THE NEW EXTREMAL CRITERION OF STABILITY

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1 UDC T. G. Stryzhak THE NEW EXTREMAL CRITERION OF STABILITY Introduction It would be rather difficult to list all the publications about the Pendulu. Throughout history, oscillations of pendulu systes have been researched by such illustrious scientists as Galileo, Newton, Euler, Huygens, as well as any others. Their contributions into studying the echaniss of pendulu otion were of great iportance in the history of ankind s discoveries and technological progress. The pendulu has been used in clocks to define tie, in devices for easuring terrestrial gravitation, as a plub line in building to define the vertical, etc. The Earth s rotation was proved with the help of the pendulu oscillations as well. Pendulu systes are, as a rule, non linear and reuire specific ethods of research. For instance, creation of the elliptical function theory by Abel, Jacoby, and Weierstrass is also connected with the research of the atheatical pendulu oscillations. Nowadays due to the growing aount of atheatics involved in the research of different sciences there has been an increasing interest in studying otions of pendulu systes. The following special ters have appeared: the pendulu law of the population igration, the pendulu law of the rhyth regulator action, the pendulu of eotions, etc. It is a well known fact that all living organiss have so called biological clocks, at the basis of which there is an oscillating syste a non linear oscillator. It is well known that the vestibular apparatus of anials and huans contains three non linear pendulu systes, which are located in three utually perpendicular planes. Oscillations are present everywhere: in the opening of a flower at the sunrise, in the growth of an ebryo, in gerinating of a grain, in the heart beating, in the work of ackhaers and dental drills, in the rise and fall of the tide. And wherever there is life, there are oscillations at the cellular level. The nature of forces causing the oscillations is variable, but the result is the sae oscillations. This inforation bears a descriptive character as we would like to attract the readers attention to the oscillation theory to convince the of the fact that oscillation processes iply soething uch deeper than what we actually know: that the source of the oscillating processes is a hidden potential force and kinetic energy.

2 Oscillations of a non linear oscillator can be forced or can have a free running character (such as the heart or aorta beating, biorhyths, etc.). In fact, the pendulu otion laws, periodic or alost periodic oscillations, are ianent in the whole physical world. A huan being receives the ain inforation about the outer world via sound and light oscillations. Pendulu systes are used to analyse various engineering tasks. For exaple, oscillations in electrical and echanical systes, rocking of ships on water, oscillations of satellites, vibration of the hull and the wings of planes, oveents of cables, chains, travelling or gantry cranes, etc. All these phenoena are explained with analogous differential euations which describe otions. At this point our introduction is copleted. In this book we shall use the beautiful algorithic atheatical language: A is given, B is to be proved. Proble Forulation The new extreal criterion of stability The stability of oscillations of a echanical or electrical syste, which is affected by low aplitude high freuency disturbances, is researched. It is shown that the effect of vibration can be replaced by the action of soe potential force. Under the action of soe disturbance a stable euilibriu position can becoe unstable, and an unstable euilibriu position can becoe stable. New positions of stable dynaic euilibriu can also appear. Conclusions about the stability can be ade with the help of the new extreal criterion for stability described in this work. The criterion for stability rests on the tie averaging operation of the canonical syste of differential euations. The new extreal criterion of stability is applied when the freuency of external disturbance is substantially higher than the natural vibration of the uiescent syste. In this work the new extreal criteria of stability is applied in the research of the stability of the pendulu systes euilibriu position. In particular, it is found that any position of the pendulu, even a horizontal one, can be ade stable via vibration of the suspension point of a pendulu. An experiental installation to check the received theoretical conclusions was created. All the conclusions were proved correct. This work illustrates siple exaples of the extreal criteria for stability application. It also describes the experient. At the end of this work the atheatical survey of the extreal criteria of stability is presented. The new extreal stability criterion definition

3 Here the oveent of a echanical or electrical syste is studied. We 1,..., n, the generalized shall define the generalized coordinates by velocities by 1,..., n and the tie by t. Let L Lt, be the kinetic potential, where L -, and T is the kinetic energy of the syste and is the potential energy. We shall find the iniu of the kinetic potential L as a function of the generalized velocities, t Lt,, s 1,..., n, in,, We shall exclude fro Lt conditions of extreu, L t s s. (1) the derivative by the necessary, s 1,..., n. () Using the operation of tie averaging t on the function t able to eliinate t, we are 1 t li t, dt. (3) has a axiu at the point 1,..., n, then the axiu corresponds to the stable position of the dynaic euilibriu of the initial oscillating syste. Meanwhile the syste is steadily oscillating at the position 1,..., n. In order to find the position of the stable state, it is first necessary to find If the function the iniu of the function Lt axiu of the function using ax in L t via variable, and then we find the by which the nae extreal criterion of stability arises. It is iportant to ention that the conditions of stability are found without using Lagrange s differential euations d L L dt s s s 1,..., n and only the knowledge of Lagrange s function is used

4 L L t Stability of the atheatical pendulu oscillations at the upper euilibriu position We shall consider oscillations at the upper euilibriu position of the atheatical pendulu with length l and ass, with the vibrating vertical point of support A. We shall define the coordinates of the centre of gravity of the pendulu by x, y, the pendulu angle fro the vertical by and the deflection of the point A of the pendulu support in a vertical direction (figure 1) by rt asin t, where a is the aplitude and ω is the freuency of the vibrations. Fig. 1. Stability of the pendulu at the upper euilibriu position For the coordinates of the centre of gravity we find the following, and for the velocities, and in L x lsin, y l cos r, x lcos, y l sin r. The kinetic and potential energies are found to have the following fors, x y l l sin r r, r t asin t g l cos r Then the Lagrange function is as follows, L l r sin r cos g l cos r (4) is found with the help of the euation

5 L l r sin l, giving the following expression in L r cos gl cos gr r t asint we get As r asint, cos a r a t where <X> is the tie average of the function X. Therefore the following is found, a in L gl cos cos, where is a dynaic analogue of potential energy. The function has its axiu at the point, if the following condition is satisfied: or gl a. The stability criterion is reduced to the following ineuality: (5) a gl. (6) Relation (6) was discovered earlier in relevant scientific work. It is obvious that it was first discovered in works [1, ], and later repeated in the works of N. Bogolyubov, P. Kapitsa, G. Stoker, K. Valeev, T. Stryzhak. Conclusions The new extreal criterion of stability is a criterion sufficient for deterining the otion stability of a echanical syste under the influence of sall aplitude high freuency paraeter oscillations. This criterion allows the calculation of stability conditions of the oscillations of a non stable echanical syste. It does not involve the use of otion euations, rather, it uses only the Lagrange s function L T. References

6 1. Hirsch P. Das Pendel it oszillierende Aufhängenpunkt. Zeitschrift fuz angewandte Matheatik und Mechanik. 193, Bd. 1. S Erdelyi A. Uber die kleinen Schwingungen eines Pendels it oszillierende Aufhängenpunkt. Zeitschrift für angewandte Matheatik und Mechanik. B Bd. 14. Heft 4. S Stryzhak T. G. Freuency Criterion of Stability. Stuttgart: ibide Verlag, 9, 111 p. ISBN Блехман И. И. Вибрационная механика. М.: Физматлит, с. ISBN Stryzhak T. G. Miniax criteria of stability. Stuttgart: ibide Verlag, 8, 84 p. ISBN

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