JOURNAL OF

SOUND AND
VIBRATION
Journal of Sound and Vibration 317 (2008) 175189
Optimal sensor placement for spatial lattice structure
based on genetic algorithms
Wei Liu

, Wei-cheng Gao, Yi Sun, Min-jian Xu

Department of Astronautic Science and Mechanics, Harbin Institute of Technology, Harbin 150001, China
Received 30 April 2007; received in revised form 12 March 2008; accepted 16 March 2008
Handling Editor: L.G. Tham
Available online 2 May 2008
Abstract
Optimal sensor placement technique plays a key role in structural health monitoring of spatial lattice structures. This
paper considers the problem of locating sensors on a spatial lattice structure with the aim of maximizing the data
information so that structural dynamic behavior can be fully characterized. Based on the criterion of optimal sensor
placement for modal test, an improved genetic algorithm is introduced to nd the optimal placement of sensors. The modal
strain energy (MSE) and the modal assurance criterion (MAC) have been taken as the tness function, respectively, so that
three placement designs were produced. The decimal two-dimension array coding method instead of binary coding method
is proposed to code the solution. Forced mutation operator is introduced when the identical genes appear via the crossover
procedure. A computational simulation of a 12-bay plain truss model has been implemented to demonstrate the feasibility
of the three optimal algorithms above. The obtained optimal sensor placements using the improved genetic algorithm are
compared with those gained by exiting genetic algorithm using the binary coding method. Further the comparison criterion
based on the mean square error between the nite element method (FEM) mode shapes and the Guyan expansion mode
shapes identied by data-driven stochastic subspace identication (SSI-DATA) method are employed to demonstrate the
advantage of the different tness function. The results showed that some innovations in genetic algorithm proposed in this
paper can enlarge the genes storage and improve the convergence of the algorithm. More importantly, the three optimal
sensor placement methods can all provide the reliable results and identify the vibration characteristics of the 12-bay plain
truss model accurately.
r 2008 Elsevier Ltd. All rights reserved.
1. Introduction
Structural modal parameter identication using measured dynamic data has received much attention over
the years because of its importance in structural model updating, structural health monitoring and structural
control. In particular, the quality of a modal parameter identication process strongly depends on the quality
of the measured response data, which further depends substantially on the numbers and locations of sensors in
the structure [1]. So determining the optimal numbers and locations of sensors is a critical issue encountered in
the construction and implementation of an effective structural health monitoring system. Its basic idea is to
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doi:10.1016/j.jsv.2008.03.026

Corresponding author. Tel. +86 451 86402713.

E-mail addresses: liuweiliuwei2005@gmail.com, liuweiliuwei2003@hit.edu.cn (W. Liu), gaoweicheng@sina.com (W.-c. Gao).

