Strain modes identification using cross-correlation function of measured dynamic strain

Abstract

Strain modes of structures reflect the distribution rule of dynamic strain and can be of service to structural health monitoring. This article presents a new method to estimate the strain modes of beam structures based on the measured dynamic strain. Since the cross-correlation matrix of the modal coordinates is a diagonal matrix, thus, the cross-correlation matrix of the measured dynamic strain is also a diagonal matrix, and the strain modes can be obtained to find the eigenvectors of the strain cross-correlation matrix. The strain modes of a simply support beam and a continuous beam subjected to various impulsive excitations are identified from the numerical simulations using the proposed method. The noise effect is also investigated in this article. The numerical results show that the proposed method can effectively identify the strain modes even with noise effect. Finally, the method is verified through the experiment of a simply supported beam under hammering excitation. The test results also show that the proposed method can estimate the strain modes of beam structures with a high accuracy.

Keywords

beam structure cross-correlation function dynamic strain modal identification strain modes

Introduction

A critical step for vibration-based structural health monitoring is to identify structural parameters such as natural frequencies, mode shapes, and damping ratios from structural responses (Chen and Xu, 2002; Doebling et al., 1996, 1998; Montalvao et al., 2006). Among the parameters of structures, the dynamic strain state and its distribution is one of the critical parameters for structural health monitoring and safety evaluation. The strain mode is inherent property in the structure and it reflects the strain distribution rule of structure under the action of dynamic loads (Li et al., 1989; Yam et al., 1996). In practical situation, it is a quite challenge to obtain the stress distribution of the structure when it is subjected to dynamic loads.

Modal analysis technology as an important method of identifying the structural parameters has been intensively studied (Ewins, 1984; He and Fu, 2001; Heylen et al., 2007; Maia and Silva, 1997). In current stage, the displacement modal analysis method is still dominant compared with the strain modal analysis and is widely used to identify the strain modes (Lee, 2007; Seo et al., 1998). The strain modes are identified essentially based on the transformation of strain–displacement relationship method. For instance, Okubo and Yamaguchi (1995) predicted the distribution of dynamic strain under operating conditions using the displacement to strain transformation matrix. Karczub and Norton (1999) identified the strain modes based on the finite differencing methods for randomly vibrating structures. Sehlstedt (2001) further based on the identified strain modes calculated the dynamic strain tensor of structures. Li et al. (2002) identified the strain modes using the Rayleigh–Ritz method and detected the structural damage using the sensitive index of the identified strain modes. Xu et al. (2007) introduced the Bayesian method to estimate strain modes based on the frequency response function. More recently, Esfandiari et al. (2010) proposed a sensitivity-based algorithm for finite element model updating using the identified strain modes. Liu et al. (2011) have also shown the strain modes can be obtained by the modal analysis of the strain frequency response functions, and it can be used for damage detection. However, in these studies, the strain mode is usually calculated from the displacement mode based on the strain–displacement relationship, which is not a direct strain modes identification method. For these methods, some numerical errors will be introduced. Furthermore, the strain mode parameters are more sensitive to the changes of local characteristics in the structure. However, these local changes of strain are hard to be identified from the displacement mode. Thus, the direct strain modes identification method based on the measured dynamic strain is more practicable.

This article presents a new method to estimate the strain modes of beam structures based on the measured dynamic strain. Since the cross-correlation matrix of the modal coordinates is a diagonal matrix, thus, the cross-correlation matrix of the measured dynamic strain is also a diagonal matrix, and the strain modes can be obtained to find the eigenvectors of the strain cross-correlation matrix. The strain modes of a simply support beam and a continuous beam subjected to various impulsive excitations are identified from the numerical simulations using the proposed method. Finally, the method is verified through the experiment of a simply supported beam under hammering excitation. Both the numerical and test results show that the proposed method can estimate the strain modes of beam structures with a high accuracy.

