Keywords

1 Introduction

A fundamental step for the Structural Health Monitoring (SHM) is the structural identification that, from a dynamic point of view, coalesces with the identification of natural frequencies, damping ratios and modal shapes.

In literature there are a lot of dynamic identification methods that can be subdivided into two main classes: Experimental Modal Analysis methods (EMA) and Operational Modal Analysis methods (OMA) [1]. The first one can identify the non-linear behavior of a structure but requires the knowledge of the structural input. This aspect is penalizing because the artificial generation of the structural input is complicated and very expensive in the in situ tests. On the other hand, the set-up of the OMA methods is very cheap and simple because these methods don’t require the knowledge, and thus the artificial generation of the structural input. Furthermore, since the structural input in the OMA method is an ambient noise due to traffic, wind, ground vibrations, use of the structure and so on, it is possible to identify the dynamic properties in the real operative conditions and the structural input is moldable as a white noise. For these advantages, in the last decades, the researchers focused their attention on the OMA methods.

Another classification of the identification methods can be done considering the domain in which they are developed, in fact OMA methods are subdivided in: time domain methods, frequency domain methods and hybrid domain methods [2]. Frequency domain methods in OMA, like Peak Picking and Half Power Bandwidth Method (PP+HP) [3] and Frequency Domain Decomposition (FDD) [4, 5], are usually based on the Power Spectral Density (PSD) estimation that can be performed by using the Welch’s method [6, 7]. The estimation of the PSD with the Welch’s method allows to have different advantages due to the decomposition of the structural output in sub-signals to which is possible to apply time windows (Tukey, Hamming, Hanning) and to assign an overlap length. However, the use of this method is difficult for a non-expert user and the choice of the sub-signals length, the kind of window to apply to every sub-signal and the overlap length can influence the results especially in terms of damping ratios estimation. Furthermore, if the modes are very close the identification of the dynamic properties might not be very accurate [3]. There are several time domain methods in literature, such as: Natural Excitation Technique (NExT) [8] that, when the excitation is an ambient vibration, it requires that the analytical form of free-vibration and the analytical form of the structural output are the same; Stochastic Subspace Identification methods (SSI) [2] divided into covariance-driven models (SSI-COV) and data-driven models (SSI-DATA); Auto Regressive Moving Average models (ARMA) [9] that are articulated into Auto Regressive (AR) step and Moving Average (MA) step. Hybrid domain methods can be developed in time-frequency domain, like methods based on the Wavelet transform [10], or they can be divided into different steps some in the time domain, others in the frequency domain. An example of the latter is the Analytical Signal Method (ASM) [11] that profits by the advantages due to the use of the analytical signal i.e. a high sensitivity to the minimum variations of the natural frequency. Due to the aforementioned advantages the analytical signal is also used for the identification of structural damage [12, 13].

In this paper, a new OMA method founded upon stochastic mechanic’s principles is proposed. It uses a Singular Value Decomposition (SVD) [14] to reproduce the relationship between the correlation functions matrix in the nodal space and correlation functions matrix in the modal space. The proposed method, called Time Domain Analytical Signal Method (TD-ASM) for the similarity with the hybrid method ASM, does not involve the difficulties due to the use of Welch’s method, in fact it is developed only in time domain. Furthermore, the analytical signals of the correlation functions are used in order to have a high precision in the frequencies identification.

2 Proposed Method: Time Domain Analytical Signal Method (TD-ASM)

2.1 Identification Algorithm for SDOF Systems

In this section, a novel OMA method is proposed. This method allows to identify natural frequency and damping coefficient of the SDOF structures enforced by ambient vibration. After the acquisition of the output process \(X\left( t \right)\), its correlation function \(R_{X} \left( \tau \right)\) is calculated. The analytical signal of the correlation function can be estimated by summing the correlation function to its Hilbert transform multiplied by the imaginary unit; in fact, the analytical signal is a complex signal in which the real part is the original function and the imaginary part is its Hilbert transform. Finally, the dynamic properties of the structural system can be estimated from the properties of the analytical signal: amplitude \(A\left( \tau \right)\) and phase \(\theta \left( \tau \right)\). The different steps of the proposed method for SDOF structures can be resumed in:

  1. (1)

    Acquisition of the structural output process \(X\left( t \right)\);

  2. (2)

    Estimation of the correlation function \(R_{X} \left( \tau \right)\);

  3. (3)

    Reconstruction of the analytical signal \(z_{X} \left( \tau \right)\);

  4. (4)

    Identification of the dynamic properties.

