Optimal Placement of Wireless Sensor Nodes for Bridge Dynamic Monitoring Based on Improved Particle Swarm Algorithm

Abstract

The issue of optimal placement of wireless sensor networks (WSNs) is one of the major challenges for dynamic monitoring of bridges. It should be solved based on combining the effective monitoring of the dynamic performance of a bridge with the energy consumption of WSNs. Thus, the relationship between the bridge modal and the energy consumption of wireless networks is derived. Optimizing sensor location is achieved by using the improved wavelet particle swarm algorithm, which overcomes the disadvantage of the conventional particle swarm algorithm by applying mutation operator to the selected particles with certain small probability. Based on the comparison between the ANSYS simulation results and the measured bridge data, as well as the energy distribution diagram of WSNs, the proposed method is shown to reflect the dynamic performance of the test bridge well. Moreover, the method effectively controls the energy consumption of WSNs and achieves reasonable optimization effects. Therefore, this method can be used for dynamic monitoring of wireless sensors in various bridge types.

1. Introduction

With the rapid development of bridge construction, the safety of the bridges is attracting more and more attention. Once completed and used, the structure or constructional elements of a bridge suffer from continuous degradation, aging of material performance, and external factors. Vehicles, wind, earthquakes, fatigue, overload, and human behaviors are several external factors that damage bridges. Therefore, monitoring and evaluating the physical conditions of bridges to determine their health information are essential. The dynamic monitoring of a bridge structure has become the primary technological means for maintaining the safety of such structures [1–3].

The dynamic monitoring of a bridge structure is based on sensor nodes which collect data on the bridge, such as current structural strains, stresses, and loads. These data are then transmitted to the monitoring center by using a wireless sensor network (WSN). In general, if more sensor nodes are used in a monitoring system, then the structural characteristics of a bridge can be described more accurately. However, because of limits on sensor quantity, economic cost, installation difficulty, and other aspects, the rationality in the number and positioning of sensor node layout is regarded as the premise for guaranteeing quality of structural monitoring [4, 5].

The characteristics of the WSN should also be considered in the actual layout process of its nodes. Excessive sensor nodes will result in oversized transmission data from the WSN. Consequently, an overload in network energy consumption will lead to delayed search speed and poor network stability [6]. Therefore, the optimal layout of sensor nodes is considered as a multiobjective optimization issue based on the aforementioned contents.

Scholars and experts worldwide have studied the optimization issue of sensor layout in bridges. Huang and Zhu considered the maximum nondiagonal elements in a MAC matrix as the objective function and optimized sensor layout according to the contribution of the optional positions of sensors to the MAC matrix [7]. Shan et al. introduced an improved adaptive genetic algorithm for the optimal layout of sensors [8]. Sun et al. adopted classical optimization criteria and the improved genetic algorithm in the form of fitness to obtain optimal layout [9]. Zhang et al. adopted the Pareto genetic algorithm and designed corresponding genetic operators and coding scheme for solving the optimal position of sensors under double criteria [10].

Based on previous researches, sensor layout optimization mainly focuses on a single optimization objective that reflects the static or dynamic characteristics of bridges. Therefore, this paper presents optimal layout conditions of sensor nodes and the optimal conditions of energy consumption of WSNs. This work also established an improved sensor node optimal layout algorithm based on the improved wavelet particle swarm optimization (PSO). The result of the ANSYS simulation analysis was compared with the measured data of the bridge structure. The energy distribution of the WSN was obtained. Lastly, the validity of the optimal layout was verified.

2. Analysis of the Sensor Layout Model for Bridge Monitoring

2.1. Bridge Modal Analysis

Bridge modal analysis is effective in the dynamic monitoring and overall evaluation of the structural state of a bridge [11].

