Experimental modal analysis of the tyre measurement tower

Experimental modal analysis of
the tyre measurement tower
P.A.R. de Schrijver
DCT 2005.112
Traineeship report
Coach(es):
Dr. Ir. A.J.C. Schmeitz
Dr. Ir. I. Lopez Arteaga
Supervisor:
Prof. Dr. Ir. H. Nijmeijer
Technische Universiteit Eindhoven
Department Mechanical Engineering
Dynamics and Control Technology Group
Eindhoven, August, 2005
Samenvatting
In dit verslag is de experimentele modale analyse van de bandenmeettoren behandeld. Er is gebruik
gemaakt van een meethamer, twee versnellingsopnemers en een Siglab-analyzer. De resultaten van
de metingen zijn verwerkt met behulp van het programma ME'scope. De eigenfrequenties van de
toren zijn gevonden bij ongeveer 13.5Hz , 22Hz en 30Hz . Deze waarden zijn geëvalueerd met
behulp van de MAC-matrix (Modal Assurance Criterion).
Tegen de verwachting in, is de eerste gevonden mode een buigmode. De oorzaak hiervan is
waarschijnlijk de vloer. Het is gebleken dat de vloer erg meetrilt met de toren in de dwarsrichting.
De vloer is echter een stuk stijver in de langsrichting, dus in deze richting trilt de vloer minder
mee. De vloer zal stijver gemaakt moeten worden in de dwarsrichting om de mogelijkheden voor
het gebruik van de toren te verbeteren. De eerste buigmode zal dan waarschijnlijk naar een hogere
frequentie verschuiven met als gevolg dat de bruikbare bandbreedte voor verticale excitatie toeneemt. Dit impliceert ook dat de aan te schaen actuator hogere excitatiefrequenties aan moet
kunnen. Het is echter wenselijk om de bruikbare bandbreedte tot minimaal 15Hz uit te breiden
omdat vrijwel alle voertuigen een eigenfrequentie rond de 15Hz hebben (de wheel-hop mode).
i
Abstract
In this report, the experimental modal analysis of the tyre measurement tower is described. In
the experiments a measurement hammer, two accelerometers and a Siglab-analyzer are used. The
results of the measurements are processed using the program ME'scope. The eigenfrequencies
of the tower are found at about 13.5Hz , 22Hz and 30Hz . These values have been evaluated
using the MAC-matrix (Modal Assurance Criterion). Against expectations, the rst mode that
is encountered, is a bending mode. The oor is probably the cause of this. If the tower is hit in
the lateral direction, the oor is also set in motion. In the longitudinal direction the oor is much
stier so in this direction the oor does not participate as much in the measurements. The oor
will have to be altered to be more sti in the lateral direction in order to improve the possibilities
for use of the tower. The rst bending mode will probably shift to a higher frequency with as
result that the available bandwidth for the vertical excitation increases. This also implies that
the actuator that must be purchased must be able to deal with higher excitation frequencies. It
is however desirable to increase the available bandwidth up to about 15Hz , because almost all
vehicles have an eigenfrequency at about 15Hz (the wheel-hop mode).
ii
List of symbols
Symbol
∆t
T
N
fs
fmax
∆f
∆F
x(t)
y(t)
X(f )
Y (f )
Sxx (f )
Sxy (f )
H1
Υ2xy (f )
Denition
sampling time
record length
record length as used by Siglab
sampling frequency
maximum frequency
spectral density
frequency resolution as 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
cross power spectrum
FRF estimator
coherence function
iii
Unit
s
s
lines
Hz
Hz
Hz
Hz
Contents
Samenvatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
1.1
1.2
1.3
The measurement tower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The role of modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Goals and report structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Experiments
2.1
2.2
Method and experimental setup . . .
2.1.1 Measuring equipment . . . .
2.1.2 The measurement Points . . .
The measurements . . . . . . . . . .
2.2.1 Input range . . . . . . . . . .
2.2.2 Frequency range . . . . . . .
2.2.3 Triggering . . . . . . . . . . .
2.2.4 Windowing . . . . . . . . . .
2.2.5 Averaging . . . . . . . . . . .
2.2.6 Frequency response functions
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3 Modal parameter estimation
3.1
3.2
3.3
Number of modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Modal parameter estimation method . . . . . . . . . . . . . . . . . . . . . . . . . .
The estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Evaluation
4.1
4.2
4.3
4.4
Evaluation of the measurements .
The mode shapes . . . . . . . . .
MAC values . . . . . . . . . . . .
Discussion . . . . . . . . . . . . .
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i
ii
iii
1
1
2
2
3
3
4
6
6
7
7
8
8
8
9
10
10
12
12
14
14
15
16
17
5 Conclusions and recommendations
18
A Estimated mode shapes
19
iv
Chapter 1
Introduction
This report is the result of an internship performed at Eindhoven University of Technology. In
this report, the experimental modal analysis of a tyre measurement tower is described. In this
introduction rst the measurement tower itself is discussed. Then the importance of the modal
analysis will be explained and nally the goals and the structure of this report are discussed.
1.1 The measurement tower
Eindhoven University of Technology has recently taken over the measuring tower from Delft University of Technology. The measurement tower can be found in the Automotive Engineering
Science (AES) laboratory and will be used to measure tyre forces and moments at dierent slip
angles, axle heights and camber angles. The measurement tower is depicted in gure 1.1.
Figure 1.1: The measurement tower in the AES laboratory
As can be seen in gure 1.1, a wheel can be mounted on the axle in the center of the tower. The
wheel can be driven by the drum that is placed under the oor. A schematic overview of the
situation can be seen in gure 1.2. The tower can be rotated around the vertical axis over the
groundplate so the tyre will slip over the drum. On top of the tower, an actuator can be mounted
that can load the tyre in the vertical direction and that can in addition oscillate up to a certain
frequency. However, currently the actuator is not available in the lab and has to be purchased.
1
The prices of dierent actuators vary a lot because of the dierent frequency ranges they can
handle. An actuator that can actuate up to 30Hz for instance is much more expensive than one
that can actuate up to 15Hz .
hydropuls
top frame
tyre
wheel frame
measuring hub
bearing
drum
drum
Figure 1.2: Schematic overview of the tower
1.2
The role of modal analysis
The fact that an actuator has to be purchased is the reason that a modal analysis is performed on
the measurement tower. In a modal analysis the dynamic behavior of the tower is investigated.
It is important to know the dynamic behavior of the tower in order to make a sensible decision
about the choice of an actuator.
With a modal analysis the modal parameters, modal frequencies, modal damping and the modal
shapes are obtained. So after the modal analysis is performed, it is known at what frequencies
the tower will resonate. The actuator has to be able to actuate at a frequency suciently below
the rst eigenfrequency of the measurement tower so the measurements are not inuenced by the
resonating tower.
1.3
Goals and report structure
The goal of this report is to determine the modal properties of the tyre measurement tower in the
AES lab, so a sensible decision can be made about the choice of an actuator for the measurement
tower.
The report is organized as follows. In chapter 2 the experiments will be discussed. This includes
all the needed equipment, the measurement method and an evaluation of the quality of the measurements. In chapter 3 the modal parameters will be determined from the measured data. In
chapter 4 an evaluation of the estimated modal parameters will take place. Finally, in chapter
5 some conclusions will be drawn and recommendations will be given for the actuator and for
further research.
2
Chapter 2
Experiments
In this chapter the experiments performed to obtain the Frequency Response Functions (FRF's) of
the measurement tower are discussed. First an explanation of the method, setup and the equipment
used for the experiments will be given, then the measurements will be discussed followed by an
evaluation of the measurements.
2.1 Method and experimental setup
For modal testing, it is necessary to measure the Frequency Response Functions. In other words:
How does the tower respond to a certain input?. The input can be realised by means of a shaker
or an impact hammer. The tower is very heavy so a shaker test is very dicult. The rst problem
is that the shaker would have to be very powerful and the second problem is the mounting of the
shaker. For these reasons it is chosen to use an impact hammer for this experiment. The response
of the tower is measured with an accelerometer. Finally, the FRF's are computed with a Siglab
analyzer.
As stated in the introduction the goal is to nd the lowest eigenfrequencies. These lowest eigenfrequencies belong to the rst modes. It is expected that the rst three modes will be the torsion,
the bending in the lateral direction (x-direction) and the bending in the longitudinal direction
(y-direction). In gure 2.1 the directions used in this project are dened.
