Qualitative study of strongly nonlinear dynamic self-oscillating system with feedback Текст научной статьи по специальности «Физика»

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, x2 =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

Substituting here X^ x2 from the system of equations (3) and simplifying, we get:

dV

....................................(5)

— = 2x,x 2 (1 - x2(k-1)) + 2ex 2 (1 - x2) dr 1 1 '

at k = 1:

= 2xx + 2sx2(1 - x2) dx 12 2 1'

dV

at k = 2: -= 2x^(1 -xj) + 2exj(1 -xj),

dx

(6) (7)

as seen from relationships (5) — (7), the affix that starts its motion in any initial point Xj(0), x2(0) ofthe plane xl, x2, with increased t tends to: a) in case of k = 1 and small values of s it tends inside-out, which indicates an instability of the origin of coordinates; b) in case of arbitrary k and s — it tends from outside to inside, which indicates a stability of the origin of coordinates.

It should be noted that in case of k = 1 the equation (3) coincides with generally known Van der Paul equation.

To study the behavior of trajectories near the origin of coordinates we would apply Lyapunov's criterion [2] to the system (3). Lyapunov's function is determined in the form:

V (*i>x 2) = 2(Yixi2 + 2Yi2xix 2 + Yix 22). (8)

Unknown coefficients yl,yl2 and y2 are determined so that the derivative dV is a negatively determined function of the form:

dT

dT=«+x22). (9)

dx

Derivative of function V (x 1, x 2) (8) has the form:

dV dx ( dx dx, ^ dx,

= + Y121 x+ l + Y2x2-^, (10)

dx dx ^ dx dx ) dx

Taking into consideration a linear part of the system (3) and comparing with the expression (10), we obtain:

Yl2 = 1 Yl - Y2 + £Yl2 = °> Yl2 +£/2 =-1 . (11)

Solving these equations relative to y1,y2 and y12 we determine:

2 + s2 2

Yi =--> Y2 = — > Y12 = 1. (12)

s s

Then, Lyapunov's function has the form:

w ^ 1 ( 2 + g2 ^ 2 2

V (Xi, x 2) = -l--Xj + 2XiX 2 — X2

2 \ s s

(13)

Form the conditions yl, y2 -y2l2 > 0, y2 > 0 it is evident that the first condition presents a possibility of two scenarios of trajectories behavior:

a) at s > 0, V(xl,x2) — is a negatively determined fUnction for any values of Xj and x2, then, the trajectory of the system is directed inside the ellipse (13);

b) at s < 0 the trajectory of the system is directed to the surface of the ellipse (13).

Now we would return to initial non-linear system (3). Using

Lyapunov's function (8), we could determine dV in the form:

dx

— Xj x2 + 2Xj x2 & x3x2,

dx

at k = 1 with correlations:

— x ^ x 2 + 2x ^ x 2 & xJ x 2

dx

(14)

(15)

As seen from expressions (14) and (15) they differ only in the first term, that is, the relationship (15) is a particular case (14).

So, according to the expressions (14) and (15) we may conclude that for sufficiently small values of Xj and x2 the derivative

dV for both cases is negative. This means that the origin of coordi-

dT

nates is a singular stable point, that is, any trajectory with an increase in time t intersects the surfaced = const from outside to inside. As seen from the correlations (13), (14) and (15), Lyapunov's functions satisfy Lyapunov's theorem.

Theorem (Lyapunov's theorem of stability) If for the system (3) there exists in the domain D a sign-determined function V, its derivative in time V, taken as a system (3), is a sign-constant function of the sign opposite to the one of the function V , then the equilibrium position is stable in Lyapu-nov's point of view.

Fig.1. Diagrams of the surfaces of Lyapunov's functions and their derivatives under different values of the parameter of feedback

Figure 1 shows the diagrams of the surfaces of Lyapunov's functions and their derivatives; it is seen that the greater the feedback factor and the degree of non-linearity, the better dynamic stability of the system on the whole.

