Time–frequency analysis and applications in time-varying/nonlinear structural systems: A state-of-the-art review

Abstract

Nonlinear dynamic behaviors of civil engineering structures have been observed not only under extreme loads but also during normal operations. Characterization of the time-varying property or nonlinearity of the structures must account for temporal evolution of the frequency and amplitude contents of nonstationary vibration responses. Neither time analysis nor frequency analysis method alone can fully describe the nonstationary characteristics. In this article, an attempt is made to review the milestone developments of time–frequency analysis in the past few decades and summarize the fundamental principles and structural engineering applications of wavelet analysis and Hilbert transform analysis in system identification, damage detection, and nonlinear modeling. This article is concluded with a brief discussion on challenges and future research directions with the application of time–frequency analysis in structural engineering.

Keywords

dynamic response Hilbert transform nonlinearity time–frequency time-varying wavelet analysis

Introduction

For a time-varying or nonlinear structure, the structural dynamic responses are often nonstationary and irregular in amplitude and frequency over time. Such responses contain rich information about the structural properties and must be characterized with instantaneous features. The extracted time-varying features can further be used for structural identification, damage detection, and structural model updating. Therefore, an effective time–frequency analysis technique is required to extract the important time-varying features of the dynamic responses.

The time and frequency domain analysis methods alone are based on the stationary assumption, which can only provide the average statistical properties in the time or frequency domain, but cannot provide the local properties in both time and frequency domains. Over the past decade, researchers have become aware of the limitation of these methods, especially for structural nonstationary response analysis. Therefore, wavelet and Hilbert transform–based methods were developed for time-varying feature extraction, structural parameter identification, damage detection, and model updating.

In this article, the time–frequency methods that are focused on wavelet transform and Hilbert transform and their applications in vibrating structures are reviewed. Since numerous literatures were developed over the past decades, it is hard to review each and every article published in this area. Therefore, the authors will focus on the recent articles related to the critical advances in these two methods and their applications to time-varying and nonlinear civil engineering structures. This review is framed as follows. Section “Time-varying/nonlinear structural system” gives the introduction of time-varying/nonlinear structural system. Section “Wavelet transform and applications in civil engineering structures” gives the wavelet analysis and applications for structural parameter identification and damage detection. Section “Hilbert transform and applications in civil engineering structures” gives the Hilbert transform analysis and applications for structural parameter identification and damage detection. Section “Hilbert transform applications to nonlinear structures” introduces the Hilbert transform applications in nonlinear structural parameter identification and model updating. In section “Discussions and prospects,” some conclusions and application prospects are drawn.

Time-varying/nonlinear structural system

The equation of motion of a time-varying single-degree-of-freedom (SDOF) structure with free vibration can be simply expressed as

\overset{\cdot\cdot}{x} (t) + 2 h_{0} (t) \overset{\cdot}{x} (t) + ω_{0}^{2} (t) x (t) = 0

(1)

in which $ω_{0} (t)$ is the time-varying natural frequency and $2 h_{0} (t)$ is the time-varying damping coefficient. Applying Hilbert transform to equation (1) yields

H [\overset{\cdot\cdot}{x}] + 2 h_{0 L} H [\overset{\cdot}{x}] + 2 H [h_{0 H}] \overset{\cdot}{x} + ω_{0 L}^{2} H [x] + H [ω_{0 H}^{2}] x = 0

(2)

in which $h_{0 L}$ and $h_{0 H}$ are, respectively, the low-pass and high-pass parts of the time-varying nonlinear damping coefficient $h_{0} (t)$ . $ω_{0 L}^{2}$ and $ω_{0 H}^{2}$ are the low-pass and high-pass parts of the new time-varying frequency $ω_{0}^{2} (t)$ .

If the structure is assumed as a time-varying structure, and its parameters are varied slower than the response itself, then both $h_{0} (t)$ and $ω_{0}^{2} (t)$ are slowly varying functions of time, $h_{0 H}$ ; $H [h_{0 H}]$ , $ω_{0 H}^{2}$ , and $H [ω_{0 H}^{2}]$ are approximately equal to zero. In this condition, the instantaneous damping coefficient $2 h_{0} (t)$ and the instantaneous natural frequency $ω_{0} (t)$ can be evaluated and expressed with (Feldman, 1994a, 1994b)

ω_{0}^{2} (t) = ω^{2} - \frac{\overset{\cdot\cdot}{A}}{A} + 2 \frac{\overset{\cdot}{A}}{A^{2}} + \frac{\overset{\cdot}{A} \overset{\cdot}{ω}}{A ω} and h_{0} (t) = - \frac{\overset{\cdot}{A}}{A} - \frac{\overset{\cdot}{ω}}{2 ω}

(3)

In equation (3), $A$ and $ω$ are the instantaneous amplitude and frequency of the response, respectively. Equation (3) reveals the relationship between the instantaneous structural parameters (instantaneous frequency and damping coefficient of the structure) and the instantaneous amplitude and frequency of the response.

For a SDOF time-varying system with forced vibration, the equation of motion can be described as

\overset{\cdot\cdot}{x} (t) + 2 h_{0} (t) \overset{\cdot}{x} (t) + ω_{0}^{2} (t) x (t) = f (t) / m

(4)

Similar to free vibration, if the varying instantaneous damping coefficient $h_{0} (t)$ and natural frequency $ω_{0}^{2} (t)$ are slow-varying time functions, the instantaneous damping coefficient $h_{0} (t)$ and natural frequency $ω_{0}^{2} (t)$ can be solved as (Wang et al., 2013a)

\begin{matrix} ω_{0}^{2} (t) = ω^{2} + \frac{fx + H [f] H [x]}{m [x^{2} + {(H [x])}^{2}]} \\ + \frac{H [f] x - fH [x]}{x^{2} + {(H [x])}^{2}} \frac{\overset{\cdot}{A}}{A ω m} - \frac{\overset{\cdot\cdot}{A}}{A} + 2 \frac{\overset{\cdot}{A}}{A^{2}} + \frac{\overset{\cdot}{A} \overset{\cdot}{ω}}{A ω} \\ h_{0} (t) = \frac{H [f] x - fH [x]}{x^{2} + {(H [x])}^{2}} \frac{1}{2 ω m} - \frac{\overset{\cdot}{A}}{A} - \frac{\overset{\cdot}{ω}}{2 ω} \end{matrix}

(5)

Equation (5) reveals the relationship between the instantaneous characteristics of the structure and the instantaneous amplitude and frequency of the response, and the excitation.

In reality, the damping coefficients and derivatives of the envelope of the analytical signal are much less than the natural frequency, so their influence can be ignored $(\frac{\overset{\cdot}{A}}{A} = \frac{\overset{\cdot\cdot}{A}}{A} = 0)$ . As a result, the instantaneous frequency of the analytical signal leads to

ω^{2} (t) = ω_{0}^{2} (t) - \frac{fx + H [f] H [x]}{m [x^{2} + {(H [x])}^{2}]}

(6)

For ambient vibration with zero mean white Gaussian excitation $f$ , the second term of equation (6) is the zero mean fast time-varying function. Knowing the instantaneous frequency and amplitude of the dynamic responses, the structural time-varying parameters can then be derived based on the above theoretical derivation.

For an n degree-of-freedom (DOF) time-varying structural system, the equation of motion is described as

M (t) \overset{\cdot\cdot}{x} (t) + C (t) \overset{\cdot}{x} (t) + K (t) x (t) = f (t)

(7)

where $M (t)$ , $C (t)$ , and $K (t)$ are time-varying mass, damping, and stiffness matrices, respectively, and $f (t)$ is the external load vector.

If a structure is still assumed as a slow time-varying structure, its parameters vary slower than the responses, then equation (7) can be transformed into modal spatial coordinate

{\overset{\cdot\cdot}{q}}_{i} (t) + 2 h_{0 i} {\overset{\cdot}{q}}_{i} (t) + ω_{0 i}^{2} q_{i} (t) = \frac{\emptyset_{i}^{T} f (t)}{M_{i}} (i = 1, 2, \dots, n)

(8)

in which $M_{i} = \emptyset_{i}^{T} M \emptyset_{i}$ is the ith modal mass, $\emptyset_{i}$ is the ith mode shape vector, and $ω_{0 i}^{2} (t)$ is the natural frequency of the ith modal response.

By decomposing the dynamic responses, each mono-component can be obtained, and the instantaneous frequency and amplitude of the mono-components can then be extracted by time–frequency analysis method, and then the structural time-varying parameters can be derived similar to a SDOF structure.

For an n DOF nonlinear system, the equation of motion can be written as

M \overset{\cdot\cdot}{x} (t) + F_{c} [\overset{\cdot}{x} (t)] + F_{s} [x (t)] = f (t)

(9)

in which $M$ is the mass matrix, $F_{c} [\overset{\cdot}{x} (t)]$ is the damping force vector, $F_{s} [x (t)]$ is the stiffness force vector, and $f (t)$ is the excitation vector.

For a weakly nonlinear structure, the nonlinear restoring force can be transformed into a multiplication form $K (t) x (t)$ with a new time-varying stiffness matrix $K (t)$ and a system solution $x (t)$ with an overlapping spectrum (Feldman, 1997). Similarly, the nonlinear damping force can also be transformed into a function of time as a multiplication: $C (t) \overset{\cdot}{x} (t)$ between the time-varying damping coefficient matrix $C (t)$ and the velocity $\overset{\cdot}{x} (t)$ . Thus, the equivalent equation of motion of equation (9) can be expressed as equation (7). Therefore, for a time-varying structure of a weakly nonlinear structure, the structural time-varying parameters can be derived from the instantaneous frequency and amplitudes of the dynamic response, which can be extracted with time-varying analysis methods. For a more complicated nonlinear structure, the nonlinear model updating methods need to be developed to identify the nonlinear structural parameters.

Wavelet transform and applications in civil engineering structures

One of the widely used time–frequency analysis methods is the wavelet transform, which has been applied to nonstationary signals in the past two decades (Chui, 1992; Flandrin, 1999; Grochenig, 2001; Wang et al., 2016a; Yin et al., 2016). Since the detailed time-varying signal features can be extracted by wavelet transform, the wavelet transform–based methods have been widely applied to structural parameter identification and damage detection. In the area of civil engineering, Staszewski (1997) developed the impulse response recovery procedure, and ridge detection procedure with wavelet transform in damping identification of dynamic systems. Ruzzene et al.’s (1997) study was based on the detected wavelet ridge to identify the natural frequency and damping ratio of a real bridge. Gurly and Kareem (1999) identified a transient random process that encountered in ocean, wind, and earthquake engineering. After then, many researchers realized that the critical issue for structural time-varying parameter identification using wavelet transform is to extract the wavelet ridges; therefore, many methods were developed to enhance the time and frequency resolution of the wavelet ridges. The wavelet ridges can then be further used to identify the structural time-varying parameters (Chen et al., 2009; Kijewski and Kareem, 2002, 2003). More recently, to track the structural time-varying parameters, online identification approaches based on adaptive wavelets were also developed (Basu et al., 2008; Dziedziech et al., 2015).

