Nonlinear oscillations of the third order systems. Part III Parametric oscillation

Journal o f Technical Physics, J. Tech. P h y s 21, 2, 253 -26 5, 1980.
Polish Academy o f Sciences, Institute o f Fundamental Technological Research, Warszawa.
NONLINEAR OSCILLATIONS OF THE THIRD ORDER SYSTEMS
PART m . PARAMETRIC OSCILLATION
N G U Y E N V A N D A O (H A N O I)
Introductio n
"he theory o f the parametric oscillation o f the second-order system has been in-
vestgated in a lot o f publications. F o r a long tim e it has played an im portant role in the
theory o f nonlinear oscillations. R ecently, in som e problem s o f the dynam ics on e can
mee. the param etric oscillation o f the third-order system [7], to the study o f w h ich this
charter is devoted (cf. [11, 12]).
n the first Section the approxim ate solution o f the m otion eq uation is constructed.
The stationary solutions are studied.
The second Section is concerned with the stability condition o f the stationary oscilla-
tioi. T he R ou th-H urw itz criteria are taken out.
The influence o f the C oulom b friction on the param etric oscillation is considered in
the Sect. 3. In this case the resonance curve has the d o sed form .
n Sect. 4 the influence o f the turbulent friction on the param etric oscillation is studied.
T h i friction limits the growth o f the parametric oscillation and causes a considerable
chaige in the rigidity o f the system investigated.
Under the influence o f the com bination friction (Sect. 5), the resonance curve is
simlar in quality to that in the case o f the C oulom b friction. It is o f closed form as w ell.
1. Construction o f A pp ro xim ate S olution
Let us consider the parametric oscillation o f the system described by the third order
diferential equation o f the form :
(1 .) x '+ Ệ x + Q 2x + Ệ Q 2x + e [k x 3 + hx3 + R (x,x ,x )-cxco s y t] = 0,
w b re Ệ ,Q ,k ,h ,c ,y are con stan ts and R (x ,x ,x ) is the function characterizing the
ncilinear friction.
W e assume that there is a resonance relation
eA = Q2(l -Tj2), ri=-^Q-
T hen, Eq. (1.1) can be written as:

Ỉ54
Nguyen Van Dao
^ h e n
1.3) f ( x , X, x) = A X + ỆẠx + k x 3 + hx3 + R (x , X, x ) .
A partial tw o -p a ram eters solu tio n o f (1.2) is fo u n d in the series:
1.4) X = ỠCOS
+ + EU1 ịa , ip,
-y/j
+ e2u2 ịa ,
ip, -yfj +
n w iich us (a , y, 0) are p erio dic fu n c tio n s o f 6 a nd w ith-th e period 2 71, an d a, ip are
u ncto ns o f time d ete rm in ed fr om th e set o f e quatio ns:
~ = eAl(a,xp )+£2A 2{ a ,y )+ •••,
; i . 5 )
. ^ = eBL(a, ỳ )+ e2B 2(a, y>) +
1 ) d eterm in e th e functio n s US, A S, B S, first w c calculate:
dx y
~dt = 2
d 2x
• I A n . CUị \ 2
asm(p + s \A X cos(p—aB1sm(p+ + £ • • 9
y2 I d2ui \
— acoscp+ —yA ! sin(p — yaB1 COSẹ?+ - Ỵ T I + 8
1.6 )
í/3A' y 3 / 3 3 ổ3Mj \ 2
- ^ - 3- = asm 9? + £ I — ~ ^ -yM 1 cosẹ> + fl-Sj SÌ1199 + —^ 3 I + £ •••’
<p = y f + y .
S ib stitu tin g E qs. (1.4), (1.6) in to (1.2) a nd co m parin g the coefficients o f e w ith eq
ỉegrtes, w e ob ta in :
Ô3U1 ô2u1 y 2 dul y 2 I y 2 \

