dispersed vibroacoustic responses of industrial products...

Dispersed vibroacoustic responses of industrial products: what are we able to predict?

L. Gagliardini, Vehicle NVH dept. PSA Peugeot-Citroën, Route de Gisy 78140 VELIZY-VILLACOUBLAY - FRANCE e-mail: [email protected]

Abstract Since todays design process of manufactured objects mainly relies on simulations, the relevance of models -ie their capability to foresee real objects properties, including dispersion of these properties- is a crucial engineering issue. When making a reproducibility test (same experiment on numerous samples), it appears that industrial objects -cars for example- measured vibroacoustic FRFs show huge dispersion. Thus, dispersed FRFs exhibit some characteristics patterns. These characteristic behaviors suggest the modeling of dynamic systems uncertainties should be possible in a rather generic manner.

Among many possible methods, a maximum of entropy non-parametric modeling –based on a random matrix theory- is preferred since it does not require any detailed knowledge of the object's uncertainties. Uncertainties are considered as a whole, whatever their root causes: modeling uncertainties, data uncertainties (material and geometry), object diversity, manufacturing process or measurement errors. Amount of uncertainties is controlled through the variance of physically consistent mass, stiffness and damping matrices. These variances may be set a priori or identified a posteriori. The theory is numerically well fitted for dynamic and vibroacoustic problems when using generalized coordinates (modes) in order to compute solutions.

The stochastic model may be built from current FE models, and independent realizations are performed during a Monte-Carlo simulation. Statistics of the dynamic responses (modes, FRFs, energies…) can then be discussed, showing some interesting properties that can be linked to many practical observations. Various use of the stochastic vibroacoustic modeling in an industrial context will finally be presented: model assessment, design assessment against targets, design sensitivity analysis.

1 Introduction

Since todays design process of manufactured objects mainly relies on simulations, the relevance of models –i.e. their capability to predict real objects properties- is a crucial engineering issue. The field of vibroacoustics is known to be especially challenging since achieving a relevant forecast is not common [1].

Any acoustic engineer or researcher knows that modeling real objects is a hard job: mostly because of a lack of knowledge of these real objects. Detailed geometry, local material properties are often ignored, resulting in prediction that highly diverge from later measurements. Updating the model generally leads to satisfactory modeling results, but with a high cost and in a time scale that is not compatible with industrial projects delay. Moreover the updated model is only valid for the single real object that was carefully analyzed.

Nevertheless, such a systematic deterministic approach is necessary to get confidence that the mathematical-mechanical tools used (typically linear solid mechanics and acoustic finite element analysis) are relevant.

Reasons for differences between experimental realizations and forecast are many. The high sensitivity of dynamic responses to small system’s disturbance is certainly one of the most important. The so-called dynamic hyper-sensitivity, related to dynamic operators’ spectral properties, has many consequences in

17

the field of noise and vibrations, from the experimental point of view as much as from the modeling point of view. This peculiar behavior gives some practical relevance to the research area of uncertainties in structural dynamic and vibroacoustics. This is one aim of this paper to characterize the dispersion of uncertain systems vibroacoustic responses [2][3].

Classically, excitation issues are separated from the structural response characterized by Frequency Response Functions (FRFs). Primary excitation mechanism are various, but surprisingly stable in practical applications. One reason may be that they are often localized and consequently sensitive to a reduced number of uncertain parameters. Another reason is that the interaction of a source mechanism with its vicinity is generally low; this is implicit when considering one system’s response as an FRF times an excitation force. Consequently, excitation dispersion will not be considered here. Excitations are deterministic data either measured (direct or inverse methods) or computed from the primary mechanism modeling (generally non-linear time-domain modeling). This stands for primary forces by opposition to interaction forces between sub-systems, involving systems FRFs and consequently showing the same kind of spread as FRFs.

The paper is illustrated using automotive examples. Nevertheless most of the conclusions that will be presented can be extended to any kind of massively manufactured objects. Manufacturing means here the production of a large amount of objects, with a quality control only ensuring their main functions. To the opposite, “hand-made” products are individually checked in details (geometry as well as physical properties). It is interesting to notice that “hand-made” population remains too low to perform statistics, while, on the other hand, manufacturing products can only be characterized from a statistical point of view.

Experimental observations of production vehicle samples will be first presented and related uncertainties will be discussed. In the light of these actual results, the engineering forecasting problem can then be posed from a probabilistic point of view. After the structural-acoustic modeling technique is briefly presented, the maximum of entropy non-parametric stochastic modeling is introduced in order to account for any kind of uncertainties taking part into the vibroacoustic FRF’s prediction process. The proposed model is assessed and specific parameters are identified; typical results of such a probabilistic model are finally shown.

To end with, practical engineering applications of probabilistic modeling are shown to demonstrate the improvement that can be provided by a stochastic approach during a vehicle’s design process.

2 Experimental observations

Results presented here are taken from three different measurement campaigns on four different production cars within a period of 6 years. Although the same kind of approach was used (FRF measurements, similar measurement points) some differences exist on the measurement procedure that was improved as well as on the frequency range that was extended.