The dynamic equilibrium equation of a bridge can be simplified as

$$\begin{array}{cc}& EI\frac{{\partial }^{\mathrm{4}}u}{\partial x^{\mathrm{4}}}+m\ddot{y}=p\left(x,t\right).\end{array}$$

(1)

This dynamic equilibrium equation belongs to nonlinear dynamic systems, wherein u represents the dynamic deflection of the beam, $EI$ is a constant that represents the flexural rigidity of the beam, m represents beam quality per unit length, and $p(x,t)$ represents the dynamic load. Suppose that the corresponding vibration model function of the beam is ${\phi }_k(x)$ , the generalized coordinate $q_k(t)$ is introduced, and the vibration mode superposition method is applied. Then, the dynamic deflection $u(x,t)$ of the beam under forced vibration can be expressed in the form of a vibration mode series. This vibration mode series can be described as follows:

$$\begin{array}{cc}& y\left(x,t\right)=\sum _{k=\mathrm{1}}^{\infty }{\phi }_k\left(x\right)q_k\left(t\right).\end{array}$$

(2)

By substituting formula (2) into formula (1) and simplifying the resulting equation, the dynamic response of the beam can be obtained as follows:

$$\begin{array}{cc}& u\left(x,t\right)=\frac{\mathrm{2}F}{ml}\hspace{0.17em}\sum _{k=\mathrm{1}}^{\infty }\frac{\mathrm{1}}{{\omega }_k^{\mathrm{2}}-{\Omega }_k^{\mathrm{2}}}\left(\sin {\Omega }_{k}t-\frac{{\Omega }_k}{{\omega }_k}\sin {\omega }_{k}t\right)\sin \frac{k\pi x}{l},\end{array}$$

(3)

where ${\omega }_k={(k\pi /l)}^{\mathrm{2}}\sqrt{EI/m}$ represents the natural vibration frequency of the bridge, ${\Omega }_k=k\pi v/l$ represents the generalized disturbance frequency, v represents the average passing speed of vehicles, l represents the beam span, and F represents the applied force of vehicles on the bridge.

The measurement of the dynamic property of a bridge primarily depends on numerous strain gauges. These strain gauges are distributed throughout the bridge space and connected with WSNs. Suppose that $\{\epsilon \}$ represents the strain of the position to be measured, n represents the quantity of the strain sensor nodes, k represents the modal number, ${\psi }_j$ represents the strain modal vector at the jth order, $\mathrm{\Psi }\in R^{n\times j}$ represents the strain modal vector matrix, $q_j$ represents the strain modal coordinate at the jth order, and $\{q\}\in R^{k\times j}$ . Then, the strain of the bridge can be expressed as follows:

$$\begin{array}{cc}& \left\{\epsilon \right\}=\sum _{l=\mathrm{1}}^m{\psi }_l{q}_l=\mathrm{\Psi }\left\{q\right\}.\end{array}$$

(4)

The least square solution of formula (5) is given as

$$\begin{array}{cc}& \left\{\bar{q}\right\}={\left[{\mathrm{\Psi }}^T\mathrm{\Psi }\right]}^{-\mathrm{1}}{\mathrm{\Psi }}^T\left(\epsilon \right).\end{array}$$

(5)

Suppose that $\{\epsilon \}=\mathrm{\Psi }\{q\}+\{\gamma \}$ , where $\{\gamma \}$ represents the Gaussian white noise with the variance of ${\sigma }^{\mathrm{2}}$ . Then, the covariance of $\{\bar{q}\}$ and $\{q\}$ can be expressed as follows:

$$\begin{array}{cc}& \left[W\right]=E\left[\left(\left\{q\right\}-\left\{\bar{q}\right\}\right){\left(\left\{q\right\}-\left\{\bar{q}\right\}\right)}^T\right]={\left[{\sigma }^{\mathrm{2}}{\mathrm{\Psi }}^T\mathrm{\Psi }\right]}^{-\mathrm{1}}\\ =\frac{\mathrm{1}}{{\sigma }^{\mathrm{2}}}{\left[Q\right]}^{-\mathrm{1}}.\end{array}$$

(6)

Suppose that $\parallel Q\parallel =\parallel {\mathrm{\Psi }}^T\mathrm{\Psi }\parallel$ . When 2-norm Q reaches the maximum value, the covariance $[W]$ becomes the minimum. Then, good estimated results of the strain modal coordinate $q_j$ can be obtained [12–14], that is, good optimal layout results of sensor nodes.

As shown in formulas (4) to (6), the applied force F on the bridge in formula (3) can only be obtained when the strain modal coordinate $q_j$ and the number of strain sensor nodes n are determined. The dynamic response of the bridge can then be obtained.

