Some problems of non-linear oscillations in systems with large static deflection of elastic elements

V o l u m e 5 , N U M B E R
HANOI 1993
Proceedings of the NCST of Vietnam, Vol. 5, No. 2 (1993) (3-20)
M echanics
SO M E PROBL EM S OF N O N -L IN E A R OSCILLATIONS
IN SYST E M S WITH LARG E STATIC DEFLE CTIO N
OF ELASTIC ELEM ENTS
N guyen Van Dao
Institute of Applied Mechanics, Hochiminh City
Summary. In this work two following problems have been examined:
1) The non-linear oscillations of electro-mechanical systems with limited power supply and
large static deflection of the elastic elements.
2) The interaction between the self-excited and parametric oscillations and also between
the self-excited and forced ones in the non-linear systems with large static deflection of the
elastic elements when the mechanisms exciting these oscillations coexist.
In both problems there is a common feature characterized by the fact that the nonlin-
eaxity of the system under consideration depends on the parameters of elastic elements and
theừ static deflection and by the appearance of the non-linear terms with different degrees
of smallness in the equations of motion. Stationary oscillations and theừ stability have been
paid special attention.
N o n -lin e ar o scilla tio n s in sy stem s w ith large sta tic deflection of e lastic elem en ts
have b een exa m in ed in [l ]. T h e sp ecificity o f these s yste m s is: Their hard ness essen tially
d epe nd s on b oth th e p ara m eters of th e ela stic elem e nt and its s ta tic d eflectio n. T his
featu re lea ds to th e ch an ge o f the a m p litu d e curve and the stab ility on it.
In th is w ork so m e rela te d prob lem s w ill be in vestig ated: T he s yste m w ith lim ited
p ow er su pp ly an d the in tera ction o f n on -lin ear oscillatio ns.
I - NONLINEAR OSCILLATIONS OF THE SYSTEM WITH LARGE
STATIC DEFLECTION OF THE ELASTIC ELEMENTS
AND LIMITED POWER SUPPLY
In th is p art th e n o n -lin ea r oscillatio ns of a m ach in e w ith r otating u nba lan ce and
large s ta tic d eflection o f th e n on -lin ear spring and lim ited pow er su pp ly are co nsidered.

T he eq u ation s of m o tio n o f th e sys te m un d er con sid eration are d ifferen t w ith tho se of
cla ssic al p rob lem [2] by th e a ppe aranc e o f th e n on -lin ea r term s w ith differen t d eg rees o f
4 NGUYEN VAN DAO
sm a lln ess. Th is feature lead s to the dep en dence o f the hardn ess o f the sy stem not only on
the p aram eters o f the ela stic elem ent b ut also on its sta tic deflection.
T h e re su lts obtained are d ifferent in b oth qu ality and q uan tity w ith th ose ob ta in ed by
K o non enk o V . o. [2].
1 . E q u a t io n s o f m o ti o n
F igu re 1 illu str ates a m achine w ith a pair of cou nterro ta ting ro tors of equal u nbalance
(so th a t horizontal co m p on ents of the centrifugal force v ectors ca nc el), isolated from the
floor by n on -lin ea r springs and d ashp ots w ith d am ping coefficient h0.
1 V 0 SL
ỷ ? f
X
Fig. 1
T h e sp rin gs su pp ortin g the m ass are assum ed to be negligible in m ass w ith a n on
linear ch ara cte ristic function :
/(u ) = cou + 0ou3; (L l-1)
w here cn is a po sitive c on sta n t, /30 is either p ositive (hard ch ara cte ristic) or n egative (soft
ch a ra cter istic). T he d eform ation of the spring in th e sta tic equ ilibrium p osition is A , and
th e sprin g force c(, A ■+■ i (lA 3 is equal to the g ravita tiona l force m 0g a cting on the m ass:
C0A + £ 0A 3 = m0 g,
w here m„ = m l + m is defined as the sum of th e m ain m ass m i and the ro ta tin g u nbalance
m asse s m, t h a t is the to tal m ass sup por ted by the sp rin gs. T he d isp la cem en t X is m easu red
from th e sta t ic equilibriu m p ositio n w ith I chosen to be p ositiv e in the upw ard d ir ection .
A ll q u a ntities - force, velocity, and accelera tion - are also p ositiv e in the upw ard direction .
T h e sy stem under co nsid eration has tw o d egrees o f freedo m and th e generalized
co or din ates X and ip c om p letely define its p osition .
T h e k inetic energy of the syste m under co nsid eratio n is
^ 1 o m

