Model updating of lattice structures: A substructure energy approach

Mechanical Systems and Signal Processing 25 (2011) 1469–1484

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Model updating of lattice structures: A substructure energy approach Hui Fang a,b, Tie Jun Wang a, Xi Chen a,c,d,n a

MOE Key Laboratory for Strength and Vibration, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China Institute of Structural Mechanics, China Academy of Engineering Physics, Mianyang 621900, China Department of Civil & Environmental Engineering, Hanyang University, Seoul 133-791, Korea d Department of Earth and Environmental Engineering, Columbia University, New York, NY 10027, USA b c

a r t i c l e i n f o

abstract

Article history: Received 28 June 2009 Received in revised form 30 October 2010 Accepted 7 January 2011 Available online 2 February 2011

Model updating is an inverse problem to identify and correct uncertain modeling parameters, which leads to better predictions of the dynamic behavior of a target structure. This study presents a direct physical property adjustment method and the substructure energy approach, which can accurately update the stiffness and mass matrices of lattice structures using incomplete eigenvectors measured from critical substructures. For validation, the proposed method is applied to update models of a mass-spring system, a two-dimensional, and a three-dimensional lattice structure. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Model updating Modal analysis Lattice structure

1. Introduction With scalable geometry and repetition characteristics, lattice structures are widely employed in aerospace, mechanical, marine, and civil systems [1]. Their structural properties depend on the actual size of the unit cell, enabling applications in both the small- and large-scale structural systems [2]. For the purpose of evaluating the mechanical integrity of a lattice structure, numerical (mathematical) model needs to be established via the use of the finite element (FE) method. In order to improve the correlation between the FE model and experimentally measured data, model updating is emerging as one of the most important topics in the area of modal testing [3], and the deduced optimal structural models can be used for structural health monitoring applications [4,5]. In essence, model updating identifies and corrects uncertain modeling parameters, and integrates the information into a FE model that could better evaluate the dynamic behaviors of a target structure. There are two broad approaches for updating the system matrices based on the type of parameters that need updating as well as the available data measured from experiment: (a) updating from modal data and (b) updating directly from the force-response measurements. This study focuses on the first category. The conventional modal-based updating approaches usually rely on either the matrix adjustment method or the physical property adjustment method [6,7]. The first method computes directly the changes to the mass and stiffness matrices, and the resulting models become ‘‘abstract representations’’ and cannot be interpreted in a physical way [8]. The second method seeks physical quantities for each individual element or associated parameter; this method is closely related to the model-based methods for damage determination, which can be used to quantify the location and extent of damage [9]. With respect to the physical property adjustment method, there are two widely used approaches: (a) iterative methods and (b) non-iterative methods. For iterative methods, spatially complete measured modes are not necessary, but there are

n

Corresponding author at: Department of Earth and Environmental Engineering, Columbia University, New York, NY 10027, USA. E-mail address: [email protected] (X. Chen).