The estimate of the optimal strain modal coordinate $q_j$ , or the required and reasonable optimal layout for the sensor nodes of the bridge structure, can be determined via bridge modal analysis. However, the node layout is not only about single-objective optimization in the actual layout process of WSN nodes. Node layout also considers the characteristics of WSN nodes. The number of actual sensor nodes in the monitoring system is limited. Excessive sensor nodes may result in oversized transmission data from the WSN. Consequently, an overload in network energy consumption may result in delayed search speed and poor network stability. Therefore, establishing an optimized and energy-efficient WSN structure is extremely important.

2.2. Energy Optimization Analysis of the WSN

In this research, the dynamic monitoring system of the structural state of the bridge for collecting the dynamic signals of the bridge is based on WSN nodes. Considering the numerous and large-scale network nodes, the entire network is divided into small subnetworks called clusters. The manager of each cluster is called the cluster head. To select the cluster head with proper quantity and optimal position [15, 16], the entire WSN adopts the hierarchical topology establishment method with optimal global energy consumption based on location information. In this manner, the WSN will have a cluster hierarchical topological structure (Figure 1), which can reduce the overall energy consumption of the network and extend its life cycle [17].

Figure 1

Transmission and retransmission of signals.

The following is the description for the computation of the overall energy consumption.

Suppose that L represents the length of the bridge and M represents the width of the bridge. The WSN nodes are randomly distributed in the $L\times M$ area. $(x_{\text{CH}},y_{\text{CH}})$ represents the coordinate of the cluster head node, wherein all cluster member nodes CP are in the Voronoi diagram, and all nodes are uniformly distributed [18]. Suppose that the quantity of cluster heads in the $L\times M$ region is $n_{\text{CH}}$ ; then the average coverage region of each cluster head node can be given as $\sqrt{(L\times M)/n_{\text{CH}}}\times \sqrt{(L\times M)/n_{\text{CH}}}$ .

Similarly, suppose that $(x_{\text{CP}},y_{\text{CP}})$   represent the coordinates of the cluster member node CP which are also randomly distributed. In the bridge deck region of $L\times M$ , the mathematical expectations of the cluster head CH and the cluster points CP are given as follows:

$$\begin{array}{cc}& E\left[x_{\text{CH}}\right]=E\left[y_{\text{CH}}\right]=E\left[x_{\text{CP}}\right]=E\left[y_{\text{CP}}\right]=\frac{\mathrm{1}}{\mathrm{2}}\times \sqrt{\frac{L\times M}{n_{\text{CP}}}},\end{array}$$

(7)

$$\begin{array}{cc}& E\left[x_{\text{CH}}^{\mathrm{2}}\right]=E\left[y_{\text{CH}}^{\mathrm{2}}\right]=E\left[x_{\text{CP}}^2\right]=E\left[y_{\text{CP}}^2\right]=\frac{L\times M}{\mathrm{3}{n}_{\text{CH}}}.\end{array}$$

(8)

The location of the gateway node is set at (0, 0). Then, the mathematical expectations of the cluster head CH and the gateway node BS at a particular distance are given as follows:

$$\begin{array}{l}E\left[d_{\text{avg}\left(\text{CH},\text{BS}\right)}^{\mathrm{2}}\right]\\ \hspace{1em}\hspace{1em}=E\left[{\left(x_{\text{BS}}-x_{\text{CH}}\right)}^{\mathrm{2}}+{\left(y_{\text{BS}}-y_{\text{CH}}\right)}^{\mathrm{2}}\right]=\frac{\mathrm{2}}{\mathrm{3}}\times L\times M.\end{array}$$

(9)

Suppose that the cluster distribution function is as follows:

$$\begin{array}{cc}& \rho \left(x,y\right)=\frac{n_{\text{CH}}}{L\times M}.\end{array}$$

(10)

Then, the mathematical expectations of cluster member nodes CP and cluster head CH on a squared distance are given as follows:

$$\begin{array}{cc}& E\left[d_{\text{avg}\left(\text{CP},\text{CH}\right)}^{\mathrm{2}}\right]=\iint \left(x^{\mathrm{2}}+y^{\mathrm{2}}\right)\rho \left(x,y\right)dx\hspace{0.17em}dy=\frac{L\times M}{\mathrm{2}\pi n_{\text{CH}}}.\end{array}$$

(11)

