Modal identification for superstructure using virtual impulse response

Abstract

In civil engineering, structural modes are identified with the assumption of stationary white noise, which cannot be satisfied in practical engineering. This article proposes a new method, which contains the virtual impulse response and eigensystem realization algorithm. The formulation of virtual impulse response is derived from the inverse Fourier transform of the ratio of the cross-power to auto-power spectral density functions of the measurement responses, which is based on the concept of frequency response function. During the formulation derivation, a single point excitation is only considered. Frequency response function would not change with different excitations and responses, which means that the excitation cannot influence frequency response function. The impulse response is pointed out to only represent the behavior of superstructure. After obtaining impulse responses, eigensystem realization algorithm is then performed to identify the modes of superstructure. The proposed method is validated by a numerical example. The results show that virtual impulse response can have much better free decayed behavior than natural excitation technique and identify very precise modal parameters for superstructure.

Keywords

eigensystem realization algorithm frequency response function modal identification superstructure virtual impulse response

Introduction

Structural modal identification has attracted many researchers to investigate the application in civil engineering, such as structural model updating (Lei et al., 2012), damage detection (Soyoz et al., 2012; Zhang et al., 2017), response estimation, and sensor optimal placement. Li et al. (2016) proposed bilinear modal superposition to estimate the bilinear structural responses. Yi et al. (2015) optimized the sensor position with the proposed the immune monkey algorithm. When performing modal identification, spurious modes and closely spaced modes would be appear due to the measurement noises. Qu et al. (2017b) proposed the method to identify spurious modes through shifting measurement data. To separate the closely spaced modes, Qu et al. (2018a) proposed the singular vector angle concept and proved the good performance of the proposed concept. To suppress measurement noise, Qu et al. (2018b) proposed a new method based on the higher order spectrum to reduce the influence of noise on frequency identification. In civil engineering, the excitation is hard to be obtained, which causes the operational modal analyses. Stochastic subspace identification method only uses the structural response to identify structural modes with assumption of stationary white noise excitation (Wu et al., 2016). Eigensystem realization algorithm (ERA) identifies modes with assumption of free decayed responses (Juang and Pappa, 1985). Recently, the frame of the blind source is utilized to identify structural modes (Nagarajaiah and Yang, 2017; Yang et al., 2015).

Impulse response is always used in modal identification with time domain methods, because it can reflect the structural dynamic properties, which cannot be satisfied in practical engineering. Random decrement method obtains the impulse response by statistic of the ambient excitation response with stationary and zero mean value properties (Ibrahim, 2001). Due to the similar expressions between the impulse response and the correlation with the condition of white excitation, natural excitation technique (NExT) uses the correlation function to replace the impulse response (James et al., 1995). Mahmood et al. (2014) used NExT-ERA to obtain structural modes and detect modal-based damage. Narazaki et al. (2018) extended NExT to reduce the measurement noise. Xu (2016) identified the damage of retaining wall structures with virtual impulse response function. In the above methods, the impulse responses would be obtained under the assumption of white and stationary noise excitation, which cannot be satisfied in practical engineering.

This article proposes an innovative method which can reduce the influence of excitation based on virtual impulse response and ERA. The drawbacks of NExT are pointed out, which exist in the assumption of stationary white noise excitation. The formulation of virtual impulse response is derived, and the reason why virtual impulse response can reduce the influence of non-ideal excitation is illustrated. The modes are identified after obtaining virtual impulse response. The method is noted to be suitable for the superstructure. The procedures to identify modes of superstructures using virtual impulse response are summarized. Finally, the efficiency of the proposed method is validated by a numerical example.

Problem description

For an n-degree-of-freedom (DOF) structure, the dynamics in discrete time domain is represented in state-space form as follows

x (k + 1) = A_{d} x (k) + B_{d} u (k)

(1)

y (k) = C_{d} x (k) + D_{d} u (k)

(2)

where k means kth time step; $A_{d} \in ℜ^{2 n \times 2 n}$ , $B_{d} \in ℜ^{2 n \times l}$ , $C_{d} \in ℜ^{m \times 2 n}$ , and $D_{d} \in ℜ^{m \times l}$ are state, excitation, measurement, and feed-through matrices in discrete time domain, respectively; $x (k) \in ℜ^{2 n \times 1}$ , $y (k) \in ℜ^{m \times 1}$ , and $u (k) \in ℜ^{l \times 1}$ are state, measurement, and excitation vectors, respectively; $2 n$ is the state matrix order.

