AIAA JOURNAL Vol. 56, No. 1, January 018 Consiering the Higher-Orer Effect of Resiual Moes in the Craig Bampton Metho Jaemin Kim, Seung-Hwan Boo, an Phill-Seung Lee Korea Avance Institute of Science an Technology, Daejeon 11, Republic of Korea DOI: 10.51/1.J055666 In this paper, the accuracy of the Craig Bampton metho, one of the most wiely use component moe synthesis methos, is improve. Consiering the higher-orer effect of resiual moes that are simply truncate in the Craig Bampton metho, the original finite element moel can be more accurately reuce. In this formulation, unknown eigenvalues are consiere as aitional generalize coorinates, which can be eliminate by employing the concept of system equivalent reuction expansion process. The new component moe synthesis is name the higher-orer Craig Bampton metho. The formulation of the higher-orer Craig Bampton metho is presente, an its improve accuracy is emonstrate through various examples. Nomenclature F = resiual flexibility matrix K = stiffness matrix M = mass matrix q = moal coorinate vector T = transformation matrix u = isplacement vector Λ = eigenvalue matrix λ = eigenvalue Φ = eigenvector matrix φ = eigenvector Ψ = constraint moe matrix I. Introuction COMPONENT moe synthesis (CMS) methos have been wiely use in finite element (FE) analysis of structural ynamics problems. CMS methos are very effective for calculating moal solutions (moe shapes an natural frequencies) of complicate an large FE moels, which usually consist of many substructures [1 5]. CMS methos have also been frequently employe to reuce the number of egrees of freeom (DOFs) of structural ynamics moels for airplane, automobile, an ship structures [1,]. After Hurty s pioneering work in 1965 [], various CMS methos have been evelope; see [ 5]. The methos have ifferent characteristics an avantages. Among them, the Craig Bampton metho has been mostly wiely use ue to its simplicity an robustness. In the Craig Bampton (CB) metho [], a structural FE moel is represente by an assemblage of substructures that are connecte through a fixe interface bounary. The moal solution of the original FE moel is synthesize by selecting only ominant moes obtaine solving the substructural eigenvalue problems, an the remaining moes not selecte are esignate as the resiual moes. To efine the ominant an resiual moes appropriately, several moe selection methos [5 7] have been well stuie. Receive 8 September 016; revision receive 18 July 017; accepte for publication 1 August 017; publishe online 19 September 017. Copyright 017 by the American Institute of Aeronautics an Astronautics, Inc. All rights reserve. All requests for copying an permission to reprint shoul be submitte to CCC at www.copyright.com; employ the ISSN 0001-15 (print) or 15-85X (online) to initiate your request. See also AIAA Rights an Permissions www.aiaa.org/ranp. *Grauate Stuent, Grauate School of Ocean Systems Engineering; oceaneng@kaist.ac.kr. Stuent Member AIAA. Postoctoral Researcher, Department of Mechanical Engineering; shboo@kaist.ac.kr. Member AIAA. Associate Professor, Department of Mechanical Engineering; phillseung@kaist.eu. Member AIAA. Recently, consiering the first-orer effect of the resiual moes, the CB metho was significantly improve by Kim an Lee [19]. The new metho is name the enhance CB metho (ECB), in which the unknown eigenvalue inclue in the formulation is replace with the multiplication of the inverse matrix of the reuce mass matrix an the reuce stiffness matrix, which are the alreay-known matrices, by aopting O'Callahan s iea [6]. Then, there is a natural question: what happens if the secon-, thir-, or higher-orer effects of the resiual moes are