Substructural damage identification using autoregressive moving average with exogenous inputs model and sparse regularization

Abstract

Substructuring approaches possess many superiorities over traditional global approaches in damage identification because large-size global structures are replaced by small and manageable substructures. This paper proposes a substructural time series model for locating and quantifying the damage in complex systems. A substructural autoregressive moving average with exogenous inputs (ARMAX) model is established to extract the frequencies and mode shapes of substructures as indicators for damage detection. The detection of structural damage is essentially an inverse problem, and the damage in structure bears sparse properties. The inverse problem of substructural damage identification is efficiently solved via sparse regularization, and structural damage can be located and quantified through the nonzero terms in the solution vector. The accuracy of the proposed method is demonstrated by the numerical simulation of a frame structure and shaking table test of a shear building structure. As the substructural properties are more sensitive to local structural damage than the global properties, the substructural ARMAX model is quite accurate and efficient to be used in the damage identification of a complex system.

Keywords

Structural health monitoring damage identification substructuring approach autoregressive moving average with exogenous inputs model sparse regularization

Introduction

Damage detection based on vibration data has been widely applied in structural health monitoring in the last few decades (Farrar and Worden, 2007; Yang et al., 2020; Peng et al., 2022; Zhang et al., 2022). These approaches normally identify structural damage through the changes in modal parameters and/or their derivatives between the intact structure and the damaged structure. Most of these approaches are performed at the global structure level because the modal parameters reflect the global properties of the entire structure. However, global measures are usually insensitive to local structural damage. Thus, the use of these damage identification methods based on global parameters is extremely limited in the detection of local structural damage (Nair et al., 2006).

In substructuring approach, the large-size structure is partitioned into several substructures, and each substructure is analyzed independently. The substructuring approach has several superiorities over the global approach. First, analyzing the substructures is easier and faster because the partitioned substructure has a much smaller size than the global structure (Weng et al., 2019). Second, the sensitivity of substructural properties to local structural damage are remarkably higher than that of the global properties because the substructuring approach concerns the local part of the structure during the damage detection process. Third, only the measurement of the local part corresponding to the target substructure is needed, and the measurement of the entire structure can be prevented (Tian et al., 2019). However, the extraction of substructural properties from global measurements using substructuring methods is challenging (Alvin and Park, 1999; Yang et al., 2022).

During the past decades, many researchers have explored the techniques to extract substructural properties. Alvin and Park (1999) extracted the substructural flexibility from global modal parameters through a force method. Doebling et al. (1998) projected the measured global flexibility properties onto the local strain energy distribution to identify the stiffness matrices of target substructure. Weng et al. (2012) disassembled the measured global modal parameters to obtain the substructural flexibility subjected to the force and displacement constraints at the interfaces. Jalali and Rideout (2022) obtained the frequency response function matrix of the main substructure by decoupling the known frequency response function of the residual subsystem from the global system frequency response function. The substructural properties have been shown to be more sensitive than that of the global properties.

The damage sensitive features are usually extracted from the vibration data through time domain and/or frequency domain approaches (Xu et al., 2009; Zhang et al., 2008), and one commonly used time domain approach is time series analysis. Time series model has been widely used in damage identification because its coefficients and residual errors are sensitive to structural damage. Sohn et al. (2000) extracted the coefficients of autoregressive model as damage-sensitive features and applied X-bar control chart for damage identification. Gul and Catbas (2011) used the fit ratios of autoregressive models with exogenous outputs for damage detection and localization. Zhu et al. (2020) derived the sensitivity of the autoregressive coefficients of autoregressive moving average model to structural damage factors, and applied sparse regularization for damage localization and quantification. Yun and Lee (1997) built ARMAX model to identify the unknown local parameters through an iterative process. Xing and Mita (2012) and Su et al. (2012) utilized single-output time series model for damage identification of shear building. Mei et al. (2016, 2019) divided a shear building into a number of parts, and identified structural damage by comparing the statistical distance between local ARMAX models.

The damage in structure are usually located in a limited number of components, and civil structure contains a large number of components. Therefore, most of the entries in damage index should be zero, and only the entries for damaged elements are nonzero. In other words, the damage index is a sparse vector (Wang et al., 2022; Xu et al., 2021). In light of sparse recovery theory, the sparse damage index can be accurately recovered through sparse regularization. Sparse regularization has gained significant attention in compressed sensing (Donoho, 2006), and has been widely used in compressive imaging (Stern, 2017) and signal processing (Vlasic and Sersic, 2022). Then, sparse regularization was introduced into the domain of structural health monitoring. Few researchers have applied sparse regularization in compressive sampling (Bao et al., 2011; Wan et al., 2022) and lost data recovery (Yang and Nagarajaiah, 2016, 2017) for sensor networks. Subsequently, sparse regularization was introduced into structural damage detection (Chen et al., 2021). Compared with the commonly used Tikhonov regularization, the damage identification accuracy is improved considerably when sparse regularization is used (Li et al., 2022).

This study proposes a new substructuring method that combines ARMAX model and sparse regularization for locating and quantifying the damage in complex systems. The large-size structure is partitioned into several small substructures, and the target substructure is analyzed independently. The multi-output ARMAX model of the independent substructure is constructed by transforming the linear structural motion equation. The acceleration responses of the interior degrees of freedom (DOFs) are regarded as outputs of the ARMAX model, and the acceleration responses of the interface DOFs and external forces are regarded as inputs of the ARMAX model. The natural frequencies and mode shapes of the substructure are directly evaluated on the basis of the autoregressive coefficients of the ARMAX model, and the modal parameters of the substructure are used for damage detection. The substructure with damage can be discovered by comparing the substructural modal parameters of each substructure before and after damage. After determining the damaged substructure, an equation that contains the changes in substructural modal parameters and substructural stiffness reduction vector is constructed. The sparse solution of the equation can be obtained through sparse regularization, and damage locations and severities are detected on the basis of the nonzero values of the solution. The numerical simulation of a frame and laboratory test of a shear building structure are used to demonstrate the effectiveness and accuracy of the proposed substructural damage identification method. The locations and severities of damage can be accurately obtained with measurement noise considered, and the sensitivity of substructural properties to local structural damage are higher than that of the global properties.