So, sufficient stability conditions are stated for the system on the whole. An increase in feedback factor and consideration of non-linearity of higher orders facilitate an improvement of the stability margin of dynamic system.

To determine possible limit cycles we would utilize Poincare Method of contact curves [3]. Consider a set of concentric circles with a center in singular point of the equations (3). Define on a plane x1x 2 a geometrical position of the points, where these circles touch the integral curves of the equations (3). This geometrical position of the points forms a contact curve. Excluding the time т from the system of equations (3), we have:

dx 2 -x f-1 +e(1 - x 2)

dx; = x2 • (16)

We should note that limit cycles at t ^ со satisfy the condition of central symmetry x 2 (-Xj,e) = -x 2 (x pe) ,since the equation (16) does not change at the value of x1 ^ -xi and x2 ^ -x2. As the origin of coordinates is a singular point, we would consider a set of concentric cycles with the center in the origin of coordinates: xj + x 22 = const.

Differential form of these cycles has the following form: dx2 _ x1 dx1 x2.

Contact curve is determined by the equation:

(1 - xf-1))x! +£(1 - x2)x2 = 0. (17)

Introducing polar coordinates, the equation (17) is transformed into:

(1 - r2№-1) cos2(k-1) 0)r cose + e(1 -r2 cos2 0)r sine = 0. (18)

Everywhere except the point r = 0, contact curve is determined by the equation:

r2k-1 cos2k-16 + r2 cos2 6sin6 - (cos 6 + s sin 6) = 0, (19) at k = 1 the relationship has the form:

r + r^—-+ —L_l = 0. (20)

sin20 ^esin20 cos From quadratic equation (20) we determine:

- 1 + 1 2 1

^ = sin2d \ sin2 2d + e sin 2d + cos2 d'

Maximum and minimum values of r with an angle 9 have the form:

t ¡3s + 2 1 ¡3s + 2

r =-1+ J-, r. =-1 -J- . (21)

m ax AI ' min AI

V s \ s

For the rest of the values of the order of non-linearity k the values of rmax and rmin may be determined as well [4]. This means that there exists a ring with the center in the origin of coordinates, containing all possible limit cycles; their boundaries being the cycles of the least rmin and the greatest rmax radii of correlations (21), touching the contact curves, determined by the equation (18). Figure 2 shows integral curves and the diagrams of phase trajectories built numerically by Mathcad 13 program package for the cases k = 1 , k = 2 and k = 3 at various values of the parameter s .

Figure 2 shows transition processes from unstable focus to self-oscillating and relaxation modes of oscillations, and corresponding limit cycles which agree well with analytical definition of the rings, containing these limit cycles.

An increase in feedback factor s and in the order of non-linearity k of differential equation (3) in the system leads to occurring of self-oscillating and relaxation vibrational processes.

Some of data above were announced in the Proceedings of the XI All-Russian Congress in Fundamental Problems of Theoretical and Applied Mechanics. Russia, Kazan, 2015.

Fig. 2.

References:

1. Mishchenko E. F. Differential Equations with Small Parameter and Relaxation Oscillations. - Moscow: Nauka, 1975. - 248 p. (in Russian).

2. Barbashin Е. А. Introduction into the Theory of Stability. - Moscow: Nauka, 1967. - 224 p. (in Russian).

3. Poincare А. Selected Works. - Moscow: Nauka, 1972. -V. 2. - 543 p. (in Russian).

4. Annakulova G. К., Igamberdiev К. А., Sattarov B. B., Abdullaeva М. 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).

