Copyright © 2008 KSAE 1229−9138/2007/038−01
International Journal of Automotive Technology, Vol. 9, No. 0, pp. 00−00 (2008)
DOI 10.1007/s12239−008−0001−1
AUTOMOTIVE STRUCTURE VIBRATION WITH COMPONENT MODE SYNTHESIS ON A MULTI-LEVEL C.-W. KIM , S. N. JUNG and J. H. CHOI 1)*
2)
3)
School of Mechanical Engineering, Konkuk University, Seoul 143-701, Korea Department of Aerospace Information Engineering, Konkuk University, Seoul 143-701, Korea Department of Mechanical Engineering, KyungHee University, Yongin-si, Gyeonggi 446-701, Korea 1)
2)
3)
(Received 14 June 2007; Revised 31 October 2007)
ABSTRACT−The recursive component mode synthesis method (RCMS) has been implemented for the finite element
analysis model of an automobile structure as an efficient free vibration analysis tool. The RCMS method is intended to obtain a better performance relative to the block Lanczos method, which is a traditional method in the industry of obtaining eigenvalues, while obtaining the acceptable accuracy. A numerical example of the automobile finite element model demonstrates the outstanding performance of RCMS compared to the block Lanczos method.
KEY WORDS : Finite element, Component mode synthesis, Automobile, Free vibration, Natural frequency, Lanczos
1. INTRODUCTION
used in indutry. The Lanczos method computes more exact eigensoltuons relative to the CMS method, but the CMS method requires less computer resources than the Lanczos method. Recently, the CMS method on multi-level has been introduced (Hetmaniuk and Lehoucq, 2005) and the recursive CMS methods (RCMS) (Kim, 2006) have been introduced for simple FEA models such as plates and shells. This paper implements the RCMS method for the vibration analysis of an automobile FEA model.
Dynamic analyses for an automobile designs have been performed to assess ride comfort and to improve the component life. Dynamic analysis is a useful method of predicting the accurate dynamic stresses of the chassis during many types of maneuvers (Bae , 2005). One way of examining the accurate dynamic stresses is to perform a dynamic analysis of the full Finite Element Analysis (FEA) model. The full FEA model of the automobile structure has often become too large to capture the accurate local stresses. As a result, the dynamic analysis of the full FEA model often requires huge computer resources. The other method of examining the dynamic stresses is to perform the modal reduction by using the dominant mode shapes which are obtained from the free vibration analysis. The full FEA equations of motion are reduced to the modal coordinate space which is represented by super imposing the dominant mode shapes. For the free vibration analysis, the Lanczos method (Lanczos, 1950; Golub and Van Loan, 1996) has been widely used in the industry for extracting partial eigensolutions of large scale structural systems. However, the Lanczos method requires large amounts of disk space. Alternatively, the component mode synthesis (CMS) has been used over the past three decades in industry (Benfield and Hruda, 1960; Craig and Bampton, 1968; Meirovitch and Hale, 1981). The CMS method computes the approximate eigensolution from the reduced size of model. Both the Lanczos and the CMS methods have been intensively et al.
*
Corresponding author.
2. MATHEMATICAL BACKGROUNDS OF COMPONENT MODE SYNTHESIS METHOD A free vibration of structure represented by a finite element (FE) model can be represented as (1) [ ]− λ [ ] { } = { 0 } where [ ] and [ ] ∈ n × n represent finite element mass and stiffness matrices, respectively, and is the FE degrees of freedom. [ ] and [ ] are symmetric and positive definite. λ is the eigenvalue, and { } is the corresponding eigenvector. In the CMS method, in order to reduce the problem size, after reordering equation (1) is rewritten as K
M
u
K
M
R
n
K
M
u
⎞ ⎧⎨ L ⎫⎬= λ ⎛ LL LS ⎞ ⎧⎨ L ⎫⎬ (2) ⎠ ⎝ SL SS ⎠ ⎩ S ⎭ SL SS ⎩ S ⎭ in which S ∈ n is a set of shared (or master) degrees of freedom, L ∈ n is a set of local (or slave) degrees of ⎛ ⎝
K LL K LS
u
M
M
u
K
u
M
M
u
K
u
u
R
R
1
2
freedom, and = 1+ 2. The shared degrees of freedom consist of interface degrees of freedom which are shared n
e-mail:
[email protected] 1
n
n
2
C.-W. KIM, S. N. JUNG and J. H. CHOI
with adjacent components, and the local degrees of freedom consist of the degrees of freedom in their own components. First, the stiffness matrix [K] is decomposed into [K]=U TDU using the block Gauss elimination. The U is in the form –1 U = I – K LL K LS 0 I
(3)
The congruent transformation of [K] and [M] with U in equation (2) results in [ K˜ ] = U T [ K ] U (4)
[ M˜ ] = UT [ M ] U
(5)
Then, we obtain the following eigenvalue problem.
⎧ ⎫ U T ⎛⎝ K LL K LS ⎞⎠ U ⎨ u L ⎬ K SL K SS ⎩ u S ⎭
Figure 1. Two level partitioning. Then, equation (14) is reduced in the form
(6)
⎛ Λ LL 0 ⎞ ⎧⎨ u L ⎫⎬= λ ⎛⎜ ILL M LS ⎞⎟ ⎧⎨ uL ⎫⎬ ⎝ 0 Λ SS ⎠ ⎩ uS ⎭ ⎝ M SL I ⎠ ⎩ u S ⎭ SS
⎛ KLL 0 ⎞ ⎧ uL ⎫ ⎛ M LL M˜ LS ⎞ ⎧ u L ⎫ ⎜ ⎟ ⎨ ⎬= λ ⎜ ⎟⎨ ⎬ ⎝ 0 K˜ SS ⎠ ⎩ u S ⎭ ⎝ M˜ SL M˜ SS ⎠ ⎩ uS ⎭
(7)
where, K˜ SS = K SS − K TLS K LL–1 K LS M˜ LS = M LS − M LL K LL–1 K LS M˜ SS = M SS − M SL M LL–1 M LS − K SL K LL–1 M LS
(8) (9)
in which M LS = V TL ( M LL – M LL K –LL1 K LS ) V S (16) Note that, because most structural problems are interested in lower frequency modes, only a few eigensolution, m, are selected to form the reduced eigenvalue problem. In other words, m1 and m2 modes are selected such that m1
