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

Journal Technical Physics, J, Tech. P ky 3 .t 21, 1, 123- 134, 1980.
Polish Academy of Sciences, Institute of Fundamental Technological Research, W arszawa.
NONLINEAR OSCILLATIONS O F THIRD ORDER SYSTEMS
PART II. NON-AUTONOM OUS SYSTEM S
N GU Y E N V A N D A O (H AN OI)
Introduction
T h is c ha p te r is devoted to the stu d y o f the o sc illa tio n o f th ir d ord e r no n -au ton om o us
system in the resonance case. Se ction 1 de als w ith the g e ne ra l theo ry. T h e a p p ro xim ate
so lutio n o f the m otio n equation is fou nd by m eans o f the asy m p to tic m ethod. B y c o n 
trast w ith the auto n om o us case [ 11] the phase \p here ha s a great in flu e nce on the a m 
p litu d e an d frequency o f o scillatio n . In S e c tio n 2 the s t a b ilit y co n d itio n o f the sta tio n
a ry o sc illa tio n is analysed. It is w ritte n in a c o m pa ct an d c om fo rta b le fo rm fo r p rac tica l
use. A sp ec ia l but im p o rta nt case— the D u ffin g case is d ealt w ith in S e ctio n 3. It has
been p ro ve d that fo r the th ird -o rde r system , the ha rd n o n lin e a r ch ara cte ristic do cs no t
e xh ib it a stab le p erio dic o sc illatio n . T h is ph en o m en on is n o t ob se rved in the second-
o rd er sy stem . Th e n ext two S e c tio ns 4, 5 are o f a m o re sp ecialize d character. T h e y
c o n ta in th e studies o f the forced o sc illa tion in the th ird o rd e r self-excited system. C e rta in
zone o f sy n c h ro n iza tion is d eterm in ed. In c o m p a riso n w it h the an alo g o u s pro b lem in
the se c o nd ord er system , the fre qu ency e ntra in m en t zone b eco m es n arro w er.
A n exp e rim en t o n the a n alo g com p uter h a s been co n d ucte d to v e rify the the o retical
resu lts.
1. Two-Parameters Solution la the Resonance Case
. Let us consider the oscillations of the system governed by the differential equation
o f the th ird o rder:
(1.1) X +ỆX + Q 2X + ỆŨ 2X = e R (x t X, 3c) +Posinyf,
here f , Q , y , PQ are con stants an d R (x , X, 3c) is the n o n lin e a r fu n ctio n o f X, X, X. I t is
assumed that there is a reasonance relation
(1 .2 ) Q 2 = y 2 + £ơ,
Pq is a sm a ll q u an tity o f the first ord e r: p 0 = eP.
The partial periodic solution with the period 2nly of Eq. (1.1) is found in the form:
(1 -3 ) X = ac os(yt + yj)+ y , yt) + e2u 2(a 9 y) , y / ) +

w here u (a, y ỉ,y t) are the pe riocỉĩc fu nction s w ith the p e rio d 271 re la tiv e ly o f y an d yt
an d a , rp are determ in ed fro m the e qu atio ns
126 N g u yen V an D a o
da
— = eAiia, V)) + s M j ( a , y j)+
(1.4) d
~ = eB.ia, v )+ e 2B 2(a, v ) +
B y su b s titu tin g (1.3), (1.4) in to (1 .1 ) we get
(1.5) EịlyíyaBí -ỆA^úny-lyịyAi +ỆaBl)cos(p
a 3«, Ô2U, 2 õu,
+ 5 ĩ^ + Ệ d^-+y a f +* vui + e ■■■
= eP sinyt + £jR(acos(p, —aysin<p, - a y 2C0 S(p)+ e2
here
R ịx , X , x ) = R ( x , x ) - ơ x — íơ x .
By co m p a rin g the coefficients o f £ in (1 .5 ), w e ha ve:
(J.6) 2 yịyaB1 -Ệ Ai)siĩ\(p-2y(yA1+ỉaBl)cos<p +
+ ^ + l J £ . + r ‘í ^ + ỉ r 1u , - r s m r,
c/3 ct ot
+ R(acos<p, — yasintp, —y 2acos(f),
9
? = yt + y.
N o w , we ex p an d the fu nction in the F o u rie r serie s:
00
(1 .7 ) /?(acos< p , - y a s in ọ ? , — y 2acosọ ?) = ^ (rlHcosn<p + r2llsinn<p).
T h e fun ctio n Ui is also fo u n d in the series:
( 1-8) Ul- ỵ (ulmcosm<p + v lmsinm<p) ,
m
w ith an a d d itio n al c o n d itio n : « ! does no t co n ta in the term s w ith a v an ish in g de n om inato r.
Substituting (1.7), (1.8) into (1.6) and equating the coefficients of sinẹ>, cosip, we obtain:
2y(yA1 + ỉaB1) = P sin ip -r^ ,

