Upload
trantu
View
221
Download
3
Embed Size (px)
Citation preview
Experimental Modal Analysis of
an Automobile Tire
J.H.A.M. Vervoort
Report No. DCT 2007.084
Bachelor final project
Coach: Dr. Ir. I. Lopez Arteaga
Supervisor: Prof. Dr. Ir. H. Nijmeijer
Eindhoven University of Technology (TU/e)
Department of Mechanical Engineering
Dynamics and Control Technology Group
Eindhoven, May, 2007
i
Abstract
This project will continue an earlier performed study on the experimental modal
analysis of an automobile tire, see reference [6]. The standard automobile tire (type
205/60 R 15) with 1.6 bar pressure is hanged in three elastic strips within a specially
designed frame, to provide a free suspended condition for the tire.
When the tire is excited at resonant frequencies, it vibrates in special shapes
called mode shapes. By understanding the mode shapes, all possible types of vibration
can be predicted, see reference [12]. The excitation is realized by an electro-dynamic
shaker and the input force of the shaker is measured by a force transducer. An
accelerometer measures the response at several points on the tire and a dynamic signal
analyzer computes the frequency response functions (FRFs).
The tire is measured with enough points to see the first six mode shapes and does
not use a grid pattern to measure the tire, but only the circumference and one cross-
section of the tire. With these two measurement lines the behavior of the complete tire is
estimated. The change of the modal amplitude in the circumference and cross-section is
used to estimate the missing points of the grid.
The first six natural frequencies of the tire are clearly visualized. The resonances
have the following natural frequencies: 114 Hz, 137 Hz, 165 Hz, 196 Hz, 228 Hz and 260
Hz. The cross-section measurements of the tire are of moderate quality. This is caused by
the less effective attachment of the accelerometer. The moderate quality of the cross-
section also causes the estimation of the complete tire to be of a moderate quality. In
future work, methods to improve the attachment of the accelerometer to the tire need to
be investigated.
ii
List of symbols
Symbol Definition Unit
T
N
fs
fmax
∆f
∆F
x(t)
y(t)
X(f)
Y(f)
Sxx(f)
Sxy(f)
Syy(f)
Hxy(f)
Υ2
xy(f)
Record length
Record length used by Siglab
Sampling frequency
Maximum frequency
Spectral density
Frequency resolution used by Siglab
Input signal in time domain
Output signal in time domain
Input signal in frequency domain
Output signal in frequency domain
Auto power spectrum (input)
Cross power spectrum
Auto power spectrum (output)
Frequency response function estimator
Coherence function
Sec
Lines
Hz
Hz
Hz
Hz
iii
Table of contents
Abstract i
List of symbols ii
Table of contents iii
1 Introduction 1
1.1 An automobile tire and modal analysis 1
1.2 Goals and outline 2
2 Experiments 3
2.1 The method and setup used for the experiments 3
2.1.1 The experiments and measuring points 4
2.2 Measuring equipment 7
2.2.1 The shaker 7
2.2.2 The force transducer 8
2.2.3 The accelerometer 8
2.2.4 The dynamic signal analyzer 9
2.3 Frequency response functions
2.3.1 Auto / cross - power spectrum
10
10
2.3.2 Coherence 11
3 Measurement quality 12
3.1 Quality improvement 12
3.1.1 Input range 12
3.1.2 Frequency range 12
3.1.3 Triggering 13
3.1.4 Windowing 13
3.1.5 Averaging 14
3.2 Quality check 14
3.2.6 Repeatability and reproducibility 15
3.2.7 Driving point measurements 15
4 Modal Analysis 16
4.1 Modal parameter estimation 16
4.1.1 Number of modes 16
4.1.2 Estimation method 18
4.2 Mode shapes 19
4.2.1 Tread side or circumference of the tire 20
4.2.2 Cross-sections of the tire 21
4.3 Estimation of the complete tire 21
5 Evaluation 24
5.1 A half or quarter circumference measurement of the tire 24
iv
5.2 A complete or half cross-section measurement of the tire 25
5.3 The influence of the accelerometer attachment and the comparison of the
different sides of the tire
26
5.4 The estimation of the complete tire 28
5.4.1 Evaluation of the complete tire modulation 29
5.4.2 Checking the first Matlab model with the data of the measured cross-
section at 90 degrees from the excitation point
35
6 Conclusion 38
7 Recommendations 40
Appendix:
A Specifications of the measurement equipment 41
A1 The shaker 41
A2 The force transducer 41
A3 The accelerometer 41
A4 The dynamic signal analyzer 41
B Measurement points on the tire 42
B1 Tire’s circumference 43
B2 Tire’s sidewall 44
B3 Tire’s cross-section at 0 degrees from the excitation point 45
B4 Tire’s cross-section at 90 degrees from the excitation point 45
B5 Tire’s cross-section at 180 degrees from the excitation point 45
C Mode indicator functions 46
D Estimated mode shapes of the tire’s circumference 49
E Estimated mode shapes of the tire’s side wall 52
F Estimated mode shapes of the tire’s cross-sections 55
G Matlab model 62
H Results of Matlab model one, direct use of measurement data 68
I Results of Matlab model two, use of ME’scope fitted data 72
J Comparison of the cross-section at 90 degrees with the complete tire model 81
8 Bibliography 82
1
Chapter 1
Introduction
The experimental modal analysis of an automobile tire is the subject of this
bachelor final project at Eindhoven University of Technology. The experimental modal
analysis is performed to obtain the modal parameters of an automobile tire. The modal
parameters can be used to tune a finite element model, which for the matter of fact is not
done in this project. The finite element model greatly aids in the design of the tire by
predicting its vibration response.
1.1 An automobile tire and modal analysis
The automobile tire is an important source of vibration and noise inside the
vehicle. To improve the tire’s influence on the vehicle, one has to know how the tire
behaves. This behavior is stored in the tire’s dynamic properties which can be researched.
When the frequency response function (FRF) of a point on the tire is measured,
one knows how this point on the tire responses too a certain frequency input.
When the tire is excited at resonant frequencies, it vibrates in special shapes
called mode shapes. Under normal circumstances, the tire will vibrate in a complex
combination of all mode shapes. By understanding the mode shapes, all possible types of
vibration can be predicted, see reference [12]. By experimental modal analysis, the mode
shapes can be measured together with the modal frequency and modal damping.
Experimental modal analysis consists of: exciting the tire with an electro-dynamic shaker,
measuring the FRFs between the excitation and numerous points on the tire, and then
using software to visualize the mode shapes.
This project will continue an earlier performed study of the experimental modal
analysis of an automobile tire, see reference [6]. The standard automobile tire (type
205/60 R 15) with 1.6 bar pressure is hanged in three elastic strips within a specially
designed frame, to provide a free suspended condition for the tire. The strips have a very
low natural frequency which ensures the free-free condition. More information about the
tire can be found in appendix B.
2
In the previous study, the tire is divided into a grid pattern that covers half of the
tire. The size of the grids determines the accuracy of the measurements. But in this
previous study the tire is measured with too little grid points to see at least six mode
shapes within the circumference of the tire. In this project the tire is measured with
enough points to see these six mode shapes and does not use the grid pattern to measure
the tire, but only the circumference and one cross-section of the tire. By these two
measurement lines the behavior of the complete tire can be predicted.
1.2 Goals and outline
To measure the tire’s first six mode shapes, the tire has to be measured with
enough points. Measuring the complete tire with enough points costs a lot of time. The
previous study revealed that by measuring only half of the tire, the behavior of the
complete tire can be predicted. This means measuring time is cut in half, but maybe it is
possible to shorten the measurement time even more. This is why the first goal is: to see
if it is possible to shorten the measurement time.
The second and most important goal is to measure the tire’s first six mode shapes
and obtaining the modal parameters, with the first goal in mind. The measurements need
to be performed in the shortest possible measurement time. This is why the complete tire
is estimated with only one cross-section and circumference.
This report is divided into the five following subjects. First, the experiments with:
the experimental setup, the measurement points, the measuring equipment and an
explanation of the estimation of the FRF, are discussed in chapter 2. Then, the
measurement quality which is an important part of experimental modal analysis is
discussed in chapter 3. After that, the modal parameter estimation and the experimental
results is discussed in chapter 4. Fourth, the evaluations of these results are discussed in
chapter 5 and finally the conclusion and recommendation are discussed in chapter 6 and
7.
3
Chapter 2
Experiments
Experiments need to be performed to obtain the FRFs needed for the modal
analysis of an automobile tire. This chapter deals with the explanation of the method and
setup used for the experiments, which measurement equipment is used, and how FRFs
are estimated.
2.1 The method and setup used for the experiments
To obtain the FRFs of an automobile tire, the response of the tire to a point
excitation has to be measured. The excitation is realized by an electro-dynamic shaker
and the input force of the shaker is measured by a force transducer. An accelerometer
measures the response at several points on the tire and a dynamic signal analyzer
computes the FRFs.
A FRF is made for every point measured on the tire. The number of measurement
points determines the accuracy of the mode shape, which is explained in the next
subsection. Each FRF represents the resonant frequencies of the tire and the modal
amplitudes of its specific point. The amplitude of the FRFs indicates the ratio of the
acceleration divided by the input force, see equation 2.1, and see also reference [1].
Response = Tire properties * Input (eq. 2.1)
The automobile tire as explained in the introduction is tested in a free condition.
This means that the tire is not attached to the ground at any of its coordinates. In practice
it is not possible to make a truly free support but it is feasible to provide a suspension
system which closely approximates to the free condition. That is why the tire is hanged in
three light elastic strips as can be seen in figure 2.1 below, see reference [1] for more
information about free supports.
The elastic strip attached at the top is thicker because it needs to hold the tire’s
weight. The two other elastic strips are used to keep the tire straight. As can be seen in
4
figure 2.1 the electro-dynamic shaker is also hanged elastically to ensure the free-free
condition of the tire.
Steel frame
Elastic strip
Accelerometer
Shaker
Force transducer
Mountingring
Tire
LaptopSiglab
Amplifier
Figure 2.1: FRF measurement setup
2.1.1 The experiments and measuring points
Three different experiments are needed to accomplish the goals of this project;
these goals are mentioned in the introduction. The first experiment investigates the
possibility to measure only a quarter of the tire’s circumference and half cross-section, to
shorten measurement time. This is done by measuring the tire’s half circumference and
comparing the results of the two quarters in it. If this is possible more measuring points
can be taken of the quarter of the tire, so a clearer image of the tire will be the result. The
same is done for the tire’s cross-section, though the cross-section is completely measured
and the two halves in it compared. The tire’s half circumference is measured with 24
points, the coordinates of these points can be found in appendix B. The tire’s cross-
5
section is measured with 11 points at 180 degrees from the excitation point; the excitation
point is point number 1 of the circumference. The coordinates of the tire’s cross-section
points can also be found in appendix B.
