Refined Frequency Domain Decomposition modal dynamic identification from earthquake-induced structural responses

Abstract

In the present work, earthquake-induced structural responses at heavy damping are considered towards the assessment of current strong-motion output-only modal parameters via a novel refined Frequency Domain Decomposition (rFDD) approach. Classical FDD assumptions of stationary white noise input and light damping no longer hold in such a case, leading to extreme operational conditions for this Frequency-Domain modal dynamic identification technique. The present non-parametric output-only rFDD procedure effectively tackles such issues, through dedicated algorithmic and computational strategies, resulting in a robust identification tool within this challenging identification context. Synthetic earthquake-induced responses are generated from a set of shear-type frames, which are taken as target structural models for the purposes of validation and comparison. Twenty-two “Far-Field” multi-component (NS, WE) seismic base excitations are taken into account from the FEMA P695 strong motion database. In particular, heavy-damped structures (in terms of output-only identification challenge) are considered, raising the modal damping ratios up to 10%. Natural frequencies, mode shapes and modal damping ratios are effectively estimated through the implemented rFDD procedure. The achieved estimates are statistically compared to the target values, by proving the effectiveness of the developed rFDD approach in working with synthetic earthquake-induced responses at concurrent heavy damping. This shall demonstrate a necessary condition for the feasibility of the developed rFDD approach towards Earthquake Engineering and Structural Health Monitoring purposes.

Appendix 1: Refined FDD theoretical background

1.1 A1.1 Estimation of natural frequencies and mode shapes by rFDD

Classical FDD theory is based on the typical input/output relationship of a general stationary random process for a n-DoF system [1]:

$$\begin{aligned} \mathbf {G}_{yy}(\omega )=\overline{\mathbf {H}}(\omega )\mathbf {G}_{xx}(\omega )\mathbf {H}^{\mathrm {T}}(\omega ) \end{aligned}$$

(3)

where $\mathbf {G}_{xx}(\omega ) \in \mathcal {R}^{r \times r}$ is the PSD matrix of the input, r being the number of input channels (references), $\mathbf {G}_{yy}(\omega ) \in \mathcal {R}^{m \times m}$ is the PSD matrix of the output, m being the number of output signals (responses or measurements), $\mathbf {H}(\omega ) \in \mathcal {R}^{m \times r}$ is the Frequency Response Function (FRF) matrix; overbar denotes complex conjugate and apex symbol ${\mathrm {T}}$ transpose.

The $\mathbf {G}_{yy}(\omega )$ matrix may be calculated by the present rFDD algorithm via an integrated PSD matrix computation, which simultaneously implements the Wiener–Khinchin and Welch’s modified periodogram methods [8–10]. By writing the FRF $\mathbf {H}(\omega )$ in pole/residue form [3], and by taking all output measurements as input references (i.e. when $m = r$), Eq. (3) can be rewritten as:

$$\begin{aligned} \mathbf {G}_{yy}(\omega _{\mathrm {Sub}})=\sum _{k=1}^n \sum _{s=1}^n \Biggl ( \dfrac{\overline{\mathbf {R}}_{k} {\mathrm {G}} _{xx}}{-\mathrm {i}\,\omega -\overline{\lambda }_{k}} + \dfrac{\mathbf {R}_{k} \mathrm {G}_{xx}}{-\mathrm {i}\,\omega -\lambda _{k}}\Biggr ) \cdot \Biggl (\dfrac{\mathbf {R}_{s}^{\mathrm {T}}}{\mathrm {i}\,\omega -\lambda _{s}} + \dfrac{\mathbf {R}_{s}^{\mathrm {H}}}{\mathrm {i}\,\omega -\overline{\lambda }_{s}}\Biggr ) \! \Bigg \vert _{\omega =\omega _{Sub}} \end{aligned}$$

(4)

where n is the number of modes, $\lambda _k = -\zeta _k \omega _k + {\mathrm {i}} \omega _{dk}$ and $\overline{\lambda }_{k}$ are the $k^{th}$ poles (in complex conjugate pairs) of the FRF, being $\zeta _{k}$ the modal damping ratio and $\omega _k$ and $\omega _{dk}$ the kth natural and damped frequency, respectively, while Hermitian apex symbol ${\mathrm {H}}$ denotes complex conjugate and transpose. Then, $\mathbf {R}_{k}=\varvec{\phi }_{k}\mathbf {\varGamma }_k^{\mathrm {T}} \in \mathcal {R}^{m \times r}$ is the residue matrix, which is obtained by the product of mode shape vector $\varvec{\phi }_{k} \in \mathcal {R}^{m \times 1}$ and modal participation factor vector $\mathbf {\varGamma }_k^{\mathrm {T}} \in \mathcal {R}^{r \times 1}$.

