Optimal sensor placement using FRFs-based clustering method

Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Contents lists available at ScienceDirect

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Journal of Sound and Vibration

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journal homepage: www.elsevier.com/locate/jsvi

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Optimal sensor placement using FRFs-based clustering method

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Shiqi Li, Heng Zhang, Shiping Liu n, Zhe Zhang

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School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

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a r t i c l e i n f o

abstract

Article history: Received 21 May 2016 Received in revised form 25 August 2016 Accepted 3 September 2016 Handling Editor: L.G. Tham

The purpose of this work is to develop an optimal sensor placement method by selecting the most relevant degrees of freedom as actual measure position. Based on observation matrix of a structure's frequency response, two optimal criteria are used to avoid the information redundancy of the candidate degrees of freedom. By using principal component analysis, the frequency response matrix can be decomposed into principal directions and their corresponding singular. A relatively small number of principal directions will maintain a system's dominant response information. According to the dynamic similarity of each degree of freedom, the k-means clustering algorithm is designed to classify the degrees of freedom, and effective independence method deletes the sensors which are redundant of each cluster. Finally, two numerical examples and a modal test are included to demonstrate the efficient of the derived method. It is shown that the proposed method provides a way to extract sub-optimal sets and the selected sensors are well distributed on the whole structure. & 2016 Elsevier Ltd. All rights reserved.

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1. Introduction

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High-performance space camera will require extremely quiet on-orbit vibration environments. The support structures of sensitive optical device must be valid for a variety of vibration disturbances including reaction wheels, cryocoolers, and other scanning components. In order to obtain accurate verification model, physical vibration tests are performed to compare the computational model with the actual structures. Since finite element models have reached a satisfying depiction of conservative modes, they can be considered as a good approximation of actual structure. Planning of sensor placement can be done by using a finite element model before a physical test. Proper pretest planning plays a vital component of any successful vibration test, especially for the case when testing modern complex aerospace structures. Optimal sensor placement (OSP) has received much attention over the years because of its importance role in many areas, such as system identification [1], vibration and noise control [2], structural health monitoring [3], model updating [4], and many other applications. In particular, the quality of dynamic model verification process strongly depends on the quality of real-time test data, which further depends substantially on selecting the proper sensor locations in the structure. Previous studies revealed that arbitrary placement might lead to unreliable results or even unidentifiable situations. In order to optimize the performance of the sensory system, various methods have been developed, such as modal kinetic energy [5], information entropy [6], Guyan reduction [7], maximizing the Fisher information matrix (FIM) [8,9], etc. These work

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Corresponding author. E-mail address: [email protected] (S. Liu).

http://dx.doi.org/10.1016/j.jsv.2016.09.004 0022-460X/& 2016 Elsevier Ltd. All rights reserved.

61 Please cite this article as: S. Li, et al., Optimal sensor placement using FRFs-based clustering method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.004i

