Parametric oscillations of dynamical systems with cubic term at the modulation depth under the influence of nonlinear frictions

CỘNG HÒA XÃ H ội- CHỦ NGHĨA VIỆT NAVI
VIỆN K H O A H Ọ C VIỆT NAM
ACTA MATHEMATICA
VIETNAMICA
T O M III
N ° 2
H À NỘI— 1978
ACTA MATHEMATICA VIETNAMlCA
TOM3, N °2 (1978)
Parametric oscillations of dynamical systems with
cubic term at the modulation depth under the
influence of nonlinear frictions.
NGUYỄN VẰN ĐẠO
National Institute of Sciences SRV
This paper deals with the influence of nonlinear frictions to the para
metric oscillations of dynamical systems described by the equation with the cubic
term at the modulation depth
X + »2X + e(cx + i x 3)cosV i + cctx* + zR{x, i) — 0, (0.1)
where 0 1, c, d, a are constants, z is a small positive parameter, R {x, x) is a non
linear function of X, X characterized the frictions considered. Three forms of non
linear frictions will be investigated heret1. 2]: the Coulomb friction, the turbulent
one and their combination.
As will be seen later in the analysis, the sign and value of parameter d
sharply change the motion picture and the stable regions.
It m ust be emphasized that the equation (o .l) describes the real physical
system s more precisely than the one in which d «= ot4'5.61 The system of type
( o .l) with linear friction was studied qualitatively by Minorsky[3] but no attempt
has yet been m ade to investigate it with the Coulomb friction, turbulent one and
their combination.
§ 1. ST ATIO NAR Y OSC ILLATIO NS AND THEIR STABILITY
Let us consider the resonant case when there exists the follow ing relation

between the frequencies
*>2 = — + CA (1.1 )
4
where A is detuning. At first, we transform the equation (0 .1 ) into standard form
by means of the formulae
X = csinB, X — ứcose. (1-2)
2
The transformed equations are
— à = — z [ a x + a x 3 + (cx + ilr3) COSVt + R(x, i ) ] cose,
2 (1.3)
<2<i> = e [ a i -f + (cx + dx3) cosV/ + R{x, i ) ] sine,
2
Y
w here 4» = 8

— t and J, 4» are slowly varying functions of t.
2
The asymptotic method of nonlinear oscillations gives in the first approx
imation the follow ing equations
Y ả = — e ^su , Y) + Ị- ~ - + sin2 ♦ J ,
2
Y = z\ \ a + "f" ơc3 + Y) “ (■?" + ^ ĩ ) COs2*]'
(1.4 )
received by averaging the right hand sides of (1.3 ) over the time, where we
designate
2%
s{a, y ) = I cose . R (asine, đcose) dữ,
2x J 2
o
2%

H{at y ) = f sine .R Usine, acose) ie.
2 tc J 2
o
(1.5)
The steady state harmonic solution corresponding to a «=e 0 , ♦ = 0 are
(~A~ + ~ t ) sin 2 *• “ “ s ^ 0’
4 8 ( 1 .6 )
■+ ” ~ r ) cos2 * 0 = -J a 0 + Y a a l + H (a0’ V).
Eliminating 4>0 gives
VV(fl0, Y ) = 0 (1.7)
The subharmonic response given by equations (1.2) is obtainable only
when it is physically stable. W e study now the stability of stationary oscillations.
Let òa and be small perturbations and set a — a0 + òa, t = + Ò*. Sub
stituting these expressions into equations ( 1.4 ), neglecting powers of òa, 0<t> above
the first and also making use of the relationships (1.6 ) yields
= + ( l" + da°) sin2**] M + ( y + - J al ) a0cos2^ồ* ị.
^2 d~ d T = £ỉ[f + í + H ' - (i + 7 ^ 1 cos2*‘]ôfl +
+ ( y + ~ al ) a0sin2<t-0ỏ* ị.
The characteristic equation of this system is given by
2 2
[? ) < ' ^ 2 + ị v — G v S ) + ^ ỉ ) ( 2 ể + ^ > 4 ^ = 0 , (1 .9 )
\ z / 2 ÒÍZ0 3 2 ốứ c
where the follow ing notation is introduced & — —r, ^
Ci>*
The stability condition is given by the Routh-Hurwitz criterion that is
— ia jS ) > 0, (Ổ + (2Ổ + 9)flỉ) — > 0. (1.10)
àũ0 òaD
In the figures presented below the darkish areas correspond to the unstable regions
where the conditions ( 1 .1 0 ) violated and the undarkish ones — to the stable
regions. Sometime the unstable branches of resonant curves are shown dotted to

indicate that they are physically unobtainable.
A s will be seen later the nonlinearity of the system under consider
ation/coefficient a/strongly influences to the maximum of amplitudes of stationary
oscillations and their stability.
§ 2 THE INFLUENCE OF COULOMB FRICTION
Let us consider the Coulpmb friction of typ e,
where
1 if X > 0,
sig ni = ị — 1 if X < 0,
•0 if i 0.
R
(x,
x) = hg
sign* (2.1)
5
In this case we have
S(«. Y) - 1 *
0 for a = 0,
1 2 hD
and the equations (1.4) become
for a =£ 0 :
— à = — £ r — A + (2 c + <ia2) ’Sin24>l,
2 L r 8 J
X <2 = E r <2 + — aa3