The second experiment takes the main goal of this project into account. The tire’s
modal parameters need to be obtained and the tire needs to be measured with enough
points to see the first six mode shapes within the desired bandwidth of 0 to 500 Hz. In the
previous study the tire is divided into a grid pattern that covers half of the tire. The size of
the grid determines the accuracy of the measurements.
It takes a lot of time to measure the complete grid. Because measurement time has
to be shortened according to the first goal, only the circumference and one cross-section
of the tire are measured. The change of the modal amplitude in the circumference and
cross-section is used to calculate the missing points of the grid. The circumference and
cross-section are measured with more points, if they are measured with the same amount
of points, as the tire would be with the use of a grid. This results in more accurate mode
shapes, or if the accuracy admits lesser points are measured and measurement time is
shortened. Section 4.3 provides more information about the use of one cross-section and
circumference.
To obtain a smooth mode shape it is important to measure the tire with enough
measurement points. The more points, the smoother the mode shapes but the more time it
costs to complete the experiment. Every mode shape can be described in terms of an
integer number of wavelengths along the circumference; the tire’s sixth mode shape has
seven wavelengths within the tire. Every wavelength should have at least six
measurement points for a smooth visualization.
Experiment one determines how in experiment two the tire’s mode shapes and
modal parameters are estimated. This can be done with the measurement of the half or
quarter circumference and the complete or half cross-section. The results of experiment
one are described in chapter 4 and discussed in chapter 5. The conclusion that could be
drawn is that the half circumference and the half cross-section have to be measured for an
accurate closure of experiment two. That is why the tire’s half circumference is measured
with 24 points, which is enough for a smooth visualization. The tire’s half cross-section is
measured with 7 points, because every mode shape has one and a half wavelengths within
6
the cross-section. The coordinates of the measurement points of the circumference and
the cross-section are presented in Appendix B. For experiment two, the cross-section at 0
degrees from the excitation point is used, because the vibrations provided by the electro-
dynamic shaker are not damped yet.
The third experiment measures a second cross-section of the tire at 90 degrees
from the excitation point. This cross-section is measured to check if the model calculates
the points on the grid correctly. If the model calculates the points correct the difference
between the calculated points and the measured points should be minimal. The
measurement points of this cross-section are also represented in Appendix B.
A chirp type signal is chosen to excite the tire. For one measurement the chirp
runs from 0 to 500 Hz in one second. After this second there is a one second pause before
a next chirp is started for the next measurement. In the pause the chirp dies out, so the
next measurement is not corrupted by the previous chirp. One point is measured 50 times
with a chirp signal and averaged. More information about the frequency range, averaging
and other measurement settings can be found in chapter 3. The software which provides
the demanded chirp and measures the response of the tire is named Siglab. More
information about this software can be found in references [2] and [3]. After the
measured data is saved, the modal parameters are calculated with a different software
program named ME’scope. More information about ME’scope can be found in reference
[4].
7
2.2 Measuring equipment
In the experiments explained above measuring equipment provides the demanded
input and output data. In this section the measuring equipment will be described to give a
better understanding of the experiments. The measuring equipment consists of: an
electro-dynamic shaker, a force transducer, an accelerometer and a dynamic signal
analyzer, as can be seen on figure 2.1 in the previous section.
2.2.1 The shaker
Figure 2.2: The electro-dynamic shaker
The electro-dynamic shaker (see figure 2.2) is used to give the tire an excitation
which is prescribed by Siglab and amplified by an amplifier. The electro-dynamic shaker
is connected through a stinger with a force transducer. A stinger is a small metal wire
which is stiff in one direction and flexible in all the other. The electro-dynamic shaker is
hanged elastically, so external noise is damped as explained in section 2.1. When
elastically hanged the electro-dynamic shaker also gives itself an excitation, but this does
not matter, because the force transducer only measures the resulting force, and the
prescribed frequency is not changed. The specifications of the electro-dynamic shaker are
represented in appendix A.
8
2.2.2 The force transducer
Figure 2.3: The force transducer, attached to the tire
The force transducer (see figure 2.3) is used to measure the force of the excitation
brought by the electro-dynamic shaker. It is attached to the tire with a screw, which fits in
the profile of the tire. Siglab powers the force transducer with an automatically chosen
constant voltage (type bias). The output signal of the force transducer runs through the
same cable and is present as a voltage modulation. The specifications of the force
transducer are represented in appendix A.
2.2.3 The accelerometer
Figure 2.4: The accelerometer
The accelerometer (see figure 2.4) is used to measure the acceleration of different
points on the tire. It is attached to the tire with a thin layer of mounting wax, or is placed
9
between the tire’s profile. It has a built-in charge amplifier and a low-impedance voltage
output. The amplifier in the accelerometer is directly fed from Siglab, using a bias type of
voltage. The specifications of the accelerometer are represented in appendix A.
2.2.4 The dynamic signal analyzer
Figure 2.5: The dynamic signal analyzer
The measurement of the force and acceleration signal is performed by the
dynamic signal analyzer. The dynamic signal analyzer is a four channel Siglab analyzer,
which samples the voltage signals coming from the force transducer and the
accelerometer. The sensitivity information of the sensors is used to convert the voltages
to equivalent force and acceleration signals. It is also used to transform these time
domain signals into a FRF. The transformation from the time domain signals into a FRF
is invisible for the user. In subsection 2.3 the basic idea of this transformation is
explained. The specifications of the dynamic signal analyzer are represented in appendix
A.
10
2.3 Frequency response functions
The measured FRFs quality directly determines the quality of modal parameters.
This is why the estimation of the FRF is an important process of modal analysis. The
estimation of this FRF is performed inside the dynamic signal analyzer and is invisible
for the user. The basic ideas of the estimation are explained in the next subsections. More
information about the estimation of the FRF can be found in references [7], [8], [9] and
[13].
2.3.1 Auto / cross - power spectrum
The in time domain recorded input and response signals x(t) and y(t) are
transformed into the frequency domain using the Fast Fourier Transformation. The
frequency domain signals, referred as X(f) and Y(f), are used to calculate the auto and
cross power spectrum. The auto spectrum of the input, referred as Sxx(f), and the cross
power spectrum of the input and output, referred as Sxy(f), are formulated in equations 2.2
and 2.3.
)()(1
)( *fXfX
TfS xx = (2.2)
)()(1
)( *fYfX
TfS xy = (2.3)
* indicates the complex conjugate and T the measured time record length
The auto power spectrum provides information about the frequencies that are
present in the input signal, and about the contribution of each component to the total
power in the signal. In contrast to the auto power spectrum, the cross power spectrum is a
complex function.
With the auto power spectrum and the cross power spectrum, the FRF [Hxy(f)] can
be estimated using equation (2.4).
11
)(
)()(
fS
fSfH
xx
xy
xy = (2.4)
The FRF contains both magnitude and phase information. The magnitude is
typically shown on a logarithmic Y axis in dB scale, and the phase is often shown on a 0
to 360 degree scale. However, since the phase often has shifts in degrees the
discontinuities can be removed with use of phase unwrapping.
2.3.2 Coherence
The coherence function is used to determine the quality of the estimated FRF. The
coherence is formed when the cross power spectrum is normalized using the
corresponding auto power spectra and is defined in equation (2.5). The coherence
function is a normalized coefficient of correlation between the excitation signal x(t) and
the response signal y(t) evaluated at each frequency.
)()(
|)(|)(
2
2
fSfS
fSf
yyxx
xy
xy =γ (2.5)
The coherence function shows for each frequency which part of the response y(t)
is caused by the input x(t). The range of the coherence function can be described by
equation 2.6. If the value of the coherence is near one a linear relation exist between the
input x(t) en the response y(t) and there is no influence of noise. In the other way if the
value is near zero the response is dominated by the noise.
20 ( ) 1xy fγ≤ ≤ (2.6)
2 1........... ( ) ( )xy nn yyS f S fγ = <<
2 0........... ( ) ( )xy nn yyS f S fγ = >>
12
Chapter 3
Measurement quality
The quality of the measurement is very important and always requires attention in
experiments. The quality depends on the sensors used, the dynamic signal analyzer and
the skill of the person doing the experiments. In the next subsections, the influence of
Siglab on the measurement quality, the reproducibility and driving point measurement are
discussed.
3.1 Quality improvement
Siglabs settings influence the accuracy of the measurement; they are represented
by the following aspects: the input range, the frequency range, triggering, windowing and
averaging. In the next subsections these aspects are explained.
3.1.1 Input range
The input range can be specified by Siglab varying from 20 mV to 10V, and can
be set for each channel. The range should be as small as possible because a smaller range
means a higher resolution.
If the input range is too large the analogue to digital conversion results in a coarse
resolution which leads to large quantization errors (quantization noise). When the range
is too small the amplitude of the incoming signal is larger than the maximum allowed
value and an overload will occur. More information about quantization can be found in
reference [12]
The type of voltage can also be specified for every channel. The force transducer
needs an external amplifier to collect data and the accelerometer receives power from
Siglab, both need a bias type of voltage.
3.1.2 Frequency range
Siglab provides two settings for the frequency range, the bandwidth and the
record length. The minimal bandwidth needed for this experiment is 300 Hz (determined
13
in an earlier performed study, see reference [6]), but the nearest bandwidth that can be set
in Siglab is 500 Hz. The record length in “lines” is set to 2048 lines. The ultimate
frequency resolution of the analysis can be calculated using equation 3.1 and 3.2, where
fmax is the maximum frequency, fs is the sampling frequency and N is the record length in
“lines”. Siglab’s sampling frequency is always given by: fs = 2.56 * fmax, further
information can be found in reference [2].
max2.56*f
FN
∆ = (3.1)
sfFN
∆ = (3.2)
3.1.3 Triggering
Triggering is used to capture the entire excitation and response signal in a
sampling window. Siglab makes four parameters relevant to ensure triggering for an
input channel namely; the threshold, slope, filter and trigger delay. The threshold is a
percentage of the peak value of the total impulse. This determines the amplitude of a
signal at which the measurement will start. The slope determines if the measurement
starts with a decaying or a growing signal. When a filter is applied all the data goes
through an anti-aliasing filter before it is triggered. When a signal is already present these
parameters ensure the measurement to start, but the trigger delay ensures that the entire
signal is monitored. For all experiments done in this project, the threshold is set to 9%
with a positive slope, with no filter and a trigger delay of -10%.