T = - m X 4- — V
2 1 2
2
m)
Im = X -h r COS <p, Zm = r sin V?.
H ence
SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC
5
T = - m 0 i 2 — mrxip sin <p H— /<p2, / = m r2. (1.1.3)
2 2
For the p oten tia l energy, the reference can be chosen at th e level o f the s tatic equ i
lib riu m p o sition:
Ư = — (A — x)2 + — (A — x)4 *f rnt)gx 4- mgr COS (p. (1-1-4)
2 4
T he L a sgran ge’s eq u atio ns give
I<p = mrx sin (p + mgr sin <pt
m0 X + c0 X + /?013 -f 3/30 A2x — 3/Ju Ax2 = mr^ sin + mrip2 COS V?.
Taking into account the driving moment L(<p) and the frictions H(<p), k0x we have
the follow ing eq ua tion s o f m otion :
Iỷ = L(yi>) — H[<p) -+- mrxsin <p + m^r sin V?,
T71()X + c0i - f / i 0i + /?ni 3 4- 3/90 A2x — 3)ổn Ax2 = mr^> sin Ip -f mr<p2 COS (p.
Supposing that A is rather large and X is enough small, so that £tlx3 is a small
quantity of second degree (s2), while 0it Ax2 is of first degree (5), where £ is a small positive
p aram eter. O b viou sly, in this case # ,A 2X is fin ite.
It is assu m ed a lso th a t — 1, ^ 1. T he friction forces, the forces YTiTtf? COS <p,
m0 /
m r< p sin y ? a n d t h e m o m e n t s mrxsin<pf mgr simp a re s u p p o s e d t o b e sm a ll q u a n tie s o f e2.
T h u s, we h ave th e follow in g eq uations of m otion :
w here
<p =

e
2
[ A /
1
(ý> ) + <
7(1
+
9
) s i n sơ ] ,
ĩ + w2 ĩ = í 7 I 2 + £2 [pỷ sin ip + p<p2 COS ip - hi — Ị3x2)
2 _ mr 2 L A 2 /3,,
f p = —- , e h = t c7 = -IS— c p = zf" >
m«, m„ m„ m 0
^ , e2Mi(yỉ>) = ị [£((£>) - #(¥?)], e2? =
m„ /
mr
T
(1 . 1 .8)
T h e eq ua tio ns (1.1.7) are different w ith those in K on onen ko V . o . w ork [2] by the
a pp eara nce o f the q ua dratic term 57X2 and by th e d egrees of sm alln ess o f th e term s. T h ese
e qu a tio ns c hara cte rise th e s ystem s w ith weak e xcitation and large s ta tic d eflection .
2. So lutio n
We lim it ou rselve s by con sid erin g th e m otion in th e reson ance r egio n, w h e Te the
frequ ency u o f th e free o scillation is near to the frequ ency n = <p o f the forced oscillation s.
W e sh all find th e solu tion o f e qua tion s (1.1.7) in the series [3]
X = a cos(<p + 0) + eui(a, \ịỉ, <p) + £2U2(a, t/>, <p) + £3 .
NGUYEN VAN DAO
w h e r e u t (a, d o n o t c o n ta in th e first h a r m o n ic s COs\pi sin t/>, \ị) = <p + Ỡ a n d a r e p e rio d ic
f u n c tio n s o f a n d (p w i th p e r i o d 27T, a n d a , 6 a re f u n c t i o n s s a tisf y in g t h e e q u a tio n s
à = £w4i(a, Ớ) + e2w42 (a,0) + .

# = UJ — n -f £#1 (a, 9) 4" e2B 2(a, Ỡ) 4"
dip
(1.2 .2 )
n =
dt
T he first e q u a tio n of (1.1.7) is th en
á n
= e2[Mi(n) +q(x + g) sin <p]
(1.2.3)
To de te rm in e th e un kn ow n functions Ay, Bx, ut we differentiate th e expression (1.2.1)
an d s u bs titu te it in to (1.1.7). We have:
( du Ì
X = —CLUJ sinrp + e< — aB 1 sin + Ảỵ COS 0 H- w — + n —— >-h
9 í „ ổu, <9u. <9u, i
+ £ I - a i ?2 sin t/> -f i42 cos + i4i ~ - L + n +
X = — auj COS
^ 4- (cư — n ) — - — 2aa>i? i] cos \ịỉ — f(u» — n)a + 2cưj4i
I L a# J L ƠỚ
. ^2u i 2 ^ U1 1
+ 2 w n - ^ - + w ^ - } +
sin
+ n 20 + 2w n 5V'3^
<9^>2 d\ịỉd<p d\ị)2 i
4- é 2 Ị ^(cj — n) —^ — 2 a u B 2 COS xp — Ị(u> — 0 ) a -— 2 + 2(iM2 sin \ịỉ+
+ {Ax^Ệ± + 5 1 -Q0 - - aSỉ) C0S - ( 2A 1 jBi + aAl-g-^- + a 5 l ^ i') *in V»+
+2^ f e +2^ l ầ +2n5i0 * +2“Si9 ^ (w n)^ +
+ (w - n )
cMi <3u r>2^2u2 d2U, 2^2u-
_ L Q2 — ^2. + 2cjQ — — + ur — —
<90 da d<p2 diọdrịì d\Ị) 2

