ABSTRACT
Identification of modal parameters from response data only is studied for structural systems under nonstationary ambient vibration. By assuming the ambient excitation to be nonstationary white noise in the form of a product model, the modal parameters of a system could be identified through the correlation method in conjunction with a technique of curve-fitting. However, the error involved in the approximate free-decay response would generally lead to a distortion in the modal parameters of identification. It is shown that, under appropriate conditions, the ambient response corresponding to nonstationary input of various types can be approximately expressed as a sum of exponential functions, so that we can use the Ibrahim time-domain method in conjunction with a channel-expansion technique to directly identify the major modes of a structural system without any additional treatment of converting the original data into the form of free vibration. To further distinguish the structural modes from non-structural modes, the concept of mode-shape coherence and confidence factor is employed. Numerical simulations, including one example of using practical excitation data, confirm the validity and robustness of the proposed method for identification of modal parameters from general nonstationary ambient response.
Keywords: Ibrahim time-domain method, Nonstationary ambient vibration, Channel-expansion technique, Mode-shape coherence and confidence factor
Introduction
Modal-parameter identification from ambient vibration data has gained considerable attention in recent years [1,2]. A variety of methods have been developed for extracting modal parameters from structures undergoing ambient vibration. Ibrahim [3] applied the random-decrement technique coupled with a time-domain parameter identification method [4] to process ambient vibration data. James et al. [5] developed the so-called natural excitation technique using the correlation technique coupled with time-domain parameter extraction. It was shown that the cross correlation between two response signals of a linear system with classical normal modes and subjected to stationary white-noise inputs is of the same form as free-vibration decay or impulse response. Chiang and Cheng [6] proposed a correlation technique for modal-parameter identification of a linear, complex-mode system subjected to stationary ambient excitation.
In the previous studies of modal-parameter identification from ambient vibration data, the assumption usually made is that the input excitation is a broadband stochastic process modeled by stationary white or filtered white noise. In the present paper, it is shown that if the input signals can be modeled as nonstationary white noise, which is a product of white noise and a deterministic time-varying function, the practical problem of insufficient data samples available for evaluating nonstationary correlation can be approximately resolved by first extracting the amplitude-modulating function from the response and then transforming the nonstationary responses into stationary ones. The correlation functions of the stationary response are treated as free vibration response, and so the Ibrahim time-domain method can then be applied to identify modal parameters of the system. However, the error involved in the approximate free-decay response would generally lead to a distortion in the modal parameters of identification. Furthermore, we propose a technique to extend the ITD method for modal-parameter identification from nonstationary ambient vibration data without using input data or any additional treatment of converting the original forced-vibration data into the form of free vibration. It is shown that the ambient response corresponding to nonstationary input of various types can be approximately expressed as a sum of exponential functions, so that we can use the ITD method to directly identify the major modes of a structural system. For the purpose of improving the accuracy of modal-parameter identification, a channel-expansion technique is also employed. Numerical simulations, including one example of using practical excitation data, will be performed to confirm the validity and robustness of the proposed method for identification of modal parameters from general nonstationary ambient vibration data.
Correlation Technique
James et al. [5] developed the so-called Natural Excitation Technique (NExT) using the correlation technique. It was shown that the cross-correlation between two response signals of a linear system with classical normal modes and subjected to white-noise inputs is of the same form as free vibration decay or impulse response. In combination with a time-domain parameter extraction scheme, such as the ITD method, this concept becomes a powerful tool for the identification analysis of structures under stationary ambient vibration. When a system is excited by stationary white noise, the cross-correlation function R.. (r) between two stationary response signals xi (t) and x. (t) can be shown to be [5]:
where (f)ir denotes the i-th component of the r-th mode shape, A.r a constant, and mr the r-th modal mass. The result above shows that R.. (r) in Eqn. (1) is a sum of complex exponential functions (modal responses), which is of the same mathematical form as the free vibration decay or the impulse response of the original system. Thus, the cross-correlation functions evaluated of responses data can be used as free vibration decay or as impulse response in time-domain modal extraction schemes so that measurement of white-noise inputs can be avoided. It is remarkable that the term (/>irA]r in Eqn. (1) will be identified as the mode-shape components. In order to eliminate the A]r term and retain the true mode-shape components <f>ir, all the measured channels are correlated against a common reference channel, say xj. The identified components then all possess the common AJr component, which can be normalized out to obtain the desired mode shape <f>ir.