The consumed energy $E_{\text{CP}}$ of cluster member nodes CP can be expressed as follows:

$$\begin{array}{l}{E}_{\text{CP}}=n_{\text{CP}}\times \left(E_{\text{elec}}+E_{\text{fs}}\times d^{\mathrm{2}}\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}=n_{\text{CP}}\times \left(E_{\text{elec}}+E\left[d_{\text{avg}\left(\text{CP},\text{CH}\right)}^{\mathrm{2}}\right]\right)\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}=\left(n-n_{\text{CH}}\right)\times \left(E_{\text{elec}}+E_{\text{fs}}\times \frac{L\times M}{\mathrm{2}\pi n_{\text{CH}}}\right).\end{array}$$

(12)

$E_{\text{CH}}$ has three parts: $E_{RX}$ (the energy that receives data from cluster member nodes), $E_{\text{pro}}$ (the energy that processes data), and $E_{TX}$ (the energy that sends data to the gateway BS), as shown in the following:

$$\begin{array}{l}{E}_{\text{CH}}=E_{RX}+n{E}_{\text{pro}}+E_{TX}\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em}=\left(n-n_{\text{CH}}\right){\text{E}}_{\text{elec}}+n{E}_{\text{pro}}\\ \hspace{1em}\hspace{1em}\hspace{0.17em}+\alpha n_{\text{CH}}\left(E_{\text{elec}}+E_{mp}\times \frac{\mathrm{4}}{\mathrm{9}}\times L^{\mathrm{2}}\times M^{\mathrm{2}}\right).\end{array}$$

(13)

The total energy $EP$ consumed during the topology formation process of the WSN is given in formula (14). $EP$ primarily consists of two parts: $E_{\text{CP}}$ (the consumed energy of the cluster points) and $E_{\text{CH}}$ (the consumed energy of the cluster head). Consider the following:

$$\begin{array}{l}EP=E_{\text{CP}}+E_{\text{CH}}=\left(n-n_{\text{CH}}\right)\times \left(E_{\text{elec}}+E_{\text{fs}}\times \frac{L\times M}{\mathrm{2}\pi n_{\text{CH}}}\right)\\ \hspace{1em}\hspace{0.17em}\hspace{1em}+\left(n-n_{\text{CH}}\right)E_{\text{elec}}+n{E}_{\text{pro}}\\ \hspace{1em}\hspace{1em}\hspace{0.17em}+\alpha n_{\text{CH}}\left(E_{\text{elec}}+E_{mp}\times \frac{\mathrm{4}}{\mathrm{9}}\times L^{\mathrm{2}}\times M^{\mathrm{2}}\right).\end{array}$$

(14)

As shown in formula (14), the number of the optimal cluster head nodes $n_{\text{CH}}$ and their sets can be obtained when the $EP$ is at its minimum. The consumed energy of the sensor network is also at its minimum during this moment.

As stated earlier, the optimal layout of sensors in bridge monitoring is essentially a multiobjective optimization issue. The number and locations of sensors do not only influence the test accuracy of several-order modals but also affect the energy and data transmission of the WSN. The present study adopts a linear weighting method to design the evaluation function. This weighting method can be described as follows:

$$\begin{array}{cc}& y={\lambda }_{\mathrm{1}}\left[W\right]+{\lambda }_{\mathrm{2}}·EP,\end{array}$$

(15)

where ${\lambda }_{\mathrm{1}}+{\lambda }_{\mathrm{2}}=\mathrm{1}$ ( ${\lambda }_{\mathrm{1}}$ and ${\lambda }_{\mathrm{2}}$ represent the weighting coefficients). Suppose that   ${\lambda }_{\mathrm{1}}=\mathrm{0.65}$ and ${\lambda }_{\mathrm{2}}=\mathrm{0.35}$ . Then, focus is put on the accuracy degree of the modal in the optimization. The optimal solution method is used to obtain the number of sensors and their coordinate positions. In this manner, the multiobjective optimization issue of sensor nodes can be solved.

3. Optimal Solution of the Layout Based on Wavelet Particle Swarm Algorithm

The wavelet particle swarm algorithm belongs to an intelligent optimization algorithm based on population search which has been developed in recent years. Derived from the simulation of bird population behaviors, this algorithm is simple and easy to perform; moreover, no gradient information is required and fewer parameters are needed. The present work used the improved particle optimization algorithm [19] to obtain optimal solutions.

