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

2]: the Coulom b friction, the turbulent one and their com bination.. T h e subharm onic response giv en by equations 1 .2 is obtainable only when it is physically stable... resonant cu

CỘNG HÒA XÃ H ộ i- CHỦ NGHĨA VIỆT NAVI

VIỆN K H O A H Ọ C VI ỆT NAM

ACTA MATHEMATICA

VIETNAMICA

T O M III

N ° 2

H À N Ộ I — 1 9 7 8

ACTA MATHEMATICA VIETNAMlCA

T O M 3, N ° 2 (1978)

P a r a m e tr ic oscillations of dynam ical sy stem s with cubic term at the m odu lation depth under the

influence of n o n lin ea r frictions.

N G U Y Ễ N VẰN Đ Ạ O National Institute o f Sciences S R V

T h is paper deals w ith the influence of nonlinear frictions to the para­ metric oscillations of dynam ical system s described by the equation w ith the cubic term at the m odulation depth

X + » 2X + e ( c x + i x 3)c o s V i + c c tx* + z R { x , i ) — 0, (0 1 )

w here 01, c, d, a are constants, z is a sm all p ositive param eter, R {x, x) is a non­ linear function of X , X characterized the frictions considered Three form s o f non­ linear frictions w ill be investigated h e r e t1 2]: the Coulom b friction, the turbulent one and their com bination.

A s w ill be seen later in the analysis, the sign and value o f parameter d

sharply chan ge the m otion picture and the stable regions.

It m ust be em phasized that the equation ( o l ) describes the real physical

system s m ore precisely than the one in w hich d «= o t 4' 5 61 T h e system o f type

( o l ) w ith linear friction w as studied qualitatively by M inorsk y[3] but no attem pt has y et been m ad e to investigate it w ith the Coulomb friction, turbulent one and their com bination.

§ 1 S T A T IO N A R Y O S C IL L A T IO N S A N D T H E IR S T A B IL IT Y

L et us consider the resonant case w hen there exists the fo llo w in g relation

b etw een the frequencies

*>2 = — + CA ( 1 1 )

4

where A is detuning A t first, w e transform the equation ( 0 1 ) into standard form

by m eans of the formulae

2

T he transform ed equations are

— à = — z [ a x + a x 3 + ( cx + i l r 3) COSVt + R ( x , i ) ] cose,

<2<i> = e [ a i -f + (c x + d x 3) cosV / + R { x , i ) ] sine,

2

Y

w h ere 4» = 8 - — t and J , 4» are slow ly v a ry in g fu n ctio n s of t.

2

T h e asym ptotic method of nonlinear oscillations giv es in the first approx­

im ation the fo llo w in g equations

Y ả = — e ^ s u , 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 tim e, w here we designate

2%

s{a, y) = I cose R (a s in e , đ co se) dữ,

o

2%

H { a t y ) = f sine R Usine, acose) ie

o

(1 5 )

T h e steady state harmonic solution corresponding to a «=e 0, ♦ = 0 are

( ~ A ~ + ~ t ) s i n 2 *• “ “ s ^ 0’

■+ ” ~ r ) cos2 * 0 = - J a 0 + Y a a l + H ( a 0’ V ).

E lim inating 4>0 gives

T h e subharm onic response giv en by equations (1 2 ) is obtainable only when it is physically stable W e study now the stability o f stationary oscillations

Let òa and be sm all perturbations and set a — a0 + òa, t = + Ò* Sub­

stituting these expressions into equations ( 1.4 ), neglecting pow ers o f òa, 0<t> above

the first and also m aking use of the relationships (1 6 ) yields

= + ( l " + d a °) sin2**] M + ( y + - J al ) a0cos2^ồ* ị

+ ( y + ~ al ) a 0sin2<t-0ỏ* ị.

The characteristic equation of this system is given by

[ ? ) < ' ^ 2 + ị v — G v S ) + ^ ỉ ) ( 2 ể + ^ > 4 ^ = 0 , ( 1 9 )

where the follo w in g notation is introduced & — —r , ^

Ci>*

The stability condition is given by the Routh-Hurwitz criterion that is

— ia jS ) > 0 , ( Ổ + ( 2 Ổ + 9 ) f l ỉ ) — > 0 ( 1 1 0 )

In the figu res presented below the darkish areas correspond to the unstable regions

w here the conditions ( 1.1 0 ) violated and the undarkish ones — to the stable regions Som etim e the unstable branches o f resonant curves are show n dotted to indicate that they are physically unobtainable.

A s w ill be seen later the nonlinearity o f the system under consider­ ation/coefficient a /stro n g ly influences to the m axim um of am plitudes of stationary oscillations and their stability.

