A new OMA method to perform structural dynamic identification: numerical and experimental investigation

The dynamic identification of a system, i.e. the estimation of its natural frequencies, damping ratios and modal shapes, is of crucial importance in many branches of engineering. It is usually performed with experimental modal analysis (EMA) methods or by using operational modal analysis (OMA) methods [1]. The first ones allow to identify the linear or nonlinear behaviour of a structural system but they require the knowledge of both the structural excitation and the structural response and thus the set-up of the in-situ tests is very difficult and expensive [1]. As far as the OMA methods are concerning, they are very attractive since they do not require the knowledge of the structural excitation and thus the set-up of the in-situ tests is very simple and cheap [2]. Moreover, since the structural input is due to the ambient vibrations (usually modelled as a white noise process [3‐5]), OMA methods allow to identify the structural system when it is under operative conditions. This kind of methods have been used to identify structural systems [6‐8], to perform structural health monitoring (SHM) [9‐13], to calibrate finite elements models [14] and to detect structural damages [15, 16]. OMA methods has been applied on historical buildings [7, 17, 18], tall buildings [19, 20], bridges [21‐23], masonry structures [24], dams [25], offshore platforms [26, 27] and other structural systems [28‐31]. Since the structural input is assumed as a white noise, OMA methods have a stochastic framework. For this reason, the dynamic parameters are usually estimated starting from the power spectral density (PSD) of the structural output process, in case of frequency domain methods, or starting from the correlation function of the structural output process in case of time domain methods. The most popular frequency domain methods are the Peak Picking method (PP) [32], usually linked with the Half Power bandwidth method (HP) [32], and the frequency domain decomposition (FDD) [33, 34]. PP and HP have been initially used as EMA methods and they have a deterministic framework since they are usually applied on the frequency response function (FRF) of the system; however, if applied on the PSD, they can be considered as OMA methods. FDD [33, 34] is based on the singular value decomposition (SVD) [4] of the PSDs matrix and it allows to identify the natural frequencies and the modal shapes of a structural system. A most recent version allows to estimate also the damping ratios but the exact computation of the latters is still an open issue [1]. Several OMA methods developed in time domain are present in literature. Among these, it is worth mentioning: Natural Excitation Technique (NExT) [35], Auto Regressive Moving Average (ARMA) [36], Time Domain Decomposition (TDD) [37], and Stochastic Subspace Identification (SSI) [38‐41]. NExT [35] exploits the auto and cross-correlation functions of the response process that can be considered as a summation of decaying sinusoids similarly to the impulse response functions (IRFs). It was initially used for EMA and then extended to OMA and thus NExT has a deterministic framework. ARMA methods [36] are based on the prediction of the current value of a time series taking into account the past values and the prediction error [1]. If there are multiple excitations, ARMA-Vectors (ARMAV) models [42] can be used. In ARMA model there are two parts: the auto-regressive part and the moving average part. The latter part causes nonlinearity in the model and thus ARMA identification is a highly nonlinear process iteratively implemented. For this reason, it is computationally intensive and difficult to apply especially for large dimension structures [1]. TDD [37] is based on a Single Degree of Freedom (SDoF) approach and, in the cases of Multi-Degree of Freedom (MDoF) systems, the filtering of the acquired signals around the natural frequencies is required. It allows to estimate the modal shapes and if linked with PP and HP it is possible to estimate also the natural frequencies and the damping ratios. Recently, Time Domain - Analytical Signal Method (TD - ASM) [8] has been introduced by the authors. It is the time domain version of Analytical Signal Method (ASM) [7] and it allows to identify the natural frequencies, the damping ratios and the modal shapes of a structural system in the cases in which the mass matrix can be expressed as the product between a constant and the identity matrix. In the other cases only the dominant modal shape can be obtained. A most recent version of TD-ASM, called TAGA [43], links TD-ASM with genetic algorithm in order to overcome the limits of TD-ASM. However, TAGA requires a very high computational burden especially in the case of large structures. SSI can be developed in two different ways, i.e. SSI covariance driven (SSI-COV) and SSI data driven (SSI-DATA). SSI-COV [40] decomposes two times the so-called Toeplitz matrix: firstly, it is decomposed into the product of observability matrix and controllability matrix and then a SVD is performed. By solving simultaneously the equations obtained from the aforementioned decompositions, it is possible to estimate the state transition matrix that characterizes the dynamic of the system under study. Finally, by performing an eigenvalue decomposition of the state transition matrix it is possible to estimate the modal parameters of the system. SSI-DATA [41] uses a QR decomposition of the data Hankel matrix and, successively, a SVD of the projection matrix can be performed. Both SSI-COV and SSI-DATA are faster than ARMAV. However, SSI in general is very difficult to be implemented [1]. Since OMA methods have a stochastic framework, they are difficult to be used by people that have not knowledge in signal analysis and stochastic dynamics. For this reason, in this paper an innovative semi-automated OMA procedure is proposed. It allows to identify the natural frequencies, the damping ratios and the modal shapes of a structural system in few steps. The proposed method is based on the filtering of the output process and, since the analytical signal can be used to identify the natural frequencies with high precision [44‐46], it is used in the proposed method to perform the identification. In order to prove the validity of the proposed method, numerical simulations and experimental tests have been performed on a 3-storey frame also considering comparisons with the most popular OMA methods.

