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During the last years several papers utilizing the Random Decrement transform as a basis for extraction of modal parameters from the response of linear systems subjected to unknown ambient loads have been presented. Although the Random Decrement technique was developed in a decade starting from the introduktion in 1968 the technique seems still to be attractive. This is probably due to the simplicity and the speed of the algorithm and the fact that the theory of the technique has been extended by introducing statistical measures such as correlation functions or spectral densities. The purpose of this paper is to present a state-of-the-art description of the Random Decrement technique where the statistical theory is outlined and examples are given. But also new results such as estimation of frequency response functions and quality assessment are introduced. Special attention is given to the theoretical background for the application of the Random Decrement transform to ambient measurements.

This article describes the work carried out within the project: Modal Analysis Based on the Random Decrement Technique - Application to Civil Engineering Structures. The project is part of the research programme: Dynamics of Structures sponsored by the Danish Technical Research Counsil. The planned contents and the requirement for the project prior to its start are described together with thee results obtained during the 3 year period of the project. The project was mainly carried out as a Ph.D project by the first author from September 1994 to August 1997 in cooperation with associate professor Rune Brincker from the above department. During the project there has been intensive cooperation with professor Sam Ibrahim, Old Dominion University, Virginia, USA. The article is finished with a reference list of the papers and reports prepared during the project.

This paper demonstrates how to use the Random Decrement (RD) technique for identification of linear structures subjected to ambient excitation. The theory behind the technique will be presented and guidelines how to choose the different variables will be given. This is done by introducing a new concept: Quality Assessment. The identification process and quality assessment is illustrated by an analysis of the ambient response of a two-span concrete bridge.

The Random Decrement (RD) technique is a simple and fast method for estimating the correlation functions of Gaussian processes. The RD functions are estimated as an averaging process in the time domain, which makes the technique simple and fast. If the RD technique is applied to stationary zero mean Gaussian distributed processes the RD functions are proportional to the correlation functions of the processes. If a linear structur is loaded by Gaussian white noise the modal parameters can be extracted from the correlation funtions of the response, only. One of the weaknesses of the RD technique is that no consistent approach to estimate the variance of the RD functions is known. Only approximate relations are available, which can only be used under special conditions. The variance of teh RD functions contains valuable information about accuracy of the estimates. Furthermore, the variance can be used as basis for a decision about how many time lags from the RD funtions should be used in the modal parameter extraction procedure. This paper suggests a new method for estimating the variance of the RD functions. The method is consistent in the sense that the accuracy of the approach is not dependent on neither the physical system nor the actual formulation of the RD technique.

This paper presents an ambient vibration study of the Vestvej Bridge. The bridge is a typically Danish two-span concrete bridge which crosses a highway. The purpose of the study is to perform a pre-investigation of the dynamic behavior to obtain information for the design of a demonstration project concerning application of vibration based inspection of bridges. The data analysis process of ambient vribration testing of bridges has traditionally been based on auto and cross spectral densities estimated using an FFT algorithm. In the pre-analysis state the spectral densities are all averaged to obtain the averaged spectral densities(ASD). From the ASD the eigenfrequencies of the structure can be identified. This information can be used in the main analysis, where alle modal parameters are extracted from the spectral densities. Due to long cabling and low response levels (small ambient loads) the response measurements might have a low signal to noise ratio. Thus, it might be difficult clearly to identify physical modes from the spectral densities. The Random Decrement (RD) technique is another method to perform the data analysis process in the time domain only. It is basically a very simple and very easily implemented technique. In this paper it is demonstrated how the RD technique can be used in the pre-analysis state in combination with the FFT algorithm, and how the technique can be used in a full analysis.

