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Analysis of the performances of automotive carbodies with different structural contrasts

CONFERENCE PAPER · SEPTEMBER 2015

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Martial Nobou Dassi

PSA Peugeot Citroen

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Laurent Gagliardini

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Charles Pezerat

Université du Maine

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François Gautier

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11th International Conference on Engineering Vibration

Ljubljana, Slovenia, 7–10 September 2015

ANALYSIS OF THE PERFORMANCES OF AUTOMOTIVE CARBODIES WITH DIFFERENT STRUCTURAL CONTRASTS

M. Nobou Dassi*1,2, A. Gaudin1, Z. Abbadi1, L. Gagliardini1, C. Pezerat2, F. Gautier2

1PSA Peugeot Citroen, Direction de la Recherche et de l’Ingenierie AvanceeRoute de Gisy 78140 Velizy, France

martial.noboudassi@mpsa.com

2Laboratoire d’Acoustique de l’Universite du Maine,Rue Aristote, 72085 Le Mans cedex 9, France

Keywords: Vibration, NVH performances, Structural contrast, Mobility, Automotive.

Abstract.  To reduce the greenhouse gas emissions produced by cars, one solution considered 

is to reduce their masses by using composite materials. However, the introduction of such

orthotropic materials can significantly modify the automotive design due to the new distribution

of local stiffnesses. These modifications may result in very different vibroacoustic behaviours

and therefore different NVH performances. Indeed, the panels can participate more greatlyto the overall stiffness of the car, reducing the ratio of frame in the architecture and therefore

changing the structural contrast between frame and panels. The objective of this work is to

investigate the NVH performances of several car body models built with different structural

contrasts.

Several strategies can be used to highlight the structural contrast of a mechanical system made

of frame and panels. In this work, this concept is developed by calculating the input point 

mobilities. This approach allows building maps which are representative of the vibrational

behaviour of the car body. On such maps, the areas corresponding to the presence of stiffeners

(frame) can be identified, and a histogram containing the input points mobilities values allows

the definition of a dynamic structural contrast indicator. This indicator can be used to evaluatethe level of contrast of any kind of models.

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1 INTRODUCTION

Since several years, the mass reduction of structures linked to CO2  emissions in the auto-

motive industry has become one of the main topics of industrial research. The main impact

of introducing lightweight materials such as composite materials in a car is the very poor per-

formance in terms of vibroacoustic behaviour. This performance is directly linked to some

structural parts of the car body. As it happens, the resistance to external efforts is convention-

ally ensured by the frame, whereas the tightness is ensured by the panels. It seems obvious

that this conventional configuration may change, due to the specific properties of the new used

materials. In this context, panels made with composites materials may for example participate

more greatly to the overall stiffness of a car.

Design an efficient architecture of car body suitable to the lightweight materials, highlight the

main parameters that control the behaviour and evaluate the resulting NVH performance are

the main topics that we are interested in. To achieve this, we will use a simplified model of 

car, which will be declined in several models with different configurations. The study of these

models will allow bringing out the concept of structural contrast.

2 THE GENERAL AUTOMOTIVE NVH

The Noise and Vibration Harshness (NVH) of a car body is generally defined as the study

and the modification of the noise and vibration characteristics of a vehicle to achieve a given

performance target. In the automotive industry, the targeted performance is related to needs

expressed by the customers, which are representative of the overall comfort feeling of the car.

The expressed needs are accurately studied by techniques like System Engineering, which allow

identification of subsystems that control the overall behaviour. This approach therefore leads

to the definition of a functional and a component architectures, which ensure the coherence

between the subsystems behaviours and the NVH performance.

2.1 The automotive NVH behaviour

During his life cycle, a car body is subjected to several kinds of excitations that generate

operating organs vibrations and noise inside the air cavity. To reduce the noise and vibrations,

Cremer [1] explains that one should control the phenomena bellow:

• the generation defined as the origin of an oscillation;

• the transmission of oscillatory energy to a passive structure;

• the propagation of the transmitted energy throughout the structural systems;

• the radiation of power from involved subsystems to the connected fluid.