consiere? However, O'Callahan s iea is invali to consier such higher-orer effects. In this stuy, we evelop a new CMS metho uner the consieration of higher-orer effects of resiual moes, leaing to further improvements in the accuracy of the CB metho. In this formulation, the new generalize coorinate vector is efine by consiering the aitional coorinates containing the unknown eigenvalues. Employing the concept of system equivalent reuction expansion process (SEREP) [7], the aitional unknowns are then eliminate. We name this the higher-orer CB (HCB) metho. In the following sections, we will briefly review the CB metho, efine the resiual flexibility, erive the formulation of the HCB metho, an present the performance of HCB metho through various numerical examples: rectangular plate, cylinrical panel, hyperboloi shell, bent pipe, an automobile wheel problems. The numerical results are compare to the existing CB an ECB methos. II. Craig Bampton Metho In the CB metho, the global (original) FE moel is assemble using N s substructures connecte through a fixe-interface bounary; see Fig. 1. The equations of motion for free vibration without amping are given by " # Ms M c M g u g K g u g 0 with M g ; " # Ks K c K g ; u g K T c K b ( us where M g an K g are the global mass an stiffness matrices, an u g is the global isplacement vector. The subscripts s an b represent the substructural an interface bounary quantities, respectively, an c represents the couple quantities between the substructures an interface bounary. The ouble ot inicates the secon-orer ifferentiation with respect to time t, i.e., t. Here, M s an K s are the block-iagonal matrices, an their iagonal component matrices are the substructural mass an stiffness matrices M i s an K i s (for i 1; ; ;N s ). 0 u b M T c ) M b (1)
0 KIM, BOO, AND LEE Fig. 1 c) Partitioning proceures in the Craig Bampton metho: global FE moel, partitione FE moels, an c) fixe-interface bounary. The global eigenvalue problem is efine as K g fφ g g j λ g j M g fφ g g j for j 1; ; ;N g () in which λ g j an fφ g g j are the global eigenvalue an eigenvector corresponing to the jth global moe, respectively, an N g is the number of DOFs in the global FE moel. Note that, in engineering practice, only a small fraction of the total eigenpairs nees to be consiere (for j 1; ; ;p, where p N g ). p In structural ynamics, the square root of the eigenvalue ( λ j) an eigenvector are interprete as a natural frequency ω j an the corresponing moe shape, respectively. Note that the eigenvectors are scale to satisfy the following mass-orthonormality conition: fφ g g T i M gfφ g g j δ ij for i an j 1; ; ;N g () where δ ij is the Kronecker elta (δ ij 1 if i j, otherwise δ ij 0). Using the eigenvectors calculate in Eq. (), the global isplacement vector u g is represente as 8 9 h i >< >= u g Φ g q g ; Φ g fφg g 1 fφ g g fφ g g Ng ; q g. (). >: >; where Φ g is the global eigenvector matrix containing the eigenvectors fφ g g i, an q g is the moal coorinate vector containing the moal coorinates q i corresponing to fφ g g i. In the CB metho, the global isplacement vector u g is represente as u g T 0 u 0 ; T 0 Φs q 1 q q Ng Ψ c qs ; u 0 I 0 b u b (5) Note that the matrices Φ s an Ψ c in the global transformation matrix T 0 are expresse in a substructural matrix form as Φ s 6 Ψ c 6 Φ 1 s 0 Φ s... 0 Φ N s s Ψ 1 c Ψ c.. Ψ N s c 7 5 with Ψ i c ; 7 5 K i s 1Kc (6) The iagonal component matrices of Φ s in Eq. (6) can be obtaine by solving the following substructural eigenvalue problems: K i s Φ i s Λ i s M i s Φ i s ; Φ i s h Φ i i Φ i r for i 1; ; ;N s (7) where Φ i s an Λ i s are the substructural eigenvector an eigenvalue matrices