数理解析研究所講究録
1129 巻 2000 年 23-28

23

Component mode synthesis method for large scale
coupledfluid interaction problem
and its supercomputing$\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}\cdot \mathrm{a}\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{S}\mathrm{t}\mathrm{i}\mathrm{c}$Ichiro Hagiwara (TIT)
A component lnode synthesis methodof estimating vibration responses
efficiently in a high-frequency range is presented. In order to analyze the response in
ratige, a high-and low-order truncatable mode superposition methodais incorporated in the reduction process included in botll the formulation of component
mode synthesis and the solving of the superimposed global equation of motion. By
truncating lower modes, it becomes possible to analyze a systeln with fewer degrees of
freedom than by the existing modal reduction methods. Frequency response analysesof several test problems of damped structures are performed and compared with themultipurpose versatile code,, to demonstrate the efficiency andversatility of the proposed method.
$\mathrm{c}\mathrm{a}\mathrm{P}\mathrm{a}\mathrm{l}3[\mathrm{e}$$\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}-\mathrm{f}_{\mathrm{L}}\cdot \mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{C}\mathrm{y}$$\mathrm{M}\mathrm{S}\mathrm{C}/\mathrm{N}\mathrm{A}\mathrm{s}\mathrm{T}\mathrm{R}\mathrm{A}\mathrm{N}$Key Words:

Component Mode Synthesis Method, Mode Superposition Techniques,
Structural Dynamics, Finite Element Method

1. Introduction
Computer simulations are widely used as design.
tools for vibration problems; in, finitement analysis is one of the most powerful tools for
dealillg with them. In automobile industries, it isapplied to the analysis of the vibration of various
components of the vehicle, . , vehicle body, engine
and power train, and chassis equipments. However,even though their problems are analyzed using supercomputers and can be assumed to be linear, the applications of finite element simulations are still limited.
For example, the finite element,analysis results forvehicle bodies can be snccessftlly applied to frequency
phenomena up to 100 Hz, but they are not applicable$\mathrm{P}^{\mathrm{a}\mathrm{r}}\mathrm{t}\mathrm{i}_{\mathrm{C}}\mathfrak{u}1\mathrm{a}\mathrm{r}$$\mathrm{i}$$\mathrm{e}.$$\mathrm{e}\mathrm{l}\mathrm{e}$in the higher frequency range, because the vehicle
bodies have very complex vibrations when their frequencies becolne high. In order to simulate suchvibrations, very fine meshes are required. This makes
them impossible to compute using computers withstandard storage memory and computing time.
Tlle component mode synthesis method is very.
promising for dealing with such large complex
tures, because, a very large number of degrees of
freedom of the model can be ’educed to a problem ofsmaller size by using the component synthesis tecl)nique,
can be computed using computer withless storage capacity. Since Hurty(t) and Craig and
BamPton presented their papers on the component
mode synthesis method, a number of researches havebeen carried out on this field. The existing methods
have been applied with success to the analyses for the
low-frequency ranges. However, the allalysis for thehigh-frequency range are yet to be accomplished.
Reduction of the number of degrees of freedom using
component mode synthesis is usually basecl on thelnode superposition or Rayleigh-Ritz nlethod. The
number ofof freedom that c\v{c}ln be eliminatedarein the case of the analysis for the high$\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{t}\{\mathrm{C}$$\cdot$$\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{l}\cdot 1$$(-,)$$\mathrm{d}\mathrm{e}_{\mathrm{a}}\sigma \mathrm{r}\mathrm{e}\mathrm{e}\mathrm{S}$$\mathrm{r}\mathrm{e}\mathrm{d}\dot{\iota}|\mathrm{c}\mathrm{e}\mathrm{c}\mathrm{l}$24fi-equency$1^{\cdot}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}$, and it leads to a loss of.
nlethod

