Reference-based combined deterministic–stochastic subspace identification for experimental and operational modal analysis
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.
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: with , and the mass, stiffness and damping matrices, respectively, the vector with nodal forces, the vector with nodal displacements, a selection matrix and t the time. By simple mathematical manipulation, the finite element model can
Theory
Suppose now that also the system matrices , , and and the noise covariance matrices , and 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: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): The solutions to this set of homogeneous ordinary differential equations arewhere and are the eigenvalue–eigenvector pairs of : 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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