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APPLICATION OF THE RANDOM DECREMENT TECHNIQUE FOR
EXPERIMENTAL DETERMINATION OF DAMPING PARAMETERS
OF BEARING STRUCTURES
M. Kölling
1, B. Resnik
1 and A. Sargsyan
2
1BeuthHS Berlin, University of Applied Sciences, Berlin, Germany
2Yerevan State University of Architecture and Construction, Armenia
ABSTRACT
Scientific cooperation of the BeuthHS Berlin, Germany and the Yerevan State University of
Architecture and Construction has led to several research projects in the field of structural
health monitoring. Within the past few years this cooperation has been extended to the field of
ambient vibration monitoring, which is based on the dynamic characteristics analysis of large
span engineering constructions like bridges. The fundamentals of this approach are based on
the unique dynamic characteristics that are a derivative of the equation of motion and can be
interpreted as a vibration signature. Knowledge and analysis of the current natural frequencies
for example with the Fast Fourier Transform algorithm (FFT) can lead to fast and reliable
conclusions about the condition of the structure. However, experience shows that control of
natural frequencies based on these methods alone cannot provide reliable detection of possible
damage of bearing structures and must necessarily be supplemented with the control of the
corresponding values of the damping coefficient. In order to calculate those parameters, a
software tool using algorithms based on the random decrement technique (RDT) has been
developed. In RDT free decay functions are extracted from ambient vibration measurements
of the structure, which subsequently can be used to determine the damping parameters
accurately, without performing expensive dynamic tests. Numerous measurements and
following evaluations in the last years have confirmed that the method is well suited for the
control and the secure interpretation of dynamic deformations arising due to natural loads
such as wind or traffic. This article presents all necessary steps of data acquisition, processing
and interpretation of damping parameters of bearing structures in order to verify the quality of
construction and building materials based on an extensive experiment with a bridge model.
INTRODUCTION
In the course of the last decades, the realization of vibration measurements has become more
and more affordable and easy to apply, irrespective of the development and adaptation of
specific measurement systems. However, the analysis and particularly the analysis about the
condition of bearing structures are still part of contemporary research (3), (7), (8), (9).
Commonly the dynamic parameters Eigenfrequency, Eigenforms and damping, allocated to
specific points in time and space, are used to analyse the structure's dynamics. To judge about
the structure's condition in the sense of On-Line-Monitoring using these parameters (e.g.
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graded in three categories, such as "good", "defective", "critical"), it can be chosen from two
principal methods.
1 The identified modal parameters are compared with critical values, calculated from
theoretical models of the bearing structure, or
2 Deviations of the "normal" behaviour of the structure dynamics, which has to be defined
by a-priori knowledge about the behaviour in different situations, are observed.
If the measurements yield critical or significant changes in the measured parameters, causes
can be found and counteractions can be taken. Also an automatic warning system can be
established, which detects irregularities at an early stage and sends out warning signals to
prevent further damage.
Unfortunately, the theoretical calculations of critical parameter values via FEM often are
highly inaccurate. Especially in solid constructions the effect of coatings and secondary
elements and the presumption of damping parameters are hard to model with sufficient
accuracy (8), (9).
The second method of data analysis is more promising, but elaborate test measurements must
be performed to achieve accurate results. In principal, it is distinguished between Forced
Vibration Testing (FVT) and Ambient Vibration Testing (AVT). In Figure 1 the difference is
illustrated with time series of the example used in this paper. In the first method the bearing
structures are artificially excited and the input force as well as the resulting vibrations are
being measured. A special case is described by one initial deformation of the bearing structure
at the starting point of the measurement and the examination and analysis of the subsequent
freely decaying vibrations. In the second method so-called "ambient", dynamic natural loads
(such as wind, seismic activity or traffic) are used as the input force and the quasi random
vibration measurements are evaluated. Within the last years the second method got more and
more popular, because no expensive test scenarios must be set up and the operational
condition of the building has to be interrupted.