Figure 2.1: The dened directions on the measurement tower
3
In order to nd both the lateral bending and the longitudinal bending, the tower has to be hit
in the lateral direction and the longitudinal direction respectively. The torsion mode can be
found in both the lateral direction and the longitudinal direction. Due to lack of time the choice
has been made to treat the lateral excitation completely and in the longitudinal direction only
a limited number of points are treated to get an indication of the longitudinal bending. In the
experiment one point was chosen as the xed hammering point, see gure 2.1, and the place of
the accelerometer was changed every measurement. The choice of the dierent measuring points
will be discussed later in this section.
2.1.1 Measuring equipment
In this subsection the measurement equipment will be treated. As mentioned earlier, three pieces
of measurement equipment are used: an impact hammer, an accelerometer and a Siglab analyzer.
The impact hammer
For the excitation of the tower an extra large hammer was used to make sure that enough energy
was put into the tower. In gure 2.2 a picture of the used hammer is shown. The impact force
of the hammer is measured with a force sensor from Kistler type 9341A. This is a piezoelectric
force transducer that is built into the hammer head. This transducer generates a charge which is
converted to a voltage by an amplier. This voltage is used in Siglab. The specications of the
force transducer are listed in table 2.1. Since for this experiment the low frequencies are important,
a rubber hammer tip has been chosen.
Figure 2.2: The hammer used for the experiments
Full scale range
Maximum frequency range
Resonance frequency
Sensor sensitivity
Amplier sensitivity
Head mass
Hammer mass
30
40
≈ 40
4.04
5
0.33
10
kN
kHz
kHz
pC/N
mV/N
kg
kg
Table 2.1: specications of the impact hammer, sensor type 9341A
4
The accelerometer
For the experiments two accelerometers were used so that two directions or points could be measured at the same time. The rst accelerometer was a Kistler accelerometer type 8628 B50. This
accelerometer has a built-in charge amplier and a low-impedance voltage output. This charge
amplier has to be fed directly from Siglab. The specications of this accelerometer are given in
Table 2.2. This accelerometer was attached to the tower using the magnet on the accelerometer.
Figure 2.3: The Kistler accelerometer
Range
Resonance frequency
Sensitivity
Mass
50
22
100
6.7
g
kHz
mV/g
g
Table 2.2: specications of the kistler accelerometer, type 8628 B50
The second accelerometer was an PCB accelerometer type 302 B03. This accelerometer also has
a built-in charge amplier and has to be fed directly from Siglab. The specications are given in
table 2.3. This accelerometer was attached to the tower by using mounting wax.
Range
Resonance frequency
Sensitivity
Mass
16
¿ 30
300
38
g
kHz
mV/g
g
Table 2.3: specications of the PCB accelerometer, type 302 B03
The dynamic signal analyzer
As mentioned before, the force and accelerations are measured with a Dynamic Signal Analyzer.
The measurements are performed by a four-channel Siglab analyzer from DSP. The properties of
this analyzer are given in table 2.4.
Siglab samples the voltage signals from the hammer and the accelerometers. The sensitivity
information of the sensors is used to calculate the values for the acceleration and the force. Siglab
also performs the transformations and the calculations so that the measured time domain signals
5
Figure 2.4: The PCB accelerometer
Frequency range
Dynamic Range
Accuracy
Maximum resolution
20
90
±0.03 + 0.02 ·
8192
f
20kHz
kHz
dB
dB
lines
Table 2.4: specications of the Siglab analyzer
can be converted into a frequency response function. More information on the working principles
of Siglab can be found in [6].
2.1.2
The measurement Points
When choosing the measurement points one has to keep in mind that there are enough points to
cover all the desired mode shapes. When too little points are chosen not all the desired mode
shapes may be visible. If too many points are chosen the calculations take an unnecessary long
time. Another very important thing is the location of the measurement points. The location
should be at that point of the structure where one expects a high response of the structure. Since
in this project the goal is to nd the rst three modes, the points that can be seen in gure 2.5,
here a model of the measurement tower (gure 2.6) is depicted, are sucient. There are 16 points
in total. On each vertical pillar of the structure there are three points: on the bottom, the middle
and the top. To see if the oor in the lab has a lot of inuence on the tower there are also
four measurement points on the oor between the screws that attach the tower to the oor. The
hammering point is chosen to be point 4, the upper right corner point on the front side.
