CA2020144C - Apparatus for and method of analyzing coupling characteristics - Google Patents

Abstract

Abstract of the Disclosure:
Characteristics of a coupled system, which consists of a plurality of unit structures coupled together, are analyzed accurately and quickly by providing, between calculating means for calculating transfer function matrices concerning unit structures and coupling means for coupling togethe transfer function matrices according to definitions of coupling, co-ordinate conversion means for converting transfer function matrices into those in the overall system.

Description

2 0 2 0 1 ~ 4 This invention relates to the analysis of characteristics of an overall structure consisting of a plurality of sub structures coupled together and, more particularly, a system for and a method of analyzing characteristics of an overall structure such as a structure vibration simulation system, which analyzes vibrations of a `i~ ''''i'~
plurality of structures either theoretically by a finite element method utilizing a computer or experimentally by . ; ;.
using a conventional FFT analyzer used for vibration analysis : ~;
and estimates vibration characteristics of an overall .
structure obtained by coupling together these structures by using a computer before actually manufacturing such overall -;
structure. .
'~ " ' '' Aspect~ of the prior art and present invention will be ;~
described by reference to the accompanying drawings, in .. ;~
which~

~ig. 1 is a block diagram showing a structure vibration si~u1ation system as an e~bodiment of the invention; ' ; f'`~'' Fig. 2 is a block dia8ram showing a structure vibration ; ;,~
s1wu1ation syste~ as an embodiment of the invention: - ;

';','.,~ " " .,~,.~.'.

~ ~.

- 2~20~4 ;~,~,.. .
- la -Pig. 3 is a view showing an example of sub structure co~bination~
Fi~. 4 is a view showing Euler angle;
Pig. 5 is a flow chart illustrating the operation of co~
ordinate conversion ~eans~
Pig. 6 is a view for explaining definition contents of an e~bodirent of the invention;
Pig, 7 is a view showing an overall syste~ transfer ;
function ~atrix in an embodi~ent of the invention;
Fig. 8 is a flow chart for explaining Ihe operation of the structure vibration simlllation syste~ as e~bodi~e~t of the invenlion; . ~i , Pig. 9 is a view for explaining si~ulation of a raiIway car; - ~. ~
Pig. 10 is a view illustrating a ~ethod of coupling .
transfer function ~atrices; .~
Pig. 11 is a block diagra~ showing a structure vibration .~ . YS
si~ulation syste~ based on a prior art suh structure ..
synthesis ~ethod; . .
Pig, 12 is a view for various definition contents of ..
explaining a structure vlbration si~ulation syste~ based on . .
a prior art structure synthesis ~ethod;

--- 2 ~ 2 0 1 ~ 4 . . ,;
..` . .-.. ~ - . ~ . .
- lb -: .. :-. , Fig. 13 is a view showing a constraint function ~atrix :;;```
based on the prior art sub structure synthesis method: and Fig. 14 is a view showing an overall system transfer function ~atrix based on prior art sub structure synthesis :~
method. ~ r..
~., ~
An overall structure characteristics analyzer will now be described in conjunction of a structure vibration ,:
simulation system. Computer-aided engineering (CAE), which is adopted in the design of structures, particularly machine structures for performing modeling and simulation of , :~ -structures with a computer before trial manufacture, has . ~ ~

'; ' . ", `' ,, . ,.~ ", " ,. ..

, ,: :' ,` ~, '.' `;.!. ';
: .""~. ~'`," ;,'' ~',' ':. . ". ., . .
'. ,' ... .., . '. ' ~ `, i '"'"~.`';'.

:.,-: . ,, i , ...... ... . . . .. . .. . . ~ .. ~-. . . ; i .. , ., .. ...... . ., .. ~ ., ., . . .... , . . ., .. .. , .. . .. ,,,, . , " , 202~144 l~een a~lra(:ling attention as powerful means for ot)laining ~ `
I rednction of dt~velopmenl time and cost. In the CAE, - :
vihr.J~ion analysis is impor~anl as a reliabilily evaluation method in the design of machine structures. As prior art mettlol3s of vihration analysis of machine structures, there are an experimental FFT (fast ~ourier transform) analysis method and a finite element method as theoretical ana]ysis method. ~urther, there is a sul) structure synthesis method (or a building block approach) as disclosed in Japanese Patent Application 63-060766. in which the experimental PFT
analysis and theoretical analysis based on the finite element method are performed witb respect to each element (sub structure) of a machine structure for analysis, and the ¦ results of the analyses are used to numerically simulate ~ ~;
vibration characteristics of the machine structure (overall I structure).
In the experimental P~T analysis method, operations of causing vibrations of a machine structure, measuring responses at this time, sampling these signals with an ~-n .. i~i~
converter, supplying sampled digital data to a minicomputer ~ 1 or mlcrocomputer. performing ~T of these data and ;;
producing transfer function between vibration application ','"i ' point and response point, are performed repeatedly for various points of the machine structure, and modal parameters 911Ch as peculier vibration frequency of the ,' Z ` _ 3 _ 2 ~ 2 ~ 1 ~ 4 , .,~ ~,:,.. '' .' str~lcture, damping ratio and vihration mode l)y ~eans of cllrve fitting (or modal analysis) This method is used as ~ -im~)orlanl means for obtaining vil)ration characteristics of the actual structure.
. ~
The finite element method. on the otiler hand, is a method of theoretical analysis utilizing a computer. Jn this method. a machine structure is thollght to be apable of being expressed by a collection of a finite number of finite elements. Relation between externally applied force and resultant deformation is obtained for each element.
These relations are used to define a displacement function concerning the relation between externally applied force to and resultant deformation of the overall machine structure.
Using this displacement function. stiffness matrix ~ K) and mass matrix ~ M ) are obtained to solve a eigen value i~
problem given as ~M ) ~x} + ~ K) (x) = (O) (1) where ( x ~ represents a displacement vector, and - a second order time differential. ~lso. the peculier vihration frequency and vibration mode of the structure are obtained. ~urther, an equation of motion given as ( M ) (x } -~ ~ C) ~x~ + ( K ~ (x } = ~fJ (2) where ( M ) represents a mass matrix, ~ C ) an damping matrix, ~ C ) = ~ t M ) + ~ ~ K ) , ~ and ~ damping ratios, ( K ) a stiffness matrix. ( f } external force, '.! . .~
. ,,.; ~"~.....

!
I ;~

202~144 ~

. :~
( x ) , ( x ~ and ~ x ) displacement, velocity and ~ acceleration vectors, is solved to obtain resporlse analysis j of each elemenl.
The sut) structllre synthesis method is one, in wbich experimental ~T analysis and theoretica] analysis based on the finite element method are performed with respect to each element (or sub structure) of the machine structure for analysis, and results of the analyses are numerically simulated. ~ specific example of this method will now be described with reference to ~ig. 9.
. . . ..
~ ig. 9 illustates a simulation concerning a railway car design. Referring to the ~igure, reference numeral 100 designates car body, 101 chassis, and 102 and 103 local bases A and B. These parts constitute elements of the railway car. Designated at 110 to 11~ are examples of vibration characteristic of car body 100. chassis 101 and i local ba9es A 102 and B 103. respectively, 121 an example of ~`
vibration characteristlc of overall system obtained by sub structure synthesis method 120. In the graphs of the ~-vibration characteristic examples, the ordinate x is taken `;~ ";
for vibratlon response. and the abscissa f for tbe .
frequency. Designated at 200 is a co-ordinate system, in which the rallway car is found This co-ordinate system ` ;~
represents a three-dimensional co-ordinate space defined by perpendicular x, y and z axes. Designated at 11 to 14. 21 ~,,ï,,':",',.~:,:,.,,,' `,:," ` :.'' ' ;~
',,.,, .-".....

. :'; ' ' '"~'' ,' ',. ,'''' ~"

2 0 2 ~ ~ 4 ~ ::
5 ~
-. ' to 24 and 31 to 34 are points of measurement se]ected in car ;~ -body 100 and chassis 101. Designated at A and B are `~
selected points of measurement in chassis lO1 and local -`
hases ~ 102 and ~ 103. Points of measurement designated by ~;
¦ like reference numerals or symbols constitute a poiat of coupling when the individual elements are coupled together. ~`
Generally, vihration response at one poinl of measurement d-~
may be examined by considering t!~e following six diff~rent ~ `
I directions as shown in co-ordinate system 200 (1) Direction (x) of the x axis, (2) Direction (y) of the y axis, (3) Direction (2) of the z axis, ~ g`

(4) Direction (p) of rotation ahout the x axis, (5) Direction (q) of rotation about the y axis, and (6) nirection (r) of rotation about the z axis.
These directions are referred to as degrees of freedome. -Thus, there are at most six deerees of freedom at one point of measurement. In a system, in which the directions p, q and r of rotation can be ignored, there are only three deerees x, y and z of freedom. ~3urther, where only a spring undergoes a vertical motion, there is only a single degree of freedom (in the sole direction x, for instance), ` ;`
Now, the determination of transfer function which is i~;
extensively used for analyzing vibration characteristics of ; ;
elements will be described.

:. . .

.' ~
~:~'~ ,''.','.

J! 2 0 2 ~ 1 4 4 fi .
If 12 points 11 ~o 34 of measurement in car body 100 ~ach have three degrees x to z of freedom, there are a total of N = 12 X 3 = 36 degr~,es of freedom. ;;~
ln this way, one or more points of measurement with a ~ -lotal of N degrees of freedom are selected in a structure, with numerals 1, 2, N provided to desi~nat~ the i individtlal degrees of freedom, and l)y setting a degree m of freedom to be a direction of response and another degree Q
of freedom to be a direction of vibration application, ~ ~t~
vibration of a predetermined waveform (the vibration being ~;
exI)ressed as disI)lacemHnt, velocity or acceleration of the pertinent point of measuIement) is applied in the direction '``
of vibration application, and vibration in the direction of response is measured. ~!.,';`."''~, In this case, the frequency spectrum of vibrations in ;~
the direction of vibrat;on application is wel1 known, and as for the frequency spectrum of vibration in the direction of vihratlon applicatlon the vibration transfer function ll between the dlrection Q of pressure application and ,,~
direction m of response can be expressed as function H
(~) of angular frequency ~ to determine H ~ ), Hm. .~
Z (~). ~ . H~.N (~). ~urther, there are characters - i that Hm, m (~) = 1 and Hm,~(~) = H~, m (~)~ which is ~;
referred to as theorum of reciprocity.

, ; `.,'.:,;~',,''', ..,., . ,~,,."
, '.'.;` ''~ '..

I

`` 2~2~44 - 7 ~

I?his transfer function llm, ~) is set as N x N matrix . .
:~: to obtain transfer function matrix (G( ) ) with respect to ~
sub structure, given as ~ P,' ( k ) ~ ` ~ ' ;'."

~;." .~ '. ' '. .
H ~" (~,2(~ . N ( ~ ) H z" (~ ) 112.2(~)"~ . , (3) ~ .
H ~,,(~ ) H ~;~ ) , Ttlen, by denoting force f~ applied in each direction ~ -of vibration application, externa] force vector F~k~ with respect to sub structure k in directlons 1 to N of vibration application is expressed as An equation of motion expressed by transfer function for each local structure is ~ G ~K) ) { p Ik) } = ~ X ~k) ~ .. -----------------(5) ;` ~
whsre ~ is external force vector given to de-gree of freedom of sub structure k, { X (k) } iS displacement vector of degree of : -. freedom of sub structure k, and G ~K) ): i9 transfer function matrix of sub ~: -structure k. .
,' :, ' ~,,' '.-,,' ~ "

ë: ' ;;

202~ 44 .~.

Tlle transfer function calculated here concernslransfP,r function matrix (Gtk ) ) of compliance (displace-ment/force), and for conversion lo transfer function matrix (11(~) ) of dynamic stiffness (force/displacement), it is conver~d to transf er function of dynamic stiffn~ss by .,i.,,, obtaining inverse matrix by using an equation --(Il(k ) ) = ~G(k ) ) 1 .................... -.------(6) In this way, an equation of motoion given as ~ ~i (Il(k ) ) ~ x~k' ) = I ~'~' ) (7) is eiven for each local slructllre.
Now, a method of obtaining an overall system equation by coupling together the individuul sub structllres will be descrii)sd.
Eirsl, a method of ohtaining a system by couplin~
toeether two sub structures ~nd producting an eqllation of i~`
motion of the system will be described. Of the two sub structures, the degrees of fresdom are classified to be those ( ~ x m ~ X m ~2~ ) ) where a further sub ; ;`
structure is coupled and those ( ~ x r (1> ) ~ { X r ~2) } ) where no furttler sub structure is coupled. Eor example, X m ~1) and Xm ~2) may be thought to he the degrees of freedom of points of measurement which show tlle same response when and only when sub structures are coupled together by bolting. Equations of motion of Individual sub structures are given as ;;

.. ....

~ . .

s 202~14~
~ 9 ~
,~;4~
~ ' i'"''' ' (I) ~1~ ~ ~1) r ~1) ~'`'`''',"''''`
11 r r ( ~ ) 11 r m t ~ ) J x r J li r ~I> ~> 1 ~I> = 1 sI> - (8) 11 m r ( O ) 11 m m ( ~J ) ~ x m l l;` m -, ~
~Z> ~Z) t2) ~ ~2) ~.. `~.,:,.
11 m m ( 1) )I I m r ( ~) ) ¦ x m J T~ m , ~2) ~2) ~ ~2) = ~ ~2)'''(9) ~ "
Tl rm ( ~ )11 rr ( b:~ ) l X r l 1; r ~

When ~x m ~1> ) and ~x m ~Z> } are coupled to~ether, the equation of molion of lbe individual sub structures are now ~1> ~1> ~1> ~I '''~
11 r r ( ~ I r m ( c~ ) ¦ x rS I ) = ¦ ~ rS I ) '' ~) ~I mr ( ~ I mm ( ~) ~ ¦ x m ¦ ~ m - P ~d ~2) ~z) ~ ~Z) ~ ~2) ' "'."~
. IT mm ~ ~ ) H mr ( ~) ) l x m J 1~` m tp ~z) ~2) ~ ~Z~ = ~ ~2~'''~ .`. ,~i':".. , H r m ( O ) H r r ( ) l x r l F r : :
"~
wllere ~ P} represents force applied by sub structure 1 to sub stru(,tllre 2 and l-p~ represents force applied by sub structure 2 to sub structure 1. By removing p by using equaitons (7) and (11) and a coupling condition .;`~
( x m ~ I ) } = I x m ~ ) } ( = x m ~ ) (12) .

`'. ''~
,~ " ,.~.
,,,:. ~',.....
. ' '''~` ,'.'''''~'~', ~,'',',' :':.
.. , ,~, .
:'.. :.,:
. , ~" .

. ,, '~ '.. ..
, , ~

- 1 o- . ~
:. . -,. -.;..
we obtain "' t~
~Irr (~)) 11 rm (~1)) O . . - .
~ 1 ) (2) (2) . ~
~I mr (~:u ) ~I m~)) + H mm (~)) H mr (~v ) . . .
O H rm (~t)) 1~ rr ( ~>) ¦ x ~ ~ ¦ F r ~ ............. -- n This equation is an equaiton of motion of the system `~ .
obtained by coupling together the two sub structures. Here, ( ~ m ) is ~ ~ m J - ( ~ m ~ { F m ~2) } and represents external force acting on coupling point ( x m ~ . The method of producin6 the equation of motion of the system obtained by coupling together two sub structures can be seen by regarding the coefficient matrix of equalion (13) as ~ 1~ rr (a)) 11rm ( ~)) ........................... ..... , ¦
~ 2~ ~2~ , . ~
', Hmr (~)) Hmm (O) '~ ~- Hmm (O) Hmr (~J) -~ ~
.................... ~ . ~2~ ~2~ `.

sub structure 1 sub:structure 2 .
Coupling between sub structures 1 and 2 ,. . .-:,.,j., An equation of the overall system is produced by , " ~

," ''- - .
~, , ~ . . ..
'~ ' . '~: , 2 0 2 ~ ~ 4 ~

~":,.: ......
'~''~
coui)lin~ logelher sub str~lctllres in lhe method described ahove in the order of input of tlle suh str~lctllres. When the overall syslem is produced by coupling together N sub r~ ~ r slrllclutes, (~1 ( o ) ) of lhe system is as sh()wn in ~ig. -`
1(). , ~
In lhis way, we ohtain lhe eq(lation of lh( overall system us (~ " ) (x) = ~) n~
This e(luation is applied in case where a degree of freedom provides response independent of other degrees of freedom.
Actually, there is a case where a certain degree of freedoM
is deI~end ~o a different degree of freedom. This relation is referred to as restrictive relation of degrees of freedom.
This constraint relation will now be described with reference to ~ig. 9. When there is a vibration in x direction of point A of measurement (i.e , Ax direction) and also there is vibration res~ponse in x direction at point 11 of mea9urement (i.e., 11x direction), Ax is referred to as independent degree of freedom, and 11x is dependent degree of freedom. The constraint relation in this case is 12 times, Concernin8 such constraint relation a constraint relatlon matrix ( r ~ is produced. Denoting displacement vector of an independent degree of freedom as ~ Xl } and ,''''': ' ', ~',, :, ,-, , ' .~
.~ "~ .

2 0 2 0 1 ~ 4 displacsment vector of a dependent degree of freedom as Ix d ) , W ~' tl.lv~
~ X ~ x I ) ........................... (15) as constraint relation e~uation.
Thus, it is necessary to substilllle lllis constraint relation into overall systeM equation (14) to obtain correct simulatioIl of response. Before describine a method of sul)stitlltion of the constraint relation into the overall system equation, a way of thinking of co-ordinate conversion utilizing this constraint relation will be described.
The introduction of the constraint relation as noted above is used in case where tllere is a dependent relation among the degrees of freedom of the total structure.
lleretofore, this constraint is utilized for the co-ordinate conversion to be described later.
In the conventional sub structure synthesis method, if a co-ordinate space for analyzing sub structure data by the finite element method or experimental FPT analysis and co-ordinate space where the overall structure is placed are different from each other, the co-ordinate space when performing the sub structure analysis and co-ordinate space where the overall structure is placed are relatred to each other by using the constraint relation amon8 degrees of freedom. ~or example, data obtained by measurin~ a certain ,~
. , ~

`:
2 0 2 0 ~
- I 3 ~
. .,- ,-Sllb structure with a certain co-ordinate system can not be nsed llireclly if the co-ordinate system is different from lhe co-(>r~ ntt~ system of the overall structurt3 incl~lding llle suh slruclure. For example wllen data is oblained with lht3 sub struct11re held in a horizonlal state and lhe Sllb s~r1lclure is coupled at an an6]e of 45 degrees it is neot3ssnry to (onvert the data to those in cas~ whert3 ~ho~ 9ub s~ructure is at an angle of 45 degrees.
An noted before the displacement vector ( x I ) of independent freedom degree at ths time of the sub structure analysis and disp]acement vector ~ x d ) of dependent ~Ub freedom degree of the ~4e~ structure in the co-ordinate space where the overall structure is placed are related using constraint relation matrix ~ r ) as {Xd ~ = ~ r ) {xl } -----(16) It is noted hefore that ~ r ) is given absolutely by the relation hetween ( x 1 ) and (x d ) ~X~r-t in ëquation (16) are in a con9tralnt relation among degrees of freedom.
Into this constraint relation the constraint relation of points and positional relation of co-ordinate systems for co-ordinate conversion are suhstituted.
Now, a method of substituting this constraint relation matrlx ~ r ) into the overall system equation will be described.
',',.,. ~'":
:: ,.,~,.~

!,~, 2 0 2 ~ 1 4 ~
, - 1 4 - -~;
.., ~
-. . ,,. :; -In the sut) structur~ syntllesis method, the constraint r(~lation is applied to overall system equation (14) as -:
follows.
Pirst, disI~laceMent vectors ~ x ~ of freedom degrees of .
Ihe overall system are divided into { x d ¦, ~ X ~ ¦ and X r ) ~ t~lat i ¦ X d ~ x) = ~ x, (17) ^`i-~ Xr Ilere, ~ x r ~ represents degrees of freedom wtIich do not .
appear in constraint relation equation (16). With this division of freedom degrees, equation (14) is changed to `~

1 ¦ 1 ¦ 1 `' . ' ~
H ~ d H 11 I-llr ~ X I = ~ 1;`l ----- ns~
H rd H rl H rr l X r l li`r .,~",, ""
Constraint relation equation (16) means that restrictive .. ~:~
forces ~ P ) and ~ P I ~ act on (~ J and { ~
Thus, when there is a constraint relation, the overall ',"'"I't','system equation is given as `~
. 1~ dd ~I dl H dr X d I F d + P d '~
H Id H 11 H lr X I = ~ F I + p I I -----~'9) H rd H rl ~I rr X r F r ,~
~ P d ~ and ( PI ~ are related as ,.~
( P ~ r ~ (P d ~ r, ,.~

2 0 2 0 1 4 ~
' - l 5 - ~
" ~ ~ . ., Riy using this relation, equation (19) is cllanged to - ~

~ d ~ I I dr I 1 ¦ ¦ 1 ~
llid ~111 Illr r ~ ~ ( 21) ~ ~
rd llrl llrr O I X r ~ F r ~.;~;
P d ~ ;
(I representing unit matrix, and T represents transposed).
TlliS e(lllati()n i9 combined with equation (16) to obtain an overall system equation including constraint relation as ~ lldd ~dl lidr T~ ~ X d ~ l~d l~ld 1l~ r r J x i = J (22) ;~
I[rd llrl llrr 1 X r 1 ~ r .
- I ~ O O P d O .
The vihration characteristics of the overail structure can ~-,, ~
be analyzed hy solving this equation (22).
~ig. 11 is a block diagram showing a prior art example ;~
of structural vibration simulation system. Referring to -~
the ~igure, designated at 51a and 51b are sub structure data ~ -`
memories for storing data of first and second sub structures, at 52a and 52b transfer function calculation ", . ~-.., means for calculatlng transfer function matrices of structure from sub structure data from sub structure data ~ ;
memories 51a and 51b (i.e " sub structure data obtained by R~i , .;~, .
measuring or analyticalky estimatine vibrations of the first and second sub structures), at 53 sub structure coupling definition data memory for storing coupling ' ' '~

~ . ... ~,, . , .. ~. . . .

2 0 2 ~
,; ... ~ : ~ .
' - 1 6 - ~`

conditions in(iicative of whether coupling is rigid or soft.
oons~raint relation matrices r~pres~nting constraint '~
~ relalions of indivil3ual degrees of freedom and co-ordinate . -,~ (,onversion matrices overlapl~ing the constraint relation ma~rices, and at 55 coupling m~,ans for collpling tr~nsfer function matrices of the first and second structures : ~ ,. . .
ot)taine~i from lransfer function calculation means 52a and ,~
52b to suh structure coupling definition data memory 53 in accordance wi~h a predetermined co~lpling condition to ~`,,",~
produce a transfer function matrix of the overall slructllre ,~
after co~pling.
Designate~i at 59 is eigen value analysis means for ' ~
analYzine eigen mode ant mode shape of the overall strllcture ,""",~!~, after coupline with transfer function matrix data from ',~
collplin~ means 55, at 60 ~ eigen value analysis result ~, , memory for storing analysis results obtained from eigen ~ ' ' value analYsis means 59, at, 61 time zone vibration ~'' ~'~',',`
~pplication data memory for strong time zone vihration application data of structure, and at 62 ~ourier analysis ~``,, means for ~ourier analysis converting the time zone ''; ~,,' vibration application data into frequency zone vibration , ~',', application data for analysis in frequency zone. ', i Desianated at 56 is frequency zone vibration application~''''1!', data memory for strong frequency zone vibration application data ohtained from ~ourier analysis means, at 57 frequency ',~
. .~. . ..

. ... -: ,.; .
. ' ' " ~
.. ~.. ,..,~
'~

2 0 2 0 1 ~ ~ :

7 ~ ~ ~
,'~
zune r~sl)ons~ analysis means for analyzing resp~nse in fr~qll~ncy zon~ at e~ch point of the struc~ure after collpling witll transfer function matrix data from coupling means 55 and frequency zone vibration application data from l70urier analysis mHans 62 (i.e., fre4uHncy zone vibration a~ll31ication data from freq~lency zone vibration al~plicalion dala memory 5f)), and at 58 frequency zonu rusponse r~sul~
memory for storing frequency zone response analysis results , . ~,, ~
obtained from frequency zone response analysis means 57.
Designated at 63 is inverse Fouriur analysis means for convHrting frequency response analysis results ohtained from frequency 20ne response analysis means 57 (i.e., contents of frequency zone response result memory 58) into time zone response analysis results, and at 64 a time zone response result memory for storing the time zone response analysis results.
Now, prlor art example will be described with reference to Pigs. 12(a) to 12(c). ~ig. 12(a) is a schematic view sbowing overall structure consisting of sub structur~ 1 and 2. Sub structure 1 ls a cantilever structure secured at one end to a wall and having point A1 of measurement at the other end. Sub structure 2 is a free-free beam structure with the opposite ends measured at points B1 and B2 of measurement and coupled to sub structure 1 in a state A1-A2 at an angle of 45 Aegrees in x direction. ~ig. 12(b) shows ~ ~"`'' 202~1~4 : :~

a system dsfinition file tSD~) as an example of sub structure data memory 51a and 51b. In this file the de~rees of freedom of the overall system and sub structures l and 2 as shown at 51a and 51b are defined. l`tlere are ;~
24 definitions of freedom degrees for the overall system.
This is so l)ecallse there are six degrees (x to z and p to r) of freedom al eac~l of four l)oints A1 A2 ~l and n2 of - -measurement. In sub structure 1 definition file 51a data for transfer function matrix [ H ~" ) for the six degrees of freedom at point A1 of measurement of sub structure 1 `~
are stored. In sub structure 2 definition file 52b data for transfer function matrix (~I ~Z~ ) for a total of 12 degrees of freedom at points B1 and L2 of measurement are stored. ^ ;~!p~$U;
~ig. 12(c) sbows a vibration application constraint data file (AL~) as an example of frequency zone vibration ,:.,!,~,S,,~,,,,~
application data memory 56 and sub structure coupling definition data memory 53. This file includes file 61 for storing definitions of the kind and size of forces applied and file 53 for defining constraint relation including co-ordinats conversion.
Definition file 61 represents obtaining frequency response (SINUSOIDAL) when external force with amp1itude of 1.0 and pha9e of 0.0 degree i9 applied to freedom degree A2x ~ ~
(x direction at point A2 of measurement). ~ ~ `s;
;, , l~ .i 2 ~ 2 ~

In constraint relation definition file 53, the number n of independent degrees of freedom and number m of dependent degrees of freedom are defined in CON n, m and are 8equentially listed. In this example, 12 freedom degrees Bla to B2x are independent, and freedom degrees Alx to A2x are .. ~St ~ "
dependent. Dependance relation is shown such that Alx is dependent to Bla, Aly to Bly and so forth. In this way, it -~
is shown here that the six degrees of freedom at point Bl of measurement are all dependent to six degrees of freedom at ;~
point Al of measurement, indicating rigid coupling at these points. Regarding tr], while sub structure 2 is held for measurement in horizontal state (Bl-B2), it is coupled to sub structure 1 in inclined state (Al-A2) at an-angle of 45 degrees with respect to x direction, its co-ordinate conversion matrix is entered using the intrinsic constraint relation. .
Fig. 13 shows the constraint relation. Constraint ;
relation matrix tr] does not contain any real constraint relation but includes only co-ordinated conversion matrix.
For example, considering the degrees of freedom in x to z direction at A2, A 2 x =0.7071 B 2 x +0.7071 B 2 y ... ... ... (23) A 2 y =0.7071 B 2 x ~0.7071 B 2 y ........ ... ... (24) i;','.
A 2 z =B 2 z ... ... ... ... ... ... ... ... ... (25) It will be seen that the degrees of freedom in x to z -- 19 -- ,~,: ~ ~, "'`'~' '"'`'`' "'"'"
` X

dire(tions at ~2 are converted to the degrees of freedom in x to z directions at A2. ~ ~-.. -I~ig. 14 shows the transfer function matrix of the overall system prodllced on lbe basis of a model as shown in ,. .... ~. . ,~
I;ig. 12. In tlle Pigllre, like reference sysmbols designale like or corresponding parts. Designated at l is a unit malrix. I?eferring to the ~igure, a basic section is a 3~J
matrix formed with respect to all (24) degrees of freedom defined by the overall system freedom degree definitions in Snl;. The matrix components in A2 Portion of the basis section are all ~0" because there is no data concerning A2 portion in the definitions of sub structures 1 and 2 in SD~
More specifically, sub structllre 2 is held in horizontal state (Bl-B2) for measurement, while no data at A2 after the coupling is present in the definitions of the sub structures.
Slnce A1 and U1 are coupled together, H ~ H ~Z) should be calculated as in equation tl5). llowever, since ;~
their co-ordinate systems are different, the two can not be 9imply added, and they have to be held as separate elements. ~
In an expanded section, 12 dependent degrees of freedom ;.,,' at A1 and A2 are arranged such that ~ r ) where the constraint relation matrix of B1 and B2 is arran8ed as in equation (22). (By Interchanging the rows and columns in'~'Ji~,,.,:,`~'"~
~ig. 11 the same form as in equation (22) is obtained. i~ ~;

"

- 2 1 ~

As shown, the matrix even of a s;mple setting as shown in ~ig. 12(a) is a matrix of 36 hy 36 degrees of freedom, and its soluti()n takes long calclllation lime. -` -~s is shown, the prior art method usin6 the constraint relation equation is used when the degrees of freedom of `-the entire structure involve a dependent relation (i.e., - ;;
when a motion of a degree of freedom depends on a motion of ~;
a different degree of freedom. In other words, no dependent relation of ttle entire degrees of freedom of a ~- `
certain sub structure to the entire degrees of freedom of a --differenl sub structure is defined.
llowever, when this method is used for the conversion of the co-ordinate system of a sub structure to the co-ordinate system of the overall structure as noted above, for co-ordiante conversion of one of two sub structures as shown in - `~
Fig, 12(a). the overall system equation is, from equation ;
t22), ~"' '',~``.' . ' .. :, ,i;..'.

- 2 0 2 0 ~
- 2 2 - .~

O O O -- I X d ~;~ d o ~T 1 l o r T X ~ = O -- (26) O O 11 r r O X r 1~ r - I r o o Pd O .;''~
where (1-1~ represents a transfer function matrix ( ~ ~Z~ ) requiring co-ordiante conversion, ~I-Irr): a transfer function malrix 11 "' requiring no co-ordinate conversion, ,~
( r ) :a constraint relation matrix for co-ordinate ~r conversion, { X d J : a displacement vector of freedom degree ~
after co-ordinate conversion~ '}
X I ~ : a displacement vector of freedom degree before co-ordinate conversion, ~ X r ) : other displacement vector of freedon degree, ( F d ) an external force vector of freedom degree ji~
, . ,.. .,: .... ~ ~ .
as subject of co-ordinate conversion, and .. ~.. `
( F r ) : other external force vector of freedom .;
de8ree. ..
Generally, the transfer function matrix (11) of the ~m~
overall structure in the overall system equation given as ~11) ( x ~ = ( f ) (27) ~ x:
19 a symmetrical matrix.
Thus, when solvin~ equation (27) using a computer, solution of ( M ) as symmetrical matrix requires only one `~
'' :~
~`~

2 0 2 0 ~

half of memory and is efficient. Por this reason, tlilis method is used boardly.
llowever, in the solution of simultaneous equations of a . ~ .
symmetrical matrix can not permit such process as interchange of rows. ~or tllis reason, a sweep-out process is adopted with the diagonal components of matrix as main point. In the solution of simultaneous linear equations of a symmetrical matrix where O is present, a small value (minimum value of the calculation accuracy) is added to permit affine solution of the equations. Ilowever, this ^~
small value e is liable to influence the accuracy of the solution. ` ^~
Therefore, when performing co-ordinate conversion of a '~
sub structure, the method of solving equation (26) using a constraint relation is liable that the diagonal components of a portion ;
O O O
O ~ I I O , '~ ;~ ' `'.. "; ',.~
O O 1~ rr . ~
in the basic section of the equation include 0. ~or this rsason, higher accuracy can be obtained hy solving the equalion not as symmelrical matrix hut as complete matrix. ; ;i;
When eonventional constrai n t relalion equations are used for co-ordinatH conversion of a sul) slructure, lhe follcwing problems.

''".. ,' ~',',~",,.''.'`''' ~C~

2 ~ 2 ~

(1) When simultaneous linear equations are solved as symmetrical matrix, no solution or merely approximate solution can be obtained. -(2) When the equations are solved as complete matrix, enormous memory capacity is necessary.
(3) Of the freedom degrees concerning co-ordinate .J`':'`''" ` ''',~',',.
conversion among the freedom degrees of the structure, those both before and after co-ordinate conversion are recognized.
Therefore, the degrees of freedom are increased that much, and this means memory capacity increase.
(4) The equation scale is increased to increase time required for solving the equations. ;~

: ,, .

',','"~':',~
~`.,"'."~

B ~

2 0 2 ~

- 25 - ~ ~
The invention provides a system for analyzing ;: .`"~, !`.''~' characteristics of an overall structure, which necessary calculation time and memory capacity are greatly reduced . .
compared to the prior art method of expressing information concerning co-ordinate conversion for each sub structure as constraint relation equation based on the sub structure synthesis method and is substituted into overall system equation to obtain solution, and which permits design efficiency increase and accuracy improvement and stabilization of solution. . ;
~',',-'' , ~'`

.`''.''.'.~"`''.'..''"''''."`'.

"~.",..~
~ . ::,::. :.:
:, . .. ~ :.. ...

.: . .. ..: .
. :. : :

2 ~ 2 0 ~

~ igs. 1 and 2 are block diagra~s showing the functions and construction of a structure vibration simulation system as one embodiment of the invention. Referring to the Pigures, designated at 51a and 51b are sub structure data memories for storing sub structure data of first and second suh structures, at 52a and 52b transfer function calculating as an example of response characteristics calculating means for calculating a transfer function matrix of structure from sub structure data fro~ sub structure data ~e~ories 51a and 51b (i.e., sub structure data obtained by ~easuring or analytically estimating vibrations of the first and second structures of Fig. 3), at 53 a sub structure coupling data definition data ~e~ory as an ex~aple of data storage ~eans for providing positional relations of the co-ordiantes of the first and second structures to an overall sysem co-ordiante syste~ as co-ordiante conversion data ln terms of euler's angles ( ~, ~ and ~) of three-dimensional co-ordinate conversion and storing coupling definition data specifying a method of coupling of tbe B , ! ~

202~1 44 , i~;., ` .

first and second structures as rigid coupIing of degrees offreedom (common use of degre~s of freedom) or soft coupling (coupIing hy spring and damper), at 54a and 54b co-ordinate conversion means for producing a three-dimensional co-ordinate conversion matrix from positional relation of co-ordinates provided by the Euler's angles ~, ~ and ~ of the three-dimensional co-ordiante conversion and converting transfer function matrix obtained hy transfer function calculating means to transfer function matrix in overall system co-ordinate system, at 55 coupling means for coupling together transfer function matrices of the first and second structures obtained by transfer function calculating means 52a and 52b according to coupling conditions determined by coupling definition data stored In sub structure coupling definition data memory 53 to produce a transfer function matrix of a ~ ood structure, at 59 eigen value analysis means for analyzing the eigen mode and mode shape of the coupled structure with transfer function matrix from coupIln8 mean9 55, at 60 ~ eigen value analysis result memory for storing analysis results obtained by eigen value analysis means 59, at 61 time zone vibration application data memory for storing vibration application data of struclure in time zone, at fi2 Pourier analysis means for convertin8 the time zone vibration appllcation data by Pourier analysis to frequency zone vibration application 2 0 2 0 ~ 4 ~

dall3 for analysis of the frequency zone, at 56 frequency zone vihration application data memory for sloring frequen(,y zone vit\ration application data ohtained from the ~ourier analysis means or supplied directly, at 57 frequency zone response analysis means for analyzing the response of the coupled structure at each point thereof in frequency zone from transfer function matrix from coupling means 55 and frequency zone vibration application data from ~ourier analysis means 62 or supplied directly (i.e., frequency zone vibration application data from frequency zone vibration data memory 56), and at 58 a frequency zone response result memory for 9 toring frequency zone response results obtained from frequency zone response ana1ysis means 57, at 63 inverse ~ourier analysis means for converting frequency response analysis results obtained from frequency zone response analysis means 57 (I.e., contents of frequency zone response reuslt memory 58), and at 64 a time zone response result memory for storing the time zone response analysis results. Referring to ~ig. 2. designated at 65 is a CPU, which has functions of the individual means noted above and controls the memories and main storage means 66 for data processing.
Now, Junctlon9 of main means in this embodlment will be descrihed.
Transfer function calculating means 52a and 52b .. ......
., . ..., ,.~ ~ .,, 20~01~ :
- 2 9 ~ - ~

determine transfer function matrices for individual elements ~;
llsing data obtained by experimental F~T analysis or finite element metllod analysis, This method of calculation is of the following five types depending on the kind of sub structure data. The transfer function that is determined ~ ; -;
is of two kinds, i.e., transfer function matrix ( G ) of compliance (displacement/force) and transfer function `~
matrix ~ of dynamic stiffness (force/displacement). Where the transfer function matrix ~ G ) of compliance is calculated in the following method, inverse matrix is ~ ~;
obtained and is converted to the transfer function of ~, dynamic stiffness as shown by i (fl) = ~ G ) ~' (51) ;
Pollowing equations (52) to (66) show respective types of , `.
transfer function calcuating means.
(a) n irect matrix input type j:*
(~1 (o) ) = ,','.. ,'.. ',',~
( - ~ Z ( M ) + j ~ ~ C ) + ~ K ) ) (~

~3 ( _ ~ 2 ( M ) + 3 ~ B ) + ~ K ) ) (~
whcrc ~ : reprssents angular frequency, `
( M ) : mass matrix, ~'r ( K ) : stiffncss matrix, ( C ) : viscosity damping matrix, and ( B ) : structure dampin~ matrix.
. ~,,, .. :..., ... ~

~'"'"""~

2 ~ 2 ~

(b) Non-constraint~ode synthesis type (real mode) ( Y ) n ( G ( ~ ) ~ = - +
~ 2 r=l '~
~ qb r } { ~b r } T ` ~
S ~ + ~ z ) --(54 mr (-- 2 + i ~ ~ r ~) r + ~) r 2 ~ 2 r~

mr ( - ~ 2 + i g r ~ r 2 + ~ 2 ) ~ ~ Z ) (55) where, ~ : represents angular frequency, n : ~ode nu~ber, ~ ~ .
m r :~odal ~ass (r = 1 to n), ..
~r : eigen value in mode r (r = 1 to n), ~.. ;
{ ~r 3 : vode vector in ~ode r (r = 1 to n), j.
~ r : ~ode viscous da~ping ratio in ~ode r(r = 1 g r : structure da~plne ratio in ~ode r ~r ~ Y ) :residual ~ass ~atrix, and ~ Z) : residual stiffness ~atrix.

,`,`,~
~,..' . . ` ,.`, ,.

'" "' ~''. ." "

2 ~ 2 ~ ~ L~

(c) Non-restrictive ~ode synthesis type (co~plex ~ode) . :
t G ( ~ ~ +
~ 2 r=l ( ~ r } ~ ~ r }I ~ r. } { ~b r } T
}
a r ( ~ ~ ~ P r ) a r ( j ~i) ~ P r ) ( - representing coniugate) ---- - - - (56) . ... ; -~ Y ) n ( G ( ~ ) ) = -~ 2 r=l + ~ Z) (57) mr ( ~ ~ 2 + 3 g r ~0 r 2 ~ ~i) r 2 ) .
where, ~ :represents angular frequency, n : ~ode nu~ber, a r : residue (coaplex va1ue) in aode r, Pr : couplex eigen value in aode r, . .
m r : rode ~ass in ~ode r, .i~
gr : wode structure da~pine ratio in ~ode r, ~r ~r coeplex ei~en value in ~ode r, ( ~r ) co~plex ~ode vector in ~ode r, ~, t Y ) : co~plex reeidual ~ass ~atrix, and .
t Z ) : co~plex regidual stiffness vatrix. ;
(d) Transfer function synthesis type ;.,~
( G ( ~ ) ) = Rational expression of ( 3 ~ ) ; (5) ' .. ; ;~
z5 ~ H ( ~ ) ) = Rational éxpression of ~ j ~ 3 --(59) :

. ,~ - . ' ,.,:
..::;

2 0 2 ~ 1 4 ~

te) Scale element ; . . .
H ( ~ ) = - ~ Z m t m : scalar mass) (60) ~I t ~ ) = j ~ c tc: ground scalar viscosity damping) `~
- (61) .
c - c '.; ``'"''.'` '' ~I t ~ ) j ~ ( _ c ~ ( c : scalar viscosity damping among degrees of freedom) (62) i~
II ( ~ ) = j b ( b : gr ~ ructure da~pind) H ( ~ ) = j [ ) t b : scalar structure damping) t64) 11 ( ~ ) = k ( k : ground scalar stiffness) (65) ~ H ( ~ ) ) = ~ ~ ( k : scalar stiffness among degrees of freedom) (66) ~ ig. 3 shows an example of sub structure coupling, llere the coupling angle is In the method of sub structure analysis, a sub structure ~ 9 analysis obtained by analysis for a sub structure unit after anotller (expressed by mode data, MKC matrlces, transfer function matrices, etc.) is syntllesized, and characteristics of the overall structure is estimated. ;i~
Characteristics of sub structure are analyze(l in a co-ordlnate system under certain conditions and are not always identical with the co-ordinate system of the overal1 system.
Accordingly, when coupling togetller structure with certain ; ~'.' '.''',' ,. .'.:~ ". ...
. . ~

. ., ; ;.

202~1~4 ~

three-dimensional angles ~ ), it is necessary to perform co-ordinate conversion from each co-ordinate . ~ ~ .
system under analysis into the co-ordinate system of the ~-overall system. ` :
The co-ordinate conversion of the local structure is defined as follows.
(1) Local co-ordinates in the neighborhood of the coupling point are referred to as co-ordinate systems a and b. ; ;~
(2) The co~pling angles are given as Euler angles, ~ `

Referring to Pig. 4, there are co-ordiante systems a (x, y, z) and b ~X: Y' , Z' ) , where ~ --- angle between Z and z ~ --- angle to X' of intersection OM between plane A
containing Z' and z and plane B containing X and Y

~ --- angle to x of intersection ON between plane C ,~ e,, containing x and y and plane A. `' In co-ordinate conversion means 54a and 54b, a three-dlmenslonal co-ordinate conversion matrlx ~ .`

.'''''"'''"'''~

--"` 2 ~ 2 ~

cos ~ cos~ cos ~ - sin ~sin T = cos ~sin~ cos ~ + cos~ sin - sin ~ cos ~
.- ......
- cos Ocos~ sin ~ -sin~ cos ~ sin ~ cos~ i-,.... ...
- cos~sin~ sin ~ + cos~ cos ~ sin ~siny (67) ~t~' ','~
sin ~sin ~ cos ...... , . ;.~, ;. . ~ ~
is obtained fro~ Euler's angles 0 , ~ and ~ concerning lo three-diuensional co-ordinate conversion defined by the sub structure coupling definition data ~e~ory 53. Nowever.
Euler's angles ~, ~ and ~ concerning three-di~ensional co-ordinate conversion defined by the sub`structure coupling ;
definition data ~euory are deter~ined by the following wethod. The co-ordinate syste~ hefore the conversion by a ~ `rh (x, y, z) and that after the conversion is denoted hy b .. ` - .
( x , y , z ) . .~.~ ,. .-.:, ,.,"
Por exa~ple. the euler's angles when angle is rotated as x and y axes about z axis without conversion thereof are 0 ~ = d and ~ = 0 or ~ ~ 0, ~ = 0 and More specifically, when z axis itself i9 unchanged, 0 and ~ + ~ = ~. Thus, cos ~ - sin ~ 0\ ~ .
T z = sin ~cos ~ 0 ¦ (68) ; :
0 0 1 1 ~ :

."..

3 4 - ; ~
;, 'g ' "~; ~ ' 202~14~
. .. ~
- 3 5 ~

. ~ .
When z axis itself is unchange, ;.
cos V O sin ~
T y = O 1 0 ~ (69) ~
- sin ~ O cos ~
More specifically, assuming a co-ordinate system of a . :.. ~ .... .
Sllb structure at a certain coupling point to be b co-ordinate system and a co-ordinate system of the coupled structure to be a co-ordinate system, x' = x b -- r' = r b ..
and x = x a ~.. r= r a in ~ig. 4, and considering conversion `~
of freedom degrees x b ~ y b, z b ~ p b q b and r b of tile b co-ordinate system to freedom degrees x a, y a z ~ p U ~ q a ~ and r a Of the a co-ordinate system, .. -q = ¦ i ~ (70) r , T O
suh~tituting ( r ~ = ~ o T ~

Y x z a = ( r ~ z b ' (71) IP'I I~ ~
. ~
,', . :.' "., . :,.:.,.:

2020~

T -' is calculated with respect to T obtained in equation (67). ~- . ,.~=
Itig. S is a flow chart of conversion of the degree (DEG) `~
into radian (RAD) by co-ordinate conversion means 54a and 54h, In step S2 T in equation (67) is obtained, and its ~, inverse matrix T -' is calculatPd in step S3. In step S4 co-ordina~e conversion matrix tP i~ produced. In Stl'p S5 dynamic stiffness matrix d~ after conversion is produced from U~ IF-') ~lH ) (Ir ~ , where (1-1) is the dynamic stiffness matrix before conversion.
Purtller, inverse matrix ~ r -~ ) to angular conversion ','r~
matrix ~ r ) of the entire sub structures is produced for directions x to z and p to r of freedom degrees defined with i~
the sub structures for elements of the 3 by 3 matrix of T .
~~. If the freedom degrees defined by the h~l- structure are x to z and p to r in the metnioned order, -~ :
T-~ O
( r^~ (72) ~-O T~
The equation of motion O O O -- I ~ X d F d . O H~1 0 rT J x~ _ O ~(26) -- I r O O ¦ X r F r shown in the prlor art example is disassembled to obtain . ,. , ~ .

2 0 2 0 ~ 4 ~
- 3 7 ~
"'.-.-'`'.'."-.,' equations. ~ ~
~ I d ) = { T~ d ) -- (73){X~ ( r) {Pd ~ = (74~ :
~Ilrr) {Xr ~ = (Fr ~ - (75) - ~ X ~ r ) ~ X i ~ = O ~ (76) I;rom 4~1uat~4~ (73) and (74) i ) ( x I ~ = ( r ~ ~ }~ d ~ -- (77) { X ~ H ~ r ) ~ F d ~ -- (78) ~ . ~
Also, from equations (76) and (78) { X d ~ = ( r ) ( ~ ) ( r ) ( F d ~ -- (79) ;~
r ) ' ( T-l 1 1 ) ~ r ) ~ X d ~ = ~ F d ~ -- (80) . . ~ r ) ~ r ) ( x d ) = ( F d ~ - (81) ~ - ~
~rom equations (75) and (81) we obtain a matrix.
( r - ~ ) T ( H li) ( r ) o J [ ~ = [ ~ ~ s ~
O ~Hrr) Xr Fr -." ~~
~ (82) `
As shown above, ~ (82) is derived from equation (23). and both the equations are equivalent. This equation (82) means that it may be solved after performing , calculation of co-ordinate conversion of Independent ' freedom degree H 1l to ( r - I ) T ( H ~ r ~
Thus, in co-ordinate conversion means 54a and 54b performs operation expressed as ( H' ( ~) ) = (r-l) T ( H (~ ( r-') (83) ~ ; ~
~where T represents transposed) with respect to transfsr ~ .; ,., ' . ' ., . ":, ~ ~ r `

2 0 2 ~ 1 4 ~
. .

function matrix (1-~ ~ ~ ) ) obtained from transfer `~
function calculating means 52a and 52b, thus deriving transfer function matrix (Il' ( ~ ) ) after co-ordinate ~ c-conversion.
Coupling means 55 synthesizes transfer function (~ ) to (Hn) of dynamic stiffness (force/displacement) of N ~ -elements obtained by transfer function calculating means 52a and 52b to produce an equation of motion based on transfer i~ ;``~`
matrix ( H ) after coupling, given as . - -X ¦ = I F ¦

~H ) (84) When coupling is performed, like freedom degree data as - ¦
noted in connection with the prior art example are added together, Where there is constraint relation among freedom degrees 8iven as (X d ) = ( r ) ~X I ~ (85? `~
where X ~ represents independent degree of freedom, X ~ ;;
dependent degree of freedom, and ( r ) cqnstraint relation matrix, the equation of motion after coupling is `~ ¦ ~f~

:, :`', ~ `." !

'. ~,,,',1~

- ` - 2 ~ 2 ~

~ :..........
~ . . ~, . ..
Hdd Hdl Hdr 1 ~ Xd. Fd - : , Hld H~l Hlr r J Xl Fl Hrd Hrl Hrr O ¦ Xr Fr - (86) ; . - `-r o o P o However, - ..... ;i ~.
_ j,~ H dl H dr _ (H l = H~, H1l H Ir . ~ X r ) : Degree of Hr~ Hrl Hrr freedom free from j~.
constraint relation (P) : Constraint force ~ `.
F d ~ ,.~
Fl : Porce vector ~ r$
F r Prequency zone response analysis ~eans 57 solves ` ;-~
..
equation (ô4) or (86) of uotion obtained by coupling ueans .
55 to obtain response value of real de8ree of freedo~
Purther, when there is no da~ping, eigen value analysis .~
vean9 59 separates the transfer function of dynavic ..
stiffness (force/displace~ent) obtained fron coupling ~eans 55 to uass ~atrix t M ) and stlffness ~atrix ( K ) using an . i~
equation .:
(H ) = - ~ 2 ( M ) ~ ( K) (87) `
and solves eigen value proble~
( M) IX~ + ~ K ) I X} = 10 ~ (88) g 2 0 2 0 1 ~
. ,.; ~.
- ~ o This i9 referred to as sub-space method. ~ ~
When there is damping, the designated frequency range is . ~-Z
....:,.; ~:
divided, motion (84) or (u6) of equation is solved for each frequency, and the response peak value is searched to obtain eigen value and eigen mode. This is referred to as frequency search process.
~ ourier analysis means 62 performs ~FT (fast Fourier transform of vibration application condition given in time zone to obtain frequency zone vibration application data, , ::
and inverse Pourier analysis means 63 performs inverse ~PT
of response analysis results obtained in frequency zone response analysis means 58 to obtain time zone response analysis relsults. ~ourier analysis means -62 and inverse ~ourier analysis means 63 are used when and only when time zone response results are necessary.
Now, the overall operation of this embodiment will be described. The structure coupling simulation system Or tllis embodiment is constituted by program control. The program i9 usually stored in an auxiliary memory (not shown) as non-volative memory, and at the time of initialization it is Ioaded in main memory 66 and executed by CPU 65.
In this system, sub structure data is stored in advance as input data in sub structure data memories 51a and 51b, co-ordinate conversion data and coupling definilion data are stored In advance in sub structure coupling data memory 53, 2020i44 , ... ... .

and vihration application data are stored in time zone data memory 61 when performing time zone analysis while they are stored in frequency zone data memory 56 wben performing frequency zone analysis. CPU 65 reads out data from memories 51a, 51b. 53, 61 and 56 according to the program, performs simulation analysis using main memory 66 and stores the results in frequency response result memory 58.
time zone response result memory 64 and eigen value analysis result memory 60. These memories are all on a magnetic disk system. Particularly, memories 58, 64 and 60 which provide results may be implemented by a line printer system.
~ urther, memories 58. 64 and 60 may be used to produce flange with flag generation program using their output data as input data. Transfer function calculating means 52 calculates transfer function using equations (52) to (66) for conversion depending on the kind of sub structure data.
When the transfer function i9 of complicance (displacement/force) type, it is converted to one of dynamic stiffnnss matrix (force/displacement) type using eqaution (51) to obtain transfer matrix of dynamic stiffness. Co~
ordinate conversion means 54 converts transfer matrix of dynamic stiffness obtained by transfer function calculating means into transfer function matrix of overall structure co-ordinate system using the equations of conversion.
Coupling means 55 synthesizes transfer function after 2020~4 - 4 2 - ~ -'",'~'.'`''.'''"'""

coupling using eqllation (84). When there is constraint relation of degrees of freedom as given by equation (85), -~
the constraint relation eqaution is also incorporated to -~
produce overall system matrix and produce matrix of equation ~ ~ i (86). ~requency response analysis means 57 solves equation (86) of motion. In case of a system without damping, eigen value analysis means 59 solves eigen value asing the sub-space process given by equat;ons (87) and (88). When there is damping, response for each frequency in the designated frequency range is solved to search peak value of response -~
and determine eigen value, the eigen value thus obtained ;~
being substituted into eigen value analysis memory 60.
~ ourier analysis means 62 converts vibration application data given in time zone into those in frequency zone, the converted data being stored in frequency zone vibration application data memory 56. Inverse Pourier analysis means 63 converts response results obtained in the frequency zone to those in the time zone, the converted data bein8 ~;
stored in time zone re9ponse result memory 64.
Now, a specific example of the invention will be descrihed with reference to the drawings. Pi8. 6 sbows a casc of application of the invention to the prior art example of ~i~. 12(a). Designated at 51a and 51b are sub structure data memories SDPl and SD~2, at 61 time zone vihration application data memory AL~, and at 53 suh , 202~14~
- 4 3 ~
.: ~ ,.'',''.
structure coupling definition data memory CD~. In memory SD~1 transfer function matrix ~ H ~ ) of sub structure 1 and six freedom degrees are defined. In memory SDF2 transfer function matrix ~ H 2 ) measured by B1-B2 co~
ordinate system of sub structure 2 and 12 freedom degrees are defined. In memory Al,P, it is defined to determine frequency response when an external force with an amplitude of 1.0 and a phase of 0.0 degree is applied to B2x. Memory CDP defines differences of angles of the co-ordinate systems of the sub structures with respect to the overall system co-ordinate systems. In ANG ~ and following, an~les of two co-ordinate systems with respect to the overall sysem co-ordinate system are defined, and then Euler's angles in the individual sub structures are shown in each lines. This example shows that the co-ordinate system of sub structure 1 is (1, O' , O , O ), same as the co-ordinale system of the overall system and that the co-ordinate system of suh structure 2 has angle9 ~, ~ and ~ of (2, with respect to the co-ordinate system of the overall syste Pollowing ~C 6 shows that there are six freedom degrees where the coupling condition is rigid couulin~, indicating that freedom degree Alx of sub structure 1 and freedom degree B1x of sub structure 2 are rigidly coupled together.
I,lkewise, freedom degrees Aly and Bly, , Alr and Blr are rigidly coupled together. This means that freedom degrees ,. !,,, ":j ~ .,.:' 2020~

~1 and B1 commonly have six deerees of freedom. - ;~ Fig. 7 is a view sbowing a transfer function matrix of the overall system in the above setting. Shown in ~a) is the result of co-ordinate conversion obtained by performing calculation of equation (83) with respect to transfer function 11~ of sub structure 2. Shown in (b) is a method of bombining the transfer functions of sub structure 1 and sub structure 2 after co-ordinate conversion to ~~
synthesize tlle transfer function matrix of the overall ~ -system. Il ~ and 11 " ~Z~ ' permit calculation of equation (13) for the same co-ordinate system. In this way, the matrix calculation is performed within 12 by 12 size, which is one-fourth the size both in rows and columns compared to the prior art 36 by 36 size shown in Pig. 10. : `
Now. ~ig. 8 is a flow cbart illustrating the operation ~ ` ?'~
of the syslem shown in ~ig. 1. Designed at 67 lo 97 are respective steps. In this proeram, in step 68 an area of ``
main memory 66 used in the program is performed. In step 69, an inllut data memory, freedom degree for ohtaining kind of r0l~1y in the metbod of analysis and and output data memory.
In step 70, co-ordinate conversion data of sub structure i';
couplin8 data memory is read out, euler's an81es ~, ~ and ~ with respect to the co-ordinate converslon for each sub structure are stored in a co-ordinate conversion table in memory, and sub structures that constitute the overall ~ `

' ',',, 202~

structure are counted.
In st~p 71, sub structure data is stored in an operating file. ~ `-ln ste,l 72, coupling definition data in sub struclure coupling data memory is read out, and if tllere is rigid ~1 coupling, relations where freedom de8rees are equivalent are p~
slored in a freedom degree tahle, If the coupling is sofl coupling, scalar element data are added. ` `;~
In step 73, local data is input to produce a table in main memory 66 at a designated position. In steps 74 and 75, ~ourier analysis means 62 at the time of time zone data is shown. In step 76, contents of input sul) structure data and load data are printed on an OUtpllt list. In step 77, layout of main memory 66 used in the system is performed.
In stpes 78 to 35, ana1ysis is performed for various kinds ,~
of analysis with various methods. In step 96, results are output. Steps 79 through 89 show response ca1culation means. In the response ana1ysis, processes in transfer funct~on ca1cu1atlng means 52a and 52b, co-ordinate conversion means 54a and 54b, coup1ing means 55 and frequency zone response ana1ysis means 57 are repeated in steps 79 to 87~ In the response ana1ysis in time zone, process of inverss f~ourier ana1ysis means 63 is performed in ~tep 89. In steps 91 to 92, eigen va1ue ana1ysis based on the sub-space method is performed, and in steps 93 to 95 ... .~
...:, . :: .

2~2~14~ :
. . , - 4 ~ -eigen value analYsis hased on frequency search method is performed. The response calculations used in steps 94 and 95 are the same as those in steps 80 to 86.
As has been described in the foregoing, in the structure vihration simulation system as an emhodiment of the invention, unlike the prior art three-diMensional co ordinate conversion system using constraint relation equation described above, a dynamic stiffness transfer matrix after co-ordinate conversion is obtained for each sub structure transfer matrix. Subsequently, a dynamic sliffness transfer function matrix is synthesized, the transfer function matrix after coupling is used to ohtain eigen value and eigen mode of tile overal1 structure, or the response ana1ysis based on the frequency zone under vibration application conditions is obtained furtller ~FT
analysis on vibration application condition is effected in time zone, in frequency zone inverse ~T analysis is perforMed on the resPonse to obtain response, and in this way dynamic characteristics of the structure after coupling are slmulated.
As has been shown, in the above embodiment for data deflning the vibration characteristics of the first and second local structures using equations (52) to (66) co~
ordinate conver9ion mean9 54a and 54be are provided between transfer function calculating means 52a and 52b and coupling 202~144 ::

means 55.
I`lmls, there is no need of adding any consLraint relation malrix to the overall syslem matrix, and it is possible to reduce necessary memory. That is, a least memory may be used for the overall system matrix. In addition, there is no possibility of occurrence of zero diagonal componenl, and it is possible to obtain accurate solution of symmetric matrix by the skyline method. (In the prior art, it was necessary to provide a differential as dyagonal component).
Thus, it is possible to ohtain vibration simulation of the structure accurately, quickly and with less memory capacity compared to the prior art.
Purther, with this embodiment a desired number of sub structures can be coupled togetber at a time and at every angle. Thus, particularly when evaluating the reliability of a complicated machlne design an overall evaluation can be obtained without performin~ complicated finite element method analysis with respect to the overall structure or experlmental analysis after trial manufacture but from structure unit data. It is thus possib1e to obtain efficiency increase of the machine design.
; While in the above embodiment eigen value analysis means 59 uses a sub-space method and frequency search method, the saMe effects can be obtained usin8 other methods. ~urther, while memories 51a, 51b, 56, 61, 60, 58 and 64 use magnetic , ,' ~
:6 :~

2020~4~:

, ~ .

disk systems, memories 58. 66 and 64 storing output results may consists of line printers or tlle line.
I~urttler, it is possible to provide ttle main memory with roles of vibration application data memory 56 for storing conlenls converted in ~ollrier conversion means 62 and frequency zone response result memory 58 for storing contents given as intermediate results of time zone analysis.
~ urther, it is possible to use any desired number of sub structure data memories 51a and 51b within the permissible memory capacity of the system.
~ urther, while sub structure coupling definition data memory 53 has two different kinds of data, i.e., co-ordiante conversion data and coupling definition data, it may be replaced with two separate memories having different functions.
~ urther, while the above embodiment has been described in conjunction with a structure vibration simulation apparatus used for the machine structure design in the flelds of power plants, traffic, universe, communication, electronlcs, devices, electric home appliances and dwellings, the invention is also applicable not only to simulation of vibrations but also to characterlstic analysis of coupled or organs consistlng of other eleMents as apparalus for analyzing characteristics of an overall ~'",',''',',',''.

''''': .'.,' .`, 20201~
_ ~ 9 _ structure. For example, the invetion is applicable to -;
numerical simulation for three-dimensional analysis of fluid and output simulation of electric circuit at the time of coupling.
Further, while in the above embodiment transfer functions are used for expressing characteristics, they are `i,i `
by no means limitative, and other metSlods of expression may ;
be adopted.
As has been shown in the foregoing, according to the invention co-ordinate conversion means is provided between -local characteristics calculating means and coupling means.
It is thus possible to obtain analysis of characteristics of the overall system, thus further reducing time for ~ i:
producing data concerning characteristics analysis, time and labor for ca1culation and thus permitting great increase of the design efficiency.

.. . ~

Claims (9)

1. An apparatus for analyzing coupling characteristics of an overall system consisting of a plurality of unit structures coupled to one another and having respective physical characteristics comprising:
a first memory for storing physical data of each said unit structure;
a plurality of calculating means each for calculating transfer function matrix of each said structure unit according to said physical data;
a second memory for storing definitions of co-ordinate conversion data and coupling concerning said structure units;
a plurality of co-ordinate conversion means for generating co-ordinate conversion matrices from said co-ordinate conversion data and converting said transfer function matrices into those in the overall system;
coupling means for coupling together said transfer function matrices in said overall system according to said definitions of coupling and generating transfer function matrices concerning said coupled system;
eigen value analysis means for analyzing eigen mode and mode shape of said coupled system according to said transfer function matrices;
a third memory for storing vibration application data in time zone with respect to each said structure unit;
Fourier analysis means for converting said vibration application data into those in frequency zone;
response analysis means for analyzing response in frequency zone concerning a point of measurement of said coupled system according to transfer function matrices from said coupling means and vibration application data in frequency zone; and inverse Fourier analysis means for converting the results of said response analysis into those in time zone,

2. The apparatus according to claim 1, wherein said first to third memories are provided on a magnetic disk unit.

3. A method of analyzing coupling characteristics comprising:
a step of calculating transfer function matrix of each of a plurality of structure units from physical data thereof;
a step of generating co-ordinate conversion matrices from co-ordinate conversion data of each said unit structure, and converting said transfer function matrices into those in an overall system;
a step of coupling together said transfer function matrices in said overall system according to definitions of coupling and generating transfer function matrices concerning a coupled system obtained by coupling together said unit structures;
a step of analyzing eigen mode and mode shape of said coupled system according to said transfer function matrices;

a step of converting vibration application data in time zone with respect to said unit structures through Fourier analysis into those in frequency zone;
a step of analyzing response to vibrations in frequency zone converting a point of measurement on said coupled system from transfer function matrices of said coupled system and vibration application data in frequency zone;
and a step of performing inverse Fourier conversion of the results of said response analysis into those in time zone.

4. The method according to claim 3, wherein said physical data Is obtained by measuring or analyzing vibrations of said unit structures.

5. The method according to claim 3, wherein said co-ordinate conversion data are three-dimensional co-ordinates of Euler angle.

6. The method according to claim 3. wherein said definitions of coupling are specified as rigid or soft coupling of said unit structures to one another.

7. The method according to claim 3, wherein said point of measurement consists of a plularity of degrees of freedom.

8. The method according to claim 3, wherein said transfer function matrices are calculated from data obtained through experimental FFT analysis of finite element process analysis.

9. The method according to claim 3, wherein said transfer function matrices consist of those of compliance (displacement/force) and dynamic stiffness (force/displaceme nt).