Modal Identification of damped vibrating systems by iterative smooth orthogonal decomposition method

Abstract

The smooth orthogonal decomposition method (SOD) is an efficient algorithm that can be used to extract modal matrix and frequencies of lightly damped vibrating systems. It uses the covariance matrices of output-only displacement and velocity responses to form a generalized eigenvalues problem (EVP). The mode shape vectors are estimated by the eigenvectors of the EVP. It is stated in this work that the accuracy of the SOD method is mainly affected by the correlation characteristic of modal coordinate responses. For the damped vibration systems, biases will be contained in the results of using the SOD. Therefore, an iterative smooth orthogonal decomposition (ISOD) method is proposed to identify modal parameters of the damped system from the covariance matrices of the displacement, velocity, and acceleration responses. The modal matrix given by the SOD method is updated in each iteration step using a transformation matrix. The transformation matrix can be efficiently computed using a set of analytical formulations. Meanwhile, natural frequencies and damping ratios are obtained by using a simple search method. The performance of the proposed ISOD method is verified by numerical and experimental studies. The results demonstrate that, by considering the correlation of modal responses, the ISOD method can be used to extract accurately the modal information of vibration systems with coupled modes.

Keywords

structural health monitoring modal identification operational modal analysis smooth decomposition method blind source separation

Introduction

Extracting structural modal information from vibration measurement data has always been an important issue in structural vibration analysis. Modal parameter identification using output responses only has been widely used in engineering applications, since input excitations are not required to be measured (Ren and Zong, 2004; Sadhu et al., 2017). To date, numerous operational modal analysis (OMA) methods have been proposed. The most commonly used methods include the stochastic subspace identification (Chen et al., 2019; Liu and Loh, 2011; Zhang et al., 2012) and the frequency domain decomposition (Brincker et al., 2000; Pioldi and Rizzi, 2017). The stochastic subspace identification (SSI) is a time-domain identification method with relatively complicated operations. The Hankel matrix is formed by rearranging the measured data, and then the modal parameters are obtained by matrix decomposition technique. Finally, the stabilization diagram or other methods are used to select the real modes and remove the spurious ones. The frequency domain decomposition (FDD) is a frequency domain identification method and particularly suitable for well-separated modal identification.

In addition, many simpler methods have been developed, such as the proper orthogonal decomposition (POD) (Feeny and Kappagantu, 1998; Feeny and Liang, 2003; Kerschen and Golinval, 2002). The POD, also known as principal component analysis (PCA) or Karhunen–Loeve (KL) decomposition, aims to reduce the multidimensional data based on singular value decomposition (SVD) so that the dominant components sorted in singular values (descending order) can retain the main energy of the data (Bellizzi and Sampaio, 2006; Quaranta et al., 2008). These studies have revealed the relationship between the linear normal modes (LNM) and the proper orthogonal modes (POM) obtained from the POD. Although the POD algorithm is very simple and efficient, it can only be applied to systems with uniform mass distribution ( $M = m I$ ) or with mass distribution known in prior. To overcome this limitation, several covariance matrices based methods have been proposed, such as the modal analysis by free vibration response only method (MAFVRO) (Rezaee et al., 2015; Wang and Cheng, 2008), nonsymmetric state-variable decomposition (Feeny and Farooq, 2008) and the smooth orthogonal decomposition (SOD) (Chelidze and Zhou, 2006; Farooq and Feeny, 2008; Hu et al., 2018). Their common feature is that the covariance matrices of response data are used to construct different generalized eigenvalue problems (EVPs) and obtain modal parameters by solving the EVPs. Among these methods, the SOD has received wide attention. Farooq et al. have applied SOD for the modal analysis of random exciting systems (Farooq and Feeny, 2008). Rezaee et al. (2013) used SOD to analyze a vehicle vibration excited by time shift non-independent inputs caused by the rough surface of the road. Rezaee et al. tried to improve the frequency identification accuracy of using SOD by improving the numerical differentiation method (Rezaee and Yam, 2015). Although they noticed that the system damping, the excitation arrangement form, and the derivative method will affect the identification accuracy of using SOD, the underlying theoretical reasons are not analyzed. This article will investigate the correlation properties of the modal coordinate responses which determine whether SOD can be accurately used to identify the modal parameters.

In recent years, more and more BSS algorithms are applied to the modal parameter identification under environmental excitations (Kerschen et al., 2007; Zhou and Chelidze, 2007). The modal parameter identification problem using the output-only responses is mathematically equivalent to a special case of the BSS problem. The physical coordinate vibration responses are $x (t) = Φ q (t)$ . The modal matrix $Φ$ and the modal coordinate responses $q (t)$ can be regarded as the mixed matrix and source signal of the BSS problem, respectively. Some commonly used BSS algorithms include the algorithm for multiple unknown signals extraction (Zhou and Chelidze, 2007), the second-order blind identification (Antoni and Chauhan, 2013), and the complexity pursuit method (Yang and Nagarajaiah, 2013). The complexity pursuit (CP) method can extract the component that has the smallest variation under any smooth linear filter processing. According to this concept, the SOD method can also be regarded as a special form of the generalized CP method. Analog to the CP, the gain of the filters corresponding to displacement and velocity defined in the SOD algorithm are $G (ω) = 1$ and $G (ω) = j ω$ , respectively. Therefore, the SOD has the general characteristics of the generalized CP method. In addition, the raw data used by the SOD correspond to the displacement, velocity, and acceleration responses of the structure (Hu et al., 2018). These raw data can be directly measured when conditions permit. Unlike the BSS algorithm which only provides the modal matrix, the SOD method can directly output the modal natural frequency (asymptotic value).

In most BSS algorithms (including POD and SOD), the modal coordinate response signals are always assumed to be uncorrelated to each other. In undamped or lightly damped vibration systems, the modal coordinate responses of each order can indeed be regarded as satisfying this assumption. However, when the damping is increased or the modes are closely-spaced, the results will be corrupted by errors. Although many studies have shown that the results of modal parameters of damped systems identified by using BSS, POD, and SOD are acceptable, they are biased estimates in nature. For this reason, this paper will study the correlation characteristics of modal coordinate responses of each order and derive the theoretical formula of the covariance matrix of modal coordinate responses at a given frequency band. Based on this analytical formula, an algorithm that iteratively rotates the smooth orthogonal matrix (SOM) obtained by SOD is proposed and obtain a more accurate mode shape. Thus, the proposed method is called iterative smooth orthogonal decomposition method (ISOD). This iterative process works similarly to the expectation-maximization (EM) algorithm that compute the most probable values of the Bayesian operational modal analysis (Li and Au, 2019). In fact, the idea of the iteration is also inspired by the EM algorithm. However, the algorithm in this paper has a higher calculation efficiency, since the EM-based algorithm needs to revisit all FFT data in each iteration, while the proposed approach only needs to calculate the initial covariance matrix, and it is not necessary to access the original data during the iteration.

