Nonlinear oscillations of third order systems. Part II Non-autonomous systems

Two-Parameters Solution la the Resonance Case... The amplitude and phase of the stationary oscillation are determined from the rela­ tio n s: Py sin y j- ỉ cosy = yi li+ffai* By elim

NONLINEAR OSCILLATIONS O F T H IR D O R D ER SYSTEM S

PART II N O N -A U TO N O M O U S SYSTEM S

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

Introduction

T h is c h a p t e r is d e vo ted to the stu d y o f the o s c illa t io n o f t h ir d o rd e r n o n -a u to n o m o u s system in th e re so n an ce case S e c tio n 1 d e als w it h the g e n e ra l th e o ry T h e a p p ro x im a te

s o lu tio n o f the m o tio n e q u a tio n is fo u n d b y m e a n s o f the a s y m p to tic m e th o d B y c o n ­ tra st w it h th e a u to n o m o u s case [11] the p h a se \p here h a s a g re a t in flu e n c e on the a m ­

p litu d e a n d fre q u e n cy o f o s c illa tio n In S e c t io n 2 the s t a b ilit y c o n d itio n o f the s ta tio n ­

a ry o s c illa t io n is an a ly se d It is w ritte n in a c o m p a c t a n d c o m fo rta b le fo r m fo r p ra c tic a l use A s p e c ia l but im p o rta n t case— the D u ffin g ca se is d e a lt w ith in S e c t io n 3 I t h a s been p ro v e d that f o r the th ird -o rd e r system , th e h a rd n o n lin e a r c h a ra c te ris tic d o cs n o t

e x h ib it a sta b le p e rio d ic o s c illa tio n T h is p h e n o m e n o n is n o t o b se rve d in the se c o n d -

o rd e r sy ste m T h e n e x t tw o S e c tio n s 4, 5 a re o f a m o re sp e c ia liz e d c h a ra c te r T h e y

c o n ta in th e stud ie s o f the fo rce d o s c illa tio n in th e t h ir d o r d e r se lf-e xc ite d system C e rta in

z o n e o f s y n c h ro n iz a tio n is d e term in e d I n c o m p a ris o n w it h the a n a lo g o u s p ro b le m in the s e c o n d o rd e r system , the fre q u e n c y e n tra in m e n t zo n e b e co m e s n a rro w e r

A n e x p e rim e n t o n the a n a lo g c o m p u te r h a s b een c o n d u c te d to v e r if y th e th e o re tica l

re su lts

1 Two-Parameters Solution la the Resonance Case

Let us consider the oscillations of the system governed by the differential equation

o f th e t h ir d o rd e r:

(1.1) X + Ệ X + Q 2X + Ệ Ũ 2X = e R ( x t X , 3c) + P o s i n y f ,

h e re f , Q , y, PQ are c o n stan ts a n d R ( x, X , 3c) is the n o n lin e a r fu n c t io n o f X, X , X I t is

assumed that there is a reasonance relation

Pq is a s m a ll q u a n tity o f the firs t o rd e r: p 0 = eP.

( 1 - 3 ) X = acos(yt + yj)+ y , yt) + e2u 2( a 9 y) , y / ) +

w h e re u (a, y ỉ,y t) are the p e rio c ỉĩc fu n c tio n s w it h the p e rio d 271 re la tiv e ly o f y a n d y t

a n d a, rp a re de term in e d fro m the e q u a tio n s

126 N guyen Van D ao

da

— = e A iia , V)) + s M j ( a , y j) +

~ = eB ia , v ) + e 2B 2(a, v ) +

B y s u b s titu tin g (1 3 ), (1 4 ) in to (1 1 ) w e get

a 3« , Ô2U, 2 õu, + 5 ĩ ^ + Ệ d ^ - +y a f + * v ui + e ■■■

= eP sin yt + £jR(acos(p, —aysin<p, - a y 2C0S(p)+ e2

here

R ị x, X , x ) = R ( x, x ) - ơ x — í ơ x

B y c o m p a rin g the coefficients o f £ in (1 5 ), w e h a v e :

(J.6) 2 y ị y a B 1 - Ệ A i ) s i ĩ \ ( p - 2 y ( y A 1+ ỉ a B l )cos<p +

+ ^ + l J £ + r ‘ í ^ + ỉ r 1u , - r s m r ,

+ R(acos<p , — y a sin tp , — y 2acos(f ), 9 ? = y t + y

N o w , w e e x p a n d the fu n c tio n in the F o u r ie r se rie s:

00

( 1 7 ) /?(a c o s< p , - y a s in ọ ? , — y2a co sọ ?) = ^ (rlHcosn<p + r2llsinn<p).