Romashin Valerij Nikolaevich, Technological Institute of the National Research Nuclear University MEPhI Candidate of Technical Sciences, Associate Professor E-mail: valeryromashin@yandex.ru; Romashin Roman Valerevich, Technological Institute of the National Research Nuclear University MEPhI

Senior Lecturer E-mail: rvromashin@mephi3.ru

Forgotten alternative to crank mechanism

Abstract: In the history of the industry of Europe, there was the mechanism with a screw surface for reception of rotation movement from linear movement. Having found new outlines in XIX century, but using the same principle, it took replaced the crank in the steam engine. Because of the big dimensions, it has ceased to be used and has been forgotten. The new design has appeared now, allowing to return to an old principle.

Keywords: crank mechanism, engine, drive, torque moment, screw surface, cylindrical wedge mechanism.

Transformation of translational motion into rotational motion is traditionally associated with a crank mechanism. They surround us everywhere and have become usual and indispensable. But why are screw surface mechanisms not used for this purpose? What obstructed their wide spread? Let's take a look at history.

The authors associate the first use of a screw surface in mechanics with Archytas, 428-365 BC. Archimedes went down in history as the author of a screw pump. The Mechanica by Hero of Alexandria considers simple mechanisms, including a screw. But all these examples describe the use of a screw surface as a transformer of rotational motion into translational motion.

The use of a water wheel in Europe in X-XV centuries was a foundation for future mechanization. Some processes required not rotational motion, but reciprocal motion of an output element, for instance, in blacksmithing, metal industry, construction and weaving. Hence, crank mechanism gained widespread.

Although, according to literature sources, another mechanism with a screw surface was used apart from the water wheel. Thus, the book of an Italian architect and engineer Giovanni Branca «Le Machine», 1629 depicts such mechanism in figures XVI, XIX, XX in the first part [1]. It transforms linear motion into rotation. The mechanism is based on a drum capable of revolving about its axis — it is installed vertically in the support blocks. Two spiral pipes installed at the angle of elevation over 45° are fixed on the cylindrical surface of the drum. Water is used as a vertically moving body. Supplied from top, under gravitation force, it runs down spiral pipes; herewith, affecting the walls of the pipes, radial component of gravitation force makes the light drum revolve. A toothed wheel is attached to it at the bottom and the revolving is transferred to burr stones as shown in figure XVI (Fig. 1).

Figure XIX depicts a cascade of three such drums, one below another (Fig. 2).

Figure XX shows the use of such machine in household — the drum is located in the cellar of the house and it puts the spinning wheel upstairs in motion (Fig. 3).

Thus, one can conclude that such mechanisms were widely applied in Europe in XVI-XVII centuries.

100 years before this Middle, or as it is also called «High» Renaissance, in one of his codices, Codex Madrid I, 1493, kept at the National Library of Spain under registration number 8937, Leonardo Da Vinci (1452-1519) described the structure of an endless screw in page 70. The mechanism is known under the code Madrid Ms. I (BNM), MSS/8937, fol. 70r [2]. The entire mechanism transforms the rotation from one plane into the rotation in the other plane. It is possible due to screw surfaces with a high angle of elevation. Let us suppose that if one makes such surfaces on a cylinder, installs a screw in such a way that it cannot move along axis, and makes the cylinder move linearly, the screw will revolve sliding along the turns. In the mechanism of Leonardo, such cylinder is convolved and forms a torus (Fig. 4). With the help of an input element — gear wheel m-n, that has projections, which correspond to pockets in the torus supported by worm rollers, starts rotating in the horizontal plane, and the output element — screw, sliding on the screw surfaces, also starts rotating but in the vertical plane. The figure does not show that the screw is fixed against the movement on the axis of the torus, but, according to the comment by Leonardo to the figure, «if one makes the endless screw rotate with the help of gear wheel m-n, and firmly keeps the screw f-s in place to make it possible for it to rotate, the rotation of the screw will undoubtedly take place with a strong force» [3]. Thus, one can assume that it is the first, from archive data, mechanism using screw surfaces with the angle of elevation over 45° to create rotation of the output element. Although, due to the difficult production, it was most probably never manufactured as well as many constructions of Leonardo that were ahead of time.