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select the optimal numbers and locations of the sensors such that the resulting measured data are most
informative, the identied modal parameters are quite accurate and the structural health monitoring system
are quite robust.
Many authors [210] have researched the optimal sensor placement problem for structural modal parameter
identication and structural health monitoring in the past few years. Kammer [2,3] presented and developed
the effective independence (EI) method, which maximizes a combination of target mode signal strength and
linear independence. The method starts with the large candidate sensor set, ranks all the sensors based on their
contributions to the determinant of a Fisher information matrix (FIM), and then eliminates the lowest ranked
sensor. The new candidate sensor set is then re-ranked and the lowest ranked sensor is again discarded. In an
iterative fashion, the initial candidate set is reduced to the desired number of locations. Lim [4] employed the
generalized Hankel matrix, a function of the system controllability and observability, to develop an approach
which can determine sensor locations based on a given rank for the system observability matrix while
satisfying modal test constraints. Papadimitriou et al. [1,5] introduced the information entropy norm as the
measure that best corresponds to the objective of structural testing which is to minimize the uncertainty in the
model parameter estimates. The optimal sensor conguration is selected as the one that minimizes the
information entropy measure since it gives a direct measure of this uncertainty. An important advantage of the
information entropy measure is that it allows us to make comparisons among sensor congurations involving
a different number of sensors in each conguration.
Genetic algorithms (GA) have also been proposed as an effective alternative [68] to the previous heuristic
algorithm, which is not guaranteed to give the optimal solution. Yao et al. [6] had taken GA as an alternative to
the EI method and the determinant of the FIM is chosen as the objective function. Worden and Burrows [7]
reviewed the recent work on sensor placement and applied the GA and the simulated annealing to determine the
optimal sensor placement in structural dynamic test. Then it described an approach to fault detection and
classication using neural networks and combinatorial optimization. Gao et al. [8] developed a new framework
of sensor placement optimization for structural health monitoring. The optimization problem is to minimize the
damage misdetection rate as well as to minimize the number of sensors by searching the optimized patterns of
sensor placement topology on the feasible region of the monitored structure. The program was applied to a
sample sensor placement problem of an aging aircraft wing. Optimized sensor placement designs are obtained.
Some comparison work can be seen in Refs. [9,10]. Larson et al. [9] made a comparison between some
actuator and sensor placement techniques including the EI method, the kinetic energy (KE) method, average
kinetic energy (AKE) and eigenvector component product (EVP). All methods proceed by sequentially
deleting the worst candidate points until the correct number of sensors is obtained. Meo and Zumpano [10]
investigated six different optimal sensor placement techniques on a bridge structure with the aim of
maximizing the data information. Three of them are based on the maximization of the FIM, one is based on
the properties of the covariance matrix coefcients, and last two are based on energetic approaches. The
results showed that the effective independence driving-point residue (EFI-DPR) method can provide an
effective method for optimal sensor placement to identify the vibration characteristics of the bridge.
The research presented in this paper is aimed to develop some optimal sensor placement techniques for
damage detection and structural health monitoring on the spatial lattice structure. Based on the criterion of
optimal sensor placement for modal test, an improved GA is introduced to nd the optimal placement of
sensors. The layout of the paper is as follows: Section 2 gives the basic theory of the improved GA including
the selection of the tness function, the presented coding system and genetic operator. Section 3 describes the
computational simulation using a 12-bay plain truss model and the presented optimization strategies are
demonstrated and compared using the modal parameters identied by the data-driven stochastic subspace
identication (SSI-DATA) method. Section 4 discussed the concerning work and the conclusions.
2. Basic theory
2.1. GA
For the sake of completeness, a brief discussion of GA will be given here. For more details, readers could
refer to the standard introduction in Ref. [11]. GA is optimization algorithm, which evolves in an analogous
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manner as the Darwinian principle of natural selection. To obtain the optimal solution for design problems,
the GA has been implemented so that it progresses in a similar way as the natural evolution of a species. It
means that the fundamental concepts of reproduction, chromosomal crossover, occasional mutation of genes
and natural selection are reected in the different stages of the GA process. The process is initiated by selecting
a number of candidate design variables either randomly or heuristically in order to create an initial
population. Then the initial population is encouraged to evolve over generations to produce new designs,
which are better or tter. The quality or tness of the designs is evaluated according to an objective function,
i.e. the tness function, which must be formulated in relation to the specic optimization problem. By
denition the optimal design corresponds to the maximum of this objective function. To implement the GA, it
is necessary to devise a general coding system for the representation of the design variables rst. Most
commonly the design variables are coded by the binary representation. Since the search for the optimal
solution proceeds with the population of design alternatives, the GA has a distinct advantage over traditional
optimization techniques, which start from a single point in the design space [7,12].
2.2. Fitness function
The tness functions presented in this paper are the modal strain energy (MSE) and the modal assurance
criterion (MAC), respectively. The objective of MSE is to nd a reduced conguration of sensor placements,
which maximizes the measure of the MSE of the structure. The reason is that the signal-to-noise ratio of the
measured response data is larger on the degree of freedoms (dofs) which have the larger MSE and it makes for
parameter identication when the sensor are placed on these locations. At the same time, the MAC matrix is
used to construct other two objective functions. The rst is the average value of all the off-diagonal elements in
MAC matrix. The second is the biggest value in all the off-diagonal elements in MAC matrix. The reason for
the selection of these tness functions is that the MAC matrix will be diagonal for an optimal sensor
placement strategy so the size of the off-diagonal elements can be taken as an indication of the tness.
Assume the mode shape matrix is U j
1
; j
2
; . . . ; j
p
(subscript p is the number of mode shape vectors)
and the number of the measured points is q, the MSE tness function can be given as
f
X
p
i1
X
p
j1
X
r2q
X
s2q
j
ri
k
rs
j
sj