For SDOF structures the proposed method is very similar to the ASM but it is simpler than the latter because the estimation of the PSD with the Welch’s method is removed. In particular, the direct estimation of the correlation function allows to overcome the difficulties introduced by the use of the Welch’s method for the evaluation of power spectral density; in fact, the choice of the time windows applied to the sub-signals and the overlap length between two successive sub-signals can influence the results, furthermore the use of the Welch’s method requires high specialized skills.

In order to introduce the proposed method in details, a linear SDOF shear-type frame with mass m, stiffness k and damping c is used. The dynamic properties to identify are: the natural frequencies of the structure \(f = \sqrt {{k \mathord{\left/ {\vphantom {k m}} \right. \kern-0pt} m}} /\left( {2\pi } \right)\) and the damping ratio \(\zeta = {c \mathord{\left/ {\vphantom {c {\left( {4\pi mf} \right)}}} \right. \kern-0pt} {\left( {4\pi mf} \right)}}\).

When the signal of the input force is not acquired and the excitation source is due to ambient vibrations, the key hypothesis of OMA is that the structure can be considered as excited by a white noise process, defined as in [15,16,17,18], and, consequently, the stochastic differential equation governing the structural motion is

$$\ddot{X}\left( t \right) + 2\zeta \omega_{0} \dot{X}\left( t \right) + \omega_{0}^{2} X\left( t \right) = W\left( t \right)$$
(1)

where \(\omega_{0}\) is the circular frequency that is equal to \(2\pi f\). Adding the initial condition to the Eq. (1), the structural response process \(X\left( t \right)\) can be obtained.

The correlation function \(R_{X} \left( \tau \right)\) of the output response process \(X\left( t \right)\) can be estimated as

$$R_{X} \left( \tau \right) = E\left[ {X\left( t \right)X\left( {t + \tau } \right)} \right] - \mu_{X}^{2}$$
(2)

where \(\mu_{X}^{{}}\) is the mean of the process \(X\left( t \right)\). Since \(X\left( t \right)\) is a zero-mean process the Eq. (2) coalesces with

$$R_{X} \left( \tau \right) = E\left[ {X\left( t \right)X\left( {t + \tau } \right)} \right] .$$
(3)

The Hilbert transform \(\hat{R}_{X} \left( \tau \right)\) of the correlation function \(R_{X} \left( \tau \right)\) is, for definition, the convolution of \(R_{X} \left( \tau \right)\) with the signal \({1 \mathord{\left/ {\vphantom {1 {\left( {\pi \, \tau } \right)}}} \right. \kern-0pt} {\left( {\pi \, \tau } \right)}}\), i.e. it is the response to \(R_{X} \left( \tau \right)\) of a linear time-invariant filter having impulse response \({1 \mathord{\left/ {\vphantom {1 {\left( {\pi \, \tau } \right)}}} \right. \kern-0pt} {\left( {\pi \, \tau } \right)}}\). Therefore \(\hat{R}_{X} \left( \tau \right)\) can be calculated as

$$\hat{R}_{X} \left( \tau \right) = \frac{1}{\pi }\wp \int\limits_{ - \infty }^{\infty } {\frac{{R_{X} \left( {\tilde{\tau }} \right)}}{{\tau - \tilde{\tau }}}} d\tilde{\tau }$$
(4)

where \(\wp\) is the principal value.