+ e
(1.2.4)
S u b stitu tin g the ex pre ssions (1.2.4) in to the eq uation (1*1.7) an d c om p aring the co
efficients of e a nd e 2 . . . w e ob tain:
+
/ d d \2 2
\n d i + ” i k ) u ‘ + w u -
•+* Ị(cư — n ) ~~Ệ q ’ ~~ 2a u Bi j COS rp — Ị(cư — n j a - ^ 1- - f 2cư Ai sin xp = 7a 2 COS2 t/>,
/ a a \2 ,
( n a 5 + “ a v;) “ ’ + " “ >+
+ — n) — 2aw B2| cos t/> — (w — sin ip —
+ 2 70Uj COS v> — /3a3 cos3 1/1 + sin Ip + pfi2 COS ip,
(1.2.5)
(1.2.6)
where R(0, Bi) = R(Ah 0) = 0.
Comparing the coefficients of the harmonics in (1.2.5) we have
(w - “ 2au) B 1 - °>
(u; — n ) a — ^ 4- 2a ;A 1 = 0,
SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC
7 dỡ A
/_ ổ Ổ \ 2 2 2 _ 2 /.
( n - — + U1 U1 = cos V*
<9yp dip
Solving these equations yields
X i= 0 , B i = 0, Ui = ^ 7(1 - jc os 2V >). (1.2.7)
Comparing the coefficients of the first harmonics sinự>, cosV> in the equation (1.2.6)
Wfc have
(w - n ) - 2ow £2 = g 2 - f - - + pH 2 COS Ớ,
(w - n ) a ^ “ + 2a>i42 = —hauj — p H 2 sin Ỡ.
0 6

F rom th ese e qu a tio ns on e ob ta in s:
ha pCi2
A2 = ——

—— sin 6 )
2 u + n
_ 1/3- 5 7 2 \ 2 pft2 „
B 2 = — [ Ẹ ~~ n ) a “ 7

* r ~ cos Ớ.
4w \2 3 CƯ / (w + n)fl
T h e eq u atio n for d eter m in atio n u3 is
/_<9 Ổ \ 2 o / 7 2 /?\ -I
r ~ k +UJ- k ) u’ + w u’ = - ( ò + ĩ ) a co sH -
( 1.2 . 8 )
H
ence
“’ = ĨS ? ( 2 + ắ ) ° 3'°,3'í- (l2-9)
Thus, in the resonance zone n « u; we have the following equations in the second
approximation
X = a cos(<p + 0) + « ( p 2 - 6^2 c o s 2^ ) . (1.2.1 0 )
W'here n, a and 0 satisfy the equations
d ii e2 [w . 1 2 • Ả
^ = — — — (u>/ia + p f i2 sin ớ ), (1.2.11)— = - — 77 [U)tia + DM sin £7
2u>0v
đo _ 1 f n ff2p fi2
— = “ We - w

-=

COS
dy9 n V 2a»a
« i) ,
where
w' = a, + ỉ r a2’ “ = 4^"S£- (L2-12)
The stationary solution of the equations (1.2.11) is determined from the relations
dCl _ da dỡ ^
d<p d<p dtp
A/i(n) + -<7tc>2asin0 = 0,
/la w 4* pH 2 s in 6 = 0, (1.2 .13)
_ 2 P^2
w e — n - e —— COS 6 = 0.
2ua
Eliminating the phase 6 from the last two equations of (1.2.13) we obtain
w (a 2, n ) = 0, (1.2.1 4 )
where
W(a2,n) = u 2a2 \eAh2 -f- 4(u;e - n)2] - e Ap2n 4. (1.2.IS)
In the resonance zone n « U/, the equation (1.2.14) gives approximately
8 NGUYEN VAN DAO
n 2 = u>2 + e2cxa2 ± e2 y j

h2(jj2 . (1.2.16)
From this relation it follows that the non-linear oscillation has:
- a hard c hara cte ristic (F ig. 2a) if Q > 0 or if
c0 > 7/?0A 2, (1.2.17)
- a soft ch ara cteristic (F ig . 2b) if
Co < 7/? 0A 2, (1.2.18)
- a linear cha racteristic (F ig. 2c) if
c# = 7/?0A 2. (1.2.19)
E lim in ating 6 from the first two eq uations of (1.2.13) w e have