These phenomena can be presented as shown in figure 1.

The figure 1  highlights the fact that some subsystems have to be accurately designed in order

to satisfy the needs expressed by the stakeholders. To ensure the NVH performance, each

subsystem shall fill specific functions. The set of functions to be filled by the subsystems defines

the functional architecture of the car.

2.2 The automotive NVH performances

The NVH performance is defined for each considered frequency range. Indeed, each fre-quency range matches a particular behaviour, and therefore a specific definition of performance.

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Figure 1: Noise sources and transfer paths on a car body.

The automotive frequencies ranges can be defined as following:

•  the Very Low Frequencies  (VLF) range, which corresponds to the zone where appear the

first global deformation modes of the car body (torsional mode, bending modes...), vehicle

suspension modes and the mounted powertrain modes. This range also correspond to the

zone where appear the human being modes, which need to be carefully avoided;

•   the Low Frequencies (LF) range, which corresponds to the zone where the car body has a

modal behaviour, with distinct and isolated resonance peaks;

•   the Mid-Frequencies   (MF) range, reveals a mixed behaviour. While some areas of the

car continue to have a modal behaviour, others have a more homogeneous behaviour

with a high level of overlap between adjacent peaks. This dual behaviour makes thisrange difficult to analyse because it involves the use of both deterministic and statistical

techniques;

•   the High Frequency   (HF) range corresponds to the zone where the response is almost

uniform throughout the car. This behaviour allows the use of assumptions like diffuse

field and energetic techniques on the overall car body.

Regarding the introduction of composites materials in the car and therefore the structural con-

trast, we are particularly interested in the NVH performance on the LF and the MF ranges.

2.2.1 The performance on the Low Frequency range

The LF range is related to a particular field of the mechanics which introduces all the de-

terministic methods and allows an accurate study of complex systems. These methods use

appropriate mathematical developments in order to predict the response of a system in terms of 

displacement or pressure field for a given and well described excitation.

To meet the expressed needs, one should control the vibrational behaviour of the car and the

pressure field in the air cavity, which represents the noise level felt by passengers. The vibro-

acoustic performance will therefore be defined as the ability to guarantee a low level of vibration

on the structure and a low level of pressure in the air cavity for a deterministic load applied to

the system.

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2.2.2 The performance on the Mid Frequency range

The MF range is characterised by a dual behaviour, representative to both LF and HF ranges.

This dual behaviour requires the introduction of energetic approaches, which allow taking into

account both stochastic and deterministic behaviours. Energy averages are then made to char-

acterize the behaviour of the structure and the cavity.

To properly design on this frequency band, one should control the energetic vibrational levels

on the structure and the energetic noise levels in the cavity. The NVH performance is therefore

defined as the ability to guarantee a low energetic vibrational level on structural subsystems and

a low energetic pressure level in the air cavity.

3 DESIGNING A SIMPLIFIED MODEL OF CAR FOR THE ANALYSIS

The models of cars presented in this section were already used by the authors in [4] and [3]

in the general context of understanding the consequences of introducing composites materials

in the automotive design. These models are simplified finite element models of a Citroen C3.

The original model is a finite element body in white as shown in Figure 2. This original model

has a mass of  284kg  and 1, 150, 000 degrees of freedom.

Figure 2: Original Finite Element Model

The simplified model car is obtained from this original model by introducing two main simpli-

fications. The simplifications aim to provide an easy-to-use model, with sufficiently represen-

tative results compare to the original car body. The first simplification consists in representing

the hollow sections by rectangular section beams, as presented in figure  3.  This simplification

reduces the number of degrees of freedom of the system (and therefore saves computation time).

It also facilitates changing the thickness and inertia (without remeshing or morphing). Applyingthis on all the car body’s hollow sections leads to the definition of the simplified model’s frame.