Autoregressive moving average with exogenous inputs model of substructures

The vibration equation of a linear structure subjected to external force can be described as

[M] {\ddot{x}} + [C] {\dot{x}} + [K] {x} = [L] {F}

(1)

where [M], [C], and [K] denote the mass, damping, and stiffness matrices, respectively. {x} denotes the structural displacement response, and the dot above {x} denotes time derivative. {F} represents the vector of external force. [L] relates the excitation locations with the corresponding DOFs of the system, in which the DOF with external forces is one and at all other locations are zeros. The entire structure is partitioned into several small substructures. To independently analyze the target substructure, equation (1) is rewritten into the partitioned matrices as (Law and Yong, 2011; Zhu et al., 2013)

[\begin{array}{c} M_{m m} & M_{m s} & 0 \\ M_{s m} & M_{s s} & M_{s r} \\ 0 & M_{r s} & M_{r r} \end{array}] {\begin{array}{c} {\dot{x}}_{m} \\ {\dot{x}}_{s} \\ {\dot{x}}_{r} \end{array}} + [\begin{array}{c} C_{m m} & C_{m s} & 0 \\ C_{s m} & C_{s s} & C_{s r} \\ 0 & C_{r s} & C_{r r} \end{array}] {\begin{array}{c} {\dot{x}}_{m} \\ {\dot{x}}_{s} \\ {\dot{x}}_{r} \end{array}} + [\begin{array}{c} K_{m m} & K_{m s} & 0 \\ K_{s m} & K_{s s} & K_{s r} \\ 0 & K_{r s} & K_{r r} \end{array}] {\begin{array}{c} x_{m} \\ x_{s} \\ x_{r} \end{array}} = {\begin{array}{c} L_{m} F_{m} \\ L_{s} F_{s} \\ L_{r} F_{r} \end{array}}

(2)

where subscripts m, s, and r denote the interior, interface, and remaining DOFs, respectively. A cantilever beam is utilized to demonstrate the substructure division process, and the right part of the cantilever beam is regarded as the target substructure, as shown in Figure 1(a). The interior and interface DOFs of the target substructure and the remaining DOFs outside the target substructure are defined in the figure.

Figure 1.

Configuration of a cantilever beam and a target substructure. (a) Global structure. (b) Target substructure.

The first line of equation (2) gives

[M_{m}] {{\ddot{x}}_{m}} + [C_{m m}] {{\dot{x}}_{m}} + [K_{m m}] {x_{m}} = [L_{m}] [F_{m}] - ([M_{m s}] {{\ddot{x}}_{s}} + [C_{m s}] {{\dot{x}}_{s}} + [K_{m s}] {x_{s}})

(3)

Equation (3) can be regarded as the motion equation of an independent linear system with mass matrix [M_mm], damping matrix [C_mm], and stiffness matrix [K_mm]. The right part of equation (3) contains two components: [L_m]{F_m} is the input force acting on the target substructure, and $- ([M_{m s}] {{\ddot{x}}_{s}} + [C_{m s}] {{\dot{x}}_{s}} + [K_{m s}] {x_{s}})$ is the force transmitted from the neighbor substructures. The force transmitted from the neighbor substructures can be treated as the external force acting on the target substructure (Zhu et al., 2013). In structural mechanics theory, deleting the row and column in the mass and stiffness matrices is equivalent to adding a fixed support in the corresponding DOF. As shown in the left part of equation (3), the components in [M], [C], and [K] corresponding to the interior DOFs remain, and the components corresponding to the other DOFs (including the interface DOFs) are deleted. Thus, the system in equation (3) can be regarded as an independent structure with fixed interface DOFs. Taking the cantilever beam in Figure 1(a) as an example, the DOFs at the interface of the target substructure are fixed, which is equivalent to placing virtual fixed supports on the interface, as illustrated in Figure 1(b).

An ARMAX model for the target substructure can be built from equation (3) in accordance with the transformation process similar to the one given in Yun and Lee (1997). The acceleration response of substructure ${{\ddot{x}}_{m}}$ is used as the output of the ARMAX model, the external input force applied on substructure {F_m} and the acceleration response of interface ${{\ddot{x}}_{s}}$ are used as the inputs. The ARMAX model can be given as

y (t) = P_{1} y (t - 1) + P_{2} y (t - 2) + Q_{0} u (t) + Q_{1} u (t - 1) + Q_{2} u (t - 2) + e (t) + H_{1} e (t - 1) + H_{2} e (t - 2)

(4)

where y(t) is the observed acceleration response vector of the substructure at time step t, that is,

y (t) = {{\ddot{x}}_{m} (t)}

. u(t) is the vector of external input force applied on the substructure and the acceleration response of the interface, that is,

u (t) = {F_{m}^{T} (t), {\ddot{x}}_{s}^{T} (t)}^{T}

. P₁ and P₂ are the autoregressive coefficient matrices related to the outputs, and Q₀, Q_1, and Q₂ are the moving average coefficient matrices associated with the inputs. e(t) is the prediction errors, and H₁ and H₂ are the unknown coefficient matrices related to the noise process. In this study, the ARMAX model parameters are estimated through the prediction error method (Ljung, 1999).

Identification of substructural modal parameters using the substructural ARMAX model

Based on the substructural ARMAX model, a matrix [G] is built using the autoregressive coefficients of the model, which can be expressed as (Huang, 2001)

[G] = [\begin{array}{c} 0 & I \\ P_{2} & P_{1} \end{array}]

(5)

where P₁ and P₂ are the autoregressive coefficient matrices, I represents an n_m × n_m unit matrix, and the number of interior DOFs in the substructure is n_m. The substructural frequencies and mode shapes can be obtained using the eigenvalues and eigenvectors of matrix [G].