In this section, the basic theoretical frame work of wavelet transform is reviewed. Particularly, for time-varying/nonlinear structures, the wavelet ridge, which represents the frequency/phase information of the signal, and wavelet energy/entropy, which represents the amplitude of the wavelet transform, are reviewed. Then, the structural system identification and damage detection methods mainly based on the concepts of the wavelet ridge and energy spectrum are reviewed.

Wavelet transform

In this section, the wavelet transform, wavelet ridge, and wavelet energy/entropy are briefly reviewed.

Continuous wavelet transform

A parent wavelet $ψ (t)$ is a square integrable function and it satisfies the admissibility condition. By dilating and translating the parent wavelet, the basis function of wavelet is defined as

ψ_{a, b} (t) = \frac{1}{\sqrt{a}} ψ (\frac{t - b}{a})

(10)

where a is the scale parameter that is inversely proportional to frequency, and b is the translation parameter corresponding to time. The continuous wavelet transform of a signal $x (t)$ is defined as

W T_{x} (a, b) = < x, ψ_{a, b} > = \frac{1}{\sqrt{a}} \int_{- \infty}^{\infty} x (t) {ψ^{*}}_{a, b} (t) dt

(11)

where $ψ^{*} (\cdot)$ is the complex conjugate of $ψ (\cdot)$ , and $W T_{x} (a, b)$ is the wavelet coefficients.

The wavelet transform is a linear transformation and it decomposes a signal using the basis of wavelet function as a “window function.” By dilating the scale a and translating the b, the wavelet coefficients are localized in a time–frequency window. It is important to know that the wavelet has to satisfy the admissibility condition, which is

\int_{- \infty}^{\infty} \frac{{| {\hat{ψ}}_{a, b} (ω) |}^{2}}{ω} d ω < \infty

(12)

in which ${\hat{ψ}}_{a, b} (ω)$ is the Fourier transform of $ψ_{a, b} (t)$ .

Multi-resolution analysis

For a special selection of the mother wavelet function $ψ (t)$ and for the discrete set of parameters, $a_{j} = 2^{- j}$ and $b_{j, k} = 2^{- j} k$ , with $j, k \in Z$ (the set of integers), the family wavelet can be written as

ψ_{j, k} (t) = 2^{j / 2} ψ (2^{j} t - k) j, k \in Z

(13)

Equation (13) constitutes an orthonormal basis of the Hilbert space $L^{2} (R)$ consisting of finite-energy signals. The correlated decimated discrete wavelet transform provides a non-redundant representation of the signal and its values constitute the coefficients in a wavelet series. These wavelet coefficients provide full information in a simple way and a direct estimation of local energies at different scales. The information can be organized in a hierarchical scheme of nested subspaces and called as multi-resolution analysis in $L^{2} (R)$ .

The space $L^{2} (R)$ of measurable function R can be decomposed as a sequence of closed subspaces defined as follows

\dots, V_{0} = V_{1} \oplus W_{1}, V_{1} = V_{2} \oplus W_{2}, V_{j} = V_{j + 1} \oplus W_{j + 1}, \dots

(14)

where $\oplus is$ the direct sum of space, V_j is the scale space, and W_j is the wavelet space of discrete orthogonal wavelet transform.

For any square integrable function $x (t) \in L^{2} (R)$ , its multi-resolution analysis can be carried out by projecting the function onto the scale and the wavelet space of different scales. Assuming that the space $L^{2} (R)$ is decomposed into the jth scale as follows

L^{2} (R) = \sum_{j = - \infty}^{J} W_{j} \oplus V_{j}

(15)

The function $x (t)$ can be orthogonally expanded as

\begin{matrix} x (t) = x_{s}^{j} (t) + x_{d}^{j} (t) = \sum_{k = - \infty}^{\infty} c_{j, k} φ_{j, k} (t) \\ + \sum_{j = - \infty}^{\infty} \sum_{k = - \infty}^{\infty} d_{j, k} ψ_{j, k} (t) \end{matrix}

(16)

in which $x_{s}^{j} (t)$ is the profile signal projected to the scale space V_j, $x_{d}^{j} (t)$ is the detail signal projected to the wavelet space W_j, $c_{j, k}$ is the scale coefficient of wavelet transform under the jth scale, $φ_{j, k} (t)$ is the scale function of wavelet transform, $d_{j, k}$ is the wavelet coefficient under the jth scale, and $ψ_{j, k} (t)$ is the wavelet function.

Multi-resolution analysis can be calculated through filter banks. Assuming that $h_{0}$ and $h_{1}$ are the impulse responses of low-pass and high-pass filter, respectively, corresponding to wavelet decomposing, and $g_{0}$ and $g_{1}$ are the impulse response of low-pass and high-pass filter, respectively, corresponding to wavelet reconstructing, the scale coefficients and wavelet coefficients can be recursively calculated by Mallat’s pyramid algorithm as follows

{\begin{matrix} c_{j + 1, k} = \sum_{m} h_{0} (m - 2 k) c_{j, m} \\ d_{j + 1, k} = \sum_{m} h_{1} (m - 2 k) c_{j, m} \end{matrix}

(17)

where $k$ is the length of wavelet coefficients and it is related to the length of signal and the number of scale.

The corresponding reconstructed algorithm of coefficient is then expressed as

c_{j - 1, m} = \sum_{k} c_{j, k} g_{0} (m - 2 k) + \sum_{k} d_{j, k} g_{1} (m - 2 k)

(18)

When the sampling frequency of signal is higher than Nyquist frequency, the sample sequence $x (n)$ of signal $x (t)$ can be considered approximately as the scale coefficients $c_{0, k}$ decomposing on the zero scale. If the sequence $x (n)$ is decomposed to the scale $j = 1$ , the scale and wavelet coefficients are expressed as $c_{1, k}$ and $d_{1, k}$ , respectively. Therefore, the original signal can be reconstructed by Mallat’s pyramid algorithm as follows

x (n) = \sum_{k} c_{1, k} g_{0}^{1} (n - 2 k) + \sum_{k} d_{1, k} g_{1}^{1} (n - 2 k)

(19)

The sequence $x (n)$ can be decomposed continuously to the higher scales, and the original signal can then be reconstructed using equations (18) and (19).

Wavelet ridges

For a mono-component signal $x (t) = A (t) \cos (φ (t))$ , it satisfies the condition of asymptotic signal, that is, the amplitude A(t) is a slower time-varying function than that of phase angle φ(t). The wavelet transform of response signal can be approximated by the first term of an asymptotic expansion using the stationary phase argument

\begin{array}{l} W T_{x} (a, b) = < x, ψ_{a, b} > = \frac{1}{2} < Z, ψ_{a, b} > \\ \approx \frac{1}{2} A (b) e^{i φ (b)} Ψ^{*} (a φ^{'} (b)) + o (| A^{'} | / | A |, | φ φ^{″} | / {| φ^{'} |}^{2}) \end{array}

(20)

where $o (\cdot)$ is the second-order small quantity, and $φ (b)$ is the phase of wavelet transform. When the scale a satisfies the following condition

a = a^{r} (b) = \frac{ω_{0}}{φ' (b)} = \frac{2 π F_{c}}{φ' (b)}

(21)

The modulus of wavelet coefficients has the local maximum value. The corresponding point (a, b) is named as wavelet ridge point. The curve (b, a^r(b)) chaining all the ridge points on the time-scale plane is called wavelet ridge. Figure 1 shows a wavelet ridge.

Figure 1.

Wavelet ridge.

According to the relationship between the scale and frequency, the translation parameter b corresponds to time, and the instantaneous frequency of the measured signal can be estimated from wavelet ridge by

f (t) = f (b) = \frac{φ' (b)}{2 π} = \frac{F_{c}}{a^{r} (b)}

(22)

As one can find from equation (22), the wavelet ridge represents the frequency or phase information of the measured signal, which can be further used for structural parameter identification.

Wavelet energy and entropy

With the multi-resolution analysis, Mallat (1989) developed the wavelet packet analysis. The wavelet packet analysis provides a complete level-by-level decomposition of signal as

x (t) = \sum_{i = 1}^{2^{j}} x_{j}^{i} (t)

(23)

in which the decomposed signal $x_{j}^{i} (t)$ can be calculated as

x_{j}^{i} (t) = \sum_{k = - \infty}^{\infty} c_{j, k}^{i} (t) ψ_{j, k}^{i} (t)

(24)

The wavelet packet coefficient can be calculated as

c_{j, k}^{i} (t) = \int_{- \infty}^{\infty} x (t) ψ_{j, k}^{i} (t) dt

(25)

A wavelet packet $ψ_{j, k}^{i} (t)$ is a function with three indices where integers i, j, and k are the modulation, scale, and translation parameters, respectively. Then, combining the wavelet energy of the ith component with the jth scale, $E_{j}^{i}$ can be defined from the wavelet packet coefficient

E_{j}^{i} = \sum_{k = 1}^{2^{- j} T} {| c_{j, k}^{i} (t) |}^{2}

(26)

The total energy $E_{j}$ of the original signal is equal to

E_{j} = \sum_{i} E_{j}^{i}

(27)

The wavelet entropy $S_{WT}$ can be further defined as the ratio of the component energy to the total energy, which is

S_{WT} = - \sum_{i} \frac{E_{j}^{i}}{E_{j}} \ln (\frac{E_{j}^{i}}{E_{j}})

(28)

To some extent, the wavelet energy and entropy can represent the amplitude of the wavelet transform. Therefore, it can provide useful information about the underlying measured signals, especially the amplitude information of the measured signal.

Wavelet transform applications

Instantaneous frequency identification of time-varying structures

As presented in section “Wavelet ridges,” the critical issue to estimate the instantaneous frequency of the measured signal is to extract the wavelet ridge. However, one should note that the instantaneous frequency of the measured signal may not be equal to the time-varying frequency of the structure as presented in section “Time-varying/nonlinear structural system.” When repeated impacts or white noise excitation are applied, frequency values of the measured signal are not distorted and approximately equal to the time-varying frequency of the structures in theory presented in section “Time-varying/nonlinear structural system.” However, this is not the case with chirp excitation or harmonic excitation. Chirp excitation has very high amplitude attached to a single frequency. Obviously, for this case, the instantaneous frequency of the measured response will be equal to the frequency of the external excitation.

