**Abstract**

Faurecia Exhaust Systems AB develops and manufactures exhaust manifolds for
the car industry.

**In**the developing process the FE-models are one of the main tools and they are becoming more and more important in the automotive development.**In**the tough competition between the manufacturers, the demands for fast development require good FE-models both from the manufacturers and their subcontractors.
The main goal of this thesis is to update an existing FE-model of a close
coupled exhaust manifold and look at the catalyst to investigate the behavior
of the catalytic converter and suggest a way to model the monolith.

The thesis combines analytical calculations with experimental measurements.
By the use of modal theory, an analytical model is updated to resemble the
experimental data.

The results of this thesis work include a description of an updated version
of the FE-model of the exhaust manifold. It also includes one example of how to
model the catalyst assembly. The model of the catalyst assembly has been
validated.

The initial Finite-Element model showed relatively large differences for
the first four natural frequencies, when compared to experimental data.
Relatively large amplitude errors were also obtained when comparing frequency
response function from the experiment and the model. An acceptable error was
obtained for the natural frequencies when comparing the experimental result and
the updated Finite-Element Model. The updating was mainly done by changing the
mass and stiffness in the welds and the converter tube.

**Introduction**

**Background**

The dynamic analysis of the exhaust manifold is today mainly performed with
a natural frequency analysis methodology. The catalyst assembly - that contains
a ceramic monolith brick, a mat holding and protecting the monolith and the
converter tube - is usually simplified in the FE-model for dynamic analysis. The
converter tube is meshed with shell elements and the mass for the mat and
monolith is distributed to the shell element by locally adjusting the density.

**Purpose and aim**

The aim is to update a FE-model of an exhaust manifold in order to be able
to evaluate how to model a FE-model of the catalyst assembly, and analyse how
this affects the result in the natural frequency analysis. The results from the
analysis shall be compared to modal measurements of a real close couple exhaust
manifold.

One exhaust manifold should be analysed and tested and the most suitable
method for representing the catalyst in a FE-model for dynamic analysis should
be presented.

**Approach**

The initial issue that has to be investigated is how good the simplified FEmodel
is. When compared with the experimental test of the exhaust manifold, some
questions arise:

·
Is the structure linear?

·
Is it already good enough?

·
What is good enough?

·
What is the error before
updating?

All of this has to be answered. A good way to do this is to perform a modal
analysis. A range of different tools can be used for this purpose and a
selection of them are used in this report, like Frequency response function
(FRF), AutoMAC, MAC, CoMAC, reciprocity and direct comparison of modes. A brief
explanation of these tools is included in chapter 2. For more information see
literature on the subject.

When good experimental data has been collected and the data quality
assessment is checked, the updating of the exhaust manifold can begin.

After the first comparison between the exhaust manifold and its FE-model
the second stage is to perform a modal analysis on a manifold without the
monolith and update the FE-model without the monolith. This is done to eliminate
any consistent errors in the material properties to avoid compensating for
these material properties faults during the evaluation of how to define the
monolith.

Since the materials used in the manifold is given and known from the
manufacturer, the updating is done in the welds. The connections, in form of
welds, between individual parts, are difficult to predict beforehand, and
hence, they are mainly responsible for the errors when comparing experimental
and theoretical data. However, this assumption implicitly means that all
modelling errors are accounted for by changing the physical properties of the
welds and it is possible that this simplification will lead to a somewhat
non-physical description of the welds true dynamic behaviour.

When having a model of an exhaust manifold, that fits the experimental
data, the creation of the catalyst model can be initiated.

When the FE-model without the monolith is updated as close as possible to
the manifold without the monolith, the chosen method to model a catalyst is
integrated in the FE-model and correlated against the manifold with the
monolith.

The last part is to verify the chosen method to model the catalyst. This is
done by using only the catalyst part of the manifold and performing a new modal
analysis on this. This data is then correlated against the data from the
updated FE-model of the

**Mounting of the test object**

The mounting of the test object can be done in several possible ways,
either the modal analyse is done when the structure is in its operating
conditions or without its operating conditions but in a somewhat similar state.

The most common condition is called the free-free condition and simulates a
free object without any boundary conditions.

To obtain an approximate free-free condition, a good way is typically to
suspend the test object in elastic cords. When the test object is heavy and the
suspension with the elastic cords is not enough, it is preferable to use
relatively soft springs to place the test object upon, if necessary together
with the elastic cords.

When making a modal analysis and the purpose is to correlate/update the
result of an analytical test, the free-free condition is preferred.