(1)
where k
rs
represents the stiffness coefcient between the rth dof and sth dof, is just the element corresponding
to the rth row and the sth column in the stiffness matrix. j
ri
represents the deformation of rth element in ith
mode and j
sj
represents the deformation of sth element in jth mode. rAq and sAq represents that r and s are all
included in the total measured point set.
The MAC can be dened as Eq. (2), which measures the correlation between mode shapes:
MAC
ij

j
T
i
j
j

2
j
T
i
j
i
j
T
j
j
j

(2)
where j
i
and j
j
represent the ith mode shape vector and the jth mode shape vector, respectively, and the
superscript T represents the transpose of the vector.
Then the MAC1 tness function is given as Eq. (3) and the MAC2 tness function is given as Eq. (4):
f 1 averageabsMAC
ij
; iaj (3)
f 1 maxabsMAC
ij
; iaj (4)
where abs ( ) represents the absolute value, average ( ) represents the average value and max ( ) represents
the maximal value.
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2.3. Coding system
Considering the characteristics of the optimal sensor placement problem, the decimal coding system is
adopted for the representation of the design variables in this paper. Because the number of the dofs is
enormous in large-scale spatial lattice structures, the requirement for the large storage space is increased to
save the optimal solutions. So the decimal two-dimension array coding method instead of binary coding
method is presented to code the solutions. If there are s sensors to place in the total n degrees of freedom, the
coding length of a string is s. Every value of the string is the dof on which the sensor is located. For example, 2
6 12 14 22 29 30 35 38 43 is a string, it denotes that sensors are located on the second, sixth, 12th, 14th, etc. 10
dofs. If the size of the initial population of individuals is m, then the decimal two-dimension array coding
method is formed as Table 1 in which the number of sensors s is 10 and the total degrees of freedom n is 44 in
this paper. (Referred to the illustrative example presented subsequently in Section 3.) To demonstrate the
advantage of the decimal two-dimension array coding method directly, two binary coding methods are
introduced here briey as Tables 2 and 3. Table 2 shows one kind of binary coding method in which the coding
length of a string is the total degrees of freedom n. If the value of the ith bit position of the string is 1, it
denotes that a sensor is located on the ith dof. In contrast, if the value of the ith bit position is 0, it denotes no
sensor is located on the ith dof [12]. Table 3 shows another kind of binary coding method in which one sensor
location is represented by a binary string then all the strings are connected in series as a total string [13]. The
length of the binary string in one sensor location is l which should ensure that 2 to the power of l is the nearest
integer to the total degrees of freedom n and larger than it. By comparison it is obvious that the dissipative
storage space of the decimal two-dimension array coding method is minimal among them. This is very
important in the optimal sensor placement problem for spatial lattice structure because the number of the dof
is enormous in the large-scale spatial lattice structure. The convergence of the proposed coding system can be
demonstrated in the next section.
2.4. Genetic operation
To implement the GA for the determination of the optimal sensor location, a number of candidate design
variables have been selected randomly as an initial population (such as Table 1). Then the reproduction
operation also called the natural selection is carried out, in which the tness of the different individual of the
population had been evaluated based on the above presented tness functions and ranked by the ratio of
individual tness to the total tness of the current population. Some new design variables that will become
parent designs in the next circulation are selected directly according to their individual tness ranking. Further
is the crossover process. Some sections of the bit-string representations of the two parent designs (arbitrary
two rows in Table 1) are swapped directly to create the two offspring design solutions. This process ensures
that design information is transferred from one generation to the next. Following crossover, the mutation
operation is introduced via the occasional switching of the bit value at a randomly selected location of the
generated strings. This action is important since it guards against the premature convergence of the design
towards a non-optimal solution [14].
In executing the optimal sensor placement searching via GA, the same location may be placed with two
sensors synchronously in the crossover process. It is impractical and must be avoided. In this paper, we
introduced the forced mutation operator to change the repeated sensor location in the generated strings. The
detailed operation process is represented and shown in Table 4. The rst two lines in Table 4 are the two
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Table 1
The decimal two-dimension array coding method (ms, s 10)
No. 1 2 3 4 5 6 7 8 9 10
Genes 1 2 3 6 12 20 28 33 36 38 44
Genes 2 2 5 8 18 24 30 35 38 40 43
y y y y y y y y y y y
Genes m 4 6 9 16 22 32 36 38 43 44