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proposed that the sensor configuration maximizing some norm of the FIM be taken as the optimal configuration, which addresses the issue of optimally locating a given number of sensors in a structure. For example, Papadinitriou et al. [10–12] introduced the information entropy as the measure of uncertainties that best corresponds to the objective of structural testing which is to minimize the uncertainty in the model parameter estimates. The objective function is the information entropy, while the discrete variables are related to the number and location of sensors. This information entropy-based measure resolved the issue related to the arbitrariness in selecting an appropriate norm for the FIM. It was shown that the information entropy depends on the determinant of the Fisher information entropy. The frequency response functions (FRFs) of mechanical and structural systems are of interest in dynamic problems in recent years [13–15]. The frequency response based OSP method is to place sensors to maintain the dynamically important information contained in the frequency response data. By using FRFs, errors from modal parameter extraction can be eliminated because modal parameter identifications are not required, and a large number of frequency data are available for selecting sensor locations. Principal component analysis (PCA) based on FRFs can be used to extract the main response feature effectively [16–19]. Researchers have shown that most of the systems energy is concentrated in the first several singular, and the corresponding principal directions show the energy distribution in a structure. To establish a performance matrix based on principal directions of FRFs provides a computationally efficient procedure to obtain a reasonably good suboptimal solution. Some degrees of freedom (DOFs) contain similar dynamic characteristic, the response information of these DOFs is believed to be redundant and the size of the system calculation increases in proportion to the total number of DOFs in the structural model [20,21]. Therefore, the selection of sensor locations from the candidate locations has become an important issue. Searching algorithms of OSP try to select sensor locations that maximize signal strength and render the target mode linearly independent. However, even for a small number of sensors, the combinatorial explosion of the number of sensor sets to explore makes an exhaustive search infeasible for practical structures. Friswell [22] proposes a clustering of sensor locations by using EFI. His numerical simulation shows that when more sensors are selected than modes of interest, the sensor locations will present clusters. K-means clustering is an automatically classification algorithm which is based on the similarity of the object attribute. Because of its simplicity and flexibility, clustering algorithm is useful in many applications. The contribution of this paper concerns the redundancy of information shared by DOFs in frequency domain. Indeed some DOFs provide useful information which may share the same information. The optimal search of the sensor configuration tries to maximize the determinant of the FIM but does not consider the linear independence of the selected locations. We therefore put forward a FRFs based clustering method for optimal sensor placement. In this paper, with the aim of handling the large finite element model of real structure, a sub-optimal methodology is proposed. We performed frequency reduction method based on PCA to extract the principal response functions. In addition, on the basis of dynamic similarity, we classified the candidate DOFs into different groups by k-means clustering. The FIM is constructed by principal directions of FRFs, and EFI is used to select the most relevant locations. This paper is organized as follows: Section 2 presents the fundamental theory of computational model and information redundancy of FRFs. The optimal methodology for OSP is discussed in Section 3. The numerical analysis of spacecraft structures is reported to testify the effectiveness of the proposed method in Section 4. In Section 5, this method is tested on a pretest investigation of filter wheel assembly. Finally, some conclusions are summarized in Section 6.

2. Background theory 2.1. Computational model In this section, the computational model of a real structure is described. The equation of motion representing the n degrees of freedom in physical coordinates can be expressed by Mx€ ðt Þ þCx_ ðt Þ þ Kxðt Þ ¼ f ðt Þ

47 49

(1)

in which M; C; and K are the mass, damping, and stiffness matrices, respectively, xðtÞ is the vector of physical displacements, and fðtÞ is the vector of external forces. Assuming that the system is subjected to a harmonic excitation, the force and displacement response can be expressed as

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f ðt Þ ¼ Fejωt

(2)

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xðt Þ ¼ Xejωt

(3)

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Substituting Eqs. (2) and (3) into Eq. (1), Eq. (1) can be expressed as the following:
 ω2 M þ iωC þK X ¼ F

(4)

We are interested in the frequency band of analysis B ¼ ½ωmin ; ωmax , in this range, the corresponding dynamic stiffness matrix is given by
 (5) ZðωÞ ¼  ω2 M þ iωC þ K Please cite this article as: S. Li, et al., Optimal sensor placement using FRFs-based clustering method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.004i

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For a computational model, mode frequency and mode shapes of characteristic equation form an efficient basis for the computation of the analytical frequency response functions. The modal based displacement frequency response can be calculated by hpq ðωÞ ¼

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3

m X

ϕpi ϕqi

ω2  ω2 þ 2iξi ωi ω i¼1 i

(6)

in which ωi is the undamped natural frequency of a system, ξi is the damping coefficient, ϕpi is the value of the ith mode at the pth output location, and ϕqi is the value of the ith mode at the qth input location. All the DOFs are assumed to be the sensor locations, and the accuracy of a single FRF computed though Eq. (6) can be obtained. By including all the DOFs in the frequency band of interest, the analytical frequency response data matrix, H, can be defined as the collection of individual frequency response matrices
3 2 h1 ðω1 Þ ⋯ h1 ωnd 6 7 ⋮ ⋱ ⋮ (7) HðωÞ ¼ 4
5 hna ðω1 Þ ⋯ hna ωnd in which ha ðωd Þ is the value of frequency response at frequency ωd with columns corresponding to the ath test location, and nd is the number of data points in the frequency range of interest, na is the number of sensor locations.