1.7)
+ sinọ? = - / o + ữ c o s ^ c c o s
/o = / ịa c o s ọ ) , —-yứsinọ?, — -~ ac o sẹ> Ị.
N)W, w e expan d th e fu n c tio n / o in the F o u rier series:
0 0
; i . 8 ) / o = J j ? [qm(a)cosm(p+pm(a)sinm(p],
m = 0
Nonlinear oscillations o f the third order systems. Part III 255
lere
2.71
q0 = — Ị fiacosq), — -y ứ sin ọ ? , — ^ị-ứcosọ?! d(p,
o '
271 I
.9) qm = — J" /Ịa c o s ẹ ? , — ~-a sin ẹ > , — ^ -aco sọ?! cosmcpdcp,
o '
. 271
pm = — Ị / Ị a c o s ạ ) , —-y ứ sin ọ ?, — —-a c o s ^ Ị smmĩpdcp.
Ố '
T he function «! satisfying Eq. (1.7) will be fou nd in the form
1.10) ^ [G„(a, yj)cosn<p + B n(a, y)sinm<p]
/ith the additional con dition that it contains no resonance terms. It will be seen later
hat this condition is equivalent to the following: the function Wi does not contain COS99,
in 99.
By substituting (1.8), (1.10) into (1.7), w e have1.10) into (1.7), w e have:
|yyGn- ! / / „ j s i n A ! 9 5 - ỈƠ„Ị cosnọ? -
Ị-^-v41 + y £ a 5 iỊco s 9 9 + Ị^2 ~ ứjBi - y M ij s i n g j = c a c o s ỹ c o s y í
00
- ) (ạmcosrn(p+pmsinm<j
- y , (<7 mCoswẹ>+/>msinwẹ>).
m = 0

By com parin g the harm onics sinọ?, COS op, one obtains:
y Al +yỆaB1 = - - ^ c o s 2 rp+q^
(1.12)
y Ệ A i - ^ - a B i = - - ^ - s i n 2 y + P i
By com paring the other harm onics, w e get:
nguy en van ưao
On solv in g Eqs. (1.13), w e ha ve:
(1 — 2 ,, -ị- accos2v>j ổ 3,
c " =
y / y £ . I
- y + £/>„ - |-j-r tứ cco s2y > + -^ -a c s i n 2 y I ỗ 3n
=
n 2
r
I t u i Y ' 1— — U(
(„*_!) Ị^+rl^Ị
From (1.12), w e have
Q 1
£ < /0 sin99> + .í2</0 cosẹ>> — -Ị-ac COS 2^-* Ạ-OCỆs in 2 y
^ = i 2 ( l 2 + £ 2) ’
( 1 . 1:)
£ < /o c os<p> — í2< /0sinẹ>> + -^-acsin2y> — ^-flccos2y>
Bl = a(Ệ2 + ũ 2) ’
wheie ( F ) is th e o perator o f the a veraging function F on tim e. By p uttin g in E q. (1.
fo fn m Eqs. (1.3) an d (1.7) a nd calcu latin g, w e h a ve th e fo llo w in g eq u a tio ns o f th e Í
approxim ation:
(1.10
da £
d t = Ệ2 + Q 2 [ 8
dip ■ E

dt = a (i2 + Q*)
ĩ-^ -(fc —£Q 2h)a3 — -^ -ứ ccos2y — -^ -^ sin 2 ^ y + i?! j ,
Ị^— (Ệ2 + & 2) A a + -^-(Ệk + Q*h)a3 + ^ - s i n 2 xp-
ac .
■ ~ —ỆQos2tp+R
.2 Y
w h ec
2Ệ
R i = <JR0COSỌỊ>4- — </?0sino9>,
y
2Ẽ
' l . n R 2 = — ( R 0c o s (p )-(R 0sinq)'),
y
i v y 2
R q = Rịacoscp, — |- a s i n (p, — — acoscp
Ttus, in the first a p p r o xim a tio n w e h ave a partial so lu tio n o f E q. (1.1) in th e for
L
2
Nonlinear oscillations o f the third order systems. Part ỈII
257
iere a and Iff are the solution o f Eqs. (1.16). The refinement o f the first approxim ation is:
9)
.Y = a c O S
0 Ệ \
Ệq„—Qnp„+ \-ịnacsm2y)— ~accos2tp jỗ3n
~Ó2{n2-ỉ)(Ệ 2 + n2Q2)
- X
X C O S /7ọ? +
Qnq„ + Ệp„ — ị^-nac cos2ĩf + y -ứcsin 2^Ị<5 3n
Q20 ? - ì ) ( Ệ 2 + n2Q 2)

