Interaction between the elements with different degrees of smallness in nonlinear oscillating systems

These elem ents have no effect on illation in the first approximation, but thev interact one with another in the second approximation.. The asym ptotic method o f nonlinear ICS [1] is u

p r o c e e d i n g s o f Ĩi-IL S i x i t i N A TION A L Co n f e r e n c e Oin M ECH AIM CS

Hanoi 3 - 5 December 1997

IN T E R A C T IO N B E T W E E N TH E ELEM EN TS

W IT H D IFFE R E N T D EG R EES OF SM ALLNESS

IN N O N L IN E A R O SC ILLA TIN G SYSTEM S.

Nguyen Van Dao Vietnam National University, Hanoi

I n t r o d u c t io n

In dynam ic svstem s there exist those elem ents characterizing friction, elasticity and excitations which

■vith different degrees o f sm allness in the differential equation o f motion These elem ents have no effect on illation in the first approximation, but thev interact one with another in the second approximation The ation o f the interaction between these elem ents is o f great interest The asym ptotic method o f nonlinear

ICS [1] is used to determine the equations for amplitudes and phases o f oscillation These equations are solved ligital computer The follow ing excitations and interactions will be considered :

ra tic n o n i i n e a r i t y a n d f o r c e d e x c ita tio n ,

netn c and se lf - excitations,

n etn c and forced excitations.

teraction between the elem ents characterizing the quadratic nonlinearity orced excitation.

the dots indicate differentiation with respect to time, oc, q , h and /? are constants, T = St and £ is a small

;ionless parameter characterizing the sm allness o f the terms behind it The parameter £ is introduced ally and used as a book - keeping d evice and w ill be set equal to unity in the final solution The quadratic term

: due to curvature or and asym m etric material nonlineanty The function ( p { ỳ ) is supposed to be a form

Let us consider a nonlinear system governed bv the differential equation

( 1 1)

(1.3)

■2 ( r ) x = í Ị c r t2 + q c o s2ộ9( r ) - £ 2 ( - A x + 2 h x + f i x 2,) (1.4)

A solution o f this equation is sought by using the asymptotic method o f nonlinear oscillation [1]

ỈCOSỚ+ £U, ( a , [ f/ ,9) + £ l u ^ ( a ỉ ự , 0 ) + £ 3 ,ớ = ạ> + ụ/,

a , ụ / , Ờ) are periodic functions with p en od 2 n with respect to both variables Ụ/ and 6 and do not

e first harm onics sin 9, COSỠ T he functions A ị ( a , ụ / ) , B ị ( a, ự / ) are periodic w ith respect to the variable : functions w ill be determ ined in the process o f approxim ation calculations.

ubstitutmg the exp ression s (1.5 ) into the equation (1.4 ) and com paring the co efficien ts o f £ w e obtain

4, Sin 6 - 2 a v ( r ) B ] COSỚ4- v 2 ( t )

tg the harm onics in (1.6) g iv es :

= 0 .

f d \

0 62

+ u = a a ] COS2 B + q c o s 2 ( p ( r ) ( 1 6 )

- -(a a 2 - i - 2 q c o s 2 y / ) c o s 2 6 - r — s in 2 w s in 2 6

( 1 7 )

( 1.8)

2 •

Comparing the coefficients of £ in (1.4) we

get-A , s i n < 9 - 2a v ( r ) B : C O SỚ + v : ( r )

ÔG

( 1 9 )

I COSỔ + A a c o s ớ T l h a v s m O - ( k r COS3 6,I

ves

a q

l A ~ 2 h v { z ) a - —-<3s i n2^ ,

3 v - ( r )

( 5 c : 3 "i , a a

T ) B Z = Atf + — — - — /? I < 3 - -a c o s 2 ự /

So, in the second approximation we have

: o s ớ + £ — — — ( c a r 4 -2 q c o s l ụ / ^ c o s l ỡ - — S i n 2 v / S i n 2 ớ

a ?