Structural modes can be identified by a similar matrix of $A_{d}$ according to the measurement data $y (k)$ only. When the measurement $y (k)$ is designated impulse response, it has the follow expression (Juang and Pappa, 1985)

y (k) = C_{d} A_{d}^{k - 1} B_{d}, (k \geq 1)

(3)

Then, the Hankel matrix with the dimension of $rH \times cH$ can be formed by $y (k)$ as follows

H_{k} = (\begin{matrix} y (k) & y (k + 1) & \dots & y (k + cH - 1) \\ y (k + 1) & y (k + 2) & \dots & y (k + cH) \\ \dots & \dots & \dots & \dots \\ y (k + rH - 1) & y (k + rH) & \dots & y (k + rH + cH - 2) \end{matrix})

(4)

Two Hankel matrices with one time step shift can generate the similar matrix ${\hat{A}}_{d}$ of $A_{d}$ as follows

{\hat{A}}_{d} = Γ_{k - 1}^{- 1} U_{k - 1}^{T} H_{k} V_{k - 1} Γ_{k - 1}^{- 1}

(5)

where $V_{k - 1}$ , $U_{k - 1}$ , and $Γ_{k - 1}$ satisfy the singular value decomposition of the Hankel matrix $H_{k - 1}$ , that is

H_{k - 1} = U_{k - 1} Γ_{k - 1}^{2} V_{k - 1}^{T}

(6)

Modal parameters, such as frequencies, damping ratios, and mode shapes, can be obtained from the similar state matrix ${\hat{A}}_{d}$ . This identification method is ERA.

However, the Hankel matrix (equation (4)) is constructed by the impulse response equation (3), which is not common in civil engineering. For the civil structures, the ambient excitation does not have the same trend as the impulse response. The ambient excitation in practical engineering looks like random signals and always is recognized as white noise. To use ERA, this white noise can be transferred to impulse response by NExT (James et al., 1995).

If the excitation is stationary white noise, the correlation function between the responses at two measurement positions can be simplified to a formula with the similar expression to the impulse response. This correlation function can still retain the structural mode properties. Therefore, the impulse response can be replaced with the correlation function. It is the basic idea of NExT. When the correlation function is expressed as follows

r_{τ} = E [y (k + τ) y {(k)}^{T}]

(7)

where $τ$ means a time lag; $E [\cdot]$ is the expectation operator; and $r_{τ}$ is the correlation function; then, $r_{τ}$ takes the place of $y (k)$ in equation (4) as follows

H_{τ} = (\begin{matrix} r_{τ} & r_{τ + 1} & \dots & r_{τ + cH - 1} \\ r_{τ + 1} & r_{τ + 2} & \dots & r_{τ + cH} \\ \dots & \dots & \dots & \dots \\ r_{τ + rH - 1} & r_{τ + rH} & \dots & r_{τ + rH + cH - 2} \end{matrix})

(8)

The mode parameters can be calculated using ERA. The combination of NExT and ERA is called to be NExT-ERA. However, if the excitation does not satisfy the assumption of stationary white noise, the simplification of the correlation function $r_{τ}$ cannot be performed. The formula of $r_{τ}$ would not have the similar expression to impulse response. If this $r_{τ}$ is still used in ERA, the identified modes would be wrong.

Modal identification by virtual impulse response

Structural modes reflect the structural dynamic properties, which are the structural intrinsical behavior, and do not have relationship with excitation and responses. The concept of frequency response function (FRF) also represents the structural properties only without the relationship with excitation and response (Yan et al., 2017). Therefore, the modes can be obtained based on FRF.

The FRF between the response at the measurement position p and the excitation at the position q is expressed as follows

F_{yu} (ω, p, q) = \frac{Y (ω, p)}{U (ω, q)}

(9)

with

U (ω, q) = \sum_{k = 0}^{K - 1} u_{q} (k) e^{- j 2 π \frac{ω}{K} k}

(10)

Y (ω, p) = \sum_{k = 0}^{K - 1} y_{p} (k) e^{- j 2 π \frac{ω}{K} k}

(11)

where $U (ω, q)$ is the Fourier transform of the excitation $u_{q}$ at the position q; $U^{*} (ω, q)$ is the complex conjugate of $U (ω, q)$ ; $Y (ω, p)$ is the Fourier transform of the response $y_{p}$ at the position p; and K is the selected time steps in time domain.

