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Section 6. Mechanics
Section 6. Mechanics
Annakulova Gulsara Kuchkarovna, Leading Scientific Researcher, Scientific Research Center for Sectoral Machine Science at the Tashkent State Technical University
E-mail: Annaqulova_g@mail.ru
Asymptotic definition of the periods of relaxation oscillation of strongly nonlinear systems with feedback
The research was carried out under financial support of the Fund of fundamental research (F2-FA-F050) of the Republic of Uzbekistan
Abstract: The problem of asymptotic approximation construction for the periods of relaxation oscillations of strongly nonlinear dynamic system with feedback is considered in the paper. Recurrent formulae to calculate with arbitrary degree of accuracy the periods of relaxation oscillations for corresponding degrees of nonlinearity of the system with feedback are derived. Keywords: relaxation oscillations, nonlinear dynamics, feedback, recurrent formula, limit cycle, stability.
Consider the self-oscillations of strongly nonlinear system of the type [1]:
mx + cx2k-1 - ax + bx 2x = 0, k = 1,2,... ,m ,c ,a ,b > 0 , (1) where m — is a parameter defining the characteristics of the object; c — a coefficient of restoring force; a ,b — the parameters giving the feedback to the system.
Let us reduce it to a standard form using the notations of di-mensionless variables:
x c I a
T=rt 'u=T r=L I b
i=. -
e =
a
yjcm I b
u + u-e(l - u2 )u = 0, (2)
where y ,l — are the parameters of transformation; e > 0 has the meaning of the coefficient of feedback.
Note that the study of numerous radio-technical schemes as well as dynamic systems with feedback leads to the systems of the second order (2) [2; 3].
Introducing the notations u = xp ic = x2, the equation (2) acquires the form:
U = x 2; x 2 =-x f-1 +£(1 - x2 )• x 2.
At large values of the parameter e> 0 the equation (3) may be reduced to the system of the form [2]:
sic = f (x, y)
y = g (x, y).
Introducing the notations:
y = f(V - 1)du + -• —, t = —,— = -1-, (5)
o e dx e e
the system of equations (3) is transformed into the form when t is written instead of t :
du uU
£, — = y---+ u;
1 dt 3
^ = -u-i. dt
(6)
We may approximately calculate the period of relaxation oscillations in the system described by the equation (6) at k = 1 :
dt
-y---+ u;
3
dy dt
(7)
Note that degenerate system:
— + u = 0;
3
(8)
y = -u,
according to [2] the relation (8) has a closed phase trajectory P0 (Fig. 1), which consists of two regions P2S1 and pS2 of slow motion and two regions S1P1 and S2P2 of fast motion. The coordinates of the points of separation S1 and S2 and points of fal P1 and P2 are as follows:
S1(-1,2/3),— (1,-2/3),—j (2,2/3),P2 (-2,-2/3). (9) Further instead of variable u we take x.
y / Ä / ^ /
/^x ßo Pi J
A 0 V 5 / *
X
/ / / / / / /
Fig. 1.
It was the theoretically proved [2], that in nondegenerate system (7) at each comparatively small value of the parameter s1 there exists a single and stable limit cycle , with ^ uniformly at sl ^ 0, and for the system (3) possible limit cycles are determined in [1]. In other words, the system described by the equation (3), at comparatively large values of the parameter s performs stable relaxation oscillation, numerically given in [1].
To construct an asymptotic approximation of the period of relaxation oscillation described by the system (6) we would make use of the theorem [2].
k-1
k-1
Asymptotic definition of the periods of relaxation oscillation of strongly nonlinear systems with feedback
Theorem. For the period Tc of the limit cycle ¡5S of nondegen-erate system (7), an asymptotic representation is true:
+ tAT + O(s'A), (10)
a. g(x,y) m=i
here the value of AmT, m=1,---,M, corresponds to a pair (Pm-1Sm ; SmPm) of adjacent sections ofthe trajectory and is calculated by the formula:
' 1 (11)
At k=1:
T « = f
■ (x2 - 1)dx = f (x2 - 1)dx + f (x2 - 1)dx
(14)
-i dx - 3 The first integral by (P2S1): - J--Jxdx = — ln2,
-2 X -2 2 1 dx i. 3
the second integral by (pS2): J — + Jxdx = — ln2, hence:
AmT = K;/A + KZsln - + K;os, s
where:
K-m =YX(S„P«x(sp), KZ =1 y] (Sm)X(SmPm) +1 YS)S] (S„),
3 6
At k = 2 :
T(2)=f
-x
T„(1) = 3 - 2ln2.
■ (x2 -l)dx = , (x2 -l)dx , (x2 -l)dx
' -x " * -x + ' '
X RS X
(15)
(16)
K» = j ,, gx^ ' y) dx +
1 r,,s,L(x ' y )g(x > y ) s
dx
i
g(x > y
dx +
h(Sm - 3Y,(Sm )lny(Sm ) + Y,(Sm )l*9 ^ )
f (x, y) g P ) sJmPJ (x, y )
x(sp ) +
PS
~r dx ~r dx . „ 3 the first integral by (P2SJ : -J — "J — = ln2-->
1 dx 1 dx
the second integral by (pS2) : J— + J— = ln2 —, hence:
2 X 2 x 8
+Y(Sm signf? S )x(SP
T0(2) = 2ln2 -0 4
(17)
J0 +1 lny{S„ )-1 lnV(Sm ) 6 2
V 2 fy (Sm )
At k = n :
rM
TW =
0
= j(x2 = j> (x2 - 1ldx + g (x2 - 1ldx, (18) A —x A —x P>si —x PS —x
,,, , f (S )/' (S )- 3 f" (S )f(2) (S ) 9 (S„ )= 1 ^ ",J'\m x V "'sign g (S m
6 f: (Sm )9(Sm
the first integral by (P2S1 ) :
~r dx ~r dx 1
J „2n—1 J 2n-1
1
1
x(sp ) =
_ g (Sm ) g (Pm )
6 ff'S )gX (Sm )- 2 ff'S )g & ) .
3 [ f(2'(Sm )]2
Sign g ( Sm ) Sign g (Sm
(12)
the second integral by (PtS2 ) :
!■ dx !■ dx 1
J 2n-1 ^ 2n-3 _ ' 1 +
2 (n — l)(n — 2) 2 (n —1)22<"—1) 2(n — 2) 11
2(n—2) >
hence:
2(n -l)(n -2) 2(n - l)22(n-1) 2(n -2)22'
1
1
1
S'(S ) =--/x (Sm} x signg' (S ),
^ ^ g2 (Sm )<p(Sm ) g JA ,
Y(Sm ) = \g(S,
f?(Sm ) fy (Sm )
(n - l)(n - 2)' (n - l)22(n-1) (n - 2)22(n-2),n > 2' (19) As seen from relation (19), at n > 2 it appears to be a recurrent formula for the period T0 of a closed trajectory ¡50 of the system (6).
Note that the field ofvelocities of the system (7) possesses the central symmetry, therefore A1T(1) =A2T(1). With formulae (12) and (9), we obtain:
) = i, ) = -3, x(sA ) = 3
y(s! ) = 1 Y,(Si ) = - ¡, KM ) =1.
= limv0(u), fi, = lim[v, (u)-lnv],
0 « J0 = 1 + JZo(u)du + Jvo (u)du.
—o 0
According to (10) for the period Ts of the limit cycle ¡3s of the system (7) an asymptotic representation is true
T£= T0 +AT + A2T + O (s4^, (13)
where T0 — is a period of closed trajectory ¡0 of the system (8), and the values AiT and A2T correspond to the pairs (P2S1;S1P1) and (pS2;S2P2) of adjacent sections of this trajectory. Determine T0(1), which is given in the form ofa closed integral according to (8) (Fig. 1). riod T^n> of the system (6) (Table 1)
Table 1.
At k of the highest order, the values of coefficients xn (S1P1 ), y4n (Sj) and S4n (Sj) may be derived from recurrent formulae:
Xn g) = 1 + g (S) = -(2n -1) + 3, %. (Si) = 2n -1 .(20) Needed to formulate an asymptotic representation of the pe-
Parameters g (x, y )
-x -x3 _x5 _x 2"_1
^(sj 2 3 oo | m 1 14 3 -(2n -1) + i
S fa) 1 3 5 2n-1
z(slpl) 1 +1 2 1+1 8 1 + ± 32 1 + 22"17
Y(si) 1 1 1 1
V(sm ) 1 1 1 1
V (Sm ) 1 3 1 3 1 3 1 3
g (x, y)
-1 _3x2 _5x4 -(2n -1)x2("-
x
x
+
Section 6. Mechanics
Table 2.