T h e fu n c tio n Ui is a lso fo u n d in the s e rie s:

m

w ith an a d d it io n a l c o n d itio n : « ! do es n o t c o n t a in the te rm s w it h a v a n is h in g d e n o m in a to r

Substituting (1.7), (1.8) into (1.6) and equating the coefficients of sinẹ>, cosip, we obtain:

2 y ( y A 1 + ỉ a B 1) = P s i n i p - r ^ , 2y(yaB1 - Ệ A i ) = P c o s y + r 21.

B y c o m p a rin g the coefficients o f the o th e r h a rm o n ic s , w e h a v e :

y 2(l - m 2) ( ỉ u lm + n tyv lm) = rlm, m í 1,

( 1 10 )

H e n c e w e o b ta in

t r lm' - m y r 2l

(1.11)

y 2( l - m 2) ( £ 2 + m 2y 2)

E q u a t io n s (1 9 ) g ive

P ( y s in y — Ệcosrp) — y f u - Ệ r 2l

A, =

2 y ( i 2 + y 2)

( 1- 12)

i >( i s i n v ’ + y c o s y ) - f / - n + y r 2t

= — —

2 y a ( ỉ 2 + y 2 )

T h e c a lc u la t io n o f h ig h e r a p p ro x im a tio n s p re se n ts n o d iffic u ltie s

Thus, in the first approximation we have:

d a P ( y s i n ụ > - ỉ C O S r p ) — y r tl - £ r 21

(1.14)

d y _ P ( f s in y + y c o s y ) — Ệrn + y r21

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

trĩ = 0

m# 1

H e re a a n d y sa tisfy E q s (1 1 4 )

The amplitude and phase of the stationary oscillation are determined from the rela­

tio n s:

P(y sin y j- ỉ cosy) = yi li+ffai*

By eliminating the phase rp one obtains:

2 Stability of Stationary Oscillation

L e t ôa, ỗrp be s m a ll p e rtu rb a tio n s an d set a = aQ + òa y = y>o + <ty w h ere a0f y) a rc the s ta tio n a ry v a lu e s o f J , y d e te rm in e d fr o m (1 1 6 ) P u ttin g th e a b o v e e xp re ssio n s in to

E q s ( 1 1 4 ) a n d m a k in g use o f th e steady-state E q s (1 1 6 ), w e o b ta in th e fo llo w in g v a r ia ­

t io n a l e q u a tio n s :

^ d T ■ W g T l - f r ' i + - r ' i X v )

( )

* - W V K K f r n - ^ O ^ + i m + f r O i v l ,

w h ere the p rim e deno tes the d e riv a tiv e w ith re sp e c t to a 0

T h e c h a ra c te ris tic e q u a tio n o f th e system ( 2 1 ) is

128 Nguyen Van Dao

■where

q 2ya0( Ị 2 + y 2) da0 [a°(yrLl

(2.3)

8y2a0( f2 + y 2) da0 f*r>

tv = r f t + r h - P 2.

T h e s ta b ility c o n d itio n is g iv e n b y the R o u t h - H u r v it z c rite rio n , th a t is :

(2.5)

Ạ- w > 0.

da0

A s w = 0 is the e q u atio n o f the re so n a n ce c u rv e , so the seco nd in e q u a lity (2 5 ) m e a n s

th a t the v e rtic a l tangencies o f the re so n a n c e c u r v e serve as a b o u n d a ry betw een the sta b le

a n d u n sta b le re g io ns T h e first in e q u a lit y (2 4 ) sets som e lim ita tio n s o n e ith e r the a m p li­

tu d e o f v ib ra tio n o r the p ara m ete rs o f the syste m in vestig ated

3 The Duffing Case

B y w a y o f e xam p le let us c o n s id e r the d u ffin g case

R ( x , X , x ) = —/9jc3.

N o w , th e m o tio n e q u a tio n tak e s the fo rm :

( 3 1 ) X + Ệ X + Q 2X + Ệ Í Ì 2 X + efix* = c P s in y f

I n th is case, we have

The relation (1.17) gives, in the first approximation, the following equation of the re­ sonance curve:

(3.2)

rn = Ệ ơ a

-4

r21 = ơya.