0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.01.002

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several disadvantages: a very expensive computation effort is required, the modal information must be obtained from the same order mode of both original and real models, and analytical and measured modes need to be consistent in scale [10]. To avoid these problems, Hu et al. [11] developed a non-iterative physical property adjustment method that can update the stiffness and mass matrices simultaneously. This method, however, requires the spatial complete modal data to be measured from the updated structure. When this method is applied to large-scale structures such as lattice structures, it is not practical to identify all of the unknown parameters in the structure at the same time since ill-conditioning and nonuniqueness in the solution of inverse problems appear as inevitable difficulties, and it is not practical to measure all degrees of freedom of the structural modes [12]1 since the measurement may be limited to a small number of locations of the actual structure. An alternative process to perform a non-iterative physical property adjustment approach is by applying one algorithm to identify the critical subsystems with uncertain modeling parameters (first step), and then implementing another algorithm to estimate the uncertain modeling parameters after the critical subsystems have been identified (second step). Several effective substructure identification methods [12–14], which are well established in the literature, may be used for the first step of identifying the critical parts (subsystems) with uncertain physical parameters. The complete structure is divided into several substructures and the analysis is concentrated on a critical subsystem with a smaller number of elements or associated uncertain parameters. After the critical parts are identified, one may utilize the modal information of these parts to estimate the correction coefficients of the uncertain modeling parameters. However, for such a second step (which is the focus of the current research), the existing non-iterative physical property adjustment methods (e.g. [11,15]) still require the knowledge of the complete modal information of the entire structure, which is not only impractical but also makes the previous substructure identification process ineffective. To the best of the authors’ knowledge, a non-iterative physical property adjustment method is not yet available, which could directly utilize the spatially incomplete modal information to update modeling parameters. In this paper, the main objective is to develop a non-iterative physical property adjustment method, the substructure energy approach (SEAp), which can accurately estimate the stiffness and mass matrix correction factors for lattice structures using spatially incomplete measurements (after the critical subsystems have been identified using established methods, e.g. [12–14]). The model updating method involves a set of linear simultaneous equations that are deduced from the energy functional of substructure models and substructure modes; the correction factors can be solved without iteration. The accuracy and effectiveness of the proposed SEAp are validated using three numerical examples, a mass-spring system that has been studied recently by many researchers using various updating approaches [8,11,15,16], a two-dimensional (2D) lattice structure and a three-dimensional (3D) lattice structure where for each element the mass and stiffness correction factors are determined from SEAp based on the incomplete measured modal information. 2. Substructure energy approach (SEAp) The structure of the beam-type lattice structure is composed of a sequence of identical unit cells repeating along the axial direction (and also across the thickness if necessary). Each cell is composed of beam elements. In the following text, the term ‘‘analytical model’’ refers to the original, unmodified model, and the term ‘‘true model’’ refers to the real structure whose information is measured at critical locations, which provides the basis for correcting the analytical model. The schematic of the lattice system with identical unit cells is shown in Fig. 1. The total energy functional of the analytical model under consideration may be expressed as

C¼

N X

cj

ð1Þ

j¼1

where cj is the energy functional of the jth substructure and N the total number of substructures. In general, the substructure energy functional may be defined as

cj ¼ 12 dTj Kj dj 12o2 dTj Mj dj dTj Fj

ð2Þ

where o2Mjdj is considered as the inertia force in addition to the loading factor. Kj and Mj denote the stiffness and the mass matrices of the jth substructure, dj and Fj denote the displacement and loading vectors for the jth substructure, respectively, and o is the circular frequency of vibration. For the analytical lattice structure, the stiffness and mass matrix of the substructure is repetitive, i.e. Kj = Ksub and Mj = Msub. Let Kej and Mej be the stiffness and mass matrices of the jth substructure of the true (experimental) model,

1 If a structural model is available, the mode may be expanded to obtain the spatially complete version; however, expanding mode shape may bring in additional error into the mode information [2].

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Fig. 1. Schematic of the system with linear periodic structure (A), and substructure unit (B).

respectively.2 They are modified from Kj and Mj: Kej ¼ Ksub 

Dj X

0 o ajz r 1

ajz kjz ,

z¼1

Mej ¼ Msub 

Dj X

ð3Þ

bjz mjz , 0 o bjz r1

z¼1

where kj,z and mj,z are the stiffness and mass of the zth element in the jth substructure in global coordinates, respectively, and their corresponding correction factors are aj,z and bj,z, respectively. Dj is the total number of ‘‘elements’’ in the jth substructure. A residual force [17] is employed to reflect the effect of modification of stiffness and mass matrices: FRe j ¼

Dj X

ðaj,z kj,z o2 bj,z mj,z Þdej

ð4Þ

z¼1

where dej denotes the substructure displacement of the true model. Substituting Eqs. (3) and (4) into Eq. (2), we obtain the substructure energy functional of the true model as e 1 2 eT e eT eT Re cej ¼ 12 deT j Ksub dj 2o dj Msub dj dj Fj dj Fj

