Dual-type sensor placement optimization by fully utilizing structural modal information

Abstract

Strain gauges and accelerometers are widely used in bridge structural health monitoring systems. Generally, the strain gauges are placed on the key locations to obtain local structural deformation information; the accelerometers are used to obtain the structural modal information. However, the modal information contained in the measured strains is not taken into account. In this article, to fully utilize the modal information contained in strains, a mode shape estimation method is proposed that the strain mode shapes of the strain locations are used to obtain the displacement mode shapes of some positions without accelerometers. At first, to simulate the practical situation, some positions with large structural deformations are selected as the strain gauge locations. Using the proposed mode shape estimation method, the displacement mode shapes of some locations without accelerometers are estimated by the strain mode shapes using the least squares method, and the locations with the smallest estimation error are finally determined as the estimated locations. Then, accelerometers are added to the existing sensor placement. Here, the modal assurance criterion is used to evaluate the distinguishability of the displacement mode shapes obtained from the strain gauges and accelerometers. The accelerometer locations that bring the smallest modal assurance criterion values are selected. In addition, a redundancy can be set to avoid the adjacent sensors containing similar modal information. Through the proposed sensor placement method, the deformation and modal information contained in the strain gauges is fully utilized; the modal information contained in the strain gauges and accelerometers is comprehensively utilized. Numerical experiments are carried out using a bridge benchmark structure to demonstrate the sensor placement method.

Keywords

accelerometer displacement mode shape estimation optimal sensor placement strain gauge structural modal information

Introduction

In structural health monitoring (SHM) systems, the quantity and quality of different kinds of measured data greatly affect the functionality of the SHM systems (Huang et al., 2017; Meo and Zumpano, 2005; Worden and Burrows, 2001; Yi et al., 2012). Structural information obtained from different measured data (i.e. strain, acceleration) can be used in different ways. The measured strains intuitively reflect the local deformations of the corresponding positions on the structure, which can be used to perform the deformation monitoring, fatigue analysis and structural damage detection (Adewuyi and Wu, 2011; Pedram et al., 2016; Sanayei et al., 2011). The displacement modal information obtained from accelerations reflects global structural vibration characteristics. Modal coordinates and mode shapes are often utilized in modal identification, damage detection and finite element (FE) model updating (Chang and Pakzad, 2014; Ching and Beck, 2004; Shi et al., 2000). Generally, in an SHM system, the number of the installed sensors is very small compared with the degrees of freedom (DOFs) of the whole structure. To obtain enough useful structural information, the number of sensors and the specific locations of the sensors need to be determined by an optimal sensor placement (OSP) method.

In previous studies, many OSP methods of accelerometers have been proposed, in which the structural parameter, modal coordinate and the mode shapes are usually taken into account. A parameter identification–based sensor placement theory was proposed by Udwadia (1994), in which the sensor locations are selected by maximizing the norm of the Fisher matrix to achieve the best identification of the structural parameters. Papadimitriou et al. (2000) and Papadimitriou (2004) used information entropy to evaluate the uncertainty of the estimated model parameters in the time domain, so that the selected sensor locations with small entropy value have an accurate performance in parameter identification. Yuen and Kuok (2015) proposed a Bayesian sensor placement algorithm in the frequency domain for modal identification, and the uncertainty of the estimated structural parameter is also ensured by a small entropy value. Referring to the identification of the modal coordinate, Kammer (1991) proposed the effective influence (EfI) method, in which the sensor locations are selected according to the quality of the estimated modal coordinates. The contributions of each sensor location to the linear independence of the mode shapes are calculated and the locations corresponding to a large norm value of the Fisher information matrix are selected. The EfI method was then extended by Kammer and Tinker (2004) for the OSP of the triaxial accelerometers. Generally, the distinguishability of the obtained mode shapes needs to be guaranteed before the modal analysis (Penny et al., 1994). Carne and Dohrmann (1995) used the modal assurance criterion (MAC) to the selected sensor locations, in which the sensor configuration with small MAC values ensures the distinguishability of the obtained mode shapes. Shi et al. (2000) selected sensor locations according to their ability to localize structural damage based on the eigenvector sensitivity method, in which the difference between the changes in the mode shapes from the FE model and the experimental results is analyzed. Once the criteria for evaluating sensor placement are determined, some intelligence algorithms are utilized to perform a more efficient search to obtain the sensor locations (Yi et al., 2015a, 2015b). Salehpour-Oskouei and Pourgol-Mohammad (2015, 2017, 2018) comprehensively used the information uncertainty and the risk index to select the sensor locations, in which the uncertainty of the estimation of the system state is reduced and the operation of the sensor system is guaranteed. Different from the OSP of the accelerometers, there are few special criteria to guide the selection of the strain gauge locations before placing strain gauges in an SHM system. When performing damage detection and fatigue analysis, the strain gauges are often used to obtain structural deformation information and welded details (Adewuyi and Wu, 2011; Pedram et al., 2016; Sanayei et al., 2011); the strain gauges are often placed on the positions with large structural deformations or uniformly distributed (Deng et al., 2015; Li et al., 2003; Matta et al., 2008).