(3.3)

w h ere w e deno te

(3 4 )

T h e firs t s ta b ility c o n d itio n (2 4 ) is o f the f o r m :

- 3 y p a l > 0,

i.e the n e ce ssa ry c o n d it io n fo r the sta b ility o f the s ta tio n a ry o s c illa t io n is :

T h u s , fo r the D u ffin g case, o n ly the soft system has the stab le p e r io d ic o sc illa tio n s

T h e re la tio n ( 3 3 ) is p lo tte d in F ig 1 fo r f = 10, Q = 1, p* = — 1 0_1 an d p * = 1 0" 5

(c u rv e 1), p * = 1.25* 1 0" 6 (c u rv e 2 ) T h e p lo ts in b o ld face c o rre s p o n d to the sta b le state o f v ib r a t io n w h e re the s ta b ility c o n d itio n (2 5 ) is v a lid

4 Forced Oscillation of Self-Excited System

I n th is S e c tio n th e fo rc e d o s c illa tio n o f th ir d -o r d e r n o n lin e a r system , w it h th e

se lf-e x c ita tio n g o v e rn e d b y the d iffe re n tia l e q u a tio n :

(4 1 ) x + ỉ ' x + ữ 2x + ỉ í ì 2x + e ( x 2 - ỉ ) x = e P s i n y t

is co n sid e re d W h e n p = 0, the system u n d e r c o n s id e ra tio n is a se lf-e xcite d o ne

I n th is case it is e asy to v e rify th a t:

' l i = - ỉ ơ a ,

t a = ữ 2( l —V2) , V =

Equation (1.17) for determining the amplitude of the stationary oscillation is now of

the f o r m :

130 Nguyen Van Dao

H e n ce w e o b ta in

V2 = 1

-I-e ữ ( A - l ) ± -I-e S y - ( 4 - i y

(Ệ2 + G 2) P 2

4 & Ệ 2 •

A

0.5

0

0 9 0 9 5 1 0 5 V

F i g 2.

T h e r e la tio n (4 4 ) is p lo tte d in F ig 2 f o r th e c a se Ỉ = Q = 1, e = 0 1 W h e n 7 = 0 , the

re sp o n se c u rv e c o n s is ts o f the p o in t A = 1, V 2 = 1 an d v2-a x is F o r s u ffic ie n tly lo w v a lu e s

o f / (see b ra n c h e s 1), th e re sp o n se c u rv e c o n s is t s o f tw o b ra n c h e s : th e firs t is a r o u n d the

p o in t V 2 = 1, A = 1 a n d the se c o n d lie s a b o v e the v2-a x is T h e b ra n c h e s 1 in F ig 2 c o r ­

re sp o n d to J = 4 /8 1 W h e n J = 4 /2 7 , th e re sp o n se c u rv e re d u ce s to a c ro s s -b ra n c h (c u rv e 2 ) F o r J > 4 /2 7 the re sp o n se c u rv e c o n s is ts o f one b r a n c h ly in g o u tsid e the c u rv e

2 (see c u rv e 3 f o r J = 8 /2 7 )

T h e s ta b ility c o n d itio n (2 4 ), (2 5 ) o f th e s ta tio n a ry s o lu tio n is

T h e in e q u a lit y (4 6 ) is satisfied o n the u p p e r b ra n ch e s o f th e re sp o n se c u rv e s lim ite d b y the v e rt ic a l tan g e n cie s I n F ig 2 the b r a n c h e s c o rre s p o n d in g to th e u n s ta b le s o lu tio n are m a rk e d b y h a tc h in g , w h e re th e in e q u a lit ie s (4 5 ), (4 6 ) a re n o t sa tisfie d

to the period of the cxciting force However, since the system considered is a self-excited

forced oscillation and the self-excited one in the zone near the resonance zone For this purpose, we represent the solution of Eq (4.1) in the form:

(4 5)

(4.6)

dA

5 The Process o f Frequency Enừaỉnm ent

w h ere the firs t term is the fo rced o s c illa t io n aD d th e seco n d o n e is the se lf-e xcite d o s c illa ­ tio n T h e r e are three u n k n o w n q u a n titie s a0 , y)0 , b0 T o d e te rm in e th e m , w e su b stitu te ( 5 1 ) in to (4 1 ) an d co m p a re the h a rm o n ic s

B y e q u a tin g the coefficients o f cos(>'f+Y>o)> s in ( y f + y ) a n d s in i? / , o n e o b ta in s:

Ệ ( ũ 2 - y 2)a0 - ePsintflo,

( 5 2) yịy2- Q 2)a0+ Eyị] - - y | a 0 = zPcosyo,

B y e lim in a tin g the p h ase rpo betw een the tw o fir s t e q u a tio n s o f (5 2 ) w e h ave

