The accuracy of the NASTRAN modal synthesis analysis was assessed by comparing it with full structure NASTRAN and nine other modal synthesis results using a nine-bay truss. A NASTRAN component mode transient response analysis was also performed on the free-free truss structure. A finite element method was developed for nonlinear vibration of beam structures subjected to harmonic excitation. Longitudinal deformation and inertia are both included in the formula. Tables show the finite element free vibration results with and without considering the effects of longitudinal deformation and inertia as well as the frequency ratios for a simply supported and a clamped beam subjected to a uniform harmonic force

Mei, C.

English

NASA Technical Reports Server

NASA-CR-173338 
19840010612 
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OF 
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by the 
NASA Scientific and Technical Information Facility 
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By 
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Fin&l r~cport 
for th0 &l~rioti t'&a\'iffi)l!b~r l~ 1982 to (/ctol:IIJ;\" 31, 1%3 
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Febw'wJ,Y'Y 1984 
(NASA-Cll- 173338) COIH'Ot<EN'l. MODE SYN'llIESIS 
A~D ~AhGh DEFLhCTlUN VlbRA11l6S GF COMPLEX 
S1'HLlCl'Uct.l:.S final Hepor.t# I ~;ov. 19t12 - J 1 
Oct. 191.33 (Uld iJOIlJ1.!lioIl Univ., NorfoJ.Y...., Va.) 
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DEPARTMENT OF ~1ECHANICAL ENGINEERING AND MECHANICS 
SCHOOL OF ENGINEERH4G 
OLD O~4INION UNIVERSITY 
NORFOLK, VIRGINIA 
COMPONENT MODE SYNTHESIS AND 
LARGE DEFLECTION VIBRATIONS OF 
C~~PLEX STRUCTURES 
By 
Chuh Mei, Principal InvE!stigator 
Final t<eport 
for the period November I, 1982 to October 31, 1983 
Prepared for the 
National Aeronautics and Space Administration 
Langley Research Center 
Hampton, Virginia 23665 
Under 
Research Grant NAGl-301 
Joseph E. Walz, Technica'i ~1onitor 
Structural Dynamics Branch (SOD) 
Submitted by the 
Old Dominion University Research Foundation 
P. O. Box 6369 
Norfolk, Virginia 
February 1984 
FOREWORD 
The materia"' presented in this final report was performed under Grant 
No .. NAG-I-30l entitled "Component f'i'lOde Synthesis and Large Oaflection Vi-
brcltions of Complex Structures. 11 This report st.mmarizes the research re-
~ullts on modal s,Yl1thesis and nonlinear forced vibrations of beMis. The 
study was performed at the NASA/Langley Research Center during the period 
frclm November 1, 1982 to October 31, 1983. The m>rk was monitored under the 
sup1ervlsion of Joseph E. Wah and D"r. Jerrold f4. Hausner, Structural 
Dynamics Branch, Structures and Dynamics Dlv"ision, NASA/Langley Research 
Center. 
ii 
COMPONENT ~10DE SYNTHESIS AND LARGE I)EFLECTION VIBRATIONS 
OF C!l4PLEX STRUCTURES 
By 
Chuh tile;* 
Part 1. DynZlmic Analysis of Large Complex Structures 
Us i ng Component t-bde Method~; in NASTRAN 
The complexity of aerospace structures has been increased enormously 
during the past decade. A new challenge has cOlJ'lfronted the structural 
dynamists by the proposed space station to be in service by the year 1990. 
It wll11 be an evolving structure (ref. 1). and it will not be possible for 
it to be ground tested because the final conf'iguration may not be known when 
the first component is put into space. The c()mponent mode method, there-
fore!, may be employed for the dyn("Jnic analyses for detet'lllining frequency, 
mode shape and transient response of such a lar9~ structure system in space. 
The NASTRAN computer program, a structu\",al analysi s tool widely used in 
the aerospace industry, contains a modal synthesis capability. Other than 
the nine-bay truss structural problem presented in the NASTRAN demonstration 
manual, little is publicly kno~m about its capabilities. Preliminary 
assE~ssment of the accuracy of the NASTRAN modal synthesis analysis is accom-
pl ished by making a comparison of the NASTRAN modal synthesis with full 
structure NASTRAN and nine other modal synthesis results (ref. 2) using the 
ninE~-bay truss SI10\>'11 in Figure 1. Figures 2-,4 show the rel ative accuracy 
obtained using the various modal synthesis procedures. The 1 imited study 
indicates that the fixed-interface method in NASTRAN, fixed-interface 
~;oclate Profes'sor', Departlnent of t4echan'ical Engineering and Mechanics. 