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selected parent design solutions. If the jth bit in the parent design solutions is chosen randomly as the cutting
position, the two new offspring design solutions will be generated by each taking the rst part from one parent
and the second from the other as shown in the second two lines in Table 4. Unfortunately, the jth bit in one
new offspring design solution (offspring 2) is the same with the ith bit in it (the italic numbers). That means
that one location has been placed with two sensors synchronously. So the forced mutation operator is
introduced to change one value of the same two numbers to the other value, which is different from the other
numbers in offspring 2. In this example, the second value 22 is changed to 28 (the bold italic underlined
numbers in the last line in Table 4), which is not included in the offspring 2 before. For the reasonable
compatibility of GA, we reduced the ratio of the natural mutation operation as a compromise. The research
results showed that the forced mutation operator obtained the expected intention and did not inuence the
convergence of the GA.
In practice, a convergence criterion must be specied in executing the GA. In this paper, a relative large
number N is selected to avoid redundant iteration. The genetic process will be stopped automatically if the best
individual in the population does not change in continuous N iteration. To sum up, the whole owchart of the
genetic search to nd the optimal sensor locations presented in this paper is shown in Fig. 1.
3. Numerical example
3.1. Analytical model
In this section, the three different optimal sensor placement techniques presented above were tested for
modal identication on a 12-bay plain truss model as shown in Fig. 2. The total size of 12-bay plain truss
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Table 2
Binary coding method 1 (ms, s 44)
No. 1 2 3 4 5 6 7 8 9 10 y 35 36 37 38 39 40 41 42 43 44
Genes 1 0 1 1 0 0 1 0 0 0 0 y 0 1 0 1 0 0 0 0 0 1
Genes 2 0 1 0 0 1 0 0 1 0 0 y 1 0 0 1 0 1 0 0 1 0
y y y y y y y y ... y y y y y y y y y y y y y
Genes m 0 0 0 1 0 1 0 0 1 0 y 0 1 0 1 0 0 0 0 1 1
Table 3
Binary coding method 2 (ms, s 10 6)
No. 1 2 3 4 5 6 7 8 9 10
Genes 1 000010 000011 000110 001100 010100 011100 100001 100100 100110 101010
Genes 2 000010 000101 001000 010010 011000 011110 100011 100110 101000 101001
y y y y y y y y y y y
Genes m 000100 000110 001001 010000 010110 100000 100100 100110 101001 101010
Table 4
Operation process of forced mutation in genetic algorithm
No. 1 2 3 y i y j y s-2 s-1 s
Individual gene pair before crossover Parent 1 2 3 6 y 17 y 22 y 36 40 44
Parent 2 4 6 9 y 22 y 25 y 38 41 44
New individual gene pair after crossover Offspring 1 2 3 6 y 17 y 25 y 38 41 44
Offspring 2 4 6 9 y 22 y 22 y 36 40 44
New individual gene pair after forced mutation Offspring 1 2 3 6 y 17 y 25 y 38 41 44
Modied offspring 2 4 6 9 y 22 y 28 y 36 40 44
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model is 4.8 m0.4 m, the number of girds is 12 1. All element sections are tubular and the dimensions are
+16 mm2 mm. The material properties are taken from Q235 steel where the elastic modulus is 210 GPa and
the density is 7850 kg/m
3
. The deadweight of the members and the ball are treated as lumped mass
concentrated at the nodes. The joints of the plain truss are hinged connection and the plain truss is hinged at
xed-point supports on the both sides. The analytical model has 24 nodes, 45 elements and 44 dofs. In order to
provide the input data for the optimal sensor placement methods, the nite element model of the 12-bay plain
truss model is developed using the universal nite element analysis package (ANSYS [15]). The vibration
properties were calculated by performing modal analysis based on the subspace iteration method. The
structural dynamic characteristics including the rst six natural frequencies and mode shapes are obtained and
shown in Table 11 and Fig. 3. It is obvious that mode shape 1 is the rst vertical bending deection, in
sequence, mode shape 2 is the second vertical bending deection, mode shape 3 is the third vertical bending
deection, mode shape 5 is the forth vertical bending deection, mode shape 6 is the fth vertical bending
deection, while mode shape 4 behaves as a coupled vibration mode shape between the third vertical bending
deection and the horizontal longitudinal deection.
3.2. Optimization results
Based on the stiffness matrix and mode shape matrix calculated by nite element method (FEM), the above
three approaches which differ only in the chosen of objective function based on GA are implemented to select
the best sensor locations. The basic parameters of GA are listed as follows: population size is 300, probability
of selection is 0.2, probability of crossover is 0.6, probability of mutation is 0.01 and the relative large number
of generations selected for convergence is 100. All the best results for the 15, 10 and ve sensor locations are
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Start
End
Input parameters
Initial Population
Fitness Evaluation
Stopping
Criterion
Yes
No
Selection
Crossover/ Forced Mutation
Mutation
Fig. 1. Flowchart of the genetic algorithm.
Fig. 2. A 12-bay plain truss model.
W. Liu et al. / Journal of Sound and Vibration 317 (2008) 175189 180

