Nonlinear oscillations of third order systems. Part 1 Autonomous systems

-N It is supposed that the characteristic equation: has a pair of cither complex or imaginary roots.. The non-critical case has been investigated in many publications see, for example [

P o tlsc k A ca d em y o f S cien ces In stitu te o f Fundam ental T ech n ological Rtxearch, W arszaw a.

N O NLIN EA R O SC ILLA TIO N S O F T H IR D O R D ER SYSTEM S

PART I A U TO N O M O U S S Y S T E M S

N G U Y E N V A N D A O (H A N O I)

Introduction

A lot of mechanical and physical problems lead to the study of oscillations in the system described by the differential equation of the third order:

(0 1 ) J t + a x + b x + c x = e f ( x , X , X, r),

where a, b, c are real constants and the function f(x, x9 3c, t) can be expanded in the form:

H

( 0 . 2 ) f ( x , x , x , t ) ~ JT elnv,f n(x, x,x).

-N

It is supposed that the characteristic equation:

has a pair of cither complex or imaginary roots We have the following definitions:

1 I f th e c h a ra c t e ris tic E q (0 3 ) h a s a n e g a tiv e r o o t a n d a p a ir o f c o m p le x ro o ts w ith

a negative real part: Ằi = — f, Ằ2 = — *7 + /í 2 , Ằ3 = — TỊ-iíỉ , then we have the non- critical case.

2 If the characteristic equation has a negative root and a pair of imaginary roots

A1 = — f , A2 = iQ> A3 = — / Í3 , th e n w e h a v e th e c r i t ic a l case I a a d d it io n , i f V ^ p ỉq ũ ,

where /7, q are integers and V is the exciting frequency, then we have the critical resonant

c a s e

The non-critical case has been investigated in many publications (see, for example [ 1 - 6 ])» but the critical case has not, to the author’s knowledge, been examined hitherto

In this work we study systematically the different kinds of nonlinear oscillations: free oscillation, self-excited oscillation, forced oscillation and the parametric one of the third order system in the critical case, by means of the asymptotic method [ 8 , 9].

In the critical case, in the generated system (fi = 0), there exists the harmonic oscillation depending on two arbitrary parameters a and Y>:

X = a c o s ( Q t + y).

Wc shall not look for the general solution of Eq (0.1) depending on three arbitrary constants, instead wc shall construct a family of two-parameters particular solutions, which has a strong stability property It attracts all solutions close to itself of Eq (0.1).

T h e a p p r o x im a t e a s y m p to t ic m e th o d is c o n v e n ie n t f o r th e in v e s t ig a t io n o f s ta tio n a ry

o s c illa t io n s a n d t h e ir s ta b ility

T o a sse ss th e v a lid it y o f the th e o ry , a series o f e x p e rim e n ts o n th e a n a lo g c o m p u te r

h ave b e en m a d e T h e th e o re tic a l re su lts show a g o o d a g re e m e n t w ith th e e x p e rim e n ta l

ones.

T h is c h a p t e r c o n s is ts o f five S e ctio n s I n th e f ir s t S e c tio n th e tw o -p a ra m e te rs s o lu t io n

o f E q ( 1 1 ) is c o n s tru c te d b y m e a n s o f the K r u lo v - B o g o liu b o v a s y m p to tic m e th o d I n the s e c o n d S e c tio n , th e D u ffin g case R - -Ộ X3 is c o n s id e re d I t h a s t u r n e d o u t th a t th e

a m p litu d e o f fre e o s c illa t io n in cre a se s i f ậ > 0 , a n d d e cre a se s i f /? < 0 T h e t h ir d S e c tio n

is d e v o te d to the s tu d y o f the se lf-e xcite d o s c illa t io n in th e t h ir d - o r d e r syste m , w h e n th e

fu n c tio n R is o f th e f o r m R = (1 - x 2)x. I t has b e e n fo u n d t h a t in the sy ste m in v e s tig a te d ,' there e x is ts a sta b le lim it cycle T h e in flu en ce o f th e C o u lo m b f r ic t io n a n d th e tu rb u le n t one o n th e s e lf-e x c ite d o s c illa t io n in the th ird o r d e r syste m , is e x a m in e d in Sects 4 a n d 5