19 2.2. Problem of information redundancy 21 The FRF data are conveniently arranged in a compound matrix, H, which can be written in either column or row vectors 23

as

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25   H ¼ X1 ; …; Xi ; …; Xnd

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or

3 Y T1 6 7 6 ⋮ 7 6 7 6 YT 7 H¼6 j 7 6 7 6 ⋮ 7 4 5 Y Tna

(8)

where Xi ; i ¼ 1; 2; …; nd , are column vectors associated with frequencies as reference data, whereas the row vectors Y Tj ; j ¼ 1; 2; …; na , are associated with sensor locations. For the computational model using FRFs, a large number of frequency points can be used as reference data. However, the column vectors in FRF matrix associated with frequencies are linearly dependent on one another. Some frequencies are believed to be redundant, only a limited number of frequencies among a large number of candidate reference data are used for practical reasons. Thus an effective rank nr can be estimated from the decrease of singular values or by inspection of principal response functions, which will carry as much information as possible. Usual sensor placement algorithms try to maximize estimate of modal responses. But they do not take into account that two DOFs may have close information, and each of them will have a similar impact individually. If their information matrices are close, then selecting both may be similar to selecting a single. For instance, let us consider analytical frequency response functions, H, the row vectors consist of na FRFs. The whole FRFs are redundant. Seen from the point of view of the selection of measurement coordinates, one problem of interest can be formulated as follows: Given a set of na FRFs, find the smallest subset of linearly independent FRFs, ns o na , which can correctly describe the dynamics of the tested structure.

45 3. Optimal methodology 47 49 51 53 55 57 59 61

Based on the above-mentioned problem, several techniques can be devised to reduce the redundant information. Firstly, PCA frequency reduction method extracts dominant response information from the original compound FRF. Then, k-means clustering algorithm maximizes the observability information matrix for each category of a structure. Finally, EFI method deletes the redundant locations in a certain group, which makes an optimal search feasible for complex structures. The optimization method is described as the following sections. 3.1. PCA based frequency reduction Principal component analysis is similar to proper orthogonal decomposition or singular value decomposition. Due to noise and nonlinear effects, the compound FRF matrix is full rank. Using PCA based frequency reduction method, an effective rank nr can be estimated from the decrease of singular values. A sufficiently independent subset of nr columns from FRF matrix H can be obtained. Using this approach, the frequency response data matrix can be decomposed as H ¼ ψSV

(9)

Please cite this article as: S. Li, et al., Optimal sensor placement using FRFs-based clustering method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.004i

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where the columns of the matrix ψ are the left singular vectors (principal directions), they represent the spatial distribution of energy. The diagonal matrix of singular values S incorporates the amplitude information, being a measure of the energy content of the FRFs. The columns of matrix V are the right singular vectors, they contain the frequency information. Most of the system's energy is concentrated in the first several singular values. The corresponding principal directions show how the energy is distributed in the structure. If the first nr singular values retained to properly characterize a system, the matrix H in Eq. (8) can be expressed by " #" # V1   S1 (10) H ¼ ψ1 ψ2 S2 V2 in which ψ 1 A Cna nr ; V1 A Cnr nd , and S1 ¼ diagðσ 1 ; σ 2 ; …; σ nr Þ. The matrix of principal response functions (PRF) [23] is

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P1 ¼ S1 V1 ¼

nr X

σ i vi

(11)

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Principal response functions are the first singular vectors scaled by their associated singular values. The original compound FRF matrix can be approximated by the reduced-rank reconstructed matrix nr X

