Assum ing th a t certain system p aram eters are sm all, th e m ethod of m ultiple scales [1] is utilized to determ ine those frequency relationships which produce in ter esting resonan
Trang 1O L L O Q U I A M A T H E M A T I C A S O C I E T A T I S J Á N O S B O L Y A I
62 D I F F E R E N T I A L E Q U A T I O N S , B U D A P E S T ( H U N G A R Y ) , 1991
Nonlinear D ifferential E quation w ith Self-E xcited
and Param etric E xcitation s
T R A N KIM CHI and N G U Y EN VAN DAO
I n tr o d u c tio n
In this paper, a single degree — o f — freedom cubic system subject sim u lta neously to a self-excited and param etric excitations is considered Assum ing
th a t certain system p aram eters are sm all, th e m ethod of m ultiple scales [1]
is utilized to determ ine those frequency relationships which produce in ter esting resonance phenom ena T he various types of resonance exhibited by the system are stu d ied and th e stab ility analysis is presented
Exactly, th is p ap e r considers those dynam ical system s whose m otion is
(1) X + l j 2x 4- e ( 3 x3 — e R ( x ) + e F { x, X) COS 'yt = 0,
where differentiation w ith respect to th e independent variable tim e (t) is
T he “negative” friction function R ( x ) is assum ed to be of th e form
where hi are positive constants; and F ( x , x ) is of form
Trang 2T R A N KIM CHI, N G U Y E N VAN DAO :re c n rn are constants If c = 0 equation (1) describes the self-excited
them considered separately has a define self-sustained oscillation The lestion arises as to w hat will occur in the system represented by (1) when lese two oscillators are coupled Does statio n ary solution (oscillation) of .) exist? Is it stable?
T he interaction of two autoperiodic oscillations of this type was first ivestigated bv M inorsky N [2] for th e system
4) X + e ( x 2 — l ) i + [1 + ( a — c x 2 ) COS 2 t ] x = 0
>y using stroboscopic m ethod He m entioned th a t the nonlinear oscillatory ystem s exhibit always an interaction between th e com ponent oscillations
md th a t perhaps the whole theory of nonlinear oscillations could be formed )n the basis of interactions
In th is paper we are interested in th e resonant oscillations of equation
; l) and th u s will concentrate only th e sta tio n a ry solutions of am plitude -
1) rri, = n = 1, 2) m = 0, n = 2, 3) m = 0, n = 3 In order to
determ ine critical frequency relationships leading to resonant oscillations in the foregoing equation, let
where k is a rational num ber and eA is th e detuning param eter.
1 M e th o d o f a n a ly sis.
A r n p litu d e -p h a se e q u a tio n s
We use the m ethod of m ultiple scales [1] to determ ine a first-order uniform expansion of the solution of (1) To th is end, we let
where, To — t and T\ = et S u b stitu tin g (6) into (1) and com paring the coefficients of equal powers of e, we ob tain
(6)
(7)
(8)
D ị x 0 + f i 2 rc 0 = 0 ,
D q X i + Í Ỷ X ị = — 2 D q D \ ' X q + /o,
Trang 3NO NL IN EAR D I F F E R E N T I A L E Q UA T IO N 6 7
/0 = — Axo - 0 x ị + h \ x0 - h ^ x ị - cx™x0 COS71.
D Q — d /d T o , D ị — d / d T ị , Í2 = Ảrỵ
The solution of (7) can be expressed as
Xo — a c o s(fỉT0 + Ip),
X Q — — a f 2 s i n ( Q ĩ b + i p ) ,
equation (8) yields
D q X1 + Q 2 x1 = 2£ìa' s i n Ộ + 2Claip' COS ệ + / 0 ,
Ộ — f i T o + 'Ip,
:re prim e denotes differentiation w ith respect to T\.
Case 1 : m — n = 1
In th is case any p artic u lar solution of (11) contains secular term s if
: 1 an d k = 1/3 E lim in atin g th e te rm th a t produce secular term s in X\,
o b tain
For k = 1:
2a' = h \ a —^ / 1 3 7 a — COS'?/;,
2'^a'ip' = A a + ^/3a3 — ^ a27 sin ip.
For k = 1/3:
2 a ' — h \ d -— / i3 7 2 a 3 + - a 2 COS 3-ỉ/>,
27 a'lp 1 — 3 A a + — - a 27 s i n3'0
Case 2: m = 0, n = 2
T h e secular term s occur w hen /c = 1 and k = 1/3 T h e am p litu d e -
phase equations now are of form:
For Jfc = 1:
Trang 4T R A N KI M CHI, N G U Y E N VAN DAO For k = 1/3:
15)
Case 3: m = 0, n = 3
In th is case th e solution of equation (11) contains secular term s if
2 S ta b ility a n a ly sis o f sta tio n a r y so lu tio n s
We now present th e response curves and stab ility analysis for the constant solutions of am p litu d e - phase equations
equations (14) and (15), also for equations (16) and (17) So, it is sufficient
to stu d y equations (12), (14) and (16) T he response curves of equations (12) an d (14) are ellipses B ut the stability region of these curves is quite different (see Fig 1, 2)
T hus, consider system (12) Its nontrivial sta tio n a ry solution: a — do,
ip — 'Ipo is determ ined from
; = 1/2 and k = 1/4
For k = 1/2:
(16)
For k — 1/4:
(17)
( 1 8 )
4hi — 3/i372aồ — cao cos'lpo — 0 ,
4 A + 3 Ị3aị — cao7SÌn-0o = 0.
Trang 5NO NLINEAR DIFFER EN TIA L EQUATION 4
6 9
Fig 1 G raph of response curve (19)
for u = I , e(3 = 0.1, e h \ = 0.001, e h3 = 0.02, EC = 0.02.