Considering the wide use of accelerometers and strain gauges in the SHM systems, sensor placement with a single type sensor cannot take full advantage of structural information. The relationship between the strain mode shape and the displacement mode shape was derived by Yam et al. (1996), which can be helpful in strain mode shape–based displacement estimation (Wang et al., 2014). In this way, displacement information contained in the strain data can be utilized. An OSP method making full utilization of the measured data from strain gauges and accelerometers is extremely meaningful in the SHM. Zhang et al. (2011) used the relationships between the displacement and the strain to estimate the responses of some locations without sensors, and the strain gauge locations and the displacement transducer locations are selected according to the best response reconstruction result. Zhang and Xu (2016) proposed a Kalman filter–based multi-type sensor method to perform response and excitation reconstruction, which can use three types of sensors (i.e. strain gauge, accelerometer and displacement transducer) to achieve the best estimation of the structural responses of some locations without sensors.

However, these works (Wang et al., 2014; Zhang and Xu, 2016; Zhang et al., 2011) all focused on the reconstruction of structural responses, and only less attention was paid to obtaining the structural modal information. The displacement mode shapes of some locations can be estimated from the strain mode shapes of the measured strains by the relationship of the strain mode shape and displacement mode shape (Pei et al., 2018). Thus, the strain data can reflect local structural deformation and be used to obtain displacement mode shapes of some locations without accelerometers at the same time. Through this way, structural modal information contained in the strain data can be fully utilized. In addition, in practice, the strain gauge locations are usually fixed on the key locations to obtain local structural deformation information. The stain gauge locations need to be determined on the positions with large structural deformations at first.

In this article, strain gauges are placed together with accelerometers to obtain as much structural information (i.e. large structural deformations and reliable displacement mode shapes) as possible, in which the structural modal information contained in the strain and acceleration data can be fully utilized. The strain mode shapes of strain gauge locations are used to obtain the desired displacement mode shapes of estimated locations without accelerometers. After the strain gauges are first placed on key locations with large local structural deformations (i.e. mid-spans in bridge structure), the least squares method is utilized to obtain the estimated displacement mode shapes. It should be noticed that if there is not enough displacement mode shape information contained in the strain mode shapes of the selected strain gauge locations, it is possible that the estimated displacement mode shapes will deviate far from the real values. Thus, in order to obtain a better estimation result, the strain gauge locations need to be adjusted slightly nearby the large structural deformation positions to contain more desired displacement mode shape information. In addition, the accuracy of the estimated displacement mode shapes is greatly affected by the estimated locations, so that the estimated locations corresponding to the best estimation quality are finally determined. Then, when selecting the accelerometer locations, a comprehensive use of the MAC and redundancy information is taken into account, so that the reliability of the combined displacement mode shapes is guaranteed. The remaining sections of this article are organized as follows: section “Theory formulation” describes the theory formulation of the displacement mode shape estimation process and the procedure of the proposed sensor placement method. Numerical investigations are presented in section “Numerical studies,” in which the proposed sensor placement method is demonstrated by applying to a bridge benchmark structure. Some conclusions are drawn in section “Conclusion.”

Theory formulation

In this article, the strain gauges are used to monitor the normal strains of the key locations in the structure. The triaxial accelerometers are utilized to measure accelerations in three directions; situations using accelerometers with fewer axes can also be solved with the proposed method. To obtain structural deformation information of the key locations, strain gauges are preliminarily placed on the positions with large structural deformations (i.e. mid-span positions in the bridge structure). Then, the strain mode shapes of the selected strain gauge locations can be used to obtain the displacement mode shapes of some positions where there are no accelerometers installed. In the proposed sensor placement method, the selection of the strain gauges and accelerometers is connected by the relationship of the strain mode shape and displacement mode shape.

Displacement mode shape estimation

The relationship of the strain mode shape and displacement mode shape is derived by the FE method. The dynamic equation of the FE model of a structure can be expressed as

\overset{\cdot\cdot}{M} δ + C \overset{\cdot}{δ} + K δ = f

(1)

where $M$ , $C$ , and $K$ represent the mass, damping, and stiffness matrixes of the structure, respectively; $f$ represents the input vector; $δ$ , $\overset{\cdot}{δ}$ , and $\overset{\cdot\cdot}{δ}$ are the nodal displacement, velocity, and acceleration vector of all nodes in the global coordinate system, respectively. Every node of the FE model has 6 DOFs corresponding to three translations and three rotations.

When strain gauges’ locations are preliminarily determined, some strain gauge locations of the FE model are selected, and the selected strains can be expressed by the nodal displacements in the local coordinate system and global coordinate system, respectively

ε_{i} = B_{i} {\bar{δ}}_{i}

(2)

ε = \tilde{B} δ

(3)

where $ε_{i}$ and $ε$ denote the strains in the $i th$ element and the whole FE model, respectively; ${\bar{δ}}_{i}$ denotes the nodal displacement vector of the $i th$ element in the local coordinate system; $B_{i}$ and $\tilde{B}$ denote the transformation matrix of the $i th$ element and the whole FE model, in the local and global coordinate systems, respectively, representing the relationship of the selected strains and the nodal displacements. It is noted that when the FE model is determined, $B_{i}$ and $\tilde{B}$ change with the different strain gauge locations, in which each row of $B_{i}$ $(\tilde{B})$ corresponds to a selected strain gauge location.