The k-th eigenvalue and eigenvector of matrix [G] is marked by λ_k and {ψ_k}, respectively. λ_k and {ψ_k} satisfy

[G] {ψ_{k}} = λ_{k} {ψ_{k}}

(6)

The size of matrix [G] is 2n_m × 2n_m, thus {ψ_k} has 2n_m components. ${ψ_{k}}^{T}$ can be expressed as ( ${ψ_{k}^{1}}^{T}$ , ${ψ_{k}^{2}}^{T}$ ) with ${ψ_{k}^{1}}$ and ${ψ_{k}^{2}}$ each having n_m components. Equation (6) can be rewritten as

[\begin{array}{c} 0 & I \\ P_{2} & P_{1} \end{array}] {\begin{array}{c} ψ_{k}^{1} \\ ψ_{k}^{2} \end{array}} = λ_{k} {\begin{array}{c} ψ_{k}^{1} \\ ψ_{k}^{2} \end{array}}

(7)

The first line of equation (7) gives ${ψ_{k}^{2}} = λ_{k} {ψ_{k}^{1}}$ , indicating that ${ψ_{k}^{1}}$ and ${ψ_{k}^{2}}$ are parallel to each other. ${ψ_{k}^{1}}$ or ${ψ_{k}^{2}}$ can be used to obtain the k-th mode shape of the substructure (Huang, 2001). Eigenvalue λ_k is usually in the form of complex number, and the components in ${ψ_{k}^{1}}$ and ${ψ_{k}^{2}}$ are usually complex because matrix [G] is not symmetric. The real mode shape can be obtained through complex-to-real conversion. Herein, ${ψ_{k}^{1}}$ is used to obtain the k-th mode shape of the substructure. The j-th component of the k-th real mode shape ${\bar{ϕ}}_{j k}$ can be obtained as

{\bar{ϕ}}_{j k} = δ | ψ_{j k}^{1} |

(8)

where

| ψ_{j k}^{1} |

is the modulus of the complex number

ψ_{j k}^{1}

, the constant δ = +1 if the phase angle of

ψ_{j k}^{1}

lies in the first or fourth quadrant, and δ = −1 if the phase angle lies in the second or third quadrant (Rainieri and Fabbrocino, 2014).

The amplitudes of mode shapes in different measured points are relative values, which may lead to incorrect results when two mode shapes are compared at different scales. On this basis, the mode shapes should be normalized as

{ϕ_{k}} = \frac{{{\bar{ϕ}}_{k}}}{\sqrt{{{\bar{ϕ}}_{k}}^{T} M_{m m} {{\bar{ϕ}}_{k}}}}

(9)

where {ϕ_k} represents the k-th normalized substructural mode shape.

Complex eigenvalue λ_k is set to a_k + ib_k, and the circular natural frequencies of the substructure are computed as (Huang, 2001)

ω_{k} = \sqrt{η_{k}^{2} + β_{k}^{2}}

(10)

where

η_{k} = \frac{1}{2 Δ t} \ln (a_{k}^{2} + b_{k}^{2}), β_{k} = \frac{1}{Δ t} \tan^{- 1} (\frac{b_{k}}{a_{k}})

(11)

where Δt denotes the sampling time interval of structural responses, and ω_k is the k-th pseudo-undamped circular natural frequency of the substructure.

Sparse damage detection using substructural modal parameters

Given the substructural modal parameters from the ARMAX model, the target substructure is analyzed independently. A damage parameter named stiffness reduction factor (SRF) is defined as (Zhu et al., 2020)

γ_{l} = \frac{{\tilde{α}}_{l} - α_{l}}{α_{l}} (l = 1,2, \dots, n_{e})

(12)

where

α_{l}

is the stiffness parameter (e.g., bending rigidity) of the l-th element in the undamaged state and is reduced to

{\tilde{α}}_{l}

in the damaged state. The value of

γ_{l}

ranges from −1 to 0.

γ_{l} = 0

means that the l-th element is intact, and

γ_{l} = - 1

indicates the l-th elemental stiffness is completely lost. n_e denotes the number of elements in the substructure.

The detection of structural damage is essentially an inverse problem, and the damage in structure can be detected by the changes of structural properties before and after damage. Accordingly, an equation that related the changes in substructural property {ΔR} and damage parameter ${γ}$ is established as

[S] {γ} = {Δ R} = {R^{D}} - {R^{U}}

(13)

where {R^D} and {R^U} represent the substructural modal parameters from the damaged and undamaged states, respectively. {R^D} and {R^U} can be estimated from the measured structural responses and external input by the substructural ARMAX model proposed in Section 3. In this paper, the substructural frequencies and mode shapes are used for damage identification. The substructural modal parameter vector {R} can be expressed as

{R} = {ω_{1}, ω_{1}, \dots, ω_{n_{k}}, {ϕ_{1}}^{T}, {ϕ_{2}}^{T}, \dots, {ϕ_{n_{k}}}^{T}}^{T}

(14)

where n_k denotes the number of available modes.

The matrix [S] in equation (13) represents the sensitivity matrix of the substructural modal parameters to the SRF, and [S] can be expressed as

[S_{k l}^{ω}] = \frac{\partial ω_{k}}{\partial γ_{l}}, [S_{k l}^{ϕ}] = \frac{\partial {ϕ_{k}}}{\partial γ_{l}}

(15)

where

[S^{ω}]

denotes the sensitivity matrix of the substructural natural frequency to the SRF, and

[S^{ϕ}]

denotes the sensitivity matrix of the substructural mode shape to the SRF. The target substructure is analyzed independently,

[S^{ω}]

and

[S^{ϕ}]

can be computed through conventional approaches, such as Fox and Kappor’s (Fox and Kapoor, 1968) and Nelson’s methods (Nelson, 1976).

The damage in structure are usually located in a limited number of elements only, that is, the damage index is a sparse vector. Therefore, the sparse solution of equation (13) can be obtained through sparse regularization. In the optimization of sparse regularization, the l1 regularized objective function can be defined as (Wright et al., 2009)

J_{1} = {‖ {R (γ)} - {R^{D}} ‖}_{2}^{2} + τ {‖ {γ} ‖}_{1}

(16)

where

{R (γ)} = [S] {γ} + {R^{U}}

{‖ {R (γ)} - {R^{D}} ‖}_{2}^{2}

is the residual norm and narrows the difference between

{R (γ)}

and

{R^{D}}

{‖ {γ} ‖}_{1}

is the l₁-norm of vector

{γ}

, and the regularization term

{‖ {γ} ‖}_{1}

forces

{γ}

to be a sparse vector. τ is the regularization parameter that balances the solution sparsity and residual norm. The value of τ should be smaller than τ_max, and τ_max is calculated as (Kim et al., 2007)

τ_{\max} = {‖ 2 {[S]}^{T} {Δ R} ‖}_{\infty}

(17)

where

{‖ 2 {[S]}^{T} {Δ R} ‖}_{\infty}

is the l_∞-norm of vector

2 {[S]}^{T} {Δ R}

. The value of regularization parameter τ has significant impact on the solution and it is set to τ = 0.01τ_max in this paper.

