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3STUDYING THE STABILITY OF THE ZERO SOLUTION OF
THE HILL EQUATION IN PARAMETRIC MOTION ACCORDING TO THE QUASI-RECTANGULAR SINE LAW
Mirzaaxmedova Jumagul Vohobjon qizi
Doctoral student of Andijan State University https://doi.org/10.5281/zenodo.8419799
Abstract. In this article, the state of stability in parametric excitation according to the quasi-rectangular sine law of motion expressed by Hill's equation is studied. The stability of changing systems and their solutions over a certain period of time was analyzed.
Keywords: quasi-rectangular sine law, parametric excitation, Hill's equation, stability, fundamental matrix.
INTRODUCTION
It is known that human life consists of movement and various processes. Where there is movement, work is done, energy is spent, and energy is generated. This, in turn, creates a process that satisfies human needs. It is important that the movement of anything or system is continuous and stable. This issue is one of the pressing issues in our developing society. As the human mind develops, the processes activated by it become more complex. These systems of actions and processes are represented in science by various functional-differential equations and systems of equations. We can check the stability of such complex systems using these equations. Several equations have been proposed by scientists to study the stability of the movement of complex systems in life. When studying the stability of motion, it is important to express it with an equation. Hill's equations are very effective for this. In the following article, the stability of the movement represented by Hill's equation in the parametric excitation according to the quasi-rectangular sine law is studied.
LITERATURE REWIEW
The equations and concepts used in this article are studied in one chapter of ^.P.MepKHH's book named "Introduction to the theory of stability of motion". The author showed that when studying the stability of non-autonomous systems, that is, parametric driven systems, it is enough to study the stability of the solutions of Hill's equations [1]. In A.X. Terr's book named "Absolute stability of linear systems with non-unique equilibrium states in critical states" the conditions of absolute stability imposed on the fundamental solutions of the equations of motion were considered [2].
RESEARCH METHODOLOGY AND DISCUSSION
Consider a simple system, the following excited motion equation described by Hill's equation
x + [8 + e^(t)]x = 0 (1)
^(t) changes with the excitation function according to the quasi-rectangular sine law (Fig. 1). The period T of the excitation function is equal to the sum of time 71 when the function ^(t) equals +1 and time T2 when ^(t) = —1. At 7\ = T2, we get a typical quasi-rectangular sine.[1]
Figure 1
In particular, with the help of equation (1), the systems whose stiffness changes from time to time are studied using the relay device. For us, this problem means not only that its solution can be used to analyze the stability of the behavior of certain systems, but also that it is necessary to construct a fundamental matrix X(T) of solutions satisfying the condition X(0) = E for one period [0,T],
det(A - pE) =
-11 -P -12
a
a„
aY.
-22 - P
a,
a
a
n2
ann -p
= 0.
shows the construction of the matrix A = X(T) of the characteristic equation and the determination of stability conditions for the solutions x = 0,x = 0.[2]
In the equation (1), the number e is equal to the pulsation depth, and the number S is equal to the square of the frequency k of natural vibrations at S > 0 and £ = 0, i.e. S = k2. We match the beginning of time t with the beginning of any period T. Then, for the first part of the period 0<t<T1, equation (1) looks like this:
x + (k2 + e)x = 0 (0<t < T1), (2)
and for the second part of the period 0 < t <T2
x + (k2 - e)x = 0 (T2<t< T). (3)
Let's look at the first equation (2). Assuming that x1 = x, x2= x as before, we reduce equation (2) to a system of two first-order equations
x1 = x2, x2 = -k^x1 (0 <t < T), (4)
here
k2 = k2 + e. (5)
(4) system is simply solved. Two linearly uncoupled solutions of this system satisfying the
condition X(0) = E are as follows:
1
x11 = cosk1t, x12 = —sink1t, x21 = -k1sink1t, x22 = cosk1t (6)
Thus, the fundamental matrix of the solution in the first part of the period is as follows
(0<t< T1). (7)
1
cosk1t —sink1t
X(T) = .