I t has b e e n p ro v e d th a t the a m p litu d e o f th e s e lf-e x c ite d o s c illa t io n d e cre a se s as th e se fric t io n s in c re a s e I n a d d it io n , the self-e xcite d o s c illa t io n c a n d is a p e a r u n d e r the a c t io n

o f a s tro n g C o u lo m b fric tio n

1 Construction o f Two-Parameters Solution Stability

L e t u s c o n s id e r th e fo llo w in g n o n lin e a r d iffe re n tia l e q u a tio n o f the t h ir d o rd e r:

( 1 1 ) 'i' + ỉ'x + ũ 2x + ỉ ũ 2x = e R( x, X, x ),

w h ere f , Í Ì a re c o n s ta n t s , £ is a s m a ll p a ra m e te rs a n d R is a n o n lin e a r f u n c t io n o f X, X, X.

W e s h a ll fin d a s o lu t io n o f E q 1.1 in the f o r m :

cp = í ĩ t + yĩ' H e re a a n d y are th e fu n c tio n s s a tis fy in g th e f o llo w in g d iffe re n tia l e q u a tio n s :

(1 3 )

S u b s titu t in g th e e x p re ss io n s (1 2 ), (1 3 ) in to ( 1 1 ) a n d c o m p a rin g th e co e ffic ie n ts o f e

w it h e q u a l d e g re e s, w e h a v e :

(1 4 ) 2í2(Q aB ị —ỈA ỵịsin q ĩ—l í ĩ i ỉ a B i + í 2 ^ i) c o s 9 ? +

T o f in d th e u n k n o w n fu n c tio n s A l, B i, Uị f r o m (1 4 ), w c firs t e x p a n d Rq in th e F o u r ie r

series:

Nonlinear oscillations o f third order systems Part I 513

00

/7=0

£0(0) = J- f Rodv - <^o>>

0 2n

g„(<j) - I R 0cosn<pd<p = 2 < / ỉ0c0SH95>,

71 0

2n

/?„(ớ) = — f R 0 sinn(pd(p — 2 ( R0 s i n n ọ ĩ> ,

71 0

in w h i c h /J = 1 , 2 , a n d < ) is t h e a v e r a g e d o p e r a t o r o n t i m e

T h e f u n c t i o n ;/x is a l s o f o u n d in t h e se r ie s:

Ml

w i t h t h e c o n d i t i o n t h a t it d o e s n o t c o n t a i n t b c t e r m s h a v i n g a z e r o d e n o m i n a t o r

Dy s u b s titu tin g (1 6 ), ( 1 7 ) in to (1 4 ) \vc o b t a in :

(1.8) 0 * 2 ( 1 - m 2) [(Ệuim+ m Q v ln) c o s m < p + (ỉv im- m í ì u

lm)smm<p]-m

00

— 2 ũ ( ữ A l + ỉ a B ị ) c o s < p + ĩ ũ ( a ^ B 1 - Ệ A ^ s i n i p = ^ [g„(a)co&n<p+hn(a)smn<p].

«» 0

B y c o m p a rin g the s in9?, COS9? te rm s in (1 8 ) w e h a v e :

i Q ị Ũ A + aỆB,) = - £ t(a),

2 i 3 ( - ^ 1+fli251) = A,(a).