Equation (16) is rewritten as

J_{1} = \frac{1}{n_{k}} \sum_{k = 1}^{n_{k}} {[\frac{ω_{k} (γ) - ω_{k}^{D}}{ω_{k}^{U}}]}^{2} + \frac{1}{n_{k} \times n_{p}} \sum_{k = 1}^{n_{k}} \sum_{j = 1}^{n_{p}} {[ϕ_{j k} (γ) - {ϕ_{j k}}^{D}]}^{2} + \frac{τ}{n_{e}} {‖ {γ} ‖}_{1}

(18)

where ω_k is the k-th natural frequency of the substructure,

ϕ_{j k}

is the k-th normalized substructural mode shape at j-th measuring point, and n_p denotes the number of measuring points. The natural frequency changes are normalized by dividing the values from undamaged state. To make the natural frequency differences, mode shape differences, and regularization term comparable, the three terms are divided by vector lengths (i.e., n_k, n_k × n_p, and n_e, respectively). The optimization problem in equation (18) can be efficiently solved using standard optimization algorithms, such as truncated Newton interior-point method (Kim et al., 2007). The locations and severities of damage can be indicated by the nonzero entries in

{γ}

Traditionally, equation (13) can also be solved through Tikhonov regularization. Compared with sparse regularization, the regularization term in the objective function should be changed as

J_{2} = \frac{1}{n_{k}} \sum_{k = 1}^{n_{k}} {[\frac{ω_{k} (γ) - ω_{k}^{D}}{ω_{k}^{U}}]}^{2} + \frac{1}{n_{k} \times n_{p}} \sum_{k = 1}^{n_{k}} \sum_{j = 1}^{n_{p}} {[ϕ_{j k} (γ) - {ϕ_{j k}}^{D}]}^{2} + \frac{τ}{n_{e}} {‖ {γ} ‖}_{2}^{2}

(19)

Numerical study: A frame structure

The numerical simulation on a frame structure is conducted to demonstrate the accuracy of the proposed approach. The frame is modeled using 45 Euler–Bernoulli beam elements with length of 2 m each, as shown in Figure 2(a). The frame model has 44 nodes and 126 DOFs. The moment of inertia of each element is 0.0031 m⁴, and the area of the cross-section is 0.046 m². The elemental material properties are set as: elastic modulus is 55 GPa, mass density is 2400 kg/m³, and Poisson’s ratio is 0.3. The Rayleigh damping matrix is calculated as $C = 0.1795 M + 3.965 \times 10^{- 4} K$ , making the first two-order modal damping ratios to be approximately 1%.

Figure 2.

Model of frame structure. (a) Global model (b) Model of the substructures.

The global frame model is disassembled into three substructures, and the model of each substructure is built independently, as shown in Figure 2(b). The submodels are also built by Euler–Bernoulli beam elements which have the same geometrical dimensions and material properties with the elements of the global model. After division, 17, 15, and 13 elements are found in the three substructures, and the interface DOFs of the substructures are fixed.

Table 1 gives two damage cases to verify the proposed substructuring approach. The stiffness reduction of element 33 (element 1 of substructure 3) is 5% for damage case 1. Subsequently, the stiffness of elements 2 (element 2 of substructure 1) and 33 are reduced by 20% for damage case 2. The frame structure is excited at node 6 with Gaussian white noise in horizontal direction. The dynamic responses of the frame are numerically simulated with the Newmark-beta method. The sampling frequency is set to 2000 Hz with a duration length of 1 s.

Table 1.

The damage cases in the numerical study of a frame.

Damage case	Description
0	Undamaged
1	Stiffness of the element 33:5% reduction
2	Stiffness of the element 2 and element 33:20% reduction

The natural frequencies and mode shapes of the three substructures can be obtained from the substructural ARMAX model. The three substructures are treated as target substructures respectively, and each target substructure is treated as independent structure for analysis. Taking substructure 3 as an example, the global acceleration responses at nodes 1 and 10 (nodes 23 and 32 of the global model) are regarded as inputs, and the global acceleration responses at nodes 2–9 and 11–14 (nodes 33–44 of the global model) are regarded as outputs of the substructural ARMAX model. A matrix [G] is constructed in accordance with equation (5) using the autoregressive coefficients of the substructural ARMAX model. The frequencies and mode shapes of substructure 3 can be identified from the eigenvalues and eigenvectors of matrix [G].

To demonstrate the accuracy of the proposed substructural ARMAX model-based method in identifying substructural modal parameters, the exact values of the modal parameters of substructure 3 are obtained by the modal analysis of the submodel in Figure 2(b). The relative errors of the identified mode shapes are evaluated using an index named Error norm, which is expressed as

E r r o r n o r m = \frac{n o r m ({ϕ_{k}}_{ex} - {ϕ_{k}}_{id})}{n o r m ({ϕ_{k}}_{ex})}

(20)

where {ϕ_k}_ex is the k-th exact mode shape calculated by the modal analysis of the submodel, and {ϕ_k}_id is the k-th mode shape identified by substructural ARMAX model coefficients. The accuracy of the substructural ARMAX model in identifying the first 10 substructural modal parameters under undamaged state (case 0) is demonstrated in Table 2. For substructure 3, the first 10 frequency relative errors are less than 1.47 × 10⁻³, and the average value of the first 10 frequency relative errors is 5.10 × 10⁻⁴. The relative errors of the first 10 mode shapes are less than 2.47 × 10⁻¹⁰, and the average relative error of the first 10 mode shapes is 6.14 × 10⁻¹¹. The average relative error of the first 10 frequencies and mode shapes for substructure 1 are 5.27 × 10⁻⁴ and 4.59 × 10⁻⁴, and those values are 7.86 × 10⁻⁴ and 2.36 × 10⁻⁹ for substructure 2. As shown in Table 2, the substructural frequencies and mode shapes are accurately identified by the autoregressive coefficients of the substructural ARMAX model.