Measured vehicle are picked up at the end of the production line during a time window smaller than a month. Measurements are performed in a workshop with common temperature conditions; depending on the season, temperatures deviate from 5°C to 10°C during the tests.

Instrumentation is specific in order to avoid any ware on the car bodies that will later be sold. It is roughly the same for all series of measurements.

2.1 Samples selection and statistics

When wishing to forecast a physical behavior one implicitly refers to posterior measurements related to a prior model. Since engineering is governed by economical constrains one single model –called the mean model- is used for one body-type project. Regarding the mean model content, the real cars then appear uncertain because of the diversity of customers’ options: powertrains, comfort equipment, safety

18 PROCEEDINGS OF ISMA2014 INCLUDING USD2014

equipment, sound insulation package … Consequently, the experimental samples are chosen considering the vehicle diversity that is supposed to be covered by the unique model used for predictions.

The number of sample used is large enough to allow basic statistics but cannot ensure the full convergence of the probability law of FRFs which would require thousands of samples. Between 20 and 40 cars were generally measured. Since dynamical response probability density functions are unknown, a quantile statistics is the most relevant [4]. Most of the results will be gathered on a single graph with the indication of the median value, 5th and 95th quantiles.

2.2 Measurement repeatability

Since a car is made of a structural part enclosing a cavity, structural FRFs as well as acoustical and vibroacoustic FRFs are to be measured in order to fully investigate the vibroacoustic behavior of the system. Various techniques where applied depending on the frequency range and on practical issues. Each measurement technique’s repeatability was evaluated.

Figure 1 shows the repeatability of structural FRFs measurement with an impact hammer, as well as the associated coherence. As expected, when coherence is low at the lowest frequencies, the FRFs dispersion is high. At higher frequencies, FRFs spread is lower than 2 dB, except around anti-resonances. Regarding the prediction problem, measurement errors may act as a masking phenomenon. In order to capture the samples uncertainties at low frequencies, the measurement procedure was improved as shown by figure 2.

Figure 3 shows the repeatability of a structural FRFs measurement using a hand-held shaker in the mid-frequency range. This kind of shaker was used in order to be able to measure input mobilities but it appears that it introduces measurement errors. The coherence drops at low frequencies and around anti-resonances; correlatively the dispersion of FRF is increasing.

Figure 4 shows the repeatability and coherence of the vibroacoustic FRFs using a reciprocal measurement. Repeatability (as well as coherence) appears to be very good over a large frequency range, covering most of the low and mid frequency range. The same kind of results is obtained for acoustic FRF. From this point of view, measurements using an acoustic source are to be preferred.

In every case, one may notice that, close to anti-resonances, results are unstable. Fortunately, low FRFs values are of minor practical interest.

2.3 Acoustic FRFs dispersion

Figure 5 shows the dispersion of an acoustic FRF measured for two different types of cars between the “same” locations in the car compartment. Similar trends can be observed: although the spread remains low over the whole frequency range, an increase can be noticed when increasing frequency. Dispersion is also larger in the dips (up to 10 dB) than at peaks (3 dB at the most).

(a) (b)

Figure 1: Repeatability of a structural FRF measurement (a) and associated coherence (b) of a classical impact hammer measurement.

KEYNOTE 19

(a) (b)

Figure 2: Repeatability of a structural FRF improved impact hammer measurement (a) and associated coherence (b)

(b)

Figure 3: Repeatability of a structural FRF measurement using a shaker (left) and associated coherence (right)

(b)

Figure 4: Repeatability of a vibroacoustic FRF reciprocal measurement (left) and associated coherence (right).

2.4 Vibroacoustic FRFs dispersion

Figure 6 again compares vibroacoustic FRF between the “same” locations for two different vehicles. A reciprocal measurement was performed with a flow velocity source at the driver’s ear location. Dispersions show the same trends for both vehicles: they increase with frequency, from 5 dB below 100 Hz to more than 10 dB above 250 Hz. It is also noticeable that the median value is not centered: it roughly follows the upper bound with a margin of about 3-5 dB. The mean distance to the lower bound is larger and more irregular due to anti-resonance sensitivity, part of which is certainly due to measurement uncertainties.

Even when observed dispersion is large, significant differences may be observed between the median values, which can be related to each vehicle’s specific characteristics.

(a)

(a)


(a) (b)

Figure 5: Acoustic FRFs dispersion. Cars A (a) et B (b).

(a) (b)

Figure 6: Vibroacoustic FRF’s dispersion. Cars A (a) and B (b)

2.5 Structural FRF dispersion

Figure 7 shows the low frequency range FRFs dispersion measured for two different cars, between an engine mount and a single degree of freedom on the windscreen. These measurements are performed using a shock hammer with the above mentioned improved method.

The dispersion appears to be very low (< 3 dB) at the lowest frequencies (<30 Hz), increasing quickly to reach value higher that 5 dB and even higher than 10 dB around anti-resonances regions.