The strain mode shapes of the selected strain gauge locations in the FE model can then be expressed by the displacement mode shapes in the local coordinate system and global coordinate system, respectively

Ψ_{i} = B_{i} {\bar{Φ}}_{i}

(4)

Ψ = \tilde{B} Φ

(5)

where $Ψ_{i}$ and $Ψ$ represent the strain mode shapes in the $i th$ element and the whole FE model, respectively; ${\bar{Φ}}_{i}$ represents the local nodal displacement mode shapes of the $i th$ element in the local coordinate system; and $Φ$ represents the nodal displacement mode shapes of the whole FE model in the global coordinate system.

From equation (4), it is noted that the strain mode shapes in the $i th$ element are only related to the nodal displacement mode shapes of the $i th$ element. Each row of $Ψ_{i}$ corresponds to a selected strain gauge location. Each row of ${\bar{Φ}}_{i}$ corresponds to 1 DOF of the FE model. As shown in equation (4), the strain mode shape $Ψ_{i}$ is a linear combination of the nodal displacement mode shapes ${\bar{Φ}}_{i}$ . Thus, the variable values of elements in $B_{i}$ represent the amount of the displacement mode shape information of the corresponding DOFs. If some variable values in $B_{i}$ are very small, there is little displacement mode shape information of the corresponding DOFs contained in the strain mode shapes $Ψ_{i}$ . Therefore, if we want to obtain the translation mode information in the x direction, the variable values corresponding to these DOFs in $B_{i}$ need to be as large as possible and the variable values corresponding to other DOFs (i.e. translation modes in the y and z directions, rotation modes in the three directions) need to be as small as possible. As a result, the strain gauge locations need to be adjusted slightly nearby the large structural deformation positions to contain more displacement mode shape information, which can be helpful to acquire a good estimation result of the displacement mode shapes.

After the strain gauge locations are determined, the strain mode shapes of these selected locations are used to estimate the displacement mode shapes of some positions where there are no accelerometers installed. These positions are defined as estimated locations. The displacement mode shapes of these estimated locations are expressed as

Φ_{e} = S Φ

(6)

where $Φ_{e}$ represents some rows of $Φ$ that need to be estimated and $S$ is the selection matrix that consists of only zeros and ones and only one nonzero element in each row, which is corresponding to the estimated locations.

In terms of the different DOFs of the FE model, equation (5) can be further expressed as

Ψ = {\tilde{B}}_{e} Φ_{e} + {\tilde{B}}_{r} Φ_{r}

(7)

where $Φ_{r}$ represents the remaining rows of $Φ$ , ${\tilde{B}}_{e}$ represents some columns of the transformation matrix $({\tilde{B}}_{e} = \tilde{B} S^{T})$ that correspond to the estimated locations, and ${\tilde{B}}_{r}$ consists of the remaining columns of $\tilde{B}$ .

The strain mode shapes obtained from the strain measurements often differ from the actual structural system outputs and the prediction error usually occurs due to the measurement noise and the FE model deviation (Beck and Katafygiotis, 1998; Shi et al., 2000). Taking the prediction error into account gives

Ψ - {\tilde{B}}_{r} Φ_{r} = {\tilde{B}}_{e} Φ_{e} + v

(8)

where $v$ represents the prediction error; the $i th$ column of $v$ is assumed with zero mean and a covariance matrix $Cov (v_{(i)}) = σ_{i} I$ . In equation (8), when the row number of ${\tilde{B}}_{e}$ is greater than or equal to the column number of ${\tilde{B}}_{e}$ , the least squares estimation of the displacement mode shapes of the selected locations can be determined

{\hat{Φ}}_{e} = {({\tilde{B}}_{e}^{T} {\tilde{B}}_{e})}^{- 1} {\tilde{B}}_{e}^{T} (Ψ - {\tilde{B}}_{r} Φ_{r})

(9)

{\hat{Φ}}_{e (i)} = {({\tilde{B}}_{e}^{T} {\tilde{B}}_{e})}^{- 1} {\tilde{B}}_{e}^{T} (Ψ_{(i)} - {\tilde{B}}_{r} Φ_{r (i)})

(10)

where ${\hat{Φ}}_{e}$ represents the obtained estimated displacement mode shapes, which can be seen as the estimation result of $Φ_{e}$ , ${\hat{Φ}}_{e (i)}$ represents the $i th$ column of ${\hat{Φ}}_{e}$ , $Ψ_{(i)}$ represents the $i th$ column of $Ψ$ , and $Φ_{r (i)}$ is the $i th$ column of $Φ_{r}$ .