2.2
The measurements
In this section the details of the experiments are treated. The quality of the measurements depends,
among other things, on the used sensors, the dynamic signal analyzer and the skill of the person
doing the experiments. But apart from the devices themselves it is also very important that the
settings of Siglab are correct for each measurement. The aspects in Siglab that inuence the
accuracy of the measurement are the input range, the frequency range, triggering, windowing and
averaging. Because it can not be seen how Siglab calculates the frequency response functions, a
short explanation of the measurement of the frequency response functions will be given.
6
Figure 2.5: The model
2.2.1
Figure 2.6: The measurement tower
Input range
In Siglab the range of the input can be specied for every channel. This range can vary from 20mV
to 10.0V . The conversion of analogue amplitude information to digital amplitude information is
based on the input range and on the number of bits of the integrated converter. The analogue
to digital conversion causes so called quantization errors. These errors are minimized if the input
range is correctly set. If the range is chosen too large, large quantization errors occur and thus
the analogue to digital conversion results in an coarse resolution. If the range is chosen to small
overloading will occur; the amplitude of the signal is larger than the maximum allowed value and
is truncated and recorded as the maximum value. So the range should be as small as possible
without the occurrence of an overload.
The type of voltage can also be specied. For the accelerometers it has to be set to bias, because
the sensors receive its power from Siglab itself. For the hammer sensor however it has to be set
to DC. This is because the sensor needs an amplier in order to be compatible with Siglab.
2.2.2
Frequency range
For the frequency range several choices have to be made. In Siglab a choice has to be made
for the bandwidth and for the record length. Based on these choices Siglab will then select the
appropriate sampling parameters. Since for this experiment especially the lowest eigenmodes are
important the bandwidth is set to 50Hz . In order to get accurate measurements the record length
N is set to 1024lines. The resolution can be calculated using equations 2.1 and 2.2, where ∆t
is the sampling time, T is the record length, fs is the sampling frequency, fmax is the maximum
frequency (one is interested in this frequency) and ∆f is the spectral distance, [1]:
fs =
1
≥ 2 · fmax
∆t
∆f =
(2.1)
1
T
(2.2)
However, Siglab's sampling frequency is always given by: fs = 2.56 · fmax . So the frequency
resolution ∆F as used by Siglab will be calculated through equations 2.3 and 2.4, where N is the
record length in lines as used by Siglab:
∆F =
fs
N
∆F = 2.56 ·
(2.3)
fmax
N
(2.4)
7
2.2.3 Triggering
To get good results it is important that the whole excitation signal and the response is captured.
This is done by pretriggering the signal. Pretriggering means that the signals are captured before
the impulse occurs so that the entire signal will be captured. In Siglab three parameters can be
set to describe the triggering namely the level, the slope and the pretrigger delay. The level is a
percentage of the peakvalue of the total impulse. This determines the amplitude of the signal at
which the measurements are started. The slope is used to specify if the measurements should be
started at a growing or decaying signal. These last two parameters tell the measurements to start
when the impulse is already present so the pretrigger delay is used to ensure that the entire signal
is captured. In this project the measurements are triggered at a channel level of 9% at a positive
slope and with a pretrigger delay of −5.0%.
2.2.4
Windowing
In this experiment two time domain windows are used: the force window for the signal from the
force transducer and the exponential window for the accelerometer signal. These windows are
applied to the time signals after they are sampled, but before Siglab calculates the Fast Fourier
Transform.
The force window
The impact force signal, measured through the force transducer in the impact hammer, usually
has an impulse like form and is very short in time compared to the total measuring time. After
the structure is hit by the hammer the force transducer can still give a measurement signal due to
noise, swinging of the hammer, from putting the hammer on the table etcetera. This noise signal
negatively inuences the measurements so this must be removed somehow. This is the reason to
use a force window. The force window is a rectangular window and is used to remove the noise
that is present in the impulse signal. The force window has a value of one in a time band around
the impulse signal and is zero elsewhere. In Siglab three parameters can be specied: the double
hit amplitude, the double hit delay time and the force window size. The width of the window is
specied in Siglab as a percentage of the total measuring time. When the structure is accidentally
hit twice, it is called a double hit. In Siglab the double hit is specied as a percentage of the rst
hit. So impulses after the rst hit with an amplitude larger than this percentage will be rejected.