To obtain the fine wavelet ridges, various wavelet ridges extraction methods are further developed. Such as the so-called “crazy climbers algorithm” based on Markov chain Monte Carlo simulations method (Carmona et al., 1997, 1998) and a simply extracting method based on the modulus of wavelet coefficients and the optimization method (Wang et al., 2013c). In their study, a time-varying cable structure was designed to test. The property of time-varying of structure was varied by applying varying tension force to change the stiffness of cable during vibration. One end of the cable was fixed using anchor and the other end was fixed to the MTS loading system. An acceleration sensor was installed vertically in mid-span of the cable to measure the acceleration. Two scenarios with the tension force changed with linear and sinusoidal functions were considered. The wavelet transform of the measured acceleration of the two cases is presented in Figure 2. The final identified instantaneous frequency of the cable compared the theoretical values and is presented in Figure 3. As presented in Figures 2 and 3, the wavelet ridges are blurred line in Figure 2; however, the fine wavelet ridges can be extracted using their method.

Figure 2.

Wavelet transform of the measured acceleration. Tension force varies (a) linearly and (b) sinusoidally.

Figure 3.

Identified instantaneous frequency of the cable with wavelet transform. Tension force varies (a) linearly and (b) sinusoidally.

Physical parameter identification of time-varying structures

As presented in section “Time-varying/nonlinear structural system,” the wavelet ridge can only be used to identify the instantaneous frequency of the time-varying structures under impulsive or ambient vibration. It cannot identify the damping ratios and other parameters, since the wavelet ridge only represents the phase information of the measured signal; in order to estimate the physical parameters of time-varying structures, the amplitude information is also required.

For parameter identification of time-varying structural systems, Wang et al. (2014a) and Dziedziech et al. (2015) expanded the time-varying parameter onto a finite set of wavelet basis sequences and transformed the time-varying problem into a time-invariant model. In their study, the time-varying physical parameters were similarly expanded at multi-scale as profile and detail signal using multi-resolution analysis, and then, the physical parameters were identified by solving linear equations using least-squares method. The detail algorithm is presented as follows.

On the sake for convenience, an SDOF system with time-varying stiffness and damping coefficient is considered. The discrete equation can be expressed as

m \overset{\cdot\cdot}{x} (n) + c (n) \overset{\cdot}{x} (n) + k (n) x (n) = f (n)

(29)

Since the stiffness at different time point constitutes a discrete serial signal, it can be decomposed at different scales using multi-resolution analysis.

If the time-varying damping and stiffness are slow time-varying functions, the energy of the slow time-varying signal mainly concentrates on the low-frequency component. Therefore, for a slow time-varying signal, through multi-resolution analysis, the original signal can be approximately reconstructed only using the scale coefficients. Then, the original stiffness can be approximately reconstructed as

k (n) \approx \sum_{i} k_{j, i} g_{0}^{j} (n - 2^{j} i)

(30)

where $k_{j, i}$ is the decomposed scale coefficient, and $g_{0}^{j} (n - 2^{j} i)$ is the impulse response of low-pass filter corresponding to wavelet reconstructing.

The same algorithm can be used to process the damping coefficient, and the result is expressed as

c (n) \approx \sum_{i} c_{j, i} g_{0}^{j} (n - 2^{j} i)

(31)

Substituting the above two equations into equation (29), one can obtain the transformed motion equation as follows

\begin{array}{l} \sum_{i} g_{0}^{j} (n - 2^{j} i) \dot{x} (n) c_{j, i} + \sum_{i} g_{0}^{j} (n - 2^{j} i) \\ x (n) k_{j, i} x (n) = f (n) - m \overset{\cdot\cdot}{x} (n) \end{array}

(32)

Setting the input load and structure response in discrete time point $n = 1 ~ N$ and substituting the response into equation (32), one can solve equation (32) and obtain the scale coefficients of time-varying stiffness and damping coefficient. Then, the time-varying stiffness and damping coefficient can be reconstructed.

For multi-DOF system, similar algorithm can be derived as presented in Wang et al. (2014a). To illustrate the above method, a single-degree system with time-varying damping coefficient and stiffness is simulated as example. The constant mass is 2800 kg, the damping coefficient $c (t) = 2.5 + 5 t$ , and the stiffness $k (t) = 250 - 2.5 t$ . The external excitation is selected as 1940 EL-Centro earthquake time history. The simulated displacement, velocity, and acceleration combined with noise-to-signal ratio of 5% white Gaussian noise are assumed as measured signals. A Tikhonov regularization method is used to solve equation (32). The wavelet scale j is selected as four, and the wavelet function is selected as db3 wavelet function. The identified stiffness and damping coefficient are presented in Figure 4. As presented in Figure 4, the time-varying stiffness and damping coefficient were effectively identified using their method.

Figure 4.

Identified physical parameters of a single-degree-of-freedom system: (a) stiffness and (b) damping coefficient.

More recently, to enhance the time and frequency resolution of the wavelet transform, some advanced wavelet analysis methods were developed. For instance, Daubechies et al. (2011) proposed a wavelet-based time-frequency reallocation method called synchrosqueezed wavelet transform to extract the instantaneous frequency and amplitude of the signal. Thakur and Wu (2011) developed the synchrosqueezed wavelet transform for non-uniform sampling data. Li and Liang (2012a, 2012b) further proposed a generalized synchrosqueezing transform to resolve the bi-dimensional smear problem and successfully applied into gearbox fault diagnosis. With the development of the wavelet transform, the synchrosqueezed wavelet transform offers better adaptability and an exact reconstruction formula for constituent components, and indeed the wavelet ridges can be extracted with relatively high resolution. To illustrate the synchrosqueezed wavelet transform, a signal defined as $x (t) = \cos (t + t^{2} + \sin (t)) + \cos (8 t)$ is considered. A white Gaussian noise with 20% noise-to-signal ratio is added. The original signal, its wavelet transform, and the extract instantaneous frequency are presented in Figure 5(a) to (c), respectively. It can be seen from Figure 5 that the synchrosqueezed wavelet transform indeed increases the resolution of the instantaneous frequency extraction.

Figure 5.

Original signal, wavelet transform, and the extract instantaneous frequency with synchrosqueezed wavelet transform: (a) original signal, (b) wavelet transform, and (c) extract instantaneous frequency with synchrosqueezed wavelet transform.

Structural damage detection

Wavelet-based methods have also been widely used for structural damage detection. In general, there are two categories for structural damage detection based on wavelet transform. The first type of method is to analyze the identified modal curvature of time-deflections or accelerations due to moving loads (Hester and González, 2012; Hou et al., 2000; Liew and Wang, 1998; Lu and Hsu, 2002; Zhu and Law, 2006). The damage in a structure leads to localized singularities in the dynamic responses. Therefore, the wavelet transform of the response can be used to detect the singularities which represent the damage in the structure. For a damaged beam structure as presented in Figure 6, it can be found that the coefficient of the wavelet transform for the modal curvature has a peak value at the damage location. With this singularity property, the damage location can be detected. However, many environmental effects such as surface roughness can also cause significant singularity values, which may lead to false identification. This is also the reason why these kinds of methods are mainly verified by numerical simulation, and it is a challenge to apply into real structural applications.

Figure 6.

Damaged beam structure and the wavelet coefficient of the modal curvature.

The second type of damage detection method is based on the change of the structural dynamic characteristics. For instances, Ovanesova and Suárez (2004) applied wavelet transform to damage detection in frame structures based on the change of the structural dynamic properties. To identify the structural damage, the wavelet energy of the structural element is commonly used as a damage index (Han et al., 2005; Ren and Sun, 2008; Sun and Chang, 2003). In the previous studies, the results show that the component wavelet energy is more sensitive to the original signal energy, and the wavelet packet energy is even more sensitive than those of wavelet energy. Therefore, in their study, the wavelet packet energy indices were proposed to indicate the structural damage.

In the previous studies, the rate of signal wavelet packet energy $Δ (E_{j})$ at the j level is defined as

Δ (E_{j}) = \sum_{i = 1}^{2^{j}} \frac{| {(E_{j}^{i})}_{d} - {(E_{j}^{i})}_{u} |}{{(E_{j}^{i})}_{u}}

(33)

where $(E_{j}^{i})_{u}$ is the wavelet packet energy as defined in equation (26) at the j level without damage, and $(E_{j}^{i})_{d}$ is the wavelet packet energy with damage. It is postulated that structural damage would affect the wavelet packet energies and subsequently alter this damage indicator.

Damage to a structure also results in the change of wavelet entropy. The damage identification problem can then be formulated as follows: given the changes in the wavelet entropy before and after the damage, to characterize the change in the wavelet entropy, the relative wavelet entropy can then be estimated.

Based on equations (26) to (28), one can define the wavelet energy ratio as $p_{j} = \frac{E_{j}^{i}}{E_{j}}$ . Suppose that there are two set wavelet energy ratio vectors $p_{j}$ and $q_{j}$ , which are the wavelet energy ratio for two segments of a signal or of two different signals. Then, the relative wavelet entropy can be defined as

S_{WT} (p | q) = - \sum_{j} p_{j} \ln (\frac{p_{j}}{q_{j}})

(34)

Equation (34) gives a measure of the degree of similarity between two probability distributions. This index can be used for structural damage detection. For the case of damage detection, two set signals might be the dynamic responses measured from the target structure before and after damage. For the case of structural heath monitoring, two set signals might be two segment dynamic response signals continuously collected from the structural heath monitoring system. When a structure is in a good condition, the relative wavelet energy distributions are almost the same so that the relative wavelet entropy is close to zero. If the structure is damaged, the relative wavelet energy distributions of structural responses before and after damage will be changed so that the value of relative wavelet entropy will not be zero. In such a way, the damage occurrence can be identified.

To verify the proposed damage detection method, a beam structure with shear connector loosening in the slab-on-girder is tested. A 1:3 scaled model was designed and constructed in the laboratory to represent Bridge No. 852 in the Pilbara region of Western Australia by Hao’s research group in the University of Western Australia (Ren et al., 2008). For simplicity, the model was reduced to a single span and three girders. Figure 7 shows the plan of the model and details of the shear connectors and accelerometers. As there are 81 shear connectors in each beam of the prototype, clusters of connectors are combined together as one, resulting in nine connectors spaced at 600-mm intervals along each girder in the scaled model. Shear connector fixity is provided by securing both ends of the shear connector thread. The top end is secured by a T-nut. Damage is simulated by loosening the specified shear connectors. When the bar is tightened into the T-nut in the slab, the beam and the slab are fully connected with the shear connector. On the contrary, when the bar is removed, there is no connection between the girder and the slab, which simulates the complete damage of specified shear connectors.