Finally, the present rFDD algorithm defines $\mathbf {G}_{xx}(\omega ) \Rightarrow {\mathrm {G}}_{xx} = \overline{\mathrm {G}}_{xx}$ for a frequency interval $\omega _{\mathrm {Sub}} \in \omega$, as adopted in Eq. (4) [8–10]. Classically, this formulation is possible since PSD matrix ${\mathbf {G}}_{xx}(\omega )$ is a real-valued and non-negative single scalar constant ${\mathrm {G}}_{xx}$ for stationary zero mean white noise input [1]. Thus, for some selected non-stationary input, as for the adopted earthquake responses, it is possible, within a good approximation, to ignore the frequency dependence of input PSD matrix $\mathbf {G}_{xx}$ for a specific frequency interval, corresponding to a subset of the frequency lines $\omega _{\mathrm {Sub}} = {\mathrm {Sub}}(\omega ) \in \omega$ of the original input spectrum $\omega$. This means that some seismic signals may be considered as “weakly stationary”, i.e. their first two statistical moments (mean and autocorrelation) do not significantly vary in time [8, 9].

By recalling the approximation due to the use of the ${\mathrm {Sub}}(\omega )$ interval, it is possible to develop the remaining part of the rFDD theoretical background [8–10]. Specifically, by applying the Heaviside partial fraction expansion theorem to Eq. (4), one can obtain [8]:

$$\begin{aligned} \mathbf {G}_{yy}(\omega _{\mathrm {Sub}}) \simeq \sum _{k=1}^n \dfrac{\mathbf {A}_{k}}{\mathrm {i}\,\omega -\lambda _{k}} + \dfrac{\mathbf {A}_{k}^{\mathrm {H}}}{-\mathrm {i}\,\omega -\overline{\lambda }_{k}} + \dfrac{\overline{\mathbf {A}}_{k}}{\mathrm {i}\,\omega -\overline{\lambda }_{k}} + \dfrac{\mathbf {A}_{k}^{\mathrm {T}}}{-\mathrm {i}\,\omega -\lambda _{k}} \Bigg \vert _{\omega =\omega _{Sub}} \end{aligned}$$

(5)

where $\mathbf {A}_{k} \in \mathcal {R}^{m \times m}$ is the Hermitian residue matrix of the PSD output corresponding to kth pole $\lambda _k$. Both for light and heavy damping, as demonstrated in [8], the pole can be expressed in approximate form as $\lambda _k = -\zeta _k \omega _k + {\mathrm {i}} \omega _{dk} \simeq -\zeta _k \omega _k + {\mathrm {i}} \omega _k$. Then, in the vicinity of the kth modal frequency, residue matrix $\mathbf {A}_{k}$ can be written as [8]:

$$\begin{aligned} \mathbf {A}_{k} = \Biggl [ \dfrac{ \mathbf {R}_k }{2 (\zeta _k \omega _k - {\mathrm {i}} \omega _k)} + \dfrac{\overline{\mathbf {R}}_k }{2 \zeta _k \omega _k} \Biggr ] {\mathrm {G}}_{xx}\mathbf {R}_{k}^{\mathrm {T}} \simeq \dfrac{ \overline{\mathbf {R}}_{k} {\mathrm {G}} _{xx} \mathbf {R}_{k}^{\mathrm {T}}}{ 2\zeta _{k}\omega _{k}} = \dfrac{ \overline{\varvec{\phi }}_{k}\mathbf {\varGamma }_k^{\mathrm {H}} {\mathrm {G}} _{xx} {\mathbf {\varGamma}}_k \varvec{\phi }_{k}^{\mathrm {T}}}{ 2\zeta _{k}\omega _{k}} = {\mathrm {d}_{k}} \overline{\varvec{\phi }}_{k} \varvec{\phi }_{k}^{\mathrm {T}} \end{aligned}$$

(6)

where only the $\overline{\mathbf {R}}_k$ term survives, since the $2 \zeta _{k}\omega _{k}$ denominator is dominant with respect to the $2 (\zeta _k \omega _k - {\mathrm {i}} \omega _k)$ one, while ${\mathrm {d}_{k}} = {\mathbf {\varGamma }}_k^{\mathrm {H}} {\mathrm {G}} _{xx} {\mathbf {\varGamma }}_k / (2 \zeta _k \omega _k )$ is a real scalar.

