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

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 e

H A N O I 1 9 9 3

P ro c e e d in g s o f th e N C S T o f Vietnam, Vol 5, No 2 (1993) (3 -2 0 )

M e c h a n i c s

S O M E P R O B L E M S O F N O N - L I N E A R O S C I L L A T I O N S

I N S Y S T E M S W I T H L A R G E S T A T I C D E F L E C T I O N

O F E L A S T I C E L E M E N T S

Ng u y e n Va n Dao

Institute of Appl i ed Mechanics, Hochiminh City

S u m m a r y 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 - l i n e a r o s c illa t io n s in s y s t e m s w it h la r g e s t a t i c d e fle c t io n o f e la s t ic e le m e n ts

h a v e b e e n e x a m in e d in [ l ] T h e s p e c ific ity o f th e s e s y s t e m s is: T h e ir h a r d n e s s e s s e n tia lly

d e p e n d s o n b o t h t h e p a r a m e te r s o f th e e la s t ic e le m e n t a n d it s s t a t i c d e fle c tio n T h is

f e a tu r e le a d s t o t h e c h a n g e o f t h e a m p lit u d e c u r v e a n d t h e s t a b ilit y o n it

In t h is w o r k s o m e r e la te d p r o b le m s w ill b e in v e s tig a te d : T h e s y s t e m w it h lim ite d

p o w e r s u p p ly a n d t h e in t e r a c t io n o f n o n - lin e a r o s c illa tio n s

I - N O N L IN E A R OSCILLATIONS OF THE SYSTEM W ITH LARGE

STATIC DEFLECTION OF THE ELASTIC ELEMENTS

A ND LIMITED POW ER SUPPLY

In t h i s p a r t t h e n o n - lin e a r o s c illa t io n s o f a m a c h in e w it h r o t a t in g u n b a la n c e an d

la r g e s t a t i c d e f le c t io n o f t h e n o n - lin e a r sp r in g a n d lim it e d p o w e r s u p p ly are c o n s id e r e d

T h e e q u a t io n s o f m o t i o n o f t h e s y s t e m u n d e r c o n s id e r a t io n are d iffe r e n t w it h th o s e o f

c la s s ic a l p r o b le m [2] b y t h e a p p e a r a n c e o f th e n o n - lin e a r te r m s w it h d iffe r e n t d e g r e e s o f

s m a lln e s s T h is fe a tu r e le a d s to th e d e p e n d e n c e o f th e h a r d n e ss o f th e s y s t e m n o t o n ly on

th e p a r a m e t e r s o f th e e la s tic e le m e n t b u t a lso on it s s t a t i c d e fle c tio n

T h e r e s u lt s o b ta in e d are d iffe r e n t in b o th q u a lity a n d q u a n tity w it h th o s e o b ta in e d by

K o n o n e n k o V o [2].

1 E q u a t i o n s o f m o t i o n

F ig u r e 1 illu s tr a te s a m a ch in e w ith a p air o f c o u n te r r o ta tin g r o to r s o f e q u a l u n b a la n c e (s o t h a t h o r iz o n ta l c o m p o n e n ts o f th e c e n tr ifu g a l fo rce v e c to r s c a n c e l) , is o la te d fro m th e flo o r b y n o n - lin e a r sp r in g s an d d a s h p o ts w ith d a m p in g c o e ffic ie n t h 0.

1 V 0 SL

X

Fig 1

T h e s p r in g s s u p p o r tin g th e m a ss are a ssu m e d to be n e g lig ib le in m a s s w ith a n o n ­ lin e a r c h a r a c t e r is t ic fu n c tio n :

/ ( u ) = cou + 0 ou3; (L l-1 )

w h e r e cn is a p o s it iv e c o n s t a n t , /3 0 is e ith e r p o s it iv e (h a r d c h a r a c t e r is t ic ) or n e g a t iv e (s o ft

c h a r a c t e r is t ic ) T h e d e fo r m a tio n o f th e sp r in g in th e s t a t i c e q u ilib r iu m p o s it io n is A , a n d

t h e s p r in g fo r c e c(, A ■+■ i (lA 3 is eq u a l to th e g r a v it a t io n a l fo rce m 0g a c tin g o n th e m a ss:

C0 A + £ 0 A 3 = m 0 g,

w h e r e m„ = m l + m is d e fin e d as th e su m o f th e m a in m a s s m i a n d th e r o t a t in g u n b a la n c e

m a s s e s m , t h a t is th e t o t a l m a s s s u p p o r te d by th e s p r in g s T h e d is p la c e m e n t X is m e a su r e d

fr o m t h e s t a t i c e q u ilib r iu m p o s itio n w ith I c h o sen t o b e p o s it iv e in th e u p w a rd d ir e c tio n

A ll q u a n t it ie s - fo r c e , v e lo c ity , an d a c c e le r a tio n - are a lso p o s itiv e in th e u p w a rd d ir e c tio n

T h e s y s t e m u n d e r c o n s id e r a tio n h a s tw o d e g r e e s o f fr e e d o m a n d th e g e n e r a liz e d

c o o r d in a t e s X a n d i p c o m p le t e ly d efin e it s p o s itio n

T h e k in e t ic e n e r g y o f th e s y s t e m u n d er c o n s id e r a tio n is

T = - m X 4 - — V

2

m )

I m = X - h r COS <p, Z m = r s i n V?.

H e n c e

SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC 5

T = - m 0 i 2 — m r x i psin <p H— /<p2, / = m r2 (1.1.3)

For t h e p o t e n t ia l en e r g y , th e r efer en ce can b e ch o sen a t th e le v el o f th e s t a t i c e q u i­ lib r iu m p o s itio n :

Ư = — (A — x ) 2 + — (A — x ) 4 *f rnt)gx 4- m g rCOS (p. (1-1-4)

T h e L a s g r a n g e ’s e q u a t io n s g iv e

I<p = m r x sin (p + m g r sin <pt

m0 X + c0 X + /?013 - f 3/30 A 2x — 3/Ju A x 2 = m r^ sin + mri p 2 COS V?.

T a kin g into a c c o u n t th e driving m o m en t L(<p) and the frictions H(<p), k 0x we have

th e fo llo w in g e q u a t io n s o f m o tio n :

I ỷ = L(yi>) — H[<p) -+- m r x s i n <p + m^r sin V?,

T71()X + c0i - f / i 0i + /?ni 3 4- 3/90 A 2x — 3)ổn A x 2 = mr^> sin Ip -f m r< p2 COS (p.

S u p p o s in g t h a t A is rather large and X is enough sm a ll, so t h a t £tlx 3 is a sm all

q u a n tity o f second d egree (s2), w h ile 0 it A x 2 is o f first degree (5), w h ere £ is a sm a ll p o sitiv e

p a r a m e te r O b v io u s ly , in th is c a se # , A 2X is fin ite

It is a s s u m e d a ls o t h a t — 1, ^ 1 T h e fr ic tio n fo r c e s, th e fo r c e s YTiTtf? COS <p,

m r < p s i n y ? a n d t h e m o m e n t s m r xs i n <p f m g r s i m p a r e s u p p o s e d t o b e s m a l l q u a n t i e s o f e 2

T h u s , w e h a v e t h e fo llo w in g e q u a tio n s o f m o tio n :

w h e r e

<p = e 2 [ A / 1 ( ý > ) + <7 ( 1 + 9) s i n s ơ ] ,

ĩ + w2 ĩ = í7I 2 + £2 [ p ỷ sin ip + p<p 2 COS ip - h i — Ị3x2)

f p = — - , e h = t c7 = -I S — c p = zf" >

^ , e 2Mi(yỉ>) = ị [£((£>) - # (¥ ? )], e 2? =

mr

T

(1.1.8)

T h e e q u a t io n s (1 1 7 ) are d iffe r e n t w it h th o s e in K o n o n e n k o V o w o rk [2] b y th e

a p p e a r a n c e o f th e q u a d r a tic te r m 5 7X2 a n d b y th e d e g r e e s o f s m a lln e s s o f t h e te r m s T h e s e

e q u a t io n s c h a r a c te r is e t h e s y s t e m s w ith w ea k e x c it a t io n an d la rg e s t a t i c d e f le c tio n

2 S o lu t io n

W e lim it o u r s e lv e s by c o n s id e r in g th e m o t io n in t h e r e s o n a n c e r e g io n , w h e T e th e

f r e q u e n c y u o f t h e free o s c illa t io n is n ear t o th e fr e q u e n c y n = <p o f t h e fo r c e d o s c illa t io n s

W e s h a ll fin d t h e s o lu t io n o f e q u a tio n s (1 1 7 ) in th e se r ie s [3]

X = a cos(<p + 0) + e ui (a, \ịỉ, <p) + £2U2(a , t/>, <p) + £3

w h e r e u t ( a , d o n o t c o n t a i n t h e f i r s t h a r m o n i c s COs \ p i s i n t/>, \ị) = <p + Ỡ a n d a r e p e r i o d i c

f u n c t i o n s o f a n d (p w i t h p e r i o d 27T, a n d a , 6 a r e f u n c t i o n s s a t i s f y i n g t h e e q u a t i o n s

à = £w4i(a, Ớ) + e 2w42 (a ,0 ) + .