For a lattice structure the displacement vector ( )e

dej 

dL dR

ð5Þ

dej

is made up of the left and right nodal displacement vectors, i.e. ð6Þ

j

where {dL}j and {dR}j denote the nodal displacement vectors on the common boundary between j  1, j+ 1, and jth substructures, respectively (see Fig. 1). The continuity condition is expressed as fdL gejþ 1 ¼ fdR gej ,

j ¼ 1,2,. . .,N

ð7Þ

dej

Apply the U-transformation [18] to and Fj (which in general contain multiple degrees of freedom), namely pffiffiffiffiffiffiffi pffiffiffiffi PN pffiffiffiffi PN iðj1Þr t ¼ ð1= NÞ k ¼ 1 e qr and Fj ¼ ð1= NÞ k ¼ 1 eiðj1Þrt f r , where t =2p/N and i ¼ 1. Substituting into Eqs. (5)–(7), we get the energy functional of the true model in U-domain

dej

cer ¼ 12 qTr Ksub qj 12o2 qTr Msub qj qTr f r qTr f Re r

ð8Þ

and the continuity condition as follows: fqR gr ¼ eikt fqL gr

ð9Þ

Substituting Eqs. (8) and (9) into Eq. (1), the total energy functional of the true model is deduced. When Fj vanishes, by the equation of variation, dCd =0, it follows that Re

ðKsub o2 Msub ÞfqL gdr ¼ f r where

Ksub

¼ TTr Ksub Tr ,

Msub

¼ TTr Msub Tr ,

ð10Þ fqL gdr

¼ TTr qdr ,

and

Re fr

Re ¼ TTr f r ,

with Tr ¼ ½ IL

eirt IR  . Here, IL and IR are the unit T

dimensions of the vectors fqL gdr and fqR gdr , respectively. pffiffiffiffi pffiffiffiffi P P iðj1Þr t e dj and f r ¼ ð1= NÞ Nj¼ 1 eiðj1Þrt Fj , f, qr ¼ ð1= N Þ N j¼1e

matrices and their orders are in agreement with the Applying the inverse U-transformation to q and expanded governing equations for free vibration of the true model can be written as

dej ¼

Np X p¼1

2

Bj,p

Dp X

ðap,z kp,z o2v bp,z mp,z Þdep þ

z¼1

Nb X

Bj,b ðKb o2v Mb Þdeb

b¼1

Throughout the paper, the superscript ‘‘e’’ is used to indicate a variable that belongs to the ‘‘true’’ (experimental) model.

the

ð11Þ

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where Bj,p ¼ ð1=NÞ

PN

r¼1

eiðjpÞrt Tr ðKsub o2 Msub Þ1 TTr is referred to as the harmonic coefficient matrix. dej and dep denote the

associated eigenvector of the jth and pth substructure, respectively, and deb denotes the associated eigenvector of the bth border substructure. Np is the total number of (critical) substructures that need updating in the system, and Nb is the total number of the border substructures. Dp is the number of elements in the pth critical substructure. Note that in the derivation above, the stiffness and mass matrices on the border of the structure are treated in a way that is similar to external loads. This simplifies the updating process of the lattice structure. According to Eq. (11), the eigenvector for the border substructure, deb , can be written as Np X Dp X

deb ¼

Nb X

Ib 

p¼1z¼1

!1 Bb,p ðap,z kp,z o2v bp,z mp,z Þdep

Hb,b ðKb o2v Mb Þ

ð12Þ

b¼1

Substitute Eq. (12) into Eq. (11), we get3

dej,z ¼

Np X Dp X

e ðap,z HKp,z bp,z HM p,z Þdp,z

ð13Þ

p¼1z¼1

where HKp,z

¼ Bj,p kp,z þ

Nb X

Bj,b Kb Ib 

b¼1

¼o

2@



2 v Mb

Bb,b Kb o

!1 

Bb,p kp,z

b¼1

0 HM p,z

Nb X

Nb X

Bj,p mp,z þ

b¼1

Bj,b Mb Ib 

Nb X

1

!1 Bb,b ðKb o

2 v Mb Þ

Bb,p mp,z A

b¼1

where dej,z is the eignvector of the zth element in the jth substructure, and dep,z is the eigenvector of the updated element (of the pth substructure). According to Eq. (13), we can directly use dep,z to solve the correction vectors of the stiffness and mass matrices a and b. By multiplying Eq. (13) by dp,z (the eigenvector of the corresponding element from the analytical model), and using new indices P and Z to replace (p, z), Eq. (13) becomes eP ¼