H ffiψ 1 P1 ¼

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σ i ui v i

(12)

i¼1

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In Eqs. (11) and (12), ui and vi are the ith left and right singular vectors, respectively. There are a variety of methods for determining the value of nr. Using singular value ratio (SVR) is a common method to keep the singular values and principal directions which correspond to 95 percent or more of the system's total energy. The SV is the normalization of singular value. The ratio of SViþ 1 and SVi contains information about the selectivity of system's total energy:

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S Viþ1 ; S Vi

1 r io nd

(13)

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In this study, a PCA based frequency reduction method is used such that column vectors associated with frequencies in the matrix are more linearly independent of one another. Principal directions of compound FRFs are always orthogonal, which makes them more robust to modeling errors and experimental noise. The reduction matrix has advantages because they are apt to be more informative and computationally efficient.

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3.2. K-means clustering

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According to the refinement of a mesh, some degrees of freedom in the important mode contain similar dynamic characteristic, and the response of these degrees of freedom is approximately equal at a certain load. K-means clustering algorithm classifies these degrees of freedom automatically based on the similarity of multiple attribute. The extracted principal direction vector ψ ¼ fφi ; i ¼ 1; …; na g is na d-dimensional data set. The clustering problem aims at partitioning this data set into K disjoint subsets, C ¼ fck ; k ¼ 1; …; Kg. The most widely used clustering criterion is the sum of the squared Euclidean distances between each data point φi and mk (cluster center) of the subset ck which contains φi . The standard measure function is calculated by

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D¼

K X X  φi  mk 2

(14)

k ¼ 1 φi A ck

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After defining the function D, the clustering algorithm can be described as follows: (1) Samples are selected from the data set as the initial cluster centers, and K is the number of the cluster centers. (2) For the rest of the non-cluster centers, they are classified into the corresponding category based on their similarity (Euclidean distance) with the cluster centers. (3) Updating the cluster center of each category. (4) Repeat the steps from (2) to (3), until the convergence of standard measure function D is realized. 3.3. Effective independence method EFI method is one of the most popular methods for optimal sensors placement in structural dynamic pretest [8]. The key point of the EFI method is to iteratively remove sensor locations that have the smallest impact on the value of the determinant of the information matrix. We consider the frequency response based EFI to maintain the dynamically important information contained in the frequency response data within the desired frequency band. The Fisher information matrix can be defined as F ¼ ψT ψ ¼

na X

φTk φk

(15)

k¼1

where ψ is the principal directions of frequency response and φk is the kth row of the matrix ψ. The sensor locations are Please cite this article as: S. Li, et al., Optimal sensor placement using FRFs-based clustering method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.004i

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Fig. 1. Optimal sensor placement method diagram.

59 61 Please cite this article as: S. Li, et al., Optimal sensor placement using FRFs-based clustering method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.004i

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selected in an iterative way, the selection procedure is based on the orthogonal projection matrix E, which is defined as
1 T ψ E ¼ ψF  1 ψ T ¼ ψ ψ T ψ

(16)

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the ith diagonal element Ei is the EFI value corresponding to the ith sensor, which represents the contribution of the ith sensor location to the matrix E. As the ith sensor is deleted, the determinant of the new FIM can be expressed as

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detðFi Þ ¼ detðFÞdetðI Ei Þ

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In the iterative process, the candidate sensors can be ranked based upon their EFI values, the candidate set will be reduced to a desired number and the determinant of the FIM can be maintained in a suboptimal way. However, a drawback of EFI method for sensor placement is that it does not give any indication of the desired number of sensors. Only using k sensors to identify the k target modes is not possible due to the presence of model uncertainty, sensor noise, and possible sensor failures. It is a physically realistic number based on the common practice of having more sensors than target modes. Kammer proposed the minimum number of required sensors by the analysis that the initial ranked the importance of each sensor location within the candidate set, but it neglects the redundancy of information among the sensors. An optimal criterion is added here in order to quantify redundancy of sensors, and the desired number of sensors can be determined finally [24]. If the kth sensor is deleted by using EFI method, the redundancy information can be evaluated by considering the elementary information matrix:

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Ik ¼ φTk φk

(17)

(18)