Then, the covariance matrix of the estimated displacement mode shape vector ${\hat{Φ}}_{e (i)}$ is given as

\begin{matrix} Cov ({\hat{Φ}}_{e (i)}) = {({\tilde{B}}_{e}^{T} {\tilde{B}}_{e})}^{- 1} {\tilde{B}}_{e}^{T} \\ Cov (Ψ_{(i)} - {\tilde{B}}_{r} Φ_{r (i)}) {({({\tilde{B}}_{e}^{T} {\tilde{B}}_{e})}^{- 1} {\tilde{B}}_{e}^{T})}^{T} \\ = {({\tilde{B}}_{e}^{T} {\tilde{B}}_{e})}^{- 1} {\tilde{B}}_{e}^{T} P^{T} P Cov (v_{(i)}) {({({\tilde{B}}_{e}^{T} {\tilde{B}}_{e})}^{- 1} {\tilde{B}}_{e}^{T})}^{T} \\ = σ_{i}^{2} (S {\tilde{B}}^{T} \tilde{B} S^{T})^{- 1} \end{matrix}

(11)

In the covariance matrix, the diagonal terms denote the uncertainty of the estimated displacement mode shape vector. The $l th$ term of the diagonal vector of $Cov ({\hat{Φ}}_{e (i)})$ corresponds to the estimation uncertainty of the $l th$ term of ${\hat{Φ}}_{e (i)}$ . The trace of the covariance matrix $Cov ({\hat{Φ}}_{e (i)})$ is used to evaluate the uncertainty of the estimation of the $i th$ column of ${\hat{Φ}}_{e}$

error ({\hat{Φ}}_{e (i)}) = σ_{i} tr (\sqrt{{(S {\tilde{B}}^{T} \tilde{B} S^{T})}^{- 1}})

(12)

where $tr$ represents the operation of calculating the trace of the matrix. Here, the square root is used as a tool to obtain the standard deviation of the estimated displacement mode shapes, which can be more intuitive. It should be noticed that the determinant is another tool to evaluate the uncertainty, which is often used in the OSP problems. In the Gaussian distribution, the determinant of covariance matrix is contained in the formula of the probability density and the information entropy. When using the information entropy to represent the uncertainty of the estimation or identification result, the determinant is a very probable tool (Papadimitriou et al., 2000; Papadimitriou, 2004; Yuen and Kuok, 2015). However, in this article, the diagonal elements of the covariance matrix $Cov ({\hat{Φ}}_{e (i)})$ correspond to the variance of each element of the estimated vector ${\hat{Φ}}_{e (i)}$ ; the square root is used to obtain the standard deviation. When only considering the concept of the variance to represent the error, we think the trace with the square root is enough. This is why we choose the trace not the determinant.

The uncertainty of the estimated displacement mode shape matrix ${\hat{Φ}}_{e}$ consisting of the displacement mode shape vectors ${\hat{Φ}}_{e (i)}$ of all mode orders can be expressed as

\begin{matrix} error ({\hat{Φ}}_{e}) = \sum_{i = 1}^{N_{m}} error ({\hat{Φ}}_{e (i)}) = \sum_{i = 1}^{N_{m}} σ_{i} tr (\sqrt{{(S {\tilde{B}}^{T} \tilde{B} S^{T})}^{- 1}}) \\ = \propto tr (\sqrt{{(S {\tilde{B}}^{T} \tilde{B} S^{T})}^{- 1}}) \end{matrix}

(13)

where $N_{m}$ is the number of the mode orders. From equation (13), it is found that the uncertainty of the estimated displacement mode shape matrix is related to the trace value, so that the trace value can represent the quality of the estimation process.

At last, the selection of the estimated locations and the uncertainty of the estimated displacement mode shape matrix are connected by the trace value of the covariance matrix. The smaller the trace value is, the smaller the uncertainty of the estimation result is. The selection matrix $S$ corresponding to the smallest trace value is selected to ensure that the estimated displacement mode shape is the most accurate. The locations where the displacement mode shapes need to be estimated can be determined from the selection matrix $S$ .

Selection of the accelerometer locations

After the strain gauge locations are determined, more triaxial accelerometers are added to the existing sensor placement. As mentioned before, the displacement mode shapes of the estimated locations are obtained by the strain mode shapes of the selected strain gauge locations. Then, the MAC is used to evaluate the distinguishability of the obtained displacement mode shapes, and the MAC values need to be smaller than 0.2 (Carne and Dohrmann, 1995). The MAC matrix of the obtained displacement mode shapes is shown as

MA C_{i, j} = \frac{{(Φ_{*, i}^{T} Φ_{*, j})}^{2}}{(Φ_{*, i}^{T} Φ_{*, i}) (Φ_{*, j}^{T} Φ_{*, j})}

(14)

where $Φ_{*, i}$ and $Φ_{*, j}$ are the $i th$ and $j th$ columns of the obtained estimated displacement mode shapes. Small values of the off-diagonal terms of the MAC matrix are expected. The maximum off-diagonal MAC term value of the estimated displacement mode shapes of the estimated locations is calculated. If the value is large (>0.2), more accelerometers are needed to make the maximum off-diagonal MAC term value of the displacement mode shapes smaller; more accelerometers can also obtain more reliable structural modal information at the same time. Each time, only one accelerometer location bringing the smallest value is added.