sin H(p
th a and y> being the solution o f Eq. (1.16).
The stationary solution o f the set (1.16) is determ ined from the equations:
.2 0)
^ - a 0sin2ĩp + ^ - c o s l i f = -— { k - Ệ Q 2h ) a ị + R y ,
l y 4 o
~ ~ a 0co$2y.'- sin 2 y = — (Ệ2 + Q 2)Aa0 + -^-(Ệk + Q4h)ao + R 2-
2 y 4 y 4 y
By elim inating the phase y), we obtain the equation for the am plitude a0 :
.21) W(ao,y) = 0
here
3 ~>QZ
.22) W(a0, y) :
ỆA+ 4 ka2° + a0( f + Q 2) V 1 + > Rl
+
0 2
c
4
Relation (1.21) is plotted in Fig. 1 for the case R = 0, -Ệ = ũ = Ỉ, c* = 0.05, kị,
—0.1 and /7* = 0 (curve 1), /7 * = 0.05 (curve 2) and /7* = 0.1 (curve 3). From this
cure, it is seen that with increacing h, the maximum of the amplitudes decreases and
le nonlinear system b ecom es harder. In Fig. 2 the resonance curves are presented for
ie case R = 0, Ệ = o = \, c* = 0.05, /?* = 0.1 and k* = 0 (curve 1), k* = - 0 .0 5
:urve 2), /c* = -0 .1 (curve 3). With decreasing k, the maximum of the amplitude
ecreases and the nonlinear system becom es softer. Ịả'* = -Qĩk, /7* — c* — p T c j-
F i g . 1
i J o u rn a l T ec hn . Ph ys. 2/80
Nguyen Van Dao
B\ contrast with the parametric oscillation in the well known second-order system,
ie riiidity of the nonlinear system and the maximum of the amplitudes of oscillation

ĩre tepend on the com bination o f the param eters h and k. The system considered is
hart system if T = £ k + Q Ah > 0 and a soft one if T < 0. I f Q = ỆQ2h — k is positive,
len tie m axim um o f am plitudes decreases w ith increasing Q.
2. S tability o f S tationary O sc illa tio n
F ist we shall con sider the stability o f the stationary solution a0 Ỷ 0 o f Eqs. (1.16).
ubsttuting in them
a = a0+ỗa, y = ip0+ồy)
'iíh í0 , xp0 being the solution of Eqs. (1.20), we have the following variational equations:
ỊỊ^-(& -£í22/7)ứẳ + aoỊ— Ị j < 5 c - (Ậ2 + Q 2)Aa0
dt
Ệ2 + Q 2
+
2. 1)
+ *tfỉc+ữ*h)al
2 y
cỉôyj
- 2 ỗ y > Ị,
I Ả-(Ệk + QVi)a0+ Ị * ì Ị ]ôa + ị j { k - Ệ Q 2h ) a l + ~ R ^ ỏ v } -
dt Ệ2 + Q 2
Tie characteristic equation of this system is:
2.2 )
yheri
2.3)
Ằ2~ZÂ + S = 0 ,
z =
5 =
ễ2 + Q
2 r ~ ( i ' - ỉ ữ 2/ i ) « ỗ + — (0 0 * 1) ' ] .
s2a
[ 2/1

4Q 2(Ệ2 + Q 2) X y
+
2 Q2
Ỵy(Ệk + L)4h) a0 -f 4Í22 (— 2
+ - ị ( k 2 + Q 6h2)aị +
3ao(fc- í f i V , ) ( f } + + « ( £ ) ( £ ) ’
+4 í ă í ă i
+
Ệ2 + w
+
a 0 \ a 0
Nonlinear oscillations o f the third order systems, Pari III
25!
( 2 .‘ )
rhe expression z can be also written in the form:
d W
e2a 0
4 Q 2(Ệ2 + Q 2) da0
h er w is o f the form (1.22). C onseq uen tly, the stability condition o f stationary solu
tioi is:
(2.3 3(k -ỆQ 2h)al + 2(a0R iy < 0,
ÔÌV
(2.0
ôa0
> 0 .
Mow, let us consider a special case o f the stability o f equilibrium a = 0, w h en th
sysem (1.16) has the form :
( 2 .')
da £ [ 3 r, c „ ca - . _ 1
= ~ỆĨ + Õ T [ 8 4 1ccos2 ^ ~ 2 ’