2y(yaB1 -ỆAi) = P cosy + r21.
B y co m p arin g the coefficients o f the other h a rm on ics, w e have:
y2(l - m 2)(ỉu lm + ntyvlm) = rlm, m í 1,
y2(l = r2m.
(1.10)
H en c e we o bta in
trlm'- m y r 2l
(1.11)
y 2( l - m 2)(£ 2+ m 2y 2)
Nonlinear oscillations o f third order systems. Pari II
127
E q u a t io n s (1.9 ) give
P (y sin y — Ệcosrp) — y f u - Ệ r2l
A, =
2y(i2 + y2)
(
1
-
12
)
i >( is in v ’ + y c o s y ) - f / - n + y r 2t
= — —
2 y a ( ỉ 2 + y 2 )
The ca lcu la tio n o f hig he r a pp rox im a tion s presents no difficulties.
Thus, in the first approximation we have:
(1.1 3 ) X = a c o s (y f + y0 ,
d a P ( y s i n ụ > - ỉ C O S r p ) — y r tl - £ r 21
~dt ~ e 2 H F + 7 ) ’
(1.14)
dy _ P ( f sin y + y c o s y ) — Ệrn + yr21

dt E 2yaịỆ2 + y 2)
The refinem ent o f the first a p pro xim atio n is :
_ . , , Í V (Sri« m Yri J c°sm<p + {mYrlm + tri„)5inm<p
(1 .1 5 )

■ ;

•
trĩ
= 0
m# 1
H ere a an d y satisfy E q s. (1.14 ).
The amplitude and phase of the stationary oscillation are determined from the rela
tio ns:
P(y sin yj-ỉ cosy) = yili+ffai*
P(fsiny + ycosy) = frn -yr2i.
By eliminating the phase rp one obtains:
(1.17) P2 = r?i+rfi.
2. Stability of Stationary Oscillation
L e t ôa, ỗrp be sm a ll pe rturb atio ns and set a = aQ + òa y = y>o + <ty where a0f y) arc
the sta tio na ry v a lu e s o f J , y determ in e d fro m (1 .16 ). P u ttin g the ab o v e exp ressio n s into
Eqs. (1 .1 4 ) an d m ak in g use o f th e steady-state E q s. (1 .1 6 ), we ob ta in the fo llow ing v a r ia 
tio n a l e q u a tio n s:
^ d T ■ W g T l - f r ' i . + - r ' i . X v ) .
(2.1)
* - W V K K f r n - ^ O ^ + i m . + f r . O i v l ,
w here the prim e denotes the d e riv ativ e w ith re sp e ct to a 0 .
T h e ch a rac te ristic equation o f the system (2 .1 ) is
(2.2) A2 + ?A + 5 = 0,
128