The second simplification introduced consists in modelling some car body panels by introduc-

ing an orthotropic behaviour. This allows as presented in figure 4  to take into account both

ribbed floors and composites materials panels behaviours. The application of this procedure

leads to the definition of the simplified model’s panels.

Assembling frame and panels allows the definition of the simplified model of car as presented

in figure 5.  This simplified model has only 200, 000 degrees of freedom.

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Figure 3: Designing the frame

Figure 4: Modelling panels on the simplified models

Figure 5: Simplified model of car body

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4 THE CAR BODY MOBILITIES MAPS

A car body system generally has a high complexity, due to the presence of different compo-

nent parts. The separation of the car body in a frame and a set of panels constitutes a first step

of sub-structuring in order to overcome this complexity. The frame in this context represents

the part that ensures resistance to the overall system, while the panels ensures tightness. To ef-

ficiently introduce the composites materials in the cars, we propose a point mobilities approach

related to the energy balance of the car to characterize the contrast between the frame and the

panels and identify the configuration which leads to the best performance compromise.

4.1 The car body point mobilities

The equation of the structure discretized by the finite element method is conventionally writ-

ten in the harmonic domain as follows:

K  +  jK η +  jωC  − ω2M 

u (ω) = F  (ω) ,   (1)

where M , C  and  K  are respectively the mass, the damping and the stiffness matrices.  K η  is the

structural damping matrix, associate to the loss factor η. ω  is the pulsation, u is the displacement

field and F  is the applied force vector.

The projection on the eigen modes basis allows writing equation  1  as presented in equation 2,

by introducing the eigen vectors matrix  φ  defined by u =  φq:φtKφ + jφtK ηφ + jωφtCφ − ω2φtM φ

q (ω) = φtF  (ω) ,   (2)

where q is the modal coordinates vector. If the damping is small enough, all projected matrices

are diagonal. The terms of the diagonal can be written as follows:

φ

t

Kφ = [k p]   (3)φtK ηφ = [η pk p]   (4)

φtCφ = [c p]   (5)

φtM φ = [m p]   (6)

Equation 2 can therefore be written as:k p(1 + jη p) + jωc p − ω2m p

q p (ω) = φt

 pF  (ω) ,   (7)

The response of the system is then written as a sum of independent modal contributions as

presented in equation 8:

u (ω) =n

 p=1

φ pφt p

k p(1 + jη p) + jωc p − ω2m p

F  (ω) ,   (8)

The input point mobility of a structure is the ratio between the velocity at the degree of freedom

M i  and the applied force at the same degree of freedom as presented in equation 9:

Y M i,M i  =  jωuM i

F M i

.   (9)

The transfer mobility of a structure is the ratio between the velocity at the degree of freedom

M i  and the applied force at another degree of freedom Qi as presented in equation 14:

Y M i,Qi =   jω uM i

F Qi

.   (10)

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The damping is generally introduced either in viscous form or as a structural damping. The

correspondence is ensured by writing at the resonance  η p  = 2ξ  p. It is also possible to introduce

the concepts of effective modal masses, effective modal forces and generalized modal pulsation

respectively defined as:

meff  p   =

φt pM φ p

φt pφ p

(11)

F eff  p   =φt pF 

φt pφ p

(12)

 ω2 p  = ω2(1 − jη p)   (13)

The mobility can therefore been written as:

Y M i,Qi (ω) =

n

 p=1 jω

 ω2 p + 2 jξ  pω pω − ω

2

φM  p   φQ

 p

m p

.   (14)

4.2 The energy balance and mobilities maps

The energy balance of a structure is done by introducing the terms of injected and dissipated

power. The injected power due to an excitation force  F  is conventionally written as presented

in equation 15:

ΠF   = Re (V∗F ) ,   (15)

where V∗ is the conjugate transpose of the velocity. For an isolated system, the dissipated power

equals the injected power. If the damping is introduced by a constant loss factor, the vibrational

energy is proportional to the average power dissipated through the structural loss factor.