The contents of this article are arranged as follows: Section 2 introduces the SOD theory and proposes a new form of the generalized eigenvalue problem; Section 3 derives the covariance matrix of the modal coordinate responses; Section 4 provides the structure of the iterative algorithm that improve the identification results of using SOD; Section 5 uses examples to validate the convergence and accuracy of the proposed algorithm.

The SOD method

The original smooth orthogonal decomposition for modal analysis (Chelidze and Zhou, 2006) was proposed to extract modal information of undamped free vibration system. The governing equation of the system is

M \overset{\cdot\cdot}{x} (t) + Kx (t) = 0

(1)

where $M \in R^{n \times n}$ and $K \in R^{n \times n}$ are the mass and stiffness matrices, respectively; $x (t) \in R^{n \times 1}$ and $\overset{\cdot\cdot}{x} (t) \in R^{n \times 1}$ are the displacement and acceleration responses of the system. Let the ensemble matrix $X = [x (t_{1}), x (t_{2}), \dots, x (t_{N})]$ denote the recorded displacement data. Similarly, $\overset{\cdot}{X} = [\overset{\cdot}{x} (t_{1}), \overset{\cdot}{x} (t_{2}), \dots, \overset{\cdot}{x} (t_{N})]$ and $\overset{\cdot\cdot}{X} = [\overset{\cdot\cdot}{x} (t_{1}), \overset{\cdot\cdot}{x} (t_{2}), \dots, \overset{\cdot\cdot}{x} (t_{N})]$ can be obtained. Then the covariance matrix of displacement, velocity and acceleration responses are given as

\begin{matrix} R_{X} = \frac{X X^{T}}{N - 1} \\ R_{\overset{\cdot}{X}} = \frac{\overset{\cdot}{X} {\overset{\cdot}{X}}^{T}}{N - 1} \\ R_{\overset{\cdot\cdot}{X}} = \frac{\overset{\cdot\cdot}{X} {\overset{\cdot\cdot}{X}}^{T}}{N - 1} \end{matrix}

(2)

The SOD method extracts mode shapes and natural frequencies from the generalized eigenvalue problem (EVP) below:

λ R_{X} ψ = R_{\overset{\cdot}{X}} ψ

(3)

where the eigenvector $ψ \in R^{n \times 1}$ is called a SOM corresponding to the smooth orthogonal value (SOV) $λ$ . This EVP can be derived from a constrained extreme value problem:

\frac{\partial λ (ψ)}{\partial ψ} = 0, λ (ψ) = \frac{ψ^{T} R_{X} ψ}{ψ^{T} R_{\overset{\cdot}{X}} ψ}

(4)

Comparing to equation (4) with the principle of CP method (Antoni et al., 2017; Yang and Nagarajaiah, 2013), it is found that the theoretical basis of the SOD is identical to that of the CP. These two residual signals of the CP method is corresponding to displacement and velocity responses of the SOD method. Therefore, the SOD method is a specific version of the CP method. The generalized EVP defined in equation (3) is expressed in the matrix form as

R_{X} Ψ Λ = R_{\overset{\cdot}{X}} Ψ

(5)

where $Λ = diag [λ_{1}, λ_{2}, \dots, λ_{n}]$ and $Ψ = [ψ_{1}, ψ_{2}, \dots, ψ_{n}]$ . It is proven that for the undamped free vibration system: (i) the inverse transpose of the SOM matrix approaches the modal matrix of the system, that is, $Φ = Ψ^{- T}$ , where $Φ$ is the modal matrix; (ii) the $i^{th}$ diagonal elements of the matrix $Λ$ are the squared frequencies, that is, $λ_{i} = ω_{i}^{2}$ .

Let $Q = [q (t_{1}), q (t_{2}), \dots, q (t_{N})] \in R^{n \times N}$ denote the ensemble matrix of modal coordinate displacement responses. And the ensemble matrix $\overset{\cdot}{Q}$ and $\overset{\cdot\cdot}{Q}$ are defined similarly. The covariance matrices of these modal coordinate responses are given as

\begin{matrix} R_{Q} = \frac{Q Q^{T}}{N - 1} \\ R_{\overset{\cdot}{Q}} = \frac{\overset{\cdot}{Q} {\overset{\cdot}{Q}}^{T}}{N - 1} \\ R_{\overset{\cdot\cdot}{Q}} = \frac{\overset{\cdot\cdot}{Q} {\overset{\cdot\cdot}{Q}}^{T}}{N - 1} \end{matrix}

(6)

It is noted that these covariance matrices are diagonal for an un-damped free vibration system. Solving the EVP of equation (5) is equivalent to eigenvalue decomposition of the matrix $R_{X}^{- 1} R_{\overset{\cdot}{X}}$ , and with $X = Φ Q$ and $\overset{\cdot}{X} = Φ \overset{\cdot}{Q}$ , one obtains

R_{X}^{- 1} R_{\overset{\cdot}{X}} = Φ^{- T} R_{Q}^{- 1} R_{\overset{\cdot}{Q}} Φ^{T} = Ψ Λ Ψ^{- 1}

(7)

From equation (7), the accuracy of the SOD method depends on how close the matrix $R_{Q}^{- 1} R_{\overset{\cdot}{Q}}$ is to a diagonal matrix. For the lightly damped vibrating system, the matrix $R_{Q}^{- 1} R_{\overset{\cdot}{Q}}$ is well approximate to a diagonal matrix. However, when the damping ratios of the system increase, a remarkable error will be contained in the identified parameters.

Taking transpose of the matrices in equation (7) and considering matrix symmetry, a new form of the SOD is obtained as

R_{\overset{\cdot}{X}} R_{X}^{- 1} = Φ R_{\overset{\cdot}{Q}} R_{Q}^{- 1} Φ^{- 1} = \tilde{Φ} Λ {\tilde{Φ}}^{- 1}

(8)

It is noted that the modal matrix $Φ$ can be directly calculated by the eigenvalue decomposition of the matrix $R_{\overset{\cdot}{X}} R_{X}^{- 1}$ , where the matrix $\tilde{Φ}$ is the eigenvector matrix. This new form avoids taking the inverse-transpose of the SOM matrix $Ψ$ to compute the modal matrix. If the covariance matrix of modal coordinate responses are diagonal, the eigenvalue decomposition can obtain accurate mode shape, that is, $Φ = \tilde{Φ}$ . Otherwise, since the matrix $R_{\overset{\cdot}{Q}} R_{Q}^{- 1}$ are not strictly diagonal, only an approximate result of mode shape can be estimated, that is, $Φ \approx \tilde{Φ}$ .