L (n) - 5(H) = 0, (1.2.20)
where
S(ũ)=H(í)) + ~ h y . (1.2.21)
The equation (1.2.14) is similar to that in the system with ideal power supply [1]. The
difference is th a t n should be satisfied the relation (1.2.20) which can be solved graphically
SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC
as shown in Fig.
9
Fig. 3
3. S ta b ility o f statio n a ry oscillations
Equations (1.2.13) can give some stationary values n = O 0 , CL = a0, 6 = 0o . To study
the stability of these values we introduce a perturbation of n, a and 0:
ỐH = Q — n 0) ổa=a — aof Ỏ6 = 6 —0Q.
10
NGUYEN VAN DAO
We denote the right-hand sides of the equations (1.2.11) by $ n (n, a, 0), $ 12(Ấ, a, 6)J a, 0)
respectively. Below, the derivatives will be calculated at the stationary values of n, a and
6 which satisfy the relations (1.2.13). We have the following variational equations:
^ - = bll6ĩì + bll6a + bliSe>
^ = fc„5n + fc„5 a + 6,3*0, (1.3.1)
dip
— - = b^ỏĩì + b^ỏa +
ay?
where
- d- ề r - . w - - £ ( i ( n ) - f f ( n ) ]
an in ’ dn L ' ' '
3 < & n / i „ w 3 <3 $ 1 1 m n u ; 3 2
3a 2/n 3a’ 13 /fi3 )a ’
L d $ 12 , < 3 $ 1 2 /in .
an m„n2 ’ ” aa 2m„n ’ f1-3.2)

6„ = - £ ( * « - n ) a , i 4l = £ ± i i = J j ( n - 2 w e) ,
■ d
_ Ổ$13 >lo
<90 2mon
-^13 1 a / \ o
a r ' t M 1-" - 1 - "
The characteristic equation of the system (1.3.1) is
A3 + DịX2 -f Đ2^ Dz == 0»
where
Di = -(6 n 4- 6ai + 6SJ ,
^ 2 =: ^11^33 ^73^33 ““ ^33^33 ^13^31 ““ ^13^31»
•^3 = ^11^33^33 ■+* ^13^31^53 + ^13^23^31 ” ^13^31^33*
The Routh-Hurwitz’s criterium of stability is
D i > 0, .D3 > 0, D1D2 — ^3 ^ 0- (1.3.3)
We have
D' z=\ ~ m - {L3-4)
m ứ n iw
As usually, it is supposed that -^-L(n) is negative and — i/(n) is positive, so that TV is
afi ail
negative. Hence, Di is always positive.
The second stability condition D3 > 0 as shown by Kononenko [2] is the most impor
tant one. This condition is equivalent to the inequality
( 6 , A , - M M) ^ j * ĩ i ( n . M ) < 0 (1.3.5)
SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC 11
w here (ft, a (n ),0 (n )) and a (n), 6 {n ) are found from th e last two e qu ation s of
(1.2.13); namely
*h = £ f [ m - s (fi)}
It is easy to verify that
2 dW
^77 "" ^33^33 ~ ơ £ fl2 ’

where w is of the form (1.2.15) and Ơ2 is a positive constant. Now, the stability condition
(1.3.5) can be represented in the form
. i [ i ( n ) - S(n)] < 0. (1.3.6)
dW
It is noted that — - > 0 is the stability condition of stationary oscillation when n is a
da2
dW /
given constant. is positive on the heavy branches of the resonant curve. The sign of
da2
the derivative
G = J y [ L ( n - s ( n ) j (1.3.7)
can be obtained by considering the relative positions of the graphs L{n) and 5(0).
For the case of the system with a hard characteristic (Fig. 3) it is clear that G is
dW
negative at points Rị, R2 and R3i so that the points Rị and i?3, where -—r is positive,
da
correspond to the stability of stationary oscillations. The point R2 corresponds to the
instability of stationary oscillations, where — z is negative.
pa*
In comparison with a system with an ideal energy source [l], the unstable branch of
the resonance curve remains the same. But the jump phenomenon occurs in a different
maner. As n is increased the amplitude of oscillation will follow the solid arrows and the
ju m p in th e a m p litu de w ill take place from p to Q. W ith a decrease in frequ en cy n the
amplitude will follow the dashed arrows and the jump will be from T to u . The points
o f collap se p and T are th e p oints of c ontac t o f the ch aracteristic L(n) and the fu nc tion s
Sin)-
For the case of the system having a soft characteristic (Fig. 4), the part of the reso
nance curve indicated by the dashed (heavy) line corresponds to the instability (stability)
o f s ta tio n ary o scilla tio n s, provided the frequency n is a given c on sta nt. O n th is part
dW f d w \ ’