Figure 7.

Tested bridge model. Layout of (a) shear connectors and (b) accelerometers.

For the damage scenario, shear connectors at S4, S5, S8, and S9 are loosen, and the rate of signal wavelet packet energy and the relative wavelet entropy were calculated based on the measured accelerations. In their study, the wavelet packet energy ratio and the relative wavelet entropy were used for structural damage detection knowing the responses of the undamaged structure. The results are presented in Figure 8. As presented in Figure 8, the two damaged elements are detected based on the wavelet packet energy method.

Figure 8.

Damage index: (a) wavelet packet energy ratio and (b) relative wavelet entropy.

As the references reviewed in this section, one can find that, although widely used in dynamic signal processing, wavelet analysis is non-adaptive to a particular data series. It depends on the introduction of a predetermined parent wavelet. Therefore, some of the highly transient features in a time signal may be lost in the process of wavelet transform. In addition, for structural parameter identification, most of researches are focused on the time-invariant modal parameters, such as natural frequencies, modal shapes, and damping ratios. Only a few literatures tried to estimate the time-varying structural parameters. However, the identified instantaneous frequencies of the measured signal are mainly not equally to the instantaneous natural frequencies of the structure. How to establish the relationship between the extracted time-varying features of the measured signal and the structural physical parameters is still a quite challenge issue. For structural damage detection, most of the researches used the wavelet transform of the measured signal, such as the changes of the wavelet packet energy. Currently, the wavelet-based methods for structural damage detection are mainly focused on the damage location and the structure is mainly considered as linear structure. It is still a challenge issue for structural damage level detection especially for a time-varying or nonlinear structure using wavelet transform.

Hilbert transform and applications in civil engineering structures

Hilbert transform–based method is another time–frequency analysis method developed for nonstationary signal analysis and has been widely applied in parameter identification and damage detection for civil engineering structures. In the Hilbert transform, the mono-components need to be decomposed from the measured signal using suitable signal decomposition processing. To decompose a nonstationary data series into a finite number of intrinsic mode functions (IMFs), empirical mode decomposition (EMD) was developed by Huang et al. (1998, 1999, 2003). The well-known Hilbert–Huang transform (HHT) combines EMD with Hilbert spectral analysis. Some HHT applications in engineering and other areas have been presented by Huang and Shen (2005) and Huang and Attoh-Okine (2005). More research about the EMD applications in signal processing can be found in Chen and Feng (2003), Yang et al. (2003a, 2003b), Yang et al. (2004), Peng et al. (2005), Shi and Law (2007), Shi et al. (2009), and Zheng et al. (2009). To solve the modal-mixing problem, Wu and Huang (2009) further developed ensemble EMD to decompose the signal.

To separate the signal for well-behaved Hilbert transform, Feldman (2006, 2008, 2011) introduced a new signal decomposition formulation, called Hilbert vibration decomposition (HVD), for nonstationary signals. The comparison study of EMD and HVD can be found in Braun and Feldman (2011). The challenges with EMD are mostly associated with the empirical nature of its decomposition process. More recently, a new signal decomposition method named as analytical mode decomposition (AMD) was developed by Chen and Wang (2012). In essence, AMD can accurately decompose a time series into two components whose Fourier spectra are non-vanishing over two mutually exclusive frequency ranges separated by a constant bisecting frequency. Wang and Chen (2013) and Wang et al. (2013b) further extended AMD to extract the vibration features of nonstationary signals.

Once the signal is decomposed into mono-components, the modal parameters can be further estimated directly from the free vibration tests or combined random decrement technique from ambient vibration tests (Wang and Chen, 2014; Wang et al., 2014b; Yang et al., 2004). For time-varying structures, knowing the mass of the structure, the time-varying natural frequency and damping coefficient can be estimated combining the equation of motion and their Hilbert transform. The essence of these methods is that a complementary imaginary part to a given real signal part is provided by the Hilbert transform, by shifting each component of the signal by a quarter of a period, and then, the Hilbert transform pair provides a method for determining the instantaneous parameters of the structural time-varying parameters at each time instance. Previous studies usually considered that the decomposed mono-components are the modal responses, and the extract instantaneous frequency of the signal is equal to the instantaneous frequency of the structure. However, the relationship between the identified signal frequencies and modes and structural time-varying parameters for nonlinear structures are still ambiguous.

During the past decade, Hilbert transform method has also been applied to structural damage detection (Cheraghi and Taheri, 2007; Pines and Salvino, 2006; Rezaei and Taheri, 2009, 2010, 2011; Wang et al., 2015a; Yan and Gao, 2006). Pines and Salvino (2006) used the instantaneous phase data obtained from the decomposed mono-component for damage detection of a three-story building. Cheraghi and Taheri (2007) further proposed a damage index called “EMD energy damage index” for structural damage detection and verified its applicability through numerical and experimental studies (Rezaei and Taheri, 2009, 2010). Wang et al. (2015a) defined a degree of nonlinearity index, which represents the damage severity of structure, based on the integrated instantaneous frequency. The degree of nonlinearity indices are obtained based on the hysteretic structure subjected to low, medium, and larger intensity earthquakes.

Hilbert transform

In this section, the Hilbert transform, Hilbert spectral analysis, signal decomposition, and the Hilbert transform applications in structural systems are reviewed in detail.

Hilbert transform and analytical signal

Hilbert transform is one of the integral transforms such as Laplace and Fourier transforms. The Hilbert transform of a function $x (t)$ is defined by an integral transform (Hahn, 1996)

H [x (t)] = \frac{1}{π} P \int_{- \infty}^{\infty} \frac{x (τ)}{t - τ} d τ

(35)

in which P indicates the Cauchy principal value around $t = τ$ . Physically, Hilbert transform is equivalent to a special kind of linear filter, where all the amplitudes of spectral components remain unchanged, but their phases are shifted by $π / 2$ . Mathematically, the Hilbert transform $H [x (t)]$ of the original function represents a convolution of $x (t)$ and $\frac{1}{π t}$ . A complex analytic signal $z (t)$ is defined as

z (t) = x (t) + jH [x (t)]

(36)

in which $H [\cdot]$ represents the Hilbert transform of the function in the bracket and $j = \sqrt{- 1}$ is an imaginary unit.

As defined in equation (36), an analytic signal is the complex signal whose imaginary part is the Hilbert transform of the real part. The analytic signal can be viewed as a vector at the origin of the complex plane having a length $a (t)$ and an angle $(t)$ . The projection on the real axis is the original signal $x (t)$ and the projection on the imaginary axis is the Hilbert transform of the original signal. The traditional representation of the analytic signal in its exponential form can be written as (Hartmann, 2004)

z (t) = a (t) e^{j (t)}

(37)

a (t) = \sqrt{x^{2} (t) + {H [x (t)]}^{2}} and θ (t) = \arctan (\frac{H [x (t)]}{x (t)})

(38)

Here, $a (t)$ is the instantaneous amplitude and $θ (t)$ is the phase function. The instantaneous frequency is simply defined as

ω (t) = \frac{d θ}{dt}

(39)

It can also be expressed into

ω (t) = \frac{x (t) \overset{\cdot}{H} [x (t)] - \overset{\cdot}{x} (t) H [x (t)]}{a^{2} (t)}

(40)

For any signal, there is a unique value of the instantaneous phase at any given time. For a nonstationary signal whose spectral contents vary with time, the instantaneous frequency plays an important role in the understanding of signal characteristics. The instantaneous amplitude, phase angle, and frequency can be expressed by a phasor rotating in the complex plane. A phasor can be viewed as a vector at the original of the complex plane having a length $a (t)$ and a phase angle $θ (t)$ with an angle speed of $ω (t)$ . The projections on the real and imaginary axle are the original signal and the Hilbert transform of the original signal, respectively.

Hilbert spectral analysis

To have a physical meaning for instantaneous frequency, Cohen (1995) presented the Hilbert transform of a mono-component function. In this case, the instantaneous characteristics agree with the intuitive meaning of the signal amplitude, phase, and frequency.

The original signal can be expressed as the real part $Re [\cdot]$ of an analytic signal

x (t) = Re [a (t) e^{j \int_{0}^{t} ω (τ) d τ}]

(41)

The Hilbert spectrum $HS (ω, t)$ of the signal x(t) is then defined by

HS (ω, t) = {\begin{matrix} a (t) ω = ω (t) \\ 0 ω \neq ω (t) \end{matrix}

(42)

The Fourier spectrum and the Hilbert spectrum of a sine frequency modulated signal defined as $x (t) = \cos [6 π t + \sin (2 π t)]$ are presented in Figure 9.

Figure 9.

Fourier and Hilbert Spectra.

Signal decomposition

The instantaneous frequency and amplitude defined in section “Hilbert transform and analytical signal” are applicable to narrowband signals or mono-component functions. For a signal with multiple components, the signal has to be decomposed into series of mono-component signals. Huang et al. (1998) first introduced the EMD process and HHT. After that, the HHT applications in many research areas have received wide attentions.

A general signal $x (t)$ can be decomposed into a summation of IMFs: $c_{i} (t) (i = 1, 2, \dots, l)$ and a residual signal $r (t)$ after EMD process

x (t) = \sum_{i = 1}^{l} c_{i} (t) + r_{l} (t)

(43)

The EMD algorithm requires the following procedures at every iteration step as presented in Figure 10: (1) estimation of all local extrema; (2) spline fitting of all local minima and maxima, ending up with two (top and bottom) extrema functions; (3) computation of the average function between maxima and minima; (4) extraction of the average from the initial signal; and (5) iteration on the residual (the sifting procedure).

Figure 10.

Block diagram of EMD.

Recently, Feldman (2006, 2008, 2011) introduced a HVD signal decomposition formulation. It includes three steps: (1) to estimate the instantaneous frequency of the largest energy vibration component, (2) to extract the envelope of the largest energy vibration component, and (3) to subtract the largest energy component from the original signal and repeat the whole process for the remaining signal. The key idea behind HVD is that the instantaneous frequency of an analytic signal is dominated by the largest energy component. Therefore, the instantaneous frequency can be extracted using a low-pass filter. Once the instantaneous frequency is detected, the so-called coherent demodulation or phase lock-in amplifier detection technique is used to extract the envelope of the largest energy vibration component. The HVD process can be represented by a block diagram as illustrated in Figure 11.

Figure 11.

Block diagram of HVD.

