Mathematical Problems in Engineering

Volume 2015, Article ID 626342, 10 pages

http://dx.doi.org/10.1155/2015/626342

## A New Optimal Sensor Placement Strategy Based on Modified Modal Assurance Criterion and Improved Adaptive Genetic Algorithm for Structural Health Monitoring

^{1}College of Defense Engineering, PLA University of Science and Technology, Nanjing 210007, China^{2}Technical Management Office of Naval Defense Engineering, Beijing 100841, China

Received 7 July 2014; Accepted 23 August 2014

Academic Editor: Zheng-Guang Wu

Copyright © 2015 Can He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Optimal sensor placement (OSP) is an important part in the structural health monitoring. Due to the ability of ensuring the linear independence of the tested modal vectors, the minimum modal assurance criterion (minMAC) is considered as an effective method and is used widely. However, some defects are present in this method, such as the low modal energy and the long computation time. A new OSP method named IAGA-MMAC is presented in this study to settle the issue. First, a modified modal assurance criterion (MMAC) is proposed to improve the modal energy of the selected locations. Then, an improved adaptive genetic algorithm (IAGA), which uses the root mean square of off-diagonal elements in the MMAC matrix as the fitness function, is proposed to enhance computation efficiency. A case study of sensor placement on a numerically simulated wharf structure is provided to verify the effectiveness of the IAGA-MMAC strategy, and two different methods are used as contrast experiments. A comparison of these strategies shows that the optimal results obtained by the IAGA-MMAC method have a high modal strain energy, a quick computational speed, and small off-diagonal elements in the MMAC matrix.

#### 1. Introduction

With the development of constructive technology, more and more large-scale structures, such as suspension bridges, television towers, sea wharfs, and high-rise buildings, have been built all over the world. However, due to harsher environment, these civil infrastructures may be damaged by strong wind force, torrential rain, severe earthquake, explosion, and other abnormal loads [1]. The traditional routine visual inspection usually ignores the tiny failures and the flaws inside the structures. Therefore, structural health monitoring (SHM) [2–4] has become an important research topic in the engineering protection field. By monitoring the force condition of the structures real-timely, SHM system can detect the anomalies in response and find the possible deterioration at early stage to ensure the safety of the structures. A typical SHM system usually includes three subsystems: sensor subsystem, data processing subsystem, and condition evaluation subsystem [5]. Sensor subsystem is a fundamental part of SHM system. Modal parameter identification, damage detection, and structural condition evaluation are all based on the data acquired from the sensors. Generally, the more sensors are placed, the more information could be obtained. Nevertheless, the number of sensors is strictly constrained by the high cost of purchase and maintenance for the sensors. For instance, there are only 23, 65, and 72 accelerometers instrumented in the Tsing Ma Bridge, Tingkau Bridge, and Jiangyin Bridge, respectively [6]. The sensors number is far less than the available positions. Therefore, how to deploy the limited sensors becomes a challenging task.

The traditional expert experience method is effective for the simple structures. Nevertheless, for a large-scale structure that has a large volume and complex geometry, it is very difficult to determine the optimal sensor locations only by experience. In the past few years, many technologies have been developed to achieve optimal sensor placement (OSP), such as modal kinetic energy method [7, 8], effective independence method [9, 10], QR decomposition method [11, 12], and information entropy method [13, 14]. Carne and Dohrmann [15] considered that distinguishing one modal vector from another is essential to realize modal parameter identification and proposed a famous OSP method named minMAC. The minMAC method includes two parts. The first section is modal assurance criterion (MAC). In the criterion, the off-diagonal element of the MAC matrix is considered to be an index for evaluating the angle of the corresponding two modal vectors. The smaller element indicates bigger angle and less correlation between two modal vectors. The second section is to achieve sensor placement using MAC. First, some locations are selected as an initial set based on experience and structural topology, and the remaining locations are considered as a candidate set. Second, the candidate locations are added one by one to the initial set and the combination that minimizes the maximum off-diagonal element of the MAC matrix is selected as the new initial set. Third, the second step is repeated till the fixed sensors number is gained.

Although minMAC is considered as an effective OSP method and used widely, some defects still exist. Firstly, the core idea of MAC is to evaluate the correlation of the modal vectors for different placement scheme, but the modal energy of the selected location cannot be guaranteed. It means that some degree of freedoms (DOFs) with low signal to noise ratio may be selected as the optimal sensor positions; this defect could decrease the precision of the acquired data and the accuracy of modal parameters identification. Secondly, the minMAC method has a large computational complexity. To determine one location, the minMAC method needs to compute the MAC matrices of the variable combinations until all candidate locations have been selected. In order to improve the computational efficiency of the minMAC method, computational intelligence technology has been used, such as simulated annealing algorithm [16], ant colony optimization algorithm [17], particle swarm optimization algorithm [18], and monkey algorithm [19]. Due to easy coding method and quick evolution velocity, genetic algorithm (GA) has become a hot research direction. Liu et al. [20] presented a modified GA using two-dimensional array coding method instead of binary coding method; generalized genetic algorithm (GGA) was proposed by Yi et al. [21] to solve the OSP problem for high-rise buildings. Javadi et al. [22] adopted a hybrid intelligent method which was based on a combination of neural network and GA. Although these modifications make great progress, there are still some defects existing in GA [23]. The main drawback is that the crossover and mutation factors are invariant during the whole iteration cycle; this defect drops the searching ability of GA.