ND X

ðcZ aZ þ dZ bZ Þ

ð14Þ

Z¼1 e where eP ¼ dTp,z dep,z , cZ ¼ dTp,z HKp,z dep,z , dZ ¼ dTp,z HM p,z dp,z , and ND ¼

E ¼ Ca þDb

PNp

p¼1

Dp . In a matrix form, ð15Þ

where C and D are the Nq  ND matrices, E is a column vector of size Nq, a and b are the column vectors of size ND. It is worth mentioning that dp,z and dep,z can be arbitrary modes of the analytical and true models, and they are not required to start from the first mode. When Nw modes from the analytical model and Nv modes from the corresponding true model are available, there are Nq = NDNwNv equations from Eq. (14). When Nq is greater than ND  2, an unweighted least-square estimation of the correction factors can be obtained as fa

b gT ¼ ð½ C D T ½ C D Þ1 ½ C D T E

ð16Þ T

That is, by using a small number of measurement locations of lattice structure, the correction factors f a b g can be readily estimated. In essence, the SEAp obtains the optimization via solving a set of equations in the leastsquare mode. As will be demonstrated below, the proposed approach works very well for structures made of identical unit cells, for those consist of non-identical unit cells, one can adopt the residual force method described in [17] to make those nonidentical units identical, which is presented in the article from Eq. (11) to Eq. (13). Comparing with the iterative methods, the present approach has the advantage of not requiring iteration and thus the possibilities of divergence and excessive computation are eliminated. Another feature of the SEAp is that it does not require to pair or scale the measured and analytical modes. Moreover, an individual SEAp equation can be weighted differently. The method does not require the complete eigenvector, allowing one to concentrate on identifying only the critical locations/substructures [12], which is the crucial feature of this method. 3 As shown in Eq. (13), the coefficient matrices HKp,z and HM p,z are relevant with the cell number, and are not influenced by the nodal points and symmetry.

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3. Finite element implementation and numerical examples Numerical examples are given below to illustrate the procedure of implementing the SEAp into FE simulations, as well as to validate the accuracy of the proposed method for correcting the stiffness and mass coefficients based on measured eigenvectors and modal frequencies. Both the analytical model and the ‘‘measured’’ true model information are generated from the same FE mesh with different sets of stiffness and mass coefficients. It is assumed that the analytical model is with a ‘‘wrong’’ set of coefficients (K and M), and the goal is to revise them from the measured (true) modal information, which is simulated from the same finite element protocol with ‘‘correct’’ coefficients (Ke and Me). Such a numerical verification procedure is widely used in literature [11,15]. In this work, it is assumed that in case of irregular difference between the analytical and true models, the critical substructures are already identified using developed techniques (e.g. [12–14]), and the main focus is to use the incomplete modal information of the critical substructures to update relevant system parameters. Three specific structural models, a Five-DOF mass-spring system, a two-dimensional (2D) beam structure, a twodimensional (2D) lattice structure, and a three-dimensional (3D) lattice structure, are chosen for the numerical demonstration. All computations in this paper are performed using Matlab. More examinations of the proposed SEAp with real dynamic laboratory tests will be carried out in future. 3.1. Five-DOF mass-spring system Consider a mass-spring system with one free end, as shown in Fig. 2, where mj denotes the mass coefficients and kj (j = 1, 2, y, 5) denotes the spring stiffness constant. The subsystem in Fig. 2B comprises one mass and one spring. The similar system was studied by many authors [8,11,15,16]. Using FE approach, the stiffness matrix and the mass matrix of the jth subsystem are, respectively " #   kj kj mj 0 Kj ¼ and Mj ¼ ð17Þ kj kj 0 0 The system stiffness matrix K and mass matrix M can be assembled as 2 3 k1 0 0 0 k1 6 k k1 þk2 k2 0 0 7 1 6 7 5 X 6 7 7 k þ k k 0 0 k Kj ¼ 6 K¼ 2 2 3 3 6 7 6 j¼1 k3 þ k4 k4 7 0 k3 4 0 5 0