Besides the distinguishability of the displacement mode shapes, redundancy information of the displacement mode shapes of the selected locations must also be focused because of the continuity of the mode shapes (Stephan, 2012). The redundancy coefficient is used to evaluate the similarity of the mode shapes at different locations, in which the Frobenius norm is utilized

R_{i, j} = 1 - \frac{{‖ Φ_{3 i, *} - Φ_{3 j, *} ‖}_{F}}{{‖ Φ_{3 i, *} ‖}_{F} + {‖ Φ_{3 j, *} ‖}_{F}}

(15)

where $R_{i, j}$ is the redundancy coefficient of the mode shapes of the $i th$ and $j th$ locations and $Φ_{3 i, *}$ and $Φ_{3 j, *}$ are the three rows of the displacement mode shapes corresponding to the $i th$ and $j th$ locations, respectively. If the value of $R_{i, j}$ is close to one, these two locations have almost the same displacement mode shape information and only one of the two locations remains. A redundancy threshold T is set to help decide whether the two corresponding locations can both remain. If the threshold value is too small, too many candidate locations are deleted that the obtained sensor placements may not satisfy the MAC well. An appropriate redundancy threshold value needs to be determined by the concrete sensor placement process.

Procedure of the optimal dual-type sensor placement method

The proposed sensor placement method can be generalized as three parts: (1) selection of strain gauge locations, (2) determination of estimated locations and displacement mode shape estimation, and (3) selection of accelerometer locations. The steps are given as follows:

Selection of strain gauge locations

Step 1. Select the key locations with large local structural deformations as the initial strain gauge locations, such as the mid-spans in the bridge structure.

Step 2. Adjust the strain gauge locations according to the variable values of elements in $B_{i}$ , and the values corresponding to DOFs of the desired displacement mode shapes need to be large enough.

Determination of the estimated locations and displacement mode shape estimation

Step 1. Obtain the strain mode shape matrix $Ψ$ corresponding to the selected strain gauge locations.

Step 2. Calculate the estimation uncertainty of different estimated location selections by equation (13). Select the best estimated locations $S$ with the smallest trace value.

Step 3. Obtain the estimated displacement mode shapes $Φ_{e}$ of the estimated locations by the estimation process from equation (9).

Selection of accelerometer locations

Step 1. Choose a redundancy threshold value T. Calculate the displacement mode shape redundancy coefficients of the estimated locations by equation (15); delete the locations with redundancy coefficients larger than T.

Step 2. Calculate the displacement mode shape redundancy coefficients of the selected locations and the remaining candidate locations; delete the remaining candidate locations with redundancy coefficients larger than T.

Step 3. From the remaining candidate locations, add one accelerometer location with the smallest maximum off-diagonal MAC value to the existing placement.

Step 4. Check the number of remaining locations. If there are remaining locations, return to step 3; otherwise, go to the next step.

Step 5. Check the maximum off-diagonal MAC value of the displacement mode shapes. If the MAC value is small enough, make the redundancy threshold value smaller, return to step 1, and restart the accelerometer location selection that no candidate locations are deleted; otherwise, go to the next step.

Step 6. Obtain the optimal sensor placements under an appropriate redundancy threshold value T.

The flowchart of the proposed sensor placement method is shown in Figure 1.

Figure 1.

The flowchart of the sensor placement method.

Numerical studies

The presented dual-type sensor placement method is applied to accomplish the optimal sensor configuration of a two-continuous-span bridge benchmark structure (Caicedo et al., 2006); the bridge benchmark structure has been widely used in different SHM investigations (Erdogan et al., 2014; Gul and Catbas, 2011). Figures 2 and 3 show the diagram and the FE model of the benchmark structure, respectively. The universal steel beam $S 3 \times 5.7$ is used in the FE model; the FE model of this bridge benchmark structure consists of 177 nodes and 182 elements. The first 10 mode shapes obtained from the modal analysis are employed here. Besides, the strain gauges and triaxial accelerometers are placed on the bridge deck.

Figure 2.

The bridge benchmark structure.

Figure 3.

FE model of the benchmark structure.

Selection of the strain gauges

In bridge structures, strain gauges are often placed on the locations with large deformations such as the mid-span of each span to meet the engineering requirements. In this article, the strain mode shapes of the selected strain gauge locations are used to estimate the displacement mode shapes of some positions where there are no accelerometers installed, so that the strain gauge locations need to contain as much desired displacement mode shape information as possible to avoid a void estimation. Thus, the strain gauges must be placed on the locations that have large deformation and much desired displacement mode shape information at the same time.