In the narrow band with spectrum lines in the vicinity of a modal frequency, the last two terms in Eq. (5) can be ignored, since their denominator terms ${\mathrm {i}}\,\omega -\overline{\lambda }_{k} = \overline{-\mathrm {i}\,\omega -\lambda _{k}} \simeq \zeta _k \omega _k + 2 {\mathrm {i}}\,\omega _k$ are bigger with respect to the first two, $-\mathrm {i}\,\omega -\overline{\lambda }_{k} = \overline{\mathrm {i}\,\omega -\lambda _{k}} \simeq \zeta _k \omega _k$. Taking this into account, and by substituting Eq. (6) into Eq. (5) one gets:

$$\begin{aligned} \mathbf {G}_{yy}(\omega _{\mathrm {Sub}}) \simeq \sum _{k=1}^n \dfrac{\mathrm {d}_{k} \overline{\varvec{\phi }}_{k} \varvec{\phi }_{k}^{\mathrm {T}}}{\mathrm {i}\,\omega -\lambda _{k}} + \dfrac{\mathrm {d}_{k} \overline{\varvec{\phi }}_{k} \varvec{\phi }_{k}^{\mathrm {T}}}{-{\mathrm {i}}\,\omega -\overline{\lambda }_{k}} \Bigg \vert _{\omega =\omega _{Sub}} = \overline{\mathbf {\varPhi }} \left\{ {\text {diag}} \left[ \mathfrak {R}{\mathrm {e}}\left( \dfrac{2 {\mathrm {d}_{k}}}{\mathrm {i}\,\omega -\lambda _{k}}\right) \right] \right\} {\mathbf {\varPhi }}^{\mathrm {T}} \Bigg \vert _{\omega =\omega _{Sub}} \end{aligned}$$

(7)

where ${\mathbf {\varPhi}}$ is the eigenvector matrix, gathering all n eigenvectors $\varvec{\phi }_{i}$ as columns.

Previous Eq. (7) represents a modal decomposition of the output PSD matrix. By taking the ${\mathbf {G}}_{yy}(\omega _{\mathrm {Sub}})$ matrix transpose, the contribution to the PSD matrix from a single mode k can be expressed as:

$$\begin{aligned} \mathbf {G}_{yy}^{\mathrm {T}}(\omega _i = \omega _{k}) \simeq \varvec{\phi }_{k} \left\{ {\text {diag}} \left[ \mathfrak {R}\mathrm {e}\left( \dfrac{2 {\mathrm {d}}_{k}}{\mathrm {i}\,\omega -\lambda _{k}}\right) \right] \right\} \varvec{\phi }_{k}^{\mathrm {H}} = \mathrm {s}_{k} \mathbf {u}_{k1} \mathbf {u}_{k1}^{\mathrm {H}} \end{aligned}$$

(8)

where this final form of the ${\mathbf {G}_{yy}}(\omega _{k})$ matrix is decomposed, using a classical SVD technique, into a set of singular values and corresponding singular vectors, as shown in Eq. (2). From the former, natural frequencies are extracted; from the latter, approximate mode shapes are obtained.

1.2 A1.2 Estimation of modal damping ratios by rFDD

The estimation of the modal damping ratios is performed by rFDD via an advanced optimization algorithm, since the use of seismic input makes it harder for the correct detection of the SDoF system associated to the identified mode, as done with classical FDD [2, 3]. Besides, the use of the following procedure automates the estimation of the modal damping ratios and does not require the interaction with an expert operator [8]:

• 1. The spectral bell identification of the qth mode of vibration is performed first with a 0.9 MAC index, selecting the ${\mathbf {S}}_Q$ subset of SVs directly from the SVD of Eq. (1):

  $$\begin{aligned} {\mathbf {S}}_Q (\omega ) = {\mathbf {S}}_Q (\omega _i = \omega _q) = \bigl [\; 0 \; \mathrm {s}_l \; \ldots \; {\mathrm {s}}_q \; \ldots \; {\mathrm {s}}_u \; 0 \; \bigr ]^{\mathrm {T}} \end{aligned}$$

  (9)

  where ${\mathrm {s}}_l$ and ${\mathrm {s}}_u$ are the lower and upper values of the spectral bell.