0

0

k4

ð18Þ

k5

and 2

m1 6 0 6 5 X 6 Mj ¼ 6 M¼ 6 0 6 0 j¼1 4 0

0 m2

0 0

0 0

0

m3

0

0

0

m4

0

0

0

3 0 0 7 7 7 0 7 7 0 7 5 m5

ð19Þ

The SEAp may update any selected coefficient(s). The mass and stiffness coefficients of the analytical model are chosen to be m1 = 675 kg and other mj = 220 kg, and all kj = 3.55e10 N/m (j = 1, y , 5). Eigenvalue analysis shows that the first five structural frequencies are 255.30, 741.00, 1154.20, 1454.40, and 1612.20 Hz. Note that in this section, the mass and stiffness coefficients of the analytical model are chosen randomly: for a lattice structure with some disordered substructures, the stiffness and mass matrices of the disordered substructure (including the ‘‘defective’’ and border substructures) may be treated as the loading vectors (residual force [17] in governing Eq. (11)) for a perfect lattice structure.

Fig. 2. Five-DOF mass-spring system (A) and subsystem unit (B).

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First, consider the case of irregular difference between the coefficients of the analytical and true models in the 3rd, 4th, and 5th subsystem, i.e., ce= (3, 4, 5) (hereafter ce indicates the substructure number of the true structure) is investigated. According to Eq. (12), the measurement locations must include the substructures with different sets of coefficients. Therefore, the measurement locations include the 3rd, 4th, and 5th subsystems, i.e., s= (3, 4, 5) (hereafter s indicates the number of measured substructure). The values of aZ are assigned by using a normal random number generator in Matlab with the mean equal to 0 and standard deviation equal to 0.2. Similarly, bZ is assigned randomly with a mean of 0 and standard deviation of 0.1. The preset values of aZ and bZ are given in Fig. 3. In what follows, we let w and v to be the mode number of interest for the analytical and the true structures, respectively. Suppose w= (1, 2) and v =(1), then six simultaneous linear equations can be established to solve for the six correction coefficients (Fig. 3). Fig. 3 shows that the estimated stiffness correction coefficients aZ (Z = 3, 4, 5) and mass correction coefficients bZ (Z= 3, 4, 5) are in good agreement with the present (true) values. The validation shows that the SEAp can handle an irregular correction properly.

Fig. 3. Comparison between the preset and estimated correction coefficients aZ and bZ of a mass-spring system with irregular changes.

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Fig. 4. Model of the 2D beam structure: 5 elements (A) and 10 elements (B).

3.2. Two-dimensional (2D) beam structure The beam under investigation is of length 1 m, width 25 mm, and thickness 50 mm. Its Young’s modulus is taken as 205 GN/m and mass density 7860 kg/m3 (or mass density per unit length 9.83 kg/m). In this scenario, we consider a special case where the analytical model to be updated and the ‘‘true’’ model have different mesh density. Following the standard formulation for a uniform 2D beam element, one can derive the element stiffness matrix Kj and element mass matrix Mj. For the analytical model, there are 5 elements and 5 nodal points (each nodal point has 3 DOFs), as shown in Fig. 4(A). For the true model, there are 10 elements and 10 nodal points (each nodal point has 3 DOFs) (Fig. 4(B)). 3.2.1. Irregular difference Consider the case that the analytical model has irregular errors in the 2nd and 3rd elements (subsystems), i.e., ce = (2, 3). The values of a and b are assigned using random numbers with mean equal to 0 and standard deviation equal to 0.25 and 0.20, respectively. The true model has 10 elements; therefore, the ‘‘defective’’ locations include the 3rd–6th elements, i.e., s= (3, 4, 5, 6). Because the analytical model and the ‘‘true’’ model have different mesh, the modes for the true structures are not employed in this scenario. By multiplying Eq. (13) by dep,z , and using new indices P and Z to replace (p, z), Eq. (13) becomes ePu ¼