Nguyen Van Dao
■where
q 2ya0(Ị2 + y 2) da0 [a°(yrLl
(2.3)
8y 2a 0( f 2 + y 2) da0 f*r>
tv = r f t + r h - P 2.
T h e sta b ility co n ditio n is given by the R o u th -H u rv itz criterio n , th a t is :
(2.5)
(2.4)
d
> 0,
Ạ- w > 0.
da0
A s w = 0 is the equation of the resonance c urv e, so the second in eq ua lity (2 .5 ) m eans
that the v ertica l tangencies o f the reso n ance cu rv e serve as a b o u n d a ry betw een the stab le
an d u n stable regions. Th e first in e q u a lity (2 .4 ) sets som e lim itatio ns on eith er the a m p li
tud e o f vib ratio n o r the param eters o f the system investigated.
3. The Duffing Case
B y w ay o f exam ple let us co nside r the du ffin g case
R(x, X, x) = —/9jc3.
N o w , the m o tio n equation take s the fo rm :
(3 .1 ) X + Ệ X + Q 2X + Ệ Í Ì 2X + efix* = c P s in y f.
In this 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 we denote
(3.4 )
T h e first stab ility co n ditio n (2 .4) is o f the fo rm :
- 3 ypal > 0 ,
Nonlinear oscillations o f third order systems. Part II 129
i.e. the necessary c o n d itio n fo r the stab ility o f the statio n ary o sc illa tio n is :
(3.5 ) ạ < 0.
T h u s, fo r the D u ffin g case, on ly the soft system has the stable p e rio d ic oscilla tio ns.
T h e rela tio n (3 .3 ) is plotted in F ig . 1 fo r f = 10, Q = 1, p* = — 1 0 _1 and p * = 10 " 5
(cu rv e 1), p * = 1.25* 1 0 " 6 (c urv e 2). T h e plo ts in b o ld face c orre sp o n d to the stable
state o f vib ra tio n w h ere the sta bility co n dition (2 .5) is v a lid.
4. Forced Oscillation of Self-Excited System
In th is S e ction the fo rced o sc illatio n o f th ird -o rd er n o n lin e a r system, w ith the
self-e xcita tion go ve rned by the differe n tial e q u a tion :
(4 .1 ) x + ỉ'x + ữ 2x + ỉ í ì 2x + e (x 2 - ỉ ) x = ePsinyt
is co n sid e red . W he n p = 0, the system u n d e r co nsid e ra tio n is a self-excite d one.
In th is case it is easy to ve rify that:
' l i = -ỉơ a ,
(4.2) r21 = a y Ịơ -l +
t a = ữ 2(l —V2), V =
Equation (1.17) for determining the amplitude of the stationary oscillation is now of
the fo rm :
(4.3) p> = + +
9 Journal Techn. Phys. 1/80
130
Nguyen Van Dao
H ence w e obtain
V2 = 1 -I-
e ữ ( A - l ) ± e S y - ( . 4 - i y
(4.4)

Q (^ + Q 2)
(Ệ2+ G 2)P 2
4 & Ệ 2 •
A
0.5
0
0 .9
0 .9 5 1 .0 5 V
F ig . 2.
Th e re la tio n (4 .4 ) is plotted in F ig . 2 fo r the case Ỉ = Q = 1, e = 0 .1 . W h en 7 = 0 , the
respo n se cu rv e con sists o f the p o in t A = 1, V2 = 1 an d v 2-a xis. F o r su fficie ntly lo w va lue s
o f / (see b ra nch es 1), the resp o nse c u rv e c o ns ists o f two b ra n ch e s: the first is a ro u n d the
po int V 2 = 1, A = 1 an d the seco nd lies a b o ve the v 2-ax is. T h e b ra n c h es 1 in F ig . 2 c o r 
resp o nd to J = 4 /81. W h e n J = 4/2 7, the respo n se c u rv e reduce s to a c ro s s-b ran ch
(cu rve 2). F o r J > 4 /2 7 the resp o nse c u rv e c o n s ists o f one bra n ch ly in g o u tside the cu rve
2 (see cu rv e 3 fo r J = 8 /2 7).
T h e sta bility c o n d itio n (2.4), (2 .5) o f the statio nary s o lu tio n is
T h e in e q u a lity (4 .6 ) is satisfied o n the u p p e r b ra n ches o f th e resp on se cu rv es lim ite d by
the v ert ic a l tangencies. In F ig . 2 the b ra n ch es co rre sp on d in g to the un sta ble so lu tio n
are m ark ed by ha tch ing , w here the in e q u alitie s (4 .5), (4 .6 ) are n o t satisfied.
In Section 4 we have investigated the periodic solution with the period 27i/y equating
to the period of the cxciting force. However, since the system considered is a self-excited
on when p =» 0 , then it is necessary to study in more detail the interaction between the
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:
(5.1 ) X =* a0cos(yt + ip0)+ bQ osQ t9
(4.5)
(4.6)
dA
5. The Process o f Frequency Enừaỉnment