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listed and compared in Tables 57. Each algorithm was used to select the best 15, 10 and ve sensor locations
placed on the 12-bay plain truss model to independently identify the modal parameters, respectively. The aim
is to determine the optimal numbers and locations of sensors, which is enough to obtain the response data and
the structural dynamic behavior of the 12-bay plain truss model thoroughly.
In order to evaluate the reliability of the above results, all the tness convergence cures of different tness
function in different measured points cases are shown as Figs. 46. It is obvious that all the maximum tness
values tend to a constant quickly and the average tness value steadily tends to the maximum tness value
along with increasing number of generations. It shows a good characteristic of convergence. Further to
demonstrate the effectiveness of the improvements in the GA, the existing GAs with binary coding method
[12] are performed and compared with the one proposed in this paper. Each method was executed 10 times
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Fig. 3. Mode shapes calculated by nite element method (FEM): (a) the rst mode shape; (b) the second mode shape; (c) the third mode
shape; (d) the forth mode shape; (e) the fth mode shape and (f) the sixth mode shape.
Table 5
Comparison of the optimal sensor locations in 15 measured points case
Fitness function Optimal sensor locations
MSE 2 5 6 8 12 18 21 22 23 35 36 37 38 42 43
MAC1 2 3 6 11 12 13 14 20 21 22 30 36 38 43 44
MAC2 1 2 4 9 16 18 19 21 22 24 34 35 38 39 42
Table 6
Comparison of the optimal sensor locations in ten measured points case
Fitness function Optimal sensor locations
MSE 2 5 18 21 22 35 37 38 42 43
MAC1 2 6 12 14 22 29 30 35 38 43
MAC2 2 5 12 22 26 27 28 34 38 39
Table 7
Comparison of the optimal sensor locations in ve measured points case
Fitness function Optimal sensor locations
MSE 5 21 22 35 37
MAC1 4 14 30 35 44
MAC2 3 12 20 26 38
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with a different stochastic initial population. Numbers of the convergence generations are compared in
Tables 810. The average number of convergence generations of different tness function in different
measured points cases using decimal two-dimension array coding method is smaller than those using binary
coding method. That means the convergence speed of the improved GA is far higher than that of binary
coding method and 2030% reduction in computational iterations is gained to reach a satisfactory solution.
At the same time, the tness function values are compared each other. They are almost identical and are
omitted in this paper.
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20 40 60 80 100 120
2.5
3
3.5
4
4.5
5
5.5
Generation
F
i
t
n
e
s
s
(
x

1
0
8
)
50 100 150
0.977
0.978
0.979
0.98
0.981
0.982
Generation
F
i
t
n
e
s
s
20 40 60 80 100
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Generation
F
i
t
n
e
s
s
maximum fitness
average fitness
maximum fitness
average fitness
maximum fitness
average fitness
Fig. 4. Fitness curves of improved genetic algorithm with different tness functions (15 sensors): (a) MSE; (b) MAC1 and (c) MAC2.
20 40 60 80 100
1.5
2
2.5
3
3.5
4
4.5
5
Generation
F
i
t
n
e
s
s
(
x

1
0
8
)
20 40 60 80 100 120
0.96
0.961
0.962
0.963
0.964
0.965
Generation
F
i
t
n
e
s
s
20 40 60 80 100
Generation
0.2
0.25
0.3
0.35
0.4
0.45
0.5
F
i
t
n
e
s
s
maximum fitness
average fitness
maximum fitness
average fitness
maximum fitness
average fitness
Fig. 5. Fitness curves of improved genetic algorithm with different tness functions (10 sensors): (a) MSE; (b) MAC1 and (c) MAC2.
20 40 60 80 100
1
1.5
2
2.5
3
3.5
Generation
F
i
t
n
e
s
s
(
x