§ 2 T H E INFLUENCE OF COULOMB FRICTION Let us consider the Coulpmb friction of t y p e ,

w here

1 i f X > 0 ,

s i g n i = ị — 1 i f X < 0,

•0 if i 0.

5

In this case w e h av e

S ( « Y) - 1 *

0 fo r a = 0,

1 2 hD

and th e eq u atio n s (1 4 ) becom e

for a =£ 0 :

— à = — £ r — A + (2c + <ia2) ’Sin24>l,

X <2 = E r <2 + — a a 3 - — (c -f d a 2) cos2<l>l,

and for <2 = 0

and th e e q u a tio n w = 0 gives

e + ^ a l

(2.2)

— a<i> = — ổ | " a H — — a a 2 -— (c + d a 2) c o s 2 + l.

The exoression (1 8 ) now takes form

16 9ỔÌ 4 ( 1 — 02 + K ; ) 2

Wia" Ỵ) = aH 2 ể + w + Tẽ 7 ~ 1

c ( 2 e+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/th e resonant curves in the case o f n egative 0 are received by mirror

reflection/ F ig s 1, 2 correspond to the n ega tive Value o f d F or the fig 1 w e

have 0 > 9 > > — 2(33/ 2 7 ^ j 2 0, nam ely, 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/ T h e param eters for the fig 2 are ^ < — 2 ^ / 2 7 Ọ ổ o < 0 : ^ = — 0 1 ,

Ổ => 0 1 , and Ọổồ = 9 • 10 Vcurve 2/, — 2 5 1 0 Vcurve 3/.

For the positive valué o f d w e have the resonant curves in f i g 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

F ig.4 re p resen ts the re s o n a n t curves in the case d = 0 for the p a ra m e te rs

• = 0 1 , Ổ = 0 1 5 and 9 ổ 0 = o/straight lines 1/, ọ ẻ l — 6 , 2 5 1 0 4/curve 2/,

?'ẻl — 12,5 1 0 V curve 3 /, Ọ ổ l = 25 10—4/curve 4 /.

§ 3 TU RBU LENT FRICTION

t is easily to sh ow th a t in th is 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

N ow the a m p litu d e <70 of s ta tio n a ry o scillation an d the freq u en cy y / o = — /

2“ ĩre related by the equation

There 9(S2 = t h 2 Such re la tio n sh ip fo r d < 0 a re sh o w n in fig 5 /p = 0 1 ,

3 7w

^ = — 0 1 5 , ổ = 0 1 5 / F o r ^ 2 = 0 w e h ave tw o cro ssin g s tra ig h t lin e s 1

V ith the sm all v alu es o f 9 — 10 2/ th e re so n a n t cu rv e consists o f th re e

banches 2 T h e first branch lies above straight line a 2 0 = — 2 Ổ / ^ , the second

p e — betw een a \ — — C!^i> and a2 c = — 2 ổ / <5> a n d th e th ird lo w e r s tra ig h t line

4 = — Ổ /9 ) T h e tw o last b ran ch es a re tig h te n e d a t th e po int a 2 0 = — Ổ /9 ),

c

= 1 — p W ith th e g ro w th o f 9 ^ 2 th e second b ra n c h becom es lo w e r and

lw er, but the first m oves up For sufficiently large values of ^ 2 the resonant

c r v e s consist o f tw o branches/see curves 3 for 5^ 2 = 0 - !/• One of w hich is above

t e s tra ig h t line a 2 0 — — 2 d ^ b an d th e o th e r is lo w e r a 2 0 = —

T h e re so n a n t c u rv e s fo r th e case d > 0 a re sh o w n in fig 6 I f ^ 6 \l2 C >

r > 0 the re s o n a n t cu rv e consists o f tw o « o arab o lic > b ra n c h e s/se e c u rv es 3 /

Vith th e g ro w th o f 9 S 2 these b ran ch es m ove aw ay F o r ^ > Ọ ố ị/2 C > 0 the

1

resonant curve has form represented by branches 2 T he param eters of the curves

in fig 6 are ^ = 0 1 5 , Ổ = 0 1 5 , p = 0 1 and 5^ 2 — o/straight lines 1/,

~ 0 • 15/curves 2/, ^ > 2 = 0 22/curves 3/

F o r co m p ariso n th e re so n a n t cu rv es in th e case d = 0 a re g iv e n in fig 7

T h e o th e r p a ra m e te rs a r e : p = 0 1 , ể = 0 1 5 an d ^ 2 “ o /s tr a ig h t lines 1/,

^ 2 “ 0 0 5 /c u rv e 2/, Ọ6-, = 0 l/c u r v e 3/, *= 0 1 5 /c u rv e 4/

§ 4 T U R B U L E N T FRICTION TO GETH ER W ITH T H E COULOMB ONE

In this section the nonlinear friction of form [2]

is in v e stig a te d , w h ere h0, h2 a re th e positive con stants.

