Calculation of eigenvalue and eigenvector derivatives with the improved Kron s substructuring method

Transcription

1 Structural Engineering and Mechanics, Vol. 36, No. 1 (21) Calculation of eigenvalue and eigenvector derivatives with the imroved Kron s substructuring method *Yong Xia 1, Shun Weng 1a, You-Lin Xu 1b and Hong-Ping Zhu 2c 1 Deartment of Civil & Structural Engineering, he Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong 2 School of Civil Engineering & Mechanics, Huazhong University of Science and echnology, Wuhan, Hubei, P.R. China (Received August 31, 29, Acceted Aril 2, 21) Abstract. For large-scale structures, the calculation of the eigensolution and the eigensensitivity is usually very time-consuming. his aer develos the Kron s substructuring method to comute the firstorder derivatives of the eigenvalues and eigenvectors with resect to the structural arameters. he global structure is divided into several substructures. he eigensensitivity of the substructures are calculated via the conventional manner, and then assembled into the eigensensitivity of the global structure by erforming some constraints on the derivative matrices of the substructures. With the roosed substructuring method, the eigenvalue and eigenvector derivatives with resect to an elemental arameter are comuted within the substructure solely which contains the element, while the derivative matrices of all other substructures with resect to the arameter are zero. Consequently this can reduce the comutation cost significantly. he roosed substructuring method is alied to the GAREUR AG-11 frame and a highway bridge, which is roved to be comutationally efficient and accurate for calculation of the eigensensitivity. he influence of the master modes and the division formations are also discussed. Keywords: substructuring method; eigensolution; eigensensitivity; model udating. 1. Introduction Finite element (FE) model udating technology has been extensively develoed in aerosace, mechanical and civil engineering. It can serve for structural modification, model tuning, and damage identification (Friswell 1995). Model udating methods are usually classified into one-ste methods and iterative methods (Brownjohn 21). he one-ste methods directly reconstruct the global stiffness matrix and the mass matrix, while the iterative methods modify the hysical arameters in the FE model iteratively to realize an otimal match between the analytical modal roerties (such as the frequencies and the modal shaes) and the measurements. he latter aroach has been becoming more oular because they allow hysical meaning of the obtained modifications and can *Corresonding author, Assistant Professor, ceyxia@olyu.edu.hk a Ph. D Student, r@olyu.edu.hk b Chair Professor, ceylxu@olyu.edu.hk c Professor, hzhu@mail.hust.edu.cn

2 38 Yong Xia, Shun Weng, You-Lin Xu and Hong-Ping Zhu reserve the symmetry, the ositive-definiteness and the sarseness in the udated matrices. However, one drawback of the iterative methods lies in that the eigensolutions of the analytical model and their associated sensitivity matrices usually need to be calculated in each iteration (Bakir 27). Eigensensitivity is usually calculated in the global structure level. Fox and Kaoor (1968) firstly utilized the modal method to determine the eigenvalue and eigenvector derivatives by considering the changes of the hysical arameters in the mass and stiffness matrices. he disadvantage of this method lies in that all modes of the system are required, which is comutationally exensive for large-scale structures. Nelson (1976) roosed a more efficient method to calculate the eigenvector derivatives by using the modal arameters of that mode solely. Lin et al. (1995, 1996b) further imroved the comutation efficiency of the Nelson s method, by combining the inverse iteration technique, the singular value decomosition theory and the model reduction technique. he Nelson s method has also been develoed to treat with the rigid body modes, the close or reeated modes by some researchers (Lin 1996a, Song 1996, Wu 27). Since calculation of the eigensensitivity usually dominants comutation time during the iterative model udating rocess, how to calculate the eigensensitivity efficiently becomes a big challenge for the researchers. he substructuring technology can be a romising solution to accelerate the calculation of the eigensensitivity for large-scale structures. In general, the substructuring methods include three stes: first, the global structure is torn into some manageable substructures according to some division criteria; second, the substructures are analyzed indeendently to obtain the designated solutions (for examle, the eigenairs and eigensolution derivatives); finally, the solutions of the substructures are assembled to obtain the roerties of the global structure by imosing constraints on the interface of the adjacent substructures (Yun et al. 1997). With the substructuring method, the eigensolutions and the eigensensitivity of the modified substructures are reeatedly analyzed, while the unmodified substructures are unchanged during the iterative model udating rocess. In addition, the substructuring method is exected to be more efficient when it is incororated with the arallel comutation (Fulton 1991) or the model reduction techniques (Choi et al. 28, Xia and Lin 24). Hurty (1965) and Craig-Bamton (1968, 2) develoed a substructuring method based on the constraint modes with the fixed-interface condition of the substructures, while MacNeal (1971) and Rubin (1975) roosed a substructuring method based on the attachment modes with the freeinterface condition. Qiu (1997) exressed the dislacement of the substructures with the combination of the fixed interface modes and the free interface modes. Based on the different boundary conditions, Heo and Ehmann (1991) and Lallemand et al. (1999) derived the eigensolution derivatives by using the fixed-interface substructuring method and the free-interface substructuring method, resectively. he constraint modes or the linked force are required beforehand to construct the eigensensitivity formula. Gabriel Kron (1968) initiated a substructuring method to study the eigensolutions of the systems with large number of variables in a iece-wise manner. he Kron s method has a concise form, and has been develoed by a few researchers (Simson 1973, Simson and abarrok 1968, Sehmi 1986). Recently, the Weng and Xia (27) roosed a modal truncation technique to transform the original Kron s substructuring eigenequation into a simlified form, and imroved the comutation efficiency of the Kron s substructuring method. Only some lower eigenmodes of the substructures are retained as the master modes in the technique, while the higher modes are discarded and comensated with the residual flexibility. he method can achieve high recision with the second-order residual flexibility, or even high-order

