Quasilinear oscillations in systems with large static deflections

However, we will concentrate... T h e ivy dashed lines in this figure correspond to the stability instability oscillations... In the resonance zone n RJU, the equation 51 gives approxim

P r o c e e d in g s o f th e In ter n a tio n a l C on flu en ce on A p p lied D y n a m ic s H a n oi, 2 0 -2 5 /1 1 /1 9 9 5

QƯASELINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC DEFLECTIONS

N g u y e n V a n D a o

V i e t n a m N at i o n a l Uni ver sity, H a n o i

A b str a c t

In m e c h a n i c a l s y s t e m s the s tati c dcflcction o f the clastic e l emen ts is usual not

a pp ea r ed in the equations of m o t i o n The reason is that either a li near mod el of the clast ic e l emen ts or their too s m a l l s t a ti c dcflccti on a s s u m p t i o n was acccptcd

In the p r e s e n t paper both nonlinear model of clastic e l e me n t s and their large s tati c

d c f i c c t i o n arc con si der ed , 30 that the n o n l i n e a r t e r m s i n the equ at i on o f m o t i o n

a p p e a r with different degrees o f s m al l n es s In this case the nonlincarxty o f the s y s ­

t e m d e p e n d s not only on the nonlinear ch ar act er i st i c o f the clastic e l e m e n t but on its s t a t i c i t f l c c t i o n The distinguishing f eature o f the s y s t e m under co n si de r at i on

is t ha t i f the clastic e l e m e n t had soft c h a r ac t er i s t i c, the nonlinear s y s t e m also be- longs to the soft OTI C If the clastic e l e m e n t has hard char acter ist ic, the s y s t e m

m a y be ei t her soft or hard or neutral type, d ep en di ng on the relationship between the p a r a m e t e r s of the clastic e l e me n t a nd i t s st at i c deflection.

The a u t o n o m o u s and n o n - a u t o n o m o u s s y s t e m have been s t u d i e d A n a l y t i c a l m e t h ­ ods in c om bi na ti on with Computer have b t t n used.

The p r ob le m of n on li near oscil lati ons o f clastic s tr uct ur es with large s tat i c d e ­

f l e c t i o n in general, and beams, plates in particul ar, m a y be studied in a s i m i l a r

s t a t i c e q u il ib r iu m p o s it io n T h Ì3 p o s it io n is c h o sen aa th e referen ce p o s itio n

W h e n 1 = 0 , t h e sp r in g force C0A + £ 0A 3 is eq u al to th e g r a v it a t io n a l force

w h e r e e is a s m a l l p o s i t i v e p a r a m e t e r In th is ca se Ax2 is fin ite.

T a k i n g in t o a c c o u n t t h e v is c o u s d a m p in g force h0x and e x c it in g force P [ t , z)

w h i c h are b o t h a s s u m e d to be s m a ll q u a n t it ie s o f sec o n d d eg ree and in tro ­

th e in flu e n c e o f th e forces on th e m o t io n of th e m a s s M can be fo u n d in th e

s e c o n d a p p r o x i m a t i o n o f th e s o lu t io n In the p r e se n t p ap er a m o re gen eral

e q u a t i o n w ill be in v e s t i g a t e d

X + u i 2 i = e - 7 12 + c2F{r, < p ( t ) , x , i ) , (4)

ìe re r is a s lo w t i m e T = et, F(r, <p[r), X, x) is t h e p e r io d ic f u n c t io n r e la t iv e ly

ip w i t h p e r io d 2 rr w h i c h ca n be r ep resen ted in t h e fo r m

NF[r,<pl x,i) = ^ 2

l a t i v e l y t o r for all fin ite values o f r W e w ill b e s p e c ia l ly in t e r e s t e d in

e s t u d y o f t h e r e s o n a n c e zon e w h e n w is n ear t o - u, w h e r e p a n d q are

here U i ( a , 0) are p e r io d ic fu n c t io n s o f 6 w i t h p e r io d 2ir w h ic h d o n o t c o n t a in

e first harmonics sinớ, C08Ổ and a, 9 satisfy the equations:

C o m p a r in g th e first h a rm o n ics sinớ and COS $ in the second e q u a tio n of (8)

w h e r e ( / ) is a v e r a g e d valu ed o n t im e o f th e fu n c tio n / We consider now

g u i s h i n g fe a tu r e o f th e s y s t e m w i t h large s t a t i c d e fle c tio n T h e param eter

a d e p e n d s o n t h e p a r a m e t e r s c0 , /?„ (sp rin g ) and A ( s t a t i c deflection)

T h e c o n s id e r e d D u ffin g e q u a t io n is m o d e le d [3] on th e c o m p u t e r for a con­

(13)

a n d e r P o 1 /

mparing the coefficient of c and c2 we obtain:

2 / \ d 2 t i ỵ d 2 U i n d 2 U i 2

= ^a2 cos2 0 — — -y (r )) — 2awBi j COS 9

+ Ị(w - ~u{r) ) a~Q~p' + 2w-Ai] «in

\jialogou sly, we can find A i, B i and u2 from the equation (18) for the

general form of the function F ( r t ip, x , x ) However, we will concentrate

1 = + 0.1 ( F i g 5 ) , - - 0.1 ( F i g 6 ) w i t h t h e in itia l valu es: t = 0 ,

e s t a t i o n a r y a m p l i t u d e s c o r r e s p o n d in g t o th e c o n s t a n t v a lu e s o f t h e fre-

ỉn c y V are p r e s e n t e d in th e F i g 7 for t h e v a lu e s m e n t io n e d a b o v e o f / , and — + 0.1 (c u r v e 1), a = 0 (cu r v e 2 ) , = - 0.1 (c u r v e 3 ) T h e

ivy (dashed) lines in this figure correspond to the stability (instability) oscillations.

m p a r i n g t h e F i g s 5, 6 a n d F ig 7 it is see n t h a t in c r e a s in g th e v e lo c i t y o f ssin g t h r o u g h th e r e s o n a n c e , th e m a x i m u m o f th e a m p l i t u d e d e c r e a s e and

s p e a k a p p e a r a fter t h e r e s o n a n c e p ea k T h e m a x i m u m o f th e a m p l i t u d e s

s t a t i o n a r y o s c i l l a t i o n s is b ig g e s t.

Fi g 5

Fig 6

I

Fig 7

Fig 8

Fig 9

\.RT n

L this part two following problems have been examined:

I T h e n o n - l i n e a r o s c il la t io n s o f e l e c t r o m e c h a n i c a l s y s t e m s w i t h lim it e d Dwer s u p p l y a n d la r g e s t a t i c d e fle c tio n o f th e e la s t ic e le m e n t s

ie non-linear terms w ith different degrees of smallness in the equations of

ap 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 io n o f th e s y s t e m un der c o n sid -

ra tio n are d iffe re n t w i t h th o se o f cla ssic a l p r o b le m [5] by th e a p p e a r a n c e o f

l e n o n - lin e a r te r m s w i t h different d e g r ee s o f s m a lln e s s T h is fe a tu r e le a d s

D t h e d e p e n d e n c e o f t h e h a r d n e s s o f th e s y s t e m n o o n ly o n th e p a r a m e t e r s

f t h e e l a s t i c e l e m e n t b u t also o n its s t a t i c d e fle c tio n

'he r e s u l t s o b t a i n e d are differen t in b o t h q u a lity a n d q u a n t it y w i t h t h o s e

b t a i n e d b y K o n o n e n k o V o [5]

E q u a tio n s of m o tio n

ig 10 ill u s t r a t e s a m a c h in e w i t h a pair o f c o u n t e r r o t a t in g ro to r s o f eq u a l

n b a l a n c e (so t h a t h o r iz o n t a l c o m p o n e n t s o f t h e c e n tr ifu g a l force v e c t o r s

w h e r e Ui[a,rp,ip) d o n o t c o n t a in th e first h a r m o n ic s cosrp, s i nrp, = <p + 0

a n d are p e r i o d i c f u n c t io n s o f rp an d <p w i t h p e r io d 2ir, a n d a, 6 are f u n c t io n s