Theory and method of strain modes identification

The displacement and strain modes

According to the modal superposition theory, the displacement and strain can be expressed using a finite number of mode shapes. The displacement and strain at any point x and at any time t can be represented by the superposition of their mode shapes weighed by the modal coordinates

ω (x, t) = \sum_{r = 1}^{m} ϕ_{r} (x) q_{r} (t)

(1)

ε (x, t) = \sum_{r = 1}^{m} ψ_{r} (x) q_{r}^{'} (t)

(2)

where $ω (x, t)$ and $ε (x, t)$ are the displacement and strain of the beam at any point x and at any time t. $ϕ_{r} (x)$ and $ψ_{r} (x)$ are the elements at the location x of the $r th$ displacement mode shape and $r th$ strain mode shape, respectively. $q_{r} (t)$ and $q_{r}' (t)$ are the corresponding $r th$ modal coordinate. The modal coordinate denotes the proportion of each mode in the whole response. For each displacement mode, there must be a corresponding strain mode. They are just different presentations for the state of energy balance. Thus, the displacement and strain have the same modal coordinate $q_{r} (t) = q_{r}' (t)$ .

For the one-dimensional beam model, the vibration differential equation is expressed as follows

\begin{matrix} \frac{\partial^{2}}{\partial x^{2}} [EI \frac{\partial^{2}}{\partial^{2} x} ω (x, t)] & + m (x) \frac{\partial^{2}}{\partial t^{2}} ω (x, t) \\ + c (x) \frac{\partial}{\partial t} ω (x, t) = f (x, t) \end{matrix}

(3)

By multiplying $ϕ_{s}$ and integrating equation, equations (1) and (3) can be combined into

\begin{array}{l} \sum_{r = 1}^{m} [\int_{0}^{l} m (x) ϕ_{r} (x) ϕ_{s} (x) {\overset{‥}{q}}_{r} (t) d x + \int_{0}^{l} c (x) ϕ_{r} (x) ϕ_{s} (x) {\dot{q}}_{r} (t) d x + \int_{0}^{l} E I ϕ_{r}^{’ ’ ’ ’} (x) ϕ_{s} (x) q_{r} (t) d x] \\ = \int_{0}^{l} x f (x, t) ϕ_{s} (x) d x \end{array}

(4)

According to the orthogonality of mode shape, the left side of this equation is not equal to zero only when $r = s$ . Thus, equation (4) can be simplified as

m_{s} {\overset{\cdot\cdot}{q}}_{s} + c_{s} {\overset{\cdot}{q}}_{s} + k_{s} q_{s} = \int_{0}^{l} f (x, t) ϕ_{s} (x) dx

(5)

where

\int_{0}^{l} m (x) ϕ_{s} (x) ϕ_{s} (x) dx = m_{s}

\int_{0}^{l} c (x) ϕ_{s} (x) ϕ_{s} (x) dx = c_{s}

\int_{0}^{l} EI ϕ_{r}^{’ ’ ’ ’} (x) ϕ_{s} (x) dx = k_{s}

By introducing the force of $f (x, t) = {F} e^{j ω t}$ , equation (5) can be solved as

q_{s} = \frac{\int_{0}^{l} {F} e^{j ω t} ϕ_{s} (x) dx}{- ω^{2} m_{s} + j ω c_{s} + k_{s}}

(6)

Therefore, equation (2) can be rewritten as

ε (x, t) = \sum_{r = 1}^{m} ψ_{r} (x) \frac{\int_{0}^{l} {F} e^{j ω t} ϕ_{r} (x) dx}{- ω^{2} m_{r} + j ω c_{r} + k_{r}}

(7)

Using beam theory, the displacement–strain relation can be expressed as

ε_{x} = - y \frac{\partial^{2} ω}{\partial x^{2}} = - y \sum_{r = 1}^{m} ϕ_{r}^{’ ’} (x) q_{r} (t)

(8)

where y is the distance from the neutral axis.

Compared with equation (2), the relation between the $r th$ displacement mode shape and the $r th$ strain mode shape can be expressed as

ϕ_{r}^{’ ’} = - \frac{1}{y} ψ_{r}

(9)

According to equation (9), it can also be found that this is the basis of strain modes identification using displacement modes.

The method based on cross-correlation function of measured dynamic strain

In this study, the strain mode shapes are estimated from the cross-correlation function of the measured dynamic strain data. According to the modal superposition theory, the strain can be expressed by equation (2). For N modes being considered and M sensors being used, the strain at any time t can be expressed in matrix form as

{ε (t)}_{Mx 1} = ψ_{MxN} {η (t)}_{Nx 1}

(10)

where $ψ_{MxN}$ and ${η (t)}_{Nx 1}$ are strain mode shapes matrix and the modal coordinates, respectively. Thus, the modal coordinate matrix can be obtained as

{η (t)}_{Nx 1} = G_{NxM} {ε (t)}_{Mx 1}

(11)

where $G_{NxM} = (ψ_{MxN}^{T} ψ_{MxN})^{- 1} ψ_{MxN}^{T}$ is the strain weight matrix, and the rank of $G_{NxM}$ cannot exceed the number of used strain sensors. It means one can use as many modes as the number of used sensors at most.