~ 2 < ( ^ 2 > ° j - SiZn °f t ^ie d erivative G (1.3.7) d epen ds on th e slo pe o f the
0 a0
characteristic, i.e. on the quantity — L(n). It is necessary to distinguish two cases:
afi
1) w hen th e ch ar acteristic is steep , i.e . -jprL(Cl) has a la rge a bsolu te value (F ig .4.)
ail
2) w h en the ch ar acteristic is gen tly slop in g, i.e. has a sm all a bso lute value
. aw
(Fig. 5).
In th e first ca se th e deriva tive G (1.3.7) w ill be n eg ativ e on the par ts P U , P T and
T Q ( F ig .4 ). T h erefore, the sta bility con ditio n (3 .6) w ill no t be sa tisfied on P T , w here
div dW
Tp-J < 0, but it will be satisfied on PU and QT; where > 0.
12
NGUYEN VAN DAO
In the second case, the derivative G (1.3.7) will be positive on the parts PU and PT
and negative on QT. Therefore the condition (3.6) is satisfied on QTP and is not satisfied
on PU (Fig. 5).
Fig. 4
II - WEAK INTERACTION BETWEEN THE SELF-EXCITED
AND PARAMETRIC OSCILLATIONS IN THE SYSTEM
WITH LARGE STATIC DEFLECTION
The present part deals with some related problems when two mechanisms exciting the
self-sustained oscillation and parametric one coexist in one system. Following the assump
tions in [l], these oscillations are weak. They appear only in the second approximation of
the solution and their interaction is weak, also.
In comparison with the classical problem on the interaction between self-excited
and parametric oscillations [4], the system under consideration has distinguishing feature
which is characterized by the fact that its non-linearity (hardness, softness) depends on
the parameters of elastic element and its static deflection. Namely, when the initial system

SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC 13
has a hard characteristic, the resonance curve may belong to a soft type. Therefore, the
results obtained are different with the classical ones both in quality and quantity.
X. E q u ation o f m o tio n and approxim ate solution
Using the assumptions and notations in the previous paper [1] we study the oscilla
tions described by the equation:
x + cU2X = C^ỊX2 - e2 [/3x3 + D(ơx2 - l ) i - ex COS 2/i], (II. 1.1)
w h e r e w, 7 , h, p, D , e, Ơ a n d V a re c o n s t a n ts , D > 0, > 0, (Jj > 0, V > 0, c > 0, Ơ > 0, e is
a small positive parameter.
It is supposed that V is in the neighbourhood of 2u> (parametric resonance). W hen
D = 0 equation (II.1.1) describes the system with parametric excitation. When e = 0,
equation (II. 1.1) represents a self-excited oscillator.
The parametric and self-excited oscillations have a comon feature that the origin
X = X = 0 is unstable. Under the resonance condition these oscillators may have a certain
interaction.
The solution of the equation (II.l.l) is found in the form
X = a cos 9 + £Ui(a, <p, 6) + e2u2(a, 92, 0) *f £3 , (II.1.2)
w here $ = - + \p, <p = 1/t and ut are periodic fun ctions of <p and 6 w ith p eriod 2tt w hich
do no t c onta in the first h arm onics COS 9 and sinớ. T he unkn own fu nctio ns a and ĩịỉ are
determined from the equations
^ = sAi[a,\f>) + £2A2 (a,\p) + . . . ,
d t (II.1.3)
d\p
It
= u - v- + eB\ (a, rp) + e2B2 (aì r/>) + .
Substituting the expressions (II.1.2) and (II.1.3) into equation (II.1.1) and comparing
the coefficients of e and e2 we obtain:
( " ả + “ế )2"1 + “2ui = 7fl2 cos2 * ~ [(w ■ 2 ) d~ d ị - 2awBl] cos6+
+ [{u ~ ^ ) a~g~p~+ 2ulÁl sinổ' (II. 1.4)
ớ ^ 2