ND X

ðcZu aZ þdZu bZ Þ

ð20Þ

Z¼1 e eT e eT K M e where ePu ¼ deT p,z dp,z , cZu ¼ dp,z Hp,z dp,z , dZu ¼ dp,z Hp,z dp,z , and ND ¼

Eu ¼ Cua þ Dub 0

0

PNp

p¼1

Dp . In a matrix form ð21Þ

0

where C and D are Nq  ND matrices, E is a column vector of size Nq, a and b are the column vectors of size ND. When Nq is greater than ND  2, an unweighted least-square estimation of the correction factors can be obtained as   T f a b g ¼ ð½ Cu Du T Cu Du Þ1 ½ Cu Du T Eu ð22Þ According to Eq. (20), the first mode for the true structure, i.e., v = (1), is utilized. A set of SEAp equations are established, which are sufficient to solve the 8 correction coefficients. Comparisons between the preset and estimated stiffness and mass correction coefficients for the irregular case are shown in Fig. 5. The maximum error is in the order of 10  5, which suggests that the SEAp works well for handling irregular variations. 3.2.2. Uniform and nearly uniform differences Suppose the differences between analytical and true models are uniform or nearly uniform. To be consistent with the above example, for the uniform scenario, we assume that all aZ = a0 =0.25, and all bZ = b0 = 0.2, for Z =3–6. For v= (1), the solutions of the estimated stiffness and mass correction coefficients are shown to be close to the assigned values in Fig. 6, with the maximum error in the order of 10  5. For the nearly uniform scenario, small random deviations with standard deviations of 0.02 and 0.01 are added to a0 and b0, respectively. Fig. 7 shows that the SEAp works well for nearly uniform differences with the maximum error in the order of 10  5. 3.3. Two-dimensional (2D) lattice structure The 2D lattice structure under investigation is composed of 10 cells along the beam length and 1 cell across the thickness. The lattice structure is modeled using 40 beam elements, and the element numbers are given in Fig. 8. For each

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Fig. 5. Comparison between the preset and estimated correction coefficients aZ and bZ of a 2D beam structure with irregular changes.

beam, its length is 1 m and its cross-section is square of 5 cm by 5 cm. The material is aluminum (with Young’s modulus 71 GPa and density 2700 kg/m3). The out of plane thickness of the structure is uniform, 5 cm. Following the standard FEM approach, the element stiffness matrix kj,z and element mass matrix mj,z can be computed locally (where j is the substructure number and z is the local element number) and then assembled to form the global P P10 P4 P4 matrices, K ¼ 10 j¼1 z ¼ 1 kj,z and M ¼ j¼1 z ¼ 1 mj,z , respectively, for the analytical model. The true model counterP P10 P4 P4 e a k and M ¼ parts are Ke ¼ 10 j¼1 z ¼ 1 j,z j,z j¼1 z ¼ 1 bj,z mj,z . There are 20 nodal points and each nodal point has 2 DOF; thus both the stiffness and mass matrices are of size 40  40. 3.3.1. Irregular difference In this scenario, we assume that the irregular differences occur between the coefficients of the analytical and true models, in multiple critical substructures ce =(6, 7, 10). The values of aZ (Z =21–28 and 37–40) and bZ are assigned using random numbers with mean equal to 0 and standard deviation equal to 0.15 and 0.25, respectively. According to Eq. (14), one takes the first 2 modes for the analytical structures, i.e., w =(1, 2) , and the first one mode for the true structures, i.e., v = (1). The measurement locations must include the substructures with different sets of coefficients, i.e., s =(6, 7, 10); thus, a total of 24 SEAp equations are established, which are sufficient to solve the 24 correction coefficients.

