Operational Modal Analysis

Alexander Narr, summer semester 2017

Operational Modal Analysis (OMA), also known as ambient modal analysis, ambient vibration testing or output-only modal analysis. It uses the dynamic response of a structure under its operating conditions to determine its modal parameters (e.g. mode shapes, eigenfrequencies and damping ratios). In contrast to Experimental Modal Analysis (EMA) where a known force is applied as an input (e.g. by an impulse hammer, shaker etc.), OMA is applied without measuring, nor explicitly applying such inputs. Instead, OMA assumes that the sources are broadband random (ambient forcing with approximately white noise characteristics). The obtained data can then be used to fully describe the linearized dynamic behaviour of the structure, which is necessary for validation, non-destructive-evaluation and structural health monitoring.

Motivation

Since EMA is often carried out in a lab environment and has the need to create or avoid certain loading scenarios, some difficulties may arise. These difficulties can be caused by the size of a structure (e.g. civil engineering, bridges etc.), or the structures loading and boundary conditions are complex and uncontrollable (e.g. wind or wave loading on buildings, aerodynamic forces acting on an airplane or the start-up of an engine). These drawbacks are overcome in operational modal analysis by replacing the need for a deterministic input by stochastic assumptions about the characteristics of the acting forces.

Tab. 1: Pros and Cons of operational modal analysis ^{[1] [2]}

Fundamental concepts

While there exist several techniques to apply OMA onto a set of data, they share three main assumptions towards the system under consideration.

• Linearity (e.g. the superposition principle is applicable)
• Stationarity (the dynamic characteristics of the structure do not change over time)
• Observability (a suitable measurement chain is being used)

An OMA system is generally characterised as in the schematic shown in figure 1. In the general case, the structure is assumed to be excited by unknown forces $//<![CDATA[ \begin{array}{l}F(\omega)\end{array} //]]>$. These unknown forces are described by the Frequency Response Functions (FRF) of the excitation system and the Fourier transformed of the white noise input $//<![CDATA[ \begin{array}{l}N(\omega)\end{array} //]]>$.

$//<![CDATA[ \begin{array}{l}F(\omega)=H_f(\omega)*N(\omega)\end{array} //]]>$ (eq. 1)

The FRF of the combined system in the frequency domain is then obtained by the product of the structural FRF $//<![CDATA[ \begin{array}{l}H_s(\omega)\end{array} //]]>$ and the FRF of the excitation system.

$//<![CDATA[ \begin{array}{l}H_c(\omega)=H_f(\omega)*H_s(\omega)\end{array} //]]>$ (eq. 2)

The measured response of the output-only system contains information about both the structural and the loading systems characteristics. The separation between modal properties and properties of the excitation system in the output signal is possible, because the structural response is narrowband and time invariant (it is said to be a Linear-Time-Invariant System (LTI)), while the excitation systems response is broadband and it may either have time varying or time invariant properties.^[2]

Data processing and modal property estimation techniques

Extraction methods can be classified either as frequency or time domain techniques, parametric or non-parametric methods. All of them are derived in a stochastic framework and use correlation functions as well as probabilities, covariances and probability density functions.

Frequency domain techniques

Peak-Picking method

The Basic Frequency domain Method (BFM) is based on the computation of auto- and cross spectra and the fact, that the systems response is dominated by only one mode in the vicinity of its resonance (see also Modal superposition). Extracting the frequencies of the resonant peaks from the power spectral density functions (PSD) of each modal coordinate gives the systems operating deflection shapes (ODS). In the absence of heavily damped and closely spaced modes, ODS are an acceptable approximation to the systems mode shapes. The obtained results are checked by the coherence functions between two simultaneously measured PSD functions, where their value at the given eigenfrequency is supposed to be close to unity for all physical modes. ^[2]

Frequency Domain Decomposition methods (FDD)

Methods of the FDD-type use a decomposition such as the single value decomposition of the spectral density matrix, where singular values contain frequency and damping information whereas singular vectors contain information about the corresponding mode shapes. They access possibilities to check how many modes are present in a certain frequency band. This is done by plotting the singular values of the spectral density matrix over the frequency band of interest. Since the SVD provides the singular values arranged in descending order, the first singular value contains information about the dominant mode at that frequency. The first singular value that becomes flat compared to the modal response is assumed to indicate the noise level whereas the first singular values that clearly rise above that level are most likely to represent the modal response.^[1]