3 Calculation of eigenvalue and eigenvector derivatives 39 residual flexibility. he imroved Kron s substructuring method is extended in this aer, to derive the first-order derivatives of the eigensolutions with resect to a structural arameter. With the roosed substructuring method, the derivatives matrices of the eigensolutions and the residual flexibility with resect to the elemental arameter are comuted within a articular substructure, while the derivative matrices of the other substructures are zero. he eigensensitivity of the global structure with resect to the elemental arameter is recovered from the derivative matrices of the articular substructure. Since the residual flexibility is symmetric and directly related to the stiffness matrix, the first-order and high-order eigensensitivity can be calculated by directly re-differentiating the eigenequation with resect to the structural arameter. o verify the effectiveness and the accuracy of the roosed technique, the eigensensitivity formula is alied into the GAREUR AG-11 structure and a highway bridge. 2. Basic theory Generally, the global structure with N degree of freedoms (DOFs) is firstly divided into NS indeendent substructures. he jth (j =1, 2,, NS) substructure with n (j) DOFs has the submatrices K () j and M () j, and the associated n (j) eigenairs as () () j () j () j = Diag[ λ 1, λ 2,, λ ( j) n ], Φ j Λ j [ Φ () j ] K () j Φ j () Λ j () φ () j () j 1 φ 2 = [,,, φ ( j) n ] () Φ () j I j = (), [ Φ () j ] M j = () (1) he eigensolutions of the substructures are diagonally assembled into the rimitive form as Λ = Diag[ Λ ( 1), Λ ( 2),, Λ ( NS) ], Φ = Diag[ Φ ( 1), Φ ( 2),, Φ ( NS) ] (2) Hereinafter, the suerscrit reresents the rimitive matrices, which are diagonally assembled from the substructures directly. he divided substructures are then reconnected by the virtual work rincile and the geometric comatibility. Kron s substructuring method makes full use of the orthogonality roerties, and transforms the eigenequation of the assembled global structure into (Sehmi 1986) Λ λi Γ z = (3) Γ τ in which Γ = [ CΦ ], and C is a rectangular connection matrix, which constraints the interface DOFs to move jointly (Sehmi 1986, urner 1983). In C matrix, each row contains two non-zero elements. For rigid connections the two elements will be 1 and 1. If the connected oints x 1 and x 2 are not rigidly connected, which has the relationshi x 1 = rx 2, the two elements in the corresonding row of matrix C will be 1 and r. Kron s substructuring method considers the connection condition by the matrix C, and has a concise form (urner 1983). τ is the internal connection forces; λ is the eigenvalue of the global structure; z is regarded as the mode articiation factor, which indicates the contribution of the eigenmodes of the substructures to the () j

