A Model to Compute and Quantify Automotive Rattle Noises

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1 A Model to Compute and Quantify Automotive Rattle Noises Ludovic Desvard ab, Nacer Hamzaoui b, Jean-Marc Duffal a a Renault, Research Department, TCR AVA 1 63, 1 avenue du Golf, 7888 Guyancourt, France b INSA de Lyon, Laboratoire Vibrations Acoustique, Bâtiment St. Exupér 5 bis avenue Jean Capelle, 6961 Villeurbanne, France ABSTRACT In an automotive cockpit, rattle noises deal with all noises due to normal contacts that radiate as annoying noises for customers. The modeling of such a phenomenon in a full realistic unit is complex. revious works were performed to define a model based on simple structures to generate and describe automotive rattle noises. Both experimental and numerical studies were carried out using this model. The aim of this paper is to outline a detailed presentation of the modeling and to perform an evaluation of this modeling using experimental data. The comparison of modeling results to reference cases and reference experiments shows good agreements. Finall an application to rattle noises is carried out and some differences between the modeling and experimental data can bee observed. Nevertheless, the classification of rattle noises seems to be reproduced using the modeling. In future works, perceptive tests should be perform to know if these differences are significant in term of perceived annoyance. 1 INTRODUCTION In an automotive cockpit, annoying noises due to unwanted contacts are divided in two main families. On one side, rattle noises coming from random normal contacts, on the other side, squeak noises coming from random friction contacts. This paper focuses on rattle noise issues. Most of automotive rattle noises come from the instrument panel. Modeling all potential contacts in such an automotive subsystem is difficult and can generate huge models. Furthermore, a time domain modeling is necessary to describe such a non-stationary phenomenon. Therefore, we decided to develop an analytical time domain modeling based on simple geometries. revious works showed that the generation mechanism of rattle noises is the impact [1]. The impact noise is described in [] as a result of five mechanisms. Considering rattle noises, the impact noise can be decomposed as an initial pressure impulse due to sudden acceleration or deceleration of rigid bodies, followed by a decaying transient due to the vibratory response of the involved structures. Computing the impact force is the main difficulty in modeling impact phenomena. A review of analytical models usually used to compute the impact force is done in [3]. An application is performed in [4]. Several papers refer to [5] when the contact is considered as hertzian. From an acoustic point of view, several papers deal with the acoustic pressure radiated from a single impact. The acoustic pressure radiated from a released sphere falling on a plate or a slab is widely investigated in [6, 7]. Usuall random impact is described using a a address: ludovic.desvard@renault.com b address: nacer.hamzaoui@insa-lyon.fr

2 statistical approach [8]. In literature, there is no study on the modeling of several impacts, thus this paper is entirely dedicated to the modeling of the acoustic pressure radiated from a randomly impacted structure. First, a detailed presentation of the analytical model is performed, with specific attention directed to the computation of the key parameters of impact noises. Then an evaluation of each computing step is carried out using numerical and experimental data. Finall we apply the modeling to predict rattle noises and we compare the results with experimental data. THE MODELING.1 resentation of the model The first aim of this modeling is to describe accurately the physical phenomena behind automotive rattle noises. To generate automotive rattle noises, we choose a model based on simple geometries and presented in Figure 1. The model is a plate impacted by an hemispheric impactor on a point M. The output of this modeling is the radiated acoustic pressure at point. Figure 1: Model based on simple structures We choose this model for several reasons: The vibratory response of a plate is fluently known for specific boundary conditions. If we consider that the size of the impactor is much smaller than the size of the plate, we can assume that only the plate radiates in the audible frequency range. The hemispheric shape of the impactor allows us to consider the contact as a point contact. Because of the difficulties to measure or compute the contact force in a full realistic car unit, the input of the model is the relative displacement between the impactor and the plate. To generate realistic automotive rattle noises, the plate and the impactor materials are the same as the ones encountered in an automotive cockpit. All these choices lead us to establish the global solving scheme presented on the Figure.

3 Figure : Global solving scheme The key parameters to compute were identified during previous works [1]. Indeed, the modeling has to compute: The relative displacement between the impactor w I ( and the plate w ( x, The contact force F ( The vibratory response of the impacted plate w& ( x, The radiated acoustic pressure ( x, z,. Relative displacement formulation To compute the relative displacement between the impactor and the plate we assume that the impactor displacement is known during all the simulation. This assumption makes the impactor displacement independent of the contact phenomenon. Therefore, we consider that the impactor base is driven and the deformation due to the contact is locally allowed at the top of the impactor as it is shown on the Figure 3. Figure 3: Explanation of the assumption of a driven impactor Whatever the impactor displacement is, its location always fluctuates around an equilibrium location that is considered as the reference location. Note that the equilibrium location of the plate is defined according to the initial conditions and, consequentl fluctuates around the initial clearance.