Suppose that a population consisting of N particles exists in an objective search space of D dimension, where particle i represents the vector of D dimension, as shown in the following:

$$\begin{array}{cc}& X_i=\left(x_{i\mathrm{1}},x_{i\mathrm{2}},\dots ,x_{iD}\right),\hspace{1em}i=\mathrm{1,2},\dots ,N.\end{array}$$

(16)

The flying speed of particle i is expressed as follows:

$$\begin{array}{cc}& V_i=\left(v_{i\mathrm{1}},v_{i\mathrm{2}},\dots ,v_{iD}\right),\hspace{1em}i=\mathrm{1,2},\dots ,N.\end{array}$$

(17)

The optimal position of particle i that has been searched thus far is described as follows:

$$\begin{array}{cc}& p_{\text{best}}=\left(p_{i\mathrm{1}},p_{i\mathrm{2}},\dots ,p_{iD}\right),\hspace{1em}i=\mathrm{1,2},\dots ,N.\end{array}$$

(18)

The optimal position of the entire particle swarm that has been searched thus far is the global extremum, as shown in the following:

$$\begin{array}{cc}& g_{\text{best}}=\left(p_{g\mathrm{1}},p_{g\mathrm{2}},\dots ,p_{gD}\right).\end{array}$$

(19)

Particles update their speeds and positions according to the following:

$$\begin{array}{cc}& V_{id}^{k+\mathrm{1}}=\omega ·V_{id}^k+c_{\mathrm{1}}·r_{\mathrm{1}}·\left(P_{id}^k-X_{id}^k\right)+c_{\mathrm{2}}·r_{\mathrm{2}}·(P_{gd}^k-X_{id}^k),\end{array}$$

(20)

$$\begin{array}{cc}& x_{id}=x_{id}+v_{id},\end{array}$$

(21)

where ω represents acceleration factors $c_{\mathrm{1}}$ and $c_{\mathrm{2}}$ and $r_{\mathrm{1}}$ and $r_{\mathrm{2}}$ represent the uniform random numbers within the scope of $[0,1]$ .

To improve the performance of the basic PSO algorithm and to avoid premature convergence of the algorithm, the mutation operation in genetic algorithms is added to the former. Hence, an improved PSO algorithm is obtained. Suppose that the mutation probability of each particle is $p\in [0,1]$ , wherein value is determined by the dimensions of the particles. Suppose that $x_i^k=(x_{i\mathrm{1}}^k,x_{i\mathrm{2}}^k,\dots ,x_{ij}^k)$ represents the selected ith mutation particle ${x_{ij}^k}_{}$ at the kth time iteration. Then, $p{x_{\max i}}_{}$ in the jth dimension of the mutation particle represents the upper limit of the search space. Similarly, $p{x_{\min i}}_{}$ is the lower limit of the mutation particle in the search space. The mutation formula is given as follows:

$$\begin{array}{cc}& \text{mut}\hspace{0.17em}\left(x_{ij}^k\right)=\{\begin{array}{cc}{x}_{ij}^k+\sigma ·\left(p{x}_{\max i}-x_{ij}^k\right),\hfill & \text{if}\hspace{0.17em}\hspace{0.17em}\sigma >\mathrm{0},\hfill \\ x_{ij}^k+\sigma ·\left(x_{ij}^k-p{x}_{\min i}\right),\hfill & \text{if}\hspace{0.17em}\hspace{0.17em}\sigma \le \mathrm{0},\hfill \end{array}\end{array}$$

(22)

where σ represents the value of the wavelet function. Then, the Morlet wavelet function is selected and its calculation formula is given as follows:

$$\begin{array}{cc}& \sigma =\frac{\mathrm{1}}{\sqrt{a}}\exp \left(-\frac{{\mathrm{\Phi }}^{\mathrm{2}}}{\mathrm{2}{a}^{\mathrm{2}}}\right)\cos \left(\mathrm{5}\left(\frac{\mathrm{\Phi }}{\mathrm{2}}\right)\right),\end{array}$$

(23)

where the value range of Φ falls between $[-\mathrm{2.5}a,\mathrm{2.5}a]$ . The calculation formula for a is given as follows:

$$\begin{array}{cc}& a=\exp \left(-\ln \left(g\right)\times {\left(\frac{k}{\text{num}}\right)}^{\xi \omega m}\right)+\ln \left(g\right),\end{array}$$