In this section, the proposed method is described in detail to estimate the modal parameters of a structural system excited by ambient vibrations and it is very fast and simple to use since only few interactions with the users are requested. In order to describe the proposed method, the MDoF system depicted in Fig. 1 has been considered.

The differential equation governing the motion of a MDoF system excited by a ground acceleration modelled as a zero-mean white noise W(t) isin which \({\varvec{M}}\) is the mass matrix, \({\varvec{C}}\) is the dissipation matrix and \({\varvec{K}}\) is the stiffness matrix. \(\varvec{\ddot{X}}_{({\text {rel}})}(t)\), \(\varvec{\dot{X}}_{({\text {rel}})}(t)\) and \({\varvec{X}}_{({\text {rel}})}(t)\) represent the structural output process expressed, respectively, in terms of acceleration, velocity and displacement relative to the ground while \({\varvec{r}}\) is the forcing location vector that, in case of ground acceleration, contains only unitary values.

Initially, the structural output process has to be acquired. In the proposed method, the latter is expressed in terms of absolute acceleration \(\varvec{\ddot{X}}(t)=\varvec{\ddot{X}}_{({\text {rel}})}(t)+{\varvec{r}}W(t)\). Once that the structural output process has been acquired, its correlation functions matrix, labeled as \({\varvec{R}}_{\varvec{\ddot{X}}}(\tau )\), can be easily calculated considering that the components of \({\varvec{R}}_{\varvec{\ddot{X}}}(\tau )\) are expressed as [4]in which \(i=1,2,\ldots ,n\), \(j=1,2,\ldots ,n\), n is the number of components of \(\varvec{\ddot{X}}(t)\), \(E[\cdot ]\) represents the stochastic mean operator, \(\ddot{X}_i(t)\) and \(\ddot{X}_j(t)\) represent, respectively, the i-th and the j-th component of the structural output process \(\varvec{\ddot{X}}(t)\), while \(\mu _{\ddot{X}_i}\) and \(\mu _{\ddot{X}_j}\) represent the mean of \(\ddot{X}_i(t)\) and the mean of \(\ddot{X}_j(t)\).

The components of the output process’ PSDs matrix can be calculated taking into account the Wiener-Khinchine relationships, i.e. by performing the Fourier transform of the correlation functions matrix’s components and dividing them by \(2\pi \), as reported in the following equation [4]From the peaks of the PSDs matrix it is possible to perform a first estimation of the structural frequencies by extracting the abscissa of each peak. This step represents one of the interactions with the users and it can be easily performed by anyone.