6v ( r )

a s i n 2^ a — — = -d w £ z A a + £'

d t 2 v

(1.10)

— — a J -f -acos2ỉ// ,

v ( r ) 6v ( r )

( 1 12)

(1 1 3 )

Stationary Oscillation :

Supposing that v ( r ) = CÙ = c o n s t and considering the staiionarv oscillation with constant am plitude a ase ụ/ w e have :

n 2 y / = h a > , - J- c o s 2 \ j / = -y a , a ^ Q (1 1 4 )

laiing the phase Ụ/ w e g e t :

- - a V

’" ) = 36

-y a

\ *

r h Z(D

From equation (1 1 5 ) it fo llo w s th a t:

£ A = Cù - 1 ~ 2 ( ũ ) - 1 ) , ỵ = — f i — — ( 1 1 6 )

a CÚ - 1 ±

Va : <736 / / : &r

(1.17)

69

peiidcnce o f the amplitude a on the external

ic y CO u presented in figure 1 for the

:ters: e'~aq = 0,063 , s : h = 0,01 , £ zy = 0,08.

The stability of nontrivial stationary solutions (a*0)

equatioa (1 1 2 ) when CO is constant can be studied by

the corresponding variational equations, which lead to

adiiion [1]

Fig 1

jse function w (1.16) is positive outside and negative inside the resonance curve, Lhe stable branch o f the lance curve ìis the upper branch, which corresponds to the upper sign before the radical in (1.17) Thus, sen tiie rwo forms o f oscillations corresponding to definite values o f CO , the form with large amplitude is

e and the forai with small amplitude is unstable.

Follow ing chapter 4 o f [1], the trivial solution a=0 o f the equation (1.12) is stable if the value CỦ does not

1 that iDtervad o f the axis CO , from which the resonance curve is rising In figure 1 the stable branches are

m by h ea w lin e s, while the unstable ones are shown by dotted lines.

The passage of the system under consideration through resonance when v { r ) is not a constant, but Lges bv the l&w : v ( r ) = 1/ 0 T e ^ l, can be exam ined by integration o f the differential equations (1.12) The

meters lire chosen as /0 = 0, Qr) — 0.009 , [ị/ 0 = 0, £ ~ h — 0.001 , S~Y = 0.01 , s~ CCCỊ = —0,024 ,

= 1, Li = 1 O' 5 (curve 1, Fig 2); / i = 2 1 0~5 (curve 2, Fig 2) ; / i = — 10 3 (curve 1, Fig.3) :

= - 2 1 0' 5 (cnirve 2, Fig.3)

From th e expression (1.12) and (1.13) one can see that the quadratic noniineanty ( a )is always to eaize the system under consideration regardless of the sign o f a . Moreover, two elem ents characterizing

dratic*' noniim earitv CDC2 and forced ex cita tio n q C O s2< p (r) c o m b in e together and act ju st lik e a parametric

n a tio n with an intensity ccq

H e system o f equations (1.12) has a trivial solution a=0, which corresponds to a pure forced oscillation ỉer the actioni o f an external excitation e q COS 2cp :

= - E — ZOSl Cp

action between the elem ents characterizing the quadratic nonlinearity ram etric excitation.

1 this paragraph the follow ing equation

= s{C0fc2 + pxcoscot) + £ 2 ( ầ x - 2 h x - p x 1) ,

id w h ere CÙ is the excitation freq u en cy

(2 1)

( 2.2)

y o f freq u en cies, 1 is a natural frequency o f the system under consideration, Ơ , P * h , p are constants,

h e solu tion o f the equation (2.1) is also found in the form (1.5) with 0 = Củĩ + Ụ/ By a similar

1 as in tlhe paragraph 1 we obtain :

= 0.

— { a a 2 + p a COS y / ) -^— ( a a 2 + p a COS ụ/) cos2ớ - - ^ 7 sin ự/', sin 2 ớ

n the setcond approximation w e have

S/9 + - - - 7 - 3( a a 2 + /7acosy/) - a a ' c o s2G - p a c o s { 2 6 - lự)

61CÙ'

? /ỉứ -f - —r p a n sin \ự + -L—r a s m 2 w ,

(2.3)

(2.4)

(2.5)

-£ • A a + ——P ' - a - y a H -D r p a a c o s ^ -r 2 — —a c o s l ỉ ư P ' n ,„

8 a ;'

Vl2á> 1 2&)

5 a ;:

8<y

1 2ứ/)3 ■

rhe nonitrivial stationary solution of equations (2.5 ) is determined from

Củ + 2 * 5 ớ s i n s i n 2 ^ = 0 .