4 4 Yong Xia, Shun Weng, You-Lin Xu and Hong-Ping Zhu eigenmodes of the global structure. he eigenvectors of the global structure Φ can be recovered by Φ = Φ { z} and removing the identical elements of Φ at the interfaces. In Eq. (2), the rimitive matrices [ Λ ] and [ Φ ] require calculating the comlete eigensolutions of all substructures, which is time-consuming. he comlete modes of each substructure are artitioned into the master art and the slave art (Weng et al. 29). he first a few eigenmodes in each substructure are retained as the master modes, while the residual higher eigenmodes are discarded as the slave modes and comensated by the first-order residual flexibility. Assuming that the subscrit m and s reresents the master and slave variables resectively, the jth substructure has m (j) master eigenairs and s (j) slave eigenairs as () j Λ s () j Λ m () j () j () j j = Diag[ λ 1, λ 2,, λ ( j) m ], Φ m () j () j () j () j () j () j () j = Diag[ λ ( j) m, λ j m,, λ, ( j) m ] Φ ( ) s = [ φ j m, φ ( ) j m,, φ ( ) ( j) m ] (4) ( ) Assembling the master eigenairs and the slave eigenairs resectively, one has + s j () φ () j () j 1 φ 2 () j = [,,, φ ( j) m ] s j Λ m ( 1) ( 2 Diag Λ m Λ ) ( NS) = [, m,, Λ m ], Φ m ( 1) ( 2 ( NS) = Diag[ Φ m, Φ m ),, Φ m ] Λ s m ( 1) ( 2 ( NS) ( 1 = Diag[ Λ s, Λ s ),, Λ s ], Φ s Diag Φ ) ( 2 ( NS) = [ s, Φ s ),, Φ s ] NS Γ m [ CΦ m ] =, Γ s = [ CΦ s ] = m () j, s = s () j, m () j + s () j = n () j, ( j = 12,,, NS) (5) j = 1 NS j = 1 Partitioning Eq. (3) according to the master and slave modes, Eq. (3) can be exanded as λi Γ m z m Λ s λi Γ s z s = Γ m Γ s τ Λ m (6) he second line of Eq. (6) gives z s = ( Λ s λi) 1 Γ s τ (7) stituting Eq. (7) into Eq. (6) results in Λ m λi Γ m Γ s ( Λ s λi) 1 Γ m Γ s z m = τ (8) he interested eigenvalues λ corresond to the lowest modes of the global structure, and are far less than the items in Λ s when the master modes are roerly chosen. Eq. (8) is aroximated as

5 Calculation of eigenvalue and eigenvector derivatives 41 Λ m λi Γ m Γ m Γ s ( Λ s ) 1 Γ s z m = τ (9) Reresenting τ with z m from the second line of Eq. (9) and substituting it into the first line, the eigenequation is simlified into Λ m Γ m Γ Γ s ) 1 Γ m ]{ z m } = λ{ z m } Γ s ( Λ s ) 1 Γ CΦ ( Λ ) 1 = [ Φ s s s s ] C (1) (11) where Φ s ( Λ s ) 1 [ Φ s ] is the first-order residual flexibility, which is reresented by diagonally assembling the stiffness matrices and the master modes of the substructures as Φ s ( Λ s ) 1 [ Φ s ] Diag ( K ( 1) ) 1 Φ ( 1) ( 1) ( Λ (12) m m ) 1 ( 1) ( [ Φ m ] ) ( K ( NS) ) 1 Φ ( NS) ( NS) ( Λ m m ) 1 ( NS) = [,,( [ Φ m ] )] In Eq. (1), the mode articiation factor { z m } leads to the eigenvectors of the global structure via the transform of Φ = Φ. he reduced eigenequation (Eq. (1)) has the size of m m { z m }, which is much smaller than that of the original one (Eq. (3)). 3. Eigenvalue derivatives with the substructuring method For the ith mode, the eigenequation (Eq. (1)) can be rewritten as Λ m Γ m Γ Γ s ) 1 Γ m ]{ z i } = λ i { z i } (13) Eq. (13) is differentiated with resect to a design arameter r as Λ m Γ m Γ Γ s ) 1 Γ m λ i I] z i Λ m Γ m Γ Γ s ) 1 Γ m λ i I] { z i } = { } (14) Pre-multilying { z i } on both sides of Eq. (14) gives { z i } Λ m Γ m Γ Γ s ) 1 Γ m λ i I] z i { z i } Λ m Γ m Γ Γ s ) 1 Γ m λ ii] { z i } = (15) Due to symmetry of Λ m Γ m Γ Γ s ) 1 Γ m λ ii], the first item in the left hand side of Eq. (15) is zero. Arranging Eq. (15), the derivative of the eigenvalue λ i with resect to the design arameter r is λ i { z i } Λ m Γ m Γ Γ s ) 1 Γ m ] = { z i } (16)