# = UJ — n -f £#1 (a, 9) 4" e 2B 2 (a, Ỡ) 4"

dip

(1.2.2)

n =

dt

T h e f i r s t e q u a t i o n o f ( 1 1 7 ) i s t h e n

á n

= e 2 [ M i ( n ) + q ( x + g) sin <p] (1.2.3)

T o d e t e r m i n e t h e u n k n o w n f u n c t i o n s Ay, B x, u t w e d i f f e r e n t i a t e t h e e x p r e s s i o n ( 1 2 1 )

a n d s u b s t i t u t e i t i n t o ( 1 1 7 ) W e h a v e :

X = —CLUJ s i nr p + e< — a B 1 sin + Ảỵ COS 0 H- w — + n —— >-h

+ £ I - a i?2 sin t/> -f i42 cos + i4i ~ - L + n +

X = — auj COS^ 4- (cư — n ) — - — 2 a a > i? i] cos \ịỉ — f(u» — n ) a + 2cư j4i

^ 2 u i 2 ^ U1 1 + 2 w n - ^ - + w ^ - } +

sin

+ n 20<9^>2 + 2w n 5 V'3 ^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+

+ { A x ^ Ệ ± + 5 1 -Q 0 - - a S ỉ ) C0S - (2A1 jBi + a A l-g-^- + a 5 l ^ i') *i n V»+

+ ( w - n ) cMi <3u _ L Q2 — ^ r>2^2u2 2 + 2cjQ — — + u r — — d 2U, 2 ^

2u-<90 da d<p 2 diọdrịì d\Ị ) 2 + e (1.2.4)

S u b s t i t u t i n g t h e e x p r e s s i o n s ( 1 2 4 ) i n t o t h e e q u a t i o n (1*1.7) a n d c o m p a r i n g t h e c o ­

e f f i c i e n t s o f e a n d e 2 w e o b t a i n :

+

/ d d \2 2

\ n d i + ” i k ) u ‘ + w u

-•+* Ị(cư — n ) ~~Ệ q ’ ~~ 2 a u B i j COS rp — Ị(cư — n j a - ^ 1- - f 2cư A i s in xp = 7a2 COS2 t/>,

/ a a \ 2 ,

( n a 5 + “ a v ; ) “ ’ + " “ >+

+ — n ) — 2a w B 2| cos t/> — (w — sin ip —

+ 2 7 0Uj COS v> — /3a3 c o s31/1 + sin Ip + p f i2 COS ip,

(1.2.5)

(1.2.6)

w h e r e R( 0, Bi ) = R ( A h 0) = 0.

C o m p a r in g t h e coefficients o f the harm onics in (1.2.5) we have

(w - “ 2au ) B 1 - °>

(u; — n ) a — ^ 4- 2 a ;A1 = 0,

SOME PROBLEMS OF NON-LINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC

/ _ ổ Ổ \ 2 2 2 _ 2 /.

( n - — + U1 U1 = cos V*

<9yp dip

S o l v i n g th e s e e q u a tio n s yield s

X i = 0 , B i = 0, U i = ^ 7 ( 1 - j c o s 2 V > ) ( 1 2 7 )

C o m p a r in g th e coefficients o f th e first harm on ics sinự>, cosV> in th e eq u a tio n (1.2.6) Wfc have

(w - n ) - 2 o w £ 2 = g 2 - f - - + p H 2 COS Ớ,

(w - n ) a ^ “ + 2a>i42 = —hauj — p H 2 s i n Ỡ.

0 6

F r o m th e s e e q u a t io n s o n e o b ta in s :

ha pCi 2

A 2 = —— -—— sin 6 )

2 u + n

B 2 = — [ Ẹ ~~ n ) a “ 7 - * r ~ c o s Ớ.