Table 2.

The exact and identified substructural frequencies and mode shapes under undamaged state. (frame structure).

Mode	Substructure 1				Substructure 2				Substructure 3
	Exact	Identified			Exact	Identified			Exact	Identified
	Natural frequency(Hz)	Natural frequency (Hz)	Relative error	Mode shape	Natural frequency (Hz)	Natural frequency (Hz)	Relative error	Mode shape	Natural frequency (Hz)	Natural frequency (Hz)	Relative error	Mode shape
	Natural frequency(Hz)	Natural frequency (Hz)	Relative error	Error norm	Natural frequency (Hz)	Natural frequency (Hz)	Relative error	Error norm	Natural frequency (Hz)	Natural frequency (Hz)	Relative error	Error norm
1	32.626	32.623	9.65 × 10⁻⁵	5.03 × 10⁻⁵	36.854	36.853	4.47 × 10⁻⁵	2.50 × 10⁻⁹	8.976	8.976	2.65 × 10⁻⁶	6.40 × 10⁻¹²
2	34.147	34.144	8.21 × 10⁻⁵	7.77 × 10⁻⁴	43.909	43.906	6.34 × 10⁻⁵	1.56 × 10⁻⁹	27.963	27.962	2.57 × 10⁻⁵	6.83 × 10⁻¹²
3	43.868	43.866	5.77 × 10⁻⁵	2.85 × 10⁻⁵	62.034	62.026	1.26 × 10⁻⁴	4.63 × 10⁻¹⁰	58.890	58.883	1.14 × 10⁻⁴	1.06 × 10⁻¹⁰
4	69.770	69.758	1.72 × 10⁻⁴	2.34 × 10⁻⁴	89.990	89.966	2.66 × 10⁻⁴	9.22 × 10⁻¹⁰	60.435	60.428	1.20 × 10⁻⁴	6.30 × 10⁻¹²
5	107.394	107.354	3.74 × 10⁻⁴	2.11 × 10⁻⁵	110.896	110.851	4.03 × 10⁻⁴	7.31 × 10⁻¹⁰	90.699	90.675	2.70 × 10⁻⁴	1.17 × 10⁻¹¹
6	110.649	110.605	4.02 × 10⁻⁴	7.43 × 10⁻⁴	152.631	152.515	7.61 × 10⁻⁴	1.06 × 10⁻⁹	128.232	128.163	5.38 × 10⁻⁴	8.29 × 10⁻¹²
7	129.485	129.428	4.43 × 10⁻⁴	1.98 × 10⁻⁴	188.935	188.716	1.16 × 10⁻³	2.24 × 10⁻⁹	147.486	147.381	7.11 × 10⁻⁴	2.47 × 10⁻¹⁰
8	182.709	182.508	1.10 × 10⁻³	3.06 × 10⁻⁴	204.014	203.739	1.35 × 10⁻³	2.95 × 10⁻⁹	152.424	152.308	7.59 × 10⁻⁴	3.36 × 10⁻¹¹
9	193.039	192.796	1.26 × 10⁻³	9.51 × 10⁻⁴	234.712	234.294	1.78 × 10⁻³	7.93 × 10⁻⁹	182.525	182.327	1.09 × 10⁻³	1.58 × 10⁻¹⁰
10	198.645	198.390	1.28 × 10⁻³	1.28 × 10⁻³	242.518	242.057	1.90 × 10⁻³	3.28 × 10⁻⁹	213.196	212.881	1.47 × 10⁻³	2.97 × 10⁻¹¹
Average			5.27 × 10⁻⁴	4.59 × 10⁻⁴			7.86 × 10⁻⁴	2.36 × 10⁻⁹			5.10 × 10⁻⁴	6.14 × 10⁻¹¹

The substructural frequencies and mode shapes under damaged state are also identified, and the identification process is similar with that of the undamaged state. The substructure with damage can be discovered by comparing the substructural modal parameters of each substructure before and after damage. For damage case 1, the damaged element (element 33 of the global model) is located in substructure 3, whereas no damage exist in substructures 1 and 2. The similarity of the mode shapes under intact state and damaged state can be given by Modal Assurance Criterion (MAC), and the MAC value is expressed as (Ewins, 2000)

M A C ({ϕ_{k}^{U}}, {ϕ_{k}^{D}}) = \frac{{| {ϕ_{k}^{U}}^{T} {ϕ_{k}^{D}} |}^{2}}{({ϕ_{k}^{U}}^{T} {ϕ_{k}^{U}}) ({ϕ_{k}^{D}}^{T} {ϕ_{k}^{D}})}

(21)

where

{ϕ_{k}^{U}}

denotes the k-th mode shape of the intact structure, and

{ϕ_{k}^{D}}

denotes the k-th mode shape with damage in the structure. Table 3 shows the identified change in substructural frequencies and mode shapes caused by damage case 1. Compared with the undamaged state, the natural frequencies of the substructure decrease when damage exist in the substructure. In damage case 1, the average frequency decrease for substructure 3 is −0.30%, and the average MAC value is 0.9965.

Table 3.

Natural frequency and mode shape changes of substructure 3 under damage case 1. (frame structure).

Mode	Natural frequency (Hz)	Difference (%)^a	Mode shape
Mode	Natural frequency (Hz)	Difference (%)^a	MAC^b
1	8.922	−0.60	1.0000
2	27.931	−0.11	1.0000
3	58.686	−0.33	0.9875
4	60.249	−0.30	0.9851
5	90.492	−0.20	0.9999
6	127.574	−0.46	0.9987
7	146.722	−0.45	0.9965
8	151.966	−0.22	0.9975
9	181.862	−0.25	0.9997
10	212.690	−0.09	0.9999
Average		−0.30	0.9965

^aThe ‘Difference (%)’ means the relative difference of the identified frequencies between the undamaged case and the damaged cases.

^bThe ‘MAC’ means the MAC value of the identified mode shapes between the undamaged case and the damaged cases.

For comparison, the frequencies and mode shapes of the global structure are obtained by the modal analysis of the global frame model in Figure 2(a), as shown in Table 4. In damage case 1, the average frequency decrease for the global structure is −0.04%, and the average MAC value is 1.0000. The modal parameters of substructure 3 have larger changes than that of the global structure. The average decrease in natural frequencies for substructure 3 is approximately seven times higher than the global natural frequencies. This finding implies that substructural modal parameters experience larger changes than that of the global modal parameters. In other words, the sensitivity of substructural properties to local structural damage are higher than that of the global properties.

