R E F E R E N C E M E T H O D IN S H I M M Y P R O B L E M

A. A . M a r t y a : y u k a n d N. V. N i k i t i n a UDC 531.36

1. The g is t of the r e f e r e n c e method, p roposed by Chaplygin [6] and based on using a different ia l in- equality, i s cons t ruc t ion of a new s y s t e m ( re fe rence sys tem) f r o m w h o s e dynamic p rope r t i e s follow those of the pr inc ipa l one. Refe rence s y s t e m s a re m o r e eas i ly analyzed, thei r r ight-hand side being monotonic with r e spec t to extradiagonal e lements and, as a rule, of a lower order .

F o r the solution of one p rac t i ca l p rob lem assoc ia ted with sh immy of guide wheels on a chass i s , it became n e c e s s a r y to obtain p r e l im ina ry informat ion about the dynamic behavior of the se l f -exc i tab le vibrat ion sy s t em before the sy s t em of equations of motion could be solved. T h e r e was no informat ion avai lable about s tabil i ty in the l inear approximat ion, s ince the c h a r a c t e r i s t i c equation had a ze ro root. The r e f e r e n c e method was used in that case . We will show he re the theorem which was used for analyzing the s tabi l i ty of the ze ro -va lued solution to the r e f e r ence s y s t e m of equations [4].

Le t the r e f e r ence s y s t e m be cons t ruc ted in the f o r m

= Y (y),yE R% (1.1)

with a continuous r ight-hand side and a local ly unique solution to the Cauchy p rob lem for any y o u r 'n .

There ex is t s a vicinity U around the point y = 0 such that y # 0, Y{y) ~ 0, and Y(0) = 0 for all y ~ U .

THEOREM. In o rder that the solution y = 0 to s y s t e m (1.1) be asympto t ica l ly s table within the y >- 0 cone, i t i s n e c e s s a r y and sufficient that the s y s t em of inequal i t ies

r ( y ) < O , y:>O (1.2)

be simultaneously sat isf ied.

2. The s y s t e m of equations descr ib ing the motion of the iner t ia l pa r t of a double wheel on a chass i s , according to Keldysh theory [2] and in the formulat ion of study [1], i s

J~," + h,~ + [t,~ + 2 (p,u + ~m + c~d~)l r + 21V'OJ~

+ (b12 + 2IV/r) 0 + blsz - - d11% ~ 2htr~ - - 2 (ar + aN) ~ - - h,X sin ~ = O;

J ' 0 + hoO + b2aO - - 21Vr -~ (1 + d~/:) "r + (b21 - - 2fcbd 2)

+ b,~z - - d2t% - - 2bq~ - - ho~C cos % = O; (2.1)

rn'z'+ bssz +(bs t + 2~N) ~p + b s ~ 0 - - d S l Z - - 2h~ - - 2a~ = 0.

The equations of nonholonomic couplings a r e , according to M. V. Keldysh,

t +'z + r;~ + vo+ V(p=o; ;p + ' o - ~ v ~ + ~ v , p - v v , = o .

The hydraul ic damper and the e las t ic e lement with a spr ing ra te k a re connected in s e r i e s here. the d a m p e r - - s p r i n g s y s t em is i ne r t i a l e s s , so t h a t

(2.2)

Since the damper torque balances the elast ic component of turbulent drag

H ~ sgn i l := k (z - z l ) (2.3)

The spr ing which develops the torque

M~: = e1~r + e~O + e13z - - gLtz, (2.4)

Inst i tute of Mechanics , Academy of Sciences of the Ukrainian SSR, Kiev. T rans l a t ed f r o m Pr ik ladnaya Mekhanika, Vol. 18, No. 11, pp. 106-110, November , 1982. Original a r t i c le submit ted September 3, 1981.

1040 0038-5298/82/1811-1040507.50 �9 1983 Plenum Publishing Corpora t ion

i s connec ted in s e r i e s with the spr ing componen t of the d a m p e r (a d r y - f r i c t i o n d a m p e r is not c o n s i d e r e d here) so that

(~ - - 7.1) : lVlx. (2.5)

Equat ion (2.3) y ie lds

i t ~--"

~- (z -- xl) (z > xO;

-- -9- ( z , - - z) (z < x,);

0 (x -- ~);

or , when X1 ~ X

~.1 = -~- I Z -- Zl I sgn (7. - - 7~0. (2.6)

We will now d e m o n s t r a t e that the s y s t e m of equat ions and r e l a t i ons (2.1)-(2.5) i s one of the fm*m

d x i dt -- E ~ + b, ] / ~ s g n a + cio ( i = I . . . . . n);

(~)

(2.D

The pos i t ion of the m e c h a n i c a l s y s t e m i s defined by ine r t i a l coo rd ina t e s 0, ~, z and noniner t i a l c o o r d i - na te s g, ~o, Xl, • r e l a t ion (2.5) indicat ing that one coo rd ina t e is redundant in this fo rmuia t ion . In the va r i ab le s

