The first equation of 35 gives the phase displacement.. We consider now the stability of stationary solution a , and determined by formulae 33.
Trang 1HỘI Cơ HỌC VIỆT NAM ■ VIỆN KHOA HỌC VIỆT NAM« i « I I I
BÒ GIÁO DỤC VÀ ĐÀO TẠŨ • BỘ KHOA HỌC CÒNG NGHỆ VÀ MỒI TRƯỜNG• I I I I I
TUYỂN TẬP
CÔNG TRÌNH KHOA HOC
P R O C E E D I N G S O F T H E F I F T H
N A T I O N A L C O N F E R E N C E O N M E C H A N I C S
( H à N ộ i 3 - 5 / X I I / 1 9 9 2 )
C O HOC DAI CƯƠNG V À ỨNG DUNG
G E N E RA L A N D A P P L I E D M E C H AN I C S
Trang 2P roceed in g s of th e F ifth National Conference on M ech anics, Vol I (38 - 45), 1993
N O N L I N E A R V I B R A T I O N O F A P E N D U L U M
W I T H A S U P P O R T IN H A R M O N I C M O T I O N
D a o đ ộ n g p h i t u y ế n của con lắc có đ i ề m treo di đ ộ n g t h i n g đ i ề u hòa t ù y ý
T h e n o n lin e a r v ib r a tio n of a pen d u lu m w hose su p p o rt u n d erg o e s arbitrary rectilinear harm onic
m o tio n is stu d ie d T he m a in atten tio n is paid to th e resonant c a ses and th e sta tio n a ry vib ration s
T h e r e so n a n t co n d itio n s are explained T he am p litu d e - freq u en cy curves are p lo tted for various
v a lu e s of p a ra m eter s and the sta b ility of v ib ra tio n is in v e stig a te d , '"he r otatin g m otion of the
p e n d u lu m an d its sta b ility ,a r e also considered.
Let us con sid er the v ib ration of a pen d u lu m co n sistin g of a n egligible w eigh t rod AM of length
i an d a lo a d Af of m ass m T he pe id u lu " ‘Tipport u n d e rg o es rectilinear harm onic m o tio n by
m e a n s of a m e c h a n ism sh ow n in Fig 1 w hen the '•rank O N o f len gth R ro ta tes around o w ith a
c o n s ta n t a n g u la r v e lo c ity n and tran slates slo tte d bar BA of len g th L alon g slides 1.1 We shall
ta k e the o r ig in of the y axis vertically up T he p o sitio n of th e p en d u lu m w ill be specified by angle
(p th a t A M m ak es w ith vertical axis (F ig l ) T he k in etic and p o te n tia l energies of the p en d u lu m ,
T and V r e sp e c tiv e ly are:
T = — [t 2 {P 2 + Jỉ2n 2 sin 2 fit — 2Rftt<p sin H í sin(6 + V?)]» J J
y = rng[(L + R c o sH t) COS 6 - t costp].
U sin g L a g r a n g e ’s e q u a tio n , taking into account th e d a m p in g force, the eq u ation of m o tio n of vhe
p e n d u lu m is o b ta in e d as
w h ere a>2 = g Ị I, h is th e dam p in g coefficien t.
We a ssu m e th a t R / t and h are sm all and w e sh a ll co n sid e r sm all vib ra tio n s o f th e pendulum
a b o u t th e v e r tic a l a x is, so th a t sin V? ~ <p — ị<p 3 / ô ) , cosy? ~ 1 — (<p 2 / 2). T h e sm allness of th e m en tio n e d q u a n titie s can be taken in to co n sid eration by in tr o d u c in g a sm all d im en sion less param eter
e , w h ich w ill be set eq u al to u n ity in the final re su lts T h u s, w e are led to consider th e follow ing
e q u a tio n o f m otion :
N G U Y Ễ N V Ă N Đ Ạ O
Viện Cơ hoc, Viện K t ì V N
S I E Q U A T IO N O F M O T IO N
(2)
( 3 )
Trang 3r SB wty c = -7 sin ỏ, D = -7 COS Í, " 1 — — , 7 = —
and a prim e denotes the derivative w ith respect to the dimensionlcsa tim e r.
(4 )
In th e following sections two resonant cases which cause intensive growth of the am plitude of
v ib r a tio n o f the pendulum w ill be studied.