Figure 8 shows the results obtained for the same FRFs over a broader frequency range, using a shaker as excitation device. First, it has to be noticed that, compared to figure 7, the dispersion is increased in the low frequency range due to measurement errors associated to a low coherence, as shown on figure 3. In the mid-range, dispersion may reach 15 to 20 dB, which is higher than what was observed for the vibroacoustic FRFs.

Figure 9a shows the input inertance at the considered engine mount location in the vertical direction. Dispersion remains low all over the range. On the other hand, it can be seen on figure 9b that, for a distant observation point, on a soft structural part, the dispersion of the FRFs overpasses 10 dB above 80 Hz.

2.6 Synthesis of observations

First of all, there are some characteristics that seems to be shared among every kind of FRFs: dispersion first increases with frequency, then stabilized between 10 and 15 dB. Median value is closer to the upper bound than to the lower and dispersions increase near anti-resonances.

KEYNOTE 21

Sharp peaks can only be seen at very low frequencies when the dispersion is small. At higher frequencies, dynamic responses appear highly dispersed, but smoother in average.

Structural FRFs may be highly dispersed especially when input and output point are distant one to each other.

Acoustic FRFs are not significantly dispersed which could be associated to the very low modal density of the car compartment. Vibroacoustic FRFs seems to be slightly less dispersed than structural FRFs.

(a) (b)

Figure 7: Low frequency structural FRF’s dispersion. Cars A (a) et B (b)

(a) (b)

Figure 8: Mid-range frequency structural FRF’s dispersion. Cars A (a) et B (b)

(a) (b)

Figure 9: Low frequency structural FRF’s dispersion. Excitation at one engine mount. (a) input inertance – (b) distant observation point


3 Engineering accounting for uncertainties

3.1 Problem background

For OEMs, engineering involves first design and later manufacturing. Uncertainties hide in both.

In the previous section, the manufacturing and assembly process uncertainties were some of the observed dispersion factors. Another one was the product diversity, since a vehicle is in fact not a sole product, but a family of products in order to meet variable customer demands. Separation of the two effects is almost impossible since the possibility that 2 cars share exactly the same parts is very unlikely. Up to now these effects were considered as a whole, globally observed at the end of the assembly line.

On the other hand, design is an evolving process which could be considered to end once the production is started and stabilized. The exact definition of the product “As Built” is eventually available when the design process is finished. In advanced design stages, the only available information is the existing drawings -or sometimes only a predecessor model- of the studied system components. The mean model is built using this information, based on explicit or implicit assumptions, leading to some modeling errors. Even uncertain and incomplete, this information is entirely controlled by the user in a data file.

Finally, the experimental assessment of the product design which ends the engineering process is subject to environmental and measuring uncertainties.

In the past, the design assessment was performed iteratively using successive prototypes stages. Improvement of the simulation techniques allowed reduction of these expensive prototypes, which also contributes to decrease the time to market of products. Assessment of the project risk not to reach the target then only relies on computations.

In such an uncertain context, and considering the observed deviation of the product ensemble, only a probabilistic modeling using the available project information seems relevant to conduct a design process based on simulation. Such a probabilistic modeling should be able to provide the probability that the performance level computed is likely to be achieved by the manufactured objects.

Figure 10 illustrates the engineering process uncertainties issues.

Figure 10: Schematics of the issues related to a simulation-based design process

KEYNOTE 23

3.2 Baysian approach of the simulation forecasting ability assessment

Bayes theorem [5] states the posterior probability that model H fits experiment E may be written as:

��|�� = �|�� (1)

where

• ��|�� is the likelihood of the experiment regarding the model. When N �� independent

observations, e�, are performed, then the likelihood of the overall experiment is:

P�E|H� = � P�e�|H��

��

• P�H� is the prior probability of the model. It can be set to 1 when a single model exists according to the state of the art.

• P�E� is the probability of observed samples: when N �� independent observations, e�, are made

P�e�� = P�e�, and then P�E� = ∏ P�e�� = P�e��

Introducing the model log-likelihood, ℒℒ��|�� = �� !" log�&'��|��(, one gets:

ℒℒ��|�� = ∑ ℒℒ�*+|��, !"+-. � !" − ℒℒ�� + ℒℒ�� !" , (2)

where ℒℒ��1|�� = 234�&'��1|��( is the log-likelihood of the independent observations

Within a constant, the log-likelihood of the model appears to be the average of the log-likelihood of the observations that were made. The later can be computed from the probability density function of the model results, [6]. In principles, any observation could be used to assess the performance of the model. Practically, FRFs measurement is the most common [6]. Modal shapes can also be considered using appropriate matching technique between measured and computed modes [7].