In order to estimate the modal shapes, the contribution of the response process near each estimated frequency has to be extracted. To do this, the response process can be filtered around each peak of the PSDs by using band-pass filters having very little bandwidth. In this way, the i-th component of the process filtered around the j-th frequency, labeled as \(\ddot{X}_i^{(j)}(t)\), can be obtained. and the correlation functions’ matrix \({\varvec{R}}^{filt}(\tau )\) can be calculated considering that its component \(R_{\ddot{X}_i^{(j)}\ddot{X}_1^{(j)}}(\tau )\) are expressed asSince the i-th component of the response process in terms of absolute acceleration is \(\ddot{X}_i(t)=W(t)+\sum _{k=1}^{n} \phi _{ik}\ddot{Q}_k(t)\), then the same component filtered around the j-th frequency can be expressed as \(\ddot{X}_i^{(j)}(t)=W^{(j)}(t)+\sum _{k=1}^{n} \phi _{ik}\ddot{Q}_k^{(j)}(t)\), being \(\ddot{Q}_k(t)\) the k-th modal acceleration. In case of well separated frequencies, only the component \(\phi _{ij}\ddot{Q}_j^{(j)}(t)\) gives a significant contribution to \(\ddot{X}_i^{(j)}(t)\) and thus the latter can be approximated as \(\ddot{X}_i^{(j)}(t) \approx \phi _{ij}\ddot{Q}_j^{(j)}(t)\). Considering this approximation, the j-th modal shape can be estimated from the components of \({\varvec{R}}^{filt}(\tau )\) calculated in \(\tau =0\) asin which \(\sigma _{\ddot{X}_i^{(j)}\ddot{X}_1^{(j)}}\) is the covariance between \(\ddot{X}_i^{(j)}(t)\) and \(\ddot{X}_1^{(j)}(t)\), \(\sigma _{\ddot{X}_1^{(j)}}^2\) is the variance of the process \(\ddot{X}_1^{(j)}(t)\), while \(R_{\ddot{Q}_j^{(j)}\ddot{Q}_j^{(j)}}(0)\) represents the autocorrelation of the process \(\ddot{Q}_j^{(j)}(t)\) calculated in \(\tau =0\), i.e. the variance \(\sigma _{\ddot{Q}_j^{(j)}}^2\).

Once the modal shapes are estimated, the correlation functions’ matrix \({\varvec{R}}_{\varvec{\ddot{X}}}(\tau )\), whose components have been defined in Eq. (2), can be decomposed asbeing \(\varvec{\tilde{\Phi }}\) the matrix containing all the estimated modal shapes. Equation (6) is very similar to the modal decomposition of the correlation functions’ matrix, i.e. \({\varvec{R}}_{\varvec{\ddot{Q}}}(\tau )=\varvec{\Phi }^{-1}{\varvec{R}}_{\varvec{\ddot{X}}}(\tau )\varvec{\Phi }^{-T}\). However, \(\varvec{\Phi }\) is normalised with respect to the mass matrix, while \(\varvec{\tilde{\Phi }}\) is normalised with respect to the identity matrix. \({\varvec{R}}_{\varvec{\ddot{Y}}}(\tau )\) is therefore different from the correlation functions’ matrix in modal space \({\varvec{R}}_{\varvec{\ddot{Q}}}(\tau )\) but the j-th component on its diagonal, labeled as \(R_{\ddot{Y}_j}(\tau )\), is proportional to the correspondent component \(R_{\ddot{Q}_j}(\tau )\). \(R_{\ddot{Y}_j}(\tau )\), as well as \(R_{\ddot{Q}_j}(\tau )\), can be considered as the correlation function of the SDoF oscillator’s response excited by a white noise process. For this reason and in order to distinguish \(R_{\ddot{Q}_j}(\tau )\) from \(R_{\ddot{Y}_j}(\tau )\) the latter is called mono-component correlation function.