For the rest of the remained sensors, the information matrix can be defined as Fk ¼ Fk  1 φTk φk ;

1 rk on

(19)

in which F0 is the original FIM, it is proposed here to quantify the information by a norm of matrix, then a normalization of Fk is considered, and the quantified redundancy of information can be expressed by   Fk  1  Ik     ; 1 ok o n (20) Rk ¼  Fk  1  þ Ik  The fundamental properties of matrix norm are     A  B r A þ kBk

(21)

The redundancy measure Rk is lower bound, Rk o1. The norm of matrix Rk will be decreased if more of the sensor locations reduced. We are interested in the number of sensors that can carry as much information as possible. Therefore, a threshold ϵ can be defined as the upper limit when the search algorithm deletes the sensor locations. Rk r ϵ;

k ¼ 1; 2; …; ni

(22)

in which ni is the DOFs number of the ith category. The value of ϵ is between 0 and 1. If this value is too high, then redundant degrees of freedom could not be deleted. If this value is too low, too many degrees of freedom are deleted. It is important to choose a suitable value of ϵ. The goal of the proposed sensor placement algorithm is to place sensors to maintain the dynamically important information contained in the frequency response data within the desired frequency band. It is obvious that for a FE model with several thousands of nodes, every candidate node will have six associated DOFs. It is impossible to test all combinations, even for a few numbers. A sub-optimal is required for handling a large model. The flow chart of the proposed method can be seen in Fig. 1.

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Fig. 2. Singular values for a simple beam.

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4. Numerical simulations and discussions

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4.1. Simple beam

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Compared with reference [22], a cantilever beam model is used to demonstrate the numerical performance of the proposed method. A finite element model of the cantilever beam is constructed with 30 elements, and the translational DOF at every node of the finite element mesh of the beam is a candidate sensor location. In this example, a length of the beam is 0.5307 m. The cross-section is a rectangle with the height is 0.0032 m and width is 0.019 m. Young's modulus and the mass density of the material are 7:1  1010 N=m2 and 2700 kg/m3, respectively. The first step calculates the FRFs depending on the geometric parameters and material properties. The system has 4 modes of vibration between 1 and 400 Hz. Frequency response reduction based on PCA is performed to extract the principal directions (pd). Plots of singular values shown in Fig. 2 indicate the rank nr ¼ 4. The contribution rate of the first four modes of the beam is 29.1 percent, 24.62 percent, 23.04 percent, 22.35 percent, and their accumulate contribution rate is up to 99.9 percent. The first four principal direction vectors as for the attribute, 30 DOFs for vertical direction of the beam are classified into four groups by using k-means clustering. Fig. 3(c) shows the extracted principal direction vectors. Every point represents the element value in principal direction vector, and the DOFs of a same category are represented by the same notation. The comparison of the global searching of all sensor locations and step searching of cluster sensors is presented in Table 1. In order to demonstrate the redundant sensors for each searching method, more sensors than modes of interest are selected. The locations for six sensors are shown in Fig. 3(a) by using global search approach, the clustering occurs for the locations 15 and 16, as well as 23 and 24. This measurement within a cluster is likely to be highly correlated and hence will represent a poor choice of locations. Fig. 3(b) presents the step searching method with the same number of sensors. Note that the cluster locations are not chosen, and the selected locations are well distributed on the beam. Thus, the new method is

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Fig. 3. Sensor selection of the cantilever beam: (a) global searching of EFI and (b) step searching of EFI and (c) the first four principal directions.

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Table 1 Comparison of the global searching and cluster searching using EFI.

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Method

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All sensors Global searching

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Clusters of sensors Step searching

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Number of sensors

Locations selected

1, 2,…, 29, 30 1, 2,…, 9 10, 11,…, 18 19, 20,…, 27 28, 29, 30

8, 15, 16, 23, 24, 30 7, 9 16 21, 24 30

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effective to extract the informative locations. However, in the proposed method, the clusters are divided and the searching approach is used for each category, which increases to calculate quantity.