The double hit delay is a percentage of the measuring time. Impulses that occur after the double
hit delay time are also rejected. In this experiment the force window size is set to 10.0%, the
double hit amplitude is set to 50% and the delay time is set to 20%.
The exponential window
In the computation of the frequency response functions the Fast Fourier Transformation is used.
This transform assumes that the signals are periodic. If the measured signals are not periodic
leakage errors will occur. The function of the exponential window is to force the signal to practically
zero at the end of the sampling window, so the transform can see the signal as periodic with a
period time as long as the measuring time.
In Siglab the exponential window decay can be specied. Here it is set to 1%. This means that at
the end of the sampling window the exponential window is 1% of the beginning value. Although
the exponential window reduces leakage errors it also has a disadvantage. It introduces articial
damping to the modes of the structure. In the post processing phase this articial damping can
be removed from the estimated modal damping.
2.2.5 Averaging
In every measurement there can be random errors. These random errors are caused by measurement noise and by the person doing the measurements. A way to remove these errors is to do more
8
measurements of the same point and average the results in the frequency domain. This improves
the accuracy of the measurements. If all the frequency response functions are of equal importance
the averaging can be linear. For this experiment the frequency response functions are measured
using ve impacts at the same location. In Siglab there are some other options available. These
are the double hit reject and the overload reject. Double hit reject means that frames with more
than one impulse are rejected and overload reject means that frames with impacts that exceeds
the measurement ranges are rejected. Furthermore the person doing the measurements can also
manually reject every frame.
2.2.6
Frequency response functions
The estimation of the frequency response functions is a process that is performed inside of the
frequency response analyzer and is therefore 'invisible' to the user. In this section the basic idea
of the estimation is presented. More information on the estimation can be found in [1].
The input and output signals x(t) and y(t) are measured using a time sampling approach. In order
to transform these signals into the frequency domain the Fast Fourier Transform is applied. The
transformed signals are X(f ) and Y (f ). These frequency domain signals are used to calculate the
auto power spectrum of the input Sxx (f ) and the cross power spectrum of the input and output
Sxy (f ).
Sxx (f ) =
1 ∗
X (f )X(f )
T
(2.5)
Sxy (f ) =
1 ∗
X (f )Y (f )
T
(2.6)
In these equations ∗ represents the complex conjugate and T is the measurement time record
length. The H1 frequency response function estimator can now be determined. This estimator is
used to estimate the frequency response function.
H1 (f ) =
Sxy (f )
Sxx (f )
(2.7)
As mentioned before the measurements are averaged in order to get more accurate results. The
quality of the averaged frequency response function estimates can be evaluated using the coherence
function Υ2xy (f ).
Υ2xy (f ) =
|Sxy (f )|2
Sxx (f )Syy (f )
(2.8)
This coherence function shows which part of the output y(t) is coming from the real input x(t) and
which part is coming from measurement noise. If there exists a linear relation between input x(t)
and output y(t) and there is not much inuence of noise then the value of the coherence function
is near one. However if the value of the coherence function is near zero the output spectrum is
dominated by noise. The coherence function can only account for random errors. Bias errors have
no inuence on the coherence function.
9
Chapter 3
Modal parameter estimation
In this chapter the actual experiments and the modal parameter estimation will be discussed. In
order to estimate these modal parameters a few steps are required. First the number of modes
in the frequency band of interest has to be estimated. Then the method of parameter estimation
has to be chosen. After this step the estimates will be calculated using the computer program
ME'scope.
3.1
Number of modes
First the number of modes will be determined. This can be done in several ways. One way to
do this is by looking at the measured frequency response functions. All the dierent frequency
response functions have to be overlaid. Every resonance peak in these functions is evidence of
at least one mode. However eigenmodes that have a nodal point at the excitation location or at
the measurement location will not appear in the frequency response function. In gure 3.1 and in
gure 3.2 the overlaid measurement results of the lateral and longitudinal direction respectively
can be seen.