Table 4.

Natural frequency and mode shape changes of the global frame structure.

Mode	Case 0	Case 1			Case 2
Mode	Natural frequency (Hz)	Natural frequency (Hz)	Difference (%)^a	MAC^b	Natural frequency (Hz)	Difference (%)^a	MAC^b
1	1.860	1.860	0.00	1.0000	1.852	−0.42	1.0000
2	6.169	6.167	−0.03	1.0000	6.145	−0.39	1.0000
3	11.188	11.180	−0.08	1.0000	11.138	−0.45	0.9999
4	20.256	20.238	−0.09	1.0000	20.168	−0.43	0.9998
5	24.726	24.717	−0.03	1.0000	24.637	−0.36	0.9986
6	26.810	26.806	−0.02	1.0000	26.723	−0.32	0.9985
7	30.843	30.816	−0.09	1.0000	30.707	−0.44	0.9995
8	35.545	35.537	−0.02	1.0000	35.249	−0.83	0.9985
9	40.433	40.429	−0.01	1.0000	40.315	−0.29	0.9881
10	41.103	41.084	−0.05	0.9999	40.950	−0.37	0.9853
Average			−0.04	1.0000		−0.43	0.9968

^aThe ‘Difference (%)’ means the relative difference of the identified frequencies between the undamaged case and the damaged cases.

^bThe ‘MAC’ means the MAC value of the identified mode shapes between the undamaged case and the damaged cases.

To identify the damage in damage case 1, the identified substructural modal parameters from the undamaged state and damage case 1 are used to establish equation (13). Only substructure 3 is analyzed for damage identification because the damaged element is located in substructure 3. Thirteen elements are found in substructure 3, and only element 1 is damaged. Thus, only one entry in the actual SRF vector is nonzero, and the other entries in the actual SRF vector are zero. In other words, the actual SRF vector is sparse. Accordingly, equation (13) is solved through sparse regularization. The identification result for substructure 3 in the first damage case is displayed in Figure 3. The SRF identified by sparse regularization under the condition of no noise is −0.05 for element 1, which agrees well with the actual value. By contrast, the SRF values are approximately zero at other elements. Tikhonov regularization is also applied to solved equation (13), and the identification result of the first damage case is displayed in Figure 3. The SRF identified by Tikhonov regularization under the condition of no noise is −0.053 for element 1, and there are several observable nonzero SRF values at other elements. The practical measurement is consistently associated with measurement noises. In this study, 5% root-mean-square random noise is added into the calculated acceleration responses. The acceleration with noise are utilized to construct the substructural ARMAX model, and the substructural frequencies and mode shapes are calculated by the autoregressive coefficients of the substructural ARMAX model. The damage identification process with noise is similar to the identification process without noise, and the identification result of the first damage case with 5% noise by sparse regularization are illustrated in Figure 3. A noticeable SRF value is found at element 1, and the SRF values elsewhere are approximately zero or smaller than the value at element 1. The SRF value is −0.042 for element 1, which is compatible with the actual damage severity of −0.05. This finding indicates that the proposed method can detect the location and severity of the single damage in the frame structure with measurement noise considered.

Figure 3.

The identified SRF values for Substructure 3 in the damage case 1. (frame structure).

The substructural frequencies and mode shapes under damaged case 2 are also identified, and only the modal parameter for substructures 1 and 3 are extracted because the damaged elements (elements 2 and 33 of the global model) are located in substructures 1 and 3, respectively. Table 5 shows the identified substructural modal parameter changes induced by damage case 2. For substructure 1, the average decrease in substructural natural frequencies is −0.56%, and the average MAC value is 0.9868. For substructure 3, the average decrease in substructural natural frequencies is −1.33%, and the average MAC value is 0.9564. In damage case 2, the average decrease in global natural frequencies is −0.43%, and the average MAC value is 0.9968, as shown in Table 4. The comparison of Tables 4 and 5 shows that the modal parameters of substructures 1 and 3 under damage case 2 have larger changes than the global modal parameters.

Table 5.

Natural frequency and mode shape changes of the substructures under damage case 2. (frame structure).

Substructure 1				Substructure 3
Mode	Natural frequency (Hz)	Difference (%)^a	Mode shape	Natural frequency (Hz)	Difference (%)^a	Mode shape
Mode	Natural frequency (Hz)	Difference (%)^a	MAC^b	Natural frequency (Hz)	Difference (%)^a	MAC^b
1	32.573	−0.15	0.9990	8.744	−2.58	0.9999
2	34.114	−0.09	0.9943	27.826	−0.49	0.9998
3	43.752	−0.26	0.9998	57.819	−1.81	0.8537
4	69.331	−0.61	0.9989	59.910	−0.86	0.8311
5	105.900	−1.35	0.9937	89.820	−0.94	0.9978
6	110.624	0.02	0.9998	125.236	−2.28	0.9739
7	127.966	−1.13	0.9949	144.544	−1.93	0.9485
8	181.487	−0.56	0.9955	150.888	−0.93	0.9658
9	191.204	−0.83	0.9526	180.278	−1.12	0.9947
10	197.149	−0.63	0.9396	212.079	−0.38	0.9989
Average		−0.56	0.9868		−1.33	0.9564

^aThe ‘Difference (%)’ means the relative difference of the identified frequencies between the undamaged case and the damaged cases.

^bThe ‘MAC’ means the MAC value of the identified mode shapes between the undamaged case and the damaged cases.