= ~nr + 5 r [7 (f 2+ fi2^ a + Ậ +
. i/r £2 + £ 2 8
<7
dtp
h i
c ■ n c a t o
— flsin2v> — - y - f c o s 2 ^ .
(2.r
or
In this case we put a = ỗữ, y = ^o + ổ y and the variational eq uation s are:
d ò a
dt
EC I y \
= “ Ị 2 c o s 0 + ^ ° /
0 =
V
+ Q2)A 4- -ysin2vj0 - f cos2y>o
ỗứ
£C • M c
—— =

1 ■ ■_ sin(2u’o 4- v) 00,
eft 2 y ý Ệ r + ữ 2
(2.0
0
£C
2y (Ệ 2 + Q 2)
- (Ệ2 + Q2)A - \/ệ2 + Q2 cos(2tPo + O)
ỗ a ,
0 = arete

V
2Ệ •
The second Eq. (2.9) yields:
cos(2 tpo + 0) — —- A y i 2 + Q 1
sin(2 Y’o + ớ) = ± ~ \ / c 2—4(Ệ2 + £}2) A 2
am therefore the first Eq. (2.9) is o f the fo rm 1:
dòa
~ dt~
^ ỹ | = f » V - 4 T F ’ + i F ) 2 * í « .
Hence, there fo llow s the stability co ndition o f equilibrium a = 0
c
M l >
2
] /ệ 2 + Q 2
8*
Nguỵetì Van Dao
2.10) rị1 < 1

, ĩ]2 > 1 H

— = = = = , ĩ] = y/2í2.
2í 2 2 | / | 2 + £ 2 2& 2 > / f 2 + í 2 2
In the figures presented the stability conditions are satisfied on the lines in b old face.
3. Th e In fluence o f C oulom b Fric tio n
Let us consider the case
3.1) R(x, X , x) = ÌĨQsig n * ,
vhere h0 is a positive constant,
•+ 1 if À '> 0,
3.2) sig n * = — 1 if X < 0,
0 if X = 0.

In this case it is easy to verify that
3.3)
71
h0 if Ũ ^ 0 ,
<i?0sinọ9) =
'o if a = 0 .
<i?0cosọ)) = 0 for all a.
N o w , Eqs. (1.16), (1.17), (1.22) are o f the form : for a 0:
da £
~dt = Ệ2+ Q
3.4)
; dt a{Ệ2 + Q 2)
Ợc — ỆQ2h)a3 ^-ứcos2y — ^£sin2y> — p ~ ^ o | »
± ( p + Q *)Aa+ — (Ệk + Q V i W + ^ s m 2 ụ ,-
3.5)
ca f- ", 2
~ f c o s 2 v +
2 Ệ ,
Ri — 77
n II
R-> = —-ho
71
V - (ỉ a + f f a V + O * L + ị « Í V + ^ h X - $
3.6)
The equation w — 0 yields:
;3.7) r 1 + ~4 Ỵ ị
2
+ -Q
2
) (Ệk* + Q*h*)a 2 + ntf2 + Q'2)a 1