As shown in equation (4), in the $i th$ beam element, how much displacement mode shape information is contained in the selected strain locations is reflected by the variable values of the transformation matrix $B_{i}$ . In addition, the relationship (equation (4)) works in the local coordinate system. Each row of $B_{i}$ corresponds to a selected strain gauge location and each column of $B_{i}$ corresponds to the displacement mode shape on 1 DOF of the FE model. In this FE model of the bridge benchmark structure, the Euler–Bernoulli theory is used in the beam elements that in each beam element the strains are only related to the displacements of the two nodes. Here, in the $i th$ beam element, one row of $B_{i}$ is shown to analyze how much displacement mode shape information is contained in a strain gauge location

\begin{matrix} B_{ij} = \frac{1}{l^{3}} [- l^{2}, 6 yl - 12 xy, 6 zl - 12 xz, 0, 6 xzl - 4 z l^{2}, 4 y l^{2} - 6 xyl, \\ l^{2}, 12 xy - 6 yl, 12 xz - 6 zl, 0, 6 xzl - 2 z l^{2}, 2 y l^{2} - 6 xyl] \end{matrix}

(16)

where $B_{ij}$ represents the $j th$ row of $B_{i}$ , which shows the relationship between the strain mode shape of the $j th$ selected strain gauge location in the $i th$ beam element and the nodal displacement mode shapes of the $i th$ beam element; l represents the length of the $i th$ beam element; x, y, and z represent the local coordinates of the $j th$ selected strain gauge location in the x, y, and z directions, respectively.

Every row of $B_{i}$ has the same formulation of $B_{ij}$ , but only the values of the coordinates x, y, and z are changed according to the specific locations. On the right-hand side of equation (16), the 12 variables are related to the 12 DOFs of the nodal displacement mode shapes. The first three variables correspond to the three DOFs’ translation modes of the beginning node; the 4th–6th variables correspond to the three DOFs’ rotation modes of the beginning node. The 7th–9th variables correspond to the three DOFs’ translation modes of the ending node; the last three variables correspond to the three DOFs’ rotation modes of the ending node. Considering the practical situation, the translation modes corresponding to the accelerometers are widely used and the rotation modes are often neglected. The strain mode shapes of the selected strain gauge locations are used to estimate the translation modes of the two nodes. When selecting strain gauge locations, the translation mode information contained in the strain gauge locations should be as much as possible. The values of the six (1st–3rd and 7th–9th) variables of $B_{ij}$ correspond to the amount of the contained translation mode information of the two nodes.

In this bridge benchmark structure, strain gauges are first placed on the four cross sections of the mid-spans, because the largest deformations usually occur at these positions. Then, the variables on the right-hand side of equation (16) are examined to evaluate the selected strain gauge locations. The cross sections are at the ends of the beam elements in this FE model of the benchmark structure. It is found that the six (1st–3rd and 7th–9th) variables of $B_{ij}$ corresponding to the translation modes all have maximum absolute values $((1 / l^{3}) [- l^{2}, - 6 yl, - 6 zl, l^{2}, 6 yl, 6 zl])$ , so these four cross sections are acceptable. Considering the form of the beam element (I beam $S 3 \times 5.7$ ), from equations (2) and (16), strain gauges are placed on the flange of the beam element to obtain large deformations and ensure that the variables in $B_{ij}$ corresponding to the translation modes have the largest values (the values of y and z). Finally, the four mid-span cross sections of the FE model are chosen to place strain gauges and the stain gauges are placed on the four corners of each cross section. As shown in Figure 4, 16 strain gauges are finally placed on the bridge benchmark model; the red lines indicate the locations of the cross sections and the red solid rectangles represent the strain gauge locations at the cross section.

Figure 4.

Locations of the 16 strain gauges.

Estimation of the displacement mode shapes

The 16 strain gauges are placed on the four mid-span cross sections, which are also at the node locations of the bridge benchmark model. From equation (4), it is clear that the strain mode shapes of some selected locations in one beam element only correspond to the displacement mode shapes of the two nodes of the corresponding element. When the strain gauge locations are on one selected cross section, which is one node location of the FE model. The strain mode shapes of cross section locations are contained in two adjacent beam elements, which relate to the displacement mode shapes of three nodes of the two adjacent beam elements. The four cross sections are related to 12 nodes of 8 beam elements. Thus, there are 12 estimation locations corresponding to 12 nodes and the displacement mode shapes of these 12 estimated locations are then estimated, which is shown in Figure 5.

Figure 5.

12 estimation node locations.

When performing the displacement mode shape estimation, uncertainty is represented by the covariance matrix shown in equation (11). The trace values of the covariance matrix of the estimated displacement mode shape vector of these 12 nodes in equation (13) are listed in Table 1. From Table 1, it can be seen that the trace values are very small (below 2% of the prediction error $σ_{i}$ ). In addition, the trace values are similar to each other, and thus the 12 nodes are all preliminarily accepted as the estimated locations. The effect of error amplification caused by the transformation matrix $B_{i}$ on the estimation result is then studied by examining the quality of the estimated displacement mode shapes. According to equation (10), $σ_{i}$ is determined as 5% and 10% of the maximum value in the ith column of the displacement mode shape matrix. Then, the estimated displacement mode shapes of these 12 nodes are obtained, which are shown in Figures 6 and 7. The red lines represent the error bar of the estimated result, which corresponds to the 95% confidence interval. It is found that when $σ_{i}$ is determined as 5% of the maximum value in the ith column of the displacement mode shape matrix, the maximum absolute estimation error is below 3%; when $σ_{i}$ is determined as 10% of the maximum value in the ith column of the displacement mode shape matrix, the maximum absolute estimation error is below 6%. The accuracy of the estimation result of the displacement mode shapes demonstrates that selecting estimated locations by the trace values is acceptable; the proposed sensor placement method to obtain more structural displacement mode shape information from the strain data is effective.