The covariance of modal responses

For the damped vibrating system, the decoupled modal coordinate responses are governed by

\overset{\cdot\cdot}{q} (t) + Ξ \overset{\cdot}{q} (t) + Λ q (t) = u (t)

(9)

where $Ξ = diag [2 ω_{1} ξ_{1}, 2 ω_{2} ξ_{2}, \dots, 2 ω_{n} ξ_{n}]$ , and $u (t) \in R^{n \times 1}$ is the modal forces. The relationship between the modal forces and the applied excitation forces $f (t)$ is

u (t) = Φ^{- 1} M^{- 1} f (t)

(10)

Also, let $U = [u (t_{1}), u (t_{2}), \dots, u (t_{N})]$ . The covariance matrix of the modal forces can be written as

R_{u} = \frac{U U^{T}}{N - 1}

(11)

Similarly, $R_{f}$ is defined as the covariance matrix of the excitation forces $f (t)$ , and it is easy to obtain

R_{u} = Φ^{- 1} M^{- 1} R_{f} M^{- 1} Φ^{- T}

(12)

Using Parseval’s theorem (Oppenheim and Schafer, 1999), the element at $i^{th}$ row and $j^{th}$ column of $R_{u}$ is given by

R_{u, ij} = \frac{1}{T} \int_{0}^{T} u_{i} (t) u_{j} (t) d t = \frac{1}{B} \int_{Ω} {\hat{u}}_{i} (ω) {\hat{u}}_{j}^{*} (ω) d ω

(13)

where $T$ is the total sampling duration; $Ω$ is the frequency integration field, and $B$ is the width of the frequency integration field; ${\hat{u}}_{i} (ω)$ is the discrete Fourier transform of the modal force $u_{i} (t)$ ; and ■* represents complex conjugation of ■. For white noises excitation, the correlation of modal coordinate responses can be derived by

R_{Q, ij} = \frac{1}{B} \int_{Ω} {\hat{q}}_{i} (ω) {\hat{q}}_{j}^{*} (ω) d ω = \frac{R_{u, ij}}{B} \cdot \int_{Ω} H_{q, i} (ω) H_{q, j}^{*} (ω) d ω

(14)

where $H_{q, i} (ω)$ is the frequency response function (FRF) of the $i^{th}$ order modal vibration given by

H_{q, i} (ω) = \frac{1}{ω_{i}^{2} - ω^{2} + 2 j ξ ω_{i} ω}

(15)

Through signal derivation the velocity and acceleration FRFs are

H_{\overset{\cdot}{q}, i} (ω) = j ω \cdot H_{q, i} (ω), H_{\overset{\cdot\cdot}{q}, i} (ω) = - ω^{2} \cdot H_{q, i} (ω)

(16)

It is shown in equation (14) that the covariance matrix $R_{Q}$ is determined by two parts:

\begin{matrix} R_{Q} = R_{u} ° D_{Q} \\ R_{\overset{\cdot}{Q}} = R_{u} ° D_{\overset{\cdot}{Q}} \\ R_{\overset{\cdot\cdot}{Q}} = R_{u} ° D_{\overset{\cdot\cdot}{Q}} \end{matrix}

(17)

where $°$ means Hadamard product (the entry-wise product) of two matrices. The element at the $i^{th}$ row and $j^{th}$ column of $D_{Q}$ is given as

\begin{matrix} D_{Q, ij} = \frac{1}{B} \int_{Ω} H_{q, i} (ω) H_{q, j}^{*} (ω) d ω = \frac{I_{q, ij} (Ω)}{B} \\ D_{\overset{\cdot}{Q}, ij} = \frac{1}{B} \int_{Ω} H_{\overset{\cdot}{q}, i} (ω) H_{\overset{\cdot}{q}, j}^{*} (ω) d ω = \frac{I_{\overset{\cdot}{q}, ij} (Ω)}{B} \\ D_{\overset{\cdot\cdot}{Q}, ij} = \frac{1}{B} \int_{Ω} H_{\overset{\cdot\cdot}{q}, i} (ω) H_{\overset{\cdot\cdot}{q}, j}^{*} (ω) d ω = \frac{I_{\overset{\cdot\cdot}{q}, ij} (Ω)}{B} \end{matrix}

(18)

The frequency integration field of the original SOD method is $Ω = [- π f_{s}, π f_{s}]$ , hence $B = 2 π f_{s}$ . The analytical formulas of integrations $I_{q, ij}$ , $I_{\overset{\cdot}{q}, ij}$ , and $I_{\overset{\cdot\cdot}{q}, ij}$ are given in the Appendix. If the recorded signals are pre-filtered by a band-passing filter with a lower bound $ω_{L}$ and an upper bound $ω_{U}$ , the flatness should be true over the selected frequency band only. Analytical computation of the integrations with frequency band $[ω_{L}, ω_{U}]$ can also be carried out. In this situation,

I_{q, ij} (Ω) = I_{q, ij} (ω_{U}) - I_{q, ij} (ω_{L})

(19)

In Figure 1, the colored areas are used to present the integrations (for $\frac{ω_{1}}{ω_{2}} = 0.8$ and $ξ_{1} = ξ_{2} = 0.15$ ). In Figure 1(a), the imaginary and real parts of the integrations of $I_{q, ij} (ω_{U})$ are shown, with the integrations field $[- ω_{U}, ω_{U}]$ . In the same manner, the imaginary and real parts of the integrations of $I_{q, ij} (ω_{L})$ are shown in Figure 1(b), with the integrations field $[- ω_{L}, ω_{L}]$ . As computed by equation (19), the integration result of the function defined in a frequency band is shown in Figure 1(c). For the image part of the integrations, the values of the negative and positive domain cancel each other out, so the integrations are real numbers.

Figure 1.

Schematic diagram of integrations: (a) $I_{q} (ω_{U})$ , (b) $I_{q} (ω_{L})$ , and (c) $I_{q} (ω_{U}) - I_{q} (ω_{L})$ .

Taking the displacement responses as an example, the modal correlation coefficient (MCC) between the $i^{th}$ and the $j^{th}$ order is defined as

ρ_{Iq, ij} = \frac{I_{q, ij} (Ω)}{\sqrt{I_{q, ii} (Ω) I_{q, jj} (Ω)}}

(20)

The MCC is commonly utilized to compute earthquake responses of civil engineering structures (Wilson et al., 1981). With the analytical formulas presented in the Appendix, fast computing of this correlation coefficient can be carried out. This coefficient can be computed for the arbitrary frequency integration domain, so these formulas have potential application prospects in earthquake engineering.