In order to be able to assess the forecasting capability of a model, the model should be able to provide the probability density function of its results associated to the uncertainties to be considered regarding the prediction purpose. In the engineering process case, the accounted uncertainties include measurement uncertainties, product diversity, manufacturing uncertainties and modeling uncertainties. In that sense, the best model regarding forecasting capabilities is the model with the highest likelihood. Of course this capability can only be assessed a posteriori, when measurements are available. Before measurements can be performed, predictions validity is related to the fulfillment of the modeling assumptions agreed during a previous assessment.

4 Structural-acoustic modeling

Figure 11 shows the schematic of the mathematical-mechanical structural-acoustic problem used to address vehicles noise and vibration issues.

In this section, we considered the discretized equations as provided by a finite element software. Such expressions are derived from a continuous approach [8].

Structural response is governed by the dynamic stiffness matrix: 567�8�9:7�8� = ;7�8� (3)

Where :7�8�is the vector of the structural displacements and ;7�8� represents the external forces.


Figure 11 : Schematic of the structural-acoustic problem

The complex dynamic stiffness is built from 3 real positive matrices (mass, stiffness and damping): 56<�8�9 = −8=5><9 + ?85@<9 + 5A79 For sake of simplicity, the case of viscoelastic damping only is considered here, but extension to structural or hysteretic damping is rather straightfoward.

In the same manner, the acoustic response of a cavity is governed by an admittance matrix:

B6C�8�DEC�8� = ;C�8� (4)

Where EC�8� is the vector of the acoustic pressures and ;C�8� represents the acoustic sources.

The complex admittance matrix is built from 3 real positive matrices (stiffness, mass and damping):

B6C�8�D = −8=BMGD + ?8B@CD + BACD. These two problems are coupled on the fluid/structure interface by a coupling matrix 5H9 such that the coupled problem takes the form:

I567�8�9 5H98=5H9 B6C�8�DJ K:7�8�EC�8�L = K;7�8�;C�8�L (5)

This problem is classically solved in a reduced form using modal generalized coordinates, such that:

:7�8� = 5M79N7�8� (6a)

EC�8� = BOCDNC�8� (6b)

Where 5M79 is the matrix of structural modal shapes, BOCD is the matrix of acoustic modal shapes, and N7�8� and NC�8� are the associated generalized coordinates.

Equation (5) then becomes:

I5P7�8�9 5Q98=5Q9 BPC�8�DJ KN7�8�NC�8�L = KR7�8�RC�8�L (7)

Where generalized matrices are such that:

5P<�8�9 = 5M79S56<�8�95M79 (8)

BPG�8�D = BOCDSB6G�8�DBOCD 5Q9 = 5M79S5H9BOCD and the generalized forces are such that R7�8� = 5M79S;7�8� and RC�8� = BOCDS;C�8�

KEYNOTE 25

Thanks to the linearity of the modal projection, the generalized stiffness matrices are related to the constitutive generalized matrices by: 5P<�8�9 = −8=5ℳ<9 + ?85U<9 + 5V79 (9a)

BPC�8�D = −8=BℳGD + ?8BUCD + BVCD. (9b)

By definition, generalized mass and stiffness matrices are diagonal and positive or semi-positive. Damping matrices are generally full matrices, except for the simplified case of proportional damping. In the whole, 7 matrices are required to describe a vibro-acoustic problem.

The model described here is the one that is currently built for any vehicle projects. As mentioned earlier, it will be referred to as the mean model. Figure 12 shows the acoustic and structural meshes of such a model.

Figure 12: Acoustic and structural finite element meshes of the mean model

5 Uncertainties modeling

As mentioned in the section 3, uncertainties causes are numerous and various during the engineering process. These uncertainties apply to all structural, acoustic and coupling operators defined in section 4. Two main ways exist to account for uncertainties in the modeling, which can be complementary [9]. In any case, large size random equation sets will have to be solved.

5.1 Solving random equations

Since vibro-acoustic phenomena are rather complex, neither input nor output probability law is normal or fitted for algebraic computations. This imposes to rely on a Monte-Carlo simulation [10] in order to access the results probability density functions that are mandatory regarding the forecasting capability of the model as shown in equations (1), (2).

In a Monte-Carlo simulation, random realizations of uncertain physical values –according to relevant probability density functions- are performed as long as the convergence of the solved system’s results statistical indicators is reached. Convergence studies showed that 500 realizations are generally enough to get a convergence of the 5th and 95th centiles. Median value converges faster but for engineering purpose bounds are often more relevant than the median value. On the other hand, the results’ probability density function convergence requires a few thousands realizations.

It may be noticed here that convergence of the solutions using a modal approach may also depend on the number of modes considered. The probabilistic approach generally leads to consider an extended modal basis by comparison to the deterministic case. Again proper convergence studies have to be performed [11].


5.2 Parametric stochastic modeling

Parametric stochastic modeling is based on a probabilistic modeling of some of the mean model parameters. It is relevant when there are no uncertainties related to the modeling and when the targeted parameters’ stochastic behavior can be characterized by their probability density function. As far as realizations of the parameters values can be provided, the uncertain modeling can be constructed by changing some parameters deterministic values to random values according to their probabilistic properties.