Using the representation of the relation (21):
g (x, y) 1 -gX (x>y)dx ?Is„fX (x,y)g(x,y)
-x i-—-= 3ln2 -1 ln3 4 -x(-x2 +1) 2 2
-x3 3 J dx =3(3ln2- 1 ln31 ^ -x(-x2 +1) 1.2 2 1
-x2-1 (2n 1) f dX = (2n 1)| 3 ln2 1 ln3J v 'i -x(-x2 +1) 1 \ 2 2 J
Table 3.
g (x, y ) r g(x, y)dx 1 f (x,y )
-x 2 -xdx 2 x2 + 3 , 2 2x2 + 9x + 6 , f-= 3 f-dx + 3 \-dx —i (x -2)(x +1)2 { x4 J! x (x -2)(x +1)2
-x3 2 -x dx 2 x2 + 3 , 2 2x2 + 9x + 6 , f-= 3 f-dx + 3 f-dx —i (x -2)(x +1)2 - x2 J! x (x -2)(x +1)2
-x5 2 -x5dx I x2 + 3, „2 2x2 + 9x + 6 , 1-= 3 1-dx + 3 \-dx —i (x - 2)(x +1)2 i x0 i x°(x - 2)(x +1)2
-x7 } =3i(x2 + 3)x 2dx + 3 j x2(2x2 + 9X + 6) dx {(x -2)(x +1)2 i —i (x -2)(x +1)2
-x2n-1 f x = 3 i(x2 + 3)x2("-3)dx + -i(x - 2)(x +1)2 -1 J 2 x2<n-3)(2x2 + 9x + 6) , +31-dx -1 (x - 2)(x +1)2
-xdx 2 xdx
" = 3 J;
Py+ x "J'(x-2)(x +1)2 3
3
f -xidx _ 2 x idx " = 3 J 'i
-4ln3--. (22)
-1x3 + X -(x-2)(x +1)2 3
(23)
For convenient integration, the element of integration of the
relation (23) is separated an aliquot and expanded in elementary
fraction, then we obtain:
2 x dx 16, i
3 f-- =--ln 3 — + 9. (24)
2
Calculated values of the integrals from the Table 3 are presented in Table 4.
Writing out the expression for the value of A1 T(2) at k = 2 according to (11) with relations (17), (20), (21) and (24), first we determine the following:
C' =a ■ CC1, = -l l = 9 ln 2 - - ln 3 - 3a - 3I0 --, hence:
2,0 0 8 3,1 2 3'° 2 6 1 0 3
9 2/11 f 9 1 2 I
AT(2) =-nosA--sin— + [ - ln 2--ln 3 - 3fi1 - 310-- |s1.
1 8 0 1 2 1 s ^ 2 6 1 0 3 J 1
Then, according to formula (13) the period Ts of the limit cycle Ps of the system (7) with consideration of the central symmetry, has the form:
Tm = 2ln2 - 3 + 9 Q„£1X-s,ln — +
* 4 4 0 1 1 s,
+ | 9ln2 —ln3 -6Q1 -610 -4]£l + O(s"/3
(25)
Universal constants Q0, Q1 and I0 entering the expressions are determined according to (12). Note that here the period is calculated relative to time. If return to time t and parameter s (see (5)),
Determine the values entering the formulae (12) of general- the equality (25) should be rewritten in the form:
ized integrals at the highest values of k, listed in the Table 2 and 3.
f dx 2 dx 2, „ 1
= J—-= (21)
3 9
= 2 ln 3 -1. J(x,y ) i y -1 x3 + x 3 3
3
r(2) = (2ln2 —)s+-Q0s3-sln- + s 4 4 s
+ | 9ln2 -1—3 - 6Qt - 6I0 - 4 U + O [s1
Table 4.
(26)
g (x, y )
g(x,y)dx
, f (x,y)
(-4
- 0
- 0
+ 0 + 0
+ 0 + 0) ln3 - 1
(-3 • 4 - 2
- 2
+ 0 + 0
+ 0 + 0) ln3 - 1
(-9 • 4 - 8
- 4
+ 0 + 0
+ 0 + 2) ln3 - 1
(-27•4 - 32
- 8
+ 0 + 0
+ 0 + 0) ln3 - 1
(-81•4 - 128
- 32
- 4 + 0
+ 0 + 2) ln3 - 1
(_3). - . 4 + 2~ )-((-1)n!+ +1) - Ç + l)((-'Hn-4' +1)-...
\ / O O
,C! +i)(H)(-iH--m)+1) +2 ((-1)(-1)!+i)((-ir' + 1) 2 2
ln3 -1
At k = 1 the system of equations (6) presents a Van-der-Paul equation.