The analytical signal is calculated as

$$z_{X} \left( \tau \right) = R_{X} \left( \tau \right) + i\hat{R}_{X} \left( \tau \right)$$
(5)

and, can be written in polar form as

$$z_{X} \left( \tau \right) = A\left( \tau \right)e^{i \, \theta \left( \tau \right)}$$
(6)

where \(A\left( \tau \right)\) is the amplitude

$$A\left( \tau \right) = \sqrt {R_{X}^{2} \left( \tau \right) + \hat{R}_{X}^{2} \left( \tau \right)}$$
(7)

and \(\theta \left( \tau \right)\) is the phase

$$\theta \left( \tau \right) = \arctan \left[ {\frac{{\text{Im} \left[ {z_{X} \left( \tau \right)} \right]}}{{\text{Re} \left[ {z_{X} \left( \tau \right)} \right]}}} \right] = \arctan \left[ {\frac{{\hat{R}_{X} \left( \tau \right)}}{{R_{X} \left( \tau \right)}}} \right] .$$
(8)

Taking into account the Euler’s formula, the analytical signal in Eq. (6) can be expressed in the form

$$z_{X} \left( \tau \right) = A\left( \tau \right)\cos \left( {\theta \left( \tau \right)} \right) + iA\left( \tau \right)\sin \left( {\theta \left( \tau \right)} \right) .$$
(9)

The correlation function of a SDOF structure enforced by a white noise can be approximated, for \(\tau > 0\), as

$$R_{X} \left( \tau \right) = Qe^{ - 2\pi f\zeta \tau } \cos \left( {2\pi \bar{f}\tau } \right)$$
(10)

and thus its Hilbert transform can be expressed as

$$\hat{R}_{X} \left( \tau \right) = Qe^{ - 2\pi f\zeta \tau } \sin \left( {2\pi \bar{f}\tau } \right)$$
(11)

where \(\bar{f} = f\sqrt {1 - \zeta_{{}}^{2} }\) is the damped frequency and \(Q\) is a constant equal to the variance of the structural response.

Replacing Eqs. (10) and (11) in the Eq. (5) it leads to

$$z_{X} \left( \tau \right) = Qe^{ - 2\pi f\zeta \tau } \cos \left( {2\pi \bar{f}\tau } \right) + iQe^{ - 2\pi f\zeta \tau } \sin \left( {2\pi \bar{f}\tau } \right)$$
(12)

and thus, using Eqs. (12) and (9) is clear that

$$A\left( \tau \right) = Qe^{ - 2\pi f\zeta \tau }$$
(13)

and

$$\theta \left( \tau \right) = 2\pi \bar{f}\tau .$$
(14)

The instantaneous damped frequency can be calculated performing the first derivative of the phase and dividing by 2π, i.e.

$$\bar{f}\left( \tau \right) = \frac{1}{2\pi }\frac{d}{d\tau }\left[ {\theta \left( \tau \right)} \right] .$$
(15)

By performing the average of the instantaneous damped frequency it is possible to obtain the damped frequency of the structure like

$$\bar{f} = E\left[ {\bar{f}\left( \tau \right)} \right] .$$
(16)

The damping ratio can be obtained from the logarithm of the amplitude that has a linear form as it can be seen from the following equation

$$\ln \left[ {A\left( \tau \right)} \right] = \ln \left[ Q \right] - 2\pi f\zeta \tau = c_{2} + c_{1} \tau .$$
(17)

From the linear form of the Eq. (17) it is clear that the angular coefficient of the amplitude’s logarithm is related to the structural damping ratio; in particular, the damping ratio is obtained as

$$\zeta = - \frac{{c_{1} }}{2\pi f} .$$
(18)

Since only the damped frequency is identified, we need to take into account that \(f = {{\bar{f}} \mathord{\left/ {\vphantom {{\bar{f}} {\sqrt {1 - \zeta_{{}}^{2} } }}} \right. \kern-0pt} {\sqrt {1 - \zeta_{{}}^{2} } }}\), and thus the Eq. (18) reverts to

$$\zeta = \sqrt {\frac{{\bar{c}_{1}^{2} }}{{1 + \bar{c}_{1}^{2} }}}$$
(19)

with \(\bar{c}_{1} = {{c_{1} } \mathord{\left/ {\vphantom {{c_{1} } {\left( {2\pi \bar{f}} \right)}}} \right. \kern-0pt} {\left( {2\pi \bar{f}} \right)}}\).