(24)

where $\xi \omega m$ represents the shape parameter, k represents the number of iterations, g represents the upper limit of a, and num represents the maximum number of iterations. The wavelet mutation formula, that is, formula (22), is improved to obtain the following:

$$\begin{array}{cc}& \text{mut}\left(x_{iD}^k\right)=\{\begin{array}{cc}g{b}_D^k+\sigma \times \left(p{x}_{\max i}-g{b}_D^k\right),\hfill & \text{if}\hspace{0.17em}\hspace{0.17em}\sigma >\mathrm{0},\hfill \\ g{b}_D^k+\sigma \times \left(g{b}_D^k-p{x}_{\min i}\right),\hfill & \text{if}\hspace{0.17em}\hspace{0.17em}\sigma \le \mathrm{0},\hfill \end{array}\end{array}$$

(25)

where $g{b}_D^k$ represents the optimal solution of the population at the kth iteration. The difference between formulas (22) and (25) is obvious. In formula (22), one particle is selected in a certain probability for wavelet mutation and this particle may fail to jump out of the local optimal node after mutation. As a result, the global optimal point will not be found. Meanwhile, formula (25) performs wavelet mutation operation on the optimal solutions of the current population, and then transmits the value after mutation to the current particle selected in a certain probability. As such, formula (25) can conduct an effective search around the current optimal solution. This formula can also guide the particle swarm to move forward the global optimal point and to reduce the local optimum possibility of the particles in each iteration.

Based on the preceding analysis, the fitness function is shown in formula (15), $y={\lambda }_{\mathrm{1}}[W]+{\lambda }_{\mathrm{2}}·EP$ . Supposing that ${\lambda }_{\mathrm{1}}=\mathrm{0.65}$ , ${\lambda }_{\mathrm{2}}=\mathrm{0.35}$ , and the acceleration factor in formula (20) is $c_{\mathrm{1}}=c_{\mathrm{2}}=\mathrm{1.4945}$ ; then the number of population particles is 80, and the evolution iteration times of the algorithm are 500. Changes in the fitness values during the evolution process are shown in Figure 2. The red dotted line in the figure represents the curve from the fundamental form of the PSO algorithm. The blue solid line represents the curve from the improved wavelet mutation PSO algorithm. Based on the experimental results, the improved algorithm is superior to the fundamental form of PSO in terms of convergence rate.

Figure 2

The change in fitness values during evolution.

Based on the relationship between cluster head nodes and member nodes of the aforementioned fitness function, the number of cluster head nodes is confirmed to be 12 and that of sensor nodes is 130. All nodes are discretely distributed throughout an 800 M bridge deck. Several sensor coordinates are shown in Table 1.

Table 1

The coordinates of several sensors.

No.	X-coordinate (m)	Y-coordinate (m)
1	−367.210	−3.61
2	−310.031	1.691
3	−227.368	4.712
4	136.811	−1.067
5	53.161	2.343
6	0.123	−0.214
7	55.324	−0.061
8	144.377	−2.347
9	231.331	−1.920
10	371.019	−0.341

4. Simulation and Experimental Analysis

In actual bridge dynamic tests, installing any number of sensor nodes to acquire the dynamic parameters of the entire bridge is not feasible. The acquisition of dynamic parameters is subject to limitation under various conditions. The finite element analysis method can be used to obtain dynamic situations under certain conditions. This research used the finite element software ANSYS to analyze the intrinsic mode of a bridge measuring 800 M long and 10 M wide. The SOLIDE185 was also adopted to establish the model. The elasticity modulus was $E=30$  GPa and Poisson's ratio was $\nu =0.2$ . The model was divided into 4,000 units and 8,822 finite element nodes, which focused on the in-plane vertical bending vibration of the bridge deck. Therefore, the concerned target was a vertical vibration modal. The initial four-order vertical vibration data were selected as the to-be-detected modal vibration data. The analysis results are shown in Figure 3. The obtained vibration data can be compared with the actual measured values.

Figure 3

The calculation result of dynamic characteristic of bridge: (a) first order, (b) second order, (c) third order, and (d) fourth order.