Old Dominion University, Norfolk Virginia 23508. 
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i'-' 5 Equal Rays ~' It Equal Baya, 5' 
Component A I Co~ponent D 
Figure 1. Truss Model. 
x 
(0) Substructured Model. 
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ORIGINJU. PAGE t~ 
Of POOR. QUALITY 
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method of Hurty (ref. 3) and the equivalent Craig-Bampton method (ref. 4) 
produce the most accurate results for a given number of degrees-of-freedom. 
Slightly better accuracy was achi-eved by the procedures introduced by 
Benfield and Hruda (ref. 5). But these htter methods (ref. 5) suffer the 
disadvantage that the modes of one substructure are not independent of the 
modes of other substructures. t40re detai'led results are documented in the 
progress report entitled, "NASTRAN Modal Synthesis Capabil ity (ref. 6).11 
A NASTRAN component mode transient rE!sponse analysis \lIas also perfonned 
Ion the free-free truss structure. A conc(mtrated force was appl ied at grid 
point 42 of component B for 0.12 seconds clOd then removed. Tables 1 and 2 
give the displ acements and stresses and av'e compared with the full structure 
NASTRAN results. It demonstrates that excellent trans'lent response can be 
j)btained using component modal synthesis. Detail resu1ts p Ot~l\P alters, sol-
ution sequence~. and computer CPU time can be found 1n reference 6. 
6 
.... -- ~ ........ - - .- -......... --. ~ .. -- '" 
Table 1. Transient ReSpOili:iti Gild Percent Error in Displacement 
~ Full Tit" !.I lUI B Bub$tr. !:::! g Fun TE'uGG I. Subeu. !Pi: s. i' Time~ 26 41 29 42 
0.0 0.0 0.0 0.0 0.0 0.0 
I 0.0015 I 0.4736243 0.478;ln5 0.02566 0.429332 C.49!.\30j4 I 
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O.OOJ1) 2.010906 2.070341 O.Cl)2S; . 2.0a!HH5 2.0~9!f3S 
0.0045 4.194056 4.193802 0,00363 4.8&3719 . 4.t.H155i3 
I 0.0060 I 8.6625,73 I 8.662563 O.OC012 8,6821.111 a.6fl:il.i@i3 ~ooo." ". . 1l.6592! 13.65901 O.€)Ol~6 n.oc6M U.61151 
0.0090 19.79509 19.19587 O~COOIO i9.Gl~il9 19.tli6&9 
Q.0105 27.01146 27.07133 0.00048 21.0~H !9 27 .09122 
too."O I 3).47588 35.41573 0.00042 35.49501 35.45')476 
. ~ 
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, Table II. The Axial Force in Elements of B Substructure 
0.009 
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0.006 
0.009 
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111 112 
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23S.10321 282~6908 
252.9327 304.93S8 
179.10®2 254.5373 
137,.4126 223.7419 
141 142 
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199.0282 1021.000 
171.9509 1021.%09 
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159.4311 
..... ~_ .IIIIIOiU-.-. 
8 
PART 2. Large Deflection Vibrations of Be,iIDlS using Finite El ement ~'lethods 
Since space structures will be mostly large, lightweight and flexible, 
large deflection analysis methods are urgently needed to study the vi-
bratory responses of complex structures. 
A finite element Inethod has been developed for nonlinear vibrations of 
be&n structures subjected to harmonic exci1;atiol1. Longitudinal deformation 
and inertia are both included in the formullation. A harmonic force matrix 
[h] was developed for a beam element for nonlinear oscillations under uni-
form harmonic excitation. Formul ation of the harmonic force matrix is based 
on the mathematical basis (ref. 7) that th~! simple harmonic fOY'ca Po CoSl.Ilt 
is simply the first order approximate sol ution of the Simple elliptic forc-
ing function SA cn(p-c,k}. Also the well known perturbation solution 
3 
- ap;2. 
4 
of a Duffi ng system q'n + q + aq3 ,. Po COSI.llT is the fi rst order 
approximate solution of the Simple elliptic; response Q = A cn(p .. ,!t). Deri-
vation of the element harmonic force and neml inear stiffness matrices are 
given in detail in progress report ent'ltled, "Finite Element Analysis of 
Nonlinear Free and Forced Vibrations of Beams (ref. 8).11 
Table 3 shows the finite element free vibration results with and with-
out considering effects of longitudinal deformation and inertia (ELDI). It 
clearly demonstt~ates the remarkable agreem€!nt between the present finite 
element with ELDI and Rayleigh .. RHz solutions. 
Table 4 shows the frequency ratios fay' a simply supported and a clamped 
beam (L/R = 100) subjected to an uniform harmonic force of Po = 2.0 (Fa = 
1322 lb/in.) and 1.0 (3277 lb/in.), respectively. It demonstrates the 
9 
__ .. __ ... c --... .. _ ~ __ -___ • ___ ,i. ,'.,.-
.... ,:' 
P'ret;} Vibrat:lolll Ft'~quency Rati09 wlw., 
.. 
for Q S1~~ly Suppo~ted ~~am with I~@wcble A:iml Ends 
=========-==-=-==-='===::::::.-:::::-=-=.:'::-=""--==~-====== 
Elliptic 
Futmetion 
Sol""Uon 
(t'ef. 9) 
-~~-
1;0 1.01£192 
2.0 1.311S 
loO 1.6257 
4.0 1. ~n{iO 
.'00 2.3501 
F1~st P1nml 
Itel'QlI.:ion Raault 
t.oa,s 1.Om1l8 
1.3203 1.:n.19 
1.629)5 1.6022 
1.9761 1.9216 
2.3396 2.2544 
With !LOt (LIe m 100) 
na,lel(1h 
JUt:!! 
SolutioQ 
(ref. 10) 
1.0601 
1.2246 
1.4513 
1.730!V 
2.0239 
First V1Ml 
Iterat1o~ Remult 
t.0613 1 061J(3)b 
1.2270 1.22M(4) 
1.4620 1.4611(4) 
1.7383 1.1315(6) 
2.03~j 2.0378(7) 
_ .... _-... - .. .~== ~~~--=' 
-"" 
.. . _ ... 
-.;. 
----
a. Effeet0 of lonm!tudinal defc~t!on and 1uer,ti8. 
b.. Numb~r in hrluilicu cl~notlIDtI the num~r of ieer&tioolll !i::O get a eOllvorgt.'!d 
solution. 
10 
" , 
.\ 
. , 
\ 
r '''i!i'/ !'!!::,~:-~~-~. ': " '." . :-.~~-.-~~~,~:."""~."",,,"~~~',"""~-,,'""""."~,,"~r~~~~~ __ '~-~~ .. ~~, . 
l:';~i 
.. 
" 
P01f'COO V!braUon r1t'(£lqMnc,y bUGS I!) 1f;J L 
for a Simply Supporttld Qnd 6 el~@d 50~ with I~ovable Ax1g1 Rodo 
==-==--===.:===-=.="'-=--==.==="===~~ 
A 
- 1.0 
:1- 2.,0 
:I: 3,,0 
% 4.0 
:I: S.O 
_r. 
i: 1.0 
:!: 2.0 
:t 3.0 
:t: 4.0 
:I: 5.0 
Simple 
lUU.ptc 
Reepoitoft 
(ref.9L. 
1.1852 
0.0472 
1.6557 
1.4003 
1.8217 
1.8413 
2.1.013 
2.2606 
2.4361 
P~rtu1f'batl@u !!~ite !lem@nt .. 
Solution Ftret Final 
Ite~t1on Result 
1.7854 1.7852 1.7656 
0.8660 0.3621 0.8460 
1.6583 1.6563 1.6512 
1.4216 1.4),02 1.3760 
1.8314 1.8226 1.8002 
1.8708 1.8453 1.7846 
2.1213 2.09SS 2.0495 
2.299S' 2.2"525 2.1619 
2.4613 2.4236 2.3432 
Clamp~d ~am Subjected to P ... 1.0 ('0 ... 3'1.77 0 
0.2118 0.2165 0.2096 0.2091 
1.4307 1.4307 1.4297 1.4297 
0.8279 0.8292 0.8:!.15 0.8203 
1.2987 1.2990 1.2942 1.2936 
1.0401 1.0433 1.0219 1.0239 
1.3232 1.3248 1.3121 1.3099 
1.2183 1.2247 1.1979 l.laOa 
1.4101 1.4142 1.3910 1.3836 
1.3938 1.1.042 1.3619 1.3457 
1.5322 1.5401 1.5016 1.l~a7'v 
With EX.DI 
Finite JUemElnt 
Find 
Reault 
1.7682(l)b 
0.7106(4) 
1.5829(1) 
1.2123(4) 
1.6743(4) 
1.5871(6) 
1.3759(6) 
1.93n(1) 
2.1337(7) 
Ib/1tto) 
0.1772(3) 
1.425J.(3) 
0.7905(4) 
1.2743(4) 
(;.9726(5) 
~.2694(5) 
1.1151(6) 
1.3197(6) 
1.2513(8) 
1.40114(0) 
" __ ~~ ___ "",-""""_7~"'''-____ """"",,,,,,,, __ ,,,,,,,,-,,,,,,, __ -,,,,,,,,,,,,,, 
--~-----'~------...-----'-.---~------,..-.",~,,-
11. Effec!;1Jl of 10l:lg:U:ud:l.rual ddo~tim,l and 1nert:f.E.I. 
b. Number in brackets denote", the uu.mber of 1teratlOl'lll to get 8 conver.:gcd 
801ut:!.ou. 
,,' 
¥Q ., fi '" 
11 
c"loseness between the earlier finite element results without ElOra the 
simple elliptic response and the perturbation solution. The present finite 
el!ement results indicate clearly that the ELDI are to reduce the nonlinear-
ity. 
Reams with various boundary conditions D including movable axial ends, 
are given in reference 8. Results obtained ~'1il1 be presented at the Second 
International Conference on Recent Advances in Structural Dynamics, to be 
he,ld P,pri 1 9-13, 1983, at the Institute of Sound Vibration Research, South-
hampton, England. 
12 
-- -' ~ """ .... +.~ .. " -. '.' ... ' 
",:~""~!.,,,!,·"t)·.h"',\ .• l".·/·.· ...... • .~~"., ... , 
REFERENCES 
1. Hedgepeth, J.M.: Survey of Futln'e Requirements for Large Space Struc-
tures. NASA CR-2621, 1976. 
2. Benfield, ~U\., Bodlcys C.S., and Moroso\'I, G.: r~dal Synthesis rqethod. 
NASA Symposium on Substl"l.!cturing Testing and Synthesis. Marshall Space 
Flight Center', Alabama, August 1972. 
3. HurtYD W.C.: Dynamic Analysis of Structural Systems Using Component 
Modes. AIM J. 3:678··685, April 1965. 
4. Craig, R.R •• Jr. and Bampton. M.C.C.: Coupl ing of Substructures for 
Dynamic Analysis. AIAA J. 6:1313-1319~ July 1968. 
5. Benfield, W.A. and Hruda, R.F.: Vibration Analysis of structures by 
Component Mode Substitution. AIM J. 9:1255-1261, July 1971. 
6. Mei. C.: NASTRAN Modal Synthesis Capabi"lity. Progress report for the 
period fI.'ovember 1, 1982·-June 30, 1983, RE~search Grant NAGl-301, June 
1983. 
7. Hsu. C.S.: On the Application of Elliptic Functions in Nonlinear Forced 
Oscillations. Quarterly Applied Mathematics. 393-4079 1960. 
8. Me;, C.: Finojte Element AnalYSis of Nonlinear Free and Forced Vibra-
tions of Be&~s. Progress report for the period November 1, 1982 to 
June 3D, 1983, Research Grant NAGl-301, ,June 1983. 
9. Ho;nowsky-Krieger, S.: The Effect of an Axial Force on the Vibration of 
Hinged Bars. J. Appl. Mech. 17:35-36, Mc!rch 1950. 
10. Ra5u. I Sop Rao, G.V., and Raju. K.K.: Effect of Longitudinal or Inplane 
Deformation and Inertia on the Large Amplitude Flexur'al Vibrations of 
Slender Bea~s and Thin Plates. Journal Sound and Vibration. 415-422. 
1976. 
13 
End of Document 


Component mode synthesis and large deflection vibrations of complex structures

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