4 .3 Contact detection The contact detection can be expressed by the equation (1) considering the impactor displacement w I ( and the plate displacement w x, y, ) at the contact location. ( t If If wi ( > w w ( w I ( x, y, ( x, y, Contact No Contact (1) It must be noticed that if the contact occurs, a contact force appears and the location of the plate changes. Consequentl to detect the next contact we will have to consider the modified location of the plate instead of the equilibrium location of the plate..4 Impact force formulation The contact force is computed using Hertz theory. This theory considers the elastic contact between two bodies allowing a local deformation at the contact area. The contact force, also called in this case the impact force, is computed using the equation (). 3 F( = n α( () α represents the squashing induced by the contact. Usuall this data is computed knowing the contact force. In our case, we want to compute the impact force, so α becomes an input of this equation. The idea is now to estimate the value of this squashing. At each step of time, the impactor location is known. When the contact is detected, the impactor location is higher than the plate location. Therefore, we assumed that the squashing is the penetration of the impactor in the plate and thus α is formulated according to the equation (3). α ( = wi ( w ( x, (3) In equation (), n is a parameter depending on the material properties and the geometrical characteristics of solids involved in the contact. For the contact between a sphere of radius R and a plate, n is computed using the equation (4). Note that none dimensions of the plate appears in this equation because of the local property of the contact. The only dimensions to consider are the curve radius of both parts, and this curve radius is equal to zero for a plate. 4 R n = 3. π K I K ( + ) (4) Finall the values impactor and the plate. K I and K depend on the material properties of respectively the 1 ν i For i = I, Ki = (5) π E i.5 Vibratory response formulation The computation of the vibratory response has to be in time domain because of the transient property of the rattle noises. First, we assume that the acoustic pressure radiated is

5 due to transverse acceleration of the plate. Thus, we have to solve the flexural equation of motion (6). w ( x, w ( x, 4 ρ. h. + λ + D. w ( x, F( x, t t = (6) Where: = x x y y ρ is the density of the plate h is the thickness of the plate λ is the generalized viscous damping 3 E. h D = is the bending stiffness 1(1 υ) E is the Young modulus of the plate υ is the oisson s ratio of the plate We decided to define an analytical modeling, so, as boundary conditions, the plate is assumed to be simply supported on its four edges. If we consider the plate displacement w, boundary conditions can be expressed with equations (7). w ( x, y) = For x, y Γ( The perimeter of the plate) w w D + υ x y = For x and y =, Ly (7) w w D + υ y x = For y and x =, Lx An initial clearance j is considered as initial conditions (Eq. 8). w ( t = ) = j w& ( t = ) = (8) w&& ( t = ) = We decided to use the modal superposition principle to solve the equation (6). For a simply supported plate the modal equation, using generalized data, is well known (9). F( a& ( + ε ω a& ( + ωa( = (9) M ( Note that, we decided to use the damping ratio ε which can be expressed as a function of the viscous modal damping λ, the generalized mass M and the natural frequency ω of (m,n) mode (Eq. 1).

6 ε λ M ω = (1) The interest of using this kind of damping, is that the damping ratio is a few varying parameter with respect to frequenc [9]. We use a Newmark integration scheme based on the Taylor s developments to solve the differential equation (9) in the time domain, [1]..6 Radiated Acoustic ressure formulation To compute the radiated acoustic pressure at the listening point, we decided to use the Rayleigh s integral. We thus make the assumption that the plate is baffled and radiates in free field. For a point, located in z > (Fig. 1), the acoustic radiated pressure can be expressed using the equation (11). ρ 1 r M ( = γ ( x, t ) ds π r c S (11) Where: ρ O is the air density S is the surface of the plate r is the distance between the listening point ant the plate γ is the transverse acceleration of the plate c is the sound velocity 3 EVALUATION As an evaluation of the relevance of the modeling, we decided to run some computations on reference cases to characterize each step of the modeling. 3.1 Evaluation of the vibratory response computation The first evaluation is the vibratory response computation. To evaluate this step of the modeling, we decided not to consider the impact phenomenon. Consequentl the plate is directly excited at one point Comparison Analytical Modeling / Numerical Modeling For this comparison, both analytical and numerical approaches use the same model. We consider a simply supported steal plate excited at point (.9;.5) with a white noise signal, and we compute the acceleration response at point (.1;.1) On the Figure 4, we compare the transfer function γ computed with the analytical F modeling to the one computed with a numerical modeling. We can note a very good agreement between both computations. It means that the time domain approach is as relevant as the frequency domain one.