+ Ỏ S a c o s ụ / + /? c o s l y / = 0

- — s = — DCC R = — a ^ O

ilently

r + / , c c o s ^ = 0

ve

:.6)

/ , cos y - / , sin ự = 0

:osy/+-Ị £ ( / ? - z + 2^a2) + 4 5 :ứ 2 sin ự/ = — ,

Z - 2 ; m 2 ) - 1 2 5 : a 2 c o s ^ / + I h c ủ R s m ụ / = 2 S a ( Z - 2 ỵ a : ) - 4 S a i ? (2.7)

.inaúơn o f y / from equations (2 7 ) and the further discussion w ill be th esa m e as in the paragraph 4.

teraction between the elem ents characterizing the param etric and xcita tions with different degrees of sm allness.

The noinlinear system under consideration is

ủ ) 0 , p , Ỗ ) 0 %D ) 0 , J 3 are constants, £ 2A = CD" — 1 and 1 is a natural frequency The term characterizing metric ex c ita tio n is o f the first order o f sm allness, w hile the terms characterizing the self-excu ed excitation

f the seccond order o f sm allness The structure o f the equation (3.1) shows that in the first approximation [1] lents orn the right o f (3.1) do not affect the oscillation In the second approximation the terms mentioned and nevv nonlinear phenom ena will occur.

71

le solution o f the equation (3 1 ) w ill be found in the form (1 5 ), where 9 = (út 4- \ị/ It is easy to find lation for determining A ị, Bị and Ui :

n < 9 - 2acủB, COSỚ + ỔT

d o

w, = p a COS 6 cos((9 - (ự).

ị the harmonics sin Ỡ,COS G in (3.2) g ives :

0 ,

7- COS lị/ — ^ — Tcos(2ớ — lị/).

6Cú

(3.2)

(3.3) (3.4)

Tie equation for A2, Ei and u2 is :

, Ơ U-,

,\n ớ - l a c ủ B - c o s ỡ + CỬ

0 22COS' 6 ) a c ủ sin <9.

= p u, c o s ( ớ - Ụ/) + A a COS 0 - /3 a ' COS3 < 9

-the coefficients of s i n ớ and COSỚ in (3 5) y ield s

— I -Daco(] - — az) + -^-^rSiĩ\2ụ/ , aB1 = -

0)-1

2<y

6<U: J a

Thus, in the second approximation w e have

O S Ớ + £■ — — r C O S ụ / — -c o s (2 ớ ~ ụ /) .

2ứJ' ' L ^ J

and \ị/ satisfy the equations :

•> r

, 6 - Củt + ụ/ ,

(3.5)

rCO s2 u/

4ú)

(3.6)

(3.7)

— ~ Dữ Cù{\ — — ữ ) -r —— sin 2^/ , a - — = - — (A + - ^ ~ ) ú f - — a ;-c o s 2 ^

(3.8) Equations (3.8) have a trivial solution a=0 The noa-trivial (a^O) stationary am plitude a 0 and phase \ị/

m ined from the equations :

i 2 ụ / ữ = D d ) \ \ - - - a 0: l J L - c o s 2 ụ / 0 = - A + — ^ ~ + i f a 2 (3.9)

:nnined from the equations

_ n i l * 2

in2ự/0 = a0 |,

id n g the phase gives

, A ) = 0 ,

,A) = , p r - T ^ o 2 + D 'ứ > £ 2

1 - —a

(3.10)

, £ 2 A = a r - 1 (3.11)

From the equations (3.10 ), (3 1 1 ) w e obtain approxim ately :

\

-(3.12)

a IS pioueu III ngure •+ lOf aic paiaiiiciciò

6 10 ”3 , £ ' D = 10~3 ,Ổ = 40 — s l f5 — 0.01 (curve 1) and Ị5 — 0 (curve 2)

4

snoring the right hand sides o f the equations (3.8) by R and Q, respectively, we have

- r D ổ a ° ’ l i r j

£

= H - a

(3.13)

s are

= £ - a 0 DỊ 1 - ^ - ư 0 j

F i^.4

<0

i r ' \ ô \ ự ) \ c u J z = ^ T ưf)2củ~ r> 0 .

(3.14)

(3.15Ì

0 2,

ỉn >- 0 , the stability conditions take the form

A o

^ ( a 0; ,A )

(3.16)

ổX2f

gure 4 the heavy branch corresponds to the stability o f stationary solution where the inequalities (3.16) are L

It is easy to prove that the trivial solution 11=0 o f the equations (3.8 ) is unstable.

metric excitations.

