Optimization Strategy for Modal Test Measurement Points of Large-Span Steel Beams Based on Improved Particle Swarm Optimization Algorithm with Random Weights

Abstract

In order to obtain better vibration response data and improve the accuracy of results in large-span steel beam modal tests, this paper proposes an optimization strategy for the arrangement of measurement points on large-span steel beams. First, an optimized arrangement of large-span steel beam measurement points was proposed based on an improved particle swarm optimization algorithm; the test function verified the superiority of the improved algorithm. Secondly, the deck of a steel tube truss girder bridge (STTGB) was taken as the research object; the computational modal analysis method was adopted to obtain the computational modal results of the bridge deck. In addition, measurement points were arranged on the bridge deck according to the uniform distribution method and the proposed optimization algorithm, and modal tests were conducted. Finally, the modal parameters of the bridge deck based on the two arrangement methods were obtained and compared to the best arrangement method for the STTGB deck. The results show that the proposed method has good efficiency in the optimal arrangement of the bridge deck measurement points and the obtained modal parameters have high accuracy. Therefore, this paper has important guiding significance for the study of structural dynamic characteristics using the distribution method based on an optimization algorithm.

1. Introduction

As bridges move towards higher heights, larger spans and more complex structural forms, structural health monitoring (SHM) systems using modal- and vibration-based methods have been widely investigated [1,2]. SHM, as a control system, monitors the state of a large civil engineering structure by measuring the structural response. Monitoring and assessing the health status of structures throughout their life cycle is essential to reduce maintenance expenses and labor costs and ensure the safe operation of structures [3,4]. As an important method of SHM, the main purpose of dynamic detection technology is to identify and determine structural dynamic parameters [5]. Through a gradual change in parameters, faults can be predicted in advance and the occurrence of major safety accidents can be prevented. These parameters are mainly obtained using experimental modal analysis technology; that is, structural response data are obtained through sensors and data acquisition equipment, and then modal parameters are obtained through parameter identification. Experimental modal analysis has always been one of the research hotspots in the field of health monitoring. Modal tests of large and complex bridge structures to capture more accurate and complete structural responses from the structure and obtain structural dynamic characteristics for condition assessments and safety warnings have become a hot issue of interest, bridging the academic and engineering communities [6,7].

The first step of modal testing is to determine the number and location of the measurement points of the bridge structure, that is, to install sensors at key parts of the structure to obtain the dynamic characteristics of the bridge and other structural data. Theoretically, the more sensors deployed, the more accurate the response data available. This is feasible for structures such as small bridges but difficult for large and complex bridges [8]. Considering the engineering environment, cost and risk assessments and other factors, the needs of SHM should be met with the lowest number of sensor measurement points. At this time, the result of the modal test depends heavily on the position and number of sensor measurement points, and the optimal number is determined and configured in the best position so as to enhance the accuracy of information acquisition in the modal test. Therefore, research on the optimal arrangement of sensor measurement points is extremely important and has important engineering value and social significance.

Scholars have done extensive research on sensor measurement point optimization. Kammer [9] proposed a famous efficient independent method in 1991 for optimizing sensor placement in large spatial structures. Liu et al. [10] used modal strain energy (MSE) and the modal assurance criterion (MAC) as fitness functions to study the optimization problem of spatial structure sensors using an improved genetic algorithm. Yin et al. [11] proposed a novel relaxation sequence algorithm. A truss structure and a rigid-frame arch bridge were used as examples to verify the effectiveness of the new algorithm in optimizing sensor placement. Chai et al. [12] proposed an improved optimal sensor placement method based on the sensitively effective independence method, which further optimized sensor placement by using the sensitivity coefficient reflecting structural damage. These studies have achieved good results in the optimized arrangement of sensor measurement points and have provided some reference for further development in this field. However, they did not carry out actual tests to verify whether the accuracy of the test results obtained based on this distribution method met actual engineering requirements after the optimization arrangement was completed.

In the optimization of measurement points, a finite element model of the structure needs to be established by finite element analysis software, and then the node data is extracted as the training data of the optimization algorithm; hence, how to effectively extract a large number of node data is vital for measurement point optimization. Liu et al. [13] studied the optimal arrangement of bridge sensors based on the single parent genetic algorithm. Taking Guiliuhe Arch Bridge as an example, they used the APDL module of ANSYS to divide the bridge into 1440 units and 1276 nodes and then extracted node data to train the optimization algorithm. Zhang et al. [14] proposed a nested stacked genetic algorithm (NSGA) to solve the sensor position optimization problem in an existing bridge health monitoring system and established the finite element fishbone model of the Wuhai Yellow River cable-stayed bridge with ANSYS (R15.0). The main beam was divided into 259 nodes and 258 elements; then, the stiffness and mass matrices were extracted from the model as data for the subsequent optimization algorithm. He et al. [15] proposed a hybrid optimization strategy named MSE-AGA in view of some defects in the existing bridge sensor arrangement methods, established a finite element model of a three-span suspension bridge with ANSYS, and optimized the sensor layout with the generated 1276 nodes as data. The sensor measurement point placement methods based on the optimization algorithm proposed by the above scholars can improve the identification ability of the actual dynamic characteristic parameters of the structure. However, in the selection of optimal measurement point training data, only the finite element model is used to select the target node data, resulting in the insufficient extraction of training data because the upper limit of selected data in this method is only about 100. Insufficient training data will have a great impact on the optimization results. Therefore, there is an urgent need for a method that can extract enough node data, effectively solving the problem of insufficient training data, is easy to operate, with fast extraction efficiency and high accuracy, and provides important basic support for the optimal arrangement of sensor measurement points. This paper requires a lot of node data in order to make sure that the sensors are arranged in the most critical positions of the structure and that the most accurate dynamic response of the structure is collected. Therefore, the tetrahedral mesh type is used and the structure is meshed with a smaller mesh size to get enough nodes.