4w \ 2 3 CƯ / (w + n)fl

T h e e q u a t io n for d e t e r m in a t io n u3 is

r ~ k +UJ- k ) u ’ + w u’ = - ( ò + ĩ ) a c o s H

-( 1 2 8 )

H ence

T h u s , in t h e re so n a n ce zon e n « u; we have the follow in g e q u a tio n s in th e second

a p p r o x im a t i o n

W'here n , a and 0 s a tis fy t h e e q u a tio n s

d ii e2 [ w 1 2 • Ả

— = - — 77[U)tia+ DM sin £7

2u>0v

— = “ We - w - -= - COS

dy9 n V 2a»a « i ) ,

w here

w' = a, + ỉ r a2’ “ = 4 ^ " S £ - (L2-12)

T h e s ta tio n a r y so lu tio n o f th e equations (1.2.11) is d eterm in ed from th e relations

A /i(n ) + -<7tc>2a s in 0 = 0,

_ 2 P ^ 2

w e — n - e —— COS 6 = 0

2 u a

E lim in a tin g th e p hase 6 from the last two eq u a tio ns o f (1.2.13) we o b ta in

w here

W (a2, n ) = u 2a2 \eAh2 -f- 4(u;e - n ) 2] - e Ap 2n 4 (1.2.IS)

In the re so n an ce zone n « U/, th e equation (1.2.14) gives a p p r o x im a tely

n 2 = u>2 + e 2 cxa 2 ± e 2 y j -h 2 (jj 2 (1.2.16)

From t h is re la tio n it follows t h a t the n o n -lin ea r oscilla tio n has:

- a h a rd c h a r a c te r is tic ( F ig 2 a ) if Q > 0 or if

- a s o f t c h a r a c te r is tic ( F ig 2 b ) if

- a lin e a r c h a r a c te r is t ic ( F ig 2c) if

E lim in a t in g 6 fr o m th e fir st tw o e q u a tio n s o f (1 2 1 3 ) w e h a v e

L (n ) - 5 (H ) = 0, (1.2.20)

w h ere

T h e e q u a tio n (1.2.14) is s im ila r to th a t in the s y s t e m w i t h ideal p o w er s u p p ly [1] T h e

d i f f e r e n c e is t h a t n s h o u l d b e s a t i s f i e d t h e r e l a t i o n (1 2 2 0 ) w h i c h c a n b e s o l v e d g r a p h i c a l l y

S O M E P R O B L E M S OF N O N - L I N E A R OSCILLATION S IN SY S T E M S W IT H L A R G E STA TIC

as s h o w n in Fig.

Fig 3

3 S t a b ilit y o f s t a t io n a r y o s c illa tio n s

E q u a t i o n s (1.2.13) can give s o m e s ta tio n a r y values n = O 0 , CL = a0, 6= 0 o To stu d y

th e s t a b i li t y o f th e s e v a lu e s w e in tro d u c e a p ertu r b a tio n o f n , a and 0:

ỐH = Q — n 0) ổ a = a — a of Ỏ6 = 6 — 0 Q.

We d e n o te the right-hand sides of th e eq u a tio n s (1.2.11) by $ n (n, a, 0), $12(Ấ, a, 6 ) J a, 0) resp ectiv ely B elo w , the derivatives will be calcu lated at th e sta tio n a ry values o f n , a and

6 w h ic h s a tis fy the relations (1.2.13) We have the following variational eq u atio n s:

^ - = bl l 6ĩì + bl l 6a + bl i Se>

^ = fc„5n + f c „ 5 a + 6 , 3 * 0 , (1.3.1)

dip

— - = b ^ ỏ ĩ ì + b ^ ỏ a + ay?

w h ere

- d - ề r - w - - £ ( i ( n ) - f f ( n ) ]

3 < & n / i „ w 3 < 3 $ 1 1 m n u ; 3 2

6 „ = - £ ( * « - n ) a , i 4l = £ ± i i = J j ( n - 2 w e) ,

<90 2m on

-^ 1 3 1 a / \ o

a r ' t M 1- " - 1 - "

T h e ch a ra c ter istic e q u a tio n o f th e s y s t e m (1.3.1) is

A3 + Dị X2 -f Đ2^ Dz == 0»

w h ere

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*

T h e R o u t h - H u r w i t z ’s criteriu m o f stability is

D i > 0, D3 > 0 D 1 D 2 — ^ 3 ^ 0 (1.3.3)

We have

A s u sually, it is s u p p o s e d t h a t -^ -L (n ) is n eg a tiv e and — i / ( n ) is p o s it iv e , so t h a t TV is

n e g a tiv e H en ce, D i is a lw a y s p o sitiv e.