* £ 2 + £ 2
1 -X / 1 2 + Q 2
VÍP V 4 -
?
1 4 f J 2
- 0 ( f / i J|(i 2 2 - A - lls) f l 2 + - ^ - / i S ,
7 1 (2
(3.8)
ố‘/c
yt = /; = e] \ /;* = E^ °
* Í 22 ’ * ~ Í 2 2 ’ 0 Í 3 2 ’
c * =
£C
Í P
V
y
2 Q ■
Nonlinear oscillations o f the third order systems. Part III 261
In Fie. 3 the dependence of a0 on i f is presented for the case Ệ = Q = 1, /z* = 0.05,
¥ — —0.1, c* = 0.05 and h0 = 2 .5- 10-3 (curve I), /? 0 = 5 • 10-3 (curve 2). H ere the
:sonance curve has a closed form. H owever, only the upper branch lim ited by the
ỉrtical taneencies corresponds to the stability o f the stationary cond ition (2.6). T he
lerease in h0 leads to the narrowing o f the resonance curve. W ith sufficiently high values
f h0, there is no stationary oscillation.
T o find the expressions (1.14) first we expan d:
00
4 V I 1
sign sin 09 = — > ——-—— sin(2;?7+ 1) 09.
7r 2m + 1
m — 0

N o w the form ulae (1.9) are o f the form :
Po = P i = 4s = 0, i# l , 3 ,
q1 = ỆAa + ~ kaz,
Pl = - A Q a - Ặ h Q 3a3,
4
g3 = - L ka3, p 3 = ~ h ũ 3a3
r\ _ 4/7 o ' o
Pin, - 0 , p 2m+1 - ^ i + T r ’ m ^
Therefore, the expressions (1.14) are:
Ho = G0 = 0,
- - T f S W T W ) ( ^ 0 + ^ - 3 ^ - f « > c o s 2 V + 3i2Ca s m 2 ,, |,
" 3 = W t f ‘ + 9Q2) I - ^ - 0 + ■§- (3fr + n3 - 3 £ c a c o s2 y - ĩ c a s i n 2 y I ,
2m+1 nQ m (m + l ) [ f 2 + (2m + l ) 2i32] ’ ^ 2m ’ ^
2m+1 7T Í32w ( w + l ) ( 2 m + 1 )[£ 2 + ( 2m + 1) 2Í 2 2] ’ 2m
Nguyen Van Dao
4. The Influence of Turb ulen t Frictio n on P ara m etric O scillatio n
N ow, we turn to the study on the case o f the turbulent friction, when R(x, X, x) has
2 form:
.1) R(x, x,x) = /?2 x 2sign x ,
lere h2 is a positive constant. It is easy to see that:
( / ỉo S Ì n ọ ? ) =
.2) — ■ 2,71
<R 0coscp) = 0,
d therefore Eqs. (1.16) take the form:
for a =£ 0:
I ? _ 2
h2y a ,
da
dt
dtp

Ệ2+ Q 2
(ỆQ2h —k)a3 + ~a co s2tp+ -^ -£sin 2 y > + ^ - h 2Qa2 ,
4 4 Í2 371 j
w = £
dt a(Ệ2 + Q2)
^ 7 ( f 2 4 - £ ? 2) z l a 4 - (£Ả ' + í 24/0 ữ 3 + - ^ - s i n 2y>
2Q
Ệ c o s 2 w + -^ -A 2 i3 2a 2| .
4Í2
By com paring with Eq. (1.16), w e have:
* , -
.4)
3ti
hi Qaz.
R2 = -Z— h 2 Qza2.
07Z
Consequently, the expression (1.22) is:
\ 2
U Ỉ
5) W(a0, y ) = U A + ~ k a l\ + q 4 a + ± h Q 2a2o + ^ f - h 2a0
Fig. 4.
Nonlinear oscillations of the third order systems. Part III 263
The equation w — 0 yields:
V2
1 + 4 (Ệ2 + Q 2) ^ k * + Q *h * )a ° + 37i(Ệ2+ Q 2) h *ũ o ±
ì /
, - v , i 2 + ữ 2 2
} ± Ệ2 + Q 7^ / A c
h* = J Ỉ * y
2 m 9 V