corresponing to the ith substructure, an the substructural eigenvector matrix Φ i s is ivie into the ominant term Φ i an resiual term Φ i r. The subscripts an r enote the ominant an resiual quantities. Note that, in the substructural eigenvalue problems, a small fraction of the total substructural eigenpairs is calculate. Using the substructural eigenvector matrices Φ i an Φ i r, the fixe-interface normal moes matrix Φ s can be reorere as Φ s Φ Φ r (8) in which T 0 an u 0 are the global transformation matrix an its generalize coorinate vector, respectively; Φ s an Ψ c are the fixeinterface normal moe an constraint moe matrices, respectively; I b is the ientity matrix for the interface bounary; q s is the moal coorinate vector corresponing to Φ s ; an u b is the interface bounary isplacement vector. in which Φ an Φ r are the ominant an resiual eigenvector matrices, respectively, an these matrices are the block-iagonal matrices, in which iagonal terms consist of the substructural eigenvector matrices, Φ i an Φ i r, escribe in Eq. (7). Substituting Eq. (8) into Eq. (5), the global isplacement vector u g is represente as
KIM, BOO, AND LEE 05 us Φ Φ u g T u 0 u 0 with T 0 r Ψ c ; u b 0 0 I 0 b 8 < : q q r u b in which q an q r are the moal coorinates vectors corresponing to Φ an Φ r, respectively. Truncating the resiual eigenvector matrix Φ r an the corresponing moal coorinate vector q r in Eq. (9), the approximate global isplacement vector u g is obtaine: u g u g us u b T 0 u 0 ; T 0 Φ 9 = ; (9) Ψ c q ; u 0 I 0 b u b (10) where T 0 an u 0 are the CB transformation matrix (N g N 0 ) an the corresponing generalize coorinate vector, respectively. N 0 is the number of DOFs in the reuce FE moel; N 0 N N b with N XN s i 1 N i where N i is the number of ominant moes of the ith substructure, an N b is the number of DOFs on the interface bounary. The overbar enotes the approximate quantity. Note that the resiual eigenvector matrix Φ r is simply truncate without any consieration. Using the transformation matrix T 0 in Eq. (10), the reuce equations of motion are M 0 u 0 K 0 u 0 0 with M 0 T T 0 M g T 0 ; K 0 T T 0 K g T 0 (11) in which M 0 an K 0 are the reuce mass an stiffness matrices ( N 0 N 0 ), respectively. Using M 0 an K 0 in Eq. (11), the reuce eigenvalue problem is given by K 0 f φ 0 g j λ 0 j M 0 f φ 0 g j for j 1; ; ; N 0 (1) an the approximate eigenvector matrix Φ 0 is efine as h i Φ 0 f φ 0 g 1 f φ 0 g f φ 0 g j for j 1; ; ; N 0 (1) where λ 0 j an f φ 0 g j are the jth approximate eigenvalue an eigenvector. The reuce isplacement vector u 0 is then represente by u 0 Φ 0 q 0 (1) where q 0 is the moal coorinate vector corresponing to Φ 0, an the approximate global eigenvector matrix Φ g is obtaine by Φ g T 0 Φ 0 ; fφ g g j T 0 f φ 0 g j for j 1; ; ; N 0 (15) III. Higher-Orer Craig Bampton Metho In this section, we erive the formulation of the higher-orer CB (HCB) metho, in which the resiual eigenvector matrix Φ r is properly consiere to construct the reuce moel more accurately. Using T 0 in Eq. (9), the equations of motion in Eq. (1) are transforme into t M 0 K 0 u 0 0 (16 I 0 Φ T ^M c M 0 T T 0 M gt 0 6 0 I r Φ T r ^M c 7 5 ; ^M T c Φ ^M T c Φ r ^M b Λ 0 0 K 0 T T 0 K gt 0 6 0 Λ r 0 7 5 (16 0 0 ^K b where M 0 an K 0 are the transforme global mass an stiffness matrices, respectively, an the component matrices in Eq. (16 are efine by ^M c M c M s Ψ c ; I Φ T M sφ ; I r Φ T r M s Φ r (17 Λ Φ T K sφ ; Λ r Φ T r K s Φ r (17 ^M b M b Ψ T c M c M T c Ψ c Ψ T c M s Ψ c (17c) ^K b K b K T c Ψ c (17) Consiering a harmonic response [ t λ], Eq. (16) can be rewritten as Λ λi 0 λφ T ^M 8 9 c q >< >= 6 0 Λ r λi r λφ T r ^M c 7 q 5 r 0 (18) >: λ ^M T c Φ λ ^M T c Φ r ^K b λ ^M b u >; b an, from the secon row in Eq. (18), the following equation is obtaine: q r λ Λ r λi r 1 Φ T r ^M c u b (19) Substituting Eq. (19) into Eq. (9), the global isplacement vector u g can be represent as u g ( us u b ) T 0 u 0 with T 0 Φ Ψ c λφ r Λ r λi r 1 Φ T r ^M c 5; u 0 0 I b ( q u b ) (0) In Eq. (0), the resiual flexibility Φ r Λ r λi r 1 Φ T r can be expane by using Taylor series [17,19,1,]: Φ r Λ r λi r 1 Φ T r F 1 λ 1 F λ i 1 F i with F i Φ r Λ i r Φ T r (1) where F i is the ith-orer resiual flexibility matrix. It shoul be note that, through the Neumann series expansion theorem [8], the expansion in Eq. (1) is vali if the eigenvalue λ is smaller than the smallest eigenvalue of Λ r, which is the resiual eigenvalue matrix for substructures. In the CB metho, this expansion is generally vali because the ominant substructural moes are selecte to reflect lower moes of the original FE moel []. Without using the resiual moes, the resiual flexibility matrix F i is inirectly calculate by F i K i s Φ Λ i ΦT ()
06 KIM, BOO, AND LEE It is important to note that, as the orer i increases, K i s an Φ Λ i ΦT rapily approach one another. This results in a loss of precision in the computation of F i. Therefore, for the precise calculation of F i, the number of significant igits use must be properly chosen. This issue will be stuie through a numerical example in Sec. IV.A. Consiering the nth-orer approximation of the resiual flexibility Φ r Λ r λi r 1 Φ T r F 1 λ 1 F λ n 1 F n () an substituting it into Eq. (0), the nth-orer approximation of the global isplacement vector u g is given by 0 0 0 where ^T n an u n are the HCB transformation matrix (N g N n ) an the corresponing generalize coorinate vector, respectively. N n is the number of DOFs in the reuce FE moel ( N n N N b n), ^Θ n is the resiual moe matrix containing the nth-orer resiual flexibility F n, an η n is the aitional coorinate vector containing the unknown eigenvalue λ n. Note that the zeroth-orer transformation matrix (n 0) is nothing but the CB transformation matrix T 0 in Eq. (10). As mentione alreay, Φ has been normalize with respect to M s. On the other han, ^Θn has an arbitrary amplitue without normalization. Thus, ^Θ n nees to be properly normalize. Otherwise, ^Θ n may prouce a baly scale transformation matrix, which results in ill-conitione reuce stiffness an mass matrices. We normalize each column of ^Θ n using its L-norm [9,0]: Θ n ^Θ n G 1 n with G n 6 kfθ n g 1 k 0 kfθ n g k... 0 kfθ n g Nb k 7 5 (5) where Θ n is the normalize resiual moe matrix containing the nthorer resiual flexibility, an fθ n g j is the jth column vector of ^Θ n. For the nth-orer HCB metho, the global isplacement vector u g can be approximate by Using T n in Eq. (6), the following reuce equations of motion are obtaine: M n u n K n u n 0 with M n T T nm g T n ; K n T T nk g T n (7) in which M n an K n are the reuce mass an stiffness matrices ( N n N n ). Note that the reuce system in Eq. (7) has larger size than the system reuce by the original CB metho in Eq. (11) ue to the use of aitional generalize coorinates. The aitional coorinates can be eliminate by employing the concept of SEREP [7], which is a DOF-base reuction metho without accuracy loss. Then, the reuce system in Eq. (7) can be further reuce, leaing to the same number of equations of motion reuce by the original CB metho in Eq. (11). However, this proceure increases computation time inevitably. From Eq. (7), the following eigenvalue problem is obtaine: K n f φ n g j λ n j M n f φ n g j for j 1; ; ; N n (8) where λ n j an fφ n g j are the eigenvalue an eigenvector, respectively. We then calculate