N o w the equations (1 4 ) become

à = — t \ — + ~ Y V + a ■+ a 3) sin2

£ |~ — <2 + — cxa3 — / — a + — a A c o s 2 t l ,

and the equation (1 7 ) takes form

4 ( 2 ^ + s v * * ) 2 4 (1 - o2 + K 2)2 1 _ 0 (1

F o r d > 0 th e re so n a n t cu rv es h av e fo rm p re se n te d in fig 8 /p = 0 1 , c =

= ^ = 0 1 5 / ' S tra ig h t lines 1 co rresp o n d = 5 ^ 2 = 0 an d c u rv e s 2,3 c o rre s ­ pond to 9S p 4- 5 ^ 2 ^ 0 — 5 ^ 2 = 5 1 0 2/icurve 2 /, s= 9 S 2 = 7 , 5 1 0 2/

cu rve 3/.

If J < 0, th en d ep en d in g on th e disp osition o f th e c u rv es

y = A ( 2 9 ô c + 9 ô 2A ) 2 , z *= A ( 2 Ổ + ^ b A )2 (4 4 )

th e re s o n a n t c u rv e s h a v e fo rm s sh o w n in fig.9 T h e c u rv e 2 co rresp o n d s to th e

case w hen 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, A 3 o f the curveổ ( 4 ,4 ) : Aỵ > — 2 (? /c5>, A 2 = A 3 = — ổ / ^ If the curves (4 4 ) have three sepa­

rated points o f intersection then the resonant curves have form ^ 4 * in fig 9 if

■^1 > — 2 ổ / ^ , A 2 < — 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 re s o n a n t cu rv es in th e case d — 0 a re re p re s e n te d in fig 1 0 fo r

p = 0 2 5 , £ = 0 1 6 , (% o = 2 1 0 ~ 2 an d 5 ^ 2 = 0 1 6 /p o in t 2 /, 9 6 2 = • 10“ 2

*3

/c u rv e 3 /, 9 ^2,= = “ TT • 1 O’*"2/c u rv e 4/, ^ 2' “ 12 10~~2/c u rv e 5/

T o compare w ith the linear friction in figs 11, 12 and 13 the am plitude —

frequency responses in the system (0 1 ) w ith linear friction R { x i ) — hx are

plitted for the case d < o / f i g - l l / , d > o/fig.12/and d — o / f i g l 3 / T h e curves in

th;se figures are presented for the case p = 0 1, Ổ = 0 15 T he other parameters

fo f ig 11 are 3 = — 0 1 5 and ỌÔ = o/curve ] / ^ = 0 03/curve 2 /, = 0 21

/cirve 3/, &Ố = 0 3/curve 4/ For fig 12 we have *5) = 0 1 5 and = o/curve 1/,

Ọt = 0 2/curve 2/, ■=» 0 3/curve 3 /, ^ = 0 45/curve 4/an d for f ig 13 : d = 0,

= o/curve 1/, = 0 27/curve 2/, ^ 7 = 0 28/curve 3 /, = 0 297/curve 4/,

w ls r e 5 S — 4e/j/o>

R E F E R E N C E S

1— Osinski z Comparison of damping of Oscillations by different kinds of f r i c t ­ ions V International Conf Nonlinear Oscillations, Kiev 1969.

2— Bulgakov B w Oscillations, (in Russian)% Moscow, 1954.

3 — Minorsky N N onlinear Oscillations D Van Nostrand, 1962.

4— Kaudcrer H Nichtlinear Mechanik Berlin, 1958.

5 — Schmidt G P a r a m e te r e r r e g te Schwingungen- Berlin, 1975.

6 — N guyen Van Dao P a r a m e t r ic O scillations o f mcchariical sy ste m s w i t h r e g a r d

f o r the incomplete elasticity o f material.

Proceedings of Hanoi Polytechnical Institute 7/1 975

7 — Bogoliubov N N and Mitropolski Yu A A s y m p to tic methods in the th eory o f

nonlinear oscillations, Moscow, 1963.

Reọu le I S D écem bre 1 9 7 7

Fig 1

Fig 2

Fig■ 4

Fig 5

F ig 6

Fig 7

Fig 8

Fig 9

F«g- 10

F ig 12

Fig Í S