The covariance of modal forces is presented as

R_{u} = Γ ρ_{u} Γ

(21)

where $Γ_{ii} = \sqrt{R_{u, ii}}$ is a diagonal matrix that represents the intensity of the $i^{th}$ modal force, and $ρ_{u}$ is the correlation coefficient matrix of modal forces. Therefore, considering equation (17), the correlation coefficient of modal coordinate responses is determined by two parts

ρ_{q, ij} = ρ_{Iq, ij} ρ_{u, ij}

(22)

The modal forces are not completely correlated in general cases (i.e. $ρ_{u, ij} < 1, i \neq j$ ). This makes the matrix $ρ_{q}$ close to a diagonal matrix. However, when the system is excited at one point (or excited by ground motion), the modal forces are completely correlated (i.e. $ρ_{u, ij} = 1$ ). Then, the SOD method will give results with considerable errors. Therefore, the arrangement of excitation forces can affect the accuracy of the SOD method, which explains the excitation arrangement effect mentioned in (Rezaee and Yam, 2015).

Besides, it is noticed that the damping level also affects the accuracy of the SOD method. To understand this effect, the correlation relationship of a pair of modal coordinate responses is shown in Figure 2, ( $ρ_{u, ij} = 1$ is considered). The areas under the dashed line and dot-dash line depict the integrations $I_{q, 11}$ and $I_{q, 22}$ , and the area under the solid line depicts the integration $I_{q, 12}$ . The ratio of natural frequencies of two modes is 0.8. The damping ratios are set as $ξ_{1} = ξ_{2} = 0.015$ and $ξ_{1} = ξ_{2} = 0.15$ as shown in Figure 2(a) and (b), respectively. For the first case, the correlation coefficient is only 0.02, which can be regarded as well-separated modes. The computed correlation coefficient for the second case is $ρ_{q, ij} = 0.676$ , revealing a high degree of correlation. Thus, the two modes are coupled with each other. In these two cases, only the damping ratio is different, which reveals that difficulty will arise for modal identification with high damping ratios.

Figure 2.

Comparison of: (a) well-separated modes ( $ω_{1} / ω_{2} = 0.8, ξ_{1} = ξ_{2} = 0.015, ρ_{q, ij} = 0.02$ ), and (b) coupled modes ( $ω_{1} / ω_{2} = 0.8, ξ_{1} = ξ_{2} = 0.15 ρ_{q, ij} = 0.676$ ).

Iterative algorithm

According to the covariance of modal coordinate responses defined in the above section, we have

R_{\dot{X}} R_{X}^{- 1} = Φ (D_{\dot{Q}} ° R_{u}) {(D_{Q} ° R_{u})}^{- 1} Φ^{- 1} = Φ P Φ^{- 1}

(23)

where

P = (D_{\dot{Q}} ° R_{u}) {(D_{Q} ° R_{u})}^{- 1}

(24)

The matrix $P$ is determined by the modal frequencies, the damping ratios and the covariance matrix of the modal forces. This matrix cannot be directly given by the SOD method, it can be derived by equations (17) and (18) if the modal parameters ( $ω, ξ, R_{u}$ ) are known. In fact, since the analytical formulas are adopted to compute the integrations (see Appendix), the computation of the matrix $P$ is very fast.

For estimating the modal frequencies and damping ratios, the covariance matrices of modal coordinate data are approximately computed as

\begin{matrix} {\tilde{R}}_{Q} = Φ_{e}^{- 1} R_{X} Φ_{e}^{- T} \\ {\tilde{R}}_{\overset{\cdot}{Q}} = Φ_{e}^{- 1} R_{\overset{\cdot}{X}} Φ_{e}^{- T} \\ {\tilde{R}}_{\overset{\cdot\cdot}{Q}} = Φ_{e}^{- 1} R_{\overset{\cdot\cdot}{X}} Φ_{e}^{- T} \end{matrix}

(25)

The initial value of $Φ_{e}$ can be set to be $\tilde{Φ}$ , and will be iteratively improved. Through the $i^{th}$ diagonal elements of these covariance matrices in equation (25), one obtains

\begin{matrix} λ_{1, i} = \frac{{\tilde{R}}_{\overset{\cdot}{Q}, ii}}{{\tilde{R}}_{Q, ii}} = \frac{I_{\overset{\cdot}{q}, ij} (Ω)}{I_{q, ij} (Ω)} \\ λ_{2, i} = \frac{{\tilde{R}}_{\overset{\cdot\cdot}{Q}, ii}}{{\tilde{R}}_{\overset{\cdot}{Q}, ii}} = \frac{I_{\overset{\cdot\cdot}{q}, ij} (Ω)}{I_{\overset{\cdot}{q}, ij} (Ω)} \end{matrix}

(26)

Equation (26) defines the relationship between the SOVs ( $λ_{1, i}, λ_{2, i})$ and ( $ω_{i}, ξ_{i})$ , since the integrations are only determined by ( $ω_{i}, ξ_{i})$ . Given ( $λ_{1, i}, λ_{2, i})$ , the search of the unknown ( $ω_{i}, ξ_{i})$ is quite fast, a simple searching method can be used (Hu et al., 2018). For the damped systems $λ_{1, i} \approx ω_{i}^{2}$ , so a good initial value of the frequency can be chosen as $\sqrt{λ_{1, i}}$ , and the initial value of the damping ratio can be set to be 0.1. The optimal values of ( $ω_{i}, ξ_{i})$ minimizes the objective function below:

cost = {(λ_{1, i} - \frac{I_{\overset{\cdot}{q}, ij}}{I_{q, ij}})}^{2} + {(λ_{2, i} - \frac{I_{\overset{\cdot\cdot}{q}, ij}}{I_{\overset{\cdot}{q}, ij}})}^{2}

(27)

After obtaining ( $ω_{i}, ξ_{i})$ , the matrices ${\tilde{D}}_{Q}$ , ${\tilde{D}}_{\overset{\cdot}{Q}}$ and ${\tilde{D}}_{\overset{\cdot\cdot}{Q}}$ (approximates of $D_{Q}$ , $D_{\overset{\cdot}{Q}}$ , and $D_{\overset{\cdot\cdot}{Q}}$ respectively) can be computed, then the covariance matrix of modal forces is estimated as

{\tilde{R}}_{u} \approx {\tilde{R}}_{\overset{\cdot\cdot}{Q}} . / {\tilde{D}}_{\overset{\cdot\cdot}{Q}}

(28)

Where “./” represents the entry-wise division of matrices. Therefore, an approximate value of the matrix $P$ can be estimated by substitute ${\tilde{D}}_{Q}$ , ${\tilde{D}}_{\overset{\cdot}{Q}}$ , ${\tilde{R}}_{u}$ into equation (24),

\tilde{P} = ({\tilde{D}}_{\dot{Q}} ° {\tilde{R}}_{u}) {({\tilde{D}}_{Q} ° {\tilde{R}}_{u})}^{- 1}

(29)