In recent years, many scholars have been interested in using swarm intelligence algorithms to solve the problem of sensor measurement point arrangement optimization in complex civil engineering fields. Inspired by the group foraging behavior of birds, fish and other creatures in nature, Eberhart and Kennedy [16] first proposed a new intelligent algorithm called the particle swarm optimization (PSO) algorithm in 1995. The algorithm has the advantages of few parameters, fast convergence, a simple program and a large search range, so it is widely used in neural network training, signal processing, the sensor arrangement of complex civil structures and other engineering fields. Fakhouri et al. [17] studied multi-vector particle swarm optimization (MVPSO) for solving single-objective optimization problems. Yu et al. [18] proposed an advanced particle swarm optimization algorithm (APSO) to solve the flutter problem of long-span bridges and compared the flutter prediction results of the proposed method with the actual test results and obtained a good agreement. Huang et al. [19] proposed a hybrid optimization algorithm combining particle swarm optimization and cuckoo search. The hybrid optimization algorithm was used to test the damage identification performance of a real steel–concrete composite bridge under temperature variation and verified its good robustness and engineering applicability. Tran et al. [20] proposed a new method for the model updating of large railway bridges using orthogonal diagonalization (OD) combined with improved particle swarm optimization (IPSO).

Scholars in various fields have solved many problems by using the advantages of the PSO algorithm; however, the standard PSO algorithm has the defects of easily falling into local optimums and premature convergence. To this end, some scholars have also proposed many improvement strategies. Jiang et al. [21] proposed an improved particle swarm optimization algorithm and simulated three benchmark functions. The results showed that IPSO has a better global optimization ability than the basic PSO algorithm. Liu et al. [22] proposed a new state assessment method for reinforced concrete (RC) bridge superstructures based on fuzzy C-mean clustering using a particle swarm optimization (FCM-PSO) algorithm. It has the advantage of the PSO algorithm in global optimization. Nguyen et al. [23] proposed a new method to identify damage in structures using particle swarm optimization (PSO) combined with artificial neural networks (ANNs), which avoids the possibility that ANNs will get trapped in local minima while seeking the best solution.

When solving specific engineering problems, especially when solving combinatorial optimization problems such as the optimal placement of sensor measurement points, the two random weights of social learning and individual cognition in the PSO algorithm will directly affect the search speed and search accuracy of the optimal solution of the sensor measurement point arrangement scheme. Considering that the search performance of the PSO algorithm depends on the balance between its global search and local improvement ability, this paper will improve the inertia weight $\omega$ in the algorithm to make up for the defect of the linear decline in $\omega$ so that the particles take into account both local and global searches in the optimization process and improve the efficiency of the algorithm. In the beginning, it is anticipated that the particles can search the whole space quickly at a fast speed to find the general position of the optimal solution and then conduct a fine search of the general position at a slow speed, finally converging to the optimal solution.

The rest of this paper is organized as follows. In Section 2, aiming at the problem that the standard particle swarm optimization algorithm is prone to falling into local optimal and premature convergence due to the influence of weight, the PSO is improved, and an improved particle swarm optimization algorithm based on random weight (IPSOARW) is proposed. In Section 3, on the basis of setting the fitness function, the optimization strategy of the number and location of measurement points is proposed for the large-span steel beam structure. In Section 4, the optimization strategy of the number and location of measurement points in Section 3 is applied to the deck of the steel tube truss girder bridge (STTGB). The number and position of measurement points in the modal test of the STTGB bridge deck are optimized by taking the vibration data of the deck as the optimization data. In Section 5, based on the optimized number and position of bridge deck measurement points obtained in Section 4, the bridge deck measurement points are arranged according to the uniform distribution method and the IPSOARW distribution method and the modal test is completed. The test results based on the two methods are compared and analyzed, and the feasibility and superiority of the measurement point arrangement strategy based on IPSOARW are verified. The purpose of this paper is to provide a reference for the dynamic test of long-span steel beam structures, save test costs and improve test efficiencies on the basis of ensuring the accuracy of the test results.