The objective of this study is to present a modified minMAC method that has a big modal energy index and a quick computational speed. A new method termed IAGA-MMAC is proposed to achieve this goal. First, a modified modal assurance criterion (MMAC) is proposed to improve the modal energy of the selected locations by constructing the new modal shape matrix. And then, an improved adaptive genetic algorithm (IAGA), which uses root mean square of off-diagonal elements in the MMAC matrix as the fitness function, is adopted to determine the optimal sensor locations. The excellent calculation efficiency and global optimization ability of IAGA assist to enhance the computational speed of IAGA-MMAC.

The remaining parts of the paper are organized as follows. Section 2 introduces the traditional MAC and presents MMAC. The IAGA-MMAC strategy is described in Section 3. Section 4 provides a case study of sensor placement on a numerically simulated wharf structure to verify the effectiveness of the IAGA-MMAC method. Section 5 is the conclusion and the future work.

#### 2. The Modified Modal Assurance Criterion

##### 2.1. Introduction of the Traditional MAC

A basic requirement to distinguish the measured modes is that the measured modal vectors must be as linearly independent as possible. MAC provides a useful criterion to evaluate the correlation of modal vectors. MAC matrix is defined as where and are the th and th column vectors in the modal shape matrix .

The off-diagonal elements in the MAC matrix express the correlation between two modal vectors. Identifying the th mode and the th mode is easy if the value of is small.

##### 2.2. Description of MMAC

The main idea of MAC is to guarantee the linear independence of the modal vectors. However, some DOFs with low energy may be selected as sensor locations using MAC; this defect could decrease the signal to noise ratio of the acquired data and make troubles in modal shape identification and structural damage detection.

As shown in 1, the element of the MAC matrix is computed using the column vectors of the modal shape matrix . In general, modal shape matrix should be composed by all orders of modes. But for the sake of reducing computation complexity, only a few modal orders are selected to compose the modal shape matrix. To improve the modal energy of the selected locations, the modal orders having large dynamical response should be selected. Therefore, how to evaluate the dynamical feature of different modal orders becomes an important work. In tradition, the first several modes are considered to have large dynamical response. Nevertheless, the modal energy of a structure is not always concentrated on the low-order modes. To some complex structures, such as high-rise building and suspension bridge, some high modal orders also have large dynamical response [24]. Therefore, it is not suitable that only the first several modes are selected to compose the modal shape matrix. In order to improve the modal energy of sensor locations, a modified MAC is presented to settle the issue in this study. In the new criterion, modal participation factor (MPF) is presented as the criterion to evaluate the dynamical response of different modal orders. And then, the new modal shape matrix is composed by the modal orders with big MPF. Finally, the modified MAC matrix is computed using .

###### 2.2.1. Modal Order Selection Using MPF

Different choices of modal order result in the various placement results. According to the requirement of OSP, the modal order having a strong dynamic response should be selected. Nevertheless, modal order is selected based on experience in the traditional methods. In order to solve this problem, MPF is presented as the evaluation criterion for the modal order selection.

The equation of motion for the DOF dynamic system is represented as where is the structural mass matrix, is the structural damping matrix, is the structural stiffness matrix, is the vector describing the excitation direction, is the modal displacement vector, and is the acceleration generated by input force.

The displacement vector can be transformed by where is the modal shape matrix and is the modal coordinate.

If , , and meet the orthogonalization of the normalization modal shape, 2 can be written as where is the th modal coordinate, is the natural frequency of the th mode, and is the fraction of critical damping of the th mode.

The MPF is defined as follows in [25] where is the MPF of the th mode in direction.

The mass normalization for the modal shape matrix is determined by

Based on 5 and 6, the MPF can be expressed as where is a vector describing the excitation in direction, and it can be computed as where is a unit vector in direction and is a constant matrix that indicates the relationship between the excitation and reference points: where , , and are the global Cartesian coordinates of the reference point and , , and are the global Cartesian coordinates of the excitation point.

MPF reflects the amplitude of dynamical response for all the modes. Therefore, the modal order that has big MPF should be selected priorly to construct the new modal shape.

Although all of the modes can be sorted based on their MPFs, there is still a question that how many modes should be selected. The modal participating mass ratio is presented as a criterion to determine the number of the modes. Modal participating mass is defined as where is the modal participating mass of the th mode.

When mass normalization has been conducted for the modal shape matrix , 10 can be transformed by

The modal participating mass ratio of the th mode can be expressed as

Wilson [25] suggested that the summation of modal participating mass ratio of the selected modes should be over 90%. This criterion is employed in this study to determine the number of modes.

###### 2.2.2. Computation of the MMAC Matrix

The dynamical response of different modal orders can be sorted by comparing their MPFs; the new modal shape matrix is constructed by the modal orders with big MPFs. Therefore, the modified MAC (MMAC) matrix can be computed as

The computation process of the modified MAC matrix is shown as Figure 1.