Figure 3.1: measured frequency response functions in the lateral direction
10
Figure 3.2: measured frequency response functions in the longitudinal direction
Phase (degrees)
Phase (degrees)
Magnitude (m/s^2/N)
Magnitude (m/s^2/N)
For the measurements in the lateral direction (gure 3.1) three distinct peaks can be identied.
One around 13Hz , one wide peak around 22Hz and one wide peak around 30Hz . After 30Hz the
measurements get a bit unclear so after this point not much can be said.
Since the measurement in the longitudinal direction is done only to nd the longitudinal bending,
only four points are measured. These points are point 1, 4, 7 and 10 in gure 2.5. The measurements in the longitudinal direction are however of very bad quality, see gure 3.2. Below 20Hz
the measurements are very unclear and nothing can be said. Then around 30Hz a wide peak can
be distinguished. The quality of the measurements is so poor that no further conclusions can be
drawn.
Figure 3.3: magnitude and phase plot in lateral direction of point 4
Figure 3.4: magnitude and phase plot in longitudinal direction of point 4
11
Another way of number of modes estimation is to look at a magnitude and phase plot of the
measurements. In gure 3.3 and 3.4 a magnitude and phase plot of point 4 when hit in the lateral
and longitudinal direction can be seen. If a certain frequency is an eigenfrequency a peak should
appear in the magnitude plot and a phase dierence of 180o should be visible in the phase plot.
So in the lateral direction at 13Hz this is clearly visible. However the other two peaks do not
have a phase dierence of 180o ; the dierence is much smaller. However, based on the magnitude
plot in the lateral direction, one could draw the conclusion that there are three resonances, one at
about 13Hz , one at about 22Hz and one at about 30Hz . The gure of the longitudinal direction
is not very clear. Above 20Hz two times a phase dierence of 360o exists. The cause is that at
that point the coherence is very bad. There is one point around 30Hz where a phase-dierence of
180o can be recognized, but there is a little discontinuity in the changing of the phase. However
this could be a measurement error and this could be evidence of only one peak instead of two, but
still not much can be said from this measurement.
A third way of number of modes estimation is to use a mode indicator function. Using ME'scope
three dierent methods can be chosen: the modal peaks function, the complex mode indicator
function and the multivariate mode indicator function. The dierences between these methods
are explained in [5]. For the longitudinal direction, however, the modal parameter estimation will
not help with drawing conclusions regarding the modes, since the quality is too poor. So from this
point on only the lateral direction is treated.
For the lateral measurements the modal peaks function is chosen. This is a very simple method
that is basically the sum of the magnitude of all measured frequency response functions. If the
mode indicator function is applied to the frequency response functions in the lateral direction, 5
modes are identied. The mode around 22Hz is split in two and this is also done for the mode
around 30Hz .
3.2
Modal parameter estimation method
The choice of the modal parameter estimation method is a very important step in the estimation
proces. There are three properties on which the choice of a method can be based. The rst
property is a 'single degree of freedom'-method or a 'multiple degree of freedom'-method, the
second property is a local or a global method and the third property is whether the measurements
are in the frequency domain or in the time domain.
For this project the global polynomial method is chosen. This is a 'multiple degree of freedom'method in the frequency domain. In this case a 'multiple degree of freedom'-method yields better
results than a 'single degree of freedom'-method because the eigenmodes are close together. The
choice to use a global method instead of a local method is made because a global method considers
all the measured frequency response functions at the same time, while the local method considers
each frequency response function at a time. So with the global method there is one estimate of each
resonance frequency and its damping which leads to better estimates than with the local method
if the quality of the measurements is high enough. In this case, the quality of the measurements
does not require a local method.
3.3
The estimates
Now that the number of modes and the method of estimation is chosen, the parameters can be
estimated using ME'scope [5]. The global polynomial method gives the best results in relatively
small frequency bands. However since the frequencies lie very close (from 13Hz to 30Hz ) it is
not necessary to divide the frequency band into smaller bands and the estimates can be made
all at once. Eigenmodes that lie outside the measurement bandwidth may have an eect on the
estimates. To account for this eect additional polynomial terms are used. The estimated modal
parameters are shown in table 3.1.