More recently, Chen and Wang (2012) developed an AMD signal decomposition theorem. Wang and Chen (2013) further extended the AMD to the time-varying vibration signal processing. Instead of selecting constant cutoff frequencies, time-varying cutoff frequencies are selected. The core of the proposed theorem is that it can analytically extract the low-pass component $s (t)$ of the original signal by selecting a time-varying cutoff frequency $ω_{c} (t)$ . The low-pass component $s (t)$ can be analytically calculated as

\begin{array}{l} s (t) = \sin (\int_{- \infty}^{t} ω_{c} (τ) d τ) H [x (t) \cos (\int_{- \infty}^{t} ω_{c} (τ) d τ)] \\ - \cos (\int_{- \infty}^{t} ω_{c} (τ) d τ) H [x (t) \sin (\int_{- \infty}^{t} ω_{c} (τ) d τ)] \end{array}

(44)

Equation (44) operates like a low-pass filter that passes any low-frequency signal $s (t)$ . The block diagram of such a low-pass filter with time-varying cutoff frequency is shown in Figure 12. To eliminate the effects of the signal discretization, the one-step, two-step, and four-step low-pass filters with time-varying cutoff frequencies were also developed by Wang et al. (2017).

Figure 12.

Block diagram of the AMD.

Hilbert transform applications

Instantaneous frequency identification of time-varying structures

As presented in section “Time-varying/nonlinear structural system,” one should note that the instantaneous frequency of the measured signal is approximately equal to the time-varying frequency of the structure only when repeated impacts or white noise excitation are applied.

For an n DOF slow time-varying or weak nonlinear system, the equation of motion can be described as equation (7) in section “Time-varying/nonlinear structural system.” Equation (7) can be transformed into modal spatial coordinate, which is presented as equation (8). In equation (8), since the external load is assumed zero mean white Gaussian noise, the instantaneous frequency of the modal response $ω_{i}^{2} (t)$ can be expressed as

ω_{i}^{2} (t) = ω_{0 i}^{2} (t) - \frac{\frac{Æ_{i}^{T} f (t)}{M_{i}} q + H [\frac{Æ_{i}^{T} f (t)}{M_{i}}] H [q]}{[q^{2} + {(H [q])}^{2}]}

(45)

For ambient vibration with zero mean excitation $f (t)$ , the second term of equation (45) is the zero mean fast time-varying function. The system’s natural frequency can be obtained from the instantaneous frequency of the analytical signal by filtering out the fast time-varying part. Therefore, the low-pass instantaneous frequency of the decomposed response is equal to the instantaneous frequency of a slowly time-varying structural system. In this study, the instantaneous frequency of a time-varying cable structure presented in section “Instantaneous frequency identification of time-varying structures” is identified and the results are presented in Figure 13. As shown in Figure 13, the slowly varying portion of the instantaneous frequency of the response is indeed approximately equal to the instantaneous frequency of the cable structure.

Figure 13.

Identified instantaneous frequency of the cable with Hilbert transform: (a) tension force varies linearly and (b) tension force varies sinusoidally.

Physical parameter identification of time-varying structures

For time-varying physical parameter identification, Shi and Law (2007), Shi et al. (2009), and Wang and Chen (2012) proposed an identification algorithm for linear, time-varying systems based on the Hilbert transform method.

For an n DOF slow time-varying system described as equation (9), the Hilbert transform of both sides of leads to

M H [\overset{\cdot\cdot}{x} (t)] + H [C (t) \overset{\cdot}{x} (t)] + H [K (t) x (t)] = H [f (t)]

(46)

When $C (t) and K (t)$ change slowly in comparison with the velocity and acceleration. Based on the Bedrosian’s theorem and introducing an analytic signal of $z (t) = x (t) + jH [x (t)]$ , equation (46) can be combined into

M (t) \overset{\cdot\cdot}{z} (t) + C (t) \overset{\cdot}{z} (t) + K (t) z (t) = g (t)

(47)

where $g (t) = f (t) + j H [f (t)]$ is the analytic signal of the external load vector.

The key of the identification algorithm is to decompose the response $x (t)$ into n IMFs by the EMD process

x (t) = \sum_{i = 1}^{n} x_{i} (t) + r (t)

(48)

The analytic signal $z (t) = \sum_{i = 1}^{n} z_{i} (t) = \sum_{i = 1}^{n} {x_{i} (t) + jH [x_{i} (t)]}$ can then be formulated, and equation (48) can be written into

M (t) \sum_{i = 1}^{n} {\overset{\cdot\cdot}{z}}_{i} (t) + C (t) \sum_{i = 1}^{n} {\overset{\cdot}{z}}_{i} (t) + K (t) \sum_{i = 1}^{n} z_{i} (t) = g (t)

(49)

By left multiplying $z_{i}^{T} (t)$ on both sides of equation (49) and using the orthogonal properties of IMF, the following equation can be obtained

\begin{array}{l} z_{i}^{T} (t) M (t) {\overset{\cdot\cdot}{z}}_{i} (t) + z_{i}^{T} (t) C (t) {\dot{z}}_{i} (t) \\ + z_{i}^{T} (t) K (t) z_{i} (t) = z_{i}^{T} (t) g (t) \end{array}

(50)

Equation (50) contains 2n equations for the evaluation of 2n time-varying parameters at any time instant t with a given mass matrix.

To illustrate the above algorithm, a single-story shear building with mass m = 1.75×105 kg and damping ratio ξ = 0.02 subjected to the 1940 El Centro ground acceleration is simulated. The stiffness of the building is suddenly reduced from k = 2.76 × 104 kN/m to 1.75 × 104 kN/m. at time instant t = 4 s. The natural frequency of the building from 0 to 4 s is 12.6 Hz and reduced to 10 Hz after t = 4 s. Given the mass and damping ratio of the building, the stiffness of the building is to be identified based on the displacement response. Figure 14 presents the identified stiffness as a function of time when the numerical displacement was used with noise-to-signal ratio of 5% Gaussian white noise.

Figure 14.

Identified stiffness of the single-story building with Hilbert transform.

Structural damage detection

Similar to the wavelet packet energy index, damage index derived from the energy of the decomposed signal was developed (Cheraghi and Taheri, 2007; Rezaei and Taheri, 2011). For the IMF extracted from the original signal, the following IMF energy index is defined as

E = \int_{0}^{t} {(IMF)}^{2} d τ

(51)

The damage index can be further defined based on the IMF energy of the health structure and the damaged structure, which is

DI = | \frac{E_{health} - E_{damage}}{E_{health}} | \times 100

(52)

Based on the above damage index, Rezaei and Taheri (2011) verified the effectiveness from numerical simulations and experimental investigations. Their results show that the above damage index could detect the damage location and quantify the damage level. However, it should be noted that the decomposition of the signal should be very carefully performed to avoid filtering out the useful frequency components. Since the damage index is related to the amplitude of the decomposed signal, any energy leakage issue during the signal decomposition procedure could cause failure of damage detection. However, since the damage index is related to the relative energy of the health and damaged structure, the data have to be available for the health structure. At the same time, the dynamic load applied to the damaged structure should be exactly the same as load acted on the health structure. Therefore, it is still a challenge issue to use the above damage index for damage detection in practice.

To evaluate the structural damage under extreme loads, such as earthquakes, hurricanes, and tornados, the time-varying properties such as instantaneous natural frequency can be used for structural damage characterization. Since the structural damage due to extreme loads is mainly due to the stiffness or boundary conditions vary rapidly or slowly over excitation duration, the corresponding characteristic properties of structures change over time. Therefore, damage indices based on the identified instantaneous frequency can be defined. Again, the instantaneous frequency of the decomposed response includes a slowly varying part and a rapidly varying part. To eliminate the rapidly varying part, the instantaneous frequency is integrated over time duration. Therefore, the phase of the decomposed response can describe the structural damage during the vibration. The damage index $D$ can be defined as

D = \sqrt{\frac{\int {[\int_{0}^{t} ω_{n} (τ) d τ - ω_{l} t]}^{2} dt}{\int {[\int_{0}^{t} ω_{n} (τ) d τ]}^{2} dt}}

(53)

in which $ω_{n} (τ)$ is the time-varying frequency of the modal response of structure performed nonlinear phenomena, while $ω_{l}$ is the constant natural frequency of the structure performed linear behavior. $ω_{l}$ can be simply identified during the oscillation with low excitation.

To illustrate the application of the above damage index, a full-scale seven-story-reinforced concrete residential building is considered. The shake table test was conducted by Panagiotou et al. (2011). The acceleration at the top story is measured and used for damage detection. Four earthquake excitations including a low-intensity earthquake EQ1, two medium intensity earthquakes EQ2 and EQ3, and a large intensity earthquake EQ4 are used as excitations. The identified instantaneous frequencies of the measured responses on the top floor with AMD theory are presented in Figure 15 (Wang et al., 2015a). From the observation, the building subjected to low-, medium-, and large-intensity earthquake excitations represents that it is minor, medium, and severely damaged. The damage indices for the building subjected to a low-intensity earthquake excitation, two medium-intensity earthquake excitations, and a large-intensity earthquake excitation are equal to 12.8%, 23.0%, 23.2%, and 39.5%, respectively.

Figure 15.

Instantaneous frequencies of the measured accelerations with various earthquake excitations.

In bridge engineering, Huang and Shen (2005) also detected the damage of bridge pier with Hilbert spectral analysis. The tested bridge is in southern Taiwan. The bridge is a two-lane pre-stressed concrete girder bridge with 12 spans each 30 m in length. The girders are simply supported between piers, and the bridge deck is continuous with 15 cm reinforced concrete over three spans. One of the tested piers is damage with significant scouring. Two accelerometers are installed on the caps of the two piers, which are used to measure the accelerations due to a moving vehicle load. The Hilbert spectra of the accelerations can be used to detect the slight decrease in frequency for the damaged pier compared to the undamaged pier.

Although some damage indices were developed based on the Hilbert transform of the measured dynamic signals, it is still a quite challenging issue for damage detection based on the measured dynamic responses. One of the inherent challenges is that the dynamic response is sensitive to the global structural properties and environmental conditions, rather than the local damage. In addition, the large civil structures are very complicated compared to small-scale structures, and the damage characteristics and behavior are still not fully understood, such as close–open crack behaviors and damping characteristics.

Hilbert transform applications to nonlinear structures

Complex nonlinear behavior of structures has been observed not only when they are subjected to extreme loads but also during the operational conditions. Characterization of the nonlinearity can provide critical diagnostic and prognostic information. Stiffness and damping force nonlinearities can introduce dynamic phenomena and behavior that are dramatically different from those predicted by the linear theory. At present, the structures can be analyzed by use of tools with assumptions of linear and stationary structural behavior. Therefore, the nonlinear response of a structural system was often overlooked and valuable information was lost. In addition, determination of the dynamic characteristics for a structure exhibiting nonlinear behavior by assuming linearity may lead to misleading results. Thus, it is critical to know whether a structure is behaving nonlinearly and to detect and estimate the impact of the nonlinearity both qualitatively and quantitatively.