3.1.4 Windowing
Windowing is the most practical solution to the leakage problem. Leakage occurs
when measured signals do not drop to zero within the measurement time interval. To
minimize the effects of leakage the measured signals are multiplied with a window
function before a Fast Fourier Transformation (FFT) is applied. Another option is to
ensure that the measured signal starts at zero and drops to zero at the end of the
measurement time interval without the use of a window.
14
In the experiments a chirp signal is used. This is a fast sweep from low to high
frequency within one sample interval of the analyzer, which is explained in subsection
2.1.1. Triggering ensures that the measured signal starts at zero and a pause between two
chirp signals ensures that the measured signal is zero at the end of the measurement time
interval. The advantage of this option is that it is not necessary to use a window and no
artificial damping is introduced to the FRFs.
3.1.5 Averaging
Every measurement is contaminated by noise or is influenced by the person doing
the experiments. The accuracy of a FRF is mainly determined by these effects. This can
be improved by averaging measurements, which reduces the effect of random errors.
However averaging does not have an effect on systematic errors as leakage.
There are different types of averaging: peak hold, exponential, linear, ect. In these
experiments a linear averaging method without overlap is used. This means that every
measured FRF has the importance, when calculating the average. Overlap can only be
used when the measured signals are mutually uncorrelated, which means that the signals
have no specific relation to one and other. All the experiments in this project use a chirp
signal which belongs to a transient class of signal where overlap is of no use, see
reference [1].
In all experiments, every point is measured 50 times. The 50 measurements are
averaged resulting in a FRF without random errors.
3.2 Quality check
During the measurement process the quality of the measured FRFs can be
evaluated using the coherence function, as explained in subsection 2.3.2. The coherence
function can only indicate the effects of random errors, the effects of bias errors can not
be indicated. Therefore additional checks are needed, to ensure that the quality of the
FRFs is sufficient. Also, certain FRFs should be re-measured from time to time, to check
that neither the structure nor the measurement system has experienced any significant
changes. The next subsections will discus these topics.
15
3.2.1 Repeatability and reproducibility
The repeatability or reproducibility of the measured FRFs is an essential check for
any modal test. Certain FRFs should be re-measured from time to time, to check if there
are some significant changes to the experimental setup (the structure and the
measurement system). If a measured FRF is not reproducible this can have several
reasons, a few primary factors are nonlinearities, insufficient clamping stiffness and
inconsistencies in the measurement directions of the excitation and response.
In this project one point (Point number 1, see appendix B) is re-measured every
time an experiment is performed. The FRF of this point is compared with earlier taken
FRFs.
3.2.2 Driving point measurements
The driving point measurement is used to get an indication of the quality of the
measurement loop. This is realized by measuring the excitation and response at the same
location on the tire. The driving point FRF has to fulfill the following requirements: all
resonances should be separated by anti-resonances, phase differences larger than 180°
cannot occur and the imaginary parts of the FRF should not change sign. If the driving
point FRF does not meet the requirements this indicates that the measurement loop is of
poor quality.
Measurement point 1 of the tire’s circumference
16
Chapter 4
Modal Analysis
In the modal analysis of a tire the modal parameters are estimated from the
measured FRFs. The estimation process of the modal parameters consists of estimating
the number of modes in the frequency band, choosing the best modal parameter
estimation method and calculating and forming the mode shapes of the tire.
The modal parameter estimation process is clarified in section 4.1 and 4.2.
Section 4.3 deals with explaining the estimation process of the complete tire (with the use
of only the circumference and one cross-section of the tire). In this chapter the results of
the estimation process are presented. The next chapter will deal with the evaluation of
these results.
4.1 Modal parameter estimation
The tire’s modal parameters are estimated by curve fitting the measured FRFs.
Curve fitting is performed by ME’scope and has a set of modal parameters as outcome.
The modal parameters contain the frequency, damping and mode shape for each
eigenmode that is identified in the desired frequency range. The curve fitting process is
completed in a two steps: determining the number of modes and applying the best
estimation method. The estimation method is used to estimate the frequency, damping
and residues in the desired frequency range.
4.1.1 Number of modes
The first step in the modal parameter estimation process is estimating the number
of modes in the desired frequency range, which start at 0 Hz and ends at 500 Hz,
explained in subsection 2.1.1. Estimating the number of modes in the measured FRFs can
be done in several ways. However, in this report there are only two methods discussed.
In the first method, all the FRFs of the tire are overlaid. In figure 4.1 all the FRFs
of the tire’s circumference are overlaid and in figure 4.2 the FRFs of the cross-section (at
17
0 degrees from the excitation point). All the FRFs have resonance peaks at the tire’s
eigenmodes frequencies, except if the measured point is a nodal point at that specific
frequency.
Figure 4.1: Overlay of the FRFs of the circumference
Figure 4.2: Overlay of the FRFs of the cross-section
18
The second method to estimate the number of modes is by using a mode indicator
function. The mode indicator function can be applied with ME’scope, which offers three
different methods, namely, the modal peaks function, the complex mode indicator
function and the multivariate mode indicator function. More information about these
three different methods can be found in references [4] and [6]. For all the experiments a
modal peaks function is applied. The basic idea behind the modal peak function is that it
sums the magnitudes of all the FRFs. This sum gives a mode indicator function where the
peaks show the resonance or eigenmode frequencies. In appendix C the mode indicator
function of the tire’s circumference and cross-section is shown. The eigenmode
frequencies at the peak values of the mode indicator function are highlighted by the
vertical green lines.
4.1.2 Estimation Method
The modal parameter estimation method estimates the frequency, damping and
residues for each eigenmode that is identified. For the estimation process different
methods can be chosen, which all have their own properties. There are three sorts of
properties, namely: a single degree of freedom or a multiple degree of freedom, a local or
a global method, and the frequency domain or the time domain. For all experiments the
global polynomial method is chosen.
Global polynomial method
The polynomial method is a multi degree of freedom method that simultaneously
estimates the modal parameters of two or more modes and uses a frequency domain curve
fitting method. The fitting method utilizes the complex trace data in the cursor band for
curve fitting. More information about the polynomial method can be found in reference
[4]. The advantage of using a multi degree of freedom method is that it gives the best
modal parameter estimates when the eigenmodes are close together and the damping is
relatively high, which is for tires.
The global method is chosen because it uses a formulation where all FRFs are
considered simultaneously and curve fitted together. When doing so, a global modal
19
frequency and damping estimate results, this is one estimate of each resonance frequency
and damping value. The global method requires a high quality of measurement data
because it is sensitive for small variations. When the quality of measurement data is high
enough the global method delivers superior results compared to the local method. More
information about the global method can be found in references [4] and [8].
With the use of ME’scope a global polynomial method can be used to fit the
measured data of the tire. The global polynomial method estimates the frequency,
damping and residues for each eigenmode that is identified. Figure 4.3 shows the
resulting modal frequencies and damping of the tire’s different eigenmodes when the
global polynomial method is used for the circumference and cross-section of the tire. The
residues which are also estimated by the global polynomial method for the tire’s
circumference and cross-section are represented in appendix C.
Figure 4.3: The resulting frequencies and damping of the different eigenmodes
4.2 Mode shapes
A mode shape is a visualization of the tire at a resonance frequency or natural
frequency. The points measured on the tire are drawn in ME’scope and the fitted data of
these measured points connected to the drawn points. The resulting mode shapes of the
circumference and cross-section of the tire are presented in the following subsections.
The tire’s circumference The tire’s cross-section
20
4.2.1 Tread side or circumference of the tire
The half tire’s circumference is measured with 24 points as explained in
subsection 2.1.1. Although only half of the tire is measured, a complete tire is drawn in
ME’scope. This means that 22 points where mirrored, corresponding point-number 2
through 23. The nine estimated mode shapes are represented in appendix D. In figure 4.4
the second mode shape of the tire can be seen. This is a mode with three modal
diameters, three wavelengths along the circumference.
Figure 4.4: the second mode shape of the tire’s circumference
Figure 4.5: From left to right, the first mode shape of the cross-section at 0, 90 and 180
degrees
21
4.2.2 Cross-sections of the tire
The tire is measured with 3 cross-sections, respectively at 0, 90 and 180 degrees
from the excitation point. The cross-section at 180 degrees from the excitation point is
completely measured with 11 points and only the half cross-section at 0 and 90 degrees
are measured with 7 and 6 points. This is because experiment one concluded that the half
cross-section measurement represents the complete cross-section measurement. This is
explained in chapter 5. In figure 4.5 the first mode shapes and in appendix F all the
resulting mode shapes of these cross-sections are shown.
4.3 Estimation of the complete tire
The mode shapes of a complete tire are normally measured with a grid of points,
close enough together to see all the desired mode shapes. Because measuring the
complete tire (to see the first six mode shapes in the tire) takes a lot of time, this project
measures only the circumference and one cross-section of the tire. Mode shapes can be
seen in the cross-section and in the circumference, as proved above in section 4.2. When
the amplitude-ratio’s between the points in the mode shapes of the cross-section or
circumference are calculated, the missing points of the grid on the tire can be estimated.
Figure 4.6 shows how these missing points are estimated.
Estimating the amplitude:
Point E
Point F
E = D * (B/A)
F = D * (C/A)
A D
B
C
E
F
B/A
C/A
Cross-section
Outline
Figure 4.6: Estimating the missing grid-points
22
Two different Matlab models estimated the amplitudes of all the points.
The first model directly uses the measurement data. The magnitudes and phases
of all the measured points are loaded, and the ratios between the magnitudes and the
differences between the phases calculated. With the magnitude-ratio and phase-difference
known, the amplitudes of all the points on the tire’s grid are estimated. The amplitudes
can be estimated for all the frequencies within the frequency range. When pictures are
made for every frequency and played in a movie, the development of the running modes
can be seen within the measured frequency range. The running mode is the superposition
of the response of all modes at a given frequency.
The second Matlab model uses the fitted data from ME’scope and does the same
as the first Matlab model. It is not possible to make a movie of the development of the
mode shapes, because the mode shape only refers to the eigenfrequency.