7 1 8 Time domain modeling Frequency domain modeling (Nastran) log(γ /F) (en db) Figure 4: Transfer function at point (.1;.1) 3.1. Comparison analytical modeling / Measurements Therefore, we decided to compare the modeling results with experimental data. In this case, we consider a simply supported steal plate directly excited at point (.9;.5) with a white noise signal, and we compute the transverse velocity response at point (.87;.55). The experimental data are recorded with a sweeping laser vibrometer that measures the transverse velocity. In the modeling, we can consider either computed natural frequencies (based on the assumption of a simply supported plate and named computed modes) or measured natural frequencies (named experimental modes). In both cases, the modal shape is computed and the damping ratio is assumed constant. The responses using computed modes (5 modes) and measured modes (13 modes) are different in term of spectral density but are quite similar in term of magnitude (Fig. 5). 1 Experiment Modeling - Computed modes Modeling - Experimental modes -1 log(v /F) (db) Figure 5: Transfer function at point (.87;.55) We can see that the modeling using the experimental modes presents a better agreement with experimental data than the modeling using computed modes. We, thus, can conclude that the plate is not exactly simply supported during the experimental measurements. We can also note that the use of experimental modes reduces the response frequency band because, in the experiment, it is difficult to measure high frequency modes.

8 3. Evaluation of the radiated acoustic pressure computation To evaluate the radiated acoustic pressure computation we decided to compare modeling results with experimental data. The case chosen is a steal plate directly excited with a 1 Hz sinus. The modeling also allows us to use an experimental excitation, to evaluate the computation of the radiated acoustic pressure we decided to use both experimental natural frequencies and experimental excitation. We choose to use the experimental excitation, because it is not exactly a 1Hz sinus. Experimentall a microphone measures the radiated acoustic pressure. We thus can perform the comparison with respect to time (Fig. 6). We can notice that there is a good agreement between the experiment and the modeling results. The phase difference clearly visible on the plot is due to initial conditions. 1.5 Modeling Experimental 1 Acoustic ressure (a) Time (sec) Figure 6: Radiated acoustic pressure in time domain This good agreement is also visible comparing the two spectrums (Fig. 7). We can observe that the modeling overestimate the acoustic pressure except at 1Hz. Note that this frequency best represents the radiated acoustic pressure. 1 8 Modeling Experiment 6 log( Ac / ) (db) Figure 7: Radiated acoustic pressure in frequency domain 3.3 Validation of the impact phenomenon In previous works [1], we performed a qualitative analysis of experimental data. The idea is now to consider a succession of several impacts and to see if we can make the same

9 conclusions. The case chosen here is an aluminum plate impacted by a polymeric impactor driven by a 5Hz sinus. Firstl we plot the relative displacement between the impactor and the impact point of the plate (Fig. 8(a)). It can be seen that the displacement of the plate is changed when the impactor impacts the plate. Secondl we can observe, on the figure 8(b), the apparition of a contact force that appears to be a succession of impulses as it was noticed experimentally (Fig. 4(b) of [1]). Finall looking at the radiated acoustic pressure evolution (Fig. 8(c)), the modeling allows us to observe impulses due to the impact followed by decaying transients. Note that this observation was also done experimentally (Fig. 5 of [1]). 7 x Impactor late Displacement (m) 4 3 Contact Force (N) Time (sec) (a) Time (sec) (b) 3.5 Acoustic ressure (a) Time (sec) (c) Figure 8: Time evolution of the computed relative displacement (a), the computed contact force (b) and the computed radiated acoustic pressure (c) 4 ALICATION TO RATTLE NOISES In previous works [1], we demonstrate that automotive rattle noises can be generated using a polymeric plate impacted by a polymeric impactor driven by a road profile induced displacement. Therefore, we measure several rattle noises using different automotive materials for both plate and impactor. To evaluate the application of the modeling to the rattle noise phenomenon we compare an experimental rattle noise with a computed one. We keep the same notations as the ones used in previous works. We thus consider a 3 polymeric plate impacted by a S polymeric impactor driven by a profile road. The comparison between the modeling and the experiment (Fig. 9) shows a global overestimation of the modeling. Note that, in the modeling, the modal scheme is defined up to 51Hz.