It can be decomposed by the eigenvalue decomposition

\tilde{P} = \tilde{V} Σ {\tilde{V}}^{- 1}

(30)

where the matrix $\tilde{V}$ can be viewed as a transformation matrix that rotates the modal matrix to the SOM matrix given by the original SOD method. Substitute equation (30) into equation (23), and compared to equation (8), one obtains

Φ \tilde{V} \approx \tilde{Φ} L

(31)

where, the matrix $L$ is a diagonal matrix to normalize $\tilde{Φ}$ . The matrix $L$ can be computed by

L = ({\tilde{Φ}}^{- 1} Φ_{e} \tilde{V}) ° I

(32)

where $I$ is the identity matrix. And finally, the updated formula for a new modal matrix is

Φ_{e} \approx \tilde{Φ} L {\tilde{V}}^{- 1}

(33)

Then the updated estimation of the modal matrix can be iteratively substituted into equation (25) till convergence. In each step, the columns of the modal matrix should be normalized so that their modulus is equal to 1. The norm of the difference between the two steps can be used as the convergence metric

E_{1} = ‖ Δ Φ_{e} ‖

(34)

where ‖■‖ is the norm of the matrix ■ (Frobenius norm is suggested). While the index $E_{1} \to 0$ represents the convergence of the iterative algorithm, it can not prove the correctness of the results. Another index to show the correctness of the obtained modal parameters is defined as

E_{2} = ‖ R_{\overset{\cdot}{X}} R_{X}^{- 1} - Φ_{e} {\tilde{P} Φ}_{e}^{- 1} ‖ + ‖ R_{\overset{\cdot\cdot}{X}} R_{\overset{\cdot}{X}}^{- 1} - Φ_{e} {\tilde{P}}_{1} Φ_{e}^{- 1} ‖

(35)

where ${\tilde{P}}_{1} = ({\tilde{D}}_{\overset{\cdot\cdot}{Q}} ° {\tilde{R}}_{u}) {({\tilde{D}}_{\dot{Q}} ° {\tilde{R}}_{u})}^{- 1}$ . If $E_{2}$ approaches zero, it indicates that the combination of these modal parameters is the right one. The flowchart of the ISOD algorithm is shown in Figure 3. The modal matrix given by the original SOD method is used to provide a initial value. Then the iterative process can improve the accuracy of the modal matrix and provide the system’s frequencies and damping ratios. The computation efficiency of the iterative process is very fast because it only involves manipulating lower-dimensional covariance matrices and analytical formulas. The memory-consumed original data is used only one time to compute the covariance matrices, and not used in the iterative process.

Figure 3.

Flowchart of the ISOD algorithm.

Numerical studies

In this section, several studies are carried out to demonstrate the accuracy and performance of the proposed method. Firstly, a SDOF vibration system is studied to illustrate the principle for determining ( $ω_{i}, ξ_{i})$ . The SDOF case is emphasized because the MDOF system can be decoupled into SDOF vibration problems when mode shapes are (approximately) known. Then, in section 5.2 a 3-DOF vibration system is investigated to show the convergence ability of the iterative SOD method. The performance of the proposed method is tested with different damping levels and different excitation scenarios.

SDOF system

The search subroutine to find the optimal frequency and damping ratio is an important component of the iterative SOD method. It aims to find the optimal $(ω_{i}, ξ_{i})$ when the $λ_{1, i}$ and $λ_{2, i}$ are known. The $i^{th}$ modal coordinate vibration is govened by

{\overset{\cdot\cdot}{q}}_{i} (t) + 2 ω_{i} ξ_{i} {\overset{\cdot}{q}}_{i} (t) + ω_{i}^{2} q_{i} (t) = u_{i} (t)

(36)

If the sampling frequency is $ω_{s}$ , and the modal force $u_{i} (t)$ is white Gaussian excitation, then the expected values of $ω_{1, i}$ can be derived by the analytical formulas given in the Appendix (The corresponded frequency band is $[ω_{L}, ω_{U}] = [0, ω_{s} / 2]$ ).

Considering $ω_{1, i} = \sqrt{λ_{1, i}}$ and $ω_{2, i} = \sqrt{λ_{2, i}}$ , for different combinations of modal frequencies and damping ratios, the ratio $ω_{1, i} / ω_{i}$ is determined by $ω_{i}$ and $ξ_{i}$ , as shown in Figure 4. In the figure, the horizontal axis is $ω_{i} / ω_{s}$ and ranges from 0 to 0. 5. This corresponds to the range of the modal frequency [0, $ω_{s} / 2$ ]. Meanwhile, the damping ratio ranges from 0 to 1. It is shown in Figure 4 that the value of $ω_{1, i}$ is slightly smaller than $ω_{i}$ for small frequency or small damping ratio. When the damping ratio approaches to 0, the energy of the response signal concentrate in a frequency line at $ω_{i}$ . Since the gain of the derivation process is $j ω$ , so the integration $I_{\overset{\cdot}{q}, ij} = ω_{i}^{2} I_{q, ij}$ . For example, if $\frac{ω_{i}}{ω_{s}} = 0.4, ξ_{i} = 0.01$ , then $\frac{ω_{1, i}}{ω_{i}} = 0.9929$ . On the other hand, it is derived in the Appendix that $I_{\overset{\cdot}{q}, ij} = ω_{i}^{2} I_{q, ij}$ when $ω_{i} / ω_{s}$ approaches 0. For example, if $\frac{ω_{i}}{ω_{s}} = 0.01, ξ_{i} = 0.8$ , then $\frac{ω_{1, i}}{ω_{i}} = 0.9898$ . The results demonstrate that the accuracy of the frequency given by the original SOD method is influenced by the damping ratio and the sampling frequency. High sample frequency and small damping ratio would benefit the SOD method.

Figure 4.

Contour of $ω_{1, i} / ω_{i}$ as a function of ( $ω_{i} / ω_{s}$ , $ξ_{i}$ ).

In addition to $ω_{1, i}$ , the value of $ω_{2, i} = \sqrt{λ_{2, i}}$ can also be derived. Conversely, if the values of $ω_{1, i}$ and $ω_{2, i}$ are given, the corresponding modal frequency $ω_{i}$ and damping ratio $ξ_{i}$ can be found out. Mathematically, the $ω_{i}$ and the $ξ_{i}$ can be regarded as two functions of $ω_{1, i}$ and $ω_{2, i}$ . One can obtain the value of $ω_{i}$ and $ξ_{i}$ for a known ( $ω_{1, i},$ $ω_{2, i}$ ) combination by a simple searching method. For example, if the true modal frequency is 0.25 $ω_{s}$ and the damping ratio is 0.3, the contour map of the logarithmic objective function computed by equation (27) is shown in Figure 5. The objective function has one extreme value which ensures that it is easy to find the optimal value by a downhill-like method. The computation of the objective function involves only two known SOVs and a set of analytical integration formulas, which ensures its efficiency.