§2 P R I N C I P A L R E S O N A N C E
We consider the case when 7 differs a little from unity We are interested in finding out what happ en s close to resonance, th a t is to say when - 1 is sm all, namely:
-7 = 1 + 2
where A is a detuning parameter.
Let us introduce in equation (3) th e variables a and rj aa follows
ip = a c o s 0, <p* = - c n a i n f l , 0 = Tff + ff (6)
here th e condition
a! cos 6 — a r / sin 0 = 0
is im p o se d T he e q u a tio n s for new v ariab les w ill be
V = •¥>+/) «in0,
w hich is a set of equations in standard form with
(7)
(8)
Trang 4f k ị a*)f sin f - o s cos3 9 + c j 1 COS 7 r + D 7 aa COS 0 • COS Tff.
In the first approxim ation the right-hand sides of (8) may be replaced by th c ừ mean values, regarding a and TJ as constants (l|:
7 o' = - ^ ( h n a + C V sin r ỹ ),
W - - j ( § « + j « ’ + C i a ««•>>)•
(1 0)
T h e s ta tio n a r y a m p litu d e a 0 and phase rjo are d eterm in ed by
hi~ia 0 + C~I 2 sin rjo = 0,
~ a 0 + ~ a ị + C~ 12 cos r?o = 0
A sim p le c a lc u la tio n elim in a tin g T) leads to the resp on se curve equation:
W ( a o t ^ ) = 0,
w = a ị ị h h * + ( S - I + ị a i y } - c ' l ' .
(11)
(1 2)
(1 3 )
T h is relation is p lo tte d in Fig 2 for the param eters:
R = 2 cm , i = IGOcm, ố = o.l&rad, g = 98 0 c m / s c c 2t h = 2.47 • /li, w = \/<7/ i = 2 4 7 , CƯ2 = 6 1 2 5 , c = 8 8 1CT3 1 hi = IQ-2 aiid ? h i = 4 l < r a.
Ft? £
L et U3 n o w discu ss th e stability of th e possible stationary regimes To do th is we stu d y the
eig e n v a lu e s of th e m atrix of coefficients o f variational equations of (10):
Trang 5I _ t C 2
~ 2 ~ I c 0 i *7o
c / A 3 , \ t C
■ j ( 5 + i ay 1 1 , l n , °
T h e equation defining these eigenvalues is
<V73A3 + * i 2a 0h x H J ~ = 0 (14)
8 d a Q
fro m w h ich w e ob ta in th e co n d itio n for a s y m p to tic s ta b ility
o a n
It is n o ted th a t fu n ctio n IV (a 0 »K2 ) is p o s itiv e (n e g a tiv e ) o u tsid e (inside) o f reson an t cu rv e and
e q u a l to te r o on it So, c o n d itio n (15) sh o w s th a t th e up p er branches of reso n a n t curves (h e a v y lin e s ) in Fig 2 correspond to sta b le s ta tio n a r y reg im e s and th e broken lines to u n stab le o n e s.
§3 P A R A M E T R I C R E S O N A N C E
It is su p p o sed th a t 7 18 a p p ro x im a tely eq u al to 2, nam ely
(I6)
T h e s o lu tio n of eq u ation (3) in th is cases is fou n d to be
= i c o s ( Ị f + a ) , !f>' = - ^ 7 sin { y + a ) (1 7 )
In th e first ap p roxim ation eq u a tio n (3) c a n be rep la ced by th e averaged ones:
* ị V = - i b * ị ( h i + D i sin 2 a ) ,
(18)
It is clear that 6 = 0 is a solution of eq u a tio n s (1 8 ), b u t aj we shall show later, it may h a p p e n
th a t th is solution is unstable and th a t th e sy stem b eg in s to vibrate spontaneously T he s ta tio n a r y
am p litu d e bo if determ ined from equation V =r a* = 0 b y elim ination a:
T h e response curvet are p lo tted in F i g i f o r D = 8 8 1 0 “ 3 and 1) h \ = 1.65 10~a, 2) h \ =» 1 7 1 0 " 3
T h i stu d y show s that only th e p ositive sign before th e rad ical (19) corresponds to th e u y m p t o t ic a l sta b ility o f nontrivial station ary vibration; and th a t th e solu tio n b =* 0 i< stable ou tside the re so n a n t curve and unstable inside it Thus, only th e h eavy lin es (F ig 3) of th« resonant curve correspond
t o s ta b ility o f vibration.