An asymptotic representation for the period oftrajectory ¡3s of the system (6) at k =1 has the form:
2 1 1
T'f = (3 - 2ln2)s + 3Q0s3 —sln- + 3 s
+ ^3ln2-ln3-2Qj -210 -3js + O fs3 |.
(27)
x
x
x
x
x
x
Qualitative study of strongly nonlinear dynamic self-oscillating system with feedback
This is a well-known Dorodnitsyn's formula for the period of relaxation oscillations of the system described by a Van-der-Paul equation [3]. Similarly an asymptotic representation for the period of the trajectory ¡3c of the system (6) could be determined at other values of k = n using the Table 4.
So, recurrent formulae allowing to calculate with arbitrary degree of accuracy the period of relaxation oscillations described by nondegenerate system obtained from strongly nonlinear dynamic system with feedback of the second order are derived; with a satisfying steady limit cycle f5c.
References:
1. Annakulova G. K., Igamberdiev K. A., Sattarov B. B., Abdullaeva M. Study of Lyapunov's Functions of Strongly Non-linear Dynamic System//Proc. of the XI All-Russian Congress in Fundamental Problems of Theoretical and Applied Mechanics. - Russia, Kazan, 20-24 August, 2015. - P. 172-175. (in Russian).
2. Mishchenko E. F., Rozov N. Kh. Differential Equations with Small Parameter and Relaxation Oscillations. - Moscow: Nauka, 1975. -247 p. (in Russian).
3. Dorodnitsyn A. A. Asymptotic Solution of a Van-der-Paul Equation//Applied Mathematics and Mechanics. - 1947. - V. 11, № 3. -P. 313-328. (in Russian).
Annakulova Gulsara Kuchkarovna, Leading Scientific Researcher, Scientific Research Center for Sectoral Machine Science at the Tashkent State Technical University E-mail: Annaqulova_g@mail.ru Igamberdiev Kerimberdi Abdullaevich, Junior Scientific Researcher Abdullaeva Makhpusa, Senior Scientific Researcher
Qualitative study of strongly nonlinear dynamic self-oscillating system with feedback
The research was carried out under financial support of the Fund of fundamental research (F2-FA-F050) of the Republic of Uzbekistan
Abstract: A problem of qualitative study of oscillations of strongly nonlinear dynamic system with feedback is considered in the paper. The behavior of trajectories of the system in the state planes and near singular points is studied by applying Lyapunov's criterion. Sufficient conditions of the stability of the system as a whole are stated. Based on Poincare method of contact curves, the possible limit cycles are determined. A transitional process from unstable focus to self-oscillating and relaxation regimes of oscillations is established, as well as the corresponding limit cycles that are consistent with the analytical determination of the rings containing these limit cycles.
Keywords: self-oscillation, non-linear dynamics, trajectory, feedback, relaxation, Lyapunov's function, stability, transitional process.
We would study the trajectories of the system (3) in the state plane:
u = x,, u = x2, x =u.
Consider self-oscillations of strongly non-linear system of the form [1]:
mx + cx2 1 - ax + bx 2x = 0, k = 1,2,... ,m,c ,a.b > 0
(1)
where m — is a parameter defining the characteristic of the object; c — is a coefficient of restoring force; a ,b — are the parameters giving feedback to the system.
Let us reduce the equation to a standard form using the notations to dimensionless variables:
T = Yt, U - — ' l
, Y-f
\ m
^a] 2,i a,£"la 1 b J Mb 4cm I b
ù + u21-1 -s(l - u 2)U = 0,
(2)
where y,l — are the parameters of transformation; s > 0 has the concept of a feedback factor.
Introduce the notations u = xp u = x 2 then, the equation (2) has the form:
'+£(1 - x1) X 2.
(3)
From the system (3) it is evident that the origin of coordinates is a singular point corresponding to the state of equilibrium of the system. Determine the stability of this singular point. If the state of equilibrium is asymptotically stable, then, with increased t the affix x1(t), x2(t) must tend to the origin of coordinates.
Consider a circle defined by the equation:
V(xp x 2) = x1 + x 2 = r2, (4)
where r — is a distance between some point and the origin of coordinates.
According to [2] the origin of coordinates is an asymptotically singular point, if V(t) is a decay functionT, tending to zero at t —^ ^.
Differentiating the equation (4), we get:
dV dx dxn
-= 2Xj —1 + 2x .
dx dx dx
fc-1