H e n c e o n e o b ta in s :

A < \ _ £ M ứ) + % i ( a) _ í< /? o s in ọ 7 > + i 3 < ^0cos(p>

» _ £ < -R o S in < p > -£ < * o C O S9>>

lV ' 2ứ a ( í J + í 3 2) i2a ( f J + £ J)

T h e c o m p a ris o n o f the o th e r h a rm o n ic s y ie ld s :

i 2 2 ( l - W 1 2 ) ( f w , B + w i 2 & lln ) = g m( a ) ,

£ 2(1 - m 2) (-r n í2 u ỉm + Ệvlm) = /fm(a ), m # 1

H e n c e it fo llo w s :

ỉ g m ( à ) - m í ì h m {a)

i h m(à )+ m ữ g m(a)

tfJ(l —ma) ( | J +m ĩÃ2)

Thus, in t h e first a p p r o x i m a t i o n w e h a v e :

where a a u d xp arc the solution of the equa ti ons:

d a _ í < / ? 0 sinẹ>>-l-.í?<y?0Cos < p >

(1.12)

dtp _ Q ( R 0sin(py — Ệ ( R 0cos(py

d t e Q a ( Ệ 2 + Q 2 )

T h e re fin e m e n t o f the first a p p ro x im a tio n is :

(I n ) r = /7rvYc(.Q/+ ,,,) + V { h m- in Q h „ ) c o s m (Q t+ ỳ ) + (Ậli,„+mQgm)únin[Q t+ '

#/* =» 0

m*\

T h e c a lc u la t io n s o f h ig h e r a p p ro x im a tio n s p re se n t n o d iffic u ltie s, b u t are ra th e r 1

a n d «irc n o t re p ro d u c e d here

W e c a n w rite the first e q u a tio n o f the syste m (1 1 2 ) a s :

dt

Obviously, the stability condition of the stationary solution a = a0y &(ơ0) = 0 (1.14) is of the form:

2 The Duffing Case

L e t u s c o n s id e r the case i? ( x , X , X) = —ậ x3 N o w , E q (1 1 ) h a s th e fo r m :

I n t h is c a se w e h a ve :

(2.2) R0 = —/?ứ 3 cos3<p = — Ỵ ứ 3 (3 cos 9 ?+cos 3ẹ?),

a n d th e re fo re

g t( a ) = - J 0 a 3 , h ^ a ) = 0 ,

(2 3 )

E q u a t io n s ( 1 1 2 ) are o f the fo rm :

Nonlinear oscillations o f third order systems Part I 515

B y in te g ra tin g E q s (2 4 ) w ith the in it ia l v a lu e s ( / 0 , a0) w e o b ta in :

(2 5 )

a 2 = - CL =

a - / ? * ( / - / 0) a l

V = V o - y fí- l n l - ^ ơ - / o )

H e n c e it fo llo w s th a t (see F ig 1 ):

1 I f /? > 0 then the a m p litu d e o f the o s c illa t io n in crease s fo r t > f0 fro m a = a0

to a ~+ 0 0.

2 I f /5 < 0 then the a m p litu d e o f o s c illa t io n d e cre a se s fro m a = (ỈQ to a 0

T h u s , 1he n o n lin e a r te rm has a g r e a t in flu e n c e o n the fo rm o f the re sp o n se c u rv e o f

free o s c illa tio n T o c h e c k the v a lid it y o f the th e o re tic a l a n a ly sis, an a n a lo g -c o m p u tc r

a n a l y s i s h a s b e e n carried o u t T ile e x a m p le s o f th e w a v e fo n n s o b tain e d o n th e a n a lo £ -

computer for E q (2 1 ), with the p a ra m e te rs Ệ = 1 0 , i2 = 1, are s h o w n in Fig 2 I n this

fig u re , the firs t w ave (a ) sh o w s the ca se p - 0 , th e se co n d (b)—(ỉ *-= - JO, a n d the t h ird (c)— ỊÌ = 10 T h e figure c o n ta in s a ll types o f o s c illa t io n s that a rc p re d icte d by the th e o ry

a „

0 3

0.2

3 Self-Excited Oscillation

I n th is S e c t io n we s h a ll stu d y th e V a n d e r P o l c a s e :

I t is easy to see th a t:

( R 0sin9?) = —

< /? o C o s < p > = 0 ,

(3.2)

(3 2 ) gn = 0, 'in , hm = 0, m # 1, 3,

Q

Hị = —aQ

T h e re fo re , th e e q u a tio n s o f the first a p p ro x im a t io n a re :

(3.4)

da

h ĩ ■* 20 F T Q 2 ) 4 - t )

E q u a t io n (3 4 ) has a sta tio n a ry so lu tio n ứ0 = 2 T h e sta b ility c o n d itio n (1 1 5 ) f o r this

s o lu t io n is sa tisfie d

Fig 3

T h e re fin e m e n t o f the first a p p ro x im a tio n is o f th e fo r m :

(3.5) X = 2cosỢ2/ + y0)+ 4 (g 2 + 9Q1) [3 cos 3(^ r + Vo) - -jjSin3(flf + Vo) j

T o v e rif y th e th e o re tica l re su lts, the o r ig in a l E q (1 1 ) w ith R fr o m ( 3 1 ) , is m o d e lle d

o n th e a n a lo g -c o m p u te r M E D A 4 1 - T C f o r th e ca se ỉ = Q = 1 , e = 0 1 T h e o sc illa c io n

d ia g ra m s a re p re se n ted in F ig 3 : d isp la c e m e n t-tim e a n d F ig 4 : p h a se p ictu re T h e

e x p e rim e n ta l re s u lts agree w e ll w ith the th e o re tic a l ones

“x

2

Nonlinear oscillations o f third order systems Part Ị 517

4 Influence of Coulomb Friction on Sclf-Exdtcd Oscillation

In t h is S e c tio n the in fluen ce o f the C o u lo m b fric tio n

o n the se lf-e xcite d V a n der P o l o s c illa tio n is e xa m in e d I n th is case th e m o tio n e q u a tio n

is o f the fo rm :

(4 2 ) ‘x +ỆX + Q 2X + ỆQ2x + £ ( x2- l ) x - f £/j0sign.v = 0

T o fin d the s o lu tio n o f E q (4 2 ), we use th e th e o ry represented in S ect 1 T a k in g in to

a c c o u n t

<sinọ? s ig n s in ẹ ? ) = 2/tt, < c o s < psig n s in <p> = 0,

we h a v e in the firs t a p p ro x im a tio n :

(4 4 )

da dt d(p di

CỆ

= Q _ eQ 1 - - - 1

F r o m E q (4 4 ), it is seen th a t by the a p p e a ra n c e o f th e C o u lo m b f r ic t io n , the o rig in

.V = X = 0 is n o t s t i l l an e q u ilib riu m o f the syste m c o n sid e re d T o c le a r u p the in flu e n ce

o f the C o u lo m b fric t io n on the self-excited o s c illa tio n , w e in vestig ate th e sta tio n a ry state

w ith the a m p litu d e determ ined b y the e q u a tio n

(4 5 ) A( 1 - A 2) -= /4 = - y > .0

T h is e q u a tio n c a n be solved g ra p h ic a lly b y c o n s id e rin g the p o in t o f in te rse ctio n o f the c u b ic c u rv e y t = A ( i - A 2)a n d the sta rig h t lin e y 2 = Ih o /jtii. F ig u r e 5 lead s to the

f o llo w in g c o n c lu s io n s :

1 W h e n h0 = 0 in the system c o n sid e re d , th e re e xist tw o sta tio n a ry state s c o rre s p o n d ­

in g to A ị = 0 (u n sta b le ) and A 2 = 1 (se lf-e xc ite d o s c illa tio n )