— (c -f da2) cos2<l>l,
2 L 2 8 4 J
and for <2 = 0
and the equation w = 0 gives
e + ^ a l

( 2 .2 )
— a ■* — £ — (2c + sin2»i>, (2.3)
2 8
— a<i> = — ổ|"aH— — aa2

— (c + da2) cos2+ l.
2 2 L 4 2 J
The exoression (1.8) now takes form
16 9ỔÌ 4 (1— 02 + K ;) 2
Wia"Ỵ) = aH
2 ể + w +
Tẽ 7 ~
1
, _ - iggg- _ OB,
c( 2e+S>aĩÝ
„ = x , M Ì Ì a , ọ&c = cA„,
2 » 4 O)2 x « 2 i « 2 » 2
Figs. 1 — 3 are obtained by plotting equation (2.5) for the positive
p = + 0 . 1/the resonant curves in the case of negative 0 are received by mirror
reflection/. Figs. 1, 2 correspond to the negative Value of d. For the fig. 1 we
have 0 > 9 > > — 2(33/27^j20, namely, c2> = — 0 . 1 , (3 = 0 . 15, and ^ ^ o / s t r a i g h t
lines 1/, I = 10—4/curve 2/, — 6 ,2 5 .10'_4/curves 3/, *2^0 = 1 2 ,5 .1 0 4/cur-
ves 4/. The parameters for the fig.2 are ^ < — 2 ^ /2 7 Ọổo<0 : ^ = — 0 . 1 ,
Ổ => 0 .1 , and Ọổồ = 9 • 10 Vcurve 2/, — 2 5 .1 0 Vcurve 3/.
For the positive valué of d we have the resonant curves in fig .3 : ^ « = 0 . 1 ,
p = 0 .1 , Ổ = 0 .1 5 and = o/straight lines 1/, 'Pổ2, — 10- "Vcurve 2/, =
= 3 6 .1 0- 4 /curve 3/•
6
Fig.4 represents the resonant curves in the case d = 0 for the parameters
• = 0.1, Ổ = 0.15 and 9 ổ 0 = o/straight lines 1/, ọẻl — 6,25.10 4/curve 2/,

?'ẻl — 12,5 .10 Vcurve 3/, Ọ ổl = 25 . 10—4/curve 4/.
§3. TURBULENT FRICTION
i? (x , x ) =» /z2i 2signx (3.1)
t is easily to show that in this case
S(-a, y ) = - L h2Y2a2
Ổ X
H {a , Y) = 0 .
21 d therefore the averaging equations (1.4) take the form
i ‘“ ” - c [ s ĩ h ’ Y 'v + ( t “ + Ỉ *3) sin24 (3 -2)
X „* = c [ A „ + ! a 0 3 _ „ + i .
Now the am plitude <70 of stationary oscillation and the frequency y / o = — /
2 “
ĩre related by the equation
„ ^ 1
+ ^ ±A|S+^ i y
1
_ _ r |L _ (3'3)
There 9(S2 = th2. Such relationship for d < 0 are shown in fig.5/p = 0 .1 ,
3 7w
^ = — 0.15, ổ = 0 .1 5 /. F or ^ 2 = 0 we have tw o crossing straight lines 1.
Vith the small values of 9 — 10 2/ the resonant curve consists of three
banches 2. The first branch lies above straight line a 20 = — 2 Ổ /^ , the second
p e — between a\ — — C !^ i> and a2c = — 2 ổ / <5> and the thirdlow er straig ht line
4 = — Ổ /9). T he tw o last branches are tightened at the point a20 = — Ổ /9),
c
= 1 — p W ith the growth of 9^2 the second branch becomes low er and
lwer, but the first moves up. For sufficiently large values of ^ 2 the resonant
crves consist of two branches/see curves 3 for 5^2 = 0- !/• One of which is above
te straigh t line a20 — — 2 d ^ b and the other is low er a20 = —
The resonant curves for the case d > 0 are shown in fig. 6. If ^ 6 \l2 C >