the eigenvectors up to the N 0 th moe an construct the following eigenvector matrix: h i Φ n f φ n g 1 f φ n g f φ n gn0 (9) Using the eigenvector matrix in Eq. (9), the transformation matrix T n is reuce as ~T n T n Φ n (0) where ~T n is the reuce transformation matrix of the HCB metho, the size of which is the same as that of T 0 (N g N 0 ). Finally, the reuce matrices constructe by the HCB metho are obtaine: ~M n ~T T nm g ~T n ; ~K n ~T T nk g ~T n (1) in which ~M n an ~K n are the reuce mass an stiffness matrices of size N 0 N 0. The reuce eigenvalue problem in the HCB metho is also efine by ~K n f ~φ n g j ~λ n j ~M n f ~φ n g j for j 1; ; ; N 0 () where ~λ n j an f ~φ n g j are the approximate eigenvalues an eigenvectors, respectively. As the orer of resiual flexibility consiere in the formulation increases, the reuce system becomes more accurate. Various orers of the HCB methos can be efine epening on the orer 0 0 0 where T n an u n are the nth-orer HCB transformation matrix (N g N n ) an the corresponing generalize coorinate vector, respectively. Fig. Rectangular plate problem (0 1 mesh, three substructures).
KIM, BOO, AND LEE 07 Table 1 Numbers of ominant moes selecte for the rectangular plate problem N N N N g 1 7 5 5 165 consiere. Here, we efine the nth-orer HCB metho (enote HCB-n), in which the nth-orer HCB transformation matrix is use. Note that HCB-0 is equivalent to the original CB metho. IV. Numerical Examples In this section, we compare the performance of the present metho (HCB) with two previous methos: the original CB metho (CB) an the enhance CB metho (ECB). Four structural problems are consiere: rectangular plate, cylinrical panel, hyperboloi shell, an bent pipe problems. The component moe synthesis methos are implemente in MATLAB, an computation is performe on a personal computer (Intel Core i7-770,.0 GHz CPU, GB RAM). The well-known frequency cutoff criterion [1] is aopte to select substructural ominant moes. To measure the accuracy of the reuce moels constructe by ifferent methos, the following relative eigenvalue errors are calculate: ξ j jλ j λ j j λ j () where ξ j is the jth relative eigenvalue error, λ j is the jth exact eigenvalue calculate from the global (original) eigenvalue problem in Eq. (), an λ j is the jth approximate eigenvalue calculate from the reuce eigenvalue problem. Note that rigi-boy moes are not consiere in measuring the accuracy. A. Rectangular Plate Problem Consier a rectangular plate with free bounary in Fig.. Its length L is 0.0 m, with B is 1.0 m, an thickness h is 0.08 m. Young s moulus E is 06 GPa, Poisson s ratio ν is 0., an ensity ρ is 7850 kg m. The plate structure is moele by a 0 1 mesh of the four-noe mixe interpolation of tensorial components (MITC) shell elements [ 5] an partitione into three substructures (N s ). The number of DOFs for this problem is 165 (N g 165). We select 5 ominant moes (N 5). The number of moes selecte in each substructure (N k ) is liste in Table 1. Using four HCB methos (HCB-0, HCB-1, HCB-, an HCB-), we construct reuce moels. The four methos are implemente with two ifferent numbers of significant igits, namely 16 an. Figures a an b present the relative eigenvalue errors obtaine by the four HCB methos using 16 an significant igits, respectively. When 16 significant igits are use for computation, the accuracy of the HCB- metho eteriorates ue to the loss of Fig. Relative eigenvalue errors for the rectangular plate problem (0 1 mesh, three substructures, N 5): 16 significant igits, an significant igits. Fig. Errors for the rectangular plate problem (0 1 mesh, three substructures, N 5): relative eigenvalue errors, an relative eigenvector errors.