12
Mode
1
2
3
4
5
Frequency [Hz]
13.5
21.8
23.4
29.2
30.4
Damping [%]
1.45
3.24
1.43
0.999
1.96
Table 3.1: eigenfrequencies and damping ratios in the lateral direction
Now that the estimated modal parameters are known, their corresponding mode shapes can be
displayed. The mode shapes are presented and evaluated in the next chapter.
13
Chapter 4
Evaluation
In this chapter the estimated modal parameters and their corresponding mode shapes are presented, analysed and evaluated.
4.1 Evaluation of the measurements
In the previous chapter it was clear that the measurements when hitting in the longitudinal
direction were very distorted. There were no clear peaks and therefore there is nothing that can
be said with certainty about the modal parameters. So the longitudinal measurements are not
taken into account.
A reason for this strange behavior in the longitudinal direction can be the structure of the oor.
The oor consists of beams in the longitudinal direction with plate material on top (see gure 4.1).
Because of the direction of the beams it could be that the oor is much more willing to be
set in motion when the tower (connected to the beams) is hit in the lateral direction which is
perpendicular to the beams than when it is hit in the longitudinal direction. Fact is that a beam
is much more sti in its axial direction than in its lateral direction. This could be a reason for the
bending mode to shift to a higher frequency and for more distorted frequency response functions
in the longitudinal direction. Perhaps the construction of the oor has to be altered in order to
improve the operation of the tower.
First the mode shapes are analysed, then the MAC (Modal Assurance Criterion) values corresponding to the mode shapes are used to investigate the interaction and similarities between the
modes.
Floor
BEAM
Tower
BEAM
Y
X
Figure 4.1: schematic drawing of the laboratory oor
14
4.2 The mode shapes
The mode shapes corresponding to the estimated modal frequencies from table 3.1 are presented in
gures 4.2 to 4.6. More detailed quad view gures of the mode shapes can be found in appendix A.
Figure 4.2: the rst mode shape at 13.5Hz
Figure 4.3:
21.8Hz
the second mode shape at
Figure 4.4: the third mode shape at 23.4Hz
Figure 4.5:
29.2Hz
the fourth mode shape at
The rst gure contains the mode shape at 13.5Hz . This mode shape is the rst bending mode of
the tower, in the lateral direction. The second mode is found at 21.8Hz . This is the rst torsion
mode of the tower. The third mode is found at 23.4Hz which appears to be a torsion mode again.
The fourth mode at 29.2Hz also seems to be dominated by torsion. And nally the fth mode at
30.4 looks a lot like the fourth mode shape but with less inuence of the oor. If one compares
these identied modes with gure 3.1 one can question if mode 2 and 3 are in fact just one mode
and the same goes for mode 4 and 5. To check whether they are truly dierent the MAC value
will be used to get a better 'view' of the modes.
15
Figure 4.6: the fth mode shape at 30.4Hz
4.3
MAC values
To get an indication of the quality of the tted mode shapes the MAC criterion can be used. MAC
stands for Modal Assurance Criterion and is a technique to determine the degree of correlation
between the dierent mode shapes [1]. The auto MAC value can be calculated with ME'scope
and it compares the estimated mode shapes with each other. The MAC value is always a value
between 0 and 1. If the MAC is 1 than the mode shapes are exactly the same and if it is 0 then
the mode shapes are completely dierent. A graphical representation of the MAC values is shown
in gure 4.7.
5
4
3
1
2
2
3
4
1
5
Figure 4.7: The MAC matrix for the estimated eigenmodes
16
In the gure it can be seen that all the values on the diagonal of the MAC matrix are equal to
one. This is expected because on the diagonal every mode shape is compared with itself.
Figure 4.7 shows that there is a fairly large correlation between mode 2 and 3 and between mode
4 and 5. This means that they are related to each other. The value between 2 and 3 is 0.630 and
the value between 4 and 5 is 0.909. As stated before mode shape 4 and 5 are very similar. In
gure 3.1 it was dicult to see two peaks so since these two have such a high MAC value these
modes are just one mode. The reason that ME'scope recognized them as two dierent modes is
probably a change in the circumstances during the measuring. In gure 3.3 the peaks for mode 2
and 3 also look like just one peak. However the MAC value of 0.630 is not high enough to draw
the conclusion that this also is only one mode although gure 3.3 would suggest that. For this
conclusion some more research should be done.