Similar to the identification process of the first damage case, the identified substructural modal parameters from the undamaged state and damage case 2 are used to establish equation (13). Only substructures 1 and 3 are analyzed for damage identification because the damaged elements are located in substructures 1 and 3. The substructures are regarded as independent structures, and the damages in substructures 1 and 3 are respectively identified. Seventeen elements are found in substructure 1, and only element 2 is damaged. Therefore, the actual damage solution should be a sparse vector. Accordingly, sparse regularization is used to identify the damage in substructure 1, and the identification result is shown in Figure 4(a). The SRF value under the condition of no noise by sparse regularization is −0.21 for element 2, and the SRF values are approximately zero at other elements. The identification result by Tikhonov regularization is also shown in Figure 4(a). The SRF value under the condition of no noise by Tikhonov regularization is −0.27 for element 2, and there are several noticeable nonzero SRF values at other elements. When 5% noise is considered, the damage in the frame can still be detected. The SRF value with noise is −0.23 for element 2, and the SRF values elsewhere are zero or relatively small. For substructure 3, sparse regularization is used for damage identification, and the identification result is shown in Figure 4(b). A noticeable SRF value is found at element 1, and the identified SRF values elsewhere are zero. The SRF value at element 1 under the condition of no noise is extremely close to the actual value of −0.20. Furthermore, the SRF value at element 1 under the condition of no noise by Tikhonov regularization is −0.24, and there are several noticeable nonzero SRF values at other elements. When 5% noise is considered, the SRF value by sparse regularization is −0.17 for element 1. This finding indicates that the proposed method can detect the locations and severities of the multiple damage in the frame structure with measurement noise considered.

Figure 4.

Damage identification results of Substructure 1 and 3 in damage case 2. (frame structure). (a) Substructure 1. (b) Substructure 3.

The study on the frame structure indicates that the substructural modal parameters can be accurately identified using the autoregressive coefficients of the substructural ARMAX model, and the sensitivity of substructural modal parameters to local structural damage are higher than that of the global modal parameters. The damage locations and severities in the frame are correctly identified using the proposed substructural damage identification method with measurement noise considered. Additionally, the comparison of the damage identification results by sparse regularization and Tikhonov regularization shows that sparse regularization provides a considerably more accurate result than the traditional Tikhonov regularization.

Experimental study: A shear building structure

In this section, the laboratory test of a shear building structure is used to verify the effectiveness of the proposed substructural damage detection approach, as displayed in Figure 5 (An et al., 2014). Each stories in the shear building structure is identical, and the height and width of each story are 210 and 260 mm, respectively. The lumped mass is 2.17 kg in each story and is composed of plastic beam, steel mass, and bolts. The model was first tested in the undamaged state. Then, the structural damage were introduced by replacing the columns of the damaged stories with slightly thinner columns, and the stiffness reduction of the damaged stories are approximately 20% (An et al., 2014). Table 6 presents the damage cases of the shear building structure. In the three single damage cases (damage cases 1, 2, and 3), the stiffness of the fourth, fifth, and sixth stories reduce by 20%, respectively. The stiffness of the fourth and sixth stories reduce by 20% for the multiple damage case (damage case 4).

Figure 5.

Photograph of the shear building structure.

Table 6.

Damage cases of the shear building structure.

Damage case		Description
Single damage	0	Undamaged
	1	k₄ is reduced by 20%
	2	k₅ is reduced by 20%
	3	k₆ is reduced by 20%
Multiple damage	4	k₄ and k₆ are reduced by 20%

The experimental model shown in Figure 5 was installed on a shaking table and was excited by band-limited white noise at the base. The horizontal responses of the shear building are measured by the six accelerometers placed on the steel masses of each story, and one accelerometer was placed on the base of the model to record the acceleration of the base. The sensitivity of the accelerometers is approximately 100 mV/g. The acceleration responses were recorded using a VIBPILOT data acquisition system, and the cutoff frequency was 15 Hz. For all damage cases, the sampling frequency was chosen as 400 Hz, and 300 s structural acceleration responses were recorded. One typical acceleration response measured at the top of the shear building is illustrated in Figure 6.

Figure 6.

Typical acceleration response of the shear building structure.

An ARMAX model for the global shear building structure can be constructed by treating the acceleration of the base as input and the acceleration response of the six floors as outputs. In accordance with equation (5), the autoregressive coefficients of the ARMAX model are used to construct a matrix [G]. The frequencies and mode shapes of the shear building can be identified using the eigenvalues and eigenvectors of matrix [G]. The identified first three global modal parameters of the shear building are displayed in Table 7. The global natural frequencies decrease when cuts are introduced into the shear building. The average frequency reduction for the four damage cases are −1.52%, −1.88%, −1.85%, and −2.90%, respectively. The averages of MAC values are 0.998, 0.996, 0.993, and 0.989 for the four damage cases, respectively.

Table 7.

The identified frequencies and mode shapes of the global shear building structure.

Mode	Case 0	Case 1			Case 2			Case 3			Case 4
Mode	Natural frequency (Hz)	Natural frequency (Hz)	Difference (%)^a	MAC^b	Natural frequency (Hz)	Difference (%)^a	MAC^b	Natural frequency (Hz)	Difference (%)^a	MAC^b	Natural frequency (Hz)	Difference (%)^a	MAC^b
1	1.715	1.704	−0.62	0.999	1.717	0.13	1.000	1.697	−1.04	0.999	1.702	−0.78	1.000
2	5.221	5.041	−3.44	0.997	5.025	−3.74	0.995	5.143	−1.49	0.997	5.008	−4.07	0.998
3	8.407	8.366	−0.49	0.996	8.238	−2.02	0.993	8.153	−3.03	0.983	8.083	−3.85	0.970
Average			−1.52	0.998		−1.88	0.996		−1.85	0.993		−2.90	0.989

^aThe ‘Difference (%)’ means the relative difference of the identified frequencies between the undamaged case and the damaged cases.

^bThe ‘MAC’ means the MAC value of the identified mode shapes between the undamaged case and the damaged cases.