(4.10

Let us consider a nonlinear system described by the differential equation

\T = s [ q cos(2Cùi + z ) + /7*cosứ)f] + £ : (Ax - I h x - /3xJ ),

Ú)2 - I ,

1 is natural frequency The terms containing q and p correspond to forced and parametric excitations, iveiv It is easv to prove that in [he case under consideration

p a 1

" ' ’ Ũ " * * - ứ

<7 c o s (2 ỵ / - x ) + — zos>y/ l c o s 2 ớ

3ứ)

~ \ q ^ ( 2 \ f / - ỵ ) + £ - s m ụ / s in 2ớ ,

and in the second approximation one has

p a

X = o. COS 6 -r e \ -t— rr C OS ự / —

g c o s (2 ụ / - ỵ ) ~ — c o s ụ / ícos2ớ

(4.3)

3 Cú

p a .

q sm(2ợ/ - x ) + ^— s \ n y j

with a and ụ/ determined by the equations

d a _ e 2

d i 2 cư

— = -1 2 / 2 a & > - L i — S i n 2 ụ / - - ^ s i r n y / - £ ) L

L

-Ì

Bv puttmg A = - - -

we have the follow ing equations for stationary solutions :

f 0 = I c o h a + R a sin 2 w + £ sin(ụ/ - j ) ,

g c = (A ~ — ) a - — y3a3 + /to c o s l ụ / -r E c o s ( ự / - x ) ,

(4.4)

(4.5)

(4.6)

o r c q u i v a le n tlv

/ 0 cosy/ - g 0 sin ^ = 0, / 0 sin ^ + g 0 cost// = 0.

From here we obtain :

lc o h a s i n y / - l p a ' - - ( A - £ - + R )

a c o s y / + E C O S7 = 0,

1 ^ - - ( a + p L - R )

a s i n [ị/ + l c ú h a C O S ụ/ - E s i n X = 0

(4.7)

( 4 8 )

T h e conditions for reality of sin If/ and C OS ụr are :

> E 2 cos: X ,

4 ũ ) 2h 2 + l p a ' - - ( A + ^ - + R )

1 'ì

Ẩ.

>

(4.9)

4 ũJzh ị / 3 a 2 - ( & + £ - - R ) > E l s in : ỵ

w, we consider two cases The first case : S vstem w ithout friction (h — 0) In this case we have

3 a z - ( A + ^ - + R )

6 a c o s ụ/ = E c o s ỵ ,

-»

- y f t r - ( À + a sin ụ/ - E sin

X-— p a 1 - (A -r —— h i?) ^ 0 and

- / f o r - ( À + — - / ? ) ^ 0

elim inating the phase ụ/ from (4.10) w e obtain the equation o f die resonance curve c

f i T , d 2 ) = 0 ,

re

- / i r - ( A + — + /? ) ! V - P a 1 - ( A + £ - - / ? )

7

f — Ị3a'- - (A -1- —— R ) = 0 , i.e if we have the resonance curve C; :

Q we have :

) a C OS lị/ = E c o s ỵ , 2 R a sin y / - E s in

i therefore COS y = 0 = 0 y = — , — ; sm 7 = ± 1 , w = ± a r c s i n — —

f £ V , " - < 1 => ư > -£ 2

4 /? 2

ư

(3a~ - ( A -Ỉ -/?) = 0 , i.e if we have the resonance curve C 3

Ì n 2 _ 2

- B a - C O

-4

ỉn we have

d therefore

4 £ :

(4.10)

4 1 1 )

4.121

(4.13)

( 4 1 4 )

(4.15)

( 4 1ÒÌ

751

r, 71 37Ĩ _ n ủ7í

so, ư 7 ^ 0, — , 7 1 , — : the curves c^, C3 do not exist, l í X = — , — , then be

E 2

curve C] there is still sem i-straight line c, in the plane ( a : , CÙ2 ) ,a 2 > - — If X = 0 ,7Z, then beside t

4Zc

Cl there is s till sem i-straight line C3 in the plane ( a 2, CD2) , a : >

4R 2 '

The second case : System with friction (h *0) W e go back to the equations (4.8) and denote

D = 4 ( D 2h 2 + - / 5 a : - ( A + — ) - R 2 ,

D,=E

-C O S ^ f

sin 7

- — B a 2 + A + — + R

2 Củh

D ;=E

2 ứ)/?

a) If D ^ 0 w e have

- C O S £

Sin J

a sin t// = —r - acosu/ = —f

1 D }~ + D

-a = ■

Z)2

(4.17)

For the case ^ = 0 we obtain the follow ing equation for the amplitude (a) and frequency ( CỦ