To obtain FRF $F_{yu} (ω, p, q)$ , it contains the Fourier transform $U (ω, q)$ of the excitation, which cannot be obtained in practical engineering. The unknown excitation $U (ω, q)$ can influence the process of obtaining an accurate FRF. Therefore, it is necessary to eliminate the influence of $U (ω, q)$ . Equation (9) can be utilized to form the following expression

\begin{matrix} {\tilde{F}}_{yy} (ω, p_{2}, p_{1}) = \frac{F_{yu} (ω, p_{2}, q)}{F_{yu} (ω, p_{1}, q)} \\ = \frac{Y (ω, p_{2})}{U (ω, q)} \cdot \frac{U (ω, q)}{Y (ω, p_{1})} = \frac{Y (ω, p_{2})}{Y (ω, p_{1})} \end{matrix}

(12)

From equation (12), ${\tilde{F}}_{yy} (ω, p_{2}, p_{1})$ is formulated by two FRFs $F_{yu} (ω, p_{1}, q)$ and $F_{yu} (ω, p_{2}, q)$ , which also do not have relationship with the response and excitation. Therefore, ${\tilde{F}}_{yy} (ω, p_{2}, p_{1})$ also reflects the structural intrinsical behavior. The Fourier transform $Y (ω, p)$ is not stable, which means that the values of $Y (ω, p)$ would vibrate along with the different selection of measurement data $y_{p}$ . If the values of $Y (ω, p)$ can be a stable, the statistical property of the different selection of the measurement data $y_{p}$ should not be changing, which can only be guaranteed by the stationary white noise excitation. To figure this problem out, equation (12) can be changed to the expression as follows

\begin{matrix} {\tilde{F}}_{yy} (ω, p_{2}, p_{1}) = \frac{Y (ω, p_{2})}{Y (ω, p_{1})} = \frac{Y^{*} (ω, p_{1}) Y (ω, p_{2})}{Y^{*} (ω, p_{1}) Y (ω, p_{1})} \\ = \frac{G_{yy} (ω, p_{2}, p_{1})}{G_{yy} (ω, p_{1})} \end{matrix}

(13)

with

G_{yy} (ω, p_{2}, p_{1}) = Y^{*} (ω, p_{1}) \cdot Y (ω, p_{2})

(14)

G_{yy} (ω, p_{1}) = Y^{*} (ω, p_{1}) \cdot Y (ω, p_{1})

(15)

where $G_{yy} (ω, p_{2}, p_{1})$ has the similar formulation to the cross-power spectral density function between the measurements at positions $p_{2}$ and $p_{1}$ when there is only one selection of measurement data; $G_{yy} (ω, p_{1})$ has the similar formulation to the auto-power spectral density function of the measurement at position $p_{1}$ for one selection of measurement data.

In equation (13), the values of the multiplication of two Fourier transforms $Y^{*} (ω, p_{1}) Y (ω, p_{2})$ actually still vibrate due to the non-stationary excitation and the different selection of measurement data. Then, $G_{yy} (ω, p_{2}, p_{1})$ and $G_{yy} (ω, p_{1})$ can be replaced by the power spectral density functions that calculate the mean of $Y^{*} (ω, p_{1}) Y (ω, p_{2})$ and $Y^{*} (ω, p_{1}) \cdot Y (ω, p_{1})$ in multiple selected data sections, respectively. Because the structure is still time invariant system, the mean of $Y^{*} (ω, p_{1}) Y (ω, p_{2})$ and $Y^{*} (ω, p_{1}) \cdot Y (ω, p_{1})$ would not change the ratio of $G_{yy} (ω, p_{2}, p_{1})$ to $G_{yy} (ω, p_{1})$ in every time region selection for the non-stationary excitation. For the factor of the different selection measurement data, the mean of $Y^{*} (ω, p_{1}) Y (ω, p_{2})$ and $Y^{*} (ω, p_{1}) \cdot Y (ω, p_{1})$ can reduce the effect of noise on power spectral density function in frequency domain. This process is better than the mean of ratio of $Y^{*} (ω, p_{1}) Y (ω, p_{2})$ to $Y^{*} (ω, p_{1}) \cdot Y (ω, p_{1})$ . Therefore, ${\tilde{F}}_{yy} (ω, p_{2}, p_{1})$ is stable, which can solve the problem of non-stationary excitation. The power spectral density functions are feasible to be obtained, which only use the measured responses at positions $p_{2}$ and $p_{1}$ .