10 9 Modeling Experiment 8 7 log( Ac / ) (db) Figure 9: Rattle noise generated with plate 3 and impactor S The differences between modeling and experiment can be probably due to several points: The difference between the experimental and the modeling modal scheme The dynamic behavior of automotive polymeric materials The damping law The investigation of all these potential sources of bias will lead to a huge study. It is necessary to keep in mind that the aim of this modeling is to predict the annoyance of an automotive rattle noise. Therefore, it could be interesting to know if the perceived annoyance is the same for both modeling and experimental rattle noise. In future works, an application of perceptive tests could be done to answer this question. Furthermore, we want to know if the relative difference between two materials could be detected using the modeling approach. To give a piece of answer, we can first compare experimental rattle noise spectrums generated with three different polymeric plates with computed rattle noises. For both plots of the figure 1, we use a S polymeric impactor. 9 8 Experiment 1 Experiment Experiment Modeling 1 Modeling Modeling log( Ac / ) (db) log( Ac / ) (db) (a) (b) Figure 1: Experimental rattle noises (a), computed rattle noises (b) Firstl we can notice that, experimentall rattle noises generated using 1 and 3 polymeric plates are close to each other. The same observation can be done with computed rattle noises. Then, if we consider the experimental rattle noise generated using polymeric plates, we can observe that the spectrum is globally under the two others, except around 45Hz and 1Hz. The computed rattle noise leads to the same observation.

11 These first observations are on the good way to confirm that the difference between two rattle noises can be detected using computed rattle noises. erceptive tests should be performed to confirm these observations. 5 CONCLUSION In this paper, a detailed presentation of a modeling to characterize rattle noises is conducted. This modeling allows the computation of the acoustic pressure radiated by a randomly repeated impacted plate. Some assumptions are made to develop an analytical and time domain modeling. An evaluation of each computation step is performed and shows good agreements in the computation of both vibratory response and radiated acoustic pressure. Furthermore, the modeling well reproduces impact phenomena. The application to rattle noises shows some differences between experimental and computed rattle noises. Future works will be performed to evaluate if these differences are significant in term of perceived annoyance. However, we show that the relative difference between two experimental rattle noises is visible computing the same rattle noises. erceptive tests will be carried out to check if the classification of rattle noises with respect to perceived annoyance is the same with modeling rattle noises as with experimental ones. 6 ACKNOWLEDGEMENTS The authors would like to thank all the acoustic team of the research department of Renault. This study has been performed thanks to the help of all the team of LVA (Laboratory of Vibration and Acoustics), of INSA in Lyon, which is gratefully acknowledged. 7 REFERENCES [1] L. Desvard, N. Hamzaoui and J.M. Duffal, Modeling and characterization of rattle noise encountered in an automotive environment, roceedings of Acoustics 8 congress in aris (8). [] A. Aka A review of impact noise, Journal of American Society of Acoustics, 64(4), (1978). [3] V. Acary and B. Brogliato, Coefficients de restitution et efforts aux impacts. Revue et comparaison des estimations analytiques, INRIA, Institution National de Recherche en Informatique et en Automatique (4). [4] C. Rajalingham and S. Rakheja, Analysis of impact force variation during collision of two bodies using a single-degree-of-freedom system model, Journal of Sound and Vibration, 9(4), (). [5] G. Goldsmith, Impact, Edward Arnold Ltd, USA, (196). [6] A. Aka Acoustic radiation from the elastic impact of a sphere with a slab, Applied Acoustics, 11 (1978). [7] A. Ross and G. Ostigu ropagation of the initial transient noise from an impacted plate, Journal of Sound and Vibration, 31, 8-4 (7). [8] C.H. Lee and K.. Byrne, Impact statistics for a simple random rattling system, Journal of Sound and Vibration, 119(3), (1987). [9] J.L. Guyader, Vibrations des milieux continues, Hermes science publication, Lavoisier edition, aris, () [1] A. Miloudi, Rayonnement acoustique des plaques par une approche temporelle: Application à la synthèse sonore de modèles vibro-acoustiques, hd thesis, Université des Sciences et de la Technologie Houari Boumédiène, (4)