Since one of the first developments of Hilbert transform applications in vibration systems is to detect the nonlinearity, and the applications of Hilbert transform for nonlinear structures have some unique features, in this section, Hilbert transform applications in nonlinear structure are reviewed.

Nonlinearity detection

To deal with the nonlinear problems in structural dynamics, the first issue is to detect or identify the structural nonlinearity. Since the nonlinearities in structures lead to the distortion of the spectra in frequency domain, the spectral distortion indicates the existence of nonlinearity. As presented in Worden and Tomlinson (2001), a nonlinear response can be expressed in terms of Voltera series expansion, higher order frequency response functions (FRFs), and higher order spectra; therefore, one can inspect the Hilbert transform of FRF to examine its abnormal behavior and detect the structural nonlinearity.

In general, the FRF matrix $G (ω)$ can be defined as

Y (ω) = G (ω) X (ω)

(54)

in which $X (ω)$ and $Y (ω)$ are Fourier transform of the excitation $X (t)$ and response $Y (t)$ , respectively. Since $G (ω)$ is a complex value, for convenience, in complex analysis, the defined Hilbert transform is a little bit different with equation (35). The imaginary unit is multiplied on the right side, which is

H [G (ω)] = - \frac{1}{i π} P \int_{- \infty}^{\infty} \frac{G (ω)}{Ω - ω} d Ω

(55)

For a linear structure, the real part of FRF $Re G (ω)$ and the imaginary part $IM G (ω)$ have the following relations: $Re G (ω) = \frac{1}{π} \int_{- \infty}^{\infty} \frac{IMG (ω)}{Ω - ω} d Ω$ and $Im G (ω) = \frac{i}{π} \int_{- \infty}^{\infty} \frac{Re G (ω)}{Ω - ω} d Ω$ , respectively. Thus, the Hilbert transform of the FRF can be solved as

H [G (ω)] = - \frac{1}{i π} P \int_{- \infty}^{\infty} \frac{G (ω)}{Ω - ω} d Ω = G (ω)

(56)

Equation (56) reveals that the FRF is equal to its Hilbert transform for linear structures. However, for nonlinear structures, the FRF deviates from its Hilbert transform. Based on this property, Chanpheng et al. (2012) presented a nonlinear index (NI) for nonlinearity detection of large civil structures

NI = \frac{\int | H [G (ω)] - G (ω) | d ω}{\int | G (ω) | d ω}

(57)

The defined NI can measure the nonlinearity in an average sense and depends on the magnitude of the ground motion. The index is verified numerically and applied in a real cable-stayed bridge subjected to various earthquake excitations (Chanpheng et al., 2012). The numerical simulations in their studies showed encouraging results for nonlinearity quantification. For the cable-stayed bridge, the NI index provides a unique signature, which is demonstrated as a good feature for structural nonlinearity detection.

Based on the FRF, the Hilbert transform describer (HTD) is also developed for nonlinearity detection (Worden and Tomlinson, 2001). Considering a signal $x (t)$ , if it is n-times differentiable, it follows

\frac{d^{n} x (t)}{d t^{n}} = \frac{i^{n}}{2 π} \int_{- \infty}^{\infty} d ω ω^{n} e^{i ω t} X (ω)

(58)

{\frac{d^{n} x (t)}{d t^{n}} |}_{t = 0} = \frac{i^{n}}{2 π} \int_{- \infty}^{\infty} d ω ω^{n} X (ω) = \frac{i^{n}}{2 π} M^{(n)}

(59)

in which $M^{(n)}$ denotes the nth spectral moment of $X (ω)$ . Replacing $X (ω)$ with FRF $G (ω)$ , the HTD is defined from the spectral moment, which is

HT D^{(n)} = 100 \frac{M_{\tilde{G}}^{(n)} - M_{G}^{(n)}}{M_{G}^{(n)}}

(60)

in which $M_{G}^{(n)} = \int_{- \infty}^{\infty} d ω ω^{n} G (ω) and M_{\tilde{G}}^{(n)} = \int_{- \infty}^{\infty} d ω ω^{n} H [G (ω)]$ .

The HTDs can separate stiffness and damping nonlinearities very effectively. For the discontinuous nonlinearities, clearance, and friction, the HTDs tend to zero at high forcing as the behavior near the discontinuities becomes less significant. The describers therefore indicate the level of forcing at which the FRF of the underlying linear system can be extracted. Therefore, the HTDs can be used to detect the types of the nonlinearity.

Nonlinear system identification

Nonlinear system identification is a dynamic inverse problem to construct the unknown functions of nonlinear restoring and damping forces. The restoring force is basically the nonlinear functions of the displacement and velocity. The nonlinear system identification determines the nonlinear restoring elastic and damping force using the responses only for free vibration and the excitation and responses for forced vibration. The nonparametric identification of the nonlinear system with Hilbert transform is basically based on the developed FREEVIB and FORCEVIB proposed by Feldman (1994a, 1994b). Both methods can obtain the stiffness and damping characteristics of the SDOF system, and identify the instantaneous modal parameters. The procedure of the methods includes the following sequence: (1) taking the Hilbert transform of the measured response and calculating the envelope and the instantaneous frequency; (2) identifying the instantaneous modal parameters; (3) filtering of the modal parameters, and scaling the smooth modal parameters; (4) drawing the backbones of the frequency, damping curves, and the force static characteristics. For the multi-DOF system, the measured response can be decomposed into many mono-components so that the FREEVIB or FORCEVIB can be applied for nonlinear system identification.

The key of the FREEVIN or FORCEVIB method is to transform the nonlinear restoring force as a function of time into a multiplication form with a new fast time-varying natural frequency and a system displacement solution, and to transform the nonlinear damping force into a function of time as a multiplication between the fast time-varying instantaneous damping coefficient and the velocity. One can further draw the backbone diagram, which is drawing of the instantaneous frequency and damping coefficient as function of the amplitude envelope of the response. Knowing the instantaneous frequency and damping coefficients, one can obtain the time-varying stiffness and damping. Based on the above concepts, a variety of methods were developed for nonparametric nonlinear system identification. For instance, Yang et al. (2003a) and Poon and Chang (2007) proposed the HHT method for complex mode identification for nonlinear systems. Pan et al. (2008) used EMD and Hilbert transform for nonlinear system identification.

Nonlinear structural modeling

In the past decade, many researchers believed that the nonparametric identification of nonlinear structure based on the signal features is sufficient, since the nonlinear characteristics are underlying the features of the signal. For the nonparametric system identification, the restoring and damping forces are considered as an a-priori unknown nonlinear function. Such methods do not require an a-priori model.

For civil engineering structures, to analyze or predict the nonlinear structural responses due to dynamic loads, a suitable model should be constructed based on the design drawings or monitored data. The finite element model is one of the popular models to be created to achieve the realistic model. However, the finite element model has too many elements. It is extremely difficult to identify all the parameters of the elements, including nonlinear parameters, in practice. However, the true fact is that the nonlinear behaviors of civil structures are mainly caused by local nonlinearities (such as gaps, friction, and joints) during the disaster dynamic loads. In addition, with the theoretical or numerical analysis of the civil structures, researchers should have a priori knowledge of the structure. Therefore, it may not be necessary to create all elements as nonlinear elements. One can calibrate the critical local parts of the structure as nonlinear model, and other elements as linear models. Then, the calibrated model based on the test data can be further used to predict the nonlinear responses due to the other severe dynamic loads.

When a nonlinear model of a structure is assumed, the identification can be turned into a parametric identification problem. The unknown parameters can be derived by minimizing the residuals between the theoretical and measured values. One also can calibrate the nonlinear structural model based on the measured responses. The purpose to calibrate the nonlinear structural model is to provide test data for validating the strategies implemented for test-analysis correlation and inverse problem solving of nonlinear structures. The test-analysis correlation and inverse problem solving for nonlinear structures must address the following issues: (1) How to select a suitable theoretical nonlinear model for different structures? (2) How to characterize the variability of an experiment? (3) How to generate additional or surrogate data sets that can increase the knowledge about the experiment? (4) How to select features that best characterize a nonlinear data set?

With the development of Hilbert transform analysis method, the nonlinear calibrated model has started to attract more attention. For many nonlinear vibrating structures, the nonlinearity can be simulated by defining hysteresis model such as Bouc–Wen model. The time-varying parameters such as instantaneous frequency and modes are extracted and can be used as objective function to calibrate the nonlinear model. Since the nonlinear characteristics of a nonlinear structure are implied in the amplitude and frequency of the responses during the oscillation, the instantaneous amplitude and frequency of the decomposed mono-component can keep the complete information of the nonlinear features and can be used to update the nonlinear model. Thus, an objective function can be defined using the residuals of the instantaneous frequency and amplitude of the decomposed mono-component between experimental structure and nonlinear model. The optimal values of the nonlinear parameters are obtained by minimizing the objective function. In addition, since the slow-varying portion of the instantaneous frequency and the amplitudes of the acceleration and displacement all slowly vary with time (compared to the oscillation of the acceleration and displacement), it is not necessary to select all measured time points for calculating instantaneous parameter residuals in the objective function. The objective function can be created using a small number of data points and a reasonable estimated result can be obtained.

With the above concerns, Wang et al. (2015b, 2016b) presented a nonlinear model updating method based on the measured responses with Hilbert transform. The parameters of the nonlinear structure can be updated by minimizing the residuals between the instantaneous frequencies and amplitudes of the nonlinear structure and those measured data. An objective function $J (θ)$ is defined as a sum of instantaneous parameter residuals at selected time points along the response time series, which can be denoted as

J (θ) = J_{ω} (θ) + J_{acc} (θ) + J_{disp} (θ)

(61)

J_{ω} (θ) = \sum_{i = 1}^{N_{m}} {\sum_{j = 1}^{N_{t}} {[\frac{ω_{j}^{i} (θ, t) - {\hat{ω}}_{j}^{i} (t)}{{\hat{ω}}_{j}^{i} (t)}]}^{2}}

(62)

J_{acc} (θ) = \sum_{i = 1}^{N_{m}} {\sum_{j = 1}^{N_{t}} β_{acc}^{i} (j) {[\frac{A_{acc, j}^{i} (θ, t) - {\hat{A}}_{acc, j}^{i} (t)}{{\hat{A}}_{acc, j}^{i} (t)}]}^{2}}

(63)

J_{disp} (θ) = \sum_{i = 1}^{N_{m}} {\sum_{j = 1}^{N_{t}} β_{disp}^{i} (j) {[\frac{A_{disp, j}^{i} (θ, t) - {\hat{A}}_{disp, j}^{i} (t)}{{\hat{A}}_{disp, j}^{i} (t)}]}^{2}}

(64)

in which $θ$ is the vector of the parameters of the nonlinear structure. $J_{ω} (θ)$ , $J_{acc} (θ)$ , and $J_{disp} (θ)$ are frequency, acceleration, and displacement residuals, respectively. $ω_{j}^{i} (θ, t)$ and $ω_{j}^{i} (t)$ are the model predicted and experimentally identified instantaneous frequency of the ith mono-component at time j, respectively. $A_{acc, j}^{i} (θ, t)$ and ${\hat{A}}_{acc, j}^{i} (t)$ are the instantaneous acceleration amplitudes of the ith mono-component at time j from model prediction and experimental identification, respectively. The time symbol j represents the time points which are selected from the instantaneous amplitudes of the acceleration responses and used for calculating instantaneous parameter residuals in the objective function. Considering the importance of displacement amplitude in the updated nonlinear model, the time series of the displacement response can be uniformly divided into many intervals, and then, the location of the peak value in each interval is selected as the selected data point location (time symbol j). $N_{m}$ and $N_{t}$ are the number of mono-components and selected measured time points. $β_{acc, j}^{i} (j)$ and $β_{disp, j}^{i} (j)$ are weight factors of acceleration and displacement responses, respectively.

To validate the effectiveness of the proposed nonlinear joint model updating method, a three-pillar switch specimen under harmonic excitation is tested in the laboratory (Wang et al., 2015b, 2016b). The test equipment and layout are presented in Figure 16. Since the cross section at the bottom of the right pillar is weaker than the cross sections at the bottom of the other two pillars, the right pillar is gradually fractured under harmonic excitation. The measured acceleration and extracted instantaneous frequency and amplitude on the left-top pillar are presented in Figure 17. The bilinear spring of the zero length joint element is defined as the bilinear material. Thus, a total of six parameters $F_{1}, E_{1}, b_{1}, F_{2}, E_{2}, and b_{2}$ are selected as the updated parameters of the nonlinear joint model. The parameters $F_{1}, E_{1}, and b_{1}$ are, respectively, the transition strength, initial tangent elastic modulus, and strain-hardening ratio for the material properties of the bilinear spring at the shear direction. Similarly, the parameters $F_{2}, E_{2}, and b_{2}$ are, respectively, the transition strength, initial tangent elastic modulus, and strain-hardening ratio for the material properties of the bilinear spring at the torsion direction. Based on the nonlinear model updating method, the above six parameters are finally obtained as $F_{1} = 254 N, E_{1} = 80, 100 Pa, b_{1} = 0.76, F_{2} = 164 N, E_{2} = 182, 000 Pa, and b_{2} = 0.83$ . The displacement, acceleration, and slow-varying portion of the instantaneous frequency estimated from the updated nonlinear model and their measured data are presented in Figure 18. The estimated responses from the updated nonlinear model are generally in good agreement with the measured responses. The calibrated nonlinear model is further validated with other severe excitation with acceptable errors.

Figure 16.

Shake test specimen: (a) details of the shake table specimen, (b) test specimen and failure location, and (c) the created model with local nonlinear element.

Figure 17.

15 selected data points of the dominant mono-component for the tested structure: (a) instantaneous amplitude of the acceleration and (b) instantaneous frequency of the acceleration.

Figure 18.

Comparison of the acceleration, displacement, and instantaneous frequency between the estimated responses based on the updated nonlinear model and the measured values: (a) acceleration, (b) displacement, and (c) instantaneous frequency.

Indeed, structural nonlinear model updating method can provide reliable models for complex nonlinear structures. However, several critical issues need to be further studied. First of all, no universal features can be applied to all types of structural nonlinearities. Second, it is quite a challenging issue to select an appropriate initial nonlinear model to describe the nonlinear structure. Therefore, it is required to develop some methods to identify the nonlinear type before the nonlinear model updating method. Finally, many objective functions need a highly nonlinear solution space. Although the reviewed objective function with a few data points of the instantaneous frequency and amplitudes can significantly reduce the amount of calculation, it is difficult to select an appropriate objective function to considering the static and dynamic responses.

Discussions and prospects

This article reviewed the time–frequency analysis of dynamic responses based on wavelet and Hilbert transforms and their applications in structural parameter identification, damage detection, and updated nonlinear model. Due to the page limit, other applications of wavelet and Hilbert transforms, such as the quantification of nonstationary responses under earthquakes and tornados, and other time–frequency analysis methods are not included.

Based on the above brief review, several concluding remarks are made on the fundamental principles and applications of wavelet and Hilbert transforms as summarized below:

For the nonstationary responses of nonlinear structures under dynamic loads, time–frequency analysis based on wavelet and Hilbert transforms can be applied to effectively extract the instantaneous frequency and amplitude if the measured responses contain low noises. The key issue is to extract the wavelet ridge in wavelet analysis and to decompose the signal into mono-components in Hilbert transform.

The instantaneous frequency extracted from the measured response of a structure may not be equal to the time-varying frequency of the structure. Under repeated impacts, white noise, or broadband excitations, the two frequencies are approximately equal. Under chirp or harmonic excitations, however, the two frequencies are different.

To identify the time-varying parameters of structures, the use of wavelet and Hilbert transforms requires the availability of all responses and excitations. This requirement limits their applications in practice. The time–frequency analysis methods basically assume that the structural parameters vary over time more slowly than the measured responses.

The energy damage indices have been developed with wavelet and Hilbert transforms when both the undamaged and damaged structures are known. To avoid false damage detection, the same loads that are applied on the undamaged and damaged structures must be exactly the same. When the responses of the undamaged structure are not available, the damage indices based on the decreasing instantaneous frequency are used for damage detection. However, these damage indices do not include the useful information on the amplitude of the responses. To detect the damage location, the singularity of the measured signal is used. However, it is quite difficult to eliminate the environmental effects. In general, it is still a quite challenging issue to detect damage locations and quantify the damage level without knowing the baseline model of the structure.

The distortion of Hilbert transform on the FRF reveals the nonlinearity of a structure under dynamic loads and it can be used for structural nonlinearity detection. However, it is quite difficult to quantify the degree of nonlinearity.

In the past decade, the nonparametric identification of nonlinear structures based on signal features is considered as sufficient. For the nonparametric system identification, the nonlinear functions of restoring and damping forces are considered a priori. FREEVIB and FORCEVIB methods have been applied for nonlinear restoring force, damping force, and backbone identification with Hilbert transform.

The characteristics of a nonlinear structure are contained in the instantaneous amplitude and frequency of structural responses under dynamic loads. The key is to decompose the responses into mono-components.

Although wavelet and Hilbert transforms have been applied to parameter identification, damage detection, and nonlinear modeling of time-varying and nonlinear civil engineering structures, there are still challenges associated with their applications as briefly summarized below:

Features more than instantaneous frequency and amplitude may be extracted from dynamic responses of time-varying and nonlinear civil engineering structures when the stationarity of the responses is known a priori.

The measured responses of structures under ambient or strong broadband excitations include several main mono-components and noise. Therefore, a reliable adaptive signal decomposition method is required to extract the mono-components and eliminate the noise.

Almost all of the time-varying parameter identification methods based on wavelet and Hilbert transforms are developed for offline implementation. In practice, online identification approaches based on adaptive wavelet and Hilbert transforms are preferred.

The energy-based damage indices derived from wavelet and Hilbert transforms are mainly based on the mathematical extraction of signals while physical damage often occurs locally in structures. When a damage model is created in structural analysis, the extracted features may be associated with the physical damage, allowing the quantification of damage level.

A nonlinear model can be calibrated with measured responses and used to analyze or predict the nonlinear responses of structures under dynamic loads. However, how to select the best nonlinear model based on the features extracted by Hilbert transform requires further study.

There is a need to quantify the statistical properties of the identified parameters from surrogate data with Hilbert transform.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was financially supported by the National Key R&D Program of China under grant no. 2017 YFC 0805100, by the National Natural Science Foundation of China under grant nos 51478159 and 51578206, by “The Fundamental Research Funds for the Central Universities,” by the Natural Science Funds for Distinguished Young Scholar of Anhui province under grant no. 1708085J06, and by the US National Science Foundation under award no. CMMI1538416.

References

Basu

Nagarajaiah

Chakraborty

(2008) Online identification of linear time-varying stiffness of structural systems by wavelet analysis. Structural Health Monitoring 7(1): 21–36.

Braun

Feldman

(2011) Decomposition of non-stationary signals into varying time scales: some aspects of the EMD and HVD methods. Mechanical Systems and Signal Processing 25(7): 2608–2630.

Carmona

Hwang

Torrésani

(1998) Practical Time-Frequency Analysis: Gabor and Wavelet Transforms With an Implementation in S (Wavelet Analysis and Its Applications). San Diego, CA: Academic Press.

Carmona

Wen

Torrésani

(1997) Characterization of signals by the ridges of their wavelet transforms. IEEE Transactions on Signal Processing 45(10): 2586–2590.

Chanpheng

Yamada

Katsuchi

et al . (2012) Nonlinear features for damage detection on large civil structures due to earthquakes. Structural Health Monitoring 11(4): 482–488.

Chen

Wang

(2012) A signal decomposition theorem with Hilbert transform and its application to narrowband time series with closely spaced frequency components. Mechanical Systems and Signal Processing 28: 258–279.

Chen

Liu

Lai

(2009) Wavelet analysis for identification of damping ratios and natural frequencies. Journal of Sound and Vibration 323(1–2): 130–147.

Chen

Feng

(2003) A technique to improve the empirical mode decomposition in the Hilbert-Huang transform. Earthquake Engineering and Engineering Vibration 2(1): 75–85.

Cheraghi

Taheri

(2007) A damage index for structural health monitoring based on the empirical mode decomposition. Journal of Mechanics of Materials and Structures 2(1): 43–61.

10.

Chui

(1992) An Introduction to Wavelets. San Diego, CA: Academic Press, 280 pp.

11.

Cohen

(1995) Time-Frequency Analysis. Upper Saddle River, NJ: Prentice Hall PTR/A Pearson Education Company, 299 pp.

12.

Daubechies

(2011) Synchrosqueezed wavelet transforms: an empirical mode decomposition-like tool. Applied and Computational Harmonic Analysis 30(2): 243–261.

13.

Dziedziech

Staszewski

Uhl

(2015) Wavelet-based modal analysis for time-variant systems. Mechanical Systems and Signal Processing 50–51: 323–337.

14.

Feldman

(1994a) Non-linear system vibration analysis using Hilbert transform – I. Free vibration analysis method “FREEVIB.” Mechanical Systems and Signal Processing 8: 119–127.

