Experimental Modal Analysis of an Automobile Tire - misure

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



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



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


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


The whole coordinate system is translate -260.00 mm in the Y-direction


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


The whole coordinate system is translate 260.00 mm in the Z-direction


1 123 0 90

2 123 0 72

3 123 0 54

4 123 0 36

5 123 0 18

6 123 0 0


The whole coordinate system is translate 260.00 mm in the Y-direction


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.

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




Mode 1 Mode 2

56

Mode 3 Mode 4

Mode 5 Mode 6

57

Mode 7 Mode 8


Mode 1 Mode 2

58

Mode 3 Mode 4

Mode 5 Mode 6

59

Mode 7 Mode 8


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);



f2o=SLm.fdxvec;




m2o=m2o(IW);

fase2o=fase2o(IW);



f3o=SLm.fdxvec;




m3o=m3o(IW);

fase3o=fase3o(IW);

ND_4o = importdata('4omtrek.vna');


f4o=SLm.fdxvec;




m4o=m4o(IW);

fase4o=fase4o(IW);

ND_5o = importdata('5bomtrek.vna');


f5o=SLm.fdxvec;


fase5o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm

.xcmeas(5).xfer))*180/pi;

m5o=m5o(IW);

fase5o=fase5o(IW);



f6o=SLm.fdxvec;




m6o=m6o(IW);

fase6o=fase6o(IW);



f7o=SLm.fdxvec;




m7o=m7o(IW);

fase7o=fase7o(IW);



f8o=SLm.fdxvec;




m8o=m8o(IW);

fase8o=fase8o(IW);



f9o=SLm.fdxvec;




m9o=m9o(IW);

fase9o=fase9o(IW);



f10o=SLm.fdxvec;


fase10o=atan2(imag(SLm.xcmeas(5).xfer),real(SL

m.xcmeas(5).xfer))*180/pi;

m10o=m10o(IW);

fase10o=fase10o(IW);



f11o=SLm.fdxvec;




m11o=m11o(IW);


63



f12o=SLm.fdxvec;


fase12o=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.

xcmeas(5).xfer))*180/pi;

m12o=m12o(IW);




f13o=SLm.fdxvec;




m13o=m13o(IW);




f14o=SLm.fdxvec;




m14o=m14o(IW);




f15o=SLm.fdxvec;




m15o=m15o(IW);




f16o=SLm.fdxvec;




m16o=m16o(IW);




f17o=SLm.fdxvec;




m17o=m17o(IW);




f18o=SLm.fdxvec;




m18o=m18o(IW);




f19o=SLm.fdxvec;




m19o=m19o(IW);




f20o=SLm.fdxvec;




m20o=m20o(IW);




f21o=SLm.fdxvec;




m21o=m21o(IW);




f22o=SLm.fdxvec;




m22o=m22o(IW);




f23o=SLm.fdxvec;




m23o=m23o(IW);




f24o=SLm.fdxvec;




m24o=m24o(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


m1d=m1d(IW);

fase1d=fase1d(IW);



f2d=SLm.fdxvec;


fase2d=atan2(imag(SLm.xcmeas(5).xfer),real(SLm


m2d=m2d(IW);

fase2d=fase2d(IW);



f3d=SLm.fdxvec;


64

fase3d=atan2(imag(SLm.xcmeas(5).xfer),real(SLm.x


m3d=m3d(IW);

fase3d=fase3d(IW);



f4d=SLm.fdxvec;




m4d=m4d(IW);

fase4d=fase4d(IW);



f5d=SLm.fdxvec;




m5d=m5d(IW);

fase5d=fase5d(IW);



f6d=SLm.fdxvec;




m6d=m6d(IW);

fase6d=fase6d(IW);



f7d=SLm.fdxvec;




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);






% 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);























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


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 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



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



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



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

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[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

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[9] P.A.R. de Schrijver, Experimental modal analysis of the tyre measurement tower,

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[10] P.W.A. Zegelaar, Modal analysis of tire in-plane vibration, SAE Paper 971101,

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[12] LDS Dactron, Basics of structural vibration testing and analysis, Application Note

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Lecture notes nr.4748

Experimental Modal Analysis of an Automobile Tire - misure

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