An approximate method for investigation of the stability of motion in the critical case

b u l l e t i n d e L ’A CADEMIE P O L O N A ISE D E S SC IE N C E S s e r i e d es science s te c hniq ues V olu m e X IX , No. 2 — 1971 THEORETICAL MECHANICS An Approximate Method for Investigation of the Stability of Motion in the Critical Case by NGUYEN VAN DAO Presented by s. ZIEMBA on November 2y 1970 Summary. The paper considers the system of two first order differential equations (1) with a small parameter. The case o f a pair o f imaginary roots in the first approximation characteristic equation (the critical case) is exam ined. T he necessary and sufficient con dition o f the stability of the zero solu tion (in the first approxim ation) is obtained. Some examples explaining the m ethod used are given. This paper deals with the stability of motion in that critical case, when the characteristic equation of system in the first approximation has a pair of imaginary roots, by the asymptotic method of nonlinear mechanics. This method is very effective in many practical problems. Let X - -Ằ y+eX (*,> ’, e), (I) y=z}.x\zY(x, yy fi), be the differential equations of the perturbation motion, where e is a small positive p a r a m e te r , Ả is a p o s it i v e c o n s t a n t . X , Y a r e t h e p o w e r f u n c t i o n s in X, y, w h o s e expansions begin with terms of degree at least two: X ( x , yiz) = x 2 (v, y )+ e X 0(v , y ) + £ 2 Xi(x,y) f , y (v, y> e ) = Y2(xty) M r 3 (*,>•) f Here Xs(x, y)> Ys(xy y) denote the terms in X, y of order s. The problem of stability of the zero solution x = >’ = 0 of system (i) can be re solved simply by the following discussion. It is notorious that, when the stability is lost, in the considered system lying on the boundary of the stability, the increasing vib rations near t o the h arm onic occur. Therefore, w e sh all estab lish the corresp o n  dence b etw een th e system ( 1 ) and som e linear system w ith co n stan t coefficients ( 2) x = ax-\-by, y = c x + d y. 17— [107] 18 Nguyen van Dao [108] The question on stability of the zero solution x=y=0 of system (2) is answered by the sign of the real part of the roots of characteristic equation (3) ơ2 - ( a + đ ) ơ + a d —bc = 0 For reduction of the system (1), lying on the boundary of the stability, to the equivalent linear form (2) we use this assumption, i.e. that the motion of the consi dered, .system in the neigbourhood of the boundary of stability arises if only for one-two periods according to the law nearing to the harmonic [1]: x = r COS 6+euỵ (r, 6)+e2 u2 (r, 0)+ , (4) y = r sin d + £ v ì (r, ớ ) + £ 2 v 2 (r, 0) + where í/<(r, ỡ), Vị(rf 9) are the periodic functions with the period In relatively 0: Ui(r, 6) = Uị(rì6+2n), Vị(r, 0)= vt(r, 6+ 2n), not containing the harmonics sin Ớ, cos 0 and r is a constant. Function 9 satisfies the equation [1]: (5) Ỗ=Ấ+£Bì(r)+e2B2 (r)+ In order to find the unknown quantities uit Vị, Aly Bị, we equate the coefficients of corresponding powers of £ in (1), using the expressions (4) for x,y. We have I dtiị ôu2 \ x = (A + £ 5 1 + £ 2 £ 2 + . . . ) Ị - r sin 0 + e - ^ - + e 2 *^0 + •••], l ÕVỵ ôv2 \ ỳ = (Ầ.+eB1+e2 B2+ ) COS +*2 ỹQ + •••] • By equating coefficients of the first power of £ we find that (6) Let (7) dux -rB l sin 0+Ả-rr- = “ Ằ.VỊ+X(r COS 0, r sin Ớ, 0), ou ÕVị rBi COS 0+X — = ỈMị + Y(r COS ỡ, r sin Ớ, 0). Ou CO X ( r COS0> rsinớ, 0) = go4- (gn COS nd-\~hn sin n6) , 11*1 00 Y (r cos ớ, r sin Ớ, 0) = CQ+ J F (CH COS n O + d n sin n d) n = 1 be the expansions of functions Xy Y, then we obtain: (8) gi = dị — 0 , rBị = -/ỉ! = Cj, ( 9) 00 Mi = Oi+ JT (P.,I cosnỡ+y»i sin MỠ), u = 2 00 «i = Oj+ £ (AaCosnP+y.jSinnfl), «*2 [ 109] In v e stig a tio n oj the S tab ility of M otion in the C ritical Case 19 where Co go „ _ C '-n h , n g' + nc 5 * ^2 1 > Pm .1 /m2 rr> Pn2~~ Ả 9 Ả 9 ™ A (/I2 — 1) ’ ™ Ả (1-A I2) (10) c „ - n g m h ' - n c , J'"‘ “ A(na- 1) • y"2 - A(1 - « J) • By linearization of the right parts of system (1) we can write: tX ( x ,y , e) = e X (r COS ỡ - f £Wi+ •• • , r sin 0 + « » ! + . . . , c)aj.x -4 £y(x,j>, e) = ^y(r cos 0+E U ị + . . . , r sin 0+et>i + . . . f e)&c.x+dy. Since in the expressions uh Vị the harmonics COS 6 sin 0 are absent, then for determination of the constants a, b, c, d, we average the