6 42 Yong Xia, Shun Weng, You-Lin Xu and Hong-Ping Zhu in which = Λ m Λ m Γ m Γ Γ s ) 1 Γ m ] Γ m Γ Γ s ) 1 Γ [( Γ m Γ s ( Λ s ) 1 Γ s ) 1 ] m Γ m Γ m Γ Γ s ) Γ m In Eq. (16), Λ m / is the diagonal assembly of the eigenvalue derivatives of the substructures, Γ and m [ Φ m ] = C is associated with the diagonal assembly of the eigenvector derivatives of the substructures. Γ [ Γ s ) 1 ]/ is the derivative matrix of the first-order residual flexibility of the substructures. Since the substructures are indeendent, the derivative matrices of the eigenvalues, the eigenvectors and the residual flexibility are only calculated in the articular substructure (for examle, the rth substructure) which contains the elemental arameter r. hese quantities in other substructures are zero. Within the rth substructure, the eigenvalue and eigenvector derivatives can be obtained by the traditional methods, such as Nelson s method (Nelson 1976). he derivative of the residual flexibility with resective to the structural arameter r is Γ Γ s ) Γ ( (17) s ( Λ s ) 1 Γ s ) 1 Γ Γ s ) Γ = Γ s ) 1 and Γ Γ s ) C Φ s( Λ s ) 1 ( [ Φ s ] ) C C Diag ( K () r ) 1 Φ () r () r m ( Λ m ) 1 () r = = ( [ Φ m ] ) C = C Diag ( K () r ) 1 () r Φ m () r ( Λ m ) 1 () r [ Φ m ] Φ () r () r m ( Λ m ) 1 Λ () r m () r ( Λ m ) 1 () r [ Φ m ] Φ () r () r m ( Λ m ) 1 Φ () r [ m ] C It is noted that, if the substructure (for examle, the jth substructure) is free, the stiffness matrix K () j is singular, and the inversion of K () j to form the residual flexibility does not exist. Consequently, the derivative of the residual flexibility of the free-free substructure is not available. In this situation, the rigid body modes, which contribute to the zero-frequency modes, are emloyed (Felia et al. 1998). he detailed rocedures on how to calculate the residual flexibility and the first-order derivatives of the free-free substructures can be found in Aendix. It is noted that the eigenvalue derivatives of the global structure with resect to an elemental arameter solely rely on the articular substructure (the rth substructure) rather than the other (18)

7 Calculation of eigenvalue and eigenvector derivatives 43 substructures. Since the size of the substructures is always smaller than that of the global structure, the comutation efficiency is imroved. his significant merit might be very attractive when it is alied to the iterative model udating. With the substructuring method, only the modified substructures are re-analyzed, while other substructures are untouched. In addition, due to the symmetric and simle form of the residual flexibility, there is no difficulty to derive the high-order derivatives of the eigenvalues by directly differentiating the eigensensitivity equation (Eq. (16)), which needs to calculate the eigenvector derivatives first. 4. Eigenvector derivatives with the substructuring method Since the ith eigenvector of the global structure can be recovered by Φ i = Φ m { } z i (19) the eigenvector derivative of the ith mode with resect to the structural arameter r can be differentiated as Φ i Φ m z { z (2) i } Φ i = + m In Eq. (2), Φ m are the eigenvectors of the master modes in the substructures, and Φ m / are the associated eigenvector derivatives of the master modes of the rth substructure. { z i } is the eigenvector of the reduced eigenequation (Eq. (1)). Once the item { z i /} is available, the eigenvector derivative of the ith mode of the global structure can be obtained. Similar to the Nelson s method, { z i /} is searated into the sum of a articular art and a homogeneous art as z i = { v i } + c i { z i } (21) where c i is a articiation factor. stituting Eq. (21) into Eq. (14) gives Λ m Γ m Γ Γ s ) 1 Λ Γ m λ ii] ({ v i } + c i { z i }) m Γ m Γ Γ s ) 1 Γ m λ i I] = { z i } (22) Since Λ m Γ m Γ Γ s ) 1 Γ m λ i I] { z i } = { }, Eq. (22) is simlified as Ψ{ v i } = { Y i } (23) in which Ψ Λ m Γ m Γ Γ s ) 1 Λ = Γ m λ i I], { Y i } m Γ m Γ Γ s ) 1 Γ m λ ii] = { z i } All the items in Ψ and {Y i } have been obtained during the calculation of the eigenvalue derivatives in the revious section. If there is no reeated root, the reduced system matrix Ψ has the size of m and the rank of (m 1). o solve Eq. (23), one sets the kth item of {v i } to be zero, and eliminates the corresonding row

8 44 Yong Xia, Shun Weng, You-Lin Xu and Hong-Ping Zhu and column of Ψ and the corresonding item of {Y i }. he full rank equation is Ψ 11 Ψ 13 1 Ψ 31 Ψ 33 where the ivot, k, is chosen at the maximum entry in { z i }. he vector {v i } can be solved from Eq. (24). Solution of c i requires the orthogonal condition of the eigenvector as v i1 Y i1 v ik = v i3 Y i3 (24) { z i } { z i } = 1 Differentiating Eq. (25) with resect to r gives { z i } { z i } { z i } { z i } = stituting Eq. (21) into Eq. (26) results in ({ v i } + c i z i ) z i { } { } + { z i } ({ v i } + c i { z i }) = (25) (26) (27) herefore, the articiation factor c i is obtained as 1 c i = -- ({ v i } { z i } + { z i } { v i }) (28) 2 Finally, the first-order derivative of { z i } with resect to the structural arameter r is z i = { v i } 1 (29) 2 -- ({ v i } { z i } + { z i } { v i }){ z i } As far as Eq. (2) concerned, the eigenvector derivatives of the global structure can be regarded as the combination of the eigenvectors Φ m and the eigenvector derivatives Φ m / of the substructures, while { z i /} and z act as the weight. Similar to the calculation of the eigenvalue derivatives, the calculation of the eigenvector derivatives of the global structure is equivalent to analyze the rth substructure and a reduced eigenequation. he rocedure of the roosed substructuring method and its advantages will be demonstrated by two numerical examles. 5. Examle 1: the GAREUR structure he first examle resented here, GAREUR AG-11 (as shown in Fig. 1(a)), serves to illustrate the rocedures of the calculation of eigensensitivity with the roosed substructuring method. he frame is modeled by 78 Euler-Bernoulli beam elements and 74 nodes, as shown in Fig. 1(a). Each node has 3 DOFs, and there are 216 DOFs in total. he Young s modulus of each element is 75 GPa and the mass density is kg/m 3. he moment of inertia of all members is.756 m 4. he cross-section areas of the vertical, horizontal and diagonal bars are.6 m 2,.4 m 2 and.3 m 2, resectively.