( V ~ ti2 + cư2 U2 = 2a^Ui COS 6 -t- F(<p, a cbs ớ, — acư sin 6 )
V
dip dỡ J
d<p
c o s« + [ ( “ - 2 ) 0 a * + 2" A»] sin 6 + R(Ai, S i ),
(II.1.5)
w here
14
NGUYEN VAN DAO
<p = Ỉ/Í, i?(0, Bi) = R (A l f 0) = 0,
Flip, X, x) = - ị^ x 3 + D[ơX1 - l)x - tx cos <p]t
F[<p, a cos 0, —acư sin 0) = F(<p, X, i)
COI 0
i>>tw *ln #
Comparing the coefficients of the harmonics in (II.1.4) we have
{U ~ 2 ) d~ ề ' 2aOjBl=0'
( Ư\ dBi
(UJ

) a —r— + 2cư^4x — 0 ,
V 2 J drp
Ị d d \ 2 2 2 2 /1
ỊI/ —— + W 7 - Ux + UJ Ui = 7 a COS V .
\ ơy? Ơ0 /
These equations give
A i = B i = 0, Ui = ^ j ( l - ì c o s 2 í ) , e = ^ t + ệ.
Comparing the coefficients of the first harmonics sin 6 and COS 6 in
/ i/\ dAo _ 3 *
[UJ —
-

2oujB2 = - a a + COS 2 0 ,
V 2 / ơt/> 2
/ i/\ dBo __ / cra2 \ e . « .
V ~ 2 / a~dĩp~ + = ^ V

T ) ~ 2
t
a = ỈB t ?
4 6w2
Solving these equations we obtain
a£> / ơ a2 \ ca .
•4’ - 2 ( ‘ 4
_ a 3 ca
a .Do = — a

cos 2 \p.
2uj 2 u
Thus, we have in the second approximation
e~ta2 í 1 _ \ V
X = a COS Ỡ + - Y ( l - 3 cos 20 j , 6 =
da e2 r / cra2 \ ca . 1
J - M 1 - 4 ) ■
dé / i/\ c2 / o s 3 ea A
a - ~ = ( w — )0 + « (

— COS 2 v>) •
dt \ 2 / 2 V tư I/ /
(11.1.6)
(11.1.7)

(11.1.8)
(II. 1.5) yields
(11.1.9)
(II.1.10)
(IL1.11)
2. S tationa ry o scilla tio ns and their sta bility
Equations (II.1.11) have a trivial solution a = 0. The non trivial stationary amplitude
da „ drp
dt ~ ' dt
SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC 15
da dip
a0 and p hase rp0 are d eter m in ed by eq uation s — = 0 = 0 or
/ ơ a 2 \
2ụẢjjD y l
-
— J = e s i n 2ự>0,
4fMjj2 ( l - M + r 4 aỉ ) = e2e cos W o.
where u = — . Eliminating the phase \p0 gives
2u
W(aoin) = 0, (II.2.1)
where
From relation (II.2.1) we find approximately:
T h is form ula is p lotted in th e figure 6 for th e p aram eters
£ ^ c ^2 ^
■—V = 1.5 • 1CT3,

= 1CT3, Ơ = 40, —-p = 0.01 (curve 1) and a = 0 (curv e 2).
2cư2 cư Q/*
Stability of the stationary solution obtained is investigated by using the variational
equations. Denoting the right hand sides of the equations (II. 1.11) by p and Q, respec

tively, we have
w here the sub script “o” m ean s th at the derivatives are calculated for sta tio na ry values a0,
yịỉQ. The stability conditions are
( £ ) . + Q .
/ 5 P \ _ / 5 P \ /5Q \ a aty(a„,M)
\d a) o \d r p ) o U y J o V d c J o 0
Hence, we have
«!>-, > 0. (11-2.4)
16 NGUYEN VAN DAO
In the figure 6, the heavy (dashed) branch corresponds to the stability (instability) of
stationary solution where the inequalities (II.2.4) are satisfied.
To study the stability of the zero solution a = 0 we introduce in (II. 1.11) new variables
y, z connected with a and Xịỉ by the relations:
y = a COS z = a s in \p . (II.2.5)
We have
dy e 2 _ ( V e20
= -~Dy - [w - - + —
dt 2 V 2 2u
dz ( V e2ơ \ e2
— = ị u — j y + —
' (II.2.6)
Dz -f ,
where non-written terms contain y and 2 with high degrees.
The origin y = z = 0 (a = 0) of this system is always unstable, because the charac
teristic equation of the linear terms of (II.2.6) has the roots with positive real part.
<0 a*
III - W EAK INTE RA CTION OF SELF -EXCITE D O SCILLATION
W ITH FORC ED ONE IN N ON LINE AR SY ST EM S
W ITH LA RGE STATIC DEFLEC TIO N
In this part the attention is concentrated on1 studying the conditions under which