1
0
8
)
20 40 60 80 100
0.908
0.91
0.912
0.914
Generation
F
i
t
n
e
s
s
20 40 60 80 100
0.05
0.1
0.15
0.2
Generation
F
i
t
n
e
s
s
maximum fitness
average fitness
maximum fitness
average fitness
maximum fitness
average fitness
Fig. 6. Fitness curves of improved genetic algorithm with different tness functions (ve sensors): (a) MSE; (b) MAC1 and (c) MAC2.
W. Liu et al. / Journal of Sound and Vibration 317 (2008) 175189 182

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3.3. Comparison study
In order to demonstrate the capability of capturing the vibration behavior of the 12-bay plain truss model
using the three optimal sensor placement techniques, the SSI-DATA method [16] is adopted to identify the
modal parameters as the measured data set. To do this, the simulated excitation that is assumed as the
independent band-limited white noises is applied to the y direction of nodes 12. Meantime, the outputs (15, 10
and ve accelerations) are collected in the above-determined optimal sensor locations, respectively. To
simulate the ambient vibration case, a 5% root mean square noise is added to the measured outputs and inputs
are not collected. Comparison criteria based on the mean square error between the FEM mode shapes and the
Guyan expansion [17] mode shapes measured at the selected sensor locations was employed to demonstrate
the feasibility of the selected optimal sensor locations [10].
First, the modal parameters are obtained by SSI-DATA method in several different sensor placement cases.
Considering the integrality of the paper, stability diagram obtained by SSI-DATA algorithm in 15 measured
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Table 8
Comparison of the convergence by different coding methods using different tness functions (15 sensors)
No. Number of convergence generations
Decimal two-dimension array coding method Binary coding method
MSE MAC1 MAC2 MSE MAC1 MAC2
1 131 162 110 141 239 179
2 127 137 154 118 278 194
3 171 145 175 133 193 188
4 121 150 116 151 157 206
5 134 183 166 197 142 188
6 136 132 128 176 141 328
7 123 141 233 145 183 137
8 164 140 152 143 157 146
9 131 162 110 139 201 156
10 131 162 110 134 249 168
Average 136.9 151.4 145.4 147.7 194 189
Table 9
Comparison of the convergence by different coding methods using different tness functions (ten sensors)
No. Number of convergence generations
Decimal two-dimension array coding method Binary coding method
MSE MAC1 MAC2 MSE MAC1 MAC2
1 134 139 129 127 172 157
2 150 141 180 131 223 135
3 131 133 137 157 158 129
4 117 136 109 128 200 173
5 126 157 180 126 202 136
6 145 146 109 121 119 219
7 129 164 123 130 160 185
8 132 158 129 141 191 147
9 134 139 129 165 190 187
10 134 139 129 132 167 180
Average 133.2 145.2 135.4 135.8 178.2 164.8
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points case using MSE tness function is given in Fig. 7. The background curve is the sum of all the auto-
spectral and cross-spectral density functions. The stabilization criteria are 1% for frequencies, 5% for
damping and 5% for mode vectors. From it the rst six natural frequencies and mode shapes can identify
distinctly. All the modal frequencies are listed in Table 11. The identied modal frequencies results are
compared with those calculated by FEM in the rst row. It is obvious that the identication results by SSI-
DATA are quite accurate. This means that the ve measured points case can obtain the modal frequencies
accurately. It is not surprising because one sensor is enough to know the frequency of the structure in theory.
Then the mean square errors between the FEM mode shapes and the Guyan expansion mode shapes, which
are identied by SSI-DATA method and then expanded by the Guyan expansion technique are calculated and
summarized in Table 12. As expected, the mean square error of the three optimal sensor placement methods is
all very small. This implies that the identied mode shapes under different sensor placement cases may be
consistent with the FEM modes shapes.
To further demonstrate the feasibility of the selected optimal sensor locations, the identied six Guyan
expansion mode shapes are shown in Figs. 813 for comparing with those calculated by FEM in Fig. 3. (The
results in 15 measured points case are omitted for the length of paper.) By comparison all the identied six
mode shapes by SSI-DATA method with different optimal sensor methods in 15 measured points case and 10
measured points case are nearly consistent with those calculated by FEM. And in ve measured points case,
the optimal sensor method with MSE tness function can identify the rst two mode shapes and the optimal
sensor methods with MAC1 and MAC2 tness function can identify the rst ve mode shapes. These results
are also expected. Table 7 shows that the optimal ve sensor points obtained by the MSE tness function
include one sensor in y direction and four sensors in x direction, and the optimal ve sensor points obtained by
the MAC1 and MAC2 tness function include four sensors in y direction and one sensor in x direction,
respectively. It is obvious that four sensors in y direction and one sensor in x direction can identify the rst ve
mode shapes at most because mode shape 4 behaves as a coupled vibration mode shape between the third
vertical bending deection and the horizontal longitudinal deection and mode shape 5 is the forth vertical
bending deection as described above. In conclusion, the two methods based on MAC tness function in GA
are better than the method based on MSE tness function in the case of placing a few sensors. With the
increasing of the number of sensors the three methods based on different tness function in GAs can all obtain
the reliable optimal sensor placement to identify the vibration characteristics of the 12-bay plain truss model
accurately. To obtain the rst six mode shapes of the 12-bay plain truss model under investigation, six sensors
including ve in y direction and one in x direction is required at least because the 12-bay plain truss model
provides the vibration characteristic as the beam with both ends built-in.
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Table 10
Comparison of the convergence by different coding methods using different tness functions (ve sensors)
No. Number of convergence generations
Decimal two-dimension array coding method Binary coding method
MSE MAC1 MAC2 MSE MAC1 MAC2
1 110 112 111 115 187 150
2 116 114 108 109 158 117
3 114 116 109 111 124 122
4 107 107 113 117 167 111
5 113 115 104 114 120 126
6 115 104 113 111 197 118
7 107 137 106 113 134 170
8 110 112 111 112 141 135
9 110 112 111 112 116 143
10 116 114 108 109 143 120
Average 111.8 114.3 109.4 112.3 148.7 131.2
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4. Conclusions
In this paper, the GA was studied and improved to nd the optimal sensor placement based on the criterion
of optimal sensor placement for modal test. Three optimal placement techniques were presented when the
MSE and the MAC had been taken as the tness function, respectively. Considering the characteristics of the
optimal sensor placement techniques in the large-scale spatial lattice structure, some innovations in GA such
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0 50 100 150 200 250
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235 240 245
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Fig. 7. Stability diagram obtained by SSI-DATA algorithm: (a) all the six modes; (b) partial enlarged result of the rst mode; (c) partial
enlarged result of the second mode; (d) partial enlarged result of the third mode and the forth mode; (e) partial enlarged result of the fth
mode and (f) partial enlarged result of the sixth mode. for a stable pole; .v for a pole with stable frequency and vector; .d for a
pole with stable frequency and damping; .f for a pole with stable frequency and + for a new pole.
W. Liu et al. / Journal of Sound and Vibration 317 (2008) 175189 185