**Excitation**

When performing a modal test, the test object has to be put in some kind of
motion/vibration. To make the test object vibrate, an excitation force has to
be used. A force transducer is monitoring the input signal and the
accelerometers monitoring the response signal. The accelerometers and the force
transducer generate the time signals used in the analysis.

Excitation can be carried out in some different ways; one way is with a
shaker that transfers vibrations to the test object through the force
transducer. Between the shaker and the force transducer a stinger is mounted to
act as a mechanical fuse and to make sure that the force from the shaker is
measured in the intended direction.

The signal feeding the shaker can either be a stepped-sine signal or a
normally distributed broadband-signal. The latter has the advantages of
exciting all frequency simultaneously and is therefore much faster.

Another way to cause/generate vibrations in the test object is by impact
excitation. This is done by using a hammer. The hammer can be equipped with a force transducer, called an impulse hammer, or the force transducer
can be separately mounted on the test object.

**Data quality assessment**

Before using any data from the modal test, some checks are important:

·
Is the driving point frequency
response free from inconsistency?

y Does the frequency response imaginary part only have peaks or dips in one
direction?

·
Is the structure linear?

y All nonlinear parts should be removed.

y Shaker excitation - different excitations level should not differ in
amplitude in a frequency response.

y Hard to test with impact excitation.

For a system to be linear it has to be additive (2.19) and homogeneous
(2.20).

Xl (r) + Xz(t) -j Yl

*(t)*+ Yz(t)
·
Does Maxwell's reciprocity
theorem hold?

y A frequency response should be equal independently which DOF, of two
possible, that is chosen as response point respective excitation point.

It is also important to check the coherence to see if it is acceptable, see
chapter (2.9) for more information on the coherence function.

**Selection of reference point**

The reference point is the point that stays fixed during the entire
measurement. The reference point is the excitation point for shaker excitation
and the accelerometer point for measurements with roving hammer.

When selecting a reference point it is important that all modes of interest
are included. This means that the reference point cannot be located near a
nodal line of any mode. To select a proper reference point, it is preferable to
study a FE-model of the mode shape before the measurements to determine which
point that is appropriate to use.

When selecting the reference point for a three dimensional object the
directions of the mode shapes must also be taken in consideration. To get a
good measurement from a three dimensional object, one can try to find a skewed
reference point that make the force excite in all three directions.

The skewed response then has component in all three directions in the
original coordinate system, which has to be calculated. A new coordinate system
is made and the components from this are then transformed in to the original
coordinate system through multiplication with a matrix made for this purpose.

**Selection of response point**

One of the purposes of the measurement is to get a good display of the mode
shapes. The response points are the points in which the response from the
structure is collected. The information can be used to describe the mode shapes
of the structure.

Since it is not possible to measure everywhere on the structure, a finite
number of points most be chosen. The quantity of points to measure depends on
the geometry of the structure.

To get as much information as possible, the points should be chosen to give
a good separation of the modes. When the measurements are done an AutoMAC
matrix can be used to investigate if the points separate the mode shapes good
enough,

**Damping**

Damping is always present in a real system and has to be calculated from
the experimental data. In Abaqus and all other FEM-software, the damping is
not calculated, therefore it must be given. The given values are taken from the
experimental measurement.

During the measurement on the catalyst alone, to verify the update, the
damping could not be calculated with the curve-fit routines. Instead the
so-called "Half-power" bandwidth method was used. This method is
normally only valid for low damping, but is a good enough approximation when
comparing FRFs in this case when the natural frequencies in the FRF are of
greater interest and the damping is only for visualization.

**Parameters used to update the FE-model**

To update the analytical model the FRF is used and the aim is to fit the
analytical natural frequency to the experimental natural frequency. The natural
frequency depends on the stiffness and mass, see equation

The mass, m is updated just by weighing the exhaust manifold and then
change the density in the welds in the model.

The stiffness, k depends on (( which is the constant for the boundary condition
and I and L that are parameters depending on the structure. E is the Young's
modulus and is a material parameter. To update the stiffness, k the material
parameter E is used.

The boundary condition changes the natural frequency, but the boundary
conditions are the same for both the analytical model and the experimental
test, the free-free condition. The rubber cord used to simulate the free-free
condition in the measurement setup, has negligible stiffness compared to the
exhaust manifold, this gives a good estimation of the free-free condition.

**Experimental test**

The measurements on the exhaust manifold are following the directives from
chapter 2. A short description of the different features and results are
included in this chapter. During the measurement with the shaker, the shaker
was fed with a random signal. In every response point there are three DOFs
corresponding to the coordinates axis. This means a total of eight response
points gives a total of 24 DOFs.