Practical Treatment of Nonstationary Data
The practical problem for evaluating non-stationary correlation is that usually very limited data samples are available in engineering practice. The problem can be resolved by first extracting the modulating function from the response if the excitation can be modeled approximately as the product model. If the excitation can be modeled as non-stationary white noise as represented by the product model with a slowly-time-varying amplitude-modulating function, according to our theoretical derivation, then the responses of the system can also be modeled approximately as a product model with the same amplitude-modulating function as that associated with the excitation itself. With the modulating function extracted, the original non-stationary responses can be transformed into stationary ones. To extract the modulating function out of the non-stationary response histories, we employ the technique of curve fitting. The time-varying standard deviation of the response histories is firstly determined using interval average and then applied curve fitting. This gives us the deterministic envelope function which describes the slow variation in the amplitude of the ambient excitations. We can thus acquire the approximate stationary responses by dividing the non-stationary responses of each DOF with the associated modulating functions obtained by curve fitting. Then the correlation functions of the stationary response data can be obtained, which are in tern treated as the free decay responses corresponding to each DOF. The modal parameters of the system can then be obtained via a time-domain mode identification method, such as the Ibrahim time-domain method (ITD).
Extension of Ibrahim Time-Domain Method for Modal-parameter Identification from Ambient Vibration Data
As mentioned above, by assuming the ambient excitation to be nonstationary white noise in the form of a product model, the modal parameters of a system could be identified through the correlation method in conjunction with a technique of curve-fitting. However, the error involved in the approximate free-decay response would generally lead to a distortion in the modal parameters of identification. In the following, we are going to extend the ITD method to identify modal parameters of a structure solely and directly from forced response data, i.e., without using input data or any additional treatment of converting responses into the form of free vibration. It is remarkable that the essence of the ITD method is to utilize the free-decay responses which are expressible as linear combinations of exponential functions, so that extraction of modal parameters can be performed using different time-delayed sampled responses. In light of this concept, we come up with a new idea that if the ambient vibration data can be approximated directly as a linear combination of exponential functions, then the ITD method may be used for modal-parameter identification directly from ambient response data. In the following, we will show that, under appropriate conditions, nonstationary ambient responses can be approximately expressed as a linear combination of exponential functions.
Modeling of Ambient Excitation
In this paper, we consider that the ambient excitation can be modeled as a nonstationary process, which is represented by a proper composition of stationary process and a deterministic time-varying function. We start by considering that a stationary process W(t) can be approximately expressed as [8]
is the power spectral density function of W(t) (if where
W(t) is white noise, SWW is a constant), and is the power spectral density function of W(t) (if
are a set of independent random variables (phases) uniformly distributed over
Note that, for each
has a similar form to vibration behavior of a mode, so herein we call it a "mode" of excitation, and the corresponding can thus be viewed as a component of "mode shape" of excitation (corresponding to a certain degree of freedom). Eq.(2) is originally used to simulate stationary Gaussian processes and to generate sample functions with prescribed power spectra. It is noted that a stationary sample function W(t), as described in Eq.(2), can be approximately i(ok t expressed as a linear combination of e , which is of a similar form to the undamped, free vibration of a discrete linear system.
To describe the time-varying nature of ambient excitation in practice, we consider the excitation to be a nonstationary process. In general, a nonstationary process can be expressed, on the basis of a stationary process W(t), as follows: 1. Product model :
where pr (t) is a deterministic envelope function (or amplitude-modulating function) used to describe the time-varying amplitude (variance) of a nonstationary process. 2. Additive model :
where a r (t) is a deterministic trend function used to describe the time-varying mean of a nonstationary process.
The time-varying functions p r (t) and a r (t) mentioned above could be further approximated as
By utilizing the above formulation as well as choosing appropriately the values of n , ak , and ak , we can describe a wide variety of nonstationary behavior in reality. Substituting both Eqs.(2) and (5) into Eqs.(3) and (4), respectively, the models can be expressed as follows:
It is apparent from Eqs.(6) and (7) that a nonstationary process in the form of either a where
It is apparent from Eqs.(6) and (7) that a nonstationary process in the form of either a product model or an additive model can also be approximately expressed as a linear combination of exponential functions. In the following, we are going to analyze the responses of a system subjected to nonstationary inputs discussed above.
Analysis of Ambient Response
Consider an -degree-of-freedom (DOF), discrete-time, linear system subjected to ambient excitation resulting from a single source, which is assumed to be a nonstationary process in the form of either Eq.(6) or (7). The displacement response, either
which is a vector of displacement corresponding to different DOF of a system can be obtained by combining the complementary solution and the particular solution as follows:
where m is the number of (real) modes of the system, and
matrix, which is the complex "modal" matrix composed of
including the 2m mode shapes of the structure as well as the
"mode shapes" of the nonstationary excitation in the form of Eq.(6). p A is a
matrix composed of
for various k.
are similar to those in Eq.(8). Note that, in Eqs.(8) and (9), some \ ‘s where the roles of
are similar to those in Eq.(8). Note that, in Eqs.(8) and (9), some \ ‘s are associated with structural modes, while the others may be associated with the excitation. Eqs.(8) and (9) show that both px(t) and a x(t) can be expressed as a linear combination of exponential functions, which signifiies that the forced-response data can be put into the form of free-vibration without any additional treatment, and the major modes of a structural system can then be obtained via a time-domain modal-identification method, such as the ITD method, as described next.