To study the power injected into the structure, we assume a force having constant spectral power

density and which is therefore frequency independent. Using the definition of the input point

mobility in a modal approach, the mean injected power ΠF  is written as follows:

ΠF ∆ω  =

 ∆ω

Re {F ∗Y ∗F } dω   (16)

=

 ∆ω

Re

  n p=1

 jω

 ω2 p + 2 jξ  pω pω − ω2

F tφ pφt pF 

m p

dω   (17)

ΠF ∆ω  =

n

 p=1 φt pF 

2

m p  ∆ω

Re   jω ω2

 p + 2 jξ  pω pω − ω2 dω.   (18)

The term in the integral tends to   π2

 when the resonance frequency is included the band  ∆ω  and

is equal to zero otherwise. The mean injected power therefore becomes:

ΠF ∆ω ≈  π

2

ωk∈∆ω

|φtkF |2mk

.   (19)

For mass normalized modes, the mean mobility at the degree of freedom M i is therefore written

as:

Y M i,M i∆ω ≈  π

2 ωk∈∆ω

φ2

k(M 

i)

∆ω   ,   (20)

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In the case of a 3D structure, the point mobility correspond to a matrix. To characterize the

excitability of a structure, it is logical to consider the mobility in the direction of maximum

excitability. This quantity calculated at each node of the model allows building of a mobilities

map, which shows the stiffness behaviour of each zone of the car. An example of mobilities

map for the frequency band [500 − 1000]Hz is presented in figure 6.

Figure 6: Mobilities map of the simplified model of car

On such a map, the areas corresponding to the presence of stiffeners (frame) can be identified

easily. We are interested in understanding the evolutions of mobilities values obtained on this

map.

Let us consider some elements of the model and compare the obtained mobilities with corre-

sponding analytical values. Studying evolution of point mobilities between the different areas

of the structure will allow highlighting a structural contrast. Cremer [1] gives the as presented

in equation 21  an analytical expression of point mobility for an infinite plate, computed in its

center:

Y Ana = 1

4

 3(1 − ν 2)

ρEh4  ,   (21)

where ν  is the Poisson ratio of the material,  ρ  is the mass density, E  is the Young modulus and

h is the thickness of the plate. Let us consider the roof, the windscreen and the rear floor pan

having different mobilities values as highlighted in figure  7.

These panels have the properties presented in table 1

E [M P a]   ρ[   tmm3 ]   ν h[mm]

Roof    2.10 e+5 7.85 e−9 0.3 0.67Windscreen   7 e+4 2.5 e−9 0.22 4.4Rear floor pan   13.65 e+6 7.85 e−9 0.3 0.67

Table 1: Selected panel’s properties

As explained during the presentation of the simplified model, some panels have an orthotropic

behaviour. The rear floor pan is one of those panels. Since the values displayed on each node

correspond to the mobilities in the direction of maximum excitability, the Young modulus used

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Figure 7: Selected panels mobilities

to calculate the analytic values is taken in the same direction. Applying these properties in

equation 21 leads to the analytical mobilities  Y Ana presented in table 2.  The table also presents

the numerical mobilities values Y Num, obtained on the built map.

Roof Windscreen Rear floor pan

Y Ana[(mmt.N 

 )1

2 ] 22.66 1.65 2.8

Y Num[(mmt.N 

 )1

2 ] 19.2 1.1 3

Table 2: Comparison of analytical and numerical mobility

The mobilities levels obtained analytically and numerically are close enough to validate the

developed approach. The small differences between the values can be attributed to the fact that

the selected panels are not infinite (as considered in the analytical approach), and the selected

panels interact with other subsystems having different mobilities levels.

Cremer also give an analytical expression of point mobility for an infinite beam. This mobility

computed in the center of the beam is written as presented in equation 22:

Y Ana =  1

4√ 

ω

1 − j

(EI )1

4 (ρA)3

4

,   (22)

where j  is the imaginary unit,  I  is the moment of inertia of the beam and  A  is the section area

of the beam. Let us consider (as done for the panels) the roof reinforcing crosspiece, the engine

support and the centre pillar as highlighted in figure 8.