The first Matlab model can be found in appendix G, the program content of the
second Matlab model is identical to the first Matlab model.
After the amplitudes are estimated, the Matlab models plot all the amplitudes on a
plane, with on the X-axis the half cross-section points, on the Y-axis the half
circumference points and on the Z axis the amplitudes. Figure 4.7 shows how the tire is
flattened. Note that only half of the tire’s circumference and cross-section is modeled.
Cro
ss-s
ectio
n
Circumference
The tires round shape is converted into a flat plane in the Matlab model
Figure 4.7: Visualization of the flattened tire.
23
The resulting mode shapes of the first and second Matlab model can be found in
Appendix H and I. Since it is not possible to visualize the movie of the first Matlab model
in the report only the running modes at the eigenfrequencies of the circumference are
represented for the first model. The eigenfrequencies of the circumference are chosen
because the circumference has the best quality measurement. Figure 4.8 shows the
visualization of the tire’s running mode at 165 Hz, which is made by the first model.
Figure 4.8: The flattened tire’s running mode at 165 Hz
24
Chapter 5
Evaluation
The goals of this project where: shortening the measurement time, measuring the
first six mode shapes and estimating the modal parameters of a tire. The first experiment
investigated if it is possible to estimate the tire’s mode shapes, measuring only a quarter
of the circumference and a half cross-section of the tire. The second experiment measures
the tire’s half circumference and half cross-section to estimate the mode shapes and
modal parameters of the complete tire. The third experiment measures a second cross-
section to check if the estimating process of the missing grid points is correct. The
detailed information about these experiments is described in subsection 2.1.1. In this
chapter the results of these experiments are evaluated and the influence of the
accelerometer attachment is discussed.
5.1 A half or quarter circumference measurement of the tire
The tire’s half circumference is measured according to experiment one, to see if it
is possible to measure only a quarter of the tire. The results of the half circumference are
presented in subsection 4.2.1.
When the half circumference of the tire is measured, a FRF for every point on the
circumference is made. It is possible to reconstruct the complete tire with a quarter of the
circumference, but not by simply mirroring the FRFs of the points. When mirroring of the
points is applied it will result in an unexpected mode shape, as can be seen in figure 5.1.
Note that the mode shape has no peak or valley on the mirror line, at 90 degrees from the
excitation point. When mirroring the quarters, one would form an unnatural peak or
valley which causes the mode shape to clash with the natural form.
25
Figure 5.1: The original and expected mode shape to the left, the mode shape with the
mirrored quarters to the right
However it is still possible to reconstruct the complete tire, out of a quarter
circumference measurement. In the original mode shape in figure 5.1 can be seen that
measurement point 4 has the same FRF as measurement point 13. The same is valid for
point 3 and 14, point 2 and 15, point 1 and point 16, point 2 and point 17, et cetera. When
the FRFs of the measured points are connected to the other points, as explained above,
the expected mode shape of the tire will be formed. However for every mode shape a
different relation between the points exist, which makes it a difficult and time-consuming
work to form the expected mode shape. Therefore it is easier and less time-consuming to
measure the tire’s half circumference.
5.2 A complete or half cross-section measurement of the tire
The tire’s cross-section is also measured according to experiment one, to see if it
is possible to measure only half of it. Measuring half of the cross-section is only possible
when the two half in the mode shape of the complete cross-section are exactly the same.
The mode shapes of the cross-section at 180 degrees from the excitation point are used to
compare the two halves in it and the results can be found in appendix B. All the mode
shapes look the same if the right or left side is mirrored, as can be seen in figure 5.2. This
26
means that the half cross-section of the tire can be measured for a modal analysis of the
complete tire.
Figure 5.2: The original mode shape to the left, the mirrored mode shape to the right
5.3 The influence of the attachment of the accelerometer and the
comparison of the different sides of the tire
The accelerometer is attached to the tire with mounting wax in the cross-section
measurements, or it is placed between the tires profile in the circumference
measurements. When the results of the circumference and cross-section measurements
are compared, it is obvious the circumference measurements have a better accuracy. This
can be seen in the quality of the mode shapes and difference in the mode indicator
function. In the circumference measurement more mode shapes are recognized and
formed as one would expect and the resonances in mode indicator function are better
separated.
A tire has two different sides, namely the tread-side and the side-wall, see figure
5.3. The differences between these two sides need to be investigated to give a better
understanding of the tire’s behavior. The influence on the measurement accuracy of the
accelerometer attachment also needs to be investigated. These two investigations are
27
combined as follows, the tire’s circumference is measured on the tread-side with
accelerometer placed between the tire’s profile and the tire’s circumference is measured
on the side-wall with the accelerometer attached to the tire with wax. In this way the
influence of the accelerometer attachment and the difference of the two tire sides are
investigated.
The results of the side wall measurement can be found in appendix E, where the
resulting mode shapes and mode indicator function are represented. The tread-side
measurement is the same measurement as that of the tire’s circumference, the results of
the tire’s circumference can be found in appendix C and D.
Both circumferences are measured under the same circumstances only the
accelerometer attachment is different. If the accelerometer attachment has no influence,
the results of both circumferences should be the same, except for the fact that the side-
walls mode shape is the opposite of the tread-side’s mode shape. This can be seen in
figure 5.3, for the blue line the point measured on the tread-side moves outward while the
point on the side-wall moves inward.
Treadside of the tire
Original shape of the tire
Sidewall of the tire
Difference in amplitude between the two possible mode shapes
Possible mode shape
Possible mode shape
Figure 5.3: Cross-section view of the tire, with two possible mode shapes.
28
When the results of the side-wall and tread-side are compared, this can be seen in
figure 5.4, the side-walls circumference is indeed the opposite of the tread-sides
circumference. It is also obvious that the tread side has better results, because it has
smoother curves. In figure 5.4 the results of the side wall are not shown in a circle
because the measurement direction is perpendicular to the radial direction of the tire’s
circle. The difference in results of two circumferences indicates that the measurements
are not performed under the same circumstances, which means that the attachment of the
accelerometer influences the quality of the measurements.
Figure 5.4: The third mode shape of the side-walls-circumference to the left and the third
mode shape of the tread-side-circumference to the right, where the dotted black line
illustrates the corresponding parts of both mode shapes.
5.4 The estimation of the complete tire
The model for the estimation of the complete tire, described in section 4.3, uses
the measured data of the half circumference and the half cross-section at 0 degrees from
the excitation point. These two ‘lines’ are used to estimate the missing points on the grid
of the tire. The grid points that are known, the ones of the circumference and cross-
section are shown in figure 5.5 as red dots. All the other points in the grid pattern are the
29
missing points that need to be estimated. If the measurement data of the red dots contain
errors, or have an abnormality, it will have an influence on the missing points in the grid
and the complete model of the tire.
The half cross-section with 7 points
Th
e h
alf
cir
cum
fere
nce w
ith
24 p
oin
ts
Grid
Flattened tire
Figure 5.5: The flattened tire from figure 4.7 divided into a grid, with 24 point in the half
circumference and 7 points in the half cross-section.
5.4.1 Evaluation of the complete tire modulation
There are two different Matlab models who estimate the complete tire. This
section discusses first the results of both Matlab models. Then they are compared to each
other. After that the second Matlab model is implemented with the data of the cross-
section at 90 degrees from the excitation point, because the data of the cross-section at 0
degrees gives strange results for this Matlab model. And finally in the next section the
best Matlab model is chosen and the estimated data of the tire’s model is checked with a
measured cross-section. The results of both Matlab models can be found in appendix H
and I.
30
The first Matlab model
The first Matlab model directly uses the measurement data to estimate the 3d
plots. When looking at the resulting running modes, two strange symptoms catch the
attention. Point 3 of the circumference only gives wrong results in the first and second
mode and point 24 on the circumference is positive in every mode shape, which it should
not. Both strange symptoms can be seen in figure 5.6 and can be caused by bad
measurement data or by a fault in the Matlab model. Considering the fact that the first
Matlab model has these two strange symptoms, the images produced are as one would
expect. The formed mode shapes have the right number of peaks and values, and the
points form a smooth wave. All mode shapes of the first Matlab model can be seen in
Appendix H.
Mode 1 Mode 2
Mode 3 Mode 4
Figure 5.6: The running modes of the tire at 114Hz, 137Hz, 165Hz, 196Hz estimated by
the first Matlab model
31
The second Matlab model
The second Matlab model uses the fitted data from ME’scope to estimate the
tire’s 3D plots. In the resulting mode shapes the cross-section and circumference are
exactly the same as the mode shapes of the separate circumference and cross-section,
given in appendix D and F. The circumference of the model acts as expected but the
cross-section does not. This can be seen in figure 5.7.
Note that only a half wavelength should be visible within the half cross-section,
but instead of a half wave length a complete wavelength is visible. This is also visible in
the mode shape of the cross-section at 0 degrees from the excitation point, as can be seen
in figure 5.8. The unexpected behavior of the cross-section is caused by the poor
attachment of the accelerometer to the tire. The poor measurement data causes ME’scope
to estimate the mode shape in an unexpected form.
Mode 1 Mode 2
Mode 3 Mode 4
Figure 5.7: The first four mode shapes of the tire estimated by the second Matlab model
with the use of the cross-section at 0 degrees from the excitation point
32
Figure 5.8: The second mode shape of the cross-section at 0 degrees from the excitation
point
Comparison of the first and second Matlab model
When the mode shapes of the second Matlab model are compared with the
running modes of the first Matlab model, the circumferences always act the same but the
cross-sections do not. The comparison can be seen in figure 5.9 for the third mode shape.
It is strange that the cross-sections of both Matlab models are not identical. This
can be caused by the measurement data of ME’scope. ME’scope calculates the natural
frequencies of the circumference and the cross-section separately instead of calculating
them together. In this way the inferior results of the cross-section will not influence the
results of circumference. Because the circumference and cross-section are calculated
separately the natural frequencies of the circumference and the cross-section are not the
same, this difference can be seen in figure 5.10. The difference of the natural frequencies
is caused by the moderate results of the cross-section data, which on its turn is caused by
the poor attachment of the accelerometer to the tire.
The first Matlab model uses for the estimation of the complete tire the
circumference and cross-section data at the same frequency. This is not possible with the
fitted data from ME’scope, because only the data of the natural frequencies is available.