A c ù 2h z + A + E l - I p a ' - R

r-77 2 2

f £ - a R 2

> W - ( A + 4 >

4 ( D 2h'~

-b) I f D = 0 ; w e have — Ba ~ — Cl)2 — 1 H ± * J R 2 — A c o J h 2

(4.18)

and sin ự /, c o s ^ ex ist only when D } = = 0 , or equivalently

Z)1 COS^ - Z)2 sin 7 = 0, Z)1 sin x + D n COS7 = 0

From here w e obtain :

co = — — s in 2 r , — B a 2 = CO.2 - 1-i - i? c o s 2 r

(4.19)

By the form ulae (4.9 ), the amplitude is restricted as :

E 2

a 2 >

4 R

(420)

7 6 '

e interaction between the elem ents characterizing quadratic noniinearity and forced, parametric and seif- itations has been studied The nonlinear system s under consideration belong to special types, where the rces have no effect on the first approximation T heừ action and interaction appear only in the second ion The amplitude and phase o f oscillation are determined by using the asym ptotic method o f nonlinear

[1] and digital computer

le role o f each elem ent shown by the equations o f the second approximation van es from system to system , lie, In the equations (1.12) the terms characterizing quadratic nonlinearity ( a ) and forced excitation qual sm allness (£■) always appeared as the product o f a and q This means that each elem ent { a , q )

lone has no e ffec t on the system under consideration and these elem ents have equal role The equations that the quadratic noniinearity (a ) gives effect only with the presence o f parametric excitation (p), while

ĩe c t s the system even without the quadratic noniinearity The equations (3.8) show that however the ation (D ) is sm aller than parametric excitation, they have an equal effect in the second approximation and eparatelv act on the system For the equation (4.5) the terms characterizing forced (q) and parametric itations are not in equality The effect o f forced excitation exists only with the presence o f parametric , w hile the e ffec t o f parametric excitation w ill exist even with the absence o f forced one Some related

can be found in publications [ 2-6] listed below.

w ledgm ents

Fhe author is srateful to Dr Tran Thi Kim Chi for numerical calculations on the digital computer This work iCially supported by the Council for natural Sciences o f Vietnam.

References

x>iskii Y u A , N guyen Van Dao Applied Asym ptotic methods in nonlinear O scillations Kluwer Academic shers, 1997.

tney L Ù , N ayfeh A.H The response o f a singie-degree-of-freedom system with quadratic and Cubic non- rides to a fundamental parametric resonance J o f sound and vibration (1988) 120 (1), 63-93

en Van Dinh Interaction between parametric and forced oscillations in fundamental resonance J of lanic*' N C N ST o f Vietnam, TOM 17, N°3, 1995.

en Van Dinh Non-linearities in a quasi-linear system subjected to external and parametric excitations of rent orders J o f M echanics, NCNST o f Vietnam , TOM 18, N° u 1996.

en Van Dirih, Tran Kim Chi Fundamental resonance in one generalized system o f Vanderpol type J of

hanics, NCNST of Vietnam, TOM 18, N° 3,1996.

Kim C hit N g u y en Van Dinh On the interaction between forced and parametric oscillations in a system with

d egrees o f freedom J o f M echanics, NCNST o f Vietnam , TOM 19, N° 1, 1997.

T Ư Ơ N G TÁC GIỮA NHỮNG PHAN TỊỬ CÓ BẬC BÉ KHÁC NHAU

TRONG CÁC HỆ DAO Đ Ộ NG PHI TUYEN.

N g u y ê n V ăn Đ ạ o

Đ ạ i h ọ c Q u ố c g i a H à N ộ i

Trong c á c hê động lực có những phần tử đặc trưng cho ma sát, tính đàn hổi và kích động Chúng xuát hiên

c bé khác nhau trong phương trình vi phân của chuyén động Những phần từ này tuy không có ảnh hường dến

>ng trong xáp xi thứ nhất, song chúng tác động qua lại vói nhau trong xáp xi thứ hai và tạo nèn những hiệu ứng /ến m ới.

Phương pháp tiêm cận của cơ hoc phi tuyến được sử dụng dô lập phương trình xác định bièn đô và pha dao sau dó các phương trình này được giải trèn máy tính Những kích đông và tương tác sau đáy đã dươc khảo sát.

1 Phi tuyến bâc hai và kích động cưởng bức.

2 Phi tuyến bậc hai và kích động thông só.

3 Kích động thông số và tự chấn.

4 K ích đ ộ n g thông số và cưởng bức.

77