2.2 Identification Algorithm for MDOF Systems

The proposed method, introduced in the previous section, requires another step if applied to the MDOF systems. As a matter of fact, the components of the correlation functions matrix are “multi-component” functions and then they have a not well-behaved Hilbert transform. Therefore, it needs a “mono-component” correlation function, such as the modal correlation function, to restore the efficiency of the Hilbert transform. In light of the above, the identification method for MDOF systems can be resumed as:

  1. (1)

    Output processes \({\mathbf{X}}\left( t \right)\) acquisition;

  2. (2)

    Correlation functions matrix \({\mathbf{R}}_{{\mathbf{X}}} \left( \tau \right)\) estimation;

  3. (3)

    Singular Value Decomposition of \({\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right)\) and modal shapes identification;

  4. (4)

    Calculation of the correlation functions matrix in the modal space \({\mathbf{R}}_{{\mathbf{Y}}} \left( \tau \right)\);

  5. (5)

    Reconstruction of the analytical signals of the auto-correlation functions in the modal space;

  6. (6)

    Frequencies and damping ratios estimation.

In order to extend the proposed method to a MDOF system, a shear-type MDOF frame with mass matrix M, stiffness matrix K and damping matrix C is used. The dynamic properties to be identified are: the modal matrix \({\varvec{\Phi}}\), the natural frequencies \(f_{i}\) and the damping ratios \(\zeta_{i}\) where \(i = 1,2, \ldots ,N\) and N is the number of degree of freedom of the system. In this case the differential equations system that governs the structural motion is

$${\mathbf{M}}{\ddot{\mathbf{X}}}\left( t \right) + {\mathbf{C}}{\dot{\mathbf{X}}}\left( t \right) + {\mathbf{KX}}\left( t \right) = - {\mathbf{MV}}W\left( t \right)$$
(20)

where V is the influence vector. The correlation functions contained into the correlation functions matrix are estimated as

$$R_{{X_{i} X_{j} }} \left( \tau \right) = E\left[ {X_{i} \left( t \right)X_{j} \left( {t + \tau } \right)} \right] - \mu_{{X_{i} }} \mu_{{X_{j} }}$$
(21)

where \(\mu_{{X_{i} }}\) and \(\mu_{{X_{j} }}\) are respectively the averages of the i-th and j-th response process.

As it is well known, the differential equation system in Eq. (20) can be expressed in the modal space pre-multiplying for the modal matrix \({\varvec{\Phi}}\) and taking into account the modal transformation

$${\mathbf{X}}\left( t \right) = {\varvec{\Phi}}{\mathbf{Y}}\left( t \right)$$
(22)

where \({\mathbf{X}}\left( t \right)\) is the response process in the nodal space and \({\mathbf{Y}}\left( t \right)\) is the response process in the modal space. The Eq. (20) expressed in the modal space thus becomes

$${\ddot{\mathbf{Y}}}\left( t \right) + {\varvec{\Lambda}}{\dot{\mathbf{Y}}}\left( t \right) + {\varvec{\Omega}}{\mathbf{Y}}\left( t \right) = - {\varvec{\Phi}}^{T} {\mathbf{MV}}W\left( t \right)$$
(23)

in which \({\varvec{\Omega}} = {\varvec{\Phi}}^{T} {\mathbf{K}}{\varvec{\Phi}}\) is a diagonal matrix containing the squares of the circular frequencies of the system and \({\varvec{\Lambda}} = {\varvec{\Phi}}^{T} {\mathbf{C}}{\varvec{\Phi}}\) is, for classically damped system, a diagonal matrix that have the i-th term on the diagonal equal to \(2\zeta_{i} \omega_{i}\).