33
Figure 5.9: To the left the 3D model of the second Matlab model and to the right the 3D
model of the first Matlab model, both 3d models are of the second mode shape
Figure 5.10: The frequencies and damping of the tire’s circumference and cross-section
Comparison of the cross-section at 0 and 90 degrees from the excitation point
The second Matlab model uses the measurement data of the cross-section at 0
degrees from the excitation point. As explained above the mode shapes of the cross-
section at 0 degrees are not as expected, and therefore also the model of the complete tire.
When the mode shapes of the cross-section at 0 degrees are compared to the ones of the
cross-section at 90 degrees, the mode shapes of the cross-section at 90 degrees are as one
would expect, which can be seen in figure 5.11. Note that in the mode shapes of the
cross-section measured at 90 degrees a half wavelength is visible in the half cross-
The tire’s circumference The tire’s cross-section
34
section, in contrast to the cross-section at 0 degrees where a complete wavelength is
visible.
Figure 5.11: To the left the second mode shape of the cross-section at 0 degrees can be
seen and to the right the second mode cross-section of the cross-section at 90 degrees
from the excitation point.
Both cross-sections are measured under the same circumstances and with the
same accelerometer attachment (namely wax). The difference can be caused by the
quality of the wax attachment to the tire, which was better when the cross-section at 90
degrees was measured.
Because the measurement of the cross-section at 90 degrees has better results, the
second Matlab model is applied with the cross-section data at 90 degrees instead of the
data from the cross-section at 0 degrees.
The resulting mode shapes of the complete tire with the use of the cross-section at
90 degrees from the excitation point are shown in appendix I. When the resulting mode
shapes of this model are compared to ones of Matlab model one, the waves within the
mode shapes are identical, as can be seen in figure 5.12.
The conclusion that can be drawn from these results is that the second Matlab
model works as expected. This means that if the cross-section at 0 degrees is measured
35
with the right quality, which is possible with a better accelerometer attachment, the
Matlab model will produce an expected mode shape of the complete tire.
Figure 5.12: To the left the 3D model of the second Matlab model with the 90 degree
cross-section and to the right the 3D model of the first Matlab model, both 3d models are
of the second mode shape.
5.4.2 Checking the first Matlab model with the data of the measured cross-
section at 90 degrees from the excitation point
The second Matlab model produces a 3D view of the mode shape of the tire and
should be used to show the first six mode shapes of the tire. But the first Matlab model
produces better results compared to the second Matlab model, even if in the second
Matlab model the cross-section data at 90 degrees is implemented. Because the second
Matlab model does not produces the images as one would expect the first Matlab model
is used to check the correctness of the model.
The first Matlab model is checked with the measurement data of the cross-section
at 90 degrees from the excitation. The model estimates the values of the complete tire,
and will also estimates the points of the cross-section at 90 degrees from the excitation
point. If the first Matlab model estimates the complete tire correctly, the estimated points
are roughly the same as the measured point of the cross-section. In appendix J the
amplitude-values of the first Matlab model and the measured cross-section at 90 degrees
can be found. When comparing the values, the estimated values are indeed roughly the
same, and have an approximately abnormality of 10% with the measured data. This
36
correspondence can be seen in figure 5.13 and 5.14 for the first and second mode shape
of the cross-section.
The cross-section at 90 degrees is measured with six points and the cross-section
at 0 degrees with 7 points. For future works it is important that the cross-section at 90
degrees is measured with more points then the cross-section at 0 degrees, because then a
better comparison can be made between the estimated and measured values.
Figure 5.13: Comparison of the measured points (the red line) and the estimated points
(the blue line) of the cross-section at 90 degrees from the excitation point, which are in
the first mode shape
37
Figure 5.14: Comparison of the measured points (the red line) and the estimated points
(the blue line) of the cross-section at 90 degrees from the excitation point, which are in
the second mode shape
38
Chapter 6
Conclusion
Experimental modal analysis is used to obtain the modal parameters of an
automobile tire. The tire is excited at one point with a predefined chirp signal which
results in a response of the whole tire. A dynamic signal analyzer computes the FRFs
from the measured input signal at this point and the response signals at different points on
the tire.
When the FRFs of all these different points are fitted with a global polynomial
method, the modal parameters are obtained. These modal parameters can later on be used
to tune a finite element model which, however, is not done is this project.
The tire’s symmetry makes it possible to measure only half of the tire because the
points above and below the excitation point have exactly the same excitation. It is also
possible to reconstruct the complete tire with a quarter of the circumference, but not by
simply mirroring the FRFs of the points. The FRFs of the measured points have to be
related in a specific order to the points that are not measured. However for every mode
shape a different relation between the points exist, which makes it a difficult and time-
consuming work to form the expected mode shapes. Therefore it is easier and less time-
consuming to measure the tire’s half circumference. The symmetric shape of the cross-
section makes it possible to measure only the half cross-section, because the two halves
have exactly the same deformation.
The quality of the modal parameters depends on the measurements accuracy. The
accuracy is determined by the quality of the measurement equipment, the settings of the
dynamic analyzer and the person doing the experiments. When good quality of the modal
parameters is assured, smooth and accurate mode shapes can be obtained.
To obtain a smooth mode shape, not only the quality of the measurement is
important, but also the number of measured points on tire. Every wavelength within the
tire should have at least six measurement points for a smooth visualization. The tire’s half
circumference is measured with 24 points; this means that the first six mode shapes can
be visualized, considering that the sixth mode shape has 7 wavelengths within the tire.
39
The tire’s half cross-section is measured with 7 points, because every mode shape of the
cross-section has two wavelengths.
For a free hanged standard automobile tire (type 205/60 R 15) with 1.6 bar
pressure, the first six mode frequencies are clearly visualized. The resonances have the
following natural frequencies: 114 Hz, 137 Hz, 165 Hz, 196 Hz, 228 Hz and 260 Hz.
Knowing the change of the FRF over the length of the circumference and cross-
section of the tire makes it possible to estimate the FRF for every point on the tire.
Plotting the FRF for every point (in the radial direction) results in a 3D image of the tire.
The estimation of the complete tire is checked with another measured cross-section. The
measured points roughly correspond with the estimation of these points, which means the
complete tire is estimated right.
With the 3D image checked, it is proven that when measuring only the
circumference and cross-section of the tire, the modal parameters of the tire can be
calculated and a prediction of the response of the whole tire can be made.
40
Chapter 7
Recommendations
A few recommendations can be made for future works or studies on the
experimental modal analysis of an automobile tire. As explained in chapter 5 the
accelerometer attachment influences the measurement results. The better the
accelerometer is attached, the better the results. Because the results of the cross-section
where inferior to the results of the tire’s circumference. The modal analysis of the tire can
only be improved if the cross-section measurement is improved. The only way this can be
done, is with a better accelerometer attachment or a complete different measuring
method.
It is important to measure all the cross-sections with the same amount of points.
In this project it was not clear with how many points the cross-section had to be
measured. Therefore the number of measurement points changed when a new cross-
section was measured. In future works it is better to measure the half cross-section for the
estimation of the complete tire with 9 points.
In future works, the tires’ measured modes can be compared with the calculations
of a FEM analysis of the tire. The comparison can give one an idea of the correctness of
the FEM analysis.
41
Appendix A
Specifications of the measurement equipment
A1: The shaker
The Ling Dynamics Systems (LDS) Shaker
System sine force peak-natural cooled: 17.8 N
Resonance frequency: 13000 Hz
Useful frequency range 5-13000 Hz
System displacement (continuous) pk-pk 5 mm
Shaker mass 1.81 kg
A2: The force transducer
The force transducer, PCB 221A04
Modelnr: 002A10
Sensitivity (±15%): 1124.1 mV/kN
Measurement Range (compression): 4.448 kN
Measurement Range (tension): 4.448 kN
Maximum static force (compression): 26.69 kN
Maximum static force (tension): 5.34 kN
Upper frequency limit: 15 kHz
Mass: 31 gm
A3: The accelerometer
The Kistler accelerometer, type 8628 B50
Range: ±50 g
Resonance frequency: 22 KHz
Sensitivity (±5%): 100 mV/g
Mass: 6.7 g
A4: The dynamic signal analyzer
The DSP Siglab dynamic signal analyzer
Modelnr: 20-42
Frequency range: 20 kHz
Dynamic range: 90 kHz
Accuracy ±0.03+0.02(f/20kHz) dB
Maximum resolution: 8912 lines
42
Appendix B
Measurement points on the tire
Data of the 205/60R15 tire
Side wall: 123.00 mm
Radius: 313.50 mm
Circumference: 1909.80 mm
Revs/km: 508
Source: Miata.net tire size calculator
Contents:
B1 – Tire’s circumference
B2 – Tire’s sidewall
B3 – Tire’s cross-section at 0 degrees from the excitation point
B4 – Tire’s cross-section at 90 degrees from the excitation point
B5 – Tire’s cross-section at 180 degrees from the excitation point
43
B1 – Tire’s circumference
The half tire’s circumference is measured with 24 points; in table B1 the coordinates of
these points are represented.