Figure 1: Example of division of FVT and AVT of test measurement
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ESTIMATION OF DAMPING PARAMETERS AND EIGENFREQUENCIES FROM
FREE DECAY FUNCTIONS
In order to extract information about the frequencies and damping loss factors of a structure
from acceleration sensors using FVT, the vibration responses of those structures must be
measured. Commonly an impulse (e.g. in the case of a bridge, a heavy loaded truck driving
down a ramp and immediately stopped) or interrupted steady-state excitation (e.g. an
emergency stop of a wind power engine) are used to create free decay functions. A typical free
decay of a vibration measurement (containing one Eigenfrequency) can be described by the
formula
𝑦𝑡 = 𝑦0 ∙ 𝑒−𝛿𝑡 sin(𝜔𝑑 𝑡 + 𝜑0) (1)
in which 𝑦𝑡 describes the actual position at time 𝑡, 𝑦0 is the initial displacement with the
phase shift 𝜑0 according to the Eigenfrequency 𝜔𝑑. The logarithmic decrement 𝛿 = 𝜔0𝐷 can
be expressed as the product of eigenfrequency and damping loss factor 𝐷.
Once the free decay functions are measured, the Eigenfrequencies can easily be determined
by applying a Fourier-Transform and detecting the peaks in the corresponding spectrum of the
acceleration time series. The estimation of the damping loss factor is somewhat more
complicated. Often the frequency domain method of the Half-Power-Bandwidth (HPB) or the
relation of the first and third local maximum of the free decay is used for the estimation. In
this work a more accurate technique, namely the fitting of a damped sinusoid function is used.
As a result of this, frequency and damping factors can be determined in one estimation
process. Another method to determine the damping loss factor is to calculate the envelope of
the free decay, which is described by the absolute value of the Hilbert-Transform of the signal
(4). The damping loss factor can be calculated from the slope of a linear fit of the
logarithmized envelope function, making the technique more stable in the case of more than
one dominant Eigenfrequency in the structure. Since the damping is frequency-dependant,
each Eigenfrequency corresponds with a certain damping loss factor and it is recommended to
apply a frequency filter to achieve a factor for the observed Eigenfrequency.
To extract free decay functions from ambient vibration measurements the Random Decrement
Technique (RDT or RD Technique) has proven to provide stable results in the course of the
latest research. The RD Technique makes it possible to extract functions similar to free decay
functions from random vibration measurements, which can be used to determine all desired
parameters and shall be described more detailed in the following section.
THE RANDOM DECREMENT TECHNIQUE
The Random Decrement Technique was developed by Cole (6) in the end of the 1960s to
analyse the dynamic response of space structures exposed to ambient loads. He wanted to
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extract the free decay from a random response measurement of the structure without any
additional and predefined excitation being applied to the system. The vibration response of a
randomly excited system consists of three parts: The response to an initial displacement, the
response to an initial velocity and the response to random input loads (2). The idea of the
RDT is that by extracting a large number of time segments from the signal starting with the
same initial value and averaging over those signals, the random part of the response will tend
to disappear from the system and solely the responses to the initial conditions will remain.
The resulting function, namely RD Function, is similar to the free decay functions of the
structure and contains the same frequencies and damping coefficients and can be determined
in the same, above-mentioned manner. The two essential parameters of the RDT are the initial
value 𝑎 (or trigger condition) and the length of the time segments 𝑙. The principal is
demonstrated in figure 2. Yet, some a-priori knowledge about the examined signal is needed.
For the trigger condition, zero-crossings can as well be chosen as a fixed value. Since in noisy
signals zero-crossings are in the scope of the noise, fixed values above the standard deviation
of the signal have been chosen for this work. Also, enough time segments must be extracted to
guarantee a sufficient averaging. For the length of the time segments it is suggested to
contain, at least, one full decay for a robust estimation of damping.
Figure 2: Principal of the Random Decrement Technique
In the course of this work, a Matlab GUI has been developed, offering all necessary
adjustments for a quick testing of the method. A vibration measurement dataset can be loaded
and a fixed value RDT is performed. Also an estimation of the damping factor by curve fitting
is applied. Trigger condition and length of the time segments can be adjusted, as well as a
passband filter, if desired. The RD Function and its spectrum and the calculated fit functions
are displayed in the GUI and can be saved. Also the original signal and the triggered points
can be plotted. Of course, the estimated damping factor and frequency are displayed as well.
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EXAMPLE MEASUREMENTS
The RD technique was successfully being tested on different kinds of buildings, using
ambient loads caused by, for example, wind or traffic. It has been proven to provide for a
practical and reliable estimation of damping parameters in several scientific papers (1), (2),
(4). In this work, RDT is tested on ambient vibration measurements of a typical bridge model.