( 5 3 ) S P 2 = - y * ) 2 + Q * ị y 2 - Q 2 + e ị l _ ^ _ * Ị Ị ’ Ị

F ro m th e th ird e q u a tio n o f (5 2 ) there fo llo w s :

a ) T h e su bcase 1 : 6 = 0

I n th is su b ca se there is no se lf-e xc ite d o s c illa t io n an d E q (5 3 ) c o in c id e s w ith (4 3 ),

w h ic h y ie ld s the a m p litu d e a0 o f th e fo rce d o s c illa tio n

b ) T h e su bcase 2

-O b v io u s ly , the self-e xcite d o s c illa t io n tak e s p la c e o n ly in th e re g io n A < 1 /2 P u ttin g in (5 3 ) a2 = 2, b = 0, w e h a ve th e e q u a tio n f o r d e te rm in in g the c o rre s p o n d in g v a lu e s

v ? , v f

H e n c e w e o b ta in

(5 5 ) v f , 2 = l ~ 2 ( ỉ ỉ + ữ I) ± Ĩ Ĩ P ĩ P T ã * ) Ý 2(£2 + G 2) F 2 S 2 ữ 2

-I t s h o u ld b e n o te d th a t i f w e re p la c e A = 1 /2 in (4 4 ), w e o b ta in th ese v a lu e s a s w e ll

Thus, in the zone aị < 2 the amplitude b of the self-excited oscillation and the am­ plitude a0 of the forced oscillation are determined by equations (5.3), (5.4)

9*

132 Nguyen Van Dao

( 5 6 ) 6* = 4 Ị l - - ặ Ị

( 5 7 ) f2? 2 = a ỗ Ị f2( ^2- y 2) 2 + í 32Ị í 32- y 2 + e Ị l - - ặ Ị Ị Ị

T h e re sp o n se curves 5 = = 2?(v) a n d A = ~ = A(v) c a lc u la te d b y the fo rm u la e (5 6 ), (5 7 ) f o r the case J = 8 /2 7 , ÍÌ = Ỉ = 1, e = 0 1 are represented in F ig 3 In th is

F ig u re , th e p lo t is a lso depicted o f the re sp o n se c u rv e (c u rv e 3, F ig 2) fo r 7 = 8 /2 7

F ig 3 sh o w s the in te ra c tio n betw een the fo rce d a n d self-excited o sc illa tio n s in the zo n e

n ear the re so n a n ce zon e F o r e x a m p le , i f w e in c re a se the e x citin g fre q u e n cy y fro m the v a lu e s n e a r 1, then w e first o b se rv e the sim u lta n e o u s o rig in o f b o th fo rce d a n d se lf­ excited o s c illa tio n s T h e fo rce d o s c illa tio n d e v e lo p s a lo n g the b ra n ch M M ị, then it

su d d e n ly ju m p s to M 2 -* A /3 F r o m A/ 3 it ju m p s to N. W h ile the self-e xcite d

o s c illa tio n chan ge s a lo n g the b ra n c h p -+ P ị. T h is o sc illa tio n d isa p p e a rs on p 2 -> p 3

a n d a p p e a rs ag a in o n the b ra n ch p 3 -+ Q. T h u s , in the e xam p le co n sid e re d , we see th a t the sh e ar fo rce d o s c illa tio n o c c u rs o n ly in the in te rv a l [v2 = Vp , V2 = vị]. O u tsid e th is

in te rv a l, there e xist sim u lta n e o u sly b o th fo rc e d a n d self-excited o sc illa tio n s O b s e rv in g the o s c illa tio n d ia g ra m w h en in c re a s in g the e x c itin g fre q u e n cy y , w e see first the b eat (b e cause y is n e a r Q). I n the in te rv a l [vị , vị] the beat d isa p p e a rs an d there is o n ly the

h a rm o n ic o s c illa tio n w ith e xcitin g fre q u e n c y In c r e a s in g y s till m o re we see that the b eat

ap p e a rs a g a in T h e in te rv a l [vị yv\] is c a lle d th e fre q u e n c y e n tra in m e n t in te rva l A s it seen in F ig 3, the fre q u e n cy e n tra in m e n t zo n e is n a rro v e r th a n that in the se c o n d -o rd e r system [10]