Trang 63 6 3 8 * y 2
Ft</ ^
§4 R O T A TIN G M O T IO N OF T H E P E N D U L U M
Assuming t h a t h a n d R / Í a r e s m a l l w c c o n s i d e r t h e r o t a t i n g m o t i o n o f t h e p e n d u l a m g o v e r n e d
b y e q u a r i o n :
w h e r e
f (<p, <p, i ) = — - Y n 2 c o s O f s i n ((5 + V?) ( 2 1 )
I t i s supposed t h a t t h e e n e r g y o f t h e s y s t e m c o n s i d e r e d i s h i g h s o t h a t w h e n e = 0 t h e p e n d u l u m
w i l l b e r o t a t i n g ( e q u a t i o n ( 2 0 ) )
W e i n t r o d u c e t h e v a r i a b l e a a n d yịỉ (2]:
<p = u( a) + a i / ( a ) COS
e = nt + ự>, v(a) = l/y/ã, ị = i/(a) (23)
an d V? is the so lu tio n of the d egen erate eq u a tio n £ = 0 if a and 0 are con stan ts; 80 th at
- a i /2( a ) s i n £ 4* Ui2 s i n V? s 0 ( 2 4 )
E q u a t i o n ( 2 2 ) i m p l y t h a t
£( 1 + a COS £) + à si n £ - ix(a) (1 + a COS ( ) =3 0 (25)
T h e secon d eq u a tio n for ( and à is o b ta in e d by s u b s titu tin g eq u ation (22) in to equation (20):
- a i / ị s in £ + + a COS £) + V COS f ] â + w3 sin V? = e f(v ? , í) (2 6 )
42
Trang 7here th e su b scrip t ua” or ( )a d en otes the d eriv a tiv e w ith resp ect to the am p litu d e a FYom th e se
e q u a tio n s we get:
à = - i - F ( < p , <p,t)[ 1 + a COS 0
V- = i/(a) - n + ^ F ( v , <fi, t) Bin t ,
w here
- A = a v s in 3 { + (1 + a COS ( ) [i/flU + a C08 0 + v cos (]•
S u b s titu tin g here ỉ/ = l / y / ã w e have
— = 2a y / a + 0 ( \ / a • a 3) (28)
We shall con sid er the p rincipal reson an t case w h en th e am plitude a takes valu es close to aQ
d e te r m in e d by
n £ 2 i / ( a 0 ) = ~ =
V a o
(2 9 ) and use th e Jacob ie exp an sion s of trigon om etric fu n c tio n s in B essel functions [3]:
oo
s i l l ( a s i n 0 = 2 ^ 2 n - i ( a ) s i n ( 2 n — 1 ) £ ,
n—i
oo
c o s ( a s in £ ) = */o(a) -f 2 E ^2rv(a) COS 2n £,
n = 1
Jm (a) = ^ ib!(m + fc)! ( f ) ’ m = 0 - 1- 2
ksz 0 '
L im itin g b y considering the v ib ration w ith sm a ll a m p litu d e a we have in the first a p p ro x im a tio n averaged eq u a tio n s of the form:
(30)
à = ^ hi / (a) ^ 1 + - " ( 1 2( a i cos \p + a 2 sin v>)
ff /Ĩ
rị) = i / ( a — n + • y n 2 ( a3 COS + c*4 sin v^),
L a V
(3 1 )
here
a 3 =
a 4 =
(3 2 )
ữ l = \ 2 ^ ° * ^ ~ 4 ^ 1 + 8*n ^’
<*2 = [2 ( ^ 0 “ ^2 ) + ị + *^3 )] c o s ^>
( i i + J3) COS 6,
(J 3 - 3 J i ) sin 6 ,
* - , - ( ! ) *
S ta tio n a r y regim es of reson an t v ib ra tio n s a* and are d eterm in ed by eq u ation s ả = = 0:
7 y n 2 (a Ị cos 0 + a j s i n v>.) * /1 ( 1 + ^ ) , v/ 0 7 y n 2 (aJ COS + a j sin ự>.) * /1 ^ 1 -f * y j
(33)
Trang 8Q* = Q i ( a = a ) , a 7*^0, n = ~~7=- — cons t
T h e so lu tio n s of th ese eq u ation s are found in the series:
a = a„ + £Oj + 0( e2),
( 3 4 )
rl>. = i>o + + °(ff ).
where a 0 satisfies relation (29) Substituting equations (34) into equations (33) we have:
ỰÕT 0 ^ n 2 ( a l0 COS V>o'+ a 2o sin V>o) = /1 ( 1 +
a = a „ + e a i = a Q + 4 s a ^ n2 ^ ( a 3o COS \ị>o + c u o s in \ị>„),
here a , 0 = Q,(a = ac) The first equation of (35) gives the phase displacement Then, the correction e ai to the stationary amplitude a n will be found.