Even when uncertain parameters directly address physical matrices, it is often convenient to project uncertainties effects in the modal space. This leads to fill the diagonal generalized mass and stiffness matrices.

From a practical point of view, uncertainties parametric modeling requires the knowledge of the probability density functions of as many parameters as required. This kind of uncertainties modeling is the most relevant for example for engine mount stiffness whose spread is often addressed in the associated technical requirements, or for rigid body dynamics [12].

Uncertainties parametric modeling is a feature of many software. The main issue of parametric modeling is the quantification of the probability density function and the ability of software to handle actual functions. It will not be later addressed here since it was never recognized as the major cause of uncertainties when modeling vehicles.

5.3 Non-parametric probabilistic structural-acoustic modeling using the maximum of entropy principle

The non-parametric probabilistic modeling is introduced in order to address the modeling uncertainties, which appeared to be the major cause of uncertainties in a simulation process. Non-parametric modeling means a specific probabilistic modeling not related to the model physical parameters. Nevertheless, such a technic should be built in accordance with fundamental physical principles.

5.3.1 Random matrices theory

Since some of the considered uncertainties are not explicitly included in the mean model, first of which are the modeling uncertainties, the non-parametric probabilistic modeling was introduced [13][14]. The non-parametric modeling considers uncertainties of any kind. A random matrix theory ensures -thanks to a maximum of entropy principle- that any kind of physically possible modification in a range is accounted.

Theoretical derivations [15] show that random positive matrices can be algebraically constructed using the available information (e.g. the mean matrices) and a single hyper-parameter, ensuring any possible uncertainty within a given range will be accounted according to the maximum of entropy principle [16]. The mean of the random matrices is the mean model matrix and the associated normalized standard deviation is the dispersion hyper-parameter.

Practically, any positive random matrix, 5W9, is constructed from the mean matrix, 5ℬ9, decomposed using a Cholesky decomposition, 5ℬ9 = 5ℒℬ9Y5ℒℬ9, and a random seed,5ZW9 as

5W9 = 5ℒℬ9[5ZW95ℒℬ9 (10)

The random seed matrix 5ZW9 is a full positive definite matrix such that �\5ZW9] = 5^9 The dispersion hyper-parameter controlling the deviation range of the seed matrix is:

_ℬ = `ab5ZW9cBdℬDbefgbBdℬDbef (11)

KEYNOTE 27

The algebraic construction of 5ZW9 realizations is described in [15].

It is to be noticed that diagonal mean matrices –such as mass or stiffness matrices- will generate full random matrices.

Semi-positive matrices (i.e. with null eigenvalues) such as the fluid stiffness matrix, or the structural stiffness matrix of unsupported elastic solids, have to be handeled with care. Null eigenfrequencies of mechanical systems are not subject to dispersions; only their responses may be dispersed.

The rectangular coupling matrix, 5Q9 is decomposed using a polar decomposition [17]: 5Q9 = 5h95ℬ9, where 5h9 is a rectangular matrix such that 5h9S5h9 = 5^9 and 5ℬ9 is a positive definite matrix. Consequently, the random coupling matrix, 5i9 , is built as:

5i9 = 5h95ℒℬ9[5Z95ℒℬ9 (12)

5.3.2 Probabilistic vibro-acoustic responses

The stochastic elasto-acoustic problem is built from the mean model by substitution of the mean matrices by random matrices. System’s response then becomes random and consequently, random generalized coordinates are introduced such that:

I5j<�8�9 5i98=5i9 BjG�8�DJ Kk<�8�kG�8�L = KR7�8�RC�8�L (13)

where 5j<�8�9 = −8=5l<9 + ?85m<9 + 5n<9 , BjG�8�D = −8=BlGD + ?8BmGD + BnGD, and each of the bold matrices is built according to Eq (10).

As mentionned earlier, excitation terms are considered to be deterministic. They could easily be turned into probabilistic in a Monte-Carlo simulation. It should be noted that when model uncertainties effect on responses involves a system inversion, excitation uncertainties would linearly apply on responses.

Random physical response may then be computed using the mean model modes for the structural and acoustic domains:

o<�8� = 5M79k<�8� (14a)

pG�8� = BOCDkG�8� (14b)

5.3.3 Probabilistic modes

Stochastic modes of the structural and acoustic parts may also be computed from the generalized random mass and stiffness matrices as the solution of the following random eigenvalue problems:

5n<95q<9 = r5l<95q<9 (15a)

BnGDBsGD = rBlGDBsGD (15b)

Modal shapes in the physical space are then obtained by expanding over the mean model modes:

5t<9 = 5M795q<9 (16a)

BuGD = BOCDBsGD (16b)

In order to perform statistics, the Stochastic Modal Assurance Criterion (SMAC) with respect to the mean modal basis is to be computed [18]. For the structural part, it expresses as:

5v>6H791,x = y5z{9|+5}{95t<9~yy5z{9|+5�<95t<9~y5z{9|+5}{95z{9+5t<9|~5�<95t<9~ = �Zl��+,+�Zl��~,~ y5q<91,xy= (17)


6 Model assessment and uncertainties quantification

6.1 Maximum of likelihood principle

The probabilistic model that has been built in section 5 is able to provide most of results currently used for industrial applications (FRFs, responses and modes) in a stochastic form. It is based on an existing mean model and is only controlled by 7 dispersion hyper-parameters -one for each constitutive matrix- noted \_]. The model can be used for predictions as soon as dispersion hyper-parameters have been identified. The model capabilities and uncertainties quantification can be performed at once when identifying these parameters using the maximum of likelihood principle.