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4.2. Secondary mirrors support vane

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The second application presented here is a support vane for space camera secondary mirrors (Fig. 4). A single cantilever plate is considered which is fixed at one edge as it is shown in Fig. 4(c). The plate contains 100 nodes (20 nodes in length and 5 nodes in width). In this example, the geometric parameters and material properties of each plate element are length a ¼0.02 m, width b¼0.02 m, thickness h¼0.005 m, Young's modulus E ¼ 1:47  1011 N=m2 , mass density ρ¼8180 kg/m3, and Poisson's ratio v ¼0.3. The FRFs matrix is calculated depending on the geometric parameters and material properties. As defined previously, it is important than an order is maintained with the numbering of the nodes and elements as this will allow easy interpretation later on. As mentioned in Section 3, in the proposed method, the PCA based frequency reduction is performed. Plots of singular values shown in Fig. 5 indicate the rank nr ¼4. The contribution rate of the first four modes of the beam is 32.26 percent, 26 percent, 22.01 percent, 18.88 percent, and their accumulate contribution rate is up to 99.15 percent. This application places 8 sensors on the plate in an attempt to capture all of the dynamically important data for specified frequency band. All of the 100 node locations are placed into the initial candidate sensor set. K-means clustering algorithm is used to divide the candidate locations. To investigate sensor placement based on different cluster number, two numerical examples with 3 clusters and 4 clusters are tested. Step searching of cluster EFI method is performed to obtain the final sensor configuration. Fig. 6 shows the divided plate and sensor placement. In both examples, the nodes marked with red circles represent the positions of selected sensors. It is apparent that most of the sensors are placed on the edge of the cantilever plate with both 3 clusters and 4 clusters. This can be attributed to the fact that the majority of the modes in the frequency range are highly localized to the edge of the surface. For the 3 clusters, clustering occurs for the locations 41 and 46, as well as 99 and 100, which generates in two adjacent groups and in the same group. When the same number of sensors is placed on the vane for the 4 clusters, the locations will be placed with much more equilibrium.

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Fig. 4. Secondary mirrors support vane: (a) FE model of secondary mirrors, (b) prototype of secondary mirrors and (c) FE of a single cantilever plate.

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Fig. 5. Singular values for a support vane.

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5. Pretest investigation

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5.1. FE model

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The pretest of a filter wheel assembly in a space camera is considered. The FE model of filter wheel assembly is shown in Fig. 7, which has 11,144 elements and 16,707 nodes. The first six modes are the rigid body modes of the filter wheel assembly, and the next mode is the rotation mode of bearing. Therefore the first flexible mode of the filter wheel assembly is from the mode 8. The low frequency modes with higher modal participation factors can give sufficient information to describe the dynamic behavior. The first three flexible modes are calculated and illustrated in Fig. 9(b). As nodes located inside the structure cannot be instrumented, the focus is on the nodes located on the surface of filter wheel.

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5.2. Pretest verification The pretest is performed to validate the efficiency of the FE model and fitness of sensor placement. The filter wheel assembly is suspended with a free–free condition by using three elastic supports, as shown in Fig. 8. For modal test of the filter wheel assembly, 36 acceleration sensors are arranged on the support frame, the outer ring and inner ring of the filter wheel structure. A force hammer is used to excite its elastic deformation. The signal acquisition device includes the LMS signal acquisition instrument and the PCB acceleration sensor, and the monitoring response signal is obtained by repeated measurements. Fig. 9(a) shows the acceleration mode shapes of the first three modes with their corresponding frequencies. The mode shapes and frequencies can be used to tune the finite element model that is presented in the previous section. In Table 2 a comparison between the frequencies obtained from FEM and the pretest identified frequencies is given.

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Fig. 6. Sensor selection of the cantilever beam with different clusters: (a) sensor locations with three clusters and (b) sensor locations with four clusters. (For interpretation of the references to color in this figure, the reader is referred to the web version of this paper.)

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Fig. 7. FE model of filter wheel assembly.