4.4
Discussion
Looking at gure 3.1 and gure 3.3 the feeling arises that there are three modes present, although
the MAC values contradict this. However, it could very well be that during the measurements the
measurement environment was changed in such a way that the frequency at 22Hz shifted a little.
So in spite of what the MAC value implies, the two modes at about 22Hz will be treated as one
mode.
The rst mode is clearly the lateral bending and the other two, around 22Hz and 30Hz , are
torsion modes. The fact that there are two torsion modes instead of only one is not what was
expected. It is possible that one of the two is the result of a resonance in the oor. Also the wide
peak in the longitudinal direction around 30Hz suggests that this is the real rst torsion mode
and the peak in the lateral direction around 22Hz is caused by the oor.
As mentioned earlier the oor in the AES laboratory plays an important role in the measurements.
Had the oor been innitely sti, then it was expected that the rst mode that would appear, would
be the torsion mode. However the rst mode that appears is the lateral bending around 14Hz . It
is very realistic to think that this is because of the construction of the oor. As mentioned before
the beams are much more willing to be set in motion in the lateral direction. Therefore the point
where the lateral bending appears could be shifted to a lower frequency. This phenomenon does
not occur in the longitudinal direction because this is the axial direction of the beams and they
are much more sti in this direction. This is probably the reason that no longitudinal bending can
be found in this frequency range. The longitudinal bending can probably be found at a frequency
higher than the 50Hz -bandwidth. If the oor was made more sti in the lateral direction probably
the lateral bending would not occur at about 14Hz and the rst mode would be the torsion mode
as was expected.
17
Chapter 5
Conclusions and recommendations
In this report an experimental modal analysis of the tyre measuring tower is performed. As can
be seen in chapter 3 and 4 the three modal frequencies are found at about 13.5Hz , 22Hz and
30Hz . So the actuator would have to be able to actuate up to about 10Hz .
However, as it now seems, the oor has a considerable eect on the measurements. If the oor
is altered to be stier in the lateral direction, the lateral bending mode will occur at a higher
frequency and the available bandwidth for vertical excitation increases. This larger bandwidth is
desired, because in research on the dynamic tyre behavior the bandwidth is desired to be as large as
possible. Since almost all vehicles have an eigenfrequency at about 15Hz (the wheel-hop mode), it
is very interesting to increase the bandwidth so that actuation up to about 15Hz is possible. This
also implies that the actuator will be more expensive because the higher the actuation frequency,
the higher the accelerations and velocities, the higher the price of the actuator.
Although there is now a good idea of where the lowest eigenmodes appear, there still is room for
improvement of the accuracy. For further research it is important that the dynamic behavior of the
oor is known. If this behavior is known perhaps a measurement when hitting in the longitudinal
direction can also be done correctly. Furthermore not every point is measured in every direction
so in order to get more accurate results this is also a point that can be considered.
Finally, in this experiment the mass of the actuator has been compensated for. However if a
real actuator is mounted on the measuring tower, it can also inuence the behavior of the tower,
because the wheel cannot move up and down without any resistance. Furthermore, the mass of the
real actuator can be dierent than the dummy mass mounted on the tower during the experiments.
This can also inuence the measurements.
18
Appendix A
Estimated mode shapes
In this appendix the estimated mode shapes are presented in quad view for more detail. The mode
shapes are estimated using the global polynomial method with four additional polynomial terms.
Figure A.1: The rst mode shape at 13.5Hz
19
Figure A.2: the second mode at 21.8Hz
20
Figure A.3: the third mode at 23.4Hz
21
Figure A.4: the fourth mode at 29.2Hz
22
Figure A.5: the fth mode at 30.4Hz
23
Bibliography
[1] B. de Kraker: A Numerical-Experimental Approach in Structural Dynamics, 2000. Lecture
notes nr. 4748
[2] J.J. Kok and M.J.G. van de Molengraft: Signaalanalyse, 2003. Lecture notes course nr. 4A250
[3] R.J.E. Merry: Experimental Modal Analysis of the H-drive. Report No. DCT 2003.78
[4] M.L.J. Verhees: Experimental Modal Analysis of a Turbine Blade. Report No. DCT 2004.120
[5] Vibrant technology Inc.: ME'ScopeVES operating manual, 2003
[6] DSP technology Inc.: Siglab User Guide, 1998
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