'he response equation is readily found to be
lere
20) L i — 7 2 ( 4 / i i - 3 / i 37 2 aỔ) 2 + (4A + 3 / 5 a ồ ) 2 - c 2Ý á ị
In order to investigate th e stability of a given statio n ary solution do,
•po of (12) it is necessary to exam ine th e characteristic roots of the linear variational system for th is solution T he characteristic equation is of form:
(21)
h e r e
7 2 A 2 - f e h i ^ X — £2 d W i
64 d a 2 L = 0 = 0 ,
(22) W i = a 2 [ 7 2 ( 4 / i i - 3/?,3 7 2 a 2 ) 2 + (4A + 3p a 2 ) 2 - c 2 7 2 a 2].
The stab ility condition is
d a 2 L \ = 0 < 0
T he response curve (19) is shown in Fig 1 where the ste ad y -sta te
am plitude ao is p lo tted against the non-dim ensional frequency ratio 7] =
H ereafter asym ptotically stable solutions are represented as a solid line while unstable solutions are represented as a broken line T he stable p a rt is th e lower portion of the ellipse bounded by th eir vertical tangents, w here th e relation (23) is satisfied
Trang 6For system (14) th e response equation is
;24)
(25) i 2 = 7 2 ( V j f t 3 7 2 a ? ) 2 + ( f +
and th e characteristic equation has form:
72A2 + ^ 72(3/i37 2a ị - 2/ii)A +
(26)
= 0 ,
1 / 2 = 0
W o — a
2 „ 2
+ = + j / f c ‘ - 1 6
T he stab ility condition is then
d a 2 1/2 = 0 > 0
These conditions are in co n trast w ith (23) Only a p a rt of the u pper branch
of the ellipse is stable when th e two relations (27), (28) are sim ultaneously satisfied (Fig 2)
2 t CJ Fig 2 G rap h of response curve (24)
for CJ = 1, e/3 = 0.1, e/ll = 0.001, e/?3 = 0.015, EC = 0.02
Trang 7For system (16) the response equation is
30) L 3 = J 2 ( 2 h i - - h ^ á ù + ^2A + 2 ^ a o i ~ c2fl0
and the characteristic equation will be
2 x 2 2 x I L 2 2 ^ , £ 2 d W 3
7 A + £ 7 2 A - / i i + a s'Y a 0 + V
(31)
= 0 ,
l 3 = 0
= a and the stab ility condition is
(32)
7 2 f 2 h\ — ^ 372a 2^Ị + Í 2 A + ^ / 5 a 2^Ị - c2a 4
8
3
7 ^ 3 7 2 aồ - / l i > 0 ,
<9a2 l 3 = 0 > 0 The response curve depends O I ! the value I = c — 3/i3Cj3 If / < 0
the response curve is an ellipse w ith th e upper stability branch and lower unstability branch If / = 0 the response curve is a p arab o la (Fig 3)
7 - 4C0
for I < 0, CƯ = 1, e/3 = 0.1, eh\ = 0.001, e / i3 = 0.01, ec = 0.03
If I > 0 the response curve is a hyperbola (Fig 4, 5) It is seen from
these figures th a t increasing th e am plitude of th e param etric excitation the stable sta tio n a ry oscillation of system considered disappears
Trang 87 2 T R A N KIM C H Ị N G U Y E N VAN DAO
r *
Fig 4■ G raph of response curve (32)
for / > 0 , UJ — 1, e/3 = 0.1, e h \ = 0.001, E h3 = 0.01, EC
4 CO'
0.04
for I > 0 , U) — 1 , £0 — 0.1, e h \ — 0.001, £/13 = 0.01, EC — 0.2
3 C o n clu d in g rem ark s
Sum m ing up, the above analysis shows th e following peculiarities of the considered nonlinear equations:
defined by inequalities like (23), (27), (28), (32) and (33) O utside these regions there are no statio n a ry solutions
Trang 9N O NL IN EA R DI FFERENTIAL EQUATION 73
2 As we know, th e sim ple self-excited system (c = 0) exhibits always :he stable statio n a ry oscillation W ith the presence of the p aram etric excita
tion (c Ỷ 0) depending on th e am plitude (m, n) of excitation th e statio n ary
bility region of sta tio n a ry oscillation also depends on th e excitation form Sometimes th e lower branch is stable (Fig 1), som etim es th e upper branch
is stable (Fig 2, 3, 4) and som etim es no branch is stable (Fig 5)
3 T he stab ility conditions like (27), (32) indicate the lower energy level below which oscillations cannot m aintain them selves (since a2 = X 2 + Q ~ 2 X 2
3/i372aồ = 4 h ỵ for th e sim ple self-excited oscillator (c = 0) corresponding to
to system (16), it follows from (27) and (32) th a t, the oscillator represented
by (1) w ith m = 0, n = 2 or m = 0, n = 3 develops less energy th a n a simple self-excited oscillator E verything happens as if th e param etric p a rt of the nonlinear oscillator absorbed th e energy developed by th e simple self-excited oscillator
R e fe r e n c e s
Wiley-Interscience, 1979
Princeton, New York-London-Toronto, 1962
T ran Kim Chi, Nguyen Van Dao
Institute of Mechanics
National Centre
for Scientific Research of Vietnam