lllodes, and approximating thetruncated.
by Eq. (4), thepart of modal displacements
placement vectol [with respect to the physicalcoordinate system can be approximated as follows.
$1^{\cdot}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{n}\mathrm{g}$$\mathrm{e}\mathrm{r}\overline{\mathrm{l}\mathrm{l}}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{c}\mathrm{y}$cleveloped a mocleWe)]
to truncate the lower ntocles as well as theisonesloss of accuracy. Hence, therange can be calcu.
in thedegreesof freedom. In this work, it
lated with fewel
in the reduction process of componentismode synthesis. The efficiency of the method isverified throngh numerical study, and it is applied to
the vibration of a vehicle body.
$11c\urcorner\backslash ;\mathrm{e}$$\mathrm{s}\iota 1\mathrm{p}\mathrm{e}1^{\cdot}\mathrm{p}_{0}\mathrm{S}\mathfrak{l}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{d}\mathrm{i}\mathrm{s}$$q_{i}$$\cdot$$\mathrm{a}\mathrm{l}$$\mathrm{t}1_{1}r\mathrm{t}\mathrm{t}$$\mathrm{e}$$\iota \mathrm{v}i\mathrm{t}\mathrm{h}_{0}n\mathrm{t}$$\mathrm{h}|\mathrm{g}^{r}11\mathrm{e}\mathrm{r}$$n\}$$\mathrm{h}\mathrm{i}\mathrm{g}\mathrm{h}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{q}\iota \mathrm{l}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{y}$$1^{\cdot}\mathrm{e}\mathrm{s}_{\beta \mathrm{S}\mathrm{e}}\mathrm{o}\mathrm{n}$$\{u\}\cong\sum_{i=n\mathrm{I}}’\{l\emptyset\iota.\}_{Q}i+\sum_{i\cdot 1}^{\prime l-}\frac{\{\phi,\}\langle\phi_{i}\tau\rangle}{\omega_{i}^{2}+2i\xi_{i}\omega_{i}\omega \mathrm{C}-\omega_{C}z}\mathrm{t}f\}l1$$\cdot$$\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{o}1^{\cdot}\mathrm{P}^{0}1^{\sim}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{c}1$

used, thelf all the $N$ eigenvectors and eigenvalues
following relations hold with respect to the physicalcoordinate svstem.
$\mathrm{a}\mathrm{l}\cdot \mathrm{e}$2. Mode Superposition Method Adopted
to the Reduction Process$([K]+i \omega_{t}[C]-\omega_{c}[2M])^{-1}=i\sum_{\iota=}^{N}\frac{\{\phi_{i}\}\langle C\beta_{i}^{7}\rangle}{\omega_{i}^{2}+2i\xi_{l}\omega_{i}\omega c-\omega_{C}^{l}}$(6)

the accuracy of the mode
In order tosuperposition technique, an effective procedure is tofor the error
in der to
add extra
.
the
by
of
the
truncation
causedproposed the addition of static response to improveaccuracy. This is effective for the compensation of
the truncated higher modes. However, it is not appli.
cable for the compensation of lower modes. Hence,
the nulnber of degrees of freedom inevitably increases
in the analysis of the high-frequency range. We have,developed an alternative compensation
response
frequency
tothe, the addition of the directmode superposition. In this section, we briefly summa.
rize the formulation of the mode superposition
method.
Finite element equations of the motion of a sys.
tem can be written as:

By substituting Eq. (6) into Eq. (7), an approximation based on the mode superposition is finallyobtained:$\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{P}^{1}\mathrm{o}\mathrm{v}\mathrm{e}$$01$$\mathrm{t}\mathrm{e}1^{-}\mathrm{m}\mathrm{S}$$\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{P}^{\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{a}\mathrm{t}}\mathrm{e}$$\mathrm{n}\backslash \mathrm{o}\mathrm{d}\mathrm{e}\mathrm{s}$$\mathrm{H}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{n}^{\{)}\mathrm{s}$$\{u\}\cong\sum_{i*l1}^{n},\{\phi i\}q_{i}+(([K]+i\omega_{c}[c]-\omega_{c}2[]\gamma \mathit{1}1)^{-^{\iota}}$$- \sum_{i\Leftarrow m}\frac{\{\phi_{i}\}\langle\phi_{i}^{T}\rangle}{\omega_{l}^{\mathrm{Z}}+2i\xi i\omega i\omega_{t}-\omega^{\mathrm{z}}c})n\}\mathrm{t}f$$\mathrm{n}\urcorner \mathrm{e}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{d}$$\omega_{c}$.

$\cdot$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{C}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{r}\mathrm{e}^{\mathrm{t}\}}4$

,

(7)

,

is a parameter of the mode superposition
By setting this parameter to the range of
frequency in which the analyses are carried out, int
proved accuracy is expected.

where

3. Formulation of$\mathrm{i}.\mathrm{e}$$[M]\{\ddot{u}1+[c]\{\dot{u}\}+[K]1u\}=\{f\}$(5)$+ \sum_{i=’ 1+1}^{N}\frac{\{\phi_{l}\}\langle\phi^{\mathfrak{s}}l\rangle}{\omega_{i}^{z2}+2i\sigma^{\mathrm{a}_{i}}\omega_{i}\omega c-\omega_{c}}\mathrm{t}’\}$$\mathrm{C}_{\mathrm{o}\mathrm{m}}\mathrm{P}^{\mathrm{O}}\mathfrak{n}\mathrm{e}\mathrm{n}\mathrm{t}$Mode Synthesis

If a system can be modeled as the assembly ofcomponents, as shown Fig. 1, the corresponding finite
element equations of motion of the system can be$i$expressed as$+[C]$(1)

where, $\{u\},\{f\},$ $[K]$ , $[C]$ ,and $[M]$ are the displacemellt
vector, the external force vector, and the stiffness,damping and mass matrices with respect to the physi.
cal coordinate system. Assuming lnodal damping orofproportional damping, the corresponding
can be transforlned to uncoupled equations
using lllodal coordinates.
$\mathrm{e}\mathrm{c}_{1}\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}$$+[_{K_{pb1}}^{K\cdot.\cdot\cdot.0}0^{\cdot}.\cdot\cdot\cdot.\cdot I\mathrm{f}_{\rho}.\cdot jK_{\rho jb}^{\cdot}1p\iota...\cdot K_{\rho}1bIc_{pj}bK_{\iota}:=$$\mathrm{n}\urcorner \mathrm{o}\mathrm{t}\mathrm{i}_{0}\mathrm{n}$$\dot{q}_{i}.+2\xi l^{\zeta\ell\rangle_{1\zeta}}.\dot{l}i+\omega 2iq_{i}=\langle\phi^{\tau}i\rangle\{\prime 1(i=1,2, \cdots, N)(2)$

are the eigenvector and modal
anddisplacement of the i-th mode. $N$ is the total numberof degrees of freedom. If the external force is anlonotonous sine wave and its circular frequency is ,the displacement response of the i-th mode can bederived from Eq. (2) as

Where,$\{\phi_{i}\}$$q_{i}$(8),

where subscripts , and indicate quantities of component and their boundaries, respectively. Masses
$pj$

$b$

$i$

$\omega_{c}$.

(3)$q_{i}= \frac{\langle\phi_{l}^{\tau}\rangle\{\gamma\}}{\omega_{i}^{2}+2i\frac{\prime}{\mathrm{b}}i\omega_{j}\omega_{C^{-\omega^{\frac{\eta}{c}}}}}$

with respect to the }’ .
The displacement vectol
systemexpressed
as theiscal coordinate
multiplied by thetion of tlle modal displacelllents
eigenvector .
$\cdot$$\{n\}$$\mathrm{p}\mathrm{h}$$\mathrm{s}\mathrm{i}$,$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{l}\cdot \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}$$q_{i}$$\mathrm{c}\mathrm{o}1^{\cdot}1^{\wedge}e\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}_{\mathrm{l}\mathrm{t}}\mathrm{g}$$\phi_{1}\cdot$(4)$\{u)=\sum_{1\approx 1}^{N}/((/J_{l}\}q_{i}$Now, superposing only

$n\uparrow-\mathrm{t}]_{1}(>1)$

to n-th$(<N)$

Fig. 1 Subdivision into$\mathrm{c}\mathrm{o}\mathrm{I}7\gamma \mathrm{p}_{0}$}$\gamma \mathrm{e}\iota 7$

ts

25

are

, and external force isunned to be(consicleredofin
Eq. (8).
effect, from Eq. (8), thetheequation ofabollt the -tll components is$\mathrm{l}\mathrm{t}\mathrm{l}\mathrm{l}\gamma \mathrm{p}e\mathrm{d}$$\mathrm{a}\mathrm{s}.\mathrm{s}$$\mathrm{J}\mathrm{I}1]01\tau \mathrm{b}_{\mathrm{o}\mathrm{t}}\gamma \mathrm{C}\mathrm{l}\mathrm{a}1^{\cdot}\mathrm{i}\mathrm{e}\mathrm{S}$$\mathrm{t}1_{\mathrm{t}\mathrm{e}_{\backslash ^{\mathrm{i}}}n}\mathrm{c}\mathrm{b}\mathrm{S}\mathrm{t}1^{\cdot}\mathrm{u}\mathrm{c}\mathrm{t}_{\dot{\llcorner}}11^{\cdot}\mathrm{e}\mathrm{s}$$\mathrm{d}\mathrm{a}\mathrm{l}27\mathrm{p}\mathrm{i}\mathrm{l}$$\mathrm{N}\mathrm{e}\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{C}\mathrm{l}\mathrm{i}\mathrm{l}\gamma \mathrm{g}$$i$$\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{t}\dot{\mathrm{l}}\mathit{0}0$$\mathrm{d}\mathrm{e}\mathrm{l}\cdot \mathrm{i}\mathrm{v}\mathrm{e}\mathrm{d}$$\{_{l\{}^{-}1=[_{C}/r]\wedge\{q^{\backslash _{J^{+}}}[\hat{Cr}]\{\hat{\prime}\}$

(20)

,

$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{l}.\mathrm{e}$(21)$[\hat{\phi}]=[\sqrt{)}\wedge\ldots- m\sqrt{)}\prime \mathfrak{j}]$$l_{(} \overline{q}\}=\langle_{l\prime}^{-\sim}\int’|\ldots q,t\rangle$$(22\rangle$$[ \hat{G}]=([\hat{K}1+i\omega_{c}[\hat{c}]-a).\frac{\prime}{\iota}[J|\wedge f])-|$

as$-. \sum_{l\iota=n}^{\prime 1}\frac{\{\dot{\phi}_{1}\}\langle\hat{\emptyset}17\rangle}{\hat{\omega}_{\mathrm{i}}^{2}+2i\overline{\xi}_{i}\hat{\omega}|\omega_{c}-\omega_{\mathrm{c}}^{2}}$$[.\mathfrak{h}I,).;]|\ddot{u}_{lj}\}+[I1\prime \mathrm{j}]r\mathrm{t}\iota l,J\mathrm{j}\}=-[I\sigma_{\rho j}b]\{u_{b}\mathrm{I}.\cdot$(9)

then apply the mode superposition
Eqs. (7) andin the previous section.
(9), thetransforming relation for red .
tion ofof degrees of
inside the$\lambda\prime 1^{\gamma}\mathrm{e}$

.

(23)

$\mathrm{n}\tau \mathrm{e}\mathrm{t}\mathrm{l}\overline{\downarrow}\mathrm{o}\mathrm{d}$$\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{c}\mathrm{l}\cdot \mathrm{i}\mathrm{b}\mathrm{e}\mathrm{d}$4. Numerical Examples$\mathrm{F}1^{\cdot}\mathrm{o}\mathrm{m}$$\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{l}$$\mathrm{t}\mathrm{l}\mathrm{C}$$\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{n}(\iota\iota 7\mathrm{l}\mathrm{b}\mathrm{e}\mathrm{r}$con7pone11ts is$\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{e}\mathrm{d}\mathrm{o}\mathrm{l}?\urcorner$$\mathrm{o}\mathrm{l}\supset \mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}$:4. 1 Analysis of simplified veh\’icle cabin
The component mode]
mentioned above isto(ver.67)(5) by modifying the NASTRAN source(DMAP; Direct Matrix
code,Inverifyorder tothe efficiency of the proposed,$\mathrm{s}\backslash .’$\mathrm{i}\mathrm{l}\iota 7\mathrm{p}$$\mathrm{n}7\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{d}$\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{s}$$\mathrm{M}\mathrm{S}\mathrm{C}/\mathrm{N}\mathrm{A}\mathrm{s}\mathrm{T}\mathrm{R}\mathrm{A}\mathrm{N}$$\mathrm{e}\mathrm{i}\urcorner \mathrm{t}\mathrm{e}\mathfrak{j}\mathrm{l}\mathrm{t}\mathrm{e}\mathrm{d}$$\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{l}\cdot \mathrm{a}\mathrm{m}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{g}\rangle$$==.[\tau]\{\overline{u}\}$(10)$1\mathrm{n}\mathrm{e}\mathrm{t}1_{10}\mathrm{d}$numerical study using a simplified model wasout. Figure 2 (a) shows the finite element mesh of themodel. Each panel of the cabin is modeled as a$\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{l}\cdot \mathrm{i}\mathrm{e}\mathrm{d}$where$[Gfl\mathrm{j}1=(([K\rho J1^{-}\omega_{c}^{2}[Mpj])^{-1}$(11)$- \sum_{i\Leftrightarrow\prime}^{;}|,,\frac{\mathrm{f}\phi_{i})\langle\phi^{i},\rangle}{(\omega_{i}^{\beta j})^{2}-(\omega c)^{2}})[Icpjb]$(12)$[\phi_{\rho j}]=[\phi,\rho n^{J}\ldots\phi_{n}^{p\mathrm{j}}]$

(
(13)
.
Substituting the transformation relation (Eq. (10))
into the globalof motion (Eq. (8)),
following equations with reduced degrees of freedom$qpi\}=\langle q_{i}^{\rho}n^{j}\ldots qn\rangle^{r}FJ$$\mathrm{t}\mathrm{l}\tau \mathrm{e}$$\mathrm{e}\mathrm{c}_{\mathit{1}^{\mathrm{u}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}}}\mathrm{o}$are obtained:

,

(14)

$[\hat{M}]\{\hat{u}..\mathrm{I}+[\hat{c}]\{\overline{u}.\}+[\hat{I\zeta}]\mathrm{t}\hat{u})=\{\overline{f}\}$where[ $\overline{M}|=[\tau 1\gamma[M][T1$

(15)
(16)
(17)
(18)

$[\hat{C}]=[r]T[C][r]$$[\overline{I\mathrm{f}}]=[\tau]^{\tau}[I\mathrm{f}][T]$$\{\hat{f}\}=[\tau]^{\tau}\{\prime \mathrm{I}$.

In Eq. (14), the degrees ofwith respect to thephysicalate system inside the components aretransformed into odal coordinates, but, the degree of
freedoln of the boundary of the components still
ains. When the model is very large,
number ofdegrees of freedo171 of the boundary are also large.
Therefore, the mode superposition
described
in section 2 is applied again in order to reduce theboundary.
number of degrees of freedom ofAssuming
dantping, eqnations of motion of
Eq. (14) can be rewritten,
respect to the modal$\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{e}\mathrm{d}_{\mathrm{o}\mathrm{m}}$$\mathrm{c}\mathrm{o}\mathrm{o}\Gamma \mathrm{d}\mathrm{i}\mathfrak{j}\gamma$$\mathrm{n}\gamma$(a)$\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{e}$$\mathrm{r}\mathrm{e}\mathrm{n}\gamma$

Flnlte element mesh$\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{d}$$\mathrm{t}\mathrm{l}?\mathrm{e}$$1\mathrm{n}\mathrm{o}\mathrm{c}^{\rceil}\mathrm{a}\mathrm{l}$$\mathrm{w}\mathrm{i}\mathrm{t}]_{\urcorner}$coordinate, as .

) $(_{l}.=1,2, \cdots,\hat{N})$ , (19)
total number of degrees of freedom at, andare the displacelnent of$\dot{\hat{q}},\cdot+2_{\mathrm{b}}\hat{\epsilon}_{\hat{\omega}_{\iota}}t..\dot{\hat{q}}j+\hat{\omega}_{\iota}\overline{q}\grave{i}2=\langle\hat{\phi}_{\iota}7.\rangle\{\hat{\gamma}$

wbere isthis stage.
$\hat{N}$$\mathrm{t}\mathrm{l}7\mathrm{e}$$\hat{q}_{i},\hat{\omega},\cdot,\overline{\xi}_{j}$$17\urcorner \mathrm{o}\mathrm{d}\mathrm{a}\mathrm{l}\mathrm{c}\mathrm{o}\mathrm{o}\mathrm{l}\cdot \mathrm{d}\mathrm{l}’ 1\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{s}$, the$\hat{\phi}_{i}$$\mathrm{c}\mathrm{i}\mathrm{l}\cdot \mathrm{c}\mathrm{u}$]$\mathrm{a}Y\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{f}1^{\cdot}\mathrm{e}\mathrm{Q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}_{\}};,$$\mathrm{t}11e$.
damping ratio, and theThen,appiying the mode stlperposition method.
}
superposing
described in the section 2of . (19) the ]]to n-tli
reduced degrees of freedom can be derived:$\mathrm{n}\urcorner \mathrm{o}\mathrm{d}\mathrm{a}\mathrm{l}$$\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{e}\mathrm{n}\iota’ \mathrm{e}\mathrm{c}\mathrm{t}01_{\}}\mathrm{r}\mathrm{e}\mathrm{S}\mathrm{P}^{\mathrm{e}}\mathrm{c}\mathrm{t}\mathrm{i}\backslash \gamma \mathrm{e}$$1\backslash ’$$‘\tau \mathrm{g}\mathrm{a}\dot{\}}$$\iota 17\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{S}$$7,$$\mathrm{a}\mathrm{l}\mathrm{Y}\mathrm{d}$$n\uparrow-$$\mathrm{f}\mathrm{o}$$\mathrm{E}\mathfrak{c}\downarrow$$\mathrm{o}\mathrm{v}|\mathrm{i}\mathrm{l}1^{\cdot}\mathrm{e}\mathrm{I}\mathrm{a}\mathrm{t}\mathrm{i}_{0\mathfrak{l}}\mathrm{t}n’ \mathrm{i}\mathrm{t}\mathrm{h}$$(|3)\mathrm{s}_{\mathrm{t}1}\mathrm{b}\mathrm{d}\mathrm{i}\backslash ’\cdot \mathrm{i}_{\mathrm{S}\mathrm{i}_{\mathrm{o}\mathrm{t}1}}$$\}$$\Gamma \mathrm{i}\mathrm{g}$.

$2$into components$\mathrm{s}\mathrm{i}1\gamma\gamma \mathrm{p}^{]\mathrm{i}\mathrm{f}\mathrm{l}\mathrm{c}}\mathrm{d}\mathrm{t}\iota\cdot n\mathrm{c}$} cabin$\backslash$$1170\mathrm{C}\mathrm{l}\mathrm{e}$]26conlponent, as shown in Fig, 2 (b). Nulnerical analy(ver.67) superelement
sis using theTable 1$\mathrm{C}_{0\Pi}\rceil \mathrm{p}_{\mathrm{c}},T1^{\cdot}\mathrm{i}\mathrm{f}_{\Gamma \mathrm{o}\mathrm{a}r}|_{\backslash \mathrm{m}}\mathrm{S}\mathrm{o}11$of the number of degrees of$\mathrm{M}\mathrm{S}\mathrm{C}/\mathrm{N}\mathrm{A}\mathrm{s}\mathrm{T}\mathrm{R}\mathrm{A}\mathrm{N}$is also carried out for comparison with theproposcd method. For both analyses computed by the
proposed method and NASTRAN, the componellt
modes with frequencies over 600 Hz are truncated in. In the case of
the process of modeusing the proposed method, component modes below
100 Hz are also truncated, and the frequency parameter of the mode superposition method is set to 250 Hz.
Figure 3 shows the frequency responses on several
nodal ints. Table 1 shows the number of degrees offor both cases. Comparison of CPU times
requirecl in this study is summarized in Table 2. In
this simple case study, CPU time required in the
proposed method in the process of reducing the$\mathrm{m}\mathrm{c}_{\mathrm{c}}arrow \mathrm{h}\mathrm{o}\mathrm{d}$$\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{l}\mathrm{P}^{\mathrm{O}}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{p}o$$\mathrm{f}\mathrm{i}\cdot \mathrm{e}\mathrm{e}\mathrm{d}\mathrm{o}\mathrm{n}\mathrm{l}$

1a

$l$

$\iota \mathrm{u}’$$\perp \mathrm{t}\circ \mathrm{C}\mathrm{l}\mathrm{e}$

‘vuuc

A

$(^{\{}, )1\backslash \mathrm{O}\mathrm{C}\mathrm{l}\mathrm{e}\mathrm{C}(L^{-}\mathrm{c}11\Gamma \mathrm{e}\mathrm{c}\mathfrak{c}1\mathrm{o}\mathrm{n})$$(L^{-}\mathrm{c}11\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{L}\mathrm{l}\mathrm{o}\mathrm{n}’$)\lfloor lCCtl\cup ll)$\omega\backslash L^{-}\mathrm{u}\iota\iota \mathrm{t}:\llcorner 1l\circ\iota_{\overline{1}}$Fig. 3$\backslash \llcorner 1/$$\mathrm{F}\mathrm{r}\mathrm{e}\mathrm{c}\downarrow^{\iota}1\mathrm{C}\iota 1\mathrm{C}\mathrm{y}$responses at several$1_{\wedge}^{\urcorner}\mathrm{o}\mathrm{d}e\mathrm{s}$$1\mathrm{t}$Utlc V$\backslash L^{-}\mathrm{u}11\mathrm{C}\llcorner\llcorner 1\cup 11f$

27$\neg_{\mathrm{r}}^{\backslash }1.\mathrm{l}_{\wedge}’)$$\cap-\wedge\sim\cdot \mathrm{s}arrow \mathrm{r}:----\mathrm{f}\cap \mathrm{D}\mathrm{r}\mathrm{f}1:-arrow$

Fig. 4(a)(c)$55\mathrm{H}\mathrm{z}$of degrees of freedom in the substructures
is longer than when using NASTRAN, because thenumber offor generating the transfor.
mation matrix is increased even though the total
nunlber of component modes is decreased. However,
the total number of degrees of freedom of the systemafter theof degrees in the substructures isreduced due to the truncation of lower component
lnodes, as shown in Table 1, and this leacls to thedecrease in the CPU time
for the eigenvalue$\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{l}\supset \mathrm{e}\mathrm{r}$$\mathrm{c}\mathrm{o}\iota 11\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{s}$$e1\mathrm{i}111\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{i}_{0}\mathrm{n}$$\mathrm{r}\mathrm{e}\mathrm{c}_{\mathrm{l}\mathrm{u}\mathrm{i}}\mathrm{r}\mathrm{e}\mathrm{d}$Finite elen\urcorner ent mesh of the vehicle body

$65\mathrm{H}\mathrm{z}$analysis of the system.

Therefore, in the present, the total CPU time needed for the proposed$\mathrm{s}\mathrm{a}\ln_{\mathrm{P}}\downarrow \mathrm{e}$method is less than half of that required when using
NASTRAN.
4. 2 Analysis of real vehicle cabin model
The method is applied to the vibration analvsis of
a 4-door sedan model. The finite elelnent lnesh of themodel is shown in Fig. 4. The total lltlllber of nodes
is 3638. Shell, solid, beam, spring, rigid, alld
vehicle bodynlass elelnents are used for modeling$1\iota\iota \mathrm{m}\mathrm{p}\mathrm{e}\mathrm{d}$$\mathrm{t}\mathrm{l}\urcorner \mathrm{e}$28of elelnents is 5391. The mesh size
The totalof this 171ode\downarrow is rather coarse for analysis in the high, but the analysis is carried out infrequency
orcler to verify the versatility of the program implemenled by the proposed method. The vehicle body
panels were divided into 18 substructures (e.g., roof
panel, floor panel, door, engine compal tment, etc. ).
Input vibration is applied to the point of the frontof the suspension. The strainright shock
distribution of the vehicle body with input frequenciesof 55, 60, 65, and 70 Hz are shown in Fig. 5.
$\mathrm{n}\mathrm{u}\mathrm{n}\rceil \mathrm{b}e\mathrm{r}$can be performed effectively using the present
method.
References$\mathrm{r}\mathrm{a}\mathrm{l}\tau \mathrm{g}\mathrm{e}$$\mathrm{a}\mathrm{b}\mathrm{s}\mathrm{o}\Gamma|3e1$

5. Conclusion
A component mode synthesis nlethod is presented
in a high-frefor analyzing the vibrationquency range. It is based on the mode superposition
nlethod that is able to truncate low modes as well as
high ones. The results of this method are comparedwith the multi-purpose code
through numerical study. Several sample solutions
show that the analysis in the high-frequency range$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{n}\tau \mathrm{S}$$\mathrm{M}\mathrm{S}\mathrm{C}/\mathrm{N}\mathrm{A}\mathrm{s}\mathrm{T}\mathrm{R}\mathrm{A}\mathrm{N}$Hurty, W.C., Dynamic Analysis of Structural
System Using Component Modes, AIAA Journal,
Vol. 3, No. 4 (1965), p. 678.
( 2\rangle Craig Jr, R.R. and Bampton, M.C.C., Coupling ofStructures for Dynamic Analysis, AIAA Journal,
Vol. 6, No. 7 (1968), p. 1313.
(3) Hansteen, O.E., On the Accuracy of the Mode,
Superposition Analysis in Structural
Earthquake Eng. Structural Dyn., Vol. 7, (1979)
p. 405.
(4) Ma, Z. and Hagiwara, I.,
of New
for Truncating
Mode-Superposition
Higher Frequency Modes,
The LowerPart 1.’ Frequency Response Analysis for
Damped System, Proc. JSME, (in Japanese), C,
Vol. 57, No. 536, (1991) p. 1148.
(5)
User’s Manual, August, (1991),-Schwendler Corporation.
(1)$\mathrm{D}\}^{f}1\rceil \mathrm{a}111\mathrm{i}\mathrm{c}\mathrm{S}$$\mathrm{D}\mathrm{e}\mathrm{v}\mathrm{e}\downarrow \mathrm{o}\mathrm{p}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{t}$$\mathrm{T}e\mathrm{c}\mathrm{h}\mathrm{n}\mathrm{i}\mathrm{c}_{[}\mathrm{u}\mathrm{e}$$\mathrm{a}\mathrm{n}\mathrm{d}/\mathrm{o}\mathrm{r}$$\mathrm{M}\mathrm{S}\mathrm{C}/\mathrm{N}\mathrm{A}\mathrm{s}\mathrm{T}\mathrm{R}\mathrm{A}\mathrm{N}$$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{N}\mathrm{e}\mathrm{e}\mathrm{l}$