The auto-correlation functions contained into the diagonal of the correlation functions matrix expressed in the modal space are

$$R_{{Y_{i} Y_{i} }} \left( \tau \right) = E\left[ {Y_{i} \left( t \right)Y_{i} \left( {t + \tau } \right)} \right] - \mu_{{Y_{i} }}^{2}$$
(24)

where \(\mu_{{Y_{i} }}\) is the average of the i-th response process in the modal space.

Since the modal matrix \({\varvec{\Phi}}\) is unknown, the proposed method takes into account the relationship between the correlation functions matrix expressed in the nodal space \({\mathbf{R}}_{{\mathbf{X}}} \left( \tau \right)\) and the correlation function matrix expressed in the modal space \({\mathbf{R}}_{{\mathbf{Y}}} \left( \tau \right)\) [14], i.e.

$${\mathbf{R}}_{{\mathbf{X}}} \left( \tau \right) = {\varvec{\Phi}}{\mathbf{R}}_{{\mathbf{Y}}} \left( \tau \right){\varvec{\Phi}}^{T}.$$
(25)

In order to decompose \({\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right)\) in the product of three matrices is possible to use a SVD as it is reported in the following equation

$${\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right) = {\mathbf{USV}}^{H} .$$
(26)

In Eq. (26) S is a diagonal matrix that contains the singular values of \({\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right)\), U and V are unitary matrices that contain respectively the left singular vectors and the right singular vectors of \({\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right)\); and the apex \(H\) denote the conjugated transpose. If \({\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right)\) is a normal matrix, i.e. if it is square and \(\left( {{\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right)} \right)^{H} {\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right) = {\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right)\left( {{\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right)} \right)^{H}\), then U = V. Since \({\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right)\) is a normal and real matrix, Eq. (26) becomes

$${\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right) = {\mathbf{USU}}^{T} .$$
(27)

If the frequencies are well separated, then \({\mathbf{R}}_{{\mathbf{Y}}} \left( 0 \right)\) is almost a diagonal matrix and thus \({\mathbf{S}} \approx {\mathbf{R}}_{{\mathbf{Y}}} \left( 0 \right)\) and \({\mathbf{U}} \approx {\varvec{\Phi}}\). In light of the above, the modal matrix \({\varvec{\Phi}}\) is estimated performing a SVD of \({\mathbf{R}}_{{\mathbf{X}}} \left( 0 \right)\) and the correlation functions matrix in the nodal space can be calculated with the inverse formula of Eq. (25), i.e.

$${\mathbf{R}}_{{\mathbf{Y}}} \left( \tau \right) = {\varvec{\Phi}}^{T} {\mathbf{R}}_{{\mathbf{X}}} \left( \tau \right){\varvec{\Phi}}$$
(28)

The analytical signals \(z_{i} \left( \tau \right)\) of the auto-correlation functions in the modal space are calculated as

$$z_{i} \left( \tau \right) = R_{{Y_{i} Y_{i} }} \left( \tau \right) + i\hat{R}_{{Y_{i} Y_{i} }} \left( \tau \right)$$
(29)

where \(\hat{R}_{{Y_{i} Y_{i} }} \left( \tau \right)\) is the Hilbert transform of \(R_{{Y_{i} Y_{i} }} \left( \tau \right)\).

For each degree of freedom of the system, the frequencies and the damping ratios can be estimated with the same procedure proposed for SDOF system. In particular, applying Eqs. (7) and (8) is possible to calculate respectively the amplitudes \(A_{i} \left( \tau \right)\) and the phases \(\theta_{i} \left( \tau \right)\) of the analytical signals, by using Eqs. (15) and (16) the damped frequencies \(\bar{f}_{i}\) can be estimated and, finally, Eqs. (17–19) can be used to estimate the damping ratios \(\zeta_{i}\) of the system.

3 OMA: From Research to Engineering Applications

As regards, the paper’s contribution is to provide a user friendly method, that can be used by people who have little to no knowledge of signal processing and stochastic analysis such as those who are responsible for the maintenance of a city’s historical buildings. To aim at this, all the aforementioned steps have been implemented into an algorithm in MatLab environment reported in the Appendix.