This paper deals with applications of the vector triggering Random Decrement technique. This technique is new and developed with the aim of minimizing estimation time and identification errors. The theory behind the technique is discussed in an accompanying paper. The results presented in this paper should be regarded as a further documentation of the technique. The key point in Random Decrement estimation is the formulation of a triggering condition. If the triggering condition is fulfilled a time segment from each measurement is picked out and averaged with previous time segments. The final result is a Random Decrement function from each measurement. In traditional Random Decrement estimation the triggering condition is a scalar condition, which should only be fulfilled in a single measurement. In vector triggering Random Decrement the triggering condition is a vector condition. The advantage of this new approach should be a reduction in estimation time without a significant loss of accuracy, since the vector triggering conditions ensure cross information between the measurements in the Random Decrement functions. The different problems with this technique is highlighted in two examples. A simulation study of a 4 degree of freedom system and the identification of a laboratory bridge model, bothloaded by white noise, is made.

The thesis describes and develops the theoretical foundations of the Random Decrement technique, while giving several examples of modal analysis of large building constructions (bridges). The connection between modal parameters and Random Decrement functions is described theoretically. The efficiency of the Random Decrement technique for the estimation of correlation functions is compared to other equivalent methods (FFT, Direct method). It is shown that the Random Decrement technique can be as much as a hundred times faster than other methods. The theory behind the Random Decrement technique is expanded to include both a vector formulation that increases speed considerably, and a new method for the prediction of the variance of the estimated Random Decrement functions. The thesis closes with a number of examples of modal analysis of bridges exposed to natural (ambient) load.

The problem of estimating frequency response functions and extracting modal parameters is the topic of this paper. A new method based on the Random Decrement technique combined with Fourier transformation and the traditionalpure Fourier transformation based approach is compared with regard to speed and quality. The basis of the new method is the Fourier transformation of the Random Decrement functions which can be used to estimate the frequency response functions. The investigations are based on load and response measurements of a laboratory model of a 3 span bridge. By applying both methods to these measurements the estimation time of the frequency response functions can be compared. The modal parameters estimated by the methods are compared. It is expected that the Random Decrement technique is faster than the traditional method based on pure Fourier Transformations. This is due to the fact that the Random Decrement technique is based on a simple controlled averaging of time segments of the load and response processes. Furthermore, the Random Decrement technique is expected to produce reliable results. The Random Decrement technique will reduce leakage, since the Fourier transformation will be applied to the Random Decrement functions, which has a natura1 decay.

This paper demonstrates how to use the Random Decrement (RD) technique for identification of linear structures subjected to ambient excitation. The theory behind the technique will be presented and guidelines how to choose the different variables will be given. This is done by introducing a new concept: Quality Assessment. The identification process and quality assessment is illustrated by an analysis of the ambient response of a two-span concrete bridge.

The present collection of MATLAB exercises has been published as a supplement to the textbook, Svingningsteori, Bind 1 and the collection of exercises in Vibration theory, Vol. 1A, Solved Problems. Throughout the exercise references are made to these books.The purpose of the MATLAB exercises is to give a better understanding of the physical problems in linear vibration theory and to surpress the mathematical analysis used to solve the problems. For this purpose the MATLAB environment is excellent. The present collection of MATLAB exercises has been published as a supplement to the textbook, Svingningsteori, Bind 1 and the collection of exercises in Vibration theory, Vol. 1A, Solved Problems. Throughout the exercise references are made to these books.The purpose of the MATLAB exercises is to give a better understanding of the physical problems in linear vibration theory and to surpress the mathematical analysis used to solve the problems. For this purpose the MATLAB environment is excellent.

Modal parameter identification from ambient responses due to a general unknown random inputs is investigated. Existing identification techniques which are based on assumptions of white noise and or stationary random inputs are utilized even though the inputs conditions are not satisfied. This is accomplished via adding. In cascade. A force cascade conversion to the structures system under consideration. The input to the force conversion system is white noise and the output of which is the actual force(s) applied to the structure. The white noise input(s) and the structures responses are then used to identify the compined system. Identification results are then sorted as either structural parameters or input force(s) characteristics.