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Fig. 6. Comparison between the preset and estimated correction coefficients aZ and bZ of a 2D beam structure with uniform changes.

Comparisons between the preset and estimated stiffness and mass correction coefficients for the irregular case are shown in Fig. 9. The maximum error is in the order of 10  5, which suggests that the SEAp works well for handling irregular variations. For similar lattice structures, although the cross-model cross-mode (CMCM) method [11,15] could also yield correct results, however, the CMCM method requires the whole eigenvector of the structure (including all substructures) to be available, which may not be practical for large structures.4 Using the SEAp, only partial information of the eigenvector is required (which includes the substructures that have difference between the analytical and true models), and then the stiffness and mass matrix correction factors can be accurately estimated. 3.3.2. Uniform and nearly uniform differences Suppose the differences between analytical and true models are uniform or nearly uniform, in this case, one may feel free to choose any critical substructure for measurement. To be consistent with the above example, we continue to examine several critical substructures, i.e., ce= s= (6, 7, 10). 4 Some researchers propose to expand the mode shapes based on the available partial mode information [2]; however, the expansion of mode shape may bring additional error into the mode information.

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Fig. 7. Comparison between the preset and estimated correction coefficients aZ and bZ of a 2D beam structure with near-uniform changes.

Fig. 8. Sketch of the 2D lattice structure (A) and cell (substructure) (B).

For the uniform scenario, assuming that all aZ = a0 = 0.2, and all bZ = b0 = 0.1, for Z =21–28 and 37–40. For w= (1–2) and v= (1), the solutions of the estimated stiffness and mass correction coefficients are shown to be close to the assigned values in Fig. 10, with the maximum error in the order of 10  5.

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Fig. 9. Comparison between the preset and estimated correction coefficients aZ and bZ of a 2D lattice structure with irregular changes.

Fig. 10. Comparison between the preset and estimated correction coefficients aZ and bZ of a 2D lattice structure with uniform changes.

For the nearly uniform scenario, small random deviations with standard deviations of 0.01 and 0.005 are added to a0 and b0, respectively. Fig. 11 shows that the SEAp works well for nearly uniform differences with the maximum error in the order of 10  5.

3.4. Three-dimensional (3D) lattice structure In this section, numerical examples are given below to demonstrate the SEAp for 3D lattice structures. The considered structure features a plate-type lattice structure [19] with elements arranged according to a cubic configuration, as shown in Fig. 12. The structure is obtained by the assembly of 9  9 identical cells, where each structural member is modeled as a three-dimensional uniform beam element (whose length is 0.1 m in the x, y, and z directions) and is distinguished by assigning an element number. Due to space limitation, here we show only the number of the members with uncertain parameters. In the analytical model, the material is assumed to be aluminum, and the cross-section area and the associated moment of inertia for all members are A= 2.83  10  5 m2 and I= 6.36  10  11 m4, respectively. Using standard FEM approach, the local element stiffness matrix kjr,z and element mass matrix mjr,z can be computed (where j and r denote the substructure number along x and y directions, respectively), and then the global matrices are P9 P9 P P P P12 assembled for the analytical model, K ¼ 9j ¼ 1 9r ¼ 1 12 z ¼ 1 kjr,z and M ¼ j¼1 r¼1 z ¼ 1 mjr,z , respectively. The true P P P P P P 9 9 12 e model counterparts are Ke ¼ 9j ¼ 1 9r ¼ 1 12 a k and M ¼ b m jr,z jr,z jr,z . There are 60 nodal points jr,z z¼1 j¼1 r¼1 z¼1 and each nodal point has 6 DOF; thus both the stiffness and mass matrices are of size 360  360. The numbers of potential critical elements are 118, 121, and 122, where element 118 is an x-axis member, element 121 is a y-axis member, and element 122 is a z-axis member, as shown in Fig. 12. We consider the following scenarios of the

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Fig. 11. Comparison between the preset and estimated correction coefficients aZ and bZ of a 2D lattice structure with near-uniform changes.