T o d e t e r m i n e th e u n k n o w n fu n c tio n s Ai , Bi, U,- we d iffe re n tia te th e e x p r e s ­

X = — CLOJ COS0 + tf I Ị(u/ — n ) — 2a w B ij COS t/i — Ị(u — n)

) m p a r in g the coefficients of the h a rm o n ic s in (42) we have

In the resonance zone n RJU, the equation (51) gives approximately

6

30 t h a t /V is n e g a t iv e H ence, D i is alw ays p o s itiv e

T h e second s ta b ility condition Ữ3 > 0 as shown by Kononenko ’5 ’ is the most im p o rta n t one T h is condition is equivalent to the inequality

(61

(62)

where = $ u (n, a(n), Ớ(Q)) and o(n), Ớ(D) are found from the last two

eq uation s of (50); nam ely

where w is of the form (52) and Ơ2 is a positive constant Now, the sta b ility

c o n d itio n (63) can be represented in the form

(64)

in d P T a n d n e g a t i v e on Q T T h e re fo r e th e c o n d it io n (6 4 ) is s a tis fie d on

3 T P and is n o t s a tisfie d on P U ( F ig 13).

Fig 12

II w e a k i n t e r a c t i o n b e t w e e n t h e s e l f - e x c i t e d a n d p a r a m e t r i c o s c i l ­

l a t i o n s IN THE SYSTEM WITH LARGE STATIC DEFLECTION

T h e p r e se n t s e c t i o n d ea ls w it h s o m e r ela ted p r o b le m s w h e n tw o m echa­nism s e x c i t i n g th e s e lf - s u s t a in e d o s c illa tio n and p a r a m e tr ic one coexist in one system Following the assum ptions in [41, these oscillations are w eak

T h e y a p p e a r o n ly in th e s e c o n d a p p r o x im a t io n of th e s o lu tio n and th eir

its sta tic deflection Nam ely, when the in it ia l system has a hard c h a r a c t e r ­

is tic , th e r e s o n a n c e c u r v e may b e lo n g to a s o ft ty p e T h e r e fo r e , th e r esu lts

o b t a i n e d are d ifferen t w ith th e c la ss ic a l o n e s b o t h in q u a lity and q u a n tity

E q u ation o f m o tio n and approxim ate solu tion

ing t h e a s s u m p t i o n s a n d n o t a tio n s in t h e p r e v io u s p a p er [4 ] w e s t u d y : o s c il la t io n s d e s c r i b e d by the eq u a tio n :

ie parametric and self-excited oscillations have a comm on feature that

i o r ig in X = X = 0 is u n s t a b le U n d er t h e r e s o n a n c e c o n d i t i o n t h e s e

:illators may have a certain interaction

ie solution of the equation (66) is found in the form

I = a COS 6 + cuxỊa, <p, 6) + c2u2(a, <p, Ớ) + e3 , (67)

lere 9 — — + Tp, <p = 1/t and u, are p e r io d ic fu n c tio n s o f <p an d 8 w it h p e r io d

<p = v t , R [ 0 , f ? i ) = i ỉ ( i 4 i , 0 ) = 0 ,

F { t p , X , i ) = — [/? x 3 + D ( ơ x 2 — l ) i — c x c o s v=»], ( 7 1 )

a cos 6, —CLU1 sin 6) = F[tp, I , i )

C O * 9

C o m p a rin g the coefficients of the first h arm o nics sin 0 and COS 9 in (70) yields