FRF is obtained by the ratio between Fourier transform of output measurement and input excitation. When the excitation is Dirac delta function whose transfer function is one, then FRF is equal to the Fourier transform of output measurement while the output is impulse response. It should be noted that ${\tilde{F}}_{yy} (ω, p_{2}, p_{1})$ is the ratio of two FRFs and can be recognized as a new FRF, which is the FRF of the superstructure. Then, inverse Fourier transform of ${\tilde{F}}_{yy} (ω, p_{2}, p_{1})$ can be executed to obtain the impulse response ${\bar{y}}_{p_{2}} (k)$ at position $p_{2}$ as follows

{\bar{y}}_{p_{2}} (k) = \frac{1}{K} \sum_{ω = 0}^{K - 1} {\tilde{F}}_{yy} (ω, p_{2}, p_{1}) e^{j 2 π \frac{k}{K} ω}

(16)

Then, the impulse response vector can be calculated as follows

\bar{y} (k) = {[{\bar{y}}_{p_{2}} (k), \dots, {\bar{y}}_{p_{m}} (k)]}^{T}

(17)

Taking equation (17) into equations (4)–(6), ERA is performed to identify modal parameters. From equations (12)–(13), $p_{1}$ is only one position. Therefore, the proposed method is only valid for modal identification of a structure subject to a single input excitation. However, it should be noted that ${\tilde{F}}_{yy} (ω, p_{2}, p_{1})$ is not the structural FRF. Then, the calculated response at position $p_{2}$ is not the structural real impulse response. Because, from the expression, dividing the cross-power spectral density function by the auto-power spectral density function reveals the relationship between the responses at positions $p_{2}$ and $p_{1}$ . It means that the position $p_{1}$ can be recognized as the reference point, and $p_{2}$ has the response relative to the reference position $p_{1}$ . Then, the identified modal parameters are not structural modes. Because the impulse response is not structural free decayed response, it is named as virtual impulse response. This property can be used to identify the modes for superstructures as shown in numerical example.

The procedure is summarized as follows:

Step 1: collect the response data from all measurement positions to calculate $Y (ω, p)$ by Fourier transform of the responses.

Step 2: use multiple selected sections of response data to calculate the mean values of $Y^{*} (ω, p_{1}) Y (ω, p_{2})$ and $Y^{*} (ω, p_{1}) Y (ω, p_{2})$ in equations (14) and (15).

Step 3: calculate ${\tilde{F}}_{yy} (ω, p_{2}, p_{1})$ by the ratio of the cross-power spectral density function to the auto-power spectral density function as shown in equation (13).

Step 4: make inverse Fourier transform of ${\tilde{F}}_{yy} (ω, p_{2}, p_{1})$ to obtain the impulse response at position $p_{2}$ relative to reference position $p_{1}$ .

Step 5: execute steps 1–5 to obtain the impulse response at other positions $p_{2}$ relative to reference position $p_{1}$ . The impulse responses at other positions are also obtained and then construct the impulse response vector $\bar{y} (k)$ in equation (17).

Step 5: replacing $y (k)$ with $\bar{y} (k)$ in equation (4), ERA is then performed to identify mode parameters.

Numerical example

To illustrate and investigate the effectiveness of the modal identification by virtual impulse response for superstructure, a numerical example from Qu et al. (2017a) is changed to 8-DOF in-plane lumped-mass model as shown in the left side of Figure 1. The mass for each floor and stiffness for each story are 1.10 × 10⁶ kg and 862.07 × 10⁶ N/m, respectively. The Rayleigh damping ratios of first two modes are 5%. The model is excited by ambient noise at the $first$ floor position. The displacement time history for the $first$ and $second$ floors is shown in Figure 2.

Before using ERA to identify modal parameters, the impulse response should be obtained. The spectral density functions in equations (14) and (15) are first solved, which are calculated by averaging several overlapped data series which are the part of the measurement. Therefore, multiple data sections of the responses should be selected. Here, the sampling frequency is 100 Hz, and the frequency resolution is set to be less than 0.01 Hz. Then, the length of data series is selected to be 16,384, which is used to perform Fourier transform. The overlapping is 50%, which means that there are 8192 data points in two adjacent data series. The whole time history series of the responses contain 100,000 points. Then, the responses can be segmented to 11 data sections as shown in Figures 3 and 4.