• 2. The identified set of SV is taken back to the time domain by Inverse Fourier Transform and normalized, by obtaining the related SDoF Auto-Correlation Function (ACF), ${\mathbf {R}}_{yy,Q}(\tau )$:

  $$\begin{aligned} {\mathbf {R}}_{yy,Q} (\tau ) = \mathcal {F}^{-1} \bigl \lbrace \mathbf {S}_Q (\omega ) \bigr \rbrace = \frac{1}{m} \sum _{j=0}^{m-1} \mathbf {S}_Q (\omega ) e^{-\frac{2 \pi \mathrm {i}}{m}jl} = \bigl [\; \mathrm {r}_l \; \ldots \; \mathrm {r}_m \; \bigr ]^{\mathrm {T}}, \;\;\; l = 1,\ldots ,m-1 \end{aligned}$$

  (10)

• 3. On the normalized SDoF ACF, two exponential decays are fitted, both on peaks and valleys of the ACF, defining terms $\hat{\mathbf {y}}_{p} = {\mathrm {A}}^p \, {\mathrm {exp}} \bigl ({\mathrm {B}}^p {\mathbf {x}}^{p} \bigr )$ and $\hat{\mathbf {y}}_{v} = {\mathrm {A}}^v \, {\mathrm {exp}} \bigl ({ \mathrm {B}}^v {\mathbf {x}}^{v} \bigr )$, respectively. There, ${\mathrm {A}}^i$ and ${\mathrm {B}}^i$ are the parameters to be fitted, while ${\mathbf {x}}^{i}$ represent the peak and valley time instant vectors (superscript i denotes peaks p, and valleys v, respectively). These parameters can be calculated by the minimization of the ${\mathrm {R}}_i$ function, wherein the points are equally weighted:

  $$\begin{aligned} \mathrm {R}^2_i(\mathrm {a}^i,\mathrm {b}^i) = \sum _{j=1}^{m} \mathrm {y}_j^i \Bigl [ \ln { \bigl ( \mathrm {y}_j^i \bigr )} - \bigl ( \mathrm {a}^i + \mathrm {b}^i \mathrm {x}_j^i \bigr ) \Bigr ]^2 \Rightarrow \left\{ \begin{aligned} \mathrm {J}_{\mathrm {a},i} = \; \dfrac{\partial \left( \mathrm {R}^2_i \right) }{\partial \, \mathrm {a}^i} = 0 \\ \mathrm {J}_{\mathrm {b},i} = \; \dfrac{\partial \left( \mathrm {R}^2_i \right) }{\partial \, \mathrm {b}^i} = 0 \\ \end{aligned}\right. \end{aligned}$$

  (11)

  where ${\mathrm {J}}_{\mathrm {a},i}$ and ${\mathrm {J}}_{\mathrm {b},i}$ are the terms of Jacobian matrix ${\mathbf {J}}$, while superscript i again denotes peaks and valleys, respectively. The fitting coefficients ${\mathrm {a}}^i$ and ${\mathrm {b}}^i$ may be computed as ${\mathrm {a}}^i = \ln {(\mathrm {A})^i}$ and ${\mathrm {b}}^i = \mathrm {B}^i$. Solving for ${\mathrm {a}}^i$ and ${\mathrm {b}}^i$, it is possible to obtain the estimates of exponential decays $\hat{\mathbf {y}}_{p}$ and $\hat{\mathbf {y}}_{v}$.