Specifically, this algorithm only requires as input the time vector and the recorded structural outputs, then automatically returns all steps necessary to provide the dynamic properties of the structure. In sequential order the aforementioned algorithm estimate: the correlation functions matrix in the nodal space, the modal shapes, the correlation functions matrix in the modal space, the analytical signals of the auto-correlation functions in the modal space, the amplitudes, the phases, the damped frequencies and the damping ratios. To assess the reliability of the method and the algorithm, several numerical simulations and an experimental test are reported as it follows.

3.1 Validation of the Proposed Algorithm Through a SDOF System

In order to prove the reliability of the identification algorithm for SDOF system, a numerical simulation on a linear SDOF shear-type frame was performed for different values of the damping coefficient \(\zeta\). In particular, the SDOF structure has a natural frequency \(f = 30{\text{ Hz}}\) and the damping ratio is variable between 0.01 and 0.10 with a step equal to 0.01. The structural input \(W\left( t \right)\) has been generated with the Shinozuka’s formula [19] and the number of the generated samples is equal to 1000. Every sample has a duration of 100 s with sampling frequency 1000 Hz. The instantaneous damped frequency \(\bar{f}\left( \tau \right)\) and the logarithm of the amplitude \(A\left( \tau \right)\) are depicted respectively in Figs. 1 and 2 for \(\zeta = 0.02\).

Fig. 1
figure 1

Instantaneous damped frequency for f = 30 Hz and \(\zeta = 0.02\)

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Fig. 2
figure 2

Amplitude’s logarithm for f = 30 Hz and \(\zeta = 0.02\)

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The results obtained by using the proposed method and PP + HP are reported, with the relative discrepancies \(\varepsilon \%\), in Tables 1 and 2. These results suggest that both methods are reliable for the estimation of the natural frequency and damping ratio in a SDOF system. However, the proposed method has less discrepancy than PP + HP, both in terms of damped frequency estimation and in terms of damping ratio estimation. In this section the proposed method is compared only with PP + HP because other automated algorithm in MatLab environment, like FDD.m [20] and SSICOV.m [21], can be used only for MDOF systems.

Table 1 Comparison among the exact damped frequency and the estimated damped frequency
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Table 2 Comparison between the exact damping ratio and the estimated ramping ratio
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3.2 Validation of the Proposed Algorithm Through a MDOF System

In order to prove the reliability of the identification algorithm for MDOF systems, a numerical simulation on a linear 3DOF shear-type system was performed at various values of the damping coefficient \(\zeta_{1}\). In particular, the range of variation of the first damping ratio is \(\zeta_{1} = 0.05 \div 0.10\) with a step equal to 0.01. \(\zeta_{2}\) and \(\zeta_{3}\) are calculated considering a Rayleigh damping. The shear-type 3DOF frame used for the numerical simulations has mass \(m_{j} = 794{\text{ kg}}\) for \(j = 1,2,3\) and stiffness \(k_{j} = 6.18 \times 10^{6} {{\text{N}} \mathord{\left/ {\vphantom {{\text{N}} {\text{m}}}} \right. \kern-0pt} {\text{m}}}\) for \(j = 1,2,3\) and thus the natural frequencies are \(f_{1} = 6.2489{\text{ Hz}}\), \(f_{2} = 17.5091{\text{ Hz}}\), \(f_{3} = 25.3014{\text{ Hz}}\), and the modal shapes are \(\phi_{{{\kern 1pt} 1}} = \left[ {\begin{array}{*{20}c} {0.328} & {0.591} & {0.737} \\ \end{array} } \right]^{T}\), \(\phi_{2} = \left[ {\begin{array}{*{20}c} {0.737} & {0.328} & { - 0.591} \\ \end{array} } \right]^{T}\), \(\phi_{3} = \left[ {\begin{array}{*{20}c} {0.591} & { - 0.737} & {0.328} \\ \end{array} } \right]^{T}\). The results in terms of damped frequencies (for each value of \(\zeta_{1} = 0.05 \div 0.01\)) evaluated by using the proposed method are compared with those obtained by SSI-COV [21] and FDD [20], as reported in Table 3.