These beams have the properties presented in table 3

E [M P a]   ρ[   tmm3 ]   ν I [mm4]   A[mm2]

Roof reinforcing crosspiece   2.10 e+5 7.80 e−9 0.3 2.78 e+3 173.32Engine support   2.10 e+5 7.80 e−9 0.3 2.32 e+2 30.84Centre pillar   2.10 e+5 7.80 e−9 0.3 3.05 e+5 720.44

Table 3: Selected beam’s properties

As explained in the case of the panels, the mobilities are computed in the direction of maximum

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Figure 8: Selected beams mobilities

excitability. The moments of inertia here presented are thus taken in the same direction. Ap-

plying these properties in equation 22 leads to the analytical mobilities values  Y Ana  presented

in table 4.  The table also presents the numerical mobilities values  Y Num, obtained on the built

map.

Roof reinforcing crosspiece Engine support Centre pillar

Y Ana[(mmt.N 

 )1

2 ] 0.72 4.91 0.076

Y Num[(mmt.N 

 )1

2 ] 0.68 4.69 0.07

Table 4: Comparison of analytical and numerical mobility

Table 4 allows validating the beam’s mobilities values obtained on the maps. The observed dif-

ferences can be attributed to the nature of the beams (not infinite as supposed in the analytical

expression), to the presence of adjacent elements, and also to the curvature.

This approach brings out the fact that an automotive structure has areas with different mobility

levels, due to local stiffness differences. These differences can be characterized by introducing

the concept of structural contrast. We are interested in defining a related indicator which helps

understanding the impact of changing materials on the vibroacoustic performances of a car.

5 LIGHTWEIGHT MODELS AND DEFINITION OF THE CONTRAST INDICATOR

5.1 Definition of lightweight models

The use of composites materials in the automotive design has significant advantages that

Giocasa presents in his work [2] as following:

• a low mass density of the final structure;

• the possibility to obtain structures with complex geometry by moulding and over-moulding;

• the use of the optimum amount of material required for the desired function;

• removing machining.

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Such materials may however introduce difficulties, due to the new distribution of local stiff-

nesses. To analyse their impacts on the design, one solution is to build a comparative analysis

with models having different design scenarios and therefore different structural contrast levels.

In order to achieve this, we propose to change the Young modulus used for the frame, from a

very low value corresponding to a polymer (E ≈ 8 ∗ 108

Pa) to a high value corresponding toa carbon fiber (E ≈ 4 ∗ 1011 Pa). Switching from one model to another, the Young modulus is

doubled while the Poisson ratio and mass density are kept constant in order to observe only the

effect of the structural contrast. Thus, the 10  built models have the frame material properties

presented in table 5.

Model   1 2 3 4 5 6 7 8 9 10E [Pa] 8.20 e5 1.64 e6 3.28 e6 6.56 e6 1.31 e7 2.63 e7 5.25 e7 1.05 e8 2.10 e8 4.20 e8

ν    0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

ρ[  kgm3 ] 7.8 e3 7.8 e3 7.8 e3 7.8 e3 7.8 e3 7.8 e3 7.8 e3 7.8 e3 7.8 e3 7.8 e3

Table 5: Frame material’s properties

5.2 Mobilities histogram and contrast indicator

For each of the defined models, a study of the structural contrast is done on the frequency

band [0 − 1000] Hz, in order to highlight the evolution of the structural contrast while the

frequency increases. A mobilities map of each model is built. Figure  9  presents mobilities

maps of the models 1, 6  and 10. These models are chosen because they allow to observe major

changes in structural contrasts.