The proposed substructural damage identification approach is applied to detect the damage in the shear building. The shear building structure is represented using the 6-DOF mass point system in Figure 7(a). The global shear building structure is separated into two substructures, and the substructural models are constructed independently, as shown in Figure 7(b). The two partitioned substructures are treated as target substructures, and each target substructure is treated as independent structure for analysis. The modal parameters of the target substructure can be identified through the autoregressive coefficients of the substructural ARMAX model. For substructure 1, the acceleration of the base and the acceleration of the third floor are regarded as inputs of the substructural ARMAX model, and the acceleration of the first and second floors are regarded as outputs of the substructural ARMAX model. For substructure 2, the acceleration of the base and the acceleration response of the third floor are regarded as inputs of the substructural ARMAX model, and the acceleration response of the fourth, fifth, and sixth floors are regarded as outputs of the substructural ARMAX model. Then, the modal parameters of the two independent substructures are identified, and the identified modal parameters of the two substructures are displayed in Table 8. The average decrease in substructural natural frequencies of substructure 1 in the four damage cases are −0.08%, −0.10%, −0.23%, and −0.11%, and those values are −3.79%, −3.80%, −3.41%, and −7.18% for substructure 2. The averages of MAC values are 0.999, 1.000, 0.999, and 0.994 for substructure 1 in the four damage cases, and those values are 0.997, 0.994, 0.991, and 0.997 for substructure 2. In the four damage cases, the changes in modal parameters of substructure 2 are larger than those in substructure 1, indicating that damage exists in substructure 2. This finding is consistent with the actual damage case, that is, all the damaged columns are in substructure 2.

Figure 7.

Model of the shear building structure. (a) Global model. (b) Model of the substructures.

Table 8.

The identified frequencies and mode shapes of the substructures. (shear building structure).

	Mode	Case 0	Case 1			Case 2			Case 3			Case 4
	Mode	Natural frequency (Hz)	Natural frequency (Hz)	Difference (%)^a	MAC^b	Natural frequency (Hz)	Difference (%)^a	MAC^b	Natural frequency (Hz)	Difference (%)^a	MAC^b	Natural frequency (Hz)	Difference (%)^a	MAC^b
Substructure 1	1	7.215	7.226	0.14	0.998	7.209	−0.09	1.000	7.193	−0.31	0.997	7.208	−0.10	0.988
	2	12.686	12.646	−0.31	1.000	12.672	−0.11	1.000	12.666	−0.16	1.000	12.670	−0.12	1.000
	Average			−0.08	0.999		−0.10	1.000		−0.23	0.999		−0.11	0.994
Substructure 2	1	3.320	3.100	−6.64	0.998	3.182	−4.17	0.999	3.294	−0.78	0.999	3.073	−7.45	0.999
	2	9.466	9.117	−3.69	0.996	9.297	−1.78	0.991	8.908	−5.89	0.986	8.585	−9.30	0.996
	3	13.913	13.765	−1.06	0.998	13.157	−5.43	0.991	13.416	−3.57	0.987	13.249	−4.77	0.995
	Average			−3.79	0.997		−3.80	0.994		−3.41	0.991		−7.18	0.997

^aThe ‘Difference (%)’ means the relative difference of the identified frequencies between the undamaged case and the damaged cases.

^bThe ‘MAC’ means the MAC value of the identified mode shapes between the undamaged case and the damaged cases.

The substructural modal parameters of substructure 2 have larger changes compared with the global modal parameters listed in Table 7. The average decrease in substructural frequencies of the four damage cases are approximately two times higher than the average decrease in global natural frequencies. This finding implies that the local damage causes larger changes in the substructural modal parameters. In other words, the sensitivity of substructural properties to local structural damage are higher than that of the global properties. Subsequently, the substructural modal parameters are utilized to identify the damages in the shear building structure.

The measured substructural modal parameters are used to establish equation (13). The damaged columns in all four damage cases are located in substructure 2. Thus, only substructure 2 is analyzed in the following sparse damage identification process. The sensitivity matrix of the substructural modal parameters to the SRF can be calculated from the submodels in Figure 7(b). Then, sparse regularization is utilized to solve equation (13). The identification results of the single damage cases (cases 1–3) are displayed in Figure 8(a)–(c), respectively. In each figure, only one noticeable identified SRF value is found at the actual damaged story, and the SRF values elsewhere are zero or relatively small. The SRF values of the actual damaged story are −0.21, −0.22, and −0.21 for the three single damage cases, respectively, which agrees well with the actual damage extent of −0.20. The identification result of the multiple damage case (case 4) is demonstrated in Figure 8(d). The identified SRF values at the first and third elements (fourth and sixth stories in the global structure) are remarkable, whereas the SRF value at the second element is approximately zero. This result is in agreement with the practical damage location in damage case 4. The SRF value is −0.21 at the first element, which agrees well with the actual damage severity of −0.20. For the third element, the identified SRF value is −0.20, indicating that the damage severity is exactly identified. This finding indicates that the proposed substructural damage identification method can identify the locations and severities of the damages in the shear building structure.

Figure 8.

The identified SRF values for Substructure 2. (shear building structure). (a) Case 1. (b) Case 2. (c) Case 3. (d) Case 4.

The laboratory test on a shear building structure shows that the changes in identified modal parameters of damaged substructures are larger than those of undamaged substructures. Thus, the substructure with damage can be discovered by comparing the identified modal parameters of each substructure before and after damage. The sensitivity of substructural modal parameters to local structural damage is higher than that of the global modal parameters. The damage identification results indicate that the proposed method can identify the locations and severities of the damage in the shear building structure.

Conclusions

In this study, a substructuring approach is proposed to identify the damage in structure. The large-size structure is divided into small substructures, and the substructural ARMAX model corresponding to the target substructure is constructed by rearranging the structural motion equation. The modal parameters of the target substructure are estimated using the autoregressive coefficients of the substructural ARMAX model. Subsequently, the substructural modal parameters are used as damage indicators, and sparse regularization is utilized to locate and quantify structural damage.

A simulated frame structure and a laboratory-tested shear building structure are utilized to verify the effectiveness of the proposed substructural damage identification approach. The modal parameters of the substructures are accurately identified using the autoregressive coefficients of the substructural ARMAX models. The modal parameters of the substructure experience larger changes induced by local damage than that of the global modal parameters, that is, the substructural modal parameters are more sensitive to local structural damage than that of the global modal parameters. The proposed structural damage identification method can accurately identify the location and severity of damage with measurement noise considered.

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 work is supported by the grants from National Key R&D Program of China (contract number: 2021YFB2600400 and 2021YFF0501001), National Natural Science Foundation of China (NSFC, contract number: 51922046 and 51838006), China Postdoctoral Science Foundation (2021M703782), Major S&T Project of China Railway Construction Corporation CO.LTD (contract number: 2020-A01 and 2022-A02), and the Research Fund of China Railway Siyuan Survey and Design Group CO.LTD (contract number: 2021K085 and 2020D006).

ORCID iD

Hong Yu

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