Since the correlation function \(R_{\ddot{Y}_j}(\tau )\) is mono-component, then it exhibits a good behavior towards the Hilbert transform. The analytical signal of the j-th component on the principal diagonal of \({\varvec{R}}_{\varvec{\ddot{Y}}}(\tau )\) can be calculated in the form [7, 8]in which \(\widehat{R}_{\ddot{Y}_j}(\tau )\) is the Hilbert transform of \(R_{\ddot{Y}_j}(\tau )\) that is calculated as [7, 8]being \({\mathscr {P}}\) the principal value. \(Z_j(\tau )\) can be expressed in polar form in terms of envelope \(A_j(\tau )\) and phase \(\theta _j(\tau )\) as [7, 8]Envelope and phase can be calculated in the form [7, 8]being \(\sigma _{\ddot{Y}_j}^2\) the variance of the process \(\ddot{Y}_j(t)\). The j-th instantaneous damped frequency \({\bar{f}}_j(\tau )\) can be calculated by performingMoreover, evaluating the mean of Eq. (12), the j-th damped frequency \(f_j\sqrt{1-\zeta _j^2}\) can be easily estimated.

The natural logarithm of Eq. (10) returns the linear form [7, 8]consequently, the j-th damping ratio isbeing \({\bar{b}}_{j}=b_{j}/(2\pi f_j \sqrt{1-\zeta _j^2})\). Once the damped frequencies and the damping ratios have been estimated, the j-th natural frequency can be trivially calculated as the ratio between the j-th damped frequency and \(\sqrt{1-\zeta _j^2}\).

The flow chart of the proposed method is depicted in Fig. 2.

In order to validate the proposed method, experimental tests have been performed considering a three-storey frame excited by a ground acceleration modelled as a broadband noise. Frequencies, damping ratios and modal shapes of the structural system have been estimated by using the proposed method, FDD and SSI and the results obtained have been compared with those related to impulsive tests. Particularly, the discrepancies between the modal parameters estimated from the broadband noise tests and those identified from the impulsive tests are calculated asin which \(p^{{\text {EST}}}\) is the value of the modal parameters estimated from the broadband noise tests while \(p^{{\text {EXP}}}\) are the correspondent modal parameters identified from the impulsive tests. Further, numerical simulations have been performed in order to check the accuracy of the proposed method. In particular, the systems obtained by the modal parameters estimated from the broadband noise tests have been used in the numerical simulations and they were excited by the same base acceleration acquired during the experimental tests. The structural response obtained from the numerical simulations has been compared with the response acquired during the broadband noise tests and the discrepancy between the latters has been calculated asin which \({\ddot{X}}^{({\text {EXP}})}_{({\text {rel}})}(t)\) represents the acquired response, \({\ddot{X}}^{({\text {NUM}})}_{({\text {rel}})}(t)\) is the response obtained from the numerical simulations while \(t_{{\text {in}}}\) and \(t_{{\text {fin}}}\) are the extremes of the time interval considered.

In this paper an innovative operational modal analysis method that allows to identify the natural frequencies, the damping ratios and the modal shapes of a structural system under ambient vibrations is proposed. It is a fast only output procedure based on the signal filtering, on the correlation function of the output process and on its Hilbert transform. It is very simple to implement and it can be easily used by any type of user since few interactions with the latter are required. To assess the accuracy of the proposed method, different experimental tests and numerical simulations have been performed on a 3-storey frame and the results obtained have been compared with those related to other OMA methods. Moreover, a numerical/experimental comparison has been performed showing that the proposed method can identify the global behaviour of the structural system with a precision similar to FDD and SSI. The simplicity of use of the proposed method with the same results obtained therefore makes this method preferable to the other OMA methods for a quick, cheap and simple monitoring.