The software used during the measurement was SingelCalc, Mobilyzer.
The data were collected in SingleCalc and afterwards analysed in Matlab using Matlab toolbox for modal analysis.

**Measurement preparation**

The chosen boundary condition for the measurement was the free-free
condition. In order to obtain the free-free condition the exhaust manifold were
suspended in soft elastic cords attached to a steel rig. See picture 3.3.

**Point selection**

In these measurements a fixed accelerometer and roving hammer were used to
be able to change the excitation points. Several different sets of points were
tested before a set was chosen for further measurements. The chosen points are
displayed in fig 3.1 and 3.2. The MAC-matrixes were used to select the best
points. The MAC of the best set of points is shown in chapter 3.3.4.

**Data quality assessment**

The collected data has to be checked before the analysis. It is good
practice to perform a data quality assessment on the collected data.

**Coherence**

The signal to noise ratio at the anti-resonances often gets too small, and
is one possible cause for the dip in some of the coherence at the
anti-resonance and should be neglected.

Below is an example of the coherence from the response point 3, DOF 7, 8
and 9.

**Reciprocity**

The reciprocity is checked by switching input and output points. The points
are located at the ends of the inlet flange point 4 and 7, see figure 3.2. The
test is performed with a modal hammer and a single axis accelerometer.

**Linearity**

The linearity is checked by using a shaker where two different measurements
are done with different amplitudes in the input signal. If the frequency
response function does not change in amplitude the exhaust manifold can be
approximated as linear.

The linearity is shown below both for exhaust manifold with and without the
monolith. The measurement on the exhaust manifold with the monolith was done
between DOF 6 and 10 and the measurement on the exhaust manifold without the
monolith was done between DOF 10 and 19.

**Quality check of measuring points**

It is important to secure good measurement points and that the modes are
correct represented in the comparison, the tool used for this is AutoMAC. An
AutoMAC is done for both the analytical and experimental data.

The AutoMAC shows the ratio of the correlation between one set of modes
against itself. Full correlation results in 1. The diagonal matrix shows the
correlation of each mode with itself, and should be fully correlated. On the
other hand the off-diagonal is the correlation between different modes and
should generate a low correlation.

**Validity check of the S-matrix.**

To check the S-matrix, data from response point 3 were measured in two
different ways. The data was collected in the skewed coordinate system and in
the reference coordinate system, see figure 3.1.

By using a distance plate the accelerometer could be placed in the
reference coordinate system, see picture 3.1 and 3.2

The data measured in the skewed coordinate system were then pre-multiplied
with the S-matrix. Comparison of the FRFs can be seen below:

The solid line - is the response from the accelerometer in the skewed
coordinate system and the dotted line ... is the response in the reference
coordinate system. The amplitude difference is due to the problem mentioned in
chapter 2.15

The difference in frequency is due to the distance plate made to get the
accelerometer in the reference coordinate system.

The dashed-dotted line - . - is the response for the skewed coordinate
system that has been pre-multiplied with the S-matrix.

The amplitude harmonizes well between the reference coordinate system and
skewed coordinate system that has been pre-multiplied with the S-matrix. This
show that the S-matrix used is valid.

**Results**

·
The responding coherence for each
measurement didn't indicate any faults in the data acquisition process.

• The reciprocity shows that Maxwell theorem is valid.

·
The FRFs in figure 3.5 and 3.6
shows that the conditions of linearity are satisfied.

·
Both MACs shows that the modes
correlates and therefore correct represented.

**Model updating**

An FE-model of the exhaust manifold is provided from the manufacturer
(Faurecia AB). The given model is modelled without comparison against
experimental measurements and with simplified modelling of the monolith. The
updating is done by changing the properties of the welds and canning
surrounding the monolith. The accelerometers/dummies have been compensated for
with lumped masses in the FE-model.

**Initial Model vs. Exhaust manifold**

Before updating the analytical model, the praxis is to compare the analytical
and experimental data sets to obtain the necessary information whether the two
data sets are close enough to each other so that a correct update is at all
possible.

The comparison is done between the initial model where the mat and monolith
is modelled by distributing the mass to the canning that surrounds the monolith
and the experimental data.

**In**order to visualise the difference between the analytical and the experimental frequency response functions, an example of the FRFs is shown in figureS .1. Direct modal comparisons between the natural frequencies are displayed in figure 5.2. The systematic deviation of the points from the ideal line indicates an error in the material properties.