Point number ρ (mm) θ (degrees) ψ (degrees)
1 313.50 90 0
2 313.50 90 7.8261
3 313.50 90 15.6522
4 313.50 90 23.4783
5 313.50 90 31.3044
6 313.50 90 39.1305
7 313.50 90 46.9566
8 313.50 90 54.7827
9 313.50 90 62.6088
10 313.50 90 70.4349
11 313.50 90 78.2610
12 313.50 90 86.0871
13 313.50 90 93.9132
14 313.50 90 101.7393
15 313.50 90 109.5654
16 313.50 90 117.3915
17 313.50 90 125.2176
18 313.50 90 133.0437
19 313.50 90 140.8698
20 313.50 90 148.6959
21 313.50 90 156.5220
21 313.50 90 164.3481
23 313.50 90 172.1742
24 313.50 90 180
44
B2 – Tire’s sidewall
The half tire’s sidewall is measured with 24 points; in table B1 the coordinates of these
points are represented. The whole coordinate system is translate ±123.00 mm in the X-
direction
Point number ρ (mm) θ (degrees) ψ (degrees)
1 260 90 0
2 260 90 7.8261
3 260 90 15.6522
4 260 90 23.4783
5 260 90 31.3044
6 260 90 39.1305
7 260 90 46.9566
8 260 90 54.7827
9 260 90 62.6088
10 260 90 70.4349
11 260 90 78.2610
12 260 90 86.0871
13 260 90 93.9132
14 260 90 101.7393
15 260 90 109.5654
16 260 90 117.3915
17 260 90 125.2176
18 260 90 133.0437
19 260 90 140.8698
20 260 90 148.6959
21 260 90 156.5220
21 260 90 164.3481
23 260 90 172.1742
24 260 90 180
45
B3 – Tire’s cross-section at 0 degrees from the excitation point
The whole coordinate system is translate -260.00 mm in the Y-direction
Point number ρ (mm) θ (degrees) ψ (degrees)
1 123 180 -90
2 123 195 -90
3 123 210 -90
4 123 225 -90
5 123 240 -90
6 123 255 -90
7 123 270 -90
B4 – Tire’s cross-section at 90 degrees from the excitation point
The whole coordinate system is translate 260.00 mm in the Z-direction
Point number ρ (mm) θ (degrees) ψ (degrees)
1 123 0 90
2 123 0 72
3 123 0 54
4 123 0 36
5 123 0 18
6 123 0 0
B5 – Tire’s cross-section at 180 degrees from the excitation point
The whole coordinate system is translate 260.00 mm in the Y-direction
Point number ρ (mm) θ (degrees) ψ (degrees)
1 123 0 90
2 123 18 90
3 123 36 90
4 123 54 90
5 123 72 90
6 123 90 90
7 123 108 90
8 123 126 90
9 123 144 90
10 123 162 90
11 123 180 90
46
Appendix C
Mode indicator functions
Contents:
C1 – The mode indicator function of the tire’s circumference
C2 – The mode indicator function of the tire’s cross-section, at 0 degrees from the
excitation point.
C1 – The mode indicator function of the tire’s circumference
47
C2 – The mode indicator function of the tire’s cross-section, at 0 degrees from the
excitation point.
48
49
Appendix D
Estimated mode shapes of the tire’s circumference
Mode 1 Mode 2
Mode 2
Mode 3 Mode 4
50
Mode 5 Mode 6
Mode 7 Mode 8
51
Mode 9
52
Appendix E
Estimated mode shapes of the tire’s side wall
Mode 1 Mode 2
Mode 3 Mode 4
53
Mode 5 Mode 6
Mode 7 Mode 8
54
The mode indicator function of the tire’s sidewall
55
Appendix F
Estimated mode shapes of the tire’s cross-sections
Contents:
F1 – mode shapes of the cross-section at 0 degrees from the excitation point
F2 – mode shapes of the cross-section at 90 degrees from the excitation point
F3 – mode shapes of the cross-section at 180 degrees from the excitation point
F1 – mode shapes of the cross-section at 0 degrees from the excitation point
Mode 1 Mode 2
56
Mode 3 Mode 4
Mode 5 Mode 6
57
Mode 7 Mode 8
F2 – mode shapes of the cross-section at 90 degrees from the excitation point
Mode 1 Mode 2
58
Mode 3 Mode 4
Mode 5 Mode 6
59
Mode 7 Mode 8
F3 – mode shapes of the cross-section at 180 degrees from the excitation point
Mode 1 Mode 2
60
Mode 3 Mode 4
Mode 5 Mode 6
61
Mode 7 Mode 8
62
Appendix G
Matlab model
clear all;%clc;close all;
%Invoeren gewenste frequentie
Gewenste_frequentie= input('Welke frequentie
wilt u laten zien? in Hz : ');
%Berekenen van het aantal Hz per punt,
%<>Gemeten data in de vna-file is verwerkt in
een matrix met 800 punten,
%<>waarin de bandbreedte loopt van 0 tot 500 Hz
Aantal_Hertz_per_punt=500/800;
%Berekenen datapunt in matrix die overeenkomt
met de ingevoerde gewenste
%frequentie
IWgeheel=Gewenste_frequentie/Aantal_Hertz_per_pu
nt;
IW=round(IWgeheel);
%%% importeren data omtrek %%%%
ND_1o = importdata('1aomtrek.vna');
assignin('base','SLm',ND_1o.('SLm'));
f1o=SLm.fdxvec;
m1o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase1o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.x
cmeas(5).xfer))*180/pi;
%uit de matrix van de magnitude en de fase wordt
alleen de magnitude en
%fase die bij de gewenste frequenctie horen
geselecteerd!
m1o=m1o(IW);
fase1o=fase1o(IW);
ND_2o = importdata('2aomtrek.vna');
assignin('base','SLm',ND_2o.('SLm'));
f2o=SLm.fdxvec;
m2o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase2o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.x
cmeas(5).xfer))*180/pi;
m2o=m2o(IW);
fase2o=fase2o(IW);
ND_3o = importdata('3aomtrek.vna');
assignin('base','SLm',ND_3o.('SLm'));
f3o=SLm.fdxvec;
m3o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase3o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.x
cmeas(5).xfer))*180/pi;
m3o=m3o(IW);
fase3o=fase3o(IW);
ND_4o = importdata('4omtrek.vna');
assignin('base','SLm',ND_4o.('SLm'));
f4o=SLm.fdxvec;
m4o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase4o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.x
cmeas(5).xfer))*180/pi;
m4o=m4o(IW);
fase4o=fase4o(IW);
ND_5o = importdata('5bomtrek.vna');
assignin('base','SLm',ND_5o.('SLm'));
f5o=SLm.fdxvec;
m5o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase5o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm
.xcmeas(5).xfer))*180/pi;
m5o=m5o(IW);
fase5o=fase5o(IW);
ND_6o = importdata('6bomtrek.vna');
assignin('base','SLm',ND_6o.('SLm'));
f6o=SLm.fdxvec;
m6o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase6o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm
.xcmeas(5).xfer))*180/pi;
m6o=m6o(IW);
fase6o=fase6o(IW);
ND_7o = importdata('7bomtrek.vna');
assignin('base','SLm',ND_7o.('SLm'));
f7o=SLm.fdxvec;
m7o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase7o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm
.xcmeas(5).xfer))*180/pi;
m7o=m7o(IW);
fase7o=fase7o(IW);
ND_8o = importdata('8omtrek.vna');
assignin('base','SLm',ND_8o.('SLm'));
f8o=SLm.fdxvec;
m8o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase8o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm
.xcmeas(5).xfer))*180/pi;
m8o=m8o(IW);
fase8o=fase8o(IW);
ND_9o = importdata('9omtrek.vna');
assignin('base','SLm',ND_9o.('SLm'));
f9o=SLm.fdxvec;
m9o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase9o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm
.xcmeas(5).xfer))*180/pi;
m9o=m9o(IW);
fase9o=fase9o(IW);
ND_10o = importdata('10omtrek.vna');
assignin('base','SLm',ND_10o.('SLm'));
f10o=SLm.fdxvec;
m10o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase10o=atan2(imag(SLm.xcmeas(5).xfer),real(SL
m.xcmeas(5).xfer))*180/pi;
m10o=m10o(IW);
fase10o=fase10o(IW);
ND_11o = importdata('11omtrek.vna');
assignin('base','SLm',ND_11o.('SLm'));
f11o=SLm.fdxvec;
m11o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase11o=atan2(imag(SLm.xcmeas(5).xfer),real(SL
m.xcmeas(5).xfer))*180/pi;
m11o=m11o(IW);
fase11o=fase11o(IW);
63
ND_12o = importdata('12omtrek.vna');
assignin('base','SLm',ND_12o.('SLm'));
f12o=SLm.fdxvec;
m12o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase12o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.
xcmeas(5).xfer))*180/pi;
m12o=m12o(IW);
fase12o=fase12o(IW);
ND_13o = importdata('13omtrek.vna');
assignin('base','SLm',ND_13o.('SLm'));
f13o=SLm.fdxvec;
m13o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase13o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.
xcmeas(5).xfer))*180/pi;
m13o=m13o(IW);
fase13o=fase13o(IW);
ND_14o = importdata('14omtrek.vna');
assignin('base','SLm',ND_14o.('SLm'));
f14o=SLm.fdxvec;
m14o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase14o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.
xcmeas(5).xfer))*180/pi;
m14o=m14o(IW);
fase14o=fase14o(IW);
ND_15o = importdata('15omtrek.vna');
assignin('base','SLm',ND_15o.('SLm'));
f15o=SLm.fdxvec;
m15o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase15o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.
xcmeas(5).xfer))*180/pi;
m15o=m15o(IW);
fase15o=fase15o(IW);
ND_16o = importdata('16omtrek.vna');
assignin('base','SLm',ND_16o.('SLm'));
f16o=SLm.fdxvec;
m16o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase16o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.
xcmeas(5).xfer))*180/pi;
m16o=m16o(IW);
fase16o=fase16o(IW);
ND_17o = importdata('17omtrek.vna');
assignin('base','SLm',ND_17o.('SLm'));
f17o=SLm.fdxvec;
m17o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase17o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.
xcmeas(5).xfer))*180/pi;
m17o=m17o(IW);
fase17o=fase17o(IW);
ND_18o = importdata('18omtrek.vna');
assignin('base','SLm',ND_18o.('SLm'));
f18o=SLm.fdxvec;
m18o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase18o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.
xcmeas(5).xfer))*180/pi;
m18o=m18o(IW);
fase18o=fase18o(IW);
ND_19o = importdata('19omtrek.vna');
assignin('base','SLm',ND_19o.('SLm'));
f19o=SLm.fdxvec;
m19o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase19o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.