& = 4 , , & ~ 0 , x a = z , &----~, x s ~ 0 , x ~ = z , x T ~ , x s = % x ,=Z1 , & 0 - - ~ e x p r e s s i o n s (2.4) and (2.5) y ie ld

1 &o = ~ (e~x~ + e~x~ + e~3x3 + kx,). (2.8)

With o = xl0--xg, we obtain f r o m e x p r e s s i o n (2.6)

1 (eu& + e,2x2+ e13x3) k . . . . ~. ( 2 . 9 ) xl~ ~ git git

With the aid of r e l a t ion (2.6), we now d i f fe ren t ia te e x p r e s s i o n (2.8)

With the aid of r e l a t ion (2.9), we then e l imina te xl0 f r o m the s y s t e m of equat ions (2.1). In the v a r i a b l e s x k (k = 1 , . . . , 8), e the equat ions of mot ion , de r ived f r o m s y s t e m (2.1), (2.10), appea r in the d i m e n s i o n l e s s Cauchy f o r m as

x, =x~+ 8 ( i : 1, 2, 3);

8

X4____EOC~jXj~ dttk ~t k F - k - - - - - - ~ ~-Vl~lsgnm Jgn a + fe-t- gli

i=i 8

�9 15:E~5]xj d21k ~,k ~ - - ~ a k + gll ~- VI ~l sgn ~; i '=I

8 8

E dJ l~; 2 xe ~ ~ - - gn x n : czhjxj, k = 7, 8;

i=1 i=t

b = (en& -!- e~2x5 + el~x6)/(k + gn) - - gn 1 / -~ V] c5 I/(k -]- gn) sgn {y.

(~.11)

1 (}4 1

The nonzero coefficients in system (2.11), with the ear l ier notation [3], are

%t = -- [b11 + 2 (pN -}- aN + cbd~)]/J -~- d, le,L - gl~J ;

%2 = - - (b~2 -~ 2h~V)/ J -~- dne~e / (gnJ) ;

~1ell . ~ 3 = -- b l J J ~- d n e l J ( g i i J ) ; a ~ = - - (hr q- 2h; ) / J + k -~ gl-----~ '

a~7 ---- 2(1 ~- a N ) / J ; ~z~s = - - 2 h ; V / J ; %1 == - - (b~ - - 2fc~,d2)/J '

' �9 = - - b 2 j J ' -~ d~el~" ; a53 ---- - - b J J ' -I- d~ len / (J g n ) , %2 - - g n j ,

~ g e l l . d21e l JJ ' gn ; ~5, = 2I (1 ~- d2) V / J ' -}- k -~ gl------1 '

a~5 = - - ho/J' + [~r + gn); a~s = 13~e~r + gn);

abs ---- 2b /J ' ; ae~ = -- (ba~ + ~ N ) + dalen/gn~ r -= - - (b~ -{- h~V) -[- da:e~Jgn , a ~ --- - - b33 3c da~e~Jgn;

a ~ ---- ~ h~; ass -= - - h~; a~, = I ; ass -~ - - h~V; obz = ~ V;

a n - - - - l ; a v e : - - l ; a7v= l; ~ a S l = ? V ;

as~ = - - l; as7 = aV; %~ = - - ~V; ~ = h~ sin)~/I; ~. = ho cos tJJ ' .

Thus , Eqs . (2.7) a r e the equa t ions of m o t i o n in the Cauehy f o r m . We wi l l nex t a n a l y z e the s t a b i l i t y of the m e c h a n i c a l s y s t e m by the r e f e r e n c e me thod .

3. We r e d u c e the l i n e a r p a r t of the f i r s t n equa t ions i n s y s t e m (2.7) to the d i agona l f o r m with r e s p e c t

to a p a r t of the v a r i a b l e s

f~

= ~ a~,~zr - - bo V-I-~ I sgn ~ (a = 1 . . . . . n) , ~=I c = l

w h e r e 7 k c and p a i a r e c o e f f i c i e n t s of d i r e c t and i n v e r s e b a s i s t r a n s f o r m a t i o n .

C o n s i d e r i n g t ha t the N c o m p l e x - c o n j u g a t e r o o t s a r e p a i r e d , we have

n

Zo - - ~,~Zo q- "~ ( ~ b ~ 1/1 a ['sgn c~ -t- ~o~C~a); k = l .

k=l

k=l

Here and henceforth a = 1, . . . . N and e =N + 1 . . . . . n.

