part of
the exhaust noise radiated by the outer shell of mu%ers. They tested the rasping noise generation on various mu%er con"gurations. This paper presents the results of an experimental study concerned with the characterization of the rasping noise. The focus here is on developing an objective criterion for the detection and the quantitative assessment of the metallic component. The acoustic signatures of a large set of exhaust systems were collected at various running conditions. Simultaneously,thetailpipesoundswereperceivedandsubjectivelyestimatedastowhether or not they include a raspy component. The measurementswere performed during more or less abrupt engine run up which led to non-stationarysignals. Three types of processing are tested here. The "rst and the second processing are based on Fourier spectral analysis. The "rst processing consists of a classical short-time Fourier transform (STFT). The second one is a tracking analysis that takes into account the periodicity of the collected signal and allows synchronization with the engine crank-shaft rotation. Thethirdprocessingconsistsofadecompositionofthesignalbyusingawaveletbasis.As this treatment is less classical and less standardized, a short review of the basic theory is made. Results of both continuous and discrete wavelet analysis are presented. The last one is then used to build a detection criterion that also allows the quanti"cation of the rasping rate within the whole tailpipe noise. In the last section, the acoustic pressure captured within the exhaust system is examined and an attempt is made to propose a physical interpretation of rasping noise occurrence. The hypothesis of non-linear interaction between resonance modes is favoured.