The cross-correlation matrix R of modal coordinates with zero lag is defined as

R = E [η η^{T}] = E [G ε ε^{T} G^{T}] = G E [ε ε^{T}] G^{T}

(12)

in which the element $R_{ij} (i = 1, 2, \dots, N; j = 1, 2, \dots, N)$ in cross-correlation matrix can be calculated as

R_{ij} = E [η_{i} (k) η_{j} (k)] = \frac{1}{k} lim_{k \to \infty} \sum_{t = 1}^{k} η_{i} (t) η_{j} (t)

(13)

If $η_{i}$ and $η_{j}$ are different modal coordinates, they must be orthogonal to each other and $R_{ij} \neq 0 (i \neq j)$ . It means that the cross-correlation matrix $R$ should be a diagonal matrix, and $E [ε ε^{T}]$ is diagonalizable with an eigenvector $G^{T}$ . Thus, the strain weight matrix $G$ can be obtained by finding the eigenvectors of $E [ε ε^{T}]$ . With the strain weight matrix $G_{MxN}$ , the strain modes can be identified as

ψ_{MxN} = {(G^{T} G)}^{- 1} G^{T}

(14)

Numerical verification

In order to verify the effectiveness of proposed method, a simply supported beam and a continuous beam under impulsive loads are simulated. In this section, the simulated strain data with or without noise effect is assumed as the measured dynamic strain data, and the strain mode shapes can then be identified based on the proposed method.

Simply supported beam

In this section, a simply supported beam under impulse force is simulated The simulated I-girder is 5 m long, 0.06 m height with a section area of 2.72 cm², and the moment of inertia of 16.4 cm⁴. The elastic modulus is assumed as 70 GPa, and the density is assumed as 2700 kg/cm³. The plan of the simulated beam is shown in Figure 1.

Figure 1.

The plan of the simulated simply supported beam (unit: cm).

The simulated beam is divided into 20 elements by 21 nodes, and the length of each element is 25 cm. The impulsive force of 100 N with 0.02 s time duration is applied at 1/4 span and 3/4 span. With 200-Hz sampling frequency, the strain time history of each node can be extracted from the finite element analysis method. In order to consider the effect of noise to the identified result, two cases are discussed:

Case1: The strain response without noise effect is assumed as the measured dynamic strain data.

Case 2: The strain response with noise effect is assumed as the measured dynamic strain data, and 5% Gaussian white noise is added to the structural strain responses.

The strain responses for the node 6 (1/4 span) without/with noise are presented in Figures 2 and 3, respectively.

Figure 2.

The strain response at the node 6 (1/4 span) without noise.

Figure 3.

The strain response at the node 6 (1/4 span) with 5% Gaussian white noise.

In this numerical simulation, since the first 3 modes are dominant, thus, only the first 3 modes are identified. The identified strain modes based on the proposed method and the comparison with the theoretical modes without/with noise effect are presented in Figures 4 and 5, respectively. To further quantify the error between the identified modes and theoretical modes, the following error index E is defined

E = \frac{{‖ M_{identified} - M_{theoretical} ‖}_{2}}{{‖ M_{theoretical} ‖}_{2}} \times 100 %

(15)

Figure 4.

The first 3 identified strain modes without noise effect and the theoretical strain modes of the simply supported beam: (a) the first mode, (b) the second mode, and (c) the third mode.

Figure 5.

The first 3 identified strain modes with noise effect and the theoretical strain modes of the simply supported beam: (a) the first mode, (b) the second mode, and (c) the third mode.

In equation (15), $M_{theoretical}$ and $M_{identified}$ represent the theoretical mode and identified mode, respectively. $‖ \dots ‖_{2}$ represents 2-norm value. The error indices for the simply supported beam with or without noise are shown in Figure 6. From Figure 6, the errors of the first 3 identified strain modes without noise effect are within 2%. It can be seen that the identified strain modes are matched with the theoretical results. Besides, the errors of the first 3 identified strain modes with noise are within 8%. It can be further found that the identified strain modes can be still identified with relatively high accuracy even with noise effect.

Figure 6.