Table 1.

Trace values of the covariance matrix of different cases.

Node	1	2	3	4	5	6	7	8	9	10	11	12
Trace value (10⁻²)	1.85	1.83	1.81	1.85	1.85	1.85	1.85	1.83	1.81	1.85	1.85	1.85

Figure 6.

Estimated displacement mode shapes on the 12 nodes under 5% noise: (a) 1st mode shape in the y direction, (b) 2nd mode shape in the y direction, (c) 3rd mode shape in the z direction, (d) 4th mode shape in the z direction, (e) 5th mode shape in the z direction, and (f) 6th mode shape in the z direction.

Figure 7.

Estimated displacement mode shapes on the 12 nodes under 10% noise: (a) 1st mode shape in the y direction, (b) 2nd mode shape in the y direction, (c) 3rd mode shape in the z direction, (d) 4th mode shape in the z direction, (e) 5th mode shape in the z direction, and (f) 6th mode shape in the z direction.

After the translation modes of the estimated locations are obtained from the displacement mode shape estimation process, the redundancy coefficients of the estimated displacement mode shape matrix are calculated. As listed in Table 2, when the nodes are adjacent, the redundancy coefficients of the corresponding displacement mode shapes are close to one, which means that the three locations share almost the same displacement mode shape information. Thus, in every three adjacent nodes, only one node remained. Last, the locations of the four mid-span cross sections are chosen as the estimated locations and the displacement mode shapes of these four locations are used for further modal evaluation.

Table 2.

The redundancy coefficients of the 12 estimated node locations.

Node	1	2	3	4	5	6	7	8	9	10	11	12
1	1.00	0.95	0.89	0.31	0.32	0.32	0.39	0.38	0.38	0.30	0.30	0.30
2	0.95	1.00	0.95	0.30	0.31	0.32	0.38	0.39	0.39	0.30	0.30	0.30
3	0.89	0.95	1.00	0.30	0.30	0.31	0.38	0.39	0.39	0.30	0.30	0.30
4	0.31	0.30	0.30	1.00	0.95	0.89	0.30	0.30	0.30	0.38	0.37	0.36
5	0.32	0.31	0.30	0.95	1.00	0.95	0.30	0.30	0.30	0.37	0.37	0.36
6	0.32	0.32	0.31	0.89	0.95	1.00	0.30	0.30	0.30	0.36	0.36	0.37
7	0.39	0.38	0.38	0.30	0.30	0.30	1.00	0.95	0.89	0.31	0.32	0.32
8	0.38	0.39	0.39	0.30	0.30	0.30	0.95	1.00	0.95	0.30	0.31	0.32
9	0.38	0.39	0.39	0.30	0.30	0.30	0.89	0.95	1.00	0.30	0.30	0.31
10	0.30	0.30	0.30	0.38	0.37	0.36	0.31	0.30	0.30	1.00	0.95	0.89
11	0.30	0.30	0.30	0.37	0.37	0.36	0.32	0.31	0.30	0.95	1.00	0.95
12	0.30	0.30	0.30	0.36	0.36	0.37	0.32	0.32	0.31	0.89	0.95	1.00

Selection of added accelerometers

The distinguishability of the displacement mode shapes of the four estimated locations must be guaranteed; the maximum off-diagonal MAC term value is used to evaluate the displacement mode shapes. More accelerometers are added to the existing sensor placement to decrease the MAC values of the corresponding displacement mode shapes and also obtain more reliable structural displacement mode shape information. In every step, only one accelerometer location that brings the smallest value of the maximum off-diagonal MAC term is selected to be added to the existing placement. Figure 8 depicts the relationship of the maximum off-diagonal MAC value and the different added accelerometer numbers. Different redundancy threshold values (i.e. 1, 0.8, 0.6, and 0.5) are taken into account here. When the threshold value is 1, no accelerometer locations will be deleted from the candidate locations due to the redundancy coefficients. An appropriate redundancy threshold value will be determined according to the maximum off-diagonal MAC term value corresponding to the displacement mode shapes of the different sensor placement.

Figure 8.

Values of the maximum off-diagonal term in the MAC matrix.