9 Calculation of eigenvalue and eigenvector derivatives 45 Fig. 1 he GAREUR frame he global structure is divided into 3 substructures (NS = 3) along the horizontal direction. o be an indeendent structure, each substructure has 26 elements and 26 nodes as shown in Fig. 1(b). he first 3 modes of each substructure are chosen as the master modes to calculate the eigensensitivity of the first 1 modes of the global structure. Without losing generality, the Young s modulus of one element in structure 2 is chosen as the design arameter and denoted as r 1 in Fig. 1(b). he eigensensitivity of the first 1 modes of the global structure with resect to r 1 can be calculated with the roosed substructuring method: ( 1 (1) Calculate the eigensolutions of each substructure as: Λ ) ( 2) ( 3) ( 1) ( 2 m Λ m Λ m Φ m Φ ) ( 3),,,, m, Φ m (m = 3), and obtain the eigensolutions of the global structure with the reduced eigenequation Eq. (1) as: λ i, { z i }, Φ i = Φ m { z i } (i = 1, 2,, 1). In this structure, structure 2 and structure 3 have identical geometry and mechanical roerties, and thus only one of them needs to be analyzed; (2) Comute the eigenvalue and eigenvector derivatives of the first 3 modes of structure 2 with resect to the arameter r 1, Λ ( 2 ) / Φ ( m, 2) /, and calculate the derivative of the 1 m 1 ( 2 residual flexibility with resect to r 1, Φ ) 2 s ( Λ ( ) s ) 1 ( 2) ( [ Φ s ] )/ 1 from Eq. (17) and Eq. (18); (3) Set the eigensolution derivatives and the residual flexibility derivatives of the other two substructures to be zero: Λ () j m / 1 = [ ], Φ () j m / 1 = [ ], Φ () j () j s ( Λ s ) 1 () j ( [ Φ s ] )/ 1 = [ ], (j = 1, 3), and construct the rimitive form of the derivative matrices as:

10 46 Yong Xia, Shun Weng, You-Lin Xu and Hong-Ping Zhu [ Λ m ] = ( 2) Λ m, 1 [ Φ m ] = ( 2) Φ m 1 Φ s ( Λ s ) 1 [( [ Φ s ] )] = ( 2) ( 2) Φ s ( Λ s ) 1 ( 2) ( [ Φ s ] ) (4) Obtain the first-order eigenvalue derivatives of the global structure λ i/ 1 (i = 1, 2, 1) with Eq. (16). (5) Calculate the first-order derivatives of { z i } with resect to the arameter r 1 { z i / 1 } from Eq. (29). (6) Form the eigenvector derivatives of the global structure with resect to the arameter r 1 according to Eq. (2) and eliminate the identical values of Φ i / at the tearing interfaces. o verify the accuracy of the roosed substructuring method in calculation of the eigensensitivity, the traditional Nelson s method is directly emloyed to calculate the eigensensitivity of the global structure without dividing the global structure into individual substructures. he results from the roosed substructuring method and the global method are comared and shown in able 1. he relative errors of the eigenvalue derivatives are less than 2%, which is sufficient for most of ractical engineering alications. Following the concet of modal assurance criterion (MAC) (Friswell and Mottershead 1995), the similarity of the eigenvector derivatives between the global method and the roosed substructuring method is denoted as Correlation of Eigenvector Derivatives (COED), and given by COED φ i φ, i 1 1 φ i φ i 1 1 = φ i φ i φ i φ i where { φ i/ 1 } reresents the eigenvector derivative obtained by the global method, and { φ i / 1 } reresents the eigenvector derivative by the roosed substructuring method. In this examle, the COED values are above.995 for all modes as listed in able 1, which indicates good accuracy in calculation of the eigenvector derivatives with the resent method. he master modes retained in the substructures affect the accuracy of the eigensensitivity calculated. Here 1 master modes and 2 master modes in each substructure are also emloyed to calculate the eigensensitivity. he relative errors of the eigenvalue derivatives are comared with the case of 3 master modes and illustrated in Fig. 2. It can be found that, more master modes can imrove the accuracy of the eigensolution derivatives, as exected. he comutational efficiency of the roosed method will be investigated in the following examle with relatively large system matrices. 2 (3)

