Reference-based combined deterministic–stochastic subspace identification for experimental and operational modal analysis

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Abstract

The modal analysis of mechanical or civil engineering structures consists of three steps: data collection, system identification and modal parameter estimation. The system identification step plays a crucial role in the quality of the modal parameters, that are derived from the identified system model, as well as in the number of modal parameters that can be determined. This explains the increasing interest in sophisticated system identification methods for both experimental and operational modal analysis. In purely operational or output-only modal analysis, absolute scaling of the obtained mode shapes is not possible and the frequency content of the ambient forces could be narrow banded so that only a limited number of modes are obtained. This drives the demand for system identification methods that take both artificial and ambient excitation into account so that the amplitude of the artificial excitation can be small compared to that of the ambient excitation. An accurate, robust and efficient system identification method that meets this requirements is combined deterministic–stochastic subspace identification. It can be used both for experimental modal analysis and for operational modal analysis with deterministic inputs. In this paper, the method is generalized to a reference-based version which is faster and, if the chosen reference outputs have the highest SNR values, more accurate than the classical algorithm. The algorithm is validated with experimental data from the Z24 bridge that overpassing the A1 highway between Bern and Zurich in Switzerland, that have been proposed as a benchmark for the assessment of system identification methods for the modal analysis of large structures. With the presented algorithm, the most complete set of modes reported so far is obtained.

Introduction

During the last six decades a lot of research effort has been spent in the development of reliable system identification algorithms for the determination of the modal parameters of a mechanical structure. This resulted at first in modal analysis techniques later named experimental modal analysis (EMA). The structure is excited by one or several measured forces, the response of the structure is recorded and a system model for the structure is identified from the input–output data. Finally, the modal parameters of the structure are extracted from the identified system model. The first of these EMA methods were single degree of freedom (SDOF) techniques like the peak picking (PP) method [1] or the circle fitting method [2], which have the main disadvantage of not being useful in the case of closely spaced modes. This drawback was later removed with the introduction of multiple degree of freedom (MDOF) methods for EMA. An overview of EMA methods is given in [3], [4], [5].

In general, these experimental methods are not suited for large civil engineering structures because the contribution of artificial excitation forces to the total response of the structure is rather low. A bridge for instance can only be excited to a limited vibration amplitude level by an artificial excitation source such as a shaker or an impact hammer, while it is almost impossible to exclude ambient excitation (also called operational loading) due to wind or traffic. Moreover, the lowest natural frequencies of large structures are usually outside the frequency band of maximum artificial excitation. Due to these restrictions, special output-only or stochastic system identification algorithms have been developed, in which the unmeasured (ambient) forces are modeled as stochastic quantities with unknown parameters but with known behavior (for instance, white noise time series with zero mean and unknown covariances). These algorithms require only the outputs (accelerations, velocities, displacements, strains, …) to be measured for the construction of a system model for the mechanical structure, so artificial excitation is not necessary. The resulting modal analysis is named operational modal analysis (OMA). A wide variety of OMA methods is available at the moment. A comparative overview can be found for instance in [6], [7]. In those comparative studies, the reference-based stochastic subspace identification (SSI/ref) method [22] and the poly-reference weighted least squares complex frequency-domain (p-LSCF) estimator [6] appear as accurate, robust and efficient system identification methods for OMA.

One of the drawbacks of OMA methods is that absolute scaling of the obtained mode shapes is not possible, i.e. the modal scaling factors are undetermined. However, scaling is necessary if one wants to extract stiffness and mass properties. Furthermore, some vibration-based damage identification methods require scaled mode shapes. Another drawback of OMA is that the frequency content of the ambient forces may be narrow banded, which makes that the number of modal parameters that can be determined from an OMA test may be limited. For these reasons, there is an increasing interest in the last few years towards combined deterministic–stochastic system identification methods where both measured and unmeasured forces are accounted for. The resulting modal analysis is called “Operational Modal Analysis with eXogenous (or deterministic) inputs” (OMAX) [6]. It permits the determination of the modal scaling factors, whereas it is still possible to have low excitation levels due to the applied artificial forces. It should be noted that absolute mode scaling is also possible by performing two different OMA tests, adding a known mass to the structure during the second test [8], [9].