Figure 5.

Contour of the objective function (logarithmic value) computed by equation (27) corresponding to optimal frequency 0.25 $ω_{s}$ and optimal damping ratio $ξ_{i} = 0.3$ .

3-DOF system

The 3-DOF system shown in Figure 6 is studied here to demonstrate the performance of the proposed iterative method. The mass in the diagram is $m = 1 kg$ , and the stiffness is $k = 1 N / m$ , respectively. The magnitude of the first lumped mass is set to be twice of other lumped masses, which proves that the SOD method is capable of identifying vibration system parameters with unevenly distributed masses. The damping is considered to be Rayleigh damping, that is, $C = α M + β K$ . Two levels of damping are considered in the latter simulations. The first one is the lightly damped case with the damping ratios of the first two modes equal to 0.02. The second one is the moderately damped case with the damping ratios of the first tow modes equal to 0.2. The theoretical modal parameters of the system are listed in Table 1.

Figure 6.

The 3-DOF vibration system.

Table 1.

Theoretical frequencies and damping ratios of the 3-DOF vibration system.

Order	Natural frequencies	Dampingratios	Dampingratios
Order	(rad/s)	(1) $α = 0.1166,$ $β = 0.0029$	(2) $α = 1.1657,$ $β = 0.0293$
1	4.2162	0.0200	0.2000
2	9.4384	0.0200	0.2000
3	10.5423	0.0210	0.2097

Three different scenarios of excitation forces are considered. Firstly, the system is excited at all masses by uncorrelated white noises with $R_{f} = I$ . The corresponding covariance matrix $R_{u}$ and correlation coefficient $ρ_{u}$ of modal forces are listed in Table 2. The correlation of the modal coordinate responses (take displacement as an example) can be derived by equation (22). The derived matrices $ρ_{Iq}$ and $ρ_{u}$ are shown in Table 3. The correlation of the modal coordinate responses is low in this excitation arrangement even for the moderately damped case, so accurate results can be expected using the original SOD method.

Table 2.

Covariance matrices and correlation coefficient of external excitation and modal forces.

Excitation scenario	$R_{f}$	$R_{u}$	$ρ_{u}$
Uncorrelated external forces	$[\begin{matrix} 1.000 & 0.000 & 0.000 \\ 0.000 & 1.000 & 0.000 \\ 0.000 & 0.000 & 1.000 \end{matrix}]$	$[\begin{matrix} 0.248 & 0.006 & 0.000 \\ 0.006 & 0.161 & - 0.002 \\ 0.000 & - 0.002 & 0.250 \end{matrix}]$	$[\begin{matrix} 1.000 & 0.028 & 0.001 \\ 0.028 & 1.000 & - 0.010 \\ 0.001 & - 0.010 & 1.000 \end{matrix}]$
Correlated modal forces	$[\begin{matrix} 5.720 & - 3.561 & 3.211 \\ - 3.561 & 5.295 & - 2.533 \\ 3.211 & - 2.533 & 3.121 \end{matrix}]$	$[\begin{matrix} 1.000 & 0.600 & 0.600 \\ 0.600 & 1.000 & 0.800 \\ 0.600 & 0.800 & 1.000 \end{matrix}]$	$[\begin{matrix} 1.000 & 0.600 & 0.600 \\ 0.600 & 1.000 & 0.800 \\ 0.600 & 0.800 & 1.000 \end{matrix}]$
Excitation on mass 2	$[\begin{matrix} 0.000 & 0.000 & 0.000 \\ 0.000 & 1.000 & 0.000 \\ 0.000 & 0.000 & 0.000 \end{matrix}]$	$[\begin{matrix} 0.242 & 0.026 & 0.028 \\ 0.026 & 0.003 & 0.003 \\ 0.028 & 0.003 & 0.003 \end{matrix}]$	$[\begin{matrix} 1.000 & 1.000 & 1.000 \\ 1.000 & 1.000 & 1.000 \\ 1.000 & 1.000 & 1.000 \end{matrix}]$

Table 3.

Correlation coefficient matrix of modal coordinate displacement.

Excitationscenario	Damping (1) $ξ = [0.02, 0.02, 0.021]$		Damping (2) $ξ = [0.2, 0.2, 0.297]$
Excitationscenario	$ρ_{Iq}$	$ρ_{q} = ρ_{Iq} ° ρ_{u}$	$ρ_{Iq}$	$ρ_{q} = ρ_{Iq} ° ρ_{u}$
Uncorrelatedexternal forces		$[\begin{matrix} 1.0000 & 0.000 & 0.000 \\ 0.000 & 1.0000 & - 0.001 \\ 0.000 & - 0.001 & 1.0000 \end{matrix}]$		$[\begin{matrix} 1.0000 & 0.005 & 0.000 \\ 0.005 & 1.0000 & - 0.009 \\ 0.000 & - 0.009 & 1.0000 \end{matrix}]$
Correlatedmodal forces	$[\begin{matrix} 1.000 & 0.002 & 0.002 \\ 0.002 & 1.000 & 0.121 \\ 0.002 & 0.121 & 1.000 \end{matrix}]$	$[\begin{matrix} 1.000 & 0.001 & 0.001 \\ 0.001 & 1.000 & 0.096 \\ 0.001 & 0.096 & 1.000 \end{matrix}]$	$[\begin{matrix} 1.000 & 0.175 & 0.143 \\ 0.175 & 1.000 & 0.932 \\ 0.143 & 0.932 & 1.000 \end{matrix}]$	$[\begin{matrix} 1.000 & 0.105 & 0.086 \\ 0.105 & 1.000 & 0.745 \\ 0.086 & 0.745 & 1.000 \end{matrix}]$
Excitationon mass 2		$[\begin{matrix} 1.000 & 0.002 & 0.002 \\ 0.002 & 1.000 & 0.121 \\ 0.002 & 0.121 & 1.000 \end{matrix}]$		$[\begin{matrix} 1.000 & 0.175 & 0.143 \\ 0.175 & 1.000 & 0.932 \\ 0.143 & 0.932 & 1.000 \end{matrix}]$

Secondly, the modal forces are considered to be correlated white noises with specific covariance matrix $R_{u}$ , as shown in Table 2, where $R_{u, 12} = R_{u, 13} = 0.6$ and $R_{u, 23} = 0.8$ . To reach this, the covariance matrix of the external excitation forces should satisfy equation (12). Although the modal force has a high correlation, for the lightly damped case, the correlation of modal response between different modes is still very low. However, for the moderately damped case, the correlation of the modal forces can not be ignored ( $ρ_{u, 23} = 0.745$ ). Therefore, the original SOD method is expected to obtain results with inevitable errors.