r > 0 the resonant curve consists of two « oarabolic > branches/see curves 3/.
Vith the grow th of 9 S 2 these branches move aw ay. F o r ^ > Ọ ố ị/2 C > 0 the
1
resonant curve has form represented by branches 2. The parameters of the curves
in fig. 6 are ^ = 0.15, Ổ = 0.15, p = 0.1 and 5^2 — o/straight lines 1/,
~ 0 • 15/curves 2/, ^ > 2 = 0 . 22/curves 3/.
F o r com parison the resonant curves in the case d = 0 are given in fig. 7.
The o ther param eters a re : p = 0.1, ể = 0 .1 5 and ^ 2 “ o /straig ht lines 1/,
^ 2 “ 0 .0 5 /curv e 2/, Ọ6-, = 0 . l/curv e 3/, *= 0 . 15/curve 4/.
§ 4 . TURBULENT FRICTION TOGETHER WITH THE COULOMB ONE
In this section the nonlinear friction of form [2]
R{x, x ) = + A2x 2) s ig n i (4.1)
is investigated, where h0, h2 are the positive constants.
N ow the equations (1.4) become
à = — t \ — + ~ Y V + a ■+ a 3) sin2
2 L 71 3 JC ' 4 8 ' J (4 2)
£ |~ — <2 + — cxa3 — / — a + — a A cos2 t l ,
2 L 2 8 \ 4 4 I J
and the equation (1.7) takes form
4 ( 2 ^ + s v * * ) 2 4(1 - o2 + K 2)2 1 _ 0 (1
a l v e + ^ a y + (Ổ + 9 ^ ) 2
F o r d > 0 the resonant curves have form presented in fig.8/p = 0.1, c =
= ^ = 0 .1 5 /.' S traight lines 1 correspond = 5^2 = 0 and curves 2,3 corres
pond to 9Sp 4- 5^2 ^ 0 — 5 ^2 = 5.10 2/icurve 2/, s= 9 S 2 = 7 ,5 .1 0 2/
curve 3/.
If J < 0, then depending on the disposition of the curves
y = A(29ôc + 9ô2A )2, z *= A (2 Ổ + ^ b A )2 (4.4 )
the resonant curves have form s show n in fig.9. T he curve 2 corresponds to the
case when there exists only a point of intersection of the curves (4.4). 'Curve 3
and point 3 correspond to the points of intersection Aỵ, A 2, A3 of the curveổ

(4 ,4 ): Aỵ > — 2(?/c5>, A 2 = A 3 = — ổ / ^ . If the curves (4.4) have three sepa
rated points of intersection then the resonant curves have form ^ 4* in fig.9 if
■^1 > — 2 ổ / ^ , A2 < — d ^ b , A z < — and form c,5 >: if A 1 > — 2 ổ / t5>,
- 2 eỉ<2>> A 2 > - e / % a 3 < - e n .
T h e resonant curves in th e case d — 0 are represented in fig.10 for
p = 0 .2 5 , £ = 0 .1 6 , (% o = 2 .1 0 ~ 2 and 5 ^ 2 = 0 .1 6/p o in t 2/, 9 6 2 = • 10“ 2
*3
/curve 3/, 9^2,== “ TT • 1 O’*"2/curve 4/, ^ 2' “ 12 . 10~~2/curve 5/.
To compare with the linear friction in figs. 11, 12 and 13 the amplitude —
frequency responses in the system (0 .1 ) with linear friction R{x. i ) — hx are
plitted for the case d < o /fig -ll/, d > o/fig.12/and d — o /fig .l3/. The curves in
th;se figures are presented for the case p = 0 .1, Ổ = 0 . 15. The other parameters
fo fig .11 are 3 = — 0.15 and ỌÔ = o/curve ] / ^ = 0 . 03/curve 2/, = 0 . 21
/cirve 3/, &Ố = 0 . 3/curve 4/. For fig.12 we have *5) = 0 .15 and = o/curve 1/,
Ọt = 0 . 2/curve 2/, ■=» 0 . 3/curve 3/, ^ = 0 45/curve 4/and for fig .13 : d = 0,
= o/curve 1/, = 0 . 27/curve 2/, ^ 7 = 0 . 28/curve 3/, = 0 . 297/curve 4/,
w ls re 5 S — 4 e/j/ o>.
REFERENCES
1— Osinski z. Comparison of damping of Oscillations by different kinds of frict
ions. V International Conf. Nonlinear Oscillations, Kiev 1969.
2— Bulgakov B.w. Oscillations, (in Russian)% Moscow, 1954.
3— Minorsky N. Nonlinear Oscillations. D. Van Nostrand, 1962.
4— Kaudcrer H. Nichtlinear Mechanik. Berlin, 1958.
5 — Schmidt G. Parametererregte Schwingungen- Berlin, 1975.
6 — Nguyen Van Dao. Parametric Oscillations of mcchariical systems with regard
for the incomplete elasticity of material.
Proceedings of Hanoi Polytechnical Institute 7/1975.
7 — Bogoliubov N.N. and Mitropolski Yu. A. Asymptotic methods in the theory of
nonlinear oscillations, Moscow, 1963.
Reọu le IS Décembre 1977

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