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1 3 5 7 9 11 13 Fig. 8. Model test setup of the filter wheel: (a) laboratory structure and (b) test system.

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Fig. 9. The first three frequencies and mode shapes: (a) test mode shape and (b) calculated mode shape using the finite element model. Table 2 Comparison of the first three mode frequencies by FEM and pretest.

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Modes

FEM frequencies (Hz)

Identification frequency (Hz)

Relative error (%)

1 2 3

228.77 297.53 331.8

218.89 306.13 338.05

4.51 2.81 1.85

Based on the presented comparison results between the FEM and pretest identification, it is decided that no further tuning of the finite element model is required. A selection of sensor locations will be necessary, this is however considered as the information redundancy of sensor placement at the stage of pretest.

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5.3. Test data example

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The test data comprise 36 FRFs from vertical excitation, at 512 frequencies in the range 151–350 Hz. In the measured band, the structure has 3 modes of vibration. Compound FRF matrix H , of size 36  512, is built up. Based on the analysis of PCA, the plots of singular values (Fig. 10) indicate the rank nr ¼2. However, the accumulate contribution rate of the first two singular values is 78.44 percent, which cannot contain the total energy of the system. It is decided to consider nr ¼7 and the corresponding accumulate contribution rate is up to 98.8 percent.

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Fig. 10. Singular values for the test PCA of filter wheel assembly. Table 3 Sensors cluster and selection of the filter wheel assembly.

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Groups

Number of sensors

Locations selected

1 2 3 4 5

1; 10; 11; 12; 13; 22; 23; 24; 35 2; 3; 14; 15; 36 4; 5; 6; 7; 16; 17; 18; 19 8; 9; 20; 21 25; 26; 27; 28; 29; 30; 31; 32; 33; 34

12; 22 14 6 8 27; 34

25 27 29 31 33 35 37 Fig. 11. Principal response functions of filter wheel assembly (a) extracted from 36 original test points and (b) extracted from 7 locations using OSP method.

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A 36  7 submatrix has been considered for subset selection of FRFs. Before the EFI method, a k-means clustering is used to classify the pretest points. According to the value of principal direction vectors, 36 test points are classified into five groups by using k-means clustering. Using the EFI method, the result of a step searching shows in Table 3. Seven locations have been selected in the order: 6, 8, 12, 14, 22, 27, 34 to get a similar vibration information that from the original 36 test points, which is described by a PRF plot (Fig. 11). Selection of a larger number of FRFs was necessary to compensate for the arbitrary initial rank estimate. As shown in the pretest example, the proposed method properly divides the candidate sensors region and effectively selects the independent sensors via the iterative removing process. Additionally, the comparison of the PRF through original sensor placement and the proposed method shows a highly efficient and accurate performance in the computational process of optimal subsets.

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6. Conclusion

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An optimal sensor placement based on FRFs clustering method has been proposed for planning of a vibration structural test of large-scare space camera structures. To overcome the problem of information redundancy between sensors, principal component analysis of a compound FRF matrix is introduced to extract a reduction submatrix, which is apt to be more informative and computationally efficient. Additionally, a k-means clustering algorithm is designed to classify the degrees of freedom. In the proposed method, the FE model of a structure is grouped as several clusters, and a cluster step searching of EFI is used to handle redundant DOFs for each cluster. Through the dividing and searching processes, the number of DOFs of the system is reduced. Hence, the most relevant locations are selected for the planning of physical vibration tests. The

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Please cite this article as: S. Li, et al., Optimal sensor placement using FRFs-based clustering method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.004i

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S. Li et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

numerical analysis and dynamic pretest show the efficiency of the present method and the selected sensors distribute more balanced on the whole structure. When more number of groups is divided, the locations tend to be more refined, which increases the amount of calculation. Moreover, the proposed method helps searching an linearly independent optimal subset, which can be used in parameter identification, model updating and many other applications.

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Appendix A. Supplementary data

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Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jsv.2016. 09.004.

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Please cite this article as: S. Li, et al., Optimal sensor placement using FRFs-based clustering method, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.09.004i