The error index for the simply supported beam with or without noise.

Continuous beam

In this section, a three-span continuous beam under impulsive loads is simulated. The total length of simulated beam is 15 m, 0.05 m height with a section area of 2.52 cm², and the moment of inertia of 10.8 cm⁴. The elastic modulus is assumed as 70 GPa, and the density is assumed as 2700 kg/cm³. The simulated beam is divided into 30 elements by 31 nodes. The plan of the simulated beam is shown in Figure 7.

Figure 7.

The plan of simulated continuous beam under impulsive load (unit: cm).

The impulsive force of 100 N with 0.02 s time duration is applied at 1/6 span and 5/6 span. Again, the simulated strain data without noise or with 5% Gaussian white noise are assumed as the measured data. The assumed measured strain responses at point 6 (1/6 span) without or with noise are presented in Figures 8 and 9, respectively. The first 3 identified strain modes compared with the theoretical modes are presented in Figures 10 and 11. The error indices for the continuous beam with or without noise are shown in Figure 12. It can be seen from Figure 12 that the errors of the first 3 identified strain modes without/with noise effect are within 3% and 8%, respectively. The identified strain modes can be still effectively identified with relatively high accuracy even with noise effect.

Figure 8.

The strain response without noise at the node 6 (1/6 span).

Figure 9.

The strain response with noise at the node 6 (1/6 span).

Figure 10.

The first 3 identified strain modes without noise effect and the theoretical strain modes of the continuous beam: (a) the first mode, (b) the second mode, and (c) the third mode.

Figure 11.

The first 3 identified strain modes with noise effect and the theoretical strain modes of the continuous beam: (a) the first mode, (b) the second mode, and (c) the third mode.

Figure 12.

The error indices for the continuous beam with or without noise.

Test verification

In order to further verify the effectiveness of the proposed method, a simply supported beam under different excitation was tested in the laboratory. The tested beam is an aluminum beam with 2.5 m full length, 100 mm width, and 20 mm thickness. Seven strain sensors are uniformly installed on the beam bottom surface with 31.25-cm interval. The plan of the experimental test is shown in Figure 13. The installed strain sensors and overall layout of the experiment test are illustrated in Figure 14.

Figure 13.

The plan of the experiment test (unit: cm).

Figure 14.

The strain sensor and overall layout of the experiment test.

Strain modes identification under single-point hammering excitation

For this case, a hammer load is applied at node 2 (1/4 span). The measured strain responses are used to identify the tested strain modes. The strain response at node 2 (1/4 span) is shown in Figure 15. The identified and theoretical strain modes are shown in Figure 16. As shown in Figure 16, the strain modes can be accurately identified based on the proposed method for a beam under single-point hammering excitation.

Figure 15.

The strain response at point 2 (1/4 span) under single-point hammering excitation.

Figure 16.

The first 3 identified and theoretical strain modes of the tested beam under single-point hammering excitation: (a) the first mode, (b) the second mode, and (c) the third mode.

Strain modes identification under random excitation

In the experiment, the strain modes are obtained from the strain responses under random excitation. The dynamic strain is measured with 200-Hz sampling frequency and 55-s time duration. The dynamic strain of measured point 2 (1/4 span) is shown in Figure 17. The strain modes are identified using the proposed method and the results are presented in Figure 18. It can be found from Figure 18 that the strain modes can be accurately identified based on the proposed method for a beam under single random excitation.

Figure 17.

The measured strain response at point 2 (1/4 span) under random excitation.

Figure 18.

The first 3 identified and theoretical strain modes of the tested beam under random excitation: (a) the first mode, (b) the second mode, and (c) the third mode.

Conclusion

In this article, a new method based on the cross-correlation function of the measured strain data is developed to identify the strain modes of beam structures. The theoretical derivation shows that the strain modes can be obtained to find the eigenvectors of the strain cross-correlation matrix. The strain modes of a simply support beam and a continuous beam subjected to various impulsive excitations are identified from the numerical simulations using the proposed method. The method is also verified through the experiment of a simply supported beam under various excitations. Based on the numerical and test results, the following conclusions can be drawn:

The strain modes can be directly identified from the measured strain data.

The proposed method can be used to identify the strain modes with a high accuracy even with 5% Gaussian white noise effect.

From the test verification, only seven strain measured points are enough to obtain the first 3 strain modes for a simply support beam.

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) received no financial support for the research, authorship, and/or publication of this article.

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