the resonance regimes of oscillations occur, on explaining the role of non-linear factors
in the formation of resonance situations of the systems with large static deflection of
elastic elements [l]. The distinguishing feature of these systems is that their non-linearity
essentially depends not only on the characteristic of the elastic element but also on the
static deflection.
The character mentioned has significant influence on the interaction of self-excited
oscillations with the forced ones, in both quantity and quality.
SOME PROBLEMS OF NON-LINEAR OS< 'ILLATIONS IN SYSTEMS WITH LARGE STATIC 17
1. E q uation o f m o tio n and construction of a pproxim ate so lu tion s
Let us consider some specificities of forced oscillation in the self-excited system whose
motion is supported by the “negative” friction. The following equation will be investigated:
Ĩ + W2 I = e~ix2 — e2 [/9x3 + D(ax2 — l )i — E sin 1/t] ,
(III. 1.1)
where the notations in [l] are utilized UJ, 7 , /?, I/, ơ, D , £ are constants, Z) > 0, 7/9 > 0,
I/ > 0, u; > 0, cr > 0 and 5 > 0 is a small positive parameter. When E = 0 equation
(III. 1.1) describes a self-excited system. When D = 0 we have a forced system with
harmonic excitation. In this paper it is assumed that D.E Ỷ 0. The question is stated
as follows: what happens in the system (III.1.1) when two mechanisms of generation of
self-excited oscillation and forced one coexist? We will be specially interested in the
stationary oscillations and their stability.
Using the asymptotic method [ 1 ] we find the approximate solutions of the equation
(III.1.1) in th e form:
I = a COS(ut + \ị>) + 5Ui(a, (p, S) + e2u2(a, <p, 6) + .,
(III.1.2)
where u, are periodic functions of <p and 0 with period 2tt which do not contain the first
harmonics sin0, COS 6, <p = 6 = 1/t + t/;, and a, v> are determined from the equations:
d\ị)
It
(III.1.3)
The intensive interaction of mechanisms of generation of the oscillations of different

nature can be observed in the resonance situations. In this connection, below the resonance
case will be considered, supposing that V is near to (Jj.
By differentiating the expression (III.1.2) and substituting the results into (III.1.1)
and by comparing the coefficients of e and e2 we have:
( d d \ 2 2 2 2 r ^dAi n n 1 -
I I/—— -f tư — U i + Uj = "ya COS u — [CJ — 1/j - — ẵ

ZOUJDI c o s 0 +
\ d<p 0 6 / L drp
■+ Ị(lư — u)a~^~ + sin ớ, (III.1-4)
^1/ —

h u> — ^ U2 -h Ct>2 u 2 = 2 a ^ u i COS Ớ + a COS Ớ, —aw sin ớ) — ị[io — I/) - — 2aa> B2j COS 0 +
-f- fcj — v\cl~~2— "+■ 2cưj42 sin Ớ *+■ -^i)» (III.1.5)
ay
where
RịO.Bt) = R(Alì0 ) = ồ )
F(<p, a cos Ớ, —auj sin 0) = F(<p} I , i) l i = a COS Ớ, i = — au> sin Ớ,
X, i ) = “ ^ x 3 — D(ơx2 — l) x 4- E sin <p.
(III.1-6)
By comparing the coefficients of the harmonics in (III.1.4) we obtain
18 NGUYEN VAN DAO
(w - ~ 2aujBi = 0,
(cj — u)a— — + 2uA\ = 0,
d\p
f d d \ 2 2 22/1
U j+ W U i = 7 f l cos 0.
These equations give
A i = 0, B ,= 0, u 1 = ^ 4 ( i - Ì c o s 2 ổ ) .
Comparing the coefficients of the first harmonics SÌĨ 1 0 , COS 0 in (III.1.5) yields

t \ d A.2 _ 3 r~i * f
(cj — Ỉ/)

2 au B 2 = - a a — £ sin
(cư — t/)fl 2 + 2cjy42 = aa;Z) (1 — <7 — ) — E COS \pt
d\p 4 /
3/9 - ^ 1
4 6ui2
From these equations we have:
aD / a2 \ E
Ả2 = — - 1 - CT—

cos ự;,
2 V 4 / a; + Ỉ/
a 2 E . .
B2 = — a + “ 7

r sin
2<J a(u> 4- y)
So, in the second approximation we have
X = a COS Ổ + f 1 — - cos , 8 = I/i -+■ v>,
2u>2 \ 3 /
da e2aD / a2 \ e2E
— =

— ( 1 - <7

COS
dt 2 V 4 / w -f Ỉ/
d i p é 1 CL 2 c 2 i ?