The last excitation arrangement is to excite the system with external force (white noise) only applied to one of the masses (mass #2). The corresponding covariance and correlation matrices are listed in Tables 2 and 3. It is shown that the modal forces are completely correlated, and the correlation of the modal coordinate responses can not be ignored, for example, $ρ_{q, 23} = 0.932$ . Therefore, the original SOD method can hardly give accurate mode shapes.

All these excitations are generated at 100 Hz and the total sampling time is 10,000 s. The response signals are recorded to form ensemble matrices $X$ , $\overset{\cdot}{X}$ , and $\overset{\cdot\cdot}{X}$ . Then, the original SOD method is used to extract the modal matrix of the system. The modal assurance criterion (MAC) is adopted to measure the accuracy of the extracted modal matrices (Pastor et al., 2012). The MAC values of the modal vectors extracted by using the SOD method and the ISOD method are given in Table 4.

Table 4.

MAC matrix of the modal matrix estimated by the SOD method and the ISOD method.

Excitationscenario	Damping (1) $ξ = [0.02, 0.02, 0.021]$		Damping (2) $ξ = [0.2, 0.2, 0.297]$
Excitationscenario	MAC_SOD	MAC_ISOD	MAC_SOD	MAC_ISOD
Uncorrelatedexternalforces	$[\begin{matrix} 1.0000 & 0.0007 & 0.0000 \\ 0.0008 & 1.0000 & 0.0000 \\ 0.0000 & 0.0001 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0007 & 0.0000 \\ 0.0008 & 1.0000 & 0.0000 \\ 0.0000 & 0.0001 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0015 & 0.0000 \\ 0.0008 & 0.9998 & 0.0003 \\ 0.0000 & 0.0000 & 0.9999 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0012 & 0.0000 \\ 0.0008 & 0.9975 & 0.0003 \\ 0.0000 & 0.0016 & 0.9994 \end{matrix}]$
Correlated modalforces	$[\begin{matrix} 1.0000 & 0.0008 & 0.0000 \\ 0.0008 & 0.9983 & 0.0022 \\ 0.0000 & 0.0010 & 0.9968 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0008 & 0.0000 \\ 0.0008 & 1.0000 & 0.0003 \\ 0.0000 & 0.0003 & 1.0000 \end{matrix}]$	$[\begin{matrix} 0.9993 & 0.0149 & 0.0056 \\ 0.0025 & 0.8547 & 0.1901 \\ 0.0002 & 0.1296 & 0.7985 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0010 & 0.0000 \\ 0.0007 & 0.9980 & 0.0084 \\ 0.0000 & 0.0030 & 0.9933 \end{matrix}]$
Excitationon mass 2	$[\begin{matrix} 1.0000 & 0.0021 & 0.0001 \\ 0.0008 & 0.9963 & 0.0026 \\ 0.0000 & 0.0023 & 0.9962 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0012 & 0.0000 \\ 0.0008 & 0.9999 & 0.0026 \\ 0.0000 & 0.0001 & 1.0000 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.7415 & 0.2714 \\ 0.0010 & 0.2040 & 0.2565 \\ 0.0000 & 0.0738 & 0.4805 \end{matrix}]$	$[\begin{matrix} 1.0000 & 0.0015 & 0.0006 \\ 0.0008 & 0.9849 & 0.0318 \\ 0.0000 & 0.0174 & 0.9711 \end{matrix}]$

For the lightly damped case, $ρ_{q}$ is always close to a diagonal matrix, so the SOD method can obtain acceptable results for all three excitation scenarios. The MACs of the initial modal matrices provided by the SOD method are greater than 0.995, which demonstrates the accuracy of the original SOD method (when applied for lightly damped systems). The decrease of the index $E_{1}$ and $E_{2}$ as the iterative step increases demonstrate the convergence of the ISOD method. The convergence curve shown in Figure 7 depicts the declining trend of the indexes. Also, the variants of the MAC of the updated modal matrices obtained in the ISOD method are plot in Figure 8. The curve for excitation with uncorrelated modal forces is omitted since it is almost a straight line. It is also shown that the ISOD gives better results than the SOD method, and it converges within 10 iteration steps.

Figure 7.

Convergence of the index $E_{1}$ and $E_{2}$ for lightly damped case with excitation applied on mass #2.

Figure 8.

MAC of the mode shape vectors updated by the ISOD method for the lightly damped case with: (a) correlated modal forces, and (b) excitation applied on mass #2.

For the moderately damped case, the SOD method provides inaccurate results due to the correlation of modal responses (when modal forces are correlated or completely correlated). When correlated modal forces are applied, the diagonal elements of the MAC of the modal matrix estimated by the SOD method are (0.9993, 0.8547, 0.7985). This indicates that the mode shape vectors of the second and the third mode given by the SOD method are inaccurate. This is because these two modes are coupled modes and the corresponding modal coordinate responses are correlated with each other, for example, $ρ_{q, 23} = 0.745$ . When excitation forces are applied on mass #2, $ρ_{q, 23} = 0.932$ , and the diagonal elements of the MAC given by the SOD method are (1.0000, 0.2040, 0.4805). This also shows that the second and third mode shapes contain remarkable errors. Although the initial modal matrices are not sufficiently accurate, the ISOD method is carried out to improve the modal matrix iteratively. The convergence of the indices $E_{1}$ and $E_{2}$ (for excitation applied on mass #2) is shown in Figure 9. The changes in the MACs of the mode shape vectors updated in each step are shown in Figure 10. The results verify that the ISOD has good convergence performance even for the cases with a biased initial modal matrix. The ISOD method converges within 100 iteration steps.

Figure 9.

Convergence of the index $E_{1}$ and $E_{2}$ for moderately damped case, with excitation applied on mass #2.

Figure 10.

MAC of the mode shape vectors updated by the ISOD method for the moderately damped case with: (a) correlated modal forces, and (b) excitation applied on mass #2.

Essentially, the ISOD algorithm is a local optimal method, and the accuracy of the initial modal matrix is important for the convergence. Consider the step numbers of the above cases, it is concluded that the closer the initial modal matrix is to the true value, the faster the convergence speed of the ISOD method. The mathematical mechanism of the convergence characteristic needs further study. However, since the modal forces in the practical structures are usually not completely correlated, even for the moderately damped case, the SOD method can provide an initial modal matrix with adequate accuracy.

To further investigate the properties of the ISOD method, three sets of Monte-Carlo Simulations are performed. Considering the moderately damped case, for each excitation scenario, the vibration system is excited 1000 times. The SOD method and ISOD method are invoked to obtain a set of estimations of the modal parameters. Then the mean values and the root mean square error (RMSE) of these parameters are computed to obtain their statistical characteristic. The results are shown in Table 5. The SOD method gives accurate results when uncorrelated external excitations are applied. The difference between the identified frequencies and the true value is very small. The RMSE of the results from the SOD method is also small, which illustrates that the uncertainties of the identified parameters are also small. In fact, the RMSE is related to the sampling length, it will decrease as the data become longer.