both parts of (1 1 ) on va- 2 2 riable 0, multiplied by — COS 0 and — sin Ớ. Wc have (12) e 6 = - — nr 0 fi nr 1% COS 0+£W i + .:., r s in 0 + e v j - h , e) COS Odd, 0 2n f *<' COS 0+eui + 9 r sin 0+evl+ , e) sin 6 do9 0 2 k Ị y ( r COS 0+ fiW j-f , r sin ớ f + e)cos6dff9 0 2k 6 r ế/=— - I Y (r cos 0+ewj + , r sin Ỡ+€1>! + , c) sin ớ rfớ. nr J 0 We can now write the system (1) in the following equivalent linear form: (13) * = ơ x + ( 6 - A )> ', ỷ = (c+ Ằ )x + d y. The condition for the asymptotic stability of the zero solution of system (13) is: (14) a+ d< 0 , (15) a d - ( b - Ằ )(c + Ả )~ Ằ 2 + o(e)> 0. For sufficiently small e the condition (15) is always satisfied. Consequently, inequa lity (14) is the unique condition of asymptotic stability of zero solution x= y = 0. If a+d>0, then the unstability takes placc. The case a+ d— 0 is doubtful. For the solution of problem on stability in this case it is necessary to write the terms with higher order in expansions (4). However* in practice it often requires the knowledge only of the form of solution in the first ap p ro x im atio n x = r COS Ớ, y = r sin 0 o r in th e se co n d ap p r o x im atio n x —r COS 0+eul9 y^rsinO+EVị. We now consider some illustrative examples. 20 Nguyen van Dao [110] Example 1. Study the stability of the zero solution of system [2] X - - y — T - ^ + y 1) X , (16) By letting we have a y = x — - ( x 2 +y2)y. x = r cosớ, >> = rsiiiỡ I - - — 2 x= - — r2x-yf . ___ _<L 2 y=x 2 Therefore, a+d= — ar2 and the zero solution of system (16) is asymptotically stable for positive a and unstable for negative a. Example 2. Let we have the equation [2]: (17) x+x=axf(x,x2)+g(x,x2). A s s u m i n g t h a t x = r s in 6 , x = r COS Ớ, w e o b ta in axf(x, x 2) + g (x, x)&Ax+Bx, where A= ~~ f [axf (x, i 2)+g(x, X2)] COS 6 d0 = — f f(r sin Ỡ, r2 COS2 Ớ,) COS2 6 (10. nr J 71 J 0 0 Consequently, the zero solution of system (17) is asymptotically stable when /4<0 and ustable when y4>0. Example 3. Consider the equation of perturbation motion in the form: (18) ỹ+ y= E V i(aỳ-by)2. One can verify that the form of solution of this equation in the first approximation does not solve the problem on stability. Therefore, we write its solution in the second approximation [1]: y=r sin O+EIJ/2 r2 (a+pCOS 29+y sin 2Ớ), (19) y = r COS 0+2ey/ 2 sin 29+y COS 20), where a2+b2 b2 — a2 ab a= r — , p~—c ’ y = -y- [Ill] _ In v e s tig a tio n o f the S ta b ility o f M otio n in the Critical Case 21 Substituting y%y from (19) into the right-hand side of Eq. (18) and lincarizating it, we find: y+y=ey/ 2 {a [r COS 0+2ey/ 2 r2( — fi sin 20+'y COS 29)]-b [r sin 0-f- +ei//2 r2(a+p COS 20+y sin 20)]}2 = i4p+z?y-f t where ( 2e 9 ) e2 r2 ab A = Mt | y lị/2 ( a y - by)2 COS ỠỊ (a1+bĩ)v/ị. From here it follows that if ab>0 then the zero solution is asymptotically stable and if ab<0, then this solution is unstable. DEPARTMENT OF MATHEMATICS, POLYTECHN1CAL INSTITUTE, HANOI, (DRV) REFERENCES [1] N. N. B og o liu b o v , Yu. A. M itro p o lsk y , Asymptotic methods in the theory of non- linear oscillations, Moskva, 1963. [2] N. G. c hcta c v , The stability of motion, Moskva, 1955. H ry3H BflH /Ị a o , npHỗ/nUKeHHUH MtTOỊỊ MCCJ1 €AOB&HHfl CTaổHiĩbHOCTH ARHtteKHfl ft KpHTMHeCKOM cjiynae CoAepxeaHHe. P a ỗ o T a KacacTca CHCTeMM flByx AHỘỘepeưuHỉUTbHMx ypaBHeHuâ nepBoro nopííA- K a [ 1] c M aiibiM nap aM C T p o M . P acc M a T p H B a eT ca cjiy n a B napbi MHHMHX K o pH cit B nepBOM npH- xapaK TCp H CTK H ccKoro ypaBHeHHfl ( k p HTHHCCKHÔ C Jiy na tt). n o jiy n e H O h c o 6 x o a « m o c M flOCTaTOHHOe yc/iO B H e CTaốH/ibHOCTH H yjiC B oro PCUICHHJI (b ncpB OM npHÕ/iHHceKHỉí). npH B OAflTC* npHMcpu AJIH tfJuiiocTpauHH BbimeynoMJCHyToro MeTO/xa. . 1971 THEORETICAL MECHANICS An Approximate Method for Investigation of the Stability of Motion in the Critical Case by NGUYEN VAN DAO Presented by s. ZIEMBA on November 2y 1970 Summary. The paper. with the stability of motion in that critical case, when the characteristic equation of system in the first approximation has a pair of imaginary roots, by the asymptotic method of nonlinear. answered by the sign of the real part of the roots of characteristic equation (3) ơ2 - ( a + đ ) ơ + a d —bc = 0 For reduction of the system (1), lying on the boundary of the stability, to the