-k1sink1t It can be seen that X(0) = E.
cosk1t
Let's go to the second part of the period (T2 < t <T). Equation (3) after replacing
k2 — £ in system (4) with kf = k2 + a has the following form:[3]
&
x1 = x2, x2 = —k2,xx (Tx <t <T). (8)
RESULTS
This is in the general solution of the system
x1 = C1cosk2(t — T-) + C2sink2(t — T1), x2 = —k2C1sink2(t — T1) + k2C2cosk2(t — T1) (9)
to determine the first eigenvalues, we choose integral constants C1 and C2. For this, the solution of (9) must coincide with the solutions of x11, x21 in system (6) at t = T1. We have
cosk1T1 = C1, — k1sink1T1 = k2C2. We put the values of C1 and C2 in these equations into (9) and find the first specific solution of equation (8) in the second part of the period T1 < t <T:
kL
x11 = cosk1T1cosk2(t — T1) — — sink1T1sink2(t — t-l),
k-2
x21 = —k2cosk1T1sink2(t — T-) — k1sink1T1cosk2(t — T1),
1 1
x12 = — sink1T1cosk2(t — T-) + — cosk1T1sink2(t — tl), k- k2
x22 = — — sink1T1sink2(t — Tj) + cosk1T1cosk2(t — T^). (10)
k1
These expressions determine the elements of the fundamental matrix X(T) in the second part of the period tl < t <T. If we put t = T in (10), we get matrix elements A = X(T). Considering akj = xkj(T), we construct the characteristic equation det(4 — pE):
xii(T) — p Xi2(T) *2i(T) X22(T)—p
We put the value of xkj(T) in (10) into this equation and, taking into account kf = k2 + e, k% = k2 — £, T — T1 = T2, directly -by correct calculations we find the following:
p2 + ap + 1 = 0, (11)
here
= 0.
a = 2
ß
sink1T1sink2T2 — cosk1T1cosk2T2
(12)
p = £/k2 = e/S. (13)
In this example, all the coefficients of the characteristic equation were obtained by direct calculations. It follows from the general theory of the Hill equation that the free term of equation (11) is equal to one.[1] In order for the motion to be stable, it is necessary and sufficient that the inequality |a| <2 is satisfied. In our example, the stability condition (simple but not asymptotic) is as follows:
sink1T1sink2T2
— cosk1T1cosk2T2
< 1. (14)
If all numbers S, e, T1 and T2 are given, it is not difficult to check this condition. Without stopping at the detailed analysis of the inequality (14), we establish the conditions for the emergence of parametric resonance only at ^ = e/5 « 1. (14) without taking into account all the participating quantities ^ higher than one and taking into account that the parametric resonance for the Hill equation already occurs in the stability region, we will have
l
lcOS(k1T1 + k2T2)l = 1.
From here
k1T1 + k2T2 =nn (n= 1,2,3,...). (15)
And now we consider the values of k1 and k2
k1 = jk2 + £ = + ii, k2 = jk2 — £ = k^1 — p.. For sufficiently small ^ = £/k2 = £/8, we have the following:
k1 = k(l + 1/2 n), k2 = k(l — 1/2 n). We substitute these values for k1 and k2 in (15) and find the following (T2 +T1 = T):
kT + 1/2 ^k(T2 — t-l) = nn, or to the accuracy of the main limits
m = (n = 1,2,3,.) (16)
where m = 2n/T is the frequency of pulsation, k =
is the frequency of specific oscillations
of the system in the absence of parametric excitations.[4] CONCLUSION
It can be seen from the expression (16) that at a sufficiently small pulsation depth, the parametric resonance £ occurs at countless values of its frequency m. (16) expression does not depend on tl and T2 parts of the period in parametric excitation according to the quasi-rectangular sine law for critical values of the pulsation frequency, and it is critical in parametric excitation according to the simple sine (cosine) law overlaps the corresponding values of the frequency. Indeed, if the Mathieu equation is written in the following form
X + (k2 + £COSMt)x = 0, where k is the frequency of natural oscillations of the system without parametric excitations, in this case, using the formula wt = t, passing to dimensionless time, we get the canonical form of this equation, where S = k2/w2 . £ is defined by the equations S = n2/4 or k2/w2 = n2/4 with a small critical point, where n = 1,2,3, .... m = 2k/n, i.e. formula (16).
To conclude this example, we should note that the stability condition (14) holds when one or all of the numbers S,S + £ = and S — £ = k% are negative. For this, it is enough to go from trigonometric functions of abstract arguments to hyperbolic functions of real quantities.
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