We consider now the stability of stationary solution a , and determined by formulae (33) For this purpose we w r i t e the variational equations for system (31) Let
a = a , 4- S a , \p = + brị).
By putting these expressions into (31) and linearizing relative to 5a and 6\p we obtain:
d tfa- '{(f )„ - f n1( a ) / 01'*-+ (a).
€ R
- s i n t/>* - f a2 COS rp.)ỏ\p>
= {L „ + ^ n 2 [ ( ^ ) u C c s ự + ( ^ ) n S m V > ] } ía +
£ i ỉ + A £ ^ 2( “ ữ 3 s*n + Q4 C0S i M i
0-w h e r e
The characteristic equation of this system is of the form
where
* ~ 7 n M [ ( 2 ) - ( ? ) ] ‘ - * - * [ ( 5 ) * ( ? ) W } - ( ! )
G = — s i n yp* + Q 2 COS ự>.) + 0 ( f 2 ),
an d a prim e d e n o te s a d erivative to a T he stab ility c o n d itio n s w ill be
(39)
A s an e x a m p le le t us con sid er the case 6 = 0 T hen
1 / a 2 \ a
Oj = «4 = 0, =
Trang 9^ n ’ v / a ; ( l - sin ĩị>o ** 2A ( l + ,
« = <i» + j i | n o J t M i a ° ~ n ã
Lt form:
ã~o (1 - ^ r ) 8Ìn ỳ o *» 2A í 1 + 2 * ),
aná eq u a tio n s (35) ar« o f th f form:
(4 1 )
Because a if sm all, the first equation of (41) show s th a t there ir e two values of rpo lying on th f first and second quadrants corresponding to two values o f cot ĩpo w ith opposite signs Therefore the one
o f s t a t i o n a r y a m p l i t u d e c o r r e s p o n d i n g t o CO* > O i i l a r g e t h a n a 0 a n d t h e o t h e r c o r r e s p o n d i n g
t o CO% $o < 0 is s m a l l e r t h a n a n T h e e x p r e s s i o n s ( 3 9 ) n o w z s t :
K = fc + 0 (a 2),
Ơ = - e ^ n 2 COS ự>o + 0 ( e 2 ) ^ ^
and the stability condition gives:
For the caat 6 = * / 2 we have
a 2 - a 3 = 0 , ữ i = - + 0 ( a ) , a 4 = - Y + o ( a ) ,
a = a „ + c a i = « „ - ^ e y n 2 a j s i n V ' o , J 4 4 j
K = eh + 0 ( a 2 ),
G = J ị R f t 2 sin 'Po + 0 ( ff2) I
a n d t h e s t a b i l i t y c o n d i t i o n is
So, in b oth cases (6 = 0 and s = ff/2 ) the sta tio n a r y v ib r a tio n w ith sm a ll am plitude is s ta b le
and that with large amplitude is unstable (see equations (41) and equations (43), (44) and (45)).
C O N C L U S I O N
1 T h e sta tio n a r y nonlinear v ib ra tio n s of th e p e n d u lu m and its sta b ility have been co n sid ered
2 T o avoid reson an ce of th e p en d u lu m , th e p a r a m eter s o f th e sy ste m considered sh o u ld be
chosen so th a t (Jj7 difers from n 3 and J f l2,or
3 T h e r o ta tin g m o tio n of the p e n d u lu m m ay o ccu r U sin g th e averaging m ethod o f n o n lin ea r
m e ch a n ics an d th« Jacob ie ex p a n sio n s, th e sm a ll “v ib ra tio n " o f th e p en d u lu m around the s ta tio n a r y
rotation and its stab ility have been studied.
R E F E R E N C E
1 B o g o liu b o v N N , M itropolBkii Y u A A s y m p to tic m e th o d s in n on lin ear v ib ration s, M oscow
1974 (in R ussian).
2 M o ise ev N N A sy m p to tic m e th o d s of n o n lin ea r m e c h a n ic s, M oscow 1981 (in R u ssia n ).
3 S m ừ n o v V I C ourse o f high m a th e m a tic s, to m 3, p a r i 2, M oscow 1956 (in R u ssia n ).