The good correlation in the MAC-matrix in figure 5.3 indicates that the
analytical mode shapes corresponds well to its experimental counterparts. The
low off-diagonal values indicate that the different mode shapes are noncorrelated.

The correlation for each measure point is displayed in the CoMAC in figure
5.4. The correlation is good. DOF 22-24 is a slightly uncorrelated but since
this is where the shaker is mounted this is not to be considered as a fault but
mere disturbance.

**Updating the manifold without the monolith**

The first updating is done on an analytical model without compensation for
mat and monolith vs. an exhaust manifold without mat and monolith. The reason
for this procedure is to get an as correct model as possible to start with.

Since the material properties of the model are fixed the updating procedure
is done by updating the material properties for the welds.

Weighing the exhaust manifold and comparing with the model in Abaqus, shows
that the analytical model weighs 145g more than the exhaust manifold. Since
modal testing is rather mass sensitive the difference in mass is compensated by
changing the density of the welds in the analytical model. To get an as correct
model as possible 2000 kg/nr' is chosen as density for the welds. This density
is rather low for welding material but since material properties are not to be
changed in this updating, and the welds are distributed over the whole
structure, changing the density is acceptable.

Comparison between exhaust manifold and analytical model with correct
weight shows that the exhaust manifold has systematically higher natural
frequencies than the analytical model. The comparison is displayed with FRFs in
figure 5.5 and direct modal comparison.

The start value of the Young's modulus of the welds in the analytical model
is 22 * 1010 Pa. This is the same value as for
ordinary steel. Since the resonance frequencies were too low compared to the
measurements done on the exhaust manifold, the model must be stiffened up to
get a better match. The analytical model is tuned in by changing the Young's
modulus of the welds. The tested values are displayed in figure 5.7.

**In**the optimization of the analytical model, a value of E= 49* lOll Pa for the Young's modulus gives the smallest difference between the natural frequencies and this is the Young's modulus for the welds that is used in the following updating procedures.

**Model with updated welds vs. Exhaust manifold**

The comparison is between exhaust manifold with mat and monolith, and
corresponding analytical model with updated welds. To investigate the influence
from the updating of the welds, the mass of the monolith is implemented at the
canning surrounding the monolith. The mass is added by increasing the density
over the area.

**In**order to visualize the difference between the analytical model and the exhaust manifold after the update of the welds, an example of the FRF plot is shown in figure 5.10. A direct modal comparison is shown in figure 5.11.

A systematic derivation of the points from the ideal line indicates an
error in the material properties . Although the error is significantly
reduced, changing the properties of the welds is not enough to get a good
update of the analytical model.

The correlation between experimental and analytical mode shapes is
displayed in a MAC-matrix in figure 5.12.

The good correlation in the diagonal matrix indicates that the analytical
mode shapes corresponds to its experimental counterparts. The low off-diagonal
values indicate that the different mode shapes are non-correlated.

The correlation for each measure point is displayed in the CoMAC in figure
5.13. As can be seen the experimental and analytical DOFs correlate well. The
slightly lower correlation in DOF 22-24 is most likely due to that this is
where the shaker is mounted and should not to be considered as a fault but mere
disturbance.

**Conclusion**

The aim of this work was to study the dynamic behaviour of an exhaust
manifold. The experimental modal analysis on the exhaust manifold showed that
the existing FE-model of the manifold had to be stiffened up to match the
experimental measurements.

The initial FE-model did not have a modeled monolith. The influence from
the monolith was implemented as increased density on the canning. We have
proved that by compensating in this way, information about how the mounted
monolith affects the exhaust manifold is left out in FE calculations.

By implementing the increased stiffness in the calculations we managed to
tune in the resonance frequencies. This shows that both the physical assembly
of the mat and monolith stiffen the exhaust manifold. A strong correlation
between the experimental and the analytical mode shapes was also found as shown
in chapter 4. However, the comparison between experimental and analytical FRFs
shows relatively large amplitude errors which should be studied closer in
future investigations.

A possible source of error in the updating is the limited frequency
resolution in experimental measurements. This error does not affect the
resonance frequencies but could influence the amplitude of the FRF, and the
estimated damping. Other possible sources of error are the modelling, the
assembly and the meshing of the FE-model.

We like to suggest the following topics as themes for future investigation.

·
The welds influence on the
exhaust manifold.

·
Different ways to model the mat
and monolith.

·
Difference in weight between
FE-model and physical structure.

·
Difference in stiffness between
FE-model and physical structure.

This comment has been removed by a blog administrator.

ReplyDelete