08 KIM, BOO, AND LEE Table Relative eigenvalue errors for the rectangular plate problem Present Moe number CB ECB HCB-1 HCB- 1 9.15E-05 5.00E-07 5.005E-09.959E-10.78E-06 8.761E-09 1.7E-10 1.950E-11.116E-0.05E-07.866E-08 8.009E-11.0E-0.197E-08 1.57E-08.506E-11 5 5.675E-0.56E-08.59E-08 9.76E-11 6 5.586E-0 1.9E-08.07E-08 9.58E-11 7 1.717E-0.960E-09 5.01E-09 1.588E-1 8.91E-0 1.19E-08.600E-07.676E-10 9.19E-0 6.508E-08.05E-07.69E-09 10.91E-0 1.E-07.067E-07 1.798E-09 11.97E-0 1.75E-07 1.79E-07.57E-10 1.866E-0 9.119E-07 7.89E-07 5.16E-10 1 9.609E-0 1.71E-07 1.77E-07.608E-10 1 6.775E-0 1.88E-06 6.818E-07 1.666E-10 15 1.66E-0 1.6E-05 7.81E-06 1.07E-09 16 1.06E-0.68E-06 1.051E-06 8.905E-10 17 5.77E-0.759E-06 1.610E-06 9.068E-11 18.5E-0 1.1E-05 8.5E-06.7E-10 19 1.67E-0 5.08E-05.08E-05 1.65E-08 0.090E-0 1.05E-05 6.66E-06 8.669E-10 1 6.0E-0 1.7E-0 1.00E-0 1.9E-08.75E-0 7.98E-0.7E-0.77E-08 7.8E-0 1.60E-0.15E-05.77E-09 7.01E-0 1.50E-0.7E-05 1.10E-09 5 6.E-0 1.69E-0 1.01E-0.8E-08 L 5 6 R 7 1 60 L 1 L L L L 5 L 6 L 7 L 8 Fig. 6 Relative eigenvalue errors for the cylinrical panel problem: N 0, an N 0. Fig. 5 Cylinrical panel problem with a istorte mesh. precision in the computation of F i. However, when significant ecimal igits are use, the accuracy eterioration phenomenon isappears. Because 16 significant igits are usually employe in engineering computations, this problem must be resolve in future work. Case Table N Number of ominant moes selecte for the cylinrical panel problem N N N 5 N 6 N 7 N N g 1 0 15 8 8 8 0 15 Fig. 7 Hyperboloi shell problem.