Nonlinear oscillations o f third order systems. Part II 131
w here the first term is the forced o sc illatio n aD d the second on e is the self-excited o s c illa 
tio n . T h ere are three un kn o w n qu an titie s a0, y)0, b0. T o determ ine the m , w e substitute
(5 .1 ) in to (4 .1) and com pa re the h a rm on ics.
By eq u a tin g the coefficients o f cos(>'f+Y>o)> s in (y f + y ) an d s in i? /, o ne ob ta ins:
Ệ (ũ2- y 2)a0 - ePsintflo,
(5.2) yịy2-Q 2)a0+ Eyị] - -y|a0 = zPcosyo,
B y elim in atin g the ph ase rpo betw een the tw o firs t eq u atio n s o f (5 .2 ) we have
( 5 .3 ) S P 2 = - y * ) 2 + Q * ị y 2 - Q 2 + e ị l _ ^ _ * Ị Ị ’ Ị .
F ro m the third eq ua tion o f (5.2 ) there fo llo w s :
a) T h e subcase 1 : 6 = 0
In this subcase there is no self-excited o sc illa tio n and E q . (5 .3 ) co in cid e s w ith (4 .3 ),
w hich yie ld s the a m plitu de a0 o f the fo rce d o sc illa tio n.
b ) T h e subcase 2
(5.4) ốJ = 4 ( l- 2 / 0 , A=ị-
O b vio u sly , the self-excited o sc illa tio n takes p la ce o n ly in the re g io n A < 1/2 . P uttin g in
(5 .3 ) a2 = 2, b = 0, we have the eq u atio n fo r d ete rm ining the c o rre sp o n din g va lu e s
v ? , v f
SP* = - Í Ì 2 + - ỷ ị ] ,
H en ce we ob ta in
(5.5) vf,2 = l ~ 2(ỉỉ + ữ I) ± ĨĨP ĩP T ã * ) Ý 2(£2 + G2)F2- S 2ữ2-
It sho u ld be n oted th at if we rep la ce A = 1 /2 in (4 .4 ), w e o bta in these va lu e s as w ell.
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* = 4Ị l- - ặ Ị .
(5 .7 ) f 2? 2 = a ỗ Ị f 2( ^ 2- y 2)2 + í3 2Ị í 3 2- y 2 + e Ị l - - ặ Ị Ị Ị .
Ố2 a2

T h e resp o nse curves 5 = = 2?(v) an d A = ~ = A(v) calc ulate d by the fo rm ulae
(5.6 ), (5 .7) fo r the case J = 8/27, ÍÌ = Ỉ = 1, e = 0.1 are represented in F ig . 3. In this
Nonlinear oscillations o f third order systems. Part II
133
Fig ure , the p lo t is also depicted o f the response c u rv e (cu rv e 3, Fig . 2) fo r 7 = 8 /2 7 .
F ig. 3 sho w s the in teraction betw een the fo rced an d self-excited oscillation s in the zone
near the reso n ance zone. F o r exa m p le, if we in crease the e xciting frequency y fro m
the valu es near 1, then we first ob serve the sim ultan e ou s o rig in o f both fo rced an d self
excited osc illatio ns. T h e fo rced osc illa tio n deve lo p s alo ng the branch M M ị, then it
sudd e n ly ju m p s to M 2 -* A /3 . F ro m A/ 3 it ju m p s to N. W hile the self-excited
o sc illatio n changes a lo ng the b ra n c h p -+ P ị. T h is o scillation d isap p ears on p 2 -> p 3
and ap p ears again on the branch p 3 -+ Q. T h u s, in the exam ple considered, we see that
the shear force d o sc illation o ccu rs o nly in the in te rva l [v2 = Vp , V2 = vị]. Outside th is
in terva l, there exist sim u lta n eo u sly bo th fo rced an d self-excited oscillatio ns. O b servin g
the o sc illa tio n diag ram w hen in cre a sin g the ex citin g freq u e n cy y, we see first the beat
(because y is n ear Q). In the in te rva l [vị , vị] the beat disap p e a rs and there is o n ly the
h a rm o nic oscillatio n w ith excitin g fre qu en cy . In cre a sin g y still m ore we see that the beat
appea rs again . T he in terv a l [vị yv\] is c a lled the freq ue n cy entrainm ent interva l. A s it
seen in F ig . 3, the frequ en cy entrainm en t zone is n arro ve r than that in the seco n d -o rd e r
system [ 10].
T o c h eck the va lid ity o f the th e oretica l results an a n alog com puter an a lysis has been
ca rried out. E xa m ple s o f the w ave fo rm s ob ta in ed by m od ellin g the o rig in al eq uatio n
(4.1) fo r the param eters Q = Ỉ = 1, e = 0.1, eP = 0.08, are show n in F ig . 4. T h e F ig u re
co ntains a ll the types o f o scilla tio n s that are pred icted by the theory.
Refererces
1. z . O sinski, Vibrations o f an one-degree o f freedom system with non-linear internal friction and relaxa
tion, Proceedings of International Conference o n N o n-linear Oscillations, III, Kiev 1963.
2. z . O sinski, G . B o y a d j i e v , The vibrations o f the system with non-linear friction and relaxation with
slowly variable coefficients, Proceedings of the F ourth Conference on Non-linear Oscillations, Praque
1967.