Q(ỆQ2h!t.—k^)a0+ ỆQ2h2
j 7 l
2
a2
Q 2 ’ ' 2Q '
This relation is plotted in Fig. 4 for the case Ệ = Q = 1, A* = 0.05, fc* = —0.1,
= 0.05 and /2 * = 0.01 (curve 1), /2* = 0.1 (curve 2). Here the change o f the coefficient
leads not only to the change in the m aximum o f the amplitudes of oscillation but
3 to the change in rigidity o f the system considered. The increase in h2 leads to a de-
ase in the m axim um o f the am plitudes and the system becom es harder.
5. Th e Influence o f C om bination F riction on th& Param e tric Oscillation
In this Section we study the influence on the parametric oscillation o f the com bina-
n friction:
1)
R(x, X, x) = (h0 + h2x 2) sig nx
N o w , Eqs. (1.16) are o f the form :
for ữ / 0:
da e
~dt = ~Ệ 2+ Í2 2
dy E
dt a(Ệ2 + Q2)
| i ( f f l ^ - f c ) a > + ^ - c o s 2v + ^ f s i n 2v > + ^ ( 4 * 0 + 3 j A . w )
ca
4Í2
£ c o s 2y +
2 4 . . I
+ —-h0+ ~ h 2Q2a2 .
7 1
Nguyen Van Dao
Th: expression (1.22) is:

3) w — ịặA + ị k a ị J + Ì Q / i + ị / i Q 3a20+ ^ - h 0+ ^ - h 2Q2a0\
Th: equation that yields the relationship between the am plitude o f oscillation and the
Lcitirg frequency will be:
.4) r = 1 +
A Q ht
4(Ệ 2 + Q 2)
371 Ệ2 + Q
7i(Ệ2 + Q 2)a 0
+
± v h p ' \ / ^ 2 + Q2)cl-
8
. (ỆQ2h * - k * ) a 20 + ~ Ệ Q 2/ĩ* a 0 + - ^ - h * 0
4 . 3 n 7ian
In this case the response curve is closed too (see Fig. 5) for Ệ = Q = 1 , /7* = 0.05,
* = -0.1, c* = 0.05, hi = 2.5* 10"3, h*2 = 0.01.
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slo\ly variable coefficients, Pro c. 4th Con ference on N o n -L in e ar O scillatio ns, Prague 1967.
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M ocow 1963. r*
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N g ' Y E n v a n D a o , N o n - l i n e a r o s c il l a t i o n s o f t h e t h i r d o r d e r s y s t e m s . P a r t I I . N o n - a u t o n o m o u s s y s te m s ,
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S t r e s z c z e n i e
N IE L 1 N 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 ắ C III. P A R A M E T R Y C Z N E D R G A N IA
N iie jsza p ra c'1 stanow i czẹsc I I I p ra c [11] i [12]. R oz patrzon o w niej p aram etryczne drgan ia n ielinio -
*go uia du trzeciego rzẹd u.
W ^ na czo no przyblizo ne ro zw iạz an ie róvvnania ru ch u o ra z p od ano w aru nki statecznosci sta cjon ar-
ĩgo rcw i^ za nia.
Z bda no w plyw tarcia kulo m bow skieg o i turb ulentnego na d rg an ia param etryczne. R o zp atrz on o
kze dgania w p rzypadku ko m bina cy jnego tarcia.
N on lin ea r oscillations o f the th ird order systems. Part 111 265
p e 3 K) M e
HEJIHHEHHLIE KOJIEEAHHil CHCTEM T PETLErO nOPflJIKA
^ ỈA C T L I I I . I IA P A M E T P M ^ E C K J iE K O J IE B A H H H
HacTOHiuan paooTa cocraB jifleT TpeTLK) ^lâCTL paốoT [1 1] H [1 2] . PaccMOTpeHbi B Heổ napaM eT pii'
He KOJie6aHHH HejniHeHH OH CHCTeMbi T p eT b ero n op H A K a.
OnpeAejieHO np H ốjm > K eH H 0 e pem eH n e y p aBH eH HH H n pu B eA C H bi VCJIOBHH yc TO H H H B ocm
noHapHoro pemeHHH.
Iic cjie flo B a H o BjiHHHH e K yjioH O B CK oro H T yp G yjieH T H oro TpeHHH H a n ap aM eT p m e cK H e KO Jie6aH HH.
PaccM O Tp eH bi TOH<e KOJieSaHHH B CJiyMae KOMốHHaiỊHOHHOrO T peH H H.
Received October 17, 1978.