—1. = (J - I/ -f a +77

-— r s i n f
dt 2w aịu + 1/)
2. Statio n a ry oscillatio n s and their sta bility
The stationary amplitude a0 and phase V’o are determined by the relations
e2 / a2\ e2E
, , t'E .1
« / - w e a 0 = - J — -— - sin \/»0,
a(w + t/)
(111.1.7)
(111.1.8)
(III. 1.9)
(111.1.10)
(111.1.11)
(III. 1.12)
(HI.2.1)
SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC 19
w here
I \ _ e2°‘ 2
From these e q ua tio ns on e o btains
t g * . = f a i f e l L , ( in .2.2)
W ( a 0,v ) = 0, (III.2.3)
W[aB, u) = Ị 4Ị1/ - u,e(a)]2 + (1 - - 7 ) 2 } - • (ni.2.4)
Equation (III.2.3) gives approximately:
r?2 = l + ^ A ± — , (III.2.5)
ƠU2 UJ \ A
w here >1 = - 2 - , J = — r , r? = — . T he form ula (III.2.5) is presented graph ically in the
4 4lj2 D2 (Jj
figure 7 for th e para m eters: cr = 40, -— = 4.10~3, = —0.01 and E = 0 (po in t I and

CJ u;2
ax is a* = 0); = 0.344.10- 3 (J = £ -) (curves 1 ), ^ = 0.486.10-3 (J = ệị) (curve 2 ),
^—Ẹ- = 0.688.10 ~3 (j = — ) (curve 3).
w2 ' 27'
4
W hen E = 0, the resonan ce curves are d egenerated into a p oint I: a* = T\2 =
4
1 + - —~ and in to an a xis Or72 (a2 = 0). For sm all valu e o f E the resonan ce cu rv es c on sist
of two parts: a closed trajectory about the point I (curve 1) and a line “1” which is near to
16a;2 D2 4
the ax is Off2 . W ith large v alu e of E , beg in ning from E2 =

( J = — ) the reso nance
27a 27
curve has only one branch.
It is easy to verify the following variational equations of the system (III.1.12) for the
sta tio na ry so lu tion a0, \ịỉ0:
dSa c2 _ / 3ơ a ^ \
= — D ( l - 6a - a0 K - u)6 yỊ),
dỏĩb r . 1 , e1 _ / ơ a ? \ . duje
H en ce, th e sta b ility co n d ition s for station ary so lution are
2 3 W (a 0,i/)
2 - ơa < 0, and


> 0.
ổaồ
(III.2.6)
20
NGUYEN v an DAO

In the figure 7 the heavy (dashed) branches correspond to the stability (instability)
of the stationary oscillations where the conditions (III.2.6) are satisfied.
CONCLUSION
1. The nonlinearity of the system under consideration depends on the parameter
3 572
a 2= -/? - — r which can be either positive if c„ > 7Ổ A2, or negative if c < 7#, A2, or
4 6w Ẩ
zero if c„ = 70n A2, where c„, >9lf are characteristics of the elastic element and A is its static
deflection. A system with hard springs may become a less hard or soft one.
2. T h e m e n tioned inter esting feature affects on b oth th e qu ality and q uan tity of the
oscillating phenomena of the classical problems [2, 4, 5].
3. T h rou g h som e p rob lem s e xam in ed in this p ap er one can see th a t th e o th er prob
le m s c o nn ected w ith the o scillation s of the ela s tic str u ctu re s w ith la rge sta tic deflection
are necessary to be investigated carefully.
This work was supported in part by the National Basic Research Program in Natural
S cien ces.
REFERENCES
1. Nguyen Van Dao. Nonlinear oscillations in systems with large static deflection of elastic
elements. Journal of Mechanics, NCST VN, No. 4, 1993.
2. Kononenko V. o . V ibrating system w ith a limited power supply. M oscow, Nauka, 1964.
3. Bogoliubov N. N., Mitropolskii Yu. A. Asymptotic methods in theory of non-linear oscilla
tions, Moscow, 1874.
4. Kononenko V. o ., Kovalchuk p. s. The action of the parametric excitation on the self-excited
system. Journal of A pplied M echanics No. 6, 1991, Kiev.
5. Minorsky N. Nonlinear Oscillations. New York, 1962.
Received 5 September, 199S