3. H. R . S rir a n g a r a ja n , p. Sriniva san, Application o f ultraspherical polynomials to forced oscillations
of a third order non-linear system, J. Sound and Vibration, 36, 4, 1974.
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non-linear systems, J. Sound and Vibration, 40, 2, 1975.
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Vibration, 37, 2, 1974.
6. A. T o n d l, Additional note on a third-order system , J. Sound and Vibration, 47, 1, 1976.
7. z . O sin ski, N gu y en V an D ao, Parametric oscillation o f an uniform beam in a rheological model\
Proceedings of Second National Conference on Mechanics, Hanoi 1977.
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Moskva 1963.
9. N gu yen V an D a o , Fundamental methods o f non-linear oscillations, Hanoi 1969.
10. H. K auderer, Nichtỉineare Mechanik, Berlin 1958.
11. N gu y e n V an D ao , a ton-linear oscillations o f the third order systems, Part I. Autonomous system s,
J. Techn. Phys., 20, 4, 1979.
134
Nguyen Van Dao
Streszczcnic
NIELINIO W E D RG A NIA U K L A D 0 W T RZ E CIE GO R Z Ẹ D U
C Z Ẹ ấớ II. N IEA U T O N O M IC Z N E UKJLADV
N iniejsza praca stanowi drugạ czẹsé pracy [11]. R ozpatrzono w niej drgania nieautonomicznei
ukladu trzcciego rzẹdư w przypadku rezonansu. W yznaczono w arunek statecznosci ustalonych drga
Zbadano szczegolny przypadek lownania Duffinga. w dalszym ciạgu rozwazono wymuszone drgar
samowzbudnego ukỉadu Irzccicgo rzẹdu. Przeprowadzono eksperymentalne badania na maszynie anal
gowej dla 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 rO nO P JM K A
MACTB II. H E A B T O H O M H b lE C H C T E M bI
HacTonmaH p a ố o T a C0CTaB/iHeT D Topyio MacTb paốoTbi [11]. B H eft paccMOTpeiibi Ko/ieốâHHH H eaB
IIOMHOH CHCTCMbi T p e T b cro n^pH A Ka B cn ryqae p e3 0 H a H c a . O n p eA e .ie H o ycjTOBHe ycroitaH B O C T H y c r a

b h b l l i h x c h K OJie6aHH H. H ccjieA O B aH 'lacTH B ift cjryM aH ypâB H eH H H i l a ộ Ộ H H r a . B A ajiB H eftuieM paccM
p e H b i B femy>KA eHHbie KO/ieốaHHH aBT O K o .ieốaT ejn,H oỉi c n e re M b i T p e T b e ro n o p a ự ỊK a. rip o B eA eH b i 3KC
pH M eH T ajibH b ie HCCjieAOBaHHfl Ha aHajioroBoft M aiiuiH e OHH n pO B epK H TeopeTim ecK H X pe3y/ifeTaTOB.
T E C H N IC A L U N IVER S ITY, HAN OI, VIETNAM.
Received October 17, 1978.