It is observed from Figure 8 that when the number of the added accelerometers is smaller than 3, the maximum off-diagonal MAC term values associated with different threshold values are identical. It indicates that the redundancy coefficients of the first several selected accelerometer locations are smaller than 0.5. It is also observed from Figure 8 that when the added accelerometer number is larger than 8, the maximum off-diagonal MAC value (with the threshold values of 1 and 0.8) decreases slightly with the increase of the added accelerometers. It can be found from Figure 8 that when considering the last several added accelerometer numbers, the maximum off-diagonal MAC value (with the threshold values of 0.6 and 0.5) increases obviously with the increase of the added accelerometers. Different threshold values produce different changing trends of the maximum off-diagonal MAC value. If a too small threshold value is chosen, the corresponding selected accelerometer locations will perform poorly under the MAC. From Figure 8, 10 accelerometers can be added with a threshold of 0.6 and only 7 accelerometers can be added with a threshold value of 0.5. The number of the added accelerometers with the threshold values of 1 and 0.8 is larger than 15. The changes in the added accelerometer numbers with different threshold values indicate that many candidate accelerometer locations are deleted because of a too small threshold value. As a result, the obtained sensor placement may perform poorly under the MAC if a too small threshold value is selected.

A comprehensive consideration of the maximum off-diagonal MAC value and redundancy information is needed to determine the appropriate threshold value and the number of selected accelerometers. While the threshold value of 0.5 has a large maximum off-diagonal MAC term value, the threshold value of 0.6 can ensure a small maximum off-diagonal MAC term value and appropriate number (no smaller than the modal orders) of added accelerometers. Thus, the redundancy threshold value of 0.6 is accepted and the number of added accelerometers is 10. As shown in Figure 9, the red lines indicate the locations of the cross sections; every four strain gauges represented by the red rectangles are placed on each cross section; the 10 accelerometer locations are denoted by the red solid circles.

Figure 9.

The final multi-type sensor placement.

To visualize the influence of the different redundancy threshold values on the obtained displacement mode shape locations, sensor placement with the threshold values of 1 and 0.6 are taken into account. It is observed from Figure 10 that the two sensor placements both add 10 accelerometers to the existing 4 estimated locations; the red solid triangles represent the 4 estimated locations and the red solid circles represent the 10 added accelerometer locations. As shown in Figure 10(a), when the threshold value is 1, some selected locations are so close to each other that they share similar displacement mode shape information, which is not acceptable. In Figure 10(b), when the threshold value is 0.6, the situation of too close selected locations is avoided. A decentralized sensor placement often has good performance when the concept of redundancy information is taken into account.

Figure 10.

The displacement mode shape locations with different threshold values: (a) without threshold and (b) with the threshold value of 0.6.

In this bridge benchmark structure, the strain gauges are placed on the cross sections of the mid-span positions. In each cross section, the strain gauges are placed on the four corners because the form of the beam is an I beam. When other structures are taken into account, if the span lengths are too long, more strain gauges need to be on the cross section of the 1/4 and 3/4 length of the spans to obtain more useful deformation information and then to be adjusted by the amount of the contained displacement mode shape information. If the form of the cross-section changes, the strain gauges must be placed on the locations corresponding to large deformations and these locations must contain as much displacement mode shape information as possible, in which the transformation matrix can still be used. After the concrete strain gauge locations are determined according to the FE model of the structure, the displacement mode shapes of the estimated locations are obtained from the proposed displacement mode shape estimation process. Then, more accelerometer locations can be selected according to the MAC and the redundancy information. It can be seen that the proposed sensor placement method is still feasible in different bridge structures.

Conclusion

This study presents an optimal dual-type sensor placement method for strain gauges and triaxial accelerometers to obtain large local structural deformations and reliable displacement mode shapes, in which the structural modal information contained in the strain and acceleration data is fully utilized. A displacement mode shape estimation method is proposed using the strain mode shapes from the strain gauge locations to obtain the displacement mode shapes of the estimated locations without accelerometers. The strain gauge locations are placed near the positions with large deformations, to obtain the local structural information of the largest deformations. In addition, to obtain the accurate displacement mode shapes of the estimated locations by the proposed displacement mode shape estimation method, the strain gauge locations also contain enough information on the desired displacement mode shapes. When determining the accelerometer locations, the MAC and redundancy information is used together to guarantee the distinguishability and small redundancies of the combined displacement mode shapes from the strain gauges and accelerometers. The effectiveness of the proposed sensor placement method is demonstrated by the numerical investigation of a bridge benchmark structure. To some extent, the small trace values of the covariance matrix of the estimated displacement mode shape vector can guarantee the accuracy of the estimation results. The maximum absolute errors of the obtained estimated displacement mode shapes under the 5% and 10% noise are below 3% and 6%, respectively, which indicates that the estimation result is acceptable and the trace values can guide the selection of the estimated locations effectively. In practice, the effect of noise on the mode shapes is small so that the mode shape estimation result will be more accurate. The redundancy threshold value in this article is finally determined as 0.6 and the MAC values are smaller than 0.2, which can guarantee both the distinguishability and the small redundancy of the combined displacement mode shapes.

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: The authors wish to acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 51625802 and 51478081), the 973 Program (Grant No. 2015CB060000), and the Foundation for High Level Talent Innovation Support Program of Dalian (Grant No. 2017RD03).

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