Table 5.

Accuracy of the frequency and damping ratio of the SOD method and the ISOD method for the damping case (2).

Excitation scenario	Order	SOD			ISOD
		Frequency			Frequency			Dampr
		Mean	Error	RMSE	Mean	Error	RMSE	Mean	Error	RMSE
Uncorrelated external forces	1	4.2090	−0.0072	0.0092	4.2162	−1.4E-5	0.0093	0.2001	0.0001	0.0023
	2	9.4027	−0.0357	0.0139	9.4402	0.0018	0.0141	0.2000	4.8E-5	0.0015
	3	10.4956	−0.0468	0.0147	10.5413	−0.0010	0.0150	0.2098	4.7E-5	0.0015
Correlated modal forces	1	4.2005	−0.0157	0.0092	4.2160	−0.0002	0.0092	0.2001	0.0001	0.0025
	2	9.1834	−0.2550	0.0125	9.4430	0.0046	0.0400	0.2003	0.0003	0.0030
	3	10.8125	0.2702	0.0141	10.5375	−0.0049	0.0481	0.2098	0.0001	0.0020
Excitation on mass 2	1	4.1846	−0.0316	0.0088	4.2166	0.0004	0.0091	0.2001	0.0001	0.0030
	2	8.6939	−0.7445	0.0118	9.4529	0.0145	0.1247	0.2000	3.6E-5	0.0065
	3	11.6583	1.1160	0.0140	10.5285	−0.0138	0.1235	0.2098	0.0001	0.0030

For the correlated modal forces, however, obvious deviations appear in the mean values of frequencies obtains by the SOD method. The deviation of the third frequency reaches 0.2702 rad/s. This indicates that the SOD method provides a biased estimation of the modal parameters. The SOD method provides worse frequency estimations for the system with excitation on mass 2 only, the deviation of the third frequency reaches 1.1160 rad/s. It again demonstrates that the accuracy of the SOD method is affected by the correlation of the modal coordinate responses, which is influenced by the correlation of the modal forces. The accuracy of the modal frequencies obtained by the ISOD method is remarkably improved. The mean values of the damping ratios are also close to the true value. The differences between the RMSEs for different excitation scenarios show that the parameters can be identified with different accuracy. The modal parameters of the system excited by uncorrelated external forces can be identified with the best accuracy.

Experimental verification

To further verify the performance of the proposed approach presented in this paper, the vibration data of a seven-story frame structure model is analyzed in this section. The total height of the structural model is 2.1 m, and each floor is 0.3 m. The steel mass blocks are set on each floor to simulate floor weights. The detailed information on this model can be found in (Li et al., 2012) and the layout of the sensors used in this paper is shown in Figure 11. Seven acceleration sensors are installed on the frame model, and the sampling frequency is set as 2000 Hz. In this study, the test signal is low-pass filtered, and the signal below 50 Hz is retained. The total duration of the signal used for modal parameter identification is 500 s.

Figure 11.

(a) The laboratory frame structure modal, and (b) the layout of the sensors.

The test data are fed in the stochastic subspace algorithm (SSI) and the proposed approach in this paper, and two sets of modal identification results are obtained. Figure 12(a) shows the MAC of the mode shapes identified by using SSI and the proposed ISOD. By comparison, the two sets of mode shapes are very close. It can be observed that the results of the first seven modes obtained by these two methods have a good match. These mode shapes identified by the two methods are shown in Figure 12(b), further verifying the accuracy of using the proposed approach for modal identification.

Figure 12.

Comparison of the mode shape results obtained by using SSI and the proposed approach: (a) the MAC value, and (b) the mode shape plots.

Based on mode shape results, we can further understand the theory of frequency and damping identification in this paper. Using $\overset{\cdot\cdot}{Q} = Φ^{- 1} \overset{\cdot\cdot}{X}$ , the acceleration modal coordinate response of each mode can be approximately extracted. The Fourier transform results of the vibration acceleration of each mode are shown in Figure 13. It can be seen from the figure that each signal is decomposed by the ISOD algorithm. To obtain accurate natural frequency and damping ratio results, the frequency and damping ratio optimization search is performed with the components of the specific frequency band of the modal vibration signal. The results obtained for specific frequency bands and ISOD are listed in Table 6 and compared with those obtained by using the SSI and the empirical wavelet transform (EWT) methods (Xin et al., 2019). The above results verify the accuracy of using the proposed approach for structural modal identification.

Figure 13.

The Fourier spectrum of the decomposed modal coordinate vibrating signals.

Table 6.

Comparison the identified frequency and damping ratios with different methods.

Mode	Frequency band (Hz)	Natural frequency (Hz)			Damping ratio (%)
Mode	Frequency band (Hz)	EWT	SSI	ISOD	EWT	SSI	ISOD
1	[2,4]	2.54	2.55	2.55	0.35	0.36	0.34
2	[6,8]	7.62	7.64	7.65	0.10	0.10	0.10
3	[11,13]	12.89	12.83	12.83	0.06	0.08	0.14
4	[17,19]	17.97	18.01	18.01	0.09	0.09	0.07
5	[22,24]	22.85	22.81	22.81	0.12	0.10	0.12
6	[26,28]	26.95	26.85	26.86	0.05	0.08	0.05
7	[28,30]	29.69	29.67	29.67	0.04	0.03	0.04

Conclusion

This study stems from the investigation of the SOD method for modal identification of damped vibration systems. It is stated in this paper that the SOD method can be regarded as a specific version of the generalized complexity pursuit method. Since the sources of the damped vibration system, that is, the modal coordinate responses, are not completely uncorrelated due to the existence of the damping, which leads to the identification error of the SOD method. The influence of the damping to the correlation characteristic of the modal coordinate responses is researched. And a set of analytical formulas is given, which enables us to calculate the expected covariance matrix of modal responses. Based on the analytical formulas, the ISOD method is proposed and identifies the mode shapes, natural frequency, and the damping ratios of the system iteratively. Numerical examples and experimental studies validate the convergence of using the ISOD method for modal identification, even for cases that the initial modal matrix given by the SOD method contains remarkable errors. The ISOD method provides more accurate results than the SOD method when convergence is assured, it can be regarded as an improved version of the original SOD method.

Footnotes

Appendix

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: The work described in this paper is supported by the National Natural Science Foundation of China (Project no. 51408177 and 51978230), and the Fundamental Research Funds for the Central Universities (PA2019GDZC0094). The financial support is gratefully acknowledged.

ORCID iDs

Zhixaing Hu

Jun Li

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