The test object in this paper is the Xiazhang Highway Bridge in Quzhou. The condition of its underbridge is shown in Figure 4(a). Based on the position relationship between cluster head nodes and cluster member nodes, the positions of 130 sensor nodes and 12 cluster head nodes discretely distributed throughout the 800 M long bridge deck were determined in this study. Free-scale MMA7361 micromechanical acceleration sensors were used to detect vibration data on the aforementioned positions (surface of the underbridge). BX120-100AA strain gauges were placed on these points for the strain test. The specific bounded strain gauge condition is shown in Figure 4(b). The WSN nodes are shown in Figure 4(c). As shown in Figure 4(d), the collection nodes have two functions. One function is used to receive data from the nodes in the network, while another function is used to transmit data to the remote monitoring center by 3G communication network. Figure 4(e) shows the installed network nodes and Figure 4(f) illustrates the interface of data collection. The tested data implemented normalization processing. The cubic-spline method rounding was used for the fitted curve. After comparison with the ANSYS analysis results, the initial four-order modal shape diagram was obtained (Figure 5).

Figure 4

(a) The underbridge condition of the Xiazhang Highway Bridge, (b) strain gauges, (c) WSN nodes, (d) the collection node box, (e) installed network nodes, and (f) the interface of data collection.

Figure 5

Comparison of ANSYS analytical curves with measured experimental data. (a) The first-order mode. (b) The second-order mode. (c) The third-order mode. (d) The fourth-order mode.

The goodness of fit between the ANSYS analysis results and the test results is relatively high, thus indicating that the distribution points have relatively high accuracy. Among the vibration modes, the first two-order vibration modes can accurately reflect the actual vibration mode despite the few measuring points. Orders 3 and 4 vibration modes decrease as the number of measuring points decreases and their goodness of fit becomes poorer. The accuracy test for the higher-order vibration modes requires an increase in the number of sensor nodes.

In the regular operation of each sensor node (with a 250 kbps transmission rate), 1 byte needs 4 μs and the transmission power of the node is 0.0004 W. When the two values are multiplied, 1.6 nJ is obtained. In this manner, the consumed energy of each sensor node per second can be measured. Figure 6 shows the energy consumption diagram of each node. Figure 7 shows the energy distribution diagram.

Figure 6

Energy distribution in scenario 1.

Figure 7

Energy consumption per node.

The relationship between energy consumption and cluster nodes is calculated by formula (15) (Figure 8). According to the fitness function, the number of cluster nodes, that is, 12, is close to the minimum value of the energy consumption curve. This result illustrates the rationalization of the proposed optimization.

Figure 8

Relationship between energy consumption of sensor network and number of cluster head nodes.

Based on formula (6), the change rate of the first $i+1$ and i orders can be expressed as follows:

$$\begin{array}{cc}& \text{RC}=\frac{{\parallel Q_{i+\mathrm{1}}\parallel }_{\mathrm{2}}-{\parallel Q_i\parallel }_{\mathrm{2}}}{{\parallel Q_i\parallel }_{\mathrm{2}}},\hspace{1em}\mathrm{1}\le i\le n-\mathrm{1},\end{array}$$

(26)

where i represents the number of computing modals and n represents the number of all modals. The curve is shown in Figure 9. Formula (26) is significant because the first i order modal information can represent itself when the increment of the two norms of the Q matrix in the first $i+1$ order modal becomes zero or a relatively small value. The first i order modal basically contains the modal information of the entire bridge information, thus providing relatively complete modal information of the optimal layout of the sensors. The modal becomes flat after seven orders, thus demonstrating that most modal information of the bridge is contained within seven orders (Figure 9). Therefore, the initial four-order modal that was selected can reveal the major dynamic parameters of the bridge.

Figure 9

Relationship between RC value and modal number.

5. Conclusion

In this paper, an optimization procedure was put forward and was applied on the dynamic monitoring project of Xiazhang Highway Bridge in Quzhou. The experimental results indicated that the accuracy of the bridge modal measurement could be improved by using a multiobjective optimal layout of sensor nodes. The entire network energy could also be reduced. On the other hand, the improved wavelet particle swarm algorithm improved the convergence rate of the optimization process. However, bridge modal information in an ideal situation was acquired by means of ANSYS simulation. The actual modal data of the bridge must have some deflection with the simulation data. The proposed method has practical value and theory value. It can be analogized to the monitoring issues of WSNs for other types of bridges.