Fig. 12. Sketch of the 3D lattice structure (A) and cell (substructure) (B).

true structure: (A) elements 118 and 121 are defective, (B) elements 118 and 122 need updating (defective), and (C) elements 121 and 122 are critical (defective). 3.4.1. Irregular difference In this subsection, we consider irregular differences occurring between the coefficients of the analytical and true models in the three true models (scenarios A–C). The values of aZ and bZ (Z= 118, 121, and 122) are assigned using random numbers with mean equal to 0 and standard deviation equal to 0.3 and 0.1, respectively. The critical and measurement locations include the substructures cex = sx = (3) and cey =sy = (1–9), respectively (subscripts x and y denote the substructure along x and y directions, respectively), for Z = 97–144. For w=(1, 2) and v= (1), the solutions of the estimated stiffness and mass correction coefficients of models A–C are shown to be close to the assigned values in Fig. 13(A–C), with the maximum error in the order of 10  9. 3.4.2. Uniform and nearly uniform differences When the differences between analytical and true models are uniform or nearly uniform, the measurement can be carried out at any substructure. To be consistent with above, several critical substructures under investigation are still cex = sx = (3) and cey =sy = (1–9), for Z =97–144.

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Fig. 13. Comparison between the preset and estimated correction coefficients aZ and bZ of a 3D lattice structure with irregular changes. Three different true structures are considered (same for Figs. 14 and 15): (A) elements 118 and 121 are defective, (B) elements 118 and 122 need updating, and (C) elements 121 and 122 are critical.

For the uniform scenario, assuming that all aZ = a0 =0.25 and all bZ = b0 = 0.05 (Z= 118, 121, and 122). For w= (1–2) and v= (1), the solutions of the estimated stiffness and mass correction coefficients agree well with the assigned values in Fig. 14(A–C), with the maximum error in the order of 10  9. For the nearly uniform scenario, small random deviations with standard deviations of 0.02 and 0.01 are added to a0 and b0, respectively. Fig. 15(A–C) shows that the SEAp provide satisfactory results with the maximum error in the order of 10  9. When the elements with error occur in another critical substructure, one may also obtain the correct results of a and b using different combinations of analytical and true mode shapes. Due to space limitation of the paper, the focus of the present paper is to introduce the substructure energy approach, and we did not show the scenarios with different choices

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Fig. 14. Comparison between the preset and estimated correction coefficients aZ and bZ of a 3D lattice structure with uniform changes.

of both critical substructures and different combinations of analytical and true mode shapes. More systematic calculations of structures involving different structural configurations, substructures, mode shapes, boundary conditions, etc., will be published elsewhere. 4. Conclusion This paper presents a new substructure energy approach that effectively and accurately estimates the stiffness and mass matrix correction factors for beam-type lattice structures based on incomplete measurements of critical substructures. The method does not require iteration and its accuracy is numerically verified by a mass-spring system, a two-dimensional (2D) lattice structure, and a three-dimensional (3D) lattice structure. The method has the potential to

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Fig. 15. Comparison between the preset and estimated correction coefficients aZ and bZ of a 3D lattice structure with near-uniform changes.

evaluate the damaged elements of beam-type (or other repetitive) lattice structures. The method may be extended to planar and three-dimensional structures in future (including those at small scale such as nanoporous materials [20–22]), with demonstrated applications to different substructures, mode shapes, boundary conditions, etc. Future work also will involve a more systematic characterization of the effect of measurement error and noise.

Acknowledgments This work was supported by the National Natural Science Foundation of China (50928601 and 11002133), National Science Foundation (CMMI-0643726), World Class University program through the National Research Foundation of Korea

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