• 4. All ACF extrema ranging from $r_0 = 90\%$ to $r_n = 30\%$ of the maximum amplitude are selected within an appropriate time window, giving the normalized ACF subset ${\mathbf {R}}_{yy,Q} (\tau _{\mathrm {sub}})$:

  $$\begin{aligned} {\mathbf {R}}_{yy,Q} (\tau _{\mathrm {sub}}) = \bigl [\; {\mathrm {r}}_0 \; \ldots \; {\mathrm {r}}_n \; \bigr ]^{\mathrm {T}} \subseteq {\mathbf {R}}_{yy,Q} (\tau ) \end{aligned}$$

  (12)

• 5. Subsequent operations of regression, according to classical EFDD procedures [2, 3], are performed and the modal damping ratio $\zeta _q$ is estimated.

• 6. Exponential decays $\hat{\mathbf {y}}_{p}$ and $\hat{\mathbf {y}}_{v}$ included in the time window of the ACF are compared with the classical damping trends, arising from SDoF free damped vibrations:

  $$\begin{aligned} {\mathbf {y}}^{p} = {\mathrm {exp}} \left( - \zeta _q \frac{\omega _{dq}}{\sqrt{1-\zeta _q^2}} {\mathbf {x}}^{p} \right) , \;\;\; {\mathbf {y}}^{v} = {\mathrm {exp}} \left( - \zeta _q \frac{\omega _{dq}}{\sqrt{1-\zeta _q^2}} {\mathbf {x}}^{v} \right) \end{aligned}$$

  (13)

  where the motion’s maximum amplitude is set equal to 1 for the normalized ACF, $\zeta _q$ is the estimated modal damping ratio (point 5) and $\omega _{dq}$ is the qth estimated damped frequency.

• 7. The exponential decays of Eq. (13) are compared, by calculating the residuals $\hat{\varvec{\varepsilon }}^i$, with the fitted decays of point 3:

  $$\begin{aligned} \hat{\varvec{\varepsilon }}^p = \mathbf {y}^{p} - \hat{\mathbf {y}}^{p}, \;\;\; \hat{\varvec{\varepsilon }}^v = {\mathbf {y}}^{v} - \hat{\mathbf {y}}^{v} \end{aligned}$$

  (14)

  These residuals are compared for each peak and valley of the ACF considered time window.

• 8. The developed algorithm works fine in order to minimize these residuals, by recalibrating the time window parameters (i.e. $r_0$ and $r_n$) and the MAC index at each step, with an iterative loop from point 1 to point 7, until a reliable estimate of $\zeta _q$ is obtained. This means that the residual entries must be less than a fixed parameter, for example $10^{-2}$. The MAC index is the first value recalibrated by the algorithm, ranging from 0.70 to 0.99, while $r_0$ and $r_n$, secondarily, shall assume values between $0.7 \le r_0 \le 1.0$ and $0.1 \le r_n \le 0.4$, by expanding or contracting the time window, until the residuals have been minimized.

The optimization algorithm for residual minimization adopts two sources of information, i.e. classical damping trends ${\mathbf {y}}^{p}$, ${\mathbf {y}}^{v}$ and exponential fitted decays $\hat{\mathbf {y}}^p$, $\hat{\mathbf {y}}^v$. The minimization of the residuals, allowing for a most effective evaluation of modal damping ratio $\zeta _q$, is measured in terms of an appropriate objective function $\mathrm {P}(\mathbf {z})$:

$$\begin{aligned} {\mathrm {P}}({ \mathbf {z}} ) = \left[ \alpha \left( \frac{{y}^p_i - \hat{{y}}^p_i}{{y}^p_i} \right) ^2, \, \left( 1 - \alpha \right) \left( \frac{{y}^v_i - \hat{{y}}^v_i}{{y}^v_i} \right) ^2 \right] ^{\mathrm {T}} = \left[ \alpha \left( \frac{\hat{\varvec{\varepsilon }}^p_i}{{y}^p_i} \right) ^2, \, \left( 1 - \alpha \right) \left( \frac{\hat{\varvec{\varepsilon }}^v_i}{{y}^v_i} \right) ^2 \right] ^{\mathrm {T}} \end{aligned}$$

(15)

where ${\mathbf {z}}$ is the ($3 \times 1$) vector including the parameters to be optimized with respect to the $i = 1,\ldots ,n$ ACF points, and $\alpha \in [0,1]$ represents a weighting coefficient, set to 0.5 in the present work. Thus, despite for the iterative approach, rather limited computing time has been only added to the computational effort of a classical EFDD algorithm, in the order of $20\%$.