T o c h e c k the v a lid it y o f the th e o re tic a l re su lts a n a n a lo g co m p u te r a n a ly sis has been

c a rrie d o u t E x a m p le s o f the w a v e fo rm s o b ta in e d b y m o d e llin g the o rig in a l e q u a tio n (4 1 ) f o r the p a ra m e te rs Q = Ỉ = 1, e = 0 1 , eP = 0.08, are sh o w n in F ig 4 T h e F ig u r e

c o n ta in s a ll the types o f o s c illa tio n s th a t are p re d ic te d b y the theory

Refererces

1 z O sin sk i, Vibrations o f an one-degree o f freed o m sy ste m with non-linear internal friction and relaxa­ tion, Proceedings o f In tern atio n al C onference o n N o n -lin e a r O scillations, III, Kiev 1963.

2 z O sin s k i, G B o y a d j i e v , The vibrations o f the s y s te m with non-linear friction and relaxation with slowly variable coefficients, Proceedings o f the F o u rth Conference on Non-linear Oscillations, P raque

1967

3 H R S r i r a n g a r a j a n , p S rin iv a s a n , A pplication o f ultraspherical polynom ials to fo rc e d oscillations

of a third order non-linear system, J Sound and V ibration, 36, 4, 1974.

4 H R S r i r a n g a r a j a n , p S rin iv a s a n , Ultraspherical polynom ials approach to the study o f third-order

non-linear systems, J Sound and V ibration, 40, 2, 1975.

5 A T o n d l , N otes on the solution o f fo r c e d oscillations o f a third-order non-linear system , J Sound a n d

Vibration, 37, 2, 1974

6 A T o n d l , Additional note on a third-order system , J Sound an d V ibration, 47, 1, 1976.

7 z O sin s k i, N g u y e n V a n D a o , Param etric oscillation o f an uniform beam in a rheological m odel\

Proceedings of Second National Conference on Mechanics, Hanoi 1977

8 N N B o g o l i u b o v , Y u a M i t r o p o l s k y , Asymptotic methods in the theory o f non-linear oscillations

M oskva 1963

9 N g u y e n V a n D a o , Fundamental m ethods o f non-linear oscillations, H anoi 1969.

10 H Kauderer, Nichtỉineare Mechanik, Berlin 1958.

11 N g u y e n V a n D a o , a ton-linear oscillations o f the th ird order systems, Part I Autonom ous syste m s,

J Techn Phys., 20, 4, 1979

134 Nguyen Van Dao

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

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

C Z Ẹ ấ ớ II N IE A U T O N O M IC Z N E UKJLADV

N in iejsza p raca stanowi drugạ czẹsé pracy [11] R o z p a trz o n o w niej d rg an ia nieautonom icznei

u klad u trzcciego rzẹdư w przypadku rezonansu W yzn aczo n o w aru n ek statecznosci ustalonych drga

Z b a d a n o szczegolny przypadek low nania Duffinga w dalszym ciạgu rozw azono w ym uszone d rg ar sam ow zbudnego ukỉad u Irzccicgo rzẹdu P rzep ro w ad zo n o ek sperym entalne b ad an ia n a maszynie anal gowej d la weryiikacji teoretycznych wynikow

p e 3 K) M e

H E J IH H E H H b lE K O /IE E A H H fl C H C T E M T P E T b E r O n O P J M K A

MACTB I I H E A B T O H O M H b lE C H C T E M b I

H acTonm aH p a ố o T a C0CTaB/iHeT D T o p y io M acTb paốoTbi [11] B H e ft paccMOTpeiibi Ko/ieốâHHH H eaB IIOMHOH CHCTCMbi T p e T b c r o n ^ p H A K a B c n ry q a e p e 3 0 H a H c a O n p e A e i e H o ycjTOBHe y c ro ita H B O C T H y c r a

b h b l l i h x c h K O Jie6aH H H H c cjieA O B aH 'lac T H B ift c jry M aH y p â B H e H H H i l a ộ Ộ H H r a B A a jiB H e ftu ieM p accM

p e H b i B fem y > K A eH H b ie K O /ieốaHHH a B T O K o ie ố a T e jn ,H o ỉi c n e r e M b i T p e T b e r o n o p a ự Ị K a r ip o B e A e H b i 3KC

p H M e H T a jib H b ie HCCjieAOBaHHfl H a aH ajioroBoft M a iiu iH e OHH n p O B e p K H T e o p e T im e c K H X pe3y/ifeTaTO B

T E C H N IC A L U N IV E R S IT Y , H A N O I, V IETN A M

Received October 17, 1978.