C o m p a r i n g the coefficients of the harmonics in (69) we have

E q u a tio n ^ -(76) have a triv ia l solution a = 0 T h e non t r iv ia l sta tio n a ry

a m p litu d e a 0 and phase \Ịi are determ ined by equations ^ = 0, — = 0 or

( ƠCL2 \

2 ụ j j j D y l -— J = c s i n2</>0,

4 ụ u (^1 - + — a ' J = í c c o s 2 t/>0 ,

sre /i 5= — E lim in a t in g the phase Tp0 gives

u;2 = 0.01 (curve l ) and a = 0 (curve 2)

a b ility of the sta tio n a ry solution obtained is investigated by u sing the

n a tio n a l eq uatio n s D eno tin g the right hand sides of the equations (76)

p and Q , resp ectively, we have

lere the su b sc rip t “o” m eans th at the d erivative s are calcu la te d for sta -

in a ry valu e s a 0, rp0. T h e sta b ility co n d itio ns are

snce, we have

the figure 14 the h eavy (dashed) b ran ch corresponds to the s t a b ilit y

n s ta b ility ) of sta tio n a ry solution where the in e q u alitie s (80) are satisfied

To stu d y the s ta b ility of the zero so lutio n a = 0 we introduce in (76) n e w

varia b le s y, z connected w ith a and rp by the relation s:

where n o n -w ritte n term s contain y and z w ith high degrees

T h e o rig in y = z = 0 (a = 0) of this system is alw ays unstable, because the c h a ra c te ristic equation of the lin e a r term s of (82) has the roots w ith

p o sitiv e real p art

F ig l ị

III w e a k i n t e r a c t i o n o f s e l f - e x c i t e d o s c i l l a t i o n w i t h f o r c e d o n e in

N O N L IN E A R SYSTEM S WITH LARGE STATIC DEFLECTION

In th is section the attention is concentrated on stu d yin g the co n d itio ns under w h ich the resonance regimes of o scillatio n s occur, on e x p la in in g the role of n o n -lin e a r factors in the form ation of resonance situ a tio n s of the system s w ith large sta tic deflection of elastic elem ents [4] T h e d is tin g u is h ­ing feature of these system s is thut their n o n -lin e a rity essentially depends not o n ly on the ch a ra cte ristic of t ie elastic clem ent but also on the static deflection

le ch a ra c te r m entioned has sig n ifican t influence on the in te ra ctio n of self­cited o scilla tio n s w ith the forced ones, in both q u a n tity and q u a lity.

E q u a t i o n o f m o t i o n and c o n s t r u c t i o n o f a p p r o x i m a t e s o l u t i o n s

t us consider some specificities of forced o scilla tio n in the self-excited stem whose m otion is supported by the “n egative” frictio n T h e follow ing

u a tio n w ill be investigated:

le re the notatio ns in [4] are utilize d u , 7, /9, V, ơ, D , E are constants,

> 0 , *7/9 > 0 , V > 0 , u > 0 , a > 0 a n d t > 0 is a s m a l l p o s i t i v e p a r a m e t e r ,

h e n E = 0 equation (83) d e s c r i b e s a self-excited s y s t e m W h e n D = 0 we ive a forced system w ith harm o nic e xcitatio n In this paper it is assum ed

at D E Ỷ 0- T h e question is stated as follows: w hat happens in the

s t e m ( 8 3 ) w h e n t w o m e c h a n i s m s o f g e n e r a t i o n o f s e l f - e x c i t e d o s c i l l a t i o n

id forced one co e x ist? We w ill be sp e cia lly interested in the sta tio n a ry

d ila t io n s and their sta b ility

sing the a sym p to tic method [1] we find the ap proxim ate so lutio ns of the

u a tio n (83) in the form :

le re u, are p erio d ic functions of ip and 8 w ith period 2ir w hich do not

n ta in the first h arm o n ics sinổ, COS 8, <p = v t, Ỡ = ut + rp, and a, 0 are iterm ined from the equations:

be in te n siv e in te ra ctio n of m echanism s of generation of the o scilla tio n s

d ifferent nature can be observed in the resonance situ a tio n s In this

in n ectio n , below the resonance case w ill be considered, supposing th at u

near to u

y d iffe re n tia tin g the expression (84) and su b stitu tin g the results in to (83)

id by co m p a rin g the coefficients of t and e2 we have: