Asymptotic method for investigation of multifrequency oscillations of quasilinear systems of second order

PROCEEDINGS
O F T H E N A T I O N A L C E N T E R F O R
S C I E N T I F I C R E S E A R C H O F V I E T N A M
V o l u m e I
N u m b e r I
H A N O I 1987
PỈUVLEEDINGS OF ĩỉìli NATIONAL CENTI
M echanics
Asymptotic method for investigation of
muitifrequency oscillations of quasilinear
systems of second order
NGUYEN VAN DAO
Institute o f m e c h an ic s
NGUYEN VAN DỈNII
H anoi p o 1V! e f h n i c a i I n s t i t u u*
Si'MMAHY
Mosl of mechanical problems are written in the form of differential
equations of second order. In this article the asymptotic method for construc
tion of 1 he solution of these equations in the general resonant cases is presented.
The 1’irst paragraph is concerned wilh the system of one degree of freedom and
the second paragraph discusses the system with several degree of freedom. Bv
means of tho nroposeđ method one can rind the approximate solutions OÍ con
sidered ( ([nations willi desirable' exactness.
Mosi ol im'cha r.ica! probh'iYis ilft’ written in llic form ol din C'lvniiai
liou.s srcrih! order. Asymptotic method i'or monof requoncy oscillations and
SOI1U' spi‘(‘ial casrs oi' multifroquency oscillations of quusiiincar second order
s v s l o m s have b e e n given in • 1 . 2 TilV* mrlhcxi lor SÍUCỈY OÍ nuiltifrequene\
osfilliitions or quasi] incar first order systems was presented in — 5 . in ill!:.
p:i]K'i- [he asymptotic method for inveslignlion of multifreq urncy oscillations of
quasi linear SYSÍCIUS or second ord.r in general resonant cases is discussed. /(>/.
i.vt IIS c o n . s id t T a (juasiiiiii a: v‘:j’lit 1 ion of Si'comi order with slow k

\ arviiiiị paraiiU'kTs:
I. l\ rnonrcnox
II. SYS I KM wrm MNCiLK UKGHKK OK h'HKKDOM
-f f (-)
Ill
whore 'T = e\ is slowly varying time. z is small positive parameter.
f.) = (01

0r). — = vk (t). F is periodic* function of (")k with period 277.
(! i
!l is s u p p o s ed ilia! 1ÌÌ (t). 0 (t). F su ffic ie n t num b,'!' o f d e r i v a tiv e s
relative] V T. 0 , \. — and that for all T in interval () < T L, the quan-
(it
litics Hi (T). c ( t ) art1 positive, There w ill be the following expansions :
clx \ T, / ^ cix \ . T, / ~ fix
F ( t . 0 . X. e) - K, ( , . 0 , X. + «F,Ị T . 0. X.
F .^ .a x . F,.« i c e . c = (C,

C ) (2)
here i'j (". X. \) a;v polyno m ials ol X.
We shall consider the resonant cast' when there exists the relation of type.
p* co ÍT! + c * V (T) = II .
<:* = ( C l c . r ). V <T) = (V , V ,) (•’>)
where* CO (T) = y
11

'~1
and i5* c *

CĨ are integers, p* =ỹt= 0 . If there

V m \ Z)
is no rohilion of ÍYỊK* (.*>) \Y(k ha ve the n o nseso n a n t case.
In nonrcsonanl case the solution of equation (1) is found under the form :
\ — a. cos 4 * + £• U1 a’ lỉ'< ©) + u2 ,/T‘ a* ©' + . . . . (Ij
here u is neriodii* functions of 4 ", 0 k with period 2~ and quantities a. lF as
lunclioiv- or Ihne are (!e!ermin(‘(i from ecịnations :
— £ -Vj (T , cl) ~r £ “ A j ( t . Li) +
(It
(IT ^
——— (-J (:) T £ ill (;, a) T • • • • (5)
(It
Stibslilulini* ()) into (1) and com pari 11 ii l iisfly the c oeffic ients of e and
!ho 11 I Lie harm onics w e obtain :
V ;i (i ( 111 CO) 1 f f I • • I I ' 1 I P I, V I / i
A, (t . a) =

-



— — I tsill ỉ (ỉ Ỉ (i()|
2mco (It m co ( 2 - ) M 1 ) I
(i
f . . . I* !■', co s'iY
J J
I) (I
Bj (t . a) =

I * • 5 *; i rosTiIM MB, . . . (!(-),., (I))
mroa i 2 “ ) r + l

i(ị)‘j +1-H)
II, (T. a. T. (-)) = Y
Ự2~)'"]
m [c o — (pco + C'J
!>.(•
)_
f . . . f I-, e-i(p'l'K-0.
.

—
II II
Lei IIs consider now more interesting resonant case, it is assumed that
I here is a resonant relation :
p * 12 {-) + c * >J ( t ) = (I. Í 2 ( t) ~ (0 ( T ) (7 )
Now. (Ill* solutioji or equation ( 1 ) is presented as
\ — ;Icos (' ■ *■ 'I ) + £ IIJ (t. cl. c “Ị" , 0) ^ lu • * ■ (^ )
T he am p litu d e a and phase '!' are delennin ecl from the- eq ua tions
—— - = £ A; ( SI. in -4- e- A, ( a. T) + . . . .
(ỈI (9)
— - — = C.J (-) - Ll (z) + £ 14, a . T ) - f £ - . . .
(It
SubstiUitinp- the expressions (ÍS). (9 ) into equation (1 ) and comparing tile
coefTicients o f £ a nd those or the h arm o nic s w e ohl:»in:
— a il Ini (') w (:)| ( 2 Ỵ-\ i<rp*T (
■ J ill ( T ) Co ( t ) ( ! t i n ( t ) ( 2 -) r‘1 L-> (
ị LCO (T ) -
I — <7~p*'-'[to( 7) — Ĩ2( t ) I
') _
”* # "U *
I I-', e ?ơ|’ 1 ' cos (£ + '!') (I (Ẹ r- 'Ỉ ) <iw, (K“)

II 0
2 CO (T )
i Ị T )(T ' I )*■ [ (<)[ - ) - L2( T ) I
J • • J !■'. (T.OO.X„.\ ) e iơ|,* ! 1 sin ( c + in (I (Ẹ+ M‘) c!0. . . . (10r Ị.
X
II 1}
E
! [ I ■(£+ ll’H- (■<>; j» r - i [ J) (Ẹ 4- 4’) + C0]
e !•', e ■ (K' + T ) d 0 1 d0,.
I! II
ill))
Ill - SYSTEM W ITH SEV ER A L D EG K K ES OF FREEDOM
Consider the ]'ollo\vintỊ equations oi second o;-(k“r:
X X
—“ Ị XI :liJ {':> cii Ị + X! ìj,j lli = eí'íj (-)’ li‘ (,‘ e)‘
i = 1 i = 1
(j = 1. 2

N)
in ( r(.\sonanl case w he n I hero exist resonan l relations
]]W Q (T ) + c!a )M T ) ^ 0 . (12)
V = (V ,

v r ). a = 1 .1 !

11. Ũ = 12,
£>„).
ilj arr fOj. p = (P,

PN). c = (C cr).

T > ( “ 1 ( ® ’ •
i • . are integers, ojj are natural iroquencies. the roots or equation ><■*
a ’*■ j ~ a ijt0* // ~ ■l-
ỉn LÍ1 is C;:se the solution oi cijiUiiions (11) will be round in I he form
X
(‘j = y z>[*> ÍI, cos + ir s) + e 1!<‘> (a. Ị 4- T , 0 ) + e2u<2> (13)
a = (n,

ar r Í = a , ' X). r = (li*.

'I'x K Í-) =((-),

0,.)
vrhorC 9«* > all' the n or m a! I'iinctions — the non iriv ia ỉ solu tio n of iiu* C‘<] uu lions
Y ' (b ij — >‘ij "') -? ’ ' ‘ = iJ- (.!• s - ■ • -

X )
i = 1
u"") arc ike periodic funclions of ; + 4*. 0 vviih period 2" and I ho aniT)i iitidc
i
11 and o huso vr arc* d e te r m in ed fro 111 e q u a tio n s :
' = sA1,'’ (T. a. T ) + E A (T. 11. T) + . . .
— —— = w , (t) — i.2s (" ) + £ B , ( t . .1. ;1 ) H . . . (11)
tit
([ c
i isAJ'*' z J- v*-'*’ a - 0 (15;
<ii
By analopv jo (he rirsl paragraph, from equations (Ji) — (15) bv C01Ì1DỈI-
rin ơ I IK' c o ci l iciciiis of £ am i llio.se o! ÍÌÌO hiiniioniv*;., w o ob ta in :
n

; V' Ơ' -' s ’ ••
£ J I
•*-u" ■ ị

)
ft ■ " I) ti j -
^ ,) \ I ' I
V (toI — 1 2 ,) — —

‘i a . t o . b 1/ ’ = — Ỷ
/—1 í/ỉ . A!
1 = 1 p
(*( )S (Ẹ
4 1) . . . d ( +
) (Ifc)
. (
(1 (
i «o , — 1 2 |)as

4-12COsA.' =
r* 4 \
a s (1 (cos M J
M d x
M
I V' Ơ' ’■ 'T' • ’
a s
Í
/V
X
II i =

i £ o'.!'4
r> j IL* ' “
1
1 Q i e -
X sill • i;. -f 'l' j (I (C| -f ll\) (1 ( -f 4 y ) (]©J . . .( ]0 ,. .
>' * 4*
\ \
r \ ul = I V ( ' ,
u •
>‘N: J ( ' x + 4* ) + c',-> 0 , + . . . + c ./' 0 r.
i [!’, Ũ, + r t)
EE?;'
(ị + 4 \ ) + c , 0
r 0 r ]
I ).(• s
col — (Pcu + Cu)~] M s (27C)
o. ,N + r
( )
i IP, (Ị, + T .)
cr 0 r]
(I (S, + 0 , +
It i = I
\
52
■u.i=I
\ V s
M
'■ ai. wv 11V
( CO
foj)

(I cp',v
( í I
1 (I t
I V I
1 TIT
sin (;, + T v).
E
l ims. Ih(‘ ;Is\ ín [>Ỉo 1 ic* iiK‘th()(l pre sen ted here gives the ap prox im ate so lu 
tions OỈ the c on sidered CM I nations in co m plicated rr so m m l cases w i i Ỉ1 f h c* (losiivd
(i (‘Ljrcr of (■ Xitel II t*ss.
1. ỉ l.H. Boro.nofioB, lO.A. ;\\i)Tpono/ibCKiiii, AcHMriTOTimecKHe MOTGUbi R Teopim
ìie .iiìH eíiH b ix K o.ieỐ aH Híí, .V\0CKBa, 1 9 6 3 .
2. i().A. .UiiTpono.ibCKnii, 1 lp o ố.ie.M b i acHMHTOTỉmecKOM T eopiỉM necT au H O H ap H bix
K o .ie ố a H n ii, H avk-a, 196-4.
?>. Ỉ3.[]. PyỐaHHIÍ, Ko.ieÕaHIIH KB33H.1HHeHHbIX CHCT6.M c 3aria3;ibIBaHHeM. 113.1,
I lavica. WocKiKi, 1969.
1. A\.H. Kyaiy.ìb, o IIOHTII IiepHOAHMeCKUX peiIieHllàx KB33H.1HHÍÌHHbl.\ CHCT6M
Iipii MHOĩOKpaTHOM p e3 0 H a n c e , 113B. A H C C C P , O T H , M ex a H H K a H MamiĩHoc-
rpơeniie N = I, 1960.
5. M H. Kymy.ib, r .ll. AmiKeeB, ỉleCTau.HOHapHbie npouecchi npn noMTH nepno-
.UPieck-nx KO.ieốaHHHx KBa3H.auHeiiHbix CHCTPM. Ko.ieốaHHa H npoHHOCTb npH
nepeMeHHbix Hanpíi>KeHHsrx, M3A. Hayh-a, MocKBa, I960.
(j. Ngu ven Van Đao, Nguven Van Đinh, Asvmploíic method for tho quasi-
lincar systems or sccond order. Proceedings of Hanoi Polvlechnical
Insiiiulc X 5 7. 1975.
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