This paper deals with applications of the vector triggering Random Decrement technique. This technique is new and developed with the aim of minimizing estimation time and identification errors. The theory behind the technique is discussed in an accompanying paper. The results presented in this paper should be regarded as a further documentation of the technique. The key point in Random Decrement estimation is the formulation of a triggering condition. If the triggering condition is fulfilled a time segment from each measurement is picked out and averaged with previous time segments. The final result is a Random Decrement function from each measurement. In traditional Random Decrement estimation the triggering condition is a scalar condition, which should only be fulfilled in a single measurement. In vector triggering Random Decrement the triggering condition is a vector condition. The advantage of this new approach should be a reduction in estimation time without a significant loss of accuracy, since the vector triggering conditions ensure cross information between the measurements in the Random Decrement functions. The different problems with this technique is highlighted in two examples. A simulation study of a 4 degree of freedom system and the identification of a laboratory bridge model, bothloaded by white noise, is made.

The topic of this paper is the estimation of modal parameters from ambient data by applying the Random Decrement technique. The data fro the Queensborough Bridge over the Fraser River in Vancouver, Canada have been applied. The loads producing the dynamic response are ambient, e. g. wind, traffic and small ground motion. The random Decrement technique is used to estimate the correlation function or the free decays from the ambient data. From these functions, the modal parameters are extracted using the Ibrahim Time domain method. The possible influence of the traffic mass load on the bridge is investigated by assuming that the response level of the bridge is dependent on the mass of the vehicle load. The eigenfrequencies of the bridge is estimated as a function of the response level. This indicates the degree of influence of the mass load on the estimated eigenfrequencies. The results of the analysis using the Random decrement technique are compared with results from an analysis based on fast Fourier transformations.

The problem of estimating frequency response functions and extracting modal parameters is the topic of this paper. A new method based on the Random Decrement technique combined with Fourier transformation and the traditionalpure Fourier transformation based approach is compared with regard to speed and quality. The basis of the new method is the Fourier transformation of the Random Decrement functions which can be used to estimate the frequency response functions. The investigations are based on load and response measurements of a laboratory model of a 3 span bridge. By applying both methods to these measurements the estimation time of the frequency response functions can be compared. The modal parameters estimated by the methods are compared. It is expected that the Random Decrement technique is faster than the traditional method based on pure Fourier Transformations. This is due to the fact that the Random Decrement technique is based on a simple controlled averaging of time segments of the load and response processes. Furthermore, the Random Decrement technique is expected to produce reliable results. The Random Decrement technique will reduce leakage, since the Fourier transformation will be applied to the Random Decrement functions, which has a natura1 decay.

This paper illustrates how correlation functions can be estimated by the random decrement technique. Several different formulations of the random decrement technique, estimating the correlation functions are considered. The speed and accuracy of the different formulations of the random decrement technique is compared with the direct approach and an approach based on Fourier and inverse Fourier transformations. The investigations are based on the simulated response of a white noise loaded 2 degree of freedom system. Special attention is given to the significance of the length of the time series and the length of the correlation functions. The accuracy of the estimates with respect to the theoretical correlation functions and the modal parameters are both investigated. The modal parameters are extracted from the correlation functions using the polyreference time domain technique.

The topic of this paper is the estimation of modal parameters from ambient data by applying the Random Decrement technique. The data from the Queensborough Bridge over the Fraser River in Vancouver, Canada have been applied. The loads producing the dynamic response are ambient, e.g. wind, traffic and small ground motion. The Random Decrement technique is used to estimate the correlation function or the free decays from the ambient data. From these functions, the modal parameters are extracted using the Ibrahim Time Domain method. The possible influence of the traffic mass load on the bridge is investigated by assuming that the response level of the bridge is dependent on the mass of the vehicle load. The eigenfrequencies of the bridge are estimated as a function of the response level. This indicates the degree of influence of the mass load on the estimated eigenfrequencies. The results of the analysis using the Random Decrement technique are compared with results from an analysis based on fast Fourier transformations.