xcmeas(5).xfer))*180/pi;
m19o=m19o(IW);
fase19o=fase19o(IW);
ND_20o = importdata('20omtrek.vna');
assignin('base','SLm',ND_20o.('SLm'));
f20o=SLm.fdxvec;
m20o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase20o=atan2(imag(SLm.xcmeas(5).xfer),real(SL
m.xcmeas(5).xfer))*180/pi;
m20o=m20o(IW);
fase20o=fase20o(IW);
ND_21o = importdata('21omtrek.vna');
assignin('base','SLm',ND_21o.('SLm'));
f21o=SLm.fdxvec;
m21o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase21o=atan2(imag(SLm.xcmeas(5).xfer),real(SL
m.xcmeas(5).xfer))*180/pi;
m21o=m21o(IW);
fase21o=fase21o(IW);
ND_22o = importdata('22omtrek.vna');
assignin('base','SLm',ND_22o.('SLm'));
f22o=SLm.fdxvec;
m22o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase22o=atan2(imag(SLm.xcmeas(5).xfer),real(SL
m.xcmeas(5).xfer))*180/pi;
m22o=m22o(IW);
fase22o=fase22o(IW);
ND_23o = importdata('23omtrek.vna');
assignin('base','SLm',ND_23o.('SLm'));
f23o=SLm.fdxvec;
m23o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase23o=atan2(imag(SLm.xcmeas(5).xfer),real(SL
m.xcmeas(5).xfer))*180/pi;
m23o=m23o(IW);
fase23o=fase23o(IW);
ND_24o = importdata('24omtrek.vna');
assignin('base','SLm',ND_1o.('SLm'));
f24o=SLm.fdxvec;
m24o=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase24o=atan2(imag(SLm.xcmeas(5).xfer),real(SL
m.xcmeas(5).xfer))*180/pi;
m24o=m24o(IW);
fase24o=fase24o(IW);
%importeren data doorsnede%
ND_1d = importdata('7doorsnede.vna');
assignin('base','SLm',ND_1d.('SLm'));
f1d=SLm.fdxvec;
m1d=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase1d=atan2(imag(SLm.xcmeas(5).xfer),real(SLm
.xcmeas(5).xfer))*180/pi;
m1d=m1d(IW);
fase1d=fase1d(IW);
ND_2d = importdata('6doorsnede.vna');
assignin('base','SLm',ND_2d.('SLm'));
f2d=SLm.fdxvec;
m2d=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase2d=atan2(imag(SLm.xcmeas(5).xfer),real(SLm
.xcmeas(5).xfer))*180/pi;
m2d=m2d(IW);
fase2d=fase2d(IW);
ND_3d = importdata('5doorsnede.vna');
assignin('base','SLm',ND_3d.('SLm'));
f3d=SLm.fdxvec;
m3d=20*log10(abs(SLm.xcmeas(1,2).xfer));
64
fase3d=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.x
cmeas(5).xfer))*180/pi;
m3d=m3d(IW);
fase3d=fase3d(IW);
ND_4d = importdata('4doorsnede.vna');
assignin('base','SLm',ND_4d.('SLm'));
f4d=SLm.fdxvec;
m4d=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase4d=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.x
cmeas(5).xfer))*180/pi;
m4d=m4d(IW);
fase4d=fase4d(IW);
ND_5d = importdata('3doorsnede.vna');
assignin('base','SLm',ND_5d.('SLm'));
f5d=SLm.fdxvec;
m5d=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase5d=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.x
cmeas(5).xfer))*180/pi;
m5d=m5d(IW);
fase5d=fase5d(IW);
ND_6d = importdata('2doorsnede.vna');
assignin('base','SLm',ND_6d.('SLm'));
f6d=SLm.fdxvec;
m6d=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase6d=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.x
cmeas(5).xfer))*180/pi;
m6d=m6d(IW);
fase6d=fase6d(IW);
ND_7d = importdata('1doorsnede.vna');
assignin('base','SLm',ND_7d.('SLm'));
f7d=SLm.fdxvec;
m7d=20*log10(abs(SLm.xcmeas(1,2).xfer));
fase7d=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.x
cmeas(5).xfer))*180/pi;
m7d=m7d(IW);
fase7d=fase7d(IW);
%%%%%%%%% Verhoudingen tussen magnitude &
verschil fase %%%%%%%%%%%%%%%
% Verhoudingen van magnitude tussen meetpunt 1o
met 2o t/m 24o
ver12m=m2o/m1o;
ver13m=m3o/m1o;
ver14m=m4o/m1o;
ver15m=m5o/m1o;
ver16m=m6o/m1o;
ver17m=m7o/m1o;
ver18m=m8o/m1o;
ver19m=m9o/m1o;
ver110m=m10o/m1o;
ver111m=m11o/m1o;
ver112m=m12o/m1o;
ver113m=m13o/m1o;
ver114m=m14o/m1o;
ver115m=m15o/m1o;
ver116m=m16o/m1o;
ver117m=m17o/m1o;
ver118m=m18o/m1o;
ver119m=m19o/m1o;
ver120m=m20o/m1o;
ver121m=m21o/m1o;
ver122m=m22o/m1o;
ver123m=m23o/m1o;
ver124m=m24o/m1o;
% Verschil van de fase tussen meetpunt 1d met
2d t/m 7d
ver12f=abs(fase2d-fase1d);
ver13f=abs(fase3d-fase1d);
ver14f=abs(fase4d-fase1d);
ver15f=abs(fase5d-fase1d);
ver16f=abs(fase6d-fase1d);
ver17f=abs(fase7d-fase1d);
% Wanneer het faseverschil kleiner is dan 90
graden,
% magnitude keer 1 anders magnitude keer -1
if ver12f<90
B12=1;
else B12=-1;
end
if ver13f<90
B13=1;
else B13=-1;
end
if ver14f<90
B14=1;
else B14=-1;
end
if ver15f<90
B15=1;
else B15=-1;
end
if ver16f<90
B16=1;
else B16=-1;
end
if ver17f<90
B17=1;
else B17=-1;
end
% Verschil van de fase tussen meetpunt 1o met
2o t/m 24o
ver12fo=abs(fase2o-fase1o);
ver13fo=abs(fase3o-fase1o);
ver14fo=abs(fase4o-fase1o);
ver15fo=abs(fase5o-fase1o);
ver16fo=abs(fase6o-fase1o);
ver17fo=abs(fase7o-fase1o);
ver18fo=abs(fase8o-fase1o);
ver19fo=abs(fase9o-fase1o);
ver110fo=abs(fase10o-fase1o);
ver111fo=abs(fase11o-fase1o);
ver112fo=abs(fase12o-fase1o);
ver113fo=abs(fase13o-fase1o);
ver114fo=abs(fase14o-fase1o);
ver115fo=abs(fase15o-fase1o);
ver116fo=abs(fase16o-fase1o);
ver117fo=abs(fase17o-fase1o);
ver118fo=abs(fase18o-fase1o);
ver119fo=abs(fase19o-fase1o);
ver120fo=abs(fase20o-fase1o);
ver121fo=abs(fase21o-fase1o);
ver122fo=abs(fase22o-fase1o);
ver123fo=abs(fase23o-fase1o);
ver124fo=abs(fase24o-fase1o);
65
if ver12fo<90
L12=1;
else L12=-1;
end
if ver13fo<90
L13=1;
else L13=-1;
end
if ver14fo<90
L14=1;
else L14=-1;
end
if ver15fo<90
L15=1;
else L15=-1;
end
if ver16fo<90
L16=1;
else L16=-1;
end
if ver17fo<90
L17=1;
else L17=-1;
end
if ver18fo<90
L18=1;
else L18=-1;
end
if ver19fo<90
L19=1;
else L19=-1;
end
if ver110fo<90
L110=1;
else L110=-1;
end
if ver111fo<90
L111=1;
else L111=-1;
end
if ver112fo<90
L112=1;
else L112=-1;
end
if ver113fo<90
L113=1;
else L113=-1;
end
if ver114fo<90
L114=1;
else L114=-1;
end
if ver115fo<90
L115=1;
else L115=-1;
end
if ver116fo<90
L116=1;
else L116=-1;
end
if ver117fo<90
L117=1;
else L117=-1;
end
if ver118fo<90
L118=1;
else L118=-1;
end
if ver119fo<90
L119=1;
else L119=-1;
end
if ver120fo<90
L120=1;
else L120=-1;
end
if ver121fo<90
L121=1;
else L121=-1;
end
if ver122fo<90
L122=1;
else L122=-1;
end
if ver123fo<90
L123=1;
else L123=-1;
end
if ver124fo<90
L124=1;
else L124=-1;
end
66
%%%%% Berekening werkelijke magnitude van alle
punten in het model %%%%%
m1o=m1o*1;
m2o=m2o*L12;
m3o=m3o*L13;
m4o=m4o*L14;
m5o=m5o*L15;
m6o=m6o*L16;
m7o=m7o*L17;
m8o=m8o*L18;
m9o=m9o*L19;
m10o=m10o*L110;
m11o=m11o*L111;
m12o=m12o*L112;
m13o=m13o*L113;
m14o=m14o*L114;
m15o=m15o*L115;
m16o=m16o*L116;
m17o=m17o*L117;
m18o=m18o*L118;
m19o=m19o*L119;
m20o=m20o*L120;
m21o=m21o*L121;
m22o=m22o*L122;
m23o=m23o*L123;
m24o=m24o*L124;
m1d=m1d*1;
m2d=m2d*B12;
m3d=m3d*B13;
m4d=m4d*B14;
m5d=m5d*B15;
m6d=m6d*B16;
m7d=m7d*B17;
m22=ver12m*m2d*L12;
m23=ver12m*m3d*L12;
m24=ver12m*m4d*L12;
m25=ver12m*m5d*L12;
m26=ver12m*m6d*L12;
m27=ver12m*m7d*L12;
m32=ver13m*m2d*L13;
m33=ver13m*m3d*L13;
m34=ver13m*m4d*L13;
m35=ver13m*m5d*L13;
m36=ver13m*m6d*L13;
m37=ver13m*m7d*L13;
m42=ver14m*m2d*L14;
m43=ver14m*m3d*L14;
m44=ver14m*m4d*L14;
m45=ver14m*m5d*L14;
m46=ver14m*m6d*L14;
m47=ver14m*m7d*L14;
m52=ver15m*m2d*L15;
m53=ver15m*m3d*L15;
m54=ver15m*m4d*L15;
m55=ver15m*m5d*L15;
m56=ver15m*m6d*L15;
m57=ver15m*m7d*L15;
m62=ver16m*m2d*L16;
m63=ver16m*m3d*L16;
m64=ver16m*m4d*L16;
m65=ver16m*m5d*L16;
m66=ver16m*m6d*L16;
m67=ver16m*m7d*L16;
m72=ver17m*m2d*L17;
m73=ver17m*m3d*L17;
m74=ver17m*m4d*L17;
m75=ver17m*m5d*L17;
m76=ver17m*m6d*L17;
m77=ver17m*m7d*L17;
m82=ver18m*m2d*L18;
m83=ver18m*m3d*L18;
m84=ver18m*m4d*L18;
m85=ver18m*m5d*L18;