15.

Feldman

(1994b) Non-linear system vibration analysis using Hilbert transform – II. Forced vibration analysis method “FORCEVIB.” Mechanical Systems and Signal Processing 8: 309–318.

16.

Feldman

(1997) Non-linear free-vibration identification via the Hilbert transform. Journal of Sound and Vibration 208(3): 475–489.

17.

Feldman

(2006) Time-varying vibration decomposition and analysis based on the Hilbert transform. Journal of Sound and Vibration 295(3–5): 518–530.

18.

Feldman

(2008) Theoretical analysis and comparison of the Hilbert transform decomposition methods. Mechanical Systems and Signal Processing 22(3): 509–519.

19.

Feldman

(2011) Hilbert Transform Applications in Mechanical Vibration. Chichester: John Wiley & Sons.

20.

Flandrin

(1999) Time-Frequency/Time-Scale Analysis. San Diego, CA: Academic Press.

21.

Grochenig

(2001) Foundations of Time-Frequency Analysis. Boston, MA: Birkhauser.

22.

Gurly

Kareem

(1999) Application of wavelet transform in earthquake, wind, and ocean engineering. Engineering Structures 21(2): 149–167.

23.

Hahn

(1996) The Hilbert Transform in Signal Processing. Norwood, MA: Artech House, 442 pp.

24.

Han

Ren

Sun

(2005) Wavelet packet based damage identification of beam structures. International Journal of Solids and Structures 42(26): 6610–6627.

25.

Hartmann

(2004) Signal, Sound and Sensation. Secaucus, NJ: American Institute of Physics, 647 pp.

26.

Hester

González

(2012) A wavelet-based damage detection algorithm based on bridge acceleration response to a vehicle. Mechanical Systems and Signal Processing 28(2): 145–166.

27.

Hou

Noori

St. Amand

(2000) Wavelet-based approach for structural damage detection. Journal of Engineering Mechanics 126(7): 677–683.

28.

Huang

Attoh-Okine

(2005) The Hilbert-Huang Transform in Engineering. Boca Raton, FL: Taylor & Francis Group.

29.

Huang

Shen

SSP

(2005) Hilbert-Huang Transform and Its Applications. Singapore: World Scientific Publishing.

30.

Huang

Shen

Long

(1999) A new view of nonlinear water waves: the Hilbert spectrum. Annual Review of Fluid Mechanics 31: 417–457.

31.

Huang

Shen

Long

et al . (1998) The empirical mode decomposition and Hilbert spectrum for nonlinear and nonstationary time series analysis. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences 454(1971): 903–995.

32.

Huang

Long

et al . (2003) A confidence limit for the empirical mode decomposition and Hilbert spectral analysis. Proceedings of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences 459(2037): 2317–2345.

33.

Kijewski

Kareem

(2002) On the presence of end effects and their melioration in wavelet-based analysis. Journal of Sound and Vibration 256(5): 980–988.

34.

Kijewski

Kareem

(2003) Wavelet transforms for system identification in civil engineering. Computer-Aided Civil and Infrastructure Engineering 18(5): 339–355.

35.

Liang

(2012a) A generalized synchrosqueezing transform for enhancing signal time–frequency representation. Signal Processing 92(9): 2264–2274.

36.

Liang

(2012b) Time–frequency signal analysis for gearbox fault diagnosis using a generalized synchrosqueezing transform. Mechanical Systems and Signal Processing 26: 205–217.

37.

Liew

Wang

(1998) Application of wavelet theory for crack identification in structures. Journal of Engineering Mechanics 124(2): 152–157.

38.

Hsu

(2002) Vibration analysis of an inhomogeneous string for damage detection by wavelet transform. International Journal of Mechanical Sciences 44(4): 745–754.

39.

Mallat

(1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11: 674–693.

40.

Ovanesova

Suárez

(2004) Applications of wavelet transforms to damage detection in frame structures. Engineering Structures 26(1): 39–49.

41.

Pan

Huang

et al . (2008) Time-frequency method for nonlinear system identification and damage detection. Structural Health Monitoring 7(2): 103–127.

42.

Panagiotou

Restrepo

Conte

(2011) Shake table test of a 7-story full scale reinforced concrete structural wall building slice phase I: rectangular wall section. Journal of Structural Engineering 137: 691–704.

43.

Peng

Tse

Chu

(2005) An improved Hilbert-Huang transform and its application in vibration signal analysis. Journal of Sound and Vibration 286(1–2): 187–205.

44.

Pines

Salvino

(2006) Structural health monitoring using empirical mode decomposition and the Hilbert phase. Journal of Sound and Vibration 294(1–2): 97–124.

45.

Poon

Chang

(2007) Identification of nonlinear elastic structures using empirical mode decomposition and nonlinear normal modes. Smart Structures and Systems 3(2): 423–437.

46.

Ren

Sun

(2008) Structural damage identification by using wavelet entropy. Engineering Structures 30(10): 2840–2849.

47.

Ren

Sun

Xia

et al . (2008) Damage identification of shear connectors with wavelet packet energy: laboratory test study. Journal of Structural Engineering 134(5): 832–841.

48.

Rezaei

Taheri

(2009) Experimental validation of a novel structural damage detection method based on empirical mode decomposition. Smart Materials and Structures 18(4): 045004.

49.

Rezaei

Taheri

(2010) Damage identification in beams using empirical mode decomposition. Structural Health Monitoring 10(3): 261–274.

50.

Rezaei

Taheri

(2011) Damage identification in beams using empirical mode decomposition. Structural Health Monitoring 10(3): 261–274.

51.

Ruzzene

Fasana

Garibaldi

et al . (1997) Natural frequencies and damping identification using wavelet transform: application to real data. Mechanical Systems and Signal Processing 11(2): 207–218.

52.

Shi

Law

(2007) Identification of linear time-varying dynamical system using Hilbert transform and EMD method. Journal of Applied Mechanics 74(2): 223–230.

53.

Shi

Law

(2009) Identification of linear time-varying MDOF dynamic systems from forced excitation using Hilbert transform and EMD method. Journal of Sound and Vibration 321(3–5): 572–589.

54.

Staszewski

(1997) Identification of damping in MDOF systems using time-scale decomposition. Journal of Sound and Vibration 203(2): 283–305.

55.

Sun

Chang

(2003) Structural degradation monitoring using covariance-driven wavelet packet signature. Structural Health Monitoring 2(4): 309–325.

56.

Thakur

(2011) Synchrosqueezing-based recovery of instantaneous frequency from nonuniform samples. SIAM Journal of Mathematical Analysis 43(5): 2078–2095.

57.

Wang

Ren

Wang

et al . (2014a) Time-varying physical parameter identification of shear type structures based on discrete wavelet transform. Smart Structures and Systems 14(5): 831–845.

58.

Wang

Tao

et al . (2016a) Measurements and analysis of non-stationary wind characteristics at Sutong Bridge in Typhoon Damrey. Journal of Wind Engineering and Industrial Aerodynamics 151: 100–106.

59.

Wang

Chen

(2012) A recursive Hilbert-Huang transform method for time-varying property identification of linear shear-type buildings under base excitations. Journal of Engineering Mechanics 138(6): 631–639.

60.

Wang

Chen

(2013) Analytical mode decomposition of a time series with decaying amplitudes and overlapping instantaneous frequencies. Smart Materials and Structures 22(9): 095003.

61.

Wang

Chen

(2014) Analytical mode decomposition with Hilbert transform for modal parameter identification of buildings under ambient vibration. Engineering Structures 59: 173–184.

62.

Wang

Geng

Ren

et al . (2015a) Damage detection of nonlinear structures with analytical mode decomposition and Hilbert transform. Smart Structures and Systems 15(1): 1–13.

63.

Wang

Ren

Chen

(2013a) Time-varying linear and nonlinear structural identification with analytical mode decomposition and Hilbert transform. Journal of Structural Engineering 139(12): 06013001 (5 pp.).

64.

Wang

Ren

Liu

(2013b) A synchrosqueezed wavelet transform Enhanced by extended analytical mode decomposition method for dynamic signal reconstruction. Journal of Sound and Vibration 332(22): 6016–6028.

65.

Wang

Ren

Wang

et al . (2013c) Instantaneous frequency identification of time-varying structures by continuous wavelet transform. Engineering Structures 52: 17–25.

66.

Wang

Xin

Ren

(2015b) Nonlinear structural model updating based on instantaneous frequencies and amplitudes of the decomposed dynamic responses. Engineering Structures 100: 189–200.

67.

Wang

Xin

Ren

(2016b) Nonlinear structural joint model updating based on instantaneous characteristics of dynamic responses. Mechanical Systems and Signal Processing 76–77: 476–496.

68.

Wang

Xin

Xing

et al . (2017) Hilbert low-pass filter of non-stationary time sequence using analytical mode decomposition. Journal of Vibration and Control 23(15): 2444–2469.

69.

Wang

Zhang

Ren

et al . (2014b) Structural modal parameter identification from forced vibration with analytical mode decomposition. Advances in Structural Engineering 17(8): 1129–1143.

70.

Worden

Tomlinson

(2001) Nonlinearity in Structural Dynamics: Detection, Identification, and Modeling. Bristol: IOP Publishing Ltd., 659 pp.

71.

Huang

(2009) Ensemble empirical mode decomposition: a noise-assisted data analysis method. Advances in Adaptive Data Analysis 1(1): 1–41.

72.

Yan

Gao

(2006) Hilbert-Huang transform-based vibration signal analysis for machine health monitoring. IEEE Transactions on Instrumentation and Measurement 55(6): 2320–2329.

73.

Yang

Lei

Huang

(2004) Identification of natural frequencies and damping of in situ tall buildings using ambient wind vibration data. Journal of Engineering Mechanics 130(5): 570–577.

74.

Yang

Lei

Pan

et al . (2003a) Identification of linear structures based on Hilbert-Huang transform. Part 2: complex modes. Earthquake Engineering & Structural Dynamics 32(10): 1533–1554.

75.

Yang

Lei

Pan

et al . (2003b) System identification of linear structures based on Hilbert-Huang spectral analysis. Part I: normal modes. Earthquake Engineering & Structural Dynamics 32(9): 1443–1467.

76.

Yin

Kareem

(2016) Synthetic turbulence: a wavelet based simulation. Probabilistic Engineering Mechanics 45: 177–187.

77.

Zheng

Shen

Dou

et al . (2009) Modal identification based on Hilbert-Huang Transform of structural response with SVD preprocessing. Acta Mechanica Sinica 25: 883–888.

78.

Zhu

Law

(2006) Wavelet-based crack identification of bridge beam from operational deflection time history. International Journal of Solids and Structures 43: 2299–2317.