Indeed, as mentionned in section 3 the likelihood –or log-likelihood- is a key quantity for the stochastic model prediction capability assessment; it can also be used to identify the dispersion parameters.

The log-likelihood is a function of the model dispersion hyper-parameters, that can be schematically defined as:

ℒℒ�δ� = 234�& ��,��*�� (18)

Where ��,� is the joint probability density function of the stochastic system response Y, for a dispersion parameter _ –by extension a dispersion hyper-parameter set \_]-, and �*�� is an actual experimental realization of the studied system response.

The maximum of likelihood principle states that the optimal dispersion parameter value is the one that provides the highest probability of the observation to be part of the model.

δ��S = Argmax� ℒℒ�δ� (19)

The use of the maximum of likelihood principle implies that the results probability density function will have to be computed –generally requiring a Monte-Carlo simulation- as many times as required by the optimization process. In the case of a multi-variable optimization –at the most 7 dispersion parameters- this can lead to intensive CPU consumption.

Nevertheless, the determination of the optimal dispersion parameters has to be conducted at least once, to be able to perform any prediction.

Depending on the point of view, this optimization process updates and assesses the stochastic model or identifies the amount of uncertainties related to the studied problem.

6.2 Log-likelihood of FRF matrices

In vibro-acoustic applications, the model’s response at observation variable point M to any (unit) excitation applied at the variable point Q is a collection of �� FRFs, often organized in a frequency dependent matrix:

��,��,>,8��,�� (20)

Since this is a multiple dimension observation of a dynamical system, one may expect some correlations to exist in-between points (due to modal shapes) and frequencies (due to modes eigenfrequencies). The actual probability of an observation to occur is the product of independent components’ own probabilities. Since independence is difficult to determine, we will consider here uncorrelated components [6][19].

The mean value of the realizations is:

��,>,8� = ��,��,>,8�� (21)

and the auto-correlation matrix of the realizations is :

KEYNOTE 29

�� = � ��,��,>,8� − ��,>,8�� ,��,>,8� − ��,>,8��S� (22)

This matrix is diagonalized in order to separate uncorrelated components. The random realization can then be written as:

��,��,>,8� = ��,>,8� + ∑ �¡,��,�¢£¡,� ¤¡,��,>,8�¡�}¥¦�¥¡�� (23)

where £¡ is an eigenvalue of the auto-correlation matrix and ¤¡ ��,>,8�is the associated eigenvalue, and

�¡,��,� = �¢§¥,¨ ��,��,>,8� − ��,>,8��S ¤¡,��,>,8� is a set of realisation of the random

uncorrelated variables, ¡,�.

The number of component considered,>¡ , can be significantly reduced, when compared to the size of the correlation problem, N¡. For each component, a probability density function,� ¥,¨, can be constructed from the Monte-Carlo simulation results using a kernel density estimation [20]. It then becomes possible to evaluate the likelihood of an experimental observation, �*��,>,8�. First, one has to compute the independent components responses:

�*��,>,8� = ��,>,8� + ∑ ¡,�*�� ¢£¡,�¤¡,��,>,8�¡�}¥¦�¥¡�� (24)

With ¡,�*�� = �¢§¥,¨ '�*��,>,8� − ��,>,8�(S¤¡,��,>,8� One can then write the probability of the observed FRF matrix, as the product of the uncorrelated components probability:

��,�'�*��,>,8�( = ∏ � ¥,¨' ¡,�*��(¡�}¥¡�� (25)

6.3 Practical application

Dispersion hyper-parameters have been identified using production vehicle measurements such as the one presented in section 2, and a vibroacoustic finite element model such as the one shown on figure 12; in the presented case, the model was built 2 years prior to the measurements.

Up to now, dispersion parameters were identified step by step to avoid solving the maximum of likelihood problem with more than 2 variables. Finally, the seven hyper-parameters were identified allowing computation of the stochastic response of the studied structure. Some results are shown on figure 13.

(a)(b)

Figure 13: measured (gray patch) and computed FRFs (5th and 95th centiles solid black lines; 1st and 99th centiles dotted black line). (a): vibroacoustic FRF; (b): acoustic FRF

As it can be observed on figure 13, the maximum of entropy generally leads to overestimate the spread of results, because any physical change of the system is considered within the range identified to cover all uncertainties related to the engineering process: design, modeling, manufacturing… Nevertheless, at least for complex objects such as cars, the maximum of likelihood assumption appears to be relevant.