m86=ver18m*m6d*L18;
m87=ver18m*m7d*L18;
m92=ver19m*m2d*L19;
m93=ver19m*m3d*L19;
m94=ver19m*m4d*L19;
m95=ver19m*m5d*L19;
m96=ver19m*m6d*L19;
m97=ver19m*m7d*L19;
m102=ver110m*m2d*L110;
m103=ver110m*m3d*L110;
m104=ver110m*m4d*L110;
m105=ver110m*m5d*L110;
m106=ver110m*m6d*L110;
m107=ver110m*m7d*L110;
m112=ver111m*m2d*L111;
m113=ver111m*m3d*L111;
m114=ver111m*m4d*L111;
m115=ver111m*m5d*L111;
m116=ver111m*m6d*L111;
m117=ver111m*m7d*L111;
m122=ver112m*m2d*L112;
m123=ver112m*m3d*L112;
m124=ver112m*m4d*L112;
m125=ver112m*m5d*L112;
m126=ver112m*m6d*L112;
m127=ver112m*m7d*L112;
m132=ver113m*m2d*L113;
m133=ver113m*m3d*L113;
m134=ver113m*m4d*L113;
m135=ver113m*m5d*L113;
m136=ver113m*m6d*L113;
m137=ver113m*m7d*L113;
m142=ver114m*m2d*L114;
m143=ver114m*m3d*L114;
m144=ver114m*m4d*L114;
m145=ver114m*m5d*L114;
m146=ver114m*m6d*L114;
m147=ver114m*m7d*L114;
m152=ver115m*m2d*L115;
m153=ver115m*m3d*L115;
m154=ver115m*m4d*L115;
m155=ver115m*m5d*L115;
m156=ver115m*m6d*L115;
m157=ver115m*m7d*L115;
m162=ver116m*m2d*L116;
m163=ver116m*m3d*L116;
m164=ver116m*m4d*L116;
m165=ver116m*m5d*L116;
m166=ver116m*m6d*L116;
m167=ver116m*m7d*L116;
m172=ver117m*m2d*L117;
m173=ver117m*m3d*L117;
m174=ver117m*m4d*L117;
m175=ver117m*m5d*L117;
m176=ver117m*m6d*L117;
m177=ver117m*m7d*L117;
m182=ver118m*m2d*L118;
m183=ver118m*m3d*L118;
m184=ver118m*m4d*L118;
m185=ver118m*m5d*L118;
m186=ver118m*m6d*L118;
m187=ver118m*m7d*L118;
m192=ver119m*m2d*L119;
m193=ver119m*m3d*L119;
m194=ver119m*m4d*L119;
m195=ver119m*m5d*L119;
m196=ver119m*m6d*L119;
m197=ver119m*m7d*L119;
m202=ver120m*m2d*L120;
m203=ver120m*m3d*L120;
m204=ver120m*m4d*L120;
m205=ver120m*m5d*L120;
m206=ver120m*m6d*L120;
m207=ver120m*m7d*L120;
m212=ver121m*m2d*L121;
m213=ver121m*m3d*L121;
m214=ver121m*m4d*L121;
m215=ver121m*m5d*L121;
m216=ver121m*m6d*L121;
m217=ver121m*m7d*L121;
m222=ver122m*m2d*L122;
m223=ver122m*m3d*L122;
m224=ver122m*m4d*L122;
m225=ver122m*m5d*L122;
m226=ver122m*m6d*L122;
m227=ver122m*m7d*L122;
m232=ver123m*m2d*L123;
m233=ver123m*m3d*L123;
m234=ver123m*m4d*L123;
m235=ver123m*m5d*L123;
m236=ver123m*m6d*L123;
m237=ver123m*m7d*L123;
m242=ver124m*m2d*L124;
m243=ver124m*m3d*L124;
m244=ver124m*m4d*L124;
m245=ver124m*m5d*L124;
m246=ver124m*m6d*L124;
m247=ver124m*m7d*L124;
67
Z1=[m1d m2d m3d m4d m5d m6d m7d;
m2o m22 m23 m24 m25 m26 m27;
m3o m32 m33 m34 m35 m36 m37;
m4o m42 m43 m44 m45 m46 m47;
m5o m52 m53 m54 m55 m56 m57;
m6o m62 m63 m64 m65 m66 m67;
m7o m72 m73 m74 m75 m76 m77;
m8o m82 m83 m84 m85 m86 m87;
m9o m92 m93 m94 m95 m96 m97;
m10o m102 m103 m104 m105 m106 m107;
m11o m112 m113 m114 m115 m116 m117;
m12o m122 m123 m124 m125 m126 m127;
m13o m132 m133 m134 m135 m136 m137;
m14o m142 m143 m144 m145 m146 m147;
m15o m152 m153 m154 m155 m156 m157;
m16o m162 m163 m164 m165 m166 m167;
m17o m172 m173 m174 m175 m176 m177;
m18o m182 m183 m184 m185 m186 m187;
m19o m192 m193 m194 m195 m196 m197;
m20o m202 m203 m204 m205 m206 m207;
m21o m212 m213 m214 m215 m216 m217;
m22o m222 m223 m224 m225 m226 m227;
m23o m232 m233 m234 m235 m236 m237;
m24o m242 m243 m244 m245 m246 m247];
%%%%%%% Plotten meetpunten
[X,Y] = meshgrid(1:1:7,1:1:24);
%Doorsnede doortrekken
figure
surf(X,Y,Z1,'FaceColor','interp',...
'EdgeColor','none',...
'FaceLighting','phong')
%daspect([5 5 1])
%axis tight
%view(-50,30)
%camlight left
xlabel('doorsnede (doorgetrokken over omtrek')
ylabel('omtrek')
zlabel(Gewenste_frequentie)
title('Doorsnede doorgetrokken over omtrek')
68
Appendix H
The results of Matlab model one
Mode 1
Mode 2
Circumference
Circumference
69
Mode 3
Mode 4
Circumference
Circumference
70
Mode 5
Mode 6
Circumference
Circumference
71
Mode 7
Mode 8
Circumference
Circumference
72
Appendix I
The results of Matlab model two
I1: With the use of the cross-section at 0 degrees from the excitation point
I2: With the use of the cross-section at 90 degrees from the excitation point
I1: Matlab model two, with the use of the cross-section at 0 degrees from the
excitation point
Mode 1
Circumference
73
Mode 2
Mode 3
Circumference
Circumference
74
Mode 4
Mode 5
Circumference
Circumference
75
Mode 6
Mode 7
Circumference
Circumference
76
Mode 8
Circumference
77
I2: Matlab model two, with the use of the cross-section at 90 degrees from the
excitation point
Mode 1
Mode 2
Circumference
Circumference
78
Mode 3
Mode 4
Circumference
Circumference
79
Mode 5
Mode 6
Circumference
Circumference
80
Mode 7
Mode 8
Circumference
Circumference
81
Appendix J
Comparison of the cross-section at 90 degrees from the excitation point with the model that
estimates the complete tire
A for 113 Hz 1 2 3 4 5 6 7
Circumference mp12 -27.8320 -27.2414 -25.9360 -21.5525 17.1704 26.5426 12.1999
Circumference mp13 -27.6977 -27.1099 -25.8108 -21.4484 17.0875 26.4144 12.1410
Cross-section -29.4296 -28.1584 -27.3910 -22.5304 20.0966 18.4512
A for 137 Hz 1 2 3 4 5 6 7
Circumference mp12 -19.4419 -19.4480 -18.8511 -16.3872 9.6419 12.6164 11.3127
Circumference mp13 22.1279 22.1348 21.4554 18.6512 -10.9740 -14.3594 -12.8756
Cross-section -22.6762 -20.9053 -20.3485 -16.4424 14.1491 13.9177
A for 165 Hz 1 2 3 4 5 6 7
Circumference mp12 28.5873 28.3762 27.8991 24.9866 -13.8638 -22.5355 -15.4311
Circumference mp13 27.5876 27.3839 26.9234 24.1128 -13.3790 -21.7475 -14.8915
Cross-section 30.3679 27.8621 27.4737 23.9632 -22.9327 -21.4432
A for 196 Hz 1 2 3 4 5 6 7
Circumference mp12 20.1025 19.5364 19.2475 17.7043 -11.5933 -17.1296 -11.8871
Circumference mp13 -23.4875 -22.8261 -22.4885 -20.6854 13.5455 20.0139 13.8887
Cross-section 20.7298 17.7863 18.2754 15.4327 17.9883 -18.0246
A for 228 Hz 1 2 3 4 5 6 7
Circumference mp12 -28.1074 -26.5135 -25.6743 -24.5273 2.6688 23.6849 17.2842
Circumference mp13 -26.1666 -24.6827 -23.9014 -22.8336 2.4845 22.0495 16.0907
Cross-section -28.5715 -26.4553 -26.4157 -24.8298 22.0248 22.1103
A for 257 Hz 1 2 3 4 5 6 7
Circumference mp12 23.6714 21.7966 20.7773 20.6010 -12.2624 20.8565 16.4330
Circumference mp13 26.8165 24.6926 23.5378 23.3382 -13.8916 23.6275 18.6163
Cross-section 22.9957 21.1626 20.9382 20.3428 18.1282 -20.2179
82
Bibliography
[1] D.J. Ewins, Modal testing: theory, practice and application, second edition, 2000.
[2] DSP Technology Inc., Siglab User Guide, 1998.
[3] DSP Technology Inc., The Siglab measurement structure.
[4] Vibrant Technology Inc., ME’scopeVES 4.0 Operating manual, 2003.
[5] Gene F. Franklin, J.David Powell, Abbas Emani-Naeini; Feedback control of
Dynamic systems, fourth edition, 2002.
[6] O. Dönmez, Experimental modal analysis of an automobile tire, Erasmus
Internship Report, 2007.
[7] R.J.E. Merry, Experimental modal analysis of the H-drive, Report No. DCT
2003.78., 2003.
[8] M.L.J. Verhees, Experimental modal analysis of a turbine blade, Report No. DCT
2004.120., 2004.
[9] P.A.R. de Schrijver, Experimental modal analysis of the tyre measurement tower,
Report No. DCT 2005.112., 2005.
[10] P.W.A. Zegelaar, Modal analysis of tire in-plane vibration, SAE Paper 971101,
Vehicle research laboratory, Delft University of Technology.
[11] P.Evitabile, Modal space – in our own little world, SEM Experimental
Techniques, 2002.
[12] LDS Dactron, Basics of structural vibration testing and analysis, Application Note
ANO11, 2003.
[13] B. de Kraker: A numerical-experimental Approach in Structural Dynamics, 2000.
Lecture notes nr.4748