As clear from the previous discussion, an experimental, operational or combined modal analysis consists of three distinct steps:

  • 1.

    data collection,

  • 2.

    system identification,

  • 3.

    determination of the modal parameters.

The collection of the data will not be treated in this paper. The reader is referred to [3], [4], [5] for an overview of measurement techniques. System identification can be defined as the construction of a mathematical system model from measured data. Usually a discrimination is made between time domain [10] and frequency-domain [11] system identification and between non-parametric system identification (for instance the determination of numerical FRF data) and parametric system identification (for instance the construction of a state-space model). The modal parameters can be determined from a free vibration analysis of the identified system model.

In this paper, the use of the combined deterministic–stochastic subspace system identification (CSI) algorithm for the modal analysis of mechanical structures is discussed. CSI can be classified as a time-domain parametric method, useful for both EMA and OMAX. The original contributions of this paper are the following. At first, the CSI algorithm of [12] is extended into a reference-based version (CSI/ref) which makes it more suitable to the modal analysis of large structures. Secondly, a robust method to calculate modal scaling factors using an identified state-space model is provided. This method takes the modal decoupling of the direct transmission term into account. Thirdly, a new criterion adopted from model reduction theory [13] is extended to stochastic and combined systems and introduced into the stabilization diagram which is a big step forward in the full automatization of both EMA and OMA(X) based on subspace identification methods.

The paper is organized as follows. In Section 2, it is indicated how vibrating structures can be modeled by means of combined deterministic–stochastic state-space models. Subspace identification of such a model is discussed in Section 3. Section 4 handles the modal analysis of the identified state-space model. The application of the methods to a real structure is presented in Section 5.

Section snippets

The deterministic discrete-time state-space model

The finite element method [14], [15] is one of the most common tools for modeling mechanical structures. In the case of a linear dynamical model, one has the following system of ordinary differential equations: Md2u(t)dt2+C2du(t)dt+Ku(t)=B2f(t)with M, C2 and K the mass, stiffness and damping matrices, respectively, f(t) the vector with nodal forces, u(t) the vector with nodal displacements, B2 a selection matrix and t the time. By simple mathematical manipulation, the finite element model can

Theory

Suppose now that also the system matrices A, B, C and D and the noise covariance matrices Q, R and S are unknown, i.e. one wants to extract a combined deterministic–stochastic state-space model for the structure from the measured data. First, the measured outputs are grouped into the following block Hankel matrix:Y0|2i-1=1jy0refy1refy2refyj-1refy1refy2refy3refyjrefyi-1refyirefyi+1refyi+j-2refyiyi+1yi+2yi+j-1yi+1yi+2yi+3yi+jy2i-1y2iy2i+1y2i+j-2=YprefYf.Similarly, the inputs are

Eigenfrequencies and damping ratios

Suppose a discrete-time state-space model (5), (6) for the structure of interest has been identified. With this model, the modal parameters of the mechanical structure can be obtained via free vibration analysis of the corresponding continuous-time model (1): dx(t)dt=Ac·x(t).The solutions to this set of homogeneous ordinary differential equations arex(t)=eλciRt(cos(λciIt)ψiR-sin(λciIt)ψiI),where λci=λciR+iλciI and ψi=ψiR+iψiI are the eigenvalue–eigenvector pairs of Ac: Acψi=λciψi.From Eq. (7)

Application: modal analysis of the Z24 bridge

As a modal analysis application of the CSI and CSI/ref algorithms, the modal analysis of the three-span Z24 bridge is considered. The Z24 bridge, overpassing the A1 highway between Bern and Zürich in Switzerland, has been subjected to several progressive damage tests in the framework of the Brite-EuRam SIMCES project [24]. Before and after each applied damage scenario, the bridge was subjected to a forced and an ambient operational vibration test. For the forced vibration tests, two vertical

Conclusion

Starting from a finite element description of a vibrating structure, a combined deterministic–stochastic discrete-time state-space model for the structure has been derived using a Kalman filter that makes use of reference outputs only. The presented algorithm for the identification of this system model is the reference-based version of the combined deterministic–stochastic subspace identification algorithm, which is particularly suited for modal analysis of large mechanical structures. Similar

Acknowledgments

The authors acknowledge the financial support by the Fund for Scientific Research of Flanders (F.W.O.), research project G.0343.04.

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