7 Probabilistic model physical properties

This section is dedicated to results obtained from the non-parametric probabilistic modeling [21]. Such results consist in statistics performed on hundreds of realizations of uncertain systems: therefore, most of it could not be achieved experimentally. Nevertheless, it will be seen that some trends fit with known practical results.

7.1 Random modes

Modes are characterized by their modal shape and eigen-frequencies; when the system is uncertain both become random functions, requiring statistical post-treatments.

7.1.1 Random modal shapes

Computation of the stochastic MAC, as defined by (17), was performed on a wide frequency range, for a vehicle structure only. The stochastic MAC indicates the shape correlation between the mean model modes and the random modes of the associated uncertain structure. In practice, the mean value provides useful information. Figure 14 shows the mean stochastic MAC, ��v>6H7,��, for the first 200 modes of a car body. It is computed for an uncertain model with a level of dispersion associated to the hyper-parameter set \_]. The lowest 80 modes are plotted first, than the remaining 120 modes.

At low frequencies, one can mostly see high values on the diagonal, which means random shapes remain close to mean modal shapes. Nevertheless, some groups of modes show correlated shapes. This is a hint of an increased sensitivity of these modes to uncertainties. From this point of view, it can be said that all modes do not behave the same regarding uncertainties. Therefore, no a priori assumptions can be made on this point at least for the lowest frequency range. In this case, the range of the 80th mode seems to be the limit above which no more relevant correlation may exist between deterministic and stochastic modes.

In the mid-frequency range, the picture appears totally shaded around the diagonal. This means random modes’ shapes are always combinations of neighbor mean modes shapes. Again, the spread depends on the considered shape.

(a) (b)

Figure 14: Stochastic MAC of a car body plotted for the first 80 modes (a) and upper 120 modes (b)

7.1.2 Random eigen-frequencies

Random eigen-frequency statistic takes some sense when it is related to a well-defined modal shape. The previous section showed this is only feasible in the low frequency range. The effect of uncertainties on a well-defined structural mode has been studied separately for the mass and stiffness operators. Figure 15

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compares the distributions of the eigen-frequency values for increasing values of the dispersion hyper-parameters.

First, it can be noticed that the mass and stiffness uncertainties do not produce the same effects. Mass uncertainties cause a centered distribution while stiffness uncertainties induce a shift of the distribution to the low frequencies. This shift appears to be much larger than the span of the dispersion. The standard deviation of the random eigen-frequency is similar for both effect and proportional to the dispersion hyper-parameter.

(b)

Figure 15: Distribution of a single random mode eigen-frequency due to stiffness (a) and mass (b) uncertainties, for various values of the associated dispersion hyper-parameter.

7.2 Random FRFs

7.2.1 FRF sensitivity to the amount of uncertainties

Figure 16 shows an example of computed random structural FRF plot (mean and 95% confidence interval) compared to the mean model FRF, when increasing the amount of uncertainties. For each of the plots, the general picture looks very similar to the results of section 2: dispersion is increasing with frequency. As expected, dispersion also increases when uncertainties are increasing. It is noticeable that for the largest amount of uncertainties, the mean model result may be outside of the random model bounds. This is related to the low frequency shift already observed in the previous section. A highly uncertain structure can not behave as the same ideal (fully certain) structure.

Figure 16: Dispersion of a single structural FRF when increasing uncertainties (from left to right). . Mean model (solid red line), stochatic model 95% confidence region (blue patch), and stochastic mean (dotted

blue line)

7.2.2 FRF sensitivity to observation points location

Figure 17 shows computed random FRFs for the same excitation point and different observation points. These plots are the results of the same Monte-Carlo simulation i.e. with the same dispersion hyper-parameter set. As it was observed in section 2, the dispersion seems to increase with the distance between observation point and excitation point. It is also higher for soft structural parts (panel) than hard structural parts (framework).

(a)


Figure 17: Dispersion of FRFs between a single excitation point and various observation locations. Same

dispersion hyper-parameter set. Mean model (solid red line), stochatic model 95% confidence region (blue patch), and stochastic mean (dotted blue line)

7.2.3 About FRF statistics

Figure 18 shows typical statistical indicators of random FRFs obtained from a Monte-Carlo simulation: median value, 95% probability bounds and 99% probability bounds, associated to different locations (one soft and one hard point) on a structure. First it can be noticed that the distribution is asymmetrical: median is not in the middle and distance between 1th centile and 5th centile is much larger than the distance between 95th and 99th centiles. This is more obvious for the soft observation point. The fact that the 95th and 99th centile are very close one to each other is surprising but convenient. It suggests that there exists an absolute upper bound for the random FRF which is not much higher that the 95th centile. On the other hand, it seems that the lower bound of FRFs modulus depends on the targeted probability level.

As a consequence, the upper bound of FRFs converges faster than the median, that converges faster than the lower bound.

In any cases, these results are far away from any known probability laws. This assesses the strategy that was chosen, not to use any a priori probability laws for the input nor the output.