11 Calculation of eigenvalue and eigenvector derivatives 47 able 1 he eigensensitivity of the GAREUR structure Mode Nelson s method Eigenvalue derivatives structuring method Relative error Correlation of Eigenvector Derivatives (COED) % % % % % % % % % %.997 Fig. 2 he accuracy of the eigenvalue derivatives with different master modes 6. Examle 2: a highway bridge A ractical bridge over the Balla Balla River in Western Australia is investigated here. he FE model of the bridge has 97 elements, 947 nodes each with 6 DOFs and 542 DOFs in total, as shown in Fig. 3 (Xia et al. 28). he global structure is firstly divided into 11 substructures along the longitudinal direction as shown in Fig. 3. he detailed information of the 11 substructures is listed in able 2. here are 5 master modes retained in each substructure to assemble the global structure. he designed elemental arameters refer to the Young s modulus of the four shell elements,

12 48 Yong Xia, Shun Weng, You-Lin Xu and Hong-Ping Zhu Fig. 3 he finite element model of the Balla Balla River Bridge able 2 he information on division formation with 11 substructures Index of structure Geometric range (m) in longitudinal direction ~5 5~1 1~15 15~2 2~25 25~3 3~35 35~4 4~45 45~5 5~54 No. elements No. nodes No. tear nodes denoted as r 1 ~r 4 in Fig. 3. he elemental arameters are intentionally located in different substructures while two arameters in one substructure. With the roosed substructuring method, the eigensensitivity of the first 2 modes of the global structure with resect to the four elemental arameters are calculated in able 3. In addition, the eigensensitivity of the first 2 modes are directly calculated with the traditional Nelson s method based on the global structure for comarison. able 3 demonstrates that, when the global structure is divided into 11 substructures and the first 5 modes are retained as master modes in each substructure, the errors of eigenvalue derivatives of the first 2 modes are less than 3%, and the COED values are over 95%. hese are accetable for most of the engineering alications, such as model udating. Here the comutational efficiency is evaluated in terms of the comutation time consumed by the CPU of comuters during the calculation of the eigensensitivity with resect to the four design elemental arameters. he division formation of the substructures does affect the comutation efficiency. o investigate the effect of the division formation, the bridge is also divided into 5 substructures, 8 substructures and 15 substructures, resectively. he information of different

13 able 3 he eigensensitivity with resect to the four design structural arameters Mode Eigenvalue derivatives Global method (1-2 ) Present method (1-2 ) r 1 r 2 r 3 r 4 Relative error COED Eigenvalue derivatives Global method (1-2 ) Present Method (1-2 ) Relative error COED Eigenvalue derivatives Global method (1-2 ) Present method (1-2 ) Relative error COED Eigenvalue derivatives Global method (1-2 ) Present method (1-2 ) Relative error % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % 1....% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % %.982 COED Calculation of eigenvalue and eigenvector derivatives with the imroved Kron s... 49

14 able 4 he information on division formation with 5 substructures Index of substructures Geometric range (m) in longitudinal direction ~1 1~2 2~3 3~4 4~54 No. elements No. nodes No. tear nodes able 5 he information on division formation with 8 substructures Index of substructure Geometric range (m) in longitudinal direction ~7 7~14 14~21 21~27 27~34 35~41 41~47 47~54 No. elements No. nodes No. tear nodes able 6 he information on division formation with 15 substructures Index of substructure Geometric range in longitudinal direction (m) 1 2 ~3 3~ ~ ~ ~ ~ ~ ~ ~ 3 1 3~ ~37.5 No. elements No. nodes No. tear nodes ~ ~ ~ ~ 54 5 Yong Xia, Shun Weng, You-Lin Xu and Hong-Ping Zhu

15 Calculation of eigenvalue and eigenvector derivatives 51 Fig. 4 he comutation time with different division formations division formations is given in able 4 ~ able 6. For the four division formations, selecting different master modes will result in different recision. Based on the criteria that the relative errors of the eigenvalue derivatives for the first 2 modes of the global structure are less than 3%, there are 8 master modes retained in each substructure with the division formation of 5 substructures, 6 master modes in 8 substructures, 5 master modes in 11 substructures, and 5 master modes in 15 substructures. he comutation time in calculation of the eigensensitivity, consumed by the global method and the roosed substructuring method with the four division schemes, are comared in Fig. 4. From Fig. 4, it can be found that: (1) Comaring with the traditional global method, the roosed substructuring method can reduce the comutation time. his is because only a articular substructure and the reduced eigenequation need to be analyzed when forming the eigensensitivity of the global structure. (2) he comutational efficiency of the substructuring method heavily deends on the substructure division. For examle, dividing the global structure into 5 substructures or 8 substructures takes longer comutation time than that of 11 substructures. his is because handling large substructures takes longer time than handling smaller substructures. However, the division formation of 15 substructures is not as efficient as that of 11 substructures. he reason is that, excessive substructures lead to a large connection matrix C and the rimitive matrices of the substructures. In that case, the transformation among these matrices will take more comutation resources. his henomenon has also been observed in calculation of the eigensolutions (Weng et al. 29). he trade-off between the number of the substructures and the size of each substructure needs further studies. Nevertheless, the division formation can be tested in advance before alying the substructuring method to the iterative model udating rocess. 7. Conclusions In this aer, the first-order eigenvalue and eigenvector derivatives have been derived based on the imroved Kron s substructuring method. he eigensensitivity equation of the global structure is assembled from the eigensensitivity of articular substructures, and the eigensolution derivatives for