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Table 11
Comparison of modal frequencies identied by SSI-DATA using different tness functions under different sensor placement cases and the
ones calculated by FEM
No. 1 2 3 4 5 6
Modal frequencies calculated by FEM (Hz) 25.12 66.27 132.24 136.26 199.73 241.67
Modal frequencies identied by SSI-DATA (Hz)
Fifteen measured points
MSE 25.11 66.29 132.23 136.27 198.38 241.71
MAC1 25.11 66.29 132.22 136.27 199.90 241.77
MAC2 25.11 66.29 132.27 136.25 200.12 241.83
Ten measured points
MSE 25.11 66.30 132.23 136.20 199.57 241.78
MAC1 25.11 66.29 132.23 136.22 199.67 241.81
MAC2 25.11 66.28 132.23 136.26 199.97 241.86
Five measured points
MSE 25.12 66.28 132.22 136.25 198.38 241.85
MAC1 25.12 66.28 132.21 136.27 199.90 241.69
MAC2 25.12 66.28 132.21 136.26 200.12 241.52
Table 12
Comparison of the mean square error between the FEM mode shape and the Guyan expansion mode shapes using different tness
functions under different sensor placement cases
Mean square error Total mean square error
First mode Second mode Third mode Fourth mode Fifth mode Sixth mode
Fifteen measured points
MSE 2.04e005 2.86e005 8.31e005 2.84e004 1.79e003 8.37e004 3.05e003
MAC1 9.31e007 6.10e006 5.95e005 1.11e004 4.92e004 3.95e004 1.06e003
MAC2 1.05e005 2.04e005 2.05e004 1.17e004 1.41e003 6.18e004 2.38e003
Ten measured points
MSE 3.47e005 1.20e004 8.72e004 5.28e004 5.44e003 9.06e003 1.61e002
MAC1 1.36e006 1.44e005 1.39e004 6.15e004 1.25e003 1.85e003 3.87e003
MAC2 2.76e005 4.61e005 1.34e004 4.39e004 3.70e003 1.26e003 5.61e003
Five measured points
MSE 7.21e006 6.68e004 1.01e002 3.28e003 1.39e002 1.33e002 4.12e002
MAC1 5.99e004 3.78e005 3.08e003 1.09e003 3.99e003 9.63e003 1.84e002
MAC2 1.90e005 5.93e005 6.55e004 5.62e003 2.89e003 1.08e002 2.00e002
Fig. 8. Mode shapes identied in 10 measured points case using MSE: (a) the rst mode shape; (b) the second mode shape; (c) the third
mode shape; (d) the forth mode shape; (e) the fth mode shape and (f) the sixth mode shape.
W. Liu et al. / Journal of Sound and Vibration 317 (2008) 175189 186