Least square frequency domain (LSFD) and least square complex frequency (LSCF) methods

LSFD and LSCF methods are both parametric methods using curve fitting techniques to minimize the error between predicted and measured PSD matrices. The LSFD method calculates an error estimate by subtracting a ratio of two polynomials from the measured spectrum for each frequency for given polynomial coefficients. The use of a Frobenius norm on the error leads to a cost function which is to be minimized w.r.t. the polynomial coefficients. Besides the choice of initial values, the method needs iterative parameter estimating algorithms, because the formulation of the cost function is nonlinear. LSFD methods have rather historical relevance and have been widely replaced by LSCF and Poly-reference LSCF estimators (e.g. PolyMAX) leading to cleared stabilization diagrams and providing opportunities for automation. LSCF methods use a modified cost function formulation to keep it linear w.r.t. the optimization parameter, leading to a significant reduction in computation time. Once the physical poles are determined, the corresponding physical mode shapes can be calculated. ^[2]

Time domain techniques

NexT- type procedures

The Natural Excitation Technique (NExT) consist of four steps: Collection of response data, calculation of auto- and cross-correlation functions from the time histories, using a time-domain modal identification scheme to estimate modal parameters by treating the correlation functions as free vibration responses and estimating the mode shape vectors. The modal parameter estimation of the correlation functions can be done by standard algorithms (i.e. ERA, Polyreference etc.) since the correlation functions are assumed as a sum of decaying sinusoids in the same form as the impulse response function. ^[2]

Stochastic Subspace Identification methods (SSI)

SSI-methods are based on the stochastic realization problem and can be separated into Covariance-Driven (SSI-COV) and Data-Driven (SSI-Data) techniques. The SSI-COV methods first step towards identifying a stochastic state-space model is to check whether all system states can be excited and controlled, dependent on an initial guess on the number of physical modes. This is done by computation of the output correlations of the data matrix where the first and last data samples are successively removed. These correlations are assembled into the so-called Toeplitz block matrix on which a single value decomposition is performed revealing the order of the system. The state matrix A and output influence matrix C can then be determined, containing the systems modal parameters.

The SSI-Data method differs from the SSI-COV method mainly by replacing the computation of the covariances in the first step by projecting the row space of the future outputs into the space of the past outputs. ^{[1] [2]}

Stabilization diagrams

The presence of noise in all experimental obtained data leads to difficulties in the determination of the correct model order. A common approach to overcome this difficulty is to overpredict the order of the model, to ensure the identification of all physical modes. This overprediction introduces spurious poles (e.g. due to the excitation system or the mathematical model) which must be separated from the physical poles. Therefore, a stabilization diagram is used to compare the results of models with different order. All poles found by the used method are classified into stable and unstable poles depending on chosen tolerances. These tolerances are checked for frequencies, damping ratios and mode shape vectors between the results of the model of order n and n-1.^[2] Figure 2 shows a typical stabilization diagram of experimentally obtained OMA data processed with a SSI-COV method. The alignment of stable poles (stable frequency, damping and mode shape vector) with increasing model order is clearly visible.^[3] The separation of physical poles from spurious mathematical ones can be accomplished by selecting only stable poles that align with increasing model order. Typically only poles that fulfil all three tolerance criterions are selected.^[2]

Literature

• Brincker, Rune, Ventura, Carlos E.: Introduction to Operational Modal Analysis. John Wiley & Sons, Inc., Chichester, 2015, p. 1-13, p. 239-279.
• Brincker, Rune: Some Elements of Operational Modal Analysis. In: Shock and Vibration 2014 (4), S. 1-11. DOI: 10.1155/2014/325839. 2014.
• Rainieri, Carlo, Fabbrocino, Giovanni.: Operational Modal Analysis of Civil Engineering Structures. Springer Science+Business Media, New York, 2014, p. 103-195.

References

1. Brincker, Rune, Ventura, Carlos E.: Introduction to Operational Modal Analysis. John Wiley & Sons, Inc., Chichester, 2015.
2. Rainieri, Carlo, Fabbrocino, Giovanni.: Operational Modal Analysis of Civil Engineering Structures. Springer Science+Business Media, New York, 2014.
3. Botz, M., Zhang, Y., Raith, M., Pnkert, K..: Operational Modal Analysis of a Wind Turbine during Installation of Rotor and Generator. International Operational Modal Analysis Conference, Ingolstadt, 2017.