When the Monte-Carlo simulation has converged enough to provide responses probability density functions, predictions of the model can be handled in a probabilistic way. This generally requires a few thousand realizations.

Figure 18: examples of FRFs mean value (gray), upper and lower bounds with a 95% probability (solid black line), upper and lower bounds with a 99% probability (dotted black line). Soft observation point

(right) and hard observation point (left)

8 Practical use of a probabilistic model results

In this section, some practical booming noise results will be presented in order to highlight the benefits of probabilistic computations within an industrial engineering process. Booming noise is the response at driver’s ear, due to the engine’s second harmonic for a 4 cylinders engine, versus engine rpm. Details of the computations and industrial implementation of the non-parametric probabilistic modeling is detailed in

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[22]. All results are obtained from current vehicle projects model, with a dispersion hyper-parameter set \_] identified from production vehicle measurements. Computations of the Monte-Carlo simulation are performed using massive parallelization such that it can be completed overnight.

Probabilistic booming simulation is performed as the product of probabilistic vibroacoustic FRFs by measured loads (3 engines mounts x 3 directions = 9 DOFs), to avoid unwanted (secondary) path effects in the analysis.

8.1 Model assessment vs measurement

In order to be comparable, measured and computed booming are both calculated as the product of vibroacoustic FRFs with measured loads (3 engines mounts x 3 directions = 9 DOFs). As mentioned earlier, the loads uncertainties are not considered here. Computed FRFs can be either deterministic (when computed from the mean model) or stochastic (when computed from the probabilistic model).

Figure 19-a compares the mean model result with the measurement. Differences of 5 to 10 dB exist all over the computed range, making the model assessment difficult. Figure 19-b presents the comparison of the measured booming with the probabilistic model results. In this case, the measured values are outside the expected bounds only in specific ranges, providing useful information to target the model’s weaknesses; in the presented case, the main discrepancy occurs around 2500 rpm.

(a) (b)

Figure 19: (a) comparison of the measured booming (black) with the mean model result.(b) comparison of the measured booming (black), with the probabilistic model results (yellow patch 95% confidence

domain)

8.2 Project assessment against a target

Figure 20-a shows the check of a booming noise simulation result against a project target. Above 2300 rpm, simulation exceeds more or less the target, without information on how critical it is. On figure 20-b, when introducing the confidence interval of the probabilistic model (5th,50th and 95th centiles), one may say that below 3000 rpm, more than 50% of the results will remain lower than the targets. On the contrary, above 3500 rpm, most of the results will exceed the target.

8.3 Design sensitivity analysis

When comparing alternative designs -for example related to mass reduction-, it may happen that depending on the frequency contradictory trends appear. Figure 21-a shows such an example, where differences, associated to a minor design change, leads to an increase of more than 3 dB, which would


lead to reject one of the design. Figure 21-b compares the mean results of the two configurations including uncertainties. It then appears that, in average, the differences remain lower than 1 dB, which could be acceptable.

(a) (b) Figure 20: (a) Mean model booming (red line) compared with a project target (black line). (b) Stochastic

booming results (yellow patch: 5th, 50th and 95th centile) compared with a project target (black line)

(a) (b)

Figure 21: Computed booming noise: (a) baseline (black) and alternative (red) designs computed for the mean model. (b) same comparison for the stochastic model’s mean result.

9 Conclusions and outlook

At the end of the paper, it seems that the question of what we are able to predict for uncertain vibroacoustic systems is rather clear: as far as a relevant modal model (e.g. finite element model) is available, any FRF or modal characteristic can be computed as a random variable. Assuming uncertainties causes are numerous and various, a simple global approach using the maximum of entropy principle is proposed. The proposed method is physically and mathematically consistent, and it allows simulating most of the -experimentally known- behaviors of uncertain structural-acoustic systems. The uncertainty level accounted can be adjusted to the model refinement level. Forecasting may then be considered from a probabilistic point of view, which is the best that we can do up to now.

The maximum of entropy uncertainties modeling constitutes a good trade-off between the results relevance and the effort to assess and identify the probabilistic model parameters. Nevertheless, the first identification of dispersion hyper-parameters requires extensive measurements on series production samples, as well as computational ressources. Once this step is overcome, a new field opens showing properties that seems to fit common knowledge, although proper experimental proof of these properties is not yet achieved.

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The results presented in this paper represent the core results of 10 years of close collaboration between academics and engineers. They are the foundation of a series of investigations where stochastic properties allow simplifications of physical behaviors that could not be developed here [23][24][25].

Acknowledgements

The author would like to thank Pr, Christian Soize and the PhD students (see references) from Université Paris-Est, who provided most of the theoretical and practical work that is presented here, the colleagues at PSA involved in the industrial implementation of the stochastic methods for their help and support over the years (see references) and finally, Pascal Bouvet and the VIBRATEC team in Lyon (France) for carefully performing most of the measurements.

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dispersed vibroacoustic responses of industrial products...

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