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as the decimal two-dimension array coding system and the forced mutation operator were proposed in this
paper to enlarge the genes storage and improve the convergence of the algorithm. A 12-bay plain truss model
was taken as the simulation example to demonstrate the feasibility of the three optimal sensor placement
algorithms presented. The optimal sensor placement for 15, 10 and ve sensors cases are obtained and studied
in detail. Some conclusions and recommendations are summarized as follows:
(1) The dissipative storage space of the proposed decimal two-dimension array coding method is far less than
the existing two kinds of binary coding methods. It is propitious for optimal sensor placement in spatial
lattice structure because of the enormous dofs.
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Fig. 9. Mode shapes identied in 10 measured points case using MAC1: (a) the rst mode shape; (b) the second mode shape; (c) the third
mode shape; (d) the forth mode shape; (e) the fth mode shape and (f) the sixth mode shape.
Fig. 10. Mode shapes identied in 10 measured points case using MAC2: (a) the rst mode shape; (b) the second mode shape; (c) the third
mode shape; (d) the forth mode shape; (e) the fth mode shape and (f) the sixth mode shape.
Fig. 11. Mode shapes identied in ve measured points case using MSE: (a) the rst mode shape; (b) the second mode shape; (c) the third
mode shape; (d) the forth mode shape; (e) the fth mode shape and (f) the sixth mode shape.
W. Liu et al. / Journal of Sound and Vibration 317 (2008) 175189 187
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(2) The proposed forced mutation operator can avoid one location placed with two sensors synchronously in
the crossover process. We can reduce the ratio of the natural mutation operation as a compromise for the
reasonable compatibility of GA.
(3) The convergences of the improved GA using different tness functions under different sensor placement
cases are all better than those of the existing GA with binary coding method. In total, 2030% reduction in
computational iterations can be gained to reach the satisfactory solutions.
(4) The modal frequencies can be accurately identied even if only ve sensors are optimally placed on the
structure. And the mean square errors between the FEM mode shapes and the Guyan expansion mode
shapes which are identied by SSI-DATA method and then expanded by the Guyan expansion technique
are all very small.
(5) By comparing the identied Guyan expansion mode shapes with those calculated by FEM, the results
obtained by the improved GA based on MAC tness function is better than those obtained using MSE
tness function in ve measured points case. With the increasing of placed sensors, the three methods
based on different tness function can all provide the reliable optimal sensor placement to identify the
vibration characteristics of the 12-bay plain truss model accurately.
(6) The GA is particularly effective in solving the combinatorial optimization problem such as optimal sensor
placement problem when the performance tradeoffs are not unbearable and when the number of
combinations is too large to preclude enumeration.
Acknowledgments
This research was funded by the National Science Foundation of China under Grant no. 50478030 and the
Key Technologies Research and Development Program of Heilongjiang Province in China under Grant no.
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Fig. 12. Mode shapes identied in ve measured points case using MAC1: (a) the rst mode shape; (b) the second mode shape; (c) the
third mode shape; (d) the forth mode shape; (e) the fth mode shape and (f) the sixth mode shape.
Fig. 13. Mode shapes identied in ve measured points case using MAC2: (a) the rst mode shape; (b) the second mode shape; (c) the
third mode shape; (d) the forth mode shape; (e) the fth mode shape and (f) the sixth mode shape.
W. Liu et al. / Journal of Sound and Vibration 317 (2008) 175189 188
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GC04C101. The authors wish to express their sincere thanks to Zhanwen Huang of Harbin Institute of
Technology for the helpful improvement in language expression of this paper.
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