2 W it h in c re a s in g h0 (0 < ỌìqItcQ) < 1 / 3 ^ 3 ) , A l in c re a se s an d A 2 d e creases

E q u a t io n (4 4 ) c a n be w ritte n in th e f o r m :

w h e re A 3 < A ì < A 2y (A 3 < 0 ) I t is e a sy to see th a t 0 ' C ^ i ) > 0> <P'(A2) <

th e re fo re th e sta tio n a ry state c o r r e s p o n d in g to A l is u n sta b le a n d th e state A 2 ii (s e lf-e x c ite d o s c illa tio n ) O b v io u s ly , in th is c a s e the C o u lo m b fric t io n d e cre ases the tude o f th e self-e xcite d o sc illa tio n

3 W it h h ig h v a lu e s o f h0(h0 > (QrtỊ3 j / 3 ) ) th e se lf-e xc ite d o s c illa t io n d isa p p e a

T h u s , d e p e n d in g on its v a lu e s, th e C o u lo m b fr ic t io n c a n e ith e r d e cre a se o r e xti the s e lf-e x c ite d o sc illa tio n

5 Influence of TarbuleDt Friction on Self-Excited Oscillation

N o w , w e go o v e r to the stu dy o f th e o s c illa t io n d e sc rib e d b y the e q u a t io n :

(5 1 ) x + Ệ x + & x + Ệ ữ2x + e ( x 2 - l ) i + e h 2x 2signx = 0 ,

w h e re in th e Jast term ch a ra c te riz e s th e t u r b u le n t fr ic t io n B y m a k in g u se o f th e m

p re se n te d in P a r 1 w e o b ta in the f o llo w in g e q u a t io n s o f th e first a p p r o x im a t io n :

* = acoscp,

~ d t =

(5 2 )

d(p

2 (Ệ2 + Q 2)

e Q

]• A =

2(ề2 + Q 2)

Considering in the first quadrant of the plane (y , A ) the points of intersection o

c u b ic c u r v e y x = A ( \ -A2)a n d the p a r a b o la )*2 = (ỉ6/3n)íĩlĩ2A2( F ig 6) o ne ca n

that:

F i g 6

1 I n th e system d e scrib e d b y E q ( 5 1 ) th e re a lw a y s e xists th e u n s ta b le station* state A = 0.

2 W h e n in c re a s in g the co e fficient h 2 f r o m z e r o , the a m p litu d e o f s e lf-e x c ite d o s c illa t i

d e cre a se s I n c o n tra s t to the case w ith th e C o u lo m b f r ic t io n , the s e lf-e x c ite d o s c illa t i here is n o t e x tin g u ish e d com pletely

Nonlinear oscillations o f third order systems Part I 519

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S t r e s z c z e n i e

N IELIN IO W E D R G A N IA U K L A D 0 W T R Z E C IE G O RZẸDƯ

CZẸỐC I Ư K LAD Y A U T O N O M IC Z N E

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w niniejszej, picrwszcj czẹáci pracy z b ad an o d ig a n ia u k la d u autonom icznego R o z p a trz o n o nielinio-

w o SỎ funkeji resty tu cy jn ej w postaci funkeji D uffinga, c h a ra k te ry sty k i tlu m ien ia w postaci ta rc ia C oulom ba

i ta rc ia tu rb u len tn eg o Z b ad an o d rg a n ia sam ow zbudne p rz y rố zn y ch rod zajach tarcia

p e 3 K> M c

H E J IH H E H H L IE K O JIE B A H R H C H C T H M T P E T b E r O nO PflJXKA

M A C T b I A B T O H O M H B IE C H C T E M L I

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nepBoft uacm paốOTbi HCCJieAyioTCH KOJie6aHHH aBTOHOMHOỈí CHCTCMbi PaccMaTpHBaioTca HCJiHHeữH0CTb ộyHKUHH BOCraHOBJieHHH B BJtfle ộyHKUHH ilK>4>4>HHa H XapaKTCpHCTHKH ACMnỘHpOBaHHH THUa TpCHHH KyjioHa H Typ6yjieHTHoro TpcHKH HccjieAyioTCH caM0B03ổy>KtuicMi>ic KOiieổaHHH ỊỤIH pa3JDFiHbix THnoB

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Received October 17, 1978.