To understand the results provided by the maps, let us introduce some parameters used in theimage theory. An image histogram is conventionally defined as the function that associates to

each intensity value the number of pixels having that value. There are several definitions of 

contrast of an image. Peli [5] presents in his work a common way to define the contrast of an

image, so that the contrasts of several images can be compared. He used the root-mean-square

contrast definition presented in equation 23.

rms =

    1

N  − 1

N n=1

(xn − x)2,   (23)

where xn  is a normalized grey-level value and  x  is the mean normalized gray level. To charac-terize the structural contrast of a car, we are interested in building similar parameters based on

a mobilities map. Let us consider at each node the trace of the real part of the mobilities matrix,

corresponding to the sum of translational mobilities as defined in equation  24.

Y i = Y xi +  Y yi +  Y zi .   (24)

This definition allows building histograms related to mobilities maps as presented in figure  9

for models 1, 6  and  10. On such histograms, the main differences between the models appear

on the nodes having low mobilities level, and corresponding to the frame. Indeed, since the

panels have not been changed, their average mobilities remain unchanged as visible from

 ≈ 3

to 200  [mm/s/N]. It is also possible to notice that the more the frame is stiffened, the more lowmobilities nodes we have. Finally, we note that the windscreen properties remain unchanged,

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and his mean mobility correspond to the peak near  1  [mm/s/N]. Since the firsts (more flexible)

models have more nodes with that mobilities level, it seems that the more stiff the model is, the

less nodes with that mobility level we have. To summarize:

• we observe four sets of nodes with the same mean mobility level. These sets can be

attributed to zones of the maps as highlighted in figure 9;

• the number of sets in the histogram (four in this case) can be used to characterized the

global behaviour of the car;

• when the contrast increases (in this design case), the number of nodes in the lowest mo-

bility set increases, the one in an intermediate set decrease and the one in the highest

mobility set remains constant.

Figure 9: Mobilities histogram of models 1, 6 and 10

To quantitatively characterize the structural contrast of the models, let us introduce the  struc-

tural contrast indicator   (SCI) defined as presented in the equation   25.   Such an indicator isdimensionless and consistent with the defined concept of an image contrast.

SC I  = {rms}2

x  .   (25)

Since we are interested in the dynamic behaviour of the system when the contrast increases, we

present in figure 10 the evolution of the SCI function of the frequency, in third octave average

for the models 1, 6  and 10.

The figure 10 give a dynamic structural contrast indicator on the band [200 − 800] Hz. The

band [0

−200] Hz is indeed dominated by a modal behaviour, with resonant nodes (so very high

mobilities) somewhere and non-resonant nodes elsewhere. The structural contrast therefore hasno physical meaning on such a band.

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Figure 10: Structural Contrast Indicator of the models function of frequency

6 CONCLUSIONS

This study focussed on the main consequences of changing materials in a car, in the global

context of automotive lightweight design. The new distribution of local stiffnesses due to the

introduction of orthotropic materials leads to models with different structural contrasts. This

concept has been highlighted, by using an input point mobility approach. Built mobilities maps

allowed studying the distribution of stiffness on the models, and defining a kind of character-

istics mobility sets. A dynamic structural contrast indicator is proposed, consistently to the

image theory. This work is continuing in order to understand the relation between the structural

contrast and the vibroacoustic performance of a car, defined in terms displacement levels on the

structure and pressure levels in the air cavity.

ACKNOWLEDGMENTS

The authors would like to thank the Marie Curie Foundation which finances this research

through the GRESIMO Initial Training Network project.

REFERENCES

[1] Lothar Cremer, M. Heckl, and B. A. T. Petersson.   Structure-borne sound structural vibra-

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[2] Alain Giocosa. Les composites dans l’industrie automobile. Techniques Ingnieur, 1999.

[3] Martial Nobou Dassi and Al. Vibro-acoustic specifications and design indicators for

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[4] Martial Nobou Dassi and Al. Conception vibro-acoustique de caisses autmobiles alleges :

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[5] Eli Peli. Contrast in complex images.   Journal of the Optical Society of America, 7(10),

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