2. PSO Algorithm

In view of the fact that the PSO is easily affected by weight, where it is easy to fall into the local optimum and converge prematurely, an IPSOARW is proposed, and the test function is set to verify the feasibility of the PSO algorithm and the superiority of IPSOARW.

2.1. Standard PSO Algorithms

PSO is a random optimization algorithm that simulates the behavior of biological groups in nature [24], and the algorithm is a computational technique to attain the optimum through group evolution [25]. In the standard PSO algorithm, the search space is $D$, and the number of particles is $n$. It is initialized to obtain a set of random particles, and then the iterative update calculation is carried out according to Equations (1) and (2) until the optimal solution is obtained, and then the iterative update calculation is stopped.

$$\begin{array}{c}{\nu }_{id}\left(t+1\right)={\nu }_{id}\left(t\right)+c_1{r}_1\left[p_{id}-x_{id}\left(t\right)\right]+c_2{r}_2\left[p_{gd}-x_{id}\left(t\right)\right]\end{array}$$

(1)

$$\begin{array}{c}{x}_{id}\left(t+1\right)=x_{id}\left(t\right)+{\nu }_{id}\left(t+1\right),1\le \mathrm{i}\le n,1\le d\le D\end{array}$$

(2)

Equation (1) is the expression of the standard (PSO) algorithm, where ${\nu }_{id}\left(t\right)$ is the d-dimensional component of the velocity vector of particle i after t iterations, $x_{id}\left(t\right)$ is the d-dimensional component of the position vector of particle i after t iterations, $p_{id}$ is the d-dimensional component of particle i at the individual best position, $p_{gd}$ is the d-dimensional component of particle i at the global best position, and ${\mathrm{c}}_1$ and ${\mathrm{c}}_2$ are learning factors, which are generally positive. Random weights $r_1$ and $r_2$ are random numbers between 0 and 1. The initial PSOA particle position update schematic is shown in Figure 1.

The evolutionary process of the PSO algorithm is as follows.

• Initialization: At $t=0$, the algorithm randomly generates $m$ particles, the dimension of each particle is $D$ and the corresponding position is $x_i\left(0\right)={[x_{i1}\left(0\right),x_{i2}\left(0\right)\dots .,x_{iD}\left(0\right)]}^T$; under the limitation of the maximum velocity, take a random value and let $v_i(t)$ be initialized as $v_i\left(0\right)={[v_{i1}\left(0\right),v_{i2}\left(0\right)\dots .,v_{iD}\left(0\right)]}^T$.
• Iterative calculation of the fitness function value: the individual optimal value of the randomly generated initial particles is $P_{best}$, the global optimal value of the initial particle swarm is $g_{best}$, and its fitness function value is $f\left(x_i\left(0\right)\right)$.
• Update time: set time $t=t+1$.
• Update velocity and position: position and velocity updates according to Equations (1) and (2), respectively.
• Update the individual optimum: compare $f\left(x_i\left(t\right)\right)$ and $P_{best}$, make $P_{best}=f\left(x_i\left(0\right)\right)$ when $f\left(x_i\left(t\right)\right)$ is better than $P_{best}$; otherwise, $P_{best}$ is left untouched.
• Update the global optimal value: In $P_{besti}\left(i=1,2,\dots ,n\right)$, select the optimal $P_{besti}$ and compare it with $g_{best}$; if $P_{besti}$ is better than $g_{best}$, then make $g_{best}=P_{besti}$; otherwise, leave it untouched. Follow the above process to (6) and output the result when the end condition is met; otherwise, return to (2).

The flow of the standard PSO algorithm is shown in Figure 2.

2.2. Improved PSOA (IPSOA)

During each generation of the search process, the particle randomly adjusts its velocity using its own experience and that of its neighbors. The mathematical model of velocity and position update for the basic PSO algorithm was introduced in Section 2.1, as shown in Equations (1) and (2). The two random weights $\left(r_1,r_2\right)$ for social learning and individual cognition in the velocity update Equation (1) are randomly generated. Therefore, these two random parameters may be too large or too small at the same time. If the two random weights are too small at the same time, the convergence speed is reduced, and the social learning and individual cognition are not fully utilized, leading to the premature convergence of the algorithm. Conversely, when the two random weights are too large at the same time, the social learning and the individual cognition are over-utilized to the extent that the particles may gradually move away from the global optimal solution. To this end, the IPSOARW is proposed, which can balance local search ability and global search ability in the search process by randomly weighing the weights.

In the IPSOARW, $\omega$ is set to a random number varying between 0 and 1, thus compensating for the linearly decreasing disadvantage of $\omega$. First, if the optimal solution is approached early in the evolution, the randomly generated value of $\omega$ may be relatively small, resulting in faster convergence. In addition, if the optimal solution is not found early in the algorithm’s operation, the linearly decreasing nature of $\omega$ allows the algorithm results to eventually converge to the optimal solution. Therefore, the $\omega$ generated by random weights can effectively solve the problem of premature convergence. The equation for generating $\omega$ by random weights is as follows.

$$\{\begin{array}{c}\omega =\mu +\sigma *N(0,1)\\ \mu ={\mu }_{\min }+({\mu }_{\max }-{\mu }_{\min })*rand(0,1)\end{array}$$

(3)

where $N(0,1)$ is a random number of standard normal distribution and $rand(0,1)$ is a random number between 0 and 1.