Table 3 Comparison between the exact damped frequencies and the estimated damped frequencies
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From these results it is apparent that all methods are performing very well, in particular the SSI-COV results have the least discrepancy, but SSI-COV is not direct as the proposed method TD-ASM, since it requires the knowledge of a parameter related to the first frequency, that is a priori unknown. This means that it needs at least a Fourier Transform of the signal to get this value, while the proposed method does not require any preliminary information of the unknown characteristics. Further, results in terms of damping ratios are compared with SSI-COV only, since the algorithm FDD.m does not allow the damping ratios’ evaluation. Also in this case the proposed method gets satisfactory results as shown in Table 4 (for each value of \(\zeta_{1} = 0.05 \div 0.01\)).

Table 4 Comparison between the exact damping ratios and the estimated damping ratios
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As regards the modal shapes, Fig. 3 reports the discrepancies of results obtained with the proposed method, FDD and SSI-COV with respect the exact ones at different values of damping ratios. Also in this case, the proposed method and SSICOV (both developed in the time domain and based on the correlation function) are more precise than FDD (developed in the frequency domain and based on the PSD) especially for the first modal shape that gives the major contribution to the total structural motion.

Fig. 3
figure 3

Discrepancy of results obtained with the proposed method TD-ASM (circular marker), FDD (hexagram marker) and SSICOV (diamond marker) with respect the exact ones at different values of damping ratios: first mode a, second mode b, third mode c

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3.3 Validation of the Proposed Algorithm Through Experimental Test

In order to prove the reliability of the proposed method on real structures, an experimental test was performed on a three-story frame. The set-up of the experimental test is reported in Fig. 4. In particular, the structure was excited by a broad-band noise from 0.01 to 80 Hz through an electro-magnetic shaker APS-ELECTRO-SAIS. The input and the output signals were recorded using piezo-electric accelerometers Brüel&Kjaer 4507 002 connected to the acquisition unit NI PXIe 1082. Some tests of 240 s with sampling frequency equal to 1000 Hz were performed and the proposed algorithm was used to obtain the modal shapes, the frequencies and the damping ratios of the structure. Since the tests are performed on a real system that has unknown dynamic properties, is impossible to report a discrepancy between the exact properties and the identified properties. However, the results obtained by the used methods are reported in Tables 5 and 6.

Fig. 4
figure 4

Experimental set-up

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Table 5 Comparison between TD_ASM.m, SSICOV.m and FDD.m in terms of damped frequencies and damping ratios
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Table 6 Comparison between TD_ASM.m, SSICOV.m and FDD.m in terms of modal shapes
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From these results, it is clear that all the used algorithms well identify the same frequencies and that the differences between the damping ratios identified with the proposed algorithm and SSICOV.m are very low.

4 Conclusions

This paper introduces an innovative ambient identification method based on the Hilbert Transform to obtain the analytical representation of the system response in terms of the correlation function. It leads to identify the modal shapes performing a SVD of the correlation function matrix in \(\tau = 0\), the frequencies by the phase of each analytical signal of the auto-correlation functions in the modal space and the damping ratios by the amplitude of each analytical signal of the auto-correlation functions in the modal space. The numerical simulations prove that for SDOF structures the proposed method can identify the dynamic properties better than the PP+HP and for MDOF structures the proposed method TD-ASM can identify very well the dynamic properties of the structural systems, especially in terms of the damping ratios. The performed experimental tests prove that the dynamic properties identified with the proposed method are similar to the properties identified by other automated algorithm and that the proposed algorithm have some advantages compared to the others. However, the greatest advantage of TD-ASM is that it is user friendly, in fact the developed MatLab function requires only the time vector and the recorded outputs and can be used also by a not-expert user. As a concluding remark, the authors wish that this approach could open the pathway for a monitoring system that is user friendly and can be used by people who have little to no knowledge of signal processing and stochastic analysis such as those who are responsible for the maintenance of a city’s historical buildings.