Then, the auto-power and cross-power spectral density functions are averaged and shown as Figure 5.

In Figure 5, the symbol “seg” means the Fourier transform of the 11 different sections of the displacement time history; mean is the average of the 11 “seg.”

The impulse responses by inverse Fourier transform of these auto-power and cross-power spectral density functions, which are the method of NExT, are shown in Figure 6.

It is obvious that the spectral density function curves are not smooth in Figure 5, and the impulse responses by the inverse Fourier transform in Figure 6 are not stable in the whole time history. The curves still vibrate after 60 s. The reason is that the mean value and the standard deviation of excitation are not stable. Therefore, it does not satisfy the assumption of white noise, which causes that the impulse response generated by NExT would not coincide with the formula of the structural real impulse response. Then, the accuracy of the identified modes by ERA would be affected.

To identify structural modes using virtual impulse response, ${\tilde{F}}_{yy} (ω, p_{2}, p_{1})$ in equation (13) between the first and second floors is calculated. Then, the inverse Fourier transform of ${\tilde{F}}_{yy} (ω, p_{2}, p_{1})$ or the virtual impulse response is obtained as shown in Figure 7.

From Figure 7, the response is typically free decayed. In the similar way, the impulse responses at other floors can be calculated and utilized into ERA, the stabilization diagram is shown in Figure 8.

It should be noted that the stabilization diagram in Figure 8 represents the modes for the structure without the $first$ floor. It is not the original structure, and changed to a new structure, which is the superstructure with upper 7 DOFs as shown in the right side of Figure 1.

In Figure 8, there are seven vertical straight lines which are corresponding to seven modes of the superstructure. The identified modes are shown in Table 1. It is obvious that virtual impulse responses can supply very precise results.

Table 1.

Identified frequencies.

	$First$	$Second$	$Third$	$Fourth$	$Fifth$	$Sixth$	$Seventh$
Physical frequency (Hz)	1.1527	3.4077	5.5138	7.3789	8.9215	10.0742	10.7867
Virtual impulse response	1.1528	3.4078	5.5138	7.3784	8.929	10.0905	10.7851
Difference ratios (%)	0.0086	0.0029	almost 0	0.0068	0.084	0.16	0.015

To verify the identified modal shapes, the modal shape comparisons are drawn in Figure 9. The legend “Real” (blue circle) means that the modal shapes are the physical modes of the superstructure. The legend “ID” (red square) means that the modal shapes are identified by virtual impulse responses.

The modal shapes comparison can be also reflected by modal assurance criterion as shown in Figure 10, where MAC means modal assurance criterion. In Figure 10(a), the values of MAC are almost equal to 1, which illustrates that the identified modal shapes are very close to the physical ones. To make the difference between the value 1 and the MAC values obvious, I-MAC is used as shown in Figure 10(b), where I is identity matrix.

It should be noted that the proposed method is suitable for superstructure. The application of the method contains the structure with base isolation, the structure excited by earthquake and some structures with the moving base floor. The modes for superstructure of the base isolation structure can be identified using this proposed method without identifying the behavior of isolators. The framework for the structural base excitation coincides with the one of the proposed method. Therefore, the proposed method has a broad scope in future engineering application. However, the proposed method is not limited to the excitation at first floor, which is obvious according to the theoretical part illustration. This topic is also valuable to be investigated in future.

Conclusion

The excitation in civil engineering cannot strictly satisfy the assumption of stationary white noise. This article proposes a new modal identification method to figure out the excitation problem, which contains virtual impulse response and ERA method. There are some conclusions as follows.

This article proposed a new method, which uses virtual impulse response and ERA. Virtual impulse response can reduce the influence of the non-idea excitation. The ERA can identify more precise mode parameters with the virtual impulse response. The drawbacks of NExT are pointed out. The non-stationary excitation makes the simplification of the correlation function hard. The formula of correlation would not be coincided with the formula of impulse response, which can result in the errors of the identified modes using ERA.