On a downscaled model, the realization of a sufficient number of measurement points and a
safe interpretation of the results can be achieved more easy that under undefined conditions.
The investigated bridge model is made from aluminium and around 2.5m long with a
maximum height of 140mm. The setup of the model and the measurements is shown in Figure
3. In the middle of the bridge model a weight was fixed with a thread. To simulate a dynamic
excitation, the weight was cut-off at the beginning of the measurement, resulting in a free
decaying vibration of the model. Three acceleration sensors were placed on the model. The
test procedure was repeated several times. Three measurements are being presented, whereas
in the second measurements some physical connections within the model were being
separated to simulate a structural defect. In the third measurement the defect was fixed to
verify the results of the first measurements.
Figure 3: Setup of the test measurements
The first and fastest step to gain information about modal properties is to perform an FFT
based analysis of the vibration measurements to identify the Eigenfrequencies of the structure
with and without defects. Figure 4 shows the results of the spectral analysis of the three
measurements in cross direction. In the Test 1 and 3 a clear peak at 4.1Hz can be identified,
describing the dominant Eigenfrequency of the model. In Test 2 the simulated defect is
resulting in two dominant frequencies, one being nearly the original Eigenfrequency at around
4.0-4.1Hz, the second at 7.4Hz. Also the peak of the new Eigenfrequency at 7.4Hz is highest
in the data of Sensor 3, being the closest to the simulated defect. This can be taken as an
example of the requirements of an Online-Monitoring-System, though the main task would be
to detect significant changes of the parameters, not the localization of potential defects.
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Figure 4: FFT Analysis of the Measurements
To make possible a separate analysis of FVT and AVT in the course of a damping estimation
under identical test conditions, the vibration time series was separated in two parts: A FVT
part with the length of 1.5s, containing the free decay and an AVT part, containing the
ambient vibrations after the decay had faded out. The principal is shown in Figure 1. Equation
(1) was simply fitted to the FVT measurements with a Matlab method to estimate the
logarithmic decrement 𝛿 (Figure 5). The AVT part was processed with the Matlab methods
performing RDT on ambient signals of 1-2min length. Subsequently the RD Functions were
fitted to equation (1) with the same fitting method (Figure 6). The RDT was applied using
time segments of 1.5s after a trigger value of 𝑎 = 2 ∙ 𝜎 were reached, with 𝜎 being the
standard deviation of the measurement signal.
For all three measurements of the FVT test series similar frequencies as well as damping
parameters are calculated. After applying the simulated damage to the model the damping
values significantly rise between 50-100% of the original value. After the recovery of the
original condition, the damping values conform to the original values. In viscous damping
systems Eigenfrequencies decrease with rising damping factors. This also could be shown by
the slight frequency drop from 4.1Hz to 4.0Hz from the first to second test.
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Figure 5: Analysis of free decays (FVT)
The AVT time series were caused by random excitation within the measurement room. In all
cases a clearly detectable free decay can be seen in the RD functions so that it is a valid
method for damping estimation for Online-Monitoring-Systems that requires further research
to achieve stable results. Though, the damping values still suffer from higher variations that in
the FVT measurements. Since a high number of time segments are needed to achieve a good
averaging, the ambient time series is too short for a stable evaluation. Also higher amplitudes
and, hence, a higher Signal-to-Noise ratio in the measurement signals of field measurements
yield more stable results.
CONCLUSION AND FUTURE PROSPECTS
A robust calculation of damping loss factors is essential for the future prospect of a real-time
Online-Monitoring-System. The RDT process must be optimized in the term of starting values
of fitting function, also measurements on big scale engineering structures (FVT and AVT)
must be made, so that an evaluation of results in the field are possible. Next steps include the
realization of an evaluation algorithm, which extracts information about the condition of a
structure based on the estimated damping values. The Wavelet transform provides the
possibility to detect significant jumps in time series. An automatic warning system could be
developed based on parameters derived from a combined algorithm of RDT estimated
damping values and a detection of a significant step in those values via wavelets.
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Figure 6: Analysis of ambient vibrations (AVT)
The project has been supported by the European Union (European Social Fund)
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