16 52 Yong Xia, Shun Weng, You-Lin Xu and Hong-Ping Zhu the reduced eigenequation are then calculated emulating the Nelson s method. wo numerical examles demonstrate that the roosed method can achieve a good accuracy when the roer master modes are retained. he division formation of the global structure should be considered with caution. oo few substructures, which result in large-size substructures, might reduce the efficiency of the substructuring method. However, excessive substructures may introduce a large transformation matrix, and accordingly cause the assembly of the substructures to the global structure timeconsuming. One should trade off the number of the substructures and the size of each substructure. Retaining more master modes in the substructures can achieve a better accuracy while cost more comutation resource. he error estimation is required for the selection of the master modes in the substructures, which will be studied in the future. Moreover, further research will focus on how to imrove the accuracy of the roosed substructuring method and imlement it to the model udating rocess. Acknowledgements he work described in this aer is jointly suorted by a grant from the Research Grants Council of the Hong Kong Secial Administrative Region, China (Project No. PolyU 5321/8E) and a grant from Natural Science Foundation, China (Project No ). References Bakir, P.G., Reynders, E. and Roeck, De G. (27), Sensitivity-based finite element model udating using constrained otimization with a trust region algorithm, J. Sound Vib., 35(1-2), Brownjohn, J.M.W., Xia, P.Q., Hao, H. and Xia, Y. (21), Civil structure condition assessment by FE model udating: methodology and case studies, Finite Elem. Anal. Des., 37(1), Choi, D., Kim, H. and Cho, M. (28), Iterative method for dynamic condensation combined with substructuring scheme, J. Sound Vib., 317(1-2), Craig, Jr. R.R. and Bamton, M.M.C. (1968), Couling of substructures for dynamic analysis, AIAA J., 6(7), Craig, Jr. R.R. (2), Couling of substructures for dynamic analysis: an overview, Proceedings of the 41st AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, AIAA , Atlanta, GA, Aril. Felia, C.A., Park, K.C. and Justino Filho, M.R. (1998), he construction of free-free flexibility matrices as generalized stiffness inverses, Comut. Struct., 68(4), Fox, R.L. and Kaoor, M.P. (1968), Rate of change of eigenvalues and eigenvectors, AIAA J., 6(12), Friswell, M.I. and Mottershead, J.E. (1995), Finite Element Model Udating in Structural Dynamics, Kluwer Academic Publishers. Fulton, R.E. (1991), Structural dynamics method for arallel suercomuters, Reort for MacNeal Schwendler Cor. Hurty, W.C. (1965), Dynamic analysis of structural systems using comonent modes, AIAA J., 3(4), Kron, G. (1963), Diakotics, Macdonald and Co., London. Lin, R.M. and Lim, M.K. (1995), Structural sensitivity analysis via reduced-order analytical model, Comut. Meth. Al. Mech. Eng., 121(1-4), Lin, R.M. and Lim, M.K. (1996a), Eigenvector derivatives of structures with rigid body modes, AIAA J.,

17 Calculation of eigenvalue and eigenvector derivatives 53 34(5), Lin, R.M., Wang, Z. and Lim, M.K. (1996b), A ractical algorithm for efficient comutation of eigenvector sensitivities, Comut. Meth. Al. Mech. Eng., 13, MacNeal, R.H. (1971), A hybrid method of comonent mode synthesis, Comut. Struct., 1(4), Nelson, R.B. (1976), Simlified calculation of eigenvector derivatives, AIAA J., 14(9), Qiu, J.B., Ying, Z.G. and Williams, F.W. (1997), Exact modal synthesis techniques using residual constraint modes, Int. J. Numer. Meth. Eng., Rubin, S. (1975), Imroved comonent-mode reresentation for structural dynamic analysis, AIAA J., 13(8), Sehmi, N.S. (1986), he Lanczos algorithm alied to Kron s method, Int. J. Numer. Meth. Eng., 23, Sehmi, N.S. (1989), Large Order Structural Eigenanalysis echniques Algorithms for Finite Element Systems, Ellis Horwood Limited, Chichester, England. Simson, A. (1973), Eigenvalue and vector sensitivities in Kron's method, J. Sound Vib., 31(1), Simson, A. and abarrok, B. (1968), On Kron s eigenvalue rocedure and related methods of frequency analysis, Quarterly Journal of Mechanics and Alied Mathematics, 21, Song, D.., Han, W.Z., Chen, S.H. and Qiu, Z.P. (1996), Simlified calculation of eigenvector derivatives with reeated eigenvalues, AIAA J., 34(4), urner, G.L. (1983), Finite Element Modeling and Dynamic structuring for Prediction of Diesel Engine Vibration, Ph.D thesis, Loughborough University of echnology. Weng, S. and Xia, Y. (27), structure method in eigensolutions and model udating for large scale structures, Proceedings of the 2nd International Conference on Structural Condition Assessment, Monitoring and Imrovement, Changsha, China. Weng, S., Xia, Y., Xu, Y.L., Zhou, X.Q. and Zhu, H.P. (29), Imroved substructuring method for eigensolutions of large-scale structures, J. Sound Vib., 323(3-5), Wu, B.S., Xu, Z.H. and Li, Z.G. (27), Imroved Nelson s method for comuting eigenvector derivatives with distinct and reeated eigenvalues, AIAA J., 45(4), Xia, Y., Hao, H., Deeks, A.J. and Zhu, X.Q. (28), Condition assessment of shear connectors in slab-girder bridges via vibration measurements, J. Bridge Eng., 13(1), Xia,Y. and Lin, R.M. (24), A new iterative order reduction (IOR) method for eigensolutions of large structures, Int. J. Numer. Meth. Eng., 59, Yun, C.B. and Lee, H.J. (1997), structural identification for damage estimation of structures, Struct. Saf., 19(1),

18 54 Yong Xia, Shun Weng, You-Lin Xu and Hong-Ping Zhu Aendix: he residual flexibility and the derivatives for the free-free substructures Without losing generality, here the residual flexibility and the derivative matrix are derived for an arbitrary structure with the stiffness and mass matrices of K and M, resectively. he free-free structure has two kinds of eigenmodes: the n r rigid body modes R and the n d deformational modes Φ d. he orthogonality of the rigid body modes satisfies R R = I (A. 1) Due to the fact that ( I RR )RR =, an orthogonal rojector associated with R can be constructed as P = I RR (A. 2) he orthogonality roerties yield the sectral decomositions as N d P = Φ d Φ d, P RR + = Φ d Φ d + RR (A. 3) i = 1 N d i = 1 Since K = N d i = 1 λ i Φ d Φ d, including the rigid body modes into the free-free stiffness matrix K similarly gives N d K RR + = λ i Φ d Φ d + RR, ( K+ RR ) 1 = ---Φ d Φ d + RR (A. 4) i = 1 N d 1 λ i = 1 i he eigenvector of K+ RR are identical to those of K, but including the rigid body modes and setting the eigenvalues of rigid body modes to be unity. he comlete eigenmodes are divided into n m master modes Φ m and n s slave modes Φ s according to the main sections of this aer. he master modes Φ m are comosed by the n r rigid body master modes R and the (n m n r ) normal master modes Φ m-r. he deformational modes include the master modes Φ m-r and the slave modes Φ s. Eq. (A.4) is equivalent to (A. 5) he residual flexibility for the free-free structure is (A. 6) Accordingly, for the fixed structure without zero-frequency modes, the rigid-body modes are vanished, and the residual flexibility is simlified as usual form Considering the mass matrix, the orthogonal condition satisfies (A. 7) Decomosing the mass matrix as M = M 1/2 M 1/2, and denoting (A. 8) (A. 9)

19 Calculation of eigenvalue and eigenvector derivatives 55 the orthogonality satisfies (A. 1) he first-order residual flexibility is reresented by (A. 11) In Eq. (A.11), ( K + Φ rφ r ) is nonsingular, and the first-order residual flexibility is obtainable. For the free-free substructure, the derivative matrix of the first-order residual flexibility with resect to r is (A. 12) Since the rigid body modes are unchanged with the modification of the hysical arameter r, the derivatives of the rigid body eigenvectors with resect to r are zero. herefore, the derivative of the first-order residual flexibility with resect to r is (A. 13) It should be noted here that, the rigid body modes are art of the master modes for the free-free substructures, when the master modes of the substructures are assembled to the reduced eigenequation (Eq. (1)). he rigid body modes are esecially considered for the free-free substructures only when calculating the residual flexibility. he design arameter r is considered as the stiffness elemental arameter in this research, and hence, the mass matrix M is assumed to be constant. If the design arameter r is mass elemental arameter, the eigensensitivity can be easily derived following the same rocedures as described above but keeing the stiffness matrix K as constant.