We now i n t r o d u c e the func t ions [5]

V~ = Zae--iOa; V~ = zaefOa; V e = ze sgn ze; w ~ asgncr, (3.2)

and d i f f e r e n t i a t e t h e m a t n o n z e r o v a l u e s of the a r g u m e n t . Not f u r t h e r c o n s i d e r i n g the d e r i v a t i v e s of func t ions

(3.2) a r e d i s c o n t i n u i t y po in t s , we obta in f r o m s y s t e m (3.1)

n

dv_____.~dt = ~ a~ -{- S [Re (~a~)COS0o -~ Im (~o~) sin 0o] (b~ ] / w s g n ~ ~- c~a);

k ~ l

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I q-5

Fig . 1

n

av____~dt = - - a~v~ + E f3 ~ (bh V 'w sgn a sgn z~ + cha sgn Ze),

k=l

n N

dWdt - - 2 E ah ~ [Re (?he)COS 0 c - Ira (?he)s in 0 e] vc s g n o + ~ k~l e=N+l

k=l c=~l

a ; ~ v ~ sgn z~ sgn o - - bo V-w,,

(3.3)

w h e r e

- - r b = l . . . . . N, N + I . . . . . n.

H e r e

On the b a s i s of s y s t e m (3.3) we now c o n s t r u c t the V a z h e v s k i i r e f e r e n c e s y s t e m

dyl Bi V-y + Sly;

n--N

d..YYdt = E K j y j - b. V y

i=t

(i----- 1 . . . . . n - - N ) .

(3.4)

w h e r e

B~ = X Sihbh;

S,~ = I IL~ 1;

k=l

Sah = max [Re (~a~) cos 0~ + Im (~.k) sin 0a]; 0a tt n

k=l k==l

Q~k = max [Re (7o~) cos O~ - - Im (?~D sin OJ. Oe

E l i m i n a t i n g Yi (i = 1, . . . . n - -N) f r o m s y s t e m (3.4), we obta in the i n e q u a l i t y

- - A V - y + B y < O ,

A = - - (BiKl%aa ... ~n--N +47 B~K~%% ... an_ N + ... ;

+ Bn--NKn--N~1% "'" an--~v--I) + ba~la~ "'" %--N;

B -~ S1Kx%a 3 ... % - N + S~K~ a1% "'" % - N + "'" + S,--NCq%"" %--.u

(3.5)

The v a l u e s of the p a r a m e t e r s m u s t s a t i s f y the i n e q u a l i t y

boat% ... an_ ~ :> B1Klcz.,a3 ... r + B~K2%% ... % - N + B n-NK,,-~,ch% "'" r (3.6)

The v e c t o r of the i n v a r i a n t y - c o n e h a s an u p p e r bound, viz .

g < A2/B%

F o r po in t s w h e r e Yio 6 D,

D ----- {g~o : gio > (Bi | f~o + Sigo)/% (i ~ I . . . . n - - N)},

the so lu t i on to the s y s t e m i s bounded and p (u(t, %), D) - - 0 a s t - - + % w h e r e u(t, u0) i s the so lu t ion to s y s t e m (3.4).

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With the d imens ion less values of the p a r a m e t e r s as follows

r e = l , J = 2 .137 ; t=0 , 213 ; R = 1; ~=0.714; c=0.285;

a = l ; b=0.261; cb----2.2; p=0,457; ~----4; e = 4 ; , ? = t ;

H = 142.49; he----- 0,556; h, = 0,835;

bit = 8.377; bl~ = 8,733; b18 -- - - 7.569;

b~.1 = 8,733; b~2 = 3,588; b2s = - - 28,15;

b a = - - 3,784; b32 --~ - - 14.08; b33 ----- 15,18;

/ ~ = - - 0,2618; N = 0.514;

V = 2.144 - - 2.466,

the roots of the n - t h - o r d e r subsys t em have a negat ive r ea l pa r t and condition (3.6) i s sat isf ied.

The g raph in Fig, 1 depicts t h e a t t rac t ion range of va r iab le y, depending on the p a r a m e t e r V and on k.

L I T E R A T U R E C I T E D

1. V . I . Goncharenko, L. G. Lobas , and N. V. Nikitina, "One formulat ion of sh immy prob lem for guide whee l s , " Pr ik t . Mekh., 1.77, No. 8, 82-88 (1981).

2. M . V . Keldysh, "Shimmy a f f r o n t wheel on th ree -whee l c h a s s i s , " Tr . Tsen t r . Aerogidrodin. Inst. , No. 564, 1-33 (1945).

3. L . G . Lobas and N. V. Nikitina, "Dependence of sh immy of guide wheels on s t ruc tu ra l p a r a m e t e r s and ve loc i ty ," Pr ik l . Mekh.,. 17, No. 7, 127-131 (1981).

4. A . A . Martynyuk and A. Yu. Obolenskii , "Stabil i ty of Vazhenski i - type autonomous s y s t e m s , " Differents. Uravn. , 1__6_6 , No. 8, 1329-1407 (1980).

5. A . A . Tikhonov, "Stabil i ty of motion under continuous pe r tu rba t ions , " Vestn. Leningr . Univ., 7, No. 1, 95-101 (1965).

6. S. A, Chaplygin, New Method of Approx imate Integrat ion (Selected Works) [in Russian], Nauka, Moscow (1976), pp. 307-360.

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