The calculation steps of IPSOARW are as follows:

• Initialize the velocity and position of each particle in the PSOA.
• Evaluate the fitness of each particle, store the current fitness value and position of each particle in the $P_{best}$ of each particle, and store the fitness value and position of the individual with the best fitness value in all $P_{best}$ in $g_{best}$.
• Update the velocity and disarrangement of the particle using the following equation:

  $$v_{i,j}(t+1)=\omega v_{i,j}(t)+c_1{r}_1[p_{i,j}-x_{i,j}(t)]+c_2{r}_2[p_{g,j}-x_{i,j}(t)]$$

  (4)

  $$x_{i,j}(t+1)=x_{i,j}(t)+v_{i,j}(t+1),j=1,2\dots ,d$$

  (5)
• Update the weights according to Equation (3).

  $$\{\begin{array}{c}\omega =\mu +\sigma *N(0,1)\\ \mu ={\mu }_{\min }+({\mu }_{\max }-{\mu }_{\min })*rand(0,1)\end{array}$$
• For each particle, compare the best position it has passed through with the fitness value, and if it is better, use it as the current best position. Compare the values of all current $P_{best}$ and $g_{best}$ and update $g_{best}$.
• When the end condition is met, the search will stop and the results will be output; otherwise, return to (3) to continue the search.

The IPSOARW flow is shown in Figure 3.

2.3. Simulation Validation of IPSOARW

In order to ensure that the PSO can be applied to the research content of this paper, a simple function test is proposed to illustrate the feasibility of the PSO algorithm. To illustrate the superiority of the improved algorithm, the IPSOARW and the standard PSO algorithm are used for the test simultaneously; finally, the results are compared and analyzed. The Six-Hump Camel-Back function [26] was tested using the standard particle swarm algorithm and the IPSOARW as follows.

$$f(x)=4x_1^2-2.1x_1^4+x_1^6/3+x_1{x}_2-4x_2^2+4x_2^4$$

(6)

where $\left|x_i\right|\le 5$; this function belongs to a multi-modal function, and its global minimum point is −1.0316285. The feature of this function is that there are many peak points near the minimum extreme point. This feature of Six-Hump Camel-Back function can well detect the convergence of the algorithm. Then, the function is solved, and the test function is programmed into the software. The test result of the algorithm function is shown in Figure 4. The parameters of the IPSOARW are shown in

Table 1. Parameter settings for the IPSOARW.

Parameters	n	Learning Factor	$\mathit{m}{\mathit{\mu }}_{\mathbf{min}}$	$\mathit{m}{\mathit{\mu }}_{\mathbf{max}}$	$\mathit{m}\mathit{\sigma }$	Number of Iterations
Value	40	2	0.5	0.8	0.2	10,000

where $m{\mu }_{\max }$ represents the maximum value of the average random weight, $m{\mu }_{\min }$ represents the minimum value of the average random weight, and $m\sigma$ represents the variance of the average random weight.

.

The calculation results show that the results obtained by the two algorithms are −1.0316285, which means that the global minimum value is found. By comparison, in Figure 4, we found that the standard PSO algorithm converges for 100 iterations and finds the global optimal value, while the IPSOARW algorithm converges for 22 iterations. The analysis shows that both algorithms can find the global optimal value of the function; the PSO algorithm is effective and accurate, and its feasibility and accuracy are verified. The number of iterations for the IPSOARW to find the optimal value is less than that of the standard PSO algorithm, which indicates that the former has faster convergence and better performance.

3. Measurement Point Optimization Strategy of Long-Span Steel Beam Based on the IPSOARW

Based on the simulation verification results in Section 2.3, the IPSOARW can be used to study the optimal arrangement strategy of measurement points in the modal test of long-span beams. The modal confidence criterion was used to optimize the number of measurement points of the long-span beam structure. The IPSOARW program is realized by MATLAB to optimize the location of the measurement points. After obtaining the modal data of the beam structure model, the optimization algorithm was applied to the optimization problem of the measurement point positions.

3.1. Modal Assurance Criterion (MAC)

As a multi-degrees of freedom linear vibration system, the dynamic equation of the long-span steel beam is

$$M\ddot{X}+C\dot{X}+KX=F\left(t\right)$$

(7)

where $C$ is the damping matrix; $M$ is the mass matrix; $K$ is the stiffness matrix; $X$ is the disarrangement vector; $F\left(t\right)$ is the excitation force.

Many experimental studies show that the damping of steel structures is generally small, resulting in a small and negligible effect of damping on the mode; the solution to this equation will be obtained by a linear combination of harmonic functions, and then an eigenvalue problem will be obtained. Its characteristic equation is

$$\begin{array}{c}\left(\mathsf{\lambda }\mathrm{M}+\mathrm{K}\right)\varphi =0\end{array}$$

(8)

where $\varphi$ is its corresponding eigenvector and $\mathsf{\lambda }$ is the eigenvalue. When the quality matrix is a diagonal matrix, its corresponding eigenvector is an orthogonal vector. On this basis, the $MAC$ is used to differentiate the eigenvalues. The independence between the modal vectors is tested by the $MAC$. The $MAC$ matrix is calculated by the following equation:

$$\begin{array}{c}MA{C}_{ij}=\frac{{\left({\Phi }_i^T{\Phi }_j\right)}^2}{\left({\Phi }_i^J{\Phi }_i\right)\left({\Phi }_j^T{\Phi }_j\right)}\end{array}$$

(9)

where $MA{C}_{ij}$ is the element of the $i$ row and $j$ column of the matrix, ${\Phi }_i$ is the $i$ order vibration vector, ${\Phi }_j$ is the $j$ order vibration vector, and ${\Phi }_i^T$ and ${\Phi }_j^T$ are transpose of vectors.

The value of the $MAC$ matrix elements reflects the similarity of the two vibration patterns, so the maximum value of the non-diagonal element $MA{C}_{ij}(i\ne j)$ of the $MAC$ matrix is used to determine whether the results of the measurement point arrangement are reasonable. The value of Equation (8) indicates the correlation between the $j$ order vector and the $i$ order vector, and the value of it takes the range $\left[0, 1\right]$. When the equation value is 1, it means that the two vectors are completely correlated. When the equation value is 0, it means that the two vectors are completely independent. Therefore, in modal tests, the smaller the value of $MA{C}_{ij}$, the more accurate the test results.

3.2. Establishment of the Fitness Function

The fitness function in the optimization procedure is also a key factor affecting the optimization arrangement of the measurement points. Therefore, it is necessary to debug and program the measurement point optimization program of the IPSOARW. The fitness function is designed according to the $MAC$ and finally applied to the optimization of the measurement point arrangement of the modal test of the long-span steel beam.

It is important to simplify the fitness function as much as possible in the design of the optimization algorithm to reduce the calculation time. A fitness function based on the $MAC$ is shown in Equation (10).

$$\begin{array}{c}f=\max \left(abs\left(MA{C}_{ij}\right)\right)\end{array}$$

(10)

where $max$ is the maximum value and $abs$ is the absolute value. The smaller the value of $MA{C}_{ij}\left(i\ne j\right)$, the better the optimization effect when optimizing the measurement point arrangement.

3.3. Acquisition of Training Data for Bridge Decks Based on IPSOARW

In order to optimize the modal test points of long-span steel beam structures, the modal data of the structure is needed, no matter the number optimization or position optimization. Therefore, the finite element model of the long-span steel beam structure was established in ANSYS (R19.2), and then modal analysis was carried out to obtain the frequencies and modes of the structure in the frequency band of interest. In the follow-up work, the IPSOARW will take the modal shape data of the structure as the training data so as to optimize the position of the structure’s modal test measuring points.

3.4. Strategy for Optimizing the Number of Measurement Points for the Modal Test of Large-Span Steel Beams

The number of measurement points is optimized by:

(1) constructing a fitness function with the minimum value of the non-diagonal elements of the $MAC$ matrix; (2) increasing the number of randomly selected measurement points and observing the change in the value of the fitness function. (3) The number of measurement points is determined by the relation curve between the number of measurement points and the fitness function value.

The steps for optimizing the number of measurement points are as follows.

• Construct the $MAC$ matrix formula.

  $$MA{C}_{ij}=\frac{{\left({\Phi }_i^T{\Phi }_j\right)}^2}{\left({\Phi }_i^J{\Phi }_i\right)\left({\Phi }_j^T{\Phi }_j\right)}$$
• Set the maximum value of the non-diagonal elements of the $MAC$ matrix to be the function $f_{max}$.
• Select one node at random from all the nodes as a measurement point and calculate the relationship between the number of measurement points and $f_{max}$, select one node at random from the remaining nodes to add a measurement point, calculate the number of measurement points at this moment as a function of $f_{max}$ and repeat the superposition of measurement points.
• Using the above iterative method, the value of the function $f_{max}$ can be derived under different numbers of measurement points.
• On this basis, the most reasonable number of measurement points is determined by plotting the value of the function $f_{max}$ against the number of measurement points.

3.5. Optimization Strategy for the Location of Measurement Points on Large-Span Steel Beams Based on the IPSOARW

Based on the known data of the structural mode and the optimized number of measuring points, the fitness function is taken as the optimal number of measuring points in the particle dimension. The structural mode data are used as the training data of the optimization algorithm.

The steps of measurement point position optimization are as follows:

• Introduce the matrix

  $$MA{C}_{ij}=\frac{{\left({\Phi }_i^T{\Phi }_j\right)}^2}{\left({\Phi }_i^J{\Phi }_i\right)\left({\Phi }_j^T{\Phi }_j\right)}$$
• Construct the fitness function with the maximum value $f_{max}$ of the off-diagonal elements of the matrix.
• Set IPSOARW parameters, such as population size, particle dimension, maximum number of iterations, learning factor, particle speed, etc.
• Calculate and obtain the modal shape data of the structure.
• Use the IPSOARW to optimize the position of the measurement point, calculate the current fitness value of each particle and determine the group extreme value and the individual extreme value of the particle. The loop iterates. When the maximum number of iterations is reached, the loop ends and the optimal value is output. The value is the node number of the optimal location of the measurement points.

4. Realization of the Optimization of the Measurement Points of the Deck Modal Test of STTGB

Taking Ganhaizi Bridge as the prototype and using the similarity theory, a scale model of an STTGB was made in the laboratory. Based on the optimization strategy of the number and position of measurement points in Chapter 3, the number and position of measurement points on the deck of the STTGB were optimized.

4.1. Acquisition of Training Data for the IPSOARW Based on Computational Modal Analysis

The bridge model is simply supported at both ends, with a deck length of 8016 mm, a width of 1503 mm and a thickness of 10 mm. The main material of the steel tube truss girder bridge is structural steel Q235, and its material properties are shown in

Table 2. Structural material parameters for the STTGB.

Structural Steel (Q235)
Density	7850 kg/m3
Young’s modulus	2 × 1011 Pa
Poisson’s ratio	0.3

. The 3D solid model of the STTGB is imported into the finite element analysis software ANSYS (19.0, Canonsburg, PA, USA). The material properties of the STTGB are set according to the parameters in Table 2. According to the actual situation of the STTGB, the bridge is supported by four concrete piers on the floor; the contact surface between the STTGB and the piers is set to the Fixed Support constraint. The tetrahedral mesh type is used to mesh the STTGB; the mesh size is set to 50 mm, the number of nodes is 572,238 and the number of cells is 285,957. The constraint position and the finite element model of the STTGB are shown in Figure 5.

Usually, long-span STTGBs are extremely sensitive to wind action due to their low structural stiffness, and wind resistance is the primary consideration in their design [8]. Thus, low-frequency modes below 100 Hz are primarily considered. The mode shapes of the long-span STTGB structure are generally based on vertical mode shapes, so the coordinates and displacements of the vertical mode shape nodes of the bridge deck will be selected as the training data for optimization. The calculated modal frequencies and vibration patterns of the STTGB deck in the vertical direction are shown in

Table 3. The calculated modal frequencies and vibration patterns of the STTGB deck.

Order	Calculated Modal Frequencies [Hz]	Modal Vibration Pattern	Description of the Vibration Pattern
1	18.037		1st order bending
2	27.455		1st order twisting
3	37.271		2nd order bending
4	53.726		2nd order twisting
5	79.738		3rd order twisting
6	85.554		4th order bending
7	95.439		Combined 2nd order bending and twisting deformation

.

4.2. Optimization Results of the Number of Bridge Deck Measurement Points

To introduce the modal confidence matrix, see Equation (9) and take Equation (10) as the fitness function. The numerical simulation was conducted to optimize the number of measuring points on the bridge deck. The first seven modal shape data points of the bridge deck were selected as the training data. Based on the optimization strategy of the number of measuring points in Section 3.4, the variation curve of the obtained value with the number of measurement points is shown in Figure 6.

As can be seen from Figure 6, when the number of measurement points increases to 28, the value of $f_{max}$ basically stops changing, and it is believed that the extra measurement points above 28 do not have much impact on the $f_{max}$. When the number of measurement points is less than 28, the value of $f_{max}$ keeps decreasing as a whole, with small fluctuations in the middle but with little impact. Therefore, based on the above analysis, it is determined that the optimal number of deck modal test points of STTGB is 28.

4.3. Optimization of Bridge Deck Measurement Points Location Based on the IPSOARW

Firstly, the modal confidence matrix in Equation (9) is introduced, and Equation (10) is used as the fitness function; then, the parameters of PSO are set. The particle dimension is set as 28 according to the optimized number of measurement points; the population size is set as 50 and the maximum number of iterations is set as 300 according to the particle dimension and structural characteristics of the STTGB deck. The calculation formula of inertia weight is shown in Equation (3), and the learning factor and particle velocity are generally empirical values. The final parameter settings are shown in

Table 4. IPSOARW parameter selection.

Population Size	Particle Dimension	Maximum Number of Iterations	Learning Factor ${\mathit{c}}_1$${\mathit{c}}_2$	Particle Velocity ${\mathit{v}}_{\mathbf{min}}$${\mathit{v}}_{\mathbf{max}}$
50	28	300	2  2	−20  20

.

With the bridge deck vibration pattern data obtained in Section 4.1 as the training data, the IPSOARW was used to optimize the location of the measurement points. When the iteration reaches the maximum iteration number of 300, the cycle ends and 28 optimal values are output, namely, the node numbers of the optimal locations of the bridge deck measurement points. The results are shown in

Table 5. Results of IPSOARW-based measurement point optimization for the bridge deck.

Number of Measurement Points	Optimization Results (Node Number)
28	364274, 386971, 356881, 369766, 45758, 380325, 357042, 38894, 374472, 364668, 103815, 358666, 39530, 374055, 6193, 360275, 374735, 363356, 358739, 370854, 363740, 358884, 39414, 374296, 362180, 368224, 361438, 364724

.

After inputting these node numbers into the finite element model of the bridge deck, the optimized distribution of measurement points on the bridge deck is obtained, as shown in Figure 7. The two-dimensional coordinates of the optimized measurement points are shown in

Table 6. Optimized position coordinates of bridge deck measurement points based on the IPSOARW.

No.	X (mm)	Y (mm)	No.	X (mm)	Y (mm)
1	1253.7	422.3	15	2482.6	0
2	6211.1	820.3	16	5057.4	898
3	6968.7	443.7	17	6640.2	49
4	3606.1	1327.3	18	7731	215.8
5	760.1	905.3	19	502.6	564.2
6	2969.8	815.9	20	7192.5	1380.8
7	1964.2	195.3	21	7886.9	671.4
8	8081.7	282	22	2404.5	818.7
9	2130	1120.8	23	6925.2	1218.3
10	6655.3	1360.3	24	354	928.7
11	4656.2	1420.1	25	896.7	1088.6
12	3973	360.6	26	4367.4	28.6
13	5514.4	645.9	27	4051.6	584.4
14	4927.2	338.1	28	1339.2	1144.6

.

The convergence characteristic of the fitness function is analyzed in the following paragraph. According to the number of iterations and the fitness function value corresponding to the number of iterations, the variation trend of the fitness function value of the optimized location of measurement points on the deck of the STTGB, along with the number of iterations, is shown in Figure 8.

The analysis of Figure 8 can be obtained as follows:

• When IPSOARW is used to optimize the location of measuring points, the fitness function value decreases with the increase in the number of iterations.
• After the number of iterations of the optimization algorithm reaches 240, when the number of iterations continues to increase, the fitness function value does not change, and the optimization algorithm reaches convergence.
• According to the criterion of modal confidence, the final convergence result of the fitness function value in the figure can reach 0.006, indicating that the proposed IPSOARW-based point distribution method of the bridge deck can meet the optimization goal of measurement point positions.

5. Validation of Optimization Strategy for STTGB Bridge Deck Measurement Points Based on Experimental Modal Analysis

An STTGB bridge deck modal test system was built, the bridge deck measurement points were arranged according to the uniform distribution method and the IPSOARW distribution method, and a modal test was carried out. The random subspace modal parameter identification method was used for parameter identification, and the modal parameters obtained based on the two distribution schemes were compared and analyzed. The feasibility and superiority of the proposed IPSOARW-based modal test measurement point arrangement were verified.

5.1. The Modal Test System of the STTGB Deck

A modal test system of the STTGB deck was constructed, as shown in Figure 9. The equipment used is described in

Table 7. Main characteristics of the equipment for the modal test of the STTGB deck.

Test System Components	Test Equipment	Number
Motivational equipment	Large impact hammer (INV9313)	1
	Rubber hammer heads	1
Collection equipment	Channel cable wires	2
	INV force sensors (110610)	1
	PCB single acceleration sensor (INV8923)	1
	Magnetic base	1
	8-channel distributed collector (INV30600C)	1
Analysis of components	DASP-V10 modal analysis software	1

.

5.2. Modal Test of Bridge Deck Based on the Uniform Distribution Method

The modal test of the bridge deck was carried out by the hammering method (single point excitation and single point response). An impact hammer was used to apply excitation at the modal reference point, and an accelerometer was used to pick up the response signal. Based on the dimensions of the STTGB deck, the deck was divided into 50 measurement points according to the 10 × 5 division method, where the length direction was divided into nine equal parts with a spacing of 890.6 mm between two adjacent points, and the width direction was divided into four equal parts with a spacing of 375.75 mm between two adjacent points. The arrangement of the measurement points is shown in Figure 10.

In the modal analysis software, the analysis frequency was set to 100 Hz, the trigger amount was set to 20 N, the number of sampling points was set to 1024, and the number of triggers was set to 3. Using the random subspace modal parameter identification method for parameter identification, the modal frequencies and vibration pattern of the bridge deck, based on the uniform distribution method, were obtained, as shown in

Table 8. Frequency and vibration pattern of the bridge deck based on the uniform distribution method.

Order	Frequency [Hz]	Modal Vibration Pattern	Description of the Vibration Pattern
1	17.042		1st order bending
2	28.156		1st order twisting
3	40.614		2nd order bending
4	51.152		2nd order twisting
5	79.151		3rd order twisting
6	93.662		Combined 1st order bending and torsion deformation

.

5.3. Modal Test of the Bridge Deck Based on the IPSOARW Distribution Method

According to the coordinates of the bridge deck measurement points analyzed in Section 4.3, which were optimized based on the IPSOARW, 28 measurement points were placed on the bridge deck. Sticky notes were placed at the locations of the points, and the point numbers were written down. To ensure that the model shape is the same as the actual bridge deck, four endpoints, 29, 30, 31 and 32, were added with no constraints.