KIM, BOO, AND LEE 09 After this point, we use only 16 significant igits in computation ue to the large amount of computation time require for significant igits. Figure a an Table present the relative eigenvalue errors in reuce moels constructe by the CB, ECB, HCB-1, an HCB- methos. Figure b presents the relative eigenvector errors efine using MAC (moal assurance criterion) [6] ζ i 1 jφ i φ i j () kφ i k k φ i k where ζ i is the ith relative eigenvector error, φ i is the ith exact eigenvector calculate from the global (original) eigenvalue problem in Eq. (), an φ i is the ith approximate eigenvector obtaine from the reuce eigenvalue problem. The results in Fig. show that the accuracy of the HCB-1 metho is similar to that of the ECB metho, an the HCB- metho provies further improve accuracy, in particular, in relatively higher moes. Case Table N Numbers of ominant moes selecte for the hyperboloi shell problem N N N 5 N 6 N 7 N 8 N N g 1 00 00 B. Cylinrical Panel Problem A cylinrical panel with free bounary is consiere as shown in Fig. 5. The length L is 0.8 m, raius R is 0.5 m, an thickness is 0.005 m. Young s moulus E is 69 GPa, Poisson s ratio ν is 0.5, an ensity ρ is 700 kg m. The cylinrical panel is moele by a 16 16 istorte mesh of finite shell elements [ 5], in which each ege is iscretize in the following ratio: L 1 L L L 16 16 15 1 1 (5) The number of DOFs is 15 (N g 15). The FE moel is partitione into seven substructures (N s 7). We consier two numerical cases with 0 an 0 ominant moes selecte (N 0 an N 0). The number of moes selecte in each substructure (N k ) is liste in Table. Figure 6 presents the relative eigenvalue errors obtaine using the CB, ECB, HCB-1, an HCB- methos. The results consistently emonstrate the improve accuracy of the HCB methos. C. Hyperboloi Shell Problem We consier a hyperboloi shell structure with free bounary in Fig. 7. The height H is.0 m, an thickness is 0.05 m. Young s moulus E is 69 GPa, Poisson s ratio ν is 0.5, an ensity ρ is 700 kg m. The misurface of this shell structure is escribe by x y z ; z ; (6) The hyperboloi shell structure is moele using a 0 0 mesh of the four-noe MITC shell elements [ 5]. The number of DOFs Thickness Fig. 8 Relative eigenvalue errors for the hyperboloi shell problem: N, an N. Fig. 9 Bent pipe problem (four substructures).
10 KIM, BOO, AND LEE Case Table 5 Numbers of ominant moes selecte for the bent pipe problem N N N N N g 1 10 10 10 10 0,78 5 5 5 5 100,78 Two numerical cases are consiere with 0 an 100 ominant moes selecte (N 0 an N 100). The number of moes selecte in each substructure (N k ) is liste in Table 5. Figure 10 presents the relative eigenvalue errors obtaine using the CB, ECB, HCB-1, an HCB- methos. From the results, the improve accuracy of the HCB methos is well observe. use is 00 (N g 00), an the FE moel is partitione into eight substructures (N s 8). Two numerical cases are consiere with an ominant moes selecte (N an N ). The number of moes selecte in each substructure (N k ) is liste in Table. Figure 8 presents the relative eigenvalue errors obtaine using the CB, ECB, HCB-1, an HCB- methos. The results emonstrate the excellent performance of the HCB methos. D. Bent Pipe Problem We consier a bent pipe structure with clampe-free bounary in Fig. 9. The lengths L 1, L, L, an L are 1.0, 15.0, 9.0, an 18.0 m, respectively, an the bent angle is 90 eg. Diameter D is 0.168 m, an thickness is 0.01 m. Young s moulus E is 0 GPa, Poisson s ratio ν is 0.5, an ensity ρ is 700 kg m. The bent pipe structure is moele using the four-noe MITC shell elements [ 5]. The number of DOFs use is,78 (N g ; 78), an the FE moel is partitione into four substructures (N s ). V. Computation Time In this section, we compare the computation times require for the CB an HCB methos. An automobile wheel with free bounary is consiere as shown in Fig. 11. The outer iameter is 0.8 m (19 in.), Young s moulus E is 10 GPa, Poisson s ratio ν is 0., an ensity ρ is 7850 kg m. The automobile wheel problem is moele using three-imensional soli finite elements, an the finite element moel is partitione into four substructures (N s ). The number of DOFs use is 5,90 (N g 5;90). We establish an error criterion, namely that the relative eigenvalue errors up to the 100th moe are less than 10, an we investigate the reuce moels obtaine by the three methos (CB, HCB-1, an HCB-) until satisfying the given error criterion. Figure 1a presents the relative eigenvalue errors when the same size of reuce moels is constructe. For the three methos, we select the same number of ominant moes, N 100, as liste in Table 6. The HCB- metho provies significantly better accuracy in the whole range of moes compare to others. Also, we can ientify that the HCB- metho is only satisfying the given error criterion. In each moel reuction metho, the number of ominant moes selecte is etermine to satisfy the criterion; see Table 7. Figure 1b shows that all of the methos satisfy the criterion. Although the original CB metho satisfies the criterion with 7000 ominant moes, the HCB- metho requires only 100 ominant moes. Fig. 10 Relative eigenvalue errors for the bent pipe problem: N 0, an N 100. Fig. 11 Automobile wheel problem (four substructures).