The formulation of the virtual impulse response is derived. It comes from FRF, which would not change with different excitations and responses. Therefore, FRF can eliminate the influence of excitation. A new function obtained by the ratio of two FRFs is given, which can be solved easily by the cross-power spectral density function divided by auto-power spectral density function. It should be noted that the new function only represents the behavior of superstructure, and a single point excitation is only considered. Because the FRFs whose ratio is the new function are referred to the first-floor response, the structure from the second floor to top floor is the superstructure. The virtual impulse response can be obtained by inverse Fourier transform of the new function, where the procedures to obtain virtual impulse response are also summarized.

Structural modal identification using virtual impulse response and ERA is validated by a numerical example. Virtual impulse response can give an ideal free decayed response for ERA, which only stands for the behavior of superstructure. The identified modal parameters are very precise according to the physical modes of superstructure.

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 research was jointly supported by National Natural Science Foundation of China (Grant Nos 51778105 and 51625802), the 973 Program (Grant No. 2015CB060000), and the Fundamental Research Funds for the Central Universities (Grant No. DUT19JC48).

ORCID iD

Chun-Xu Qu

References

Ibrahim

(2001) Efficient random decrement computation for identification of ambient responses. In: Proceedings of SPIE the international society for optical engineering, Kissimmee, FL, 5–8 February.

James

Carne

Lauffer

(1995) The natural excitation technique (NExT) for modal parameter extraction from operating structures. Modal Analysis 10(4): 260–277.

Juang

Pappa

(1985) An eigensystem realization algorithm for modal parameter identification and model reduction. Journal of Guidance, Control, and Dynamics 8(5): 620–627.

Lei

Wang

Shen

(2012) Update the finite element model of Canton Tower based on direct matrix updating with incomplete modal data. Smart Structures and Systems 10(4–5): 471–483.

Huo

, et al. (2016) Equivalent bilinear elastic single degree of freedom system of multi-degree of freedom structure with negative stiffness. Journal of Sound and Vibration 365: 1–14.

Hao

Xia

, et al. (2017) Damage detection of shear connectors in bridge structures with transmissibility in frequency domain. International Journal of Structural Stability and Dynamics 14(1): 1350061.

Mahmood

SMF

Haritos

Gad

, et al. (2014) A multi-reference-based mode selection approach for the implementation of NExT-ERA in modal-based damage detection. Structural Control & Health Monitoring 21(8): 1137–1153.

Nagarajaiah

Yang

(2017) Modeling and harnessing sparse and low-rank data structure: a new paradigm for structural dynamics, identification, damage detection, and health monitoring. Structural Control & Health Monitoring 24(1): e1851.

Narazaki

Hoskere

Spencer

(2018) Free vibration-based system identification using temporal cross-correlations. Structural Control & Health Monitoring 25(8): e2207.

10.

Huo

, et al. (2017a) Optimum value of negative stiffness and additional damping in civil structures. Journal of Structural Engineering 143(8): 04017068.

11.

, et al. (2018a) Closely spaced modes identification through modified frequency domain decomposition. Measurement 128: 388–392.

12.

Yang

, et al. (2017b) Spurious mode distinguish by eigensystem realization algorithm with improved stabilization diagram. Structural Engineering and Mechanics 63(6): 743–750.

13.

Zhou

, et al. (2018b) Frequency identification of practical bridges through higher order spectrum. Journal of Aerospace Engineering 31(3): 04018018.

14.

Wang

Chen

, et al. (2016) Application of stochastic subspace identification for stay cables with an alternative stabilization diagram and hierarchical sifting process. Structural Control & Health Monitoring 23(9): 1194–1213.

15.

(2016) Damage identification investigation of retaining wall structures based on a virtual impulse response function. Shock and Vibration 2016: 1346939.

16.

Yan

Zhao

Sun

, et al. (2017) Transmissibility-based system identification for structural health monitoring: fundamentals, approaches, and applications. Mechanical Systems and Signal Processing 117: 453–482.

17.

Yang

Nagarajaiah

, et al. (2015) Real-time output-only identification of time-varying cable tension from accelerations via complexity pursuit. Journal of Structural Engineering 142(1): 04015083.

18.

Zhang

(2015) Health monitoring sensor placement optimization for Canton Tower using immune monkey algorithm. Structural Control & Health Monitoring 22(1): 123–138.

19.

Zhang

Hao

, et al. (2017) Damage detection in bridge structures under moving loads with phase trajectory change of multi-type vibration measurements. Mechanical Systems and Signal Processing 87: 410–425.