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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)

REFERENCES

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Estimation using free decays and ambient vibration tests. Mechanical Systems and

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Application to Civil Engineering Structures. University of Aalborg, 1997

4. Dande, H. A.: Panel Damping Loss Factor Estimation Using the Random Decrement

Technique. University of Kansas, USA, 2010

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