T h e s e c o n d s ta b ilit y c o n d it io n D 3 > 0 as shown by K o n o n e n k o [2] is th e m o s t im p o r­

t a n t on e T h i s c o n d it io n is eq u iv a len t to th e 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 h e r e (ft, a ( n ) , 0 ( n ) ) an d a ( n ) , 6 {n ) are fo u n d fr o m th e la s t tw o e q u a t io n s o f (1.2.13); n a m e ly

* h = £ f [ m - s (fi)}

It is easy to verify t h a t

2 d W

^77 "" ^33^33 ~ ơ £ fl2 ’

w here w is o f th e form (1.2.15) and Ơ2 is a p o s itiv e c o n s ta n t N ow, th e s ta b ilit y c o n d itio n (1 3 5 ) can be r e p r e s e n te d in th e fo rm

i [ i ( n ) - S(n)] < 0 (1.3.6)

d W

It is n o te d th a t — - > 0 is th e sta b ility c o n d ition of sta tio n a r y o scilla tio n w h e n n is a

d a2

given c o n s ta n t is p o s it iv e on the h ea v y branches o f the resonan t curve T h e sign of

d a 2

t h e d eriv a tiv e

G = J y [ L ( n - s ( n ) j (1.3.7)

can b e o b ta in e d by co n sid e rin g th e relative p o sitio n s o f th e graphs L {n ) and 5 ( 0 )

For th e ca se o f th e s y s t e m w ith a hard ch aracteristic (F ig 3) it is clear t h a t G is

d W

n e g a tiv e a t p o in ts R ị , R 2 and R 3i so th a t the p o in ts R ị and i?3, w h ere -— r is p o s itiv e ,

da

co rresp o n d to th e s ta b ility o f s ta tio n a ry o scillations T h e p oin t R 2 co r r e s p o n d s to the

in s t a b ility of s ta tio n a r y o scilla tio n s, w here — z is n egative.

pa*

In c o m p a r is o n w ith a s y s t e m w ith an ideal energy source [l], th e u n s t a b le branch of

th e re so n a n ce cu rv e rem a in s the sam e B u t th e ju m p p h e n o m en o n o cc u r s in a different

m an er A s n is increa sed th e a m p litu d e of oscillation will follow th e solid arrows and the

j u m p in t h e a m p lit u d e w ill ta k e p la c e fr o m p t o Q. W ith a d e c r e a se in fr e q u e n c y n th e

a m p litu d e will fo llo w th e d a shed arrows and th e ju m p w ill be from T t o u T h e p o in ts

o f c o lla p s e p a n d T are t h e p o in t s o f c o n ta c t o f th e c h a r a c te r is tic L (n ) a n d t h e f u n c t io n s

Sin)-For th e ca se o f th e s y s t e m having a soft ch a ra cteristic (Fig 4 ), th e p art o f th e reso­

n a n c e cu rv e in d ic a te d by th e dashed (h ea v y ) line co r resp o n d s t o the in s ta b ility (s ta b ility )

o f s t a t io n a r y o s c illa t io n s , p r o v id e d th e fr e q u e n c y n is a g iv e n c o n s t a n t O n t h is p a r t

~ 2 < ( ^ 2 > ° j - SiZn ° f t ^ie d e r iv a tiv e G (1 3 7 ) d e p e n d s o n t h e s lo p e o f th e

ch a ra c ter is tic , i e on th e q u a n tity — L (n ) It is necessa ry t o d istin g u ish tw o cases:

afi

1) w h e n t h e c h a r a c t e r is t ic is s t e e p , i e -jprL(Cl) h a s a la r g e a b s o lu t e v a lu e ( F i g 4 )

ai l

2) w h e n t h e c h a r a c t e r is t ic is g e n tly s lo p in g , i.e h a s a s m a ll a b s o lu t e v a lu e

(F ig 5).

In t h e first c a s e t h e d e r iv a tiv e G (1 3 7 ) w ill be n e g a t iv e o n t h e p a r t s P U , P T a n d

T Q ( F i g 4 ) T h e r e fo r e , t h e s t a b ilit y c o n d it io n ( 3 6 ) w ill n o t b e s a tis f ie d o n P T , w h e r e

Tp-J < 0, b u t it w ill b e s a tisfied on P U and Q T; w h ere > 0.