KIM, BOO, AND LEE 11 Table 8 Specific computation times for the automobile wheel problem in Fig. 1b Fig. 1 Relative eigenvalue errors for the automobile wheel problem with the error criterion. Table 8 presents the computation times require. The results show that the HCB- metho prouces the reuce moel satisfying the criterion with less computation time than the CB metho, even though using 70 times fewer moes. In aition, we investigate the computation times for calculating 100 eigenpairs in the global an reuce moels of the automobile wheel problem. Table 9 shows the computation times. The reuce moel requires.1 s, whereas the global moel requires 15.95 s. The Table 6 Numbers of ominant moes selecte for the automobile wheel problem in Fig. 1a Metho N N N N N g CB 5 5 5 5 100 5,90 HCB-1 5 5 5 5 100 5,90 HCB- 5 5 5 5 100 5,90 Items Solving the substructural eigenvalue problems K i s Φ i s Λ i s Ms i Φs i Methos CB HCB-1 HCB- (N 7000) (N 500) (N 100) 101.07 16.0.0 Calculating the constraint moe 95.78 95.78 95.78 matrix Ψ c Calculating the resiual moe 05.1 7.61 a matrix Θ n Conucting the transformation 918.76 698.1 778.7 proceure Conucting the SEREP 97.5 169.7 proceure a Solving the reuce eigenvalue 1.51.78.1 problem Total 50.1 115.9 095.0 a Items only require in the HCB methos. Table 9 Computation times for calculating 100 eigenpairs for the wheel problem Computation time Moel CPU time, s Ratio, % Global moel 15.95 100.00 Reuce moel (by the HCB- metho).1 1.8 results emonstrate the well-known avantage of reuce-orer moels from the computational point of view. VI. Conclusions In this stuy, a new component moe synthesis (CMS) metho was evelope. The metho is base on the well-known Craig Bampton (CB) metho. Because the higher-orer effects of resiual moes is consiere in the formulation, it is name the higher-orer Craig Bampton (HCB) metho. In the HCB metho, the unknown coefficients in the resiual flexibility are consiere as aitional generalize coorinates, which are eliminate in the final formulation. Through numerical examples, the performance of the HCB metho was emonstrate. The numerical results were compare with the original CB metho (CB) an the enhance CB metho (ECB). We observe that the HCB metho coul construct the reuce-orer moels with significantly improve accuracy. The computational efficiency of the HCB metho was also teste. In future works, it will be valuable to improve the computational efficiency of the HCB methos. Acknowlegments This research was supporte by a grant (KCG-01-015-01) through the Disaster an Safety Management Institute fune by Korea Coast Guar of Korean government. Table 7 Numbers of ominant moes selecte for the automobile wheel problem in Fig. 1b Metho N N N N N g CB 1500 1500 1500 1500 6000 5,90 HCB-1 15 15 15 15 500 5,90 HCB- 5 5 5 5 100 5,90 References [1] Craig, R. R., an Kurila, A. J., Funamentals of Structural Dynamics, Wiley, 006, pp. 51 56. [] Bathe, K. J., Finite element proceure, Prentice Hall, 006, pp. 875 895. [] Hurty, W. C., Dynamic Analysis of Structural Systems Using Component Moes, AIAA Journal, Vol., No., 1965, pp. 678 685. oi:10.51/.97
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