The coordinates of the optimized measurement points are input into the modal analysis software, and then the three adjacent measurement points are connected to form multiple triangles to form a bridge deck model, resulting in a point-line surface model of the bridge deck based on the IPSOARW, as shown in Figure 11, with the nodes on the model corresponding to the measurement points on the bridge deck.

The software parameter settings are the same as those in Section 5.2. The random subspace mode parameter identification method is used for parameter identification, and the modal frequency and vibration pattern of the bridge deck based on the IPSOARW distribution method are shown in

Table 9. Modal frequency and vibration pattern of the bridge deck based on the IPSOARW distribution method.

Order	Frequency [Hz]	Modal Vibration Pattern	Description of the Vibration Pattern
1	17.031		1st order bending
2	28.091		1st order twisting
3	40.774		2nd order bending
4	52.112		2nd order twisting
5	79.120		3rd order twisting
6	90.814		4th order bending
7	96.546		Combined 1st-order bending and torsion deformation

.

5.4. Comparison of the Experiments Results of the Two-Measurement Point Arrangement Method

The analysis in Section 4.2 shows that the number of measurement points of the IPSOARW distribution method is 28 and the number of measurement points of the uniform pointing method U (a, b) is 50. In comparison, the number of measurement points optimized by the proposed improved particle swarm algorithm is reduced by 44%, indicating that the IPSOARW is more efficient than the traditional uniform distribution method. A comparison of the frequencies of the bridge deck based on the two methods is shown in

Table 10. Comparison of the frequency of the bridge deck based on two methods.

Order	Frequency [Hz]			Error [%]		Mean Error [%]
	FEA	U (a, b)	IPSOARW	U (a, b)	IPSOARW	U (a, b)	IPSOARW
1	18.037	17.042	17.031	5.5	5.6
2	27.455	28.156	28.091	2.6	2.3
3	37.271	40.614	40.774	8.9	9.4
4	53.726	51.152	52.112	4.8	3.0	5.33	4.06
5	79.738	79.151	79.120	0.7	0.8
6	85.554	93.662	90.814	9.5	6.1
7	95.439	-	96.546	-	1.2

.

As shown in Table 10, it can be obtained that:

• The modal test on the bridge deck based on the uniform distribution method identified a total of six orders of modes, and compared to the FEA, the modal frequencies around 90 Hz were lost and a lost mode condition occurred, while the test on the bridge deck based on the IPSOARW distribution method identified a total of seven orders of modes and no lost modes.
• The mean error between the frequency obtained by the uniform distribution method and the FEA frequency of the bridge deck was 5.33%, while that of the bridge deck based on the IPSOARW distribution method was 4.06%. Obviously, the modal parameters obtained by the IPSOARW distribution method are consistent with the FEA modal parameters, and the modes of each order are basically the same.

In conclusion, in the aspect of mode shape identification, the distribution method based on the IPSOARW can identify the modes in the dense segment of frequency, and it is not easy to lose the modes. Therefore, the distribution method based on the IPSOARW has high feasibility in the arrangement of the modal measurement points of the bridge deck.

6. Conclusions

In this paper, an improved particle swarm optimization algorithm based on random weights (IPSOARW) was proposed to optimize the arrangement of modal measurement points on the deck of a steel tube truss girder bridge (STTGB). The calculated modal analysis results of the finite element model of the bridge deck were used as the training data, and the number and location of measurement points of the bridge deck were optimized using the IPSOARW. According to the experimental data, the proposed IPSOARW has faster convergence speed and better solution accuracy and can significantly improve the efficiency of the optimal arrangement of measurement points. This work will provide a new idea for optimizing the arrangement of modal measurement points of bridges and has certain reference significance. The specific conclusions are as follows:

• The number of measurement points of the bridge deck based on the uniform distribution method was 50, while the number based on the IPSOARW was 28. By contrast, the number of points was reduced by 44%. The results show that the proposed IPSOARW can effectively improve the test efficiency of the STTGB and reduce the number of sensor arrangements, which will save a lot of test costs.
• In the modal frequencies obtained based on the uniform distribution method, the mode around 90 Hz was lost, while the IPSOARW did not lose the modes. The results show that the proposed IPSOARW can effectively identify the modes in the frequency-dense segment and has the advantage of not losing the modes easily.
• The average error between the frequency obtained by the uniform distribution method and the finite element analysis frequency of the bridge deck was 5.33%, while that based on the IPSOARW distribution method was 4.06%. The results show that the proposed IPSOARW is suitable for the optimal arrangement of measurement points of the STTGB deck and that the obtained modal parameters have high accuracy. The modal parameters identified based on this optimized arrangement method have high accuracy and can accurately reflect the actual dynamic characteristics of the bridge deck.
• The experimental examples used in this paper are idealized without considering the influence of environmental factors such as wind load and vehicle load. Therefore, these factors will be taken into account in our next work to further test the applicability and effectiveness of the proposed method.