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MATHEMATICAL MODELLING OF MOUNTAIN SHOCKS AND EARTHQUAKES RELATED TO VOLCANISM Rakhimov R.Kh., Jalilov M.L., Makhsudov A.U.
05.14.01 ЭНЕРГЕТИЧЕСКИЕ СИСТЕМЫ И КОМПЛЕКСЫ
COMPLEX POWER SYSTEMS
DOI: 10.33693/2313-223X-2020-7-3-57-61
Mathematical modelling of mountain shocks and earthquakes related
to volcanism
R.Kh. Rakhimov1' а ©, M.L. Jalilov2' b ©, A.U. Makhsudov1' с ©
1 Institute of Materials Science, SPA "Physics-Sun", Academy of Science of Uzbekistan, Tashkent, Republic of Uzbekistan
2 Fergana branch of the Tashkent University of Information Technologies named after Muhammad Al-Khorazmiy,
Fergana, Republic of Uzbekistan
a E-mail: rustam-shsul@yandex.com b E-mail: mamatiso2015@yandex.ru с Е-mail: asaduz50@rambler.ru
Abstract. In article occurrence of earthquakes and mountain blows and their communication by volcanic processes occurring in a kernel is analyzed. Mathematical modeling is resulted, uniting occurring processes in a kernel, occurrence P-longitudinal shock waves and the 5-intensity before earthquakes. In the given work it is considered, how by means of mathematical modeling it is possible to create model of occurring events and to untangle communication of seismic signatures of pushes arising from seismic processes. Such method of modeling will allow to create the three dimensional image of earth crust and to show in interaction of tectonic plates as the forces creating and pushing the formed break change in due course. For this purpose it is necessary to enter the seismic given districts that the model corresponded to supervision of how the plate is deformed to and during time, and after earthquake. It will help to draw conclusions on what forces operate on plate border - plates and as it is deformed, handing over the fluctuation information outside and as in things in common one plate dives into a hot viscous cloak of the Earth. In it to a floor the fused layer firm breeds exude and behave in the unexpected image, therefore the understanding of general dynamics of a status of a kernel can help to define communication between pressure along a break before earthquake. The problem of influence of mobile loadings on layers arises from a kernel of the earth a striking power of boiling magma, a surface top a piecewise homogeneous two-layer plate-plate the running wave along a x axis with constant speed V0 normal loading extends. The blows which are starting with a kernel of the Earth from an event вулканизма, creating running waves in earth crust it is described by the total formula (17). The mathematical concept of interpretation can be applied to concept of occurring events of a kernel of definition of striking power P-waves, intensity 5-waves and places at forecasting of natural accidents for the Earth.
Key words: gravity, energy, volcanism, analysis, approximate equation, vibrations, two-layer plate, stress, deformation, equations of oscillation
f \
FOR CITATION: Rakhimov R.Kh., Jalilov M.L., Makhsudov A.U. Mathematical modelling of mountain shocks and earthquakes related to volcanism. Computational nanotechnology. 2020. Vol. 7. No. 3. Pp. 57-61. DOI: 10.33693/2313-223X-2020-7-3-57-61
V J
DOI: 10.33693/2313-223X-2020-7-3-57-61
Математическое моделирование ударных волн и интенсивности до землетрясения
Р.Х. Рахимов1, а ©, М.Л. Джалилов2, b ©, А.У. Максудов1, с ©
1 Институт материаловедения Научно-производственного объединения «Физика-Солнце» Академии наук Республики Узбекистан,
г. Ташкент, Республика Узбекистан
2 Ферганский филиал Ташкентского университета информационных технологий имени Мухаммада Ал-Хоразмий,
г. Фергана, Республика Узбекистан
a E-mail: rustam-shsul@yandex.com b E-mail: mamatiso2015@yandex.ru с Е-mail: asaduz50@rambler.ru
Аннотация. В статье анализируется возникновение землетрясений и горных ударов и их связи происходящими в ядре вулканическими процессами. Приводится математическое моделирование, объединяя происходящих процессов в ядре, возникновения Р-продольных ударных волн и их 5-интенсивности до землетрясений. В данной работе рассмотрена, как при помощи математического моделирования можно создать модель происходящих событий и распутать связь сейсмических сигнатур толчков возникающих от сейсмических процессов. Такой метод моделирования позволит создать трехмерное изображение земной коры и показать во взаимодействии тектонических плит как изменяются со временем силы, создающие и толкающие образованный разлом. Для этого необходимо ввести сейсмические данные местности, чтобы модель соответствовала наблюдениям того, как плита деформируется до и во время, и после землетрясения. Это поможет сделать выводы о том, какие силы действуют на границе пластины-плиты, и как она деформируется, передавая информацию колебания наружу, и как в точке соприкосновения одна пластина ныряет в горячую вязкую мантию Земли. В этом полу расплавленном слое твердые породы сочатся и ведут себя неожиданным образом, поэтому понимание общей динамики состояния ядра может помочь определить связь между давлением вдоль разлома до и после землетрясения. Задача воздействия подвижных нагрузок на пласты возникает из ядра земли ударной силой кипящей магмы, топа поверхности кусочно-однородной двухслойной пластины-плиты распространяется бегущая волна вдоль оси x с постоянной скоростью нормальная нагрузка. Удары, исходящие из ядра Земли от происходящего вулканизма, создающие бегущие волны в земной коре описывается итоговой формулой (17). Математическая концепция интерпретации могут быть применены для понятия происходящих событий в ядре определения ударной силы Р-волны, интенсивности 5-волны и места при прогнозировании природных катастроф на Земле.
Ключевые слова: гравитация, энергия, вулканизм, анализ, приближенный, колебания, двухслойная пластинка, напряжения, деформация, уравнения колебания
ССЫЛКА НА СТАТЬЮ: Рахимов Р.Х., Джалилов М.Л., Максудов А.У. Математическое моделирование ударных волн и интенсивности до землетрясения // Computational nanotechnology. 2020. Т. 7. № 3. С. 57-61. DOI: 10.33693/2313-223X-2020-7-3-57-61
INTRODUCTION
The destructive property of all earthly disasters is not their gigantic energy, destroying material and cultural values that take human lives, but their suddenness and the seeming unpredictability of earthquakes. Why earthquakes occur and what is their mechanism, and how to predict it?
The occurring movements in the Earth's core creates changes in the spatio-temporal geophysical fields themselves, due to the thermal and gravitational effect, both the temperature and pressure in the Earth's core increase, and at depth catastrophic natural disasters occur subsequently on the Earth's crust.
The enormous pressure of the overlying thicknesses of plates and crust prevents the melting of rocks.
Sometimes there are underground explosions or tremors associated with the movement of magma, which are considered as "volcanic" earthquakes. Near volcanoes, seismographs are installed, recording the first tremors before the eruption (sometimes eruptions begin before the tremors). With the growth of rock activity, closely spaced seismographs begin to record the fluctuations of "volcanic tremors". It is a high-frequency vibration with an oscillation period generally less than 0.5 s. The first 'tremor' is expressed weakly and lasts no more than 2 hours, and increases as the activity
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Vol. 7. № 3. 2020
Rakhimov R.Kh., Jalilov M.L., Makhsudov A.U.
of the volcano increases. The strength of the earthquake depends on the duration of the period during which the anomaly is observed. For detecting the shake of the Earth's crust has different methods. Japanese seismologist Vadati invented one of the methods in which 12 or more seismographs are installed on one profile. At stations in New Zealand and the United States, another method was tested, based on the recording of P-wave intakes from individual seismic tremors within the estimated area of anomalous velocities occurring in the Earth's core.
All processes occurring in the core of the Earth are similar to those in the Sun, where there is a vortex twisting of magnetic drills, emissions of plasma flows, the release of elementary charged particles, etc., but without thermonuclear fusion. Shock waves of high-energy boiling magma generate high amplitude perturbations in the Earth's surface, which contributes to strong earthquakes, which are accompanied by aftershocks.
To determine these processes, it is possible to assume the Earth's layers as an infinitely piecewise homogeneous two-layer plate, on which the normal gravitational load and internal pressure from the yard act, and to describe by mathematical modeling. In real conditions, the destruction of rocks is usually accompanied by shock loads, which creates P- and S-waves. P-waves determine the magnitude, and S- the intensity of the earthquake. There are some works of algorithms where the P-wave spectrum is used for at least four (Allen, 2003) and (Kanamori, 2005) stations.
Let us consider the solution of oscillations of an infinite two-layer plate under the action of a normal load, the shock wave of the occurrence of volcanism, applied to the surface of this plate, as on the earth layers.
THE EFFECT OF NORMAL LOAD
ON AN INFINITE PIECEWISE HOMOGENEOUS
TWO-LAYER PLATE
This work considers that mathematical modeling will help improve the model of events and unravel the relationship of seismic signatures of tremors. This method of modeling will create a three-dimensional image of the Earth's crust and show the interactions between tectonic plates, as the forces that create and push the fault change over time. To do this, seismic terrain data must be entered so that the model matches observations of how the plate deforms before and during and after an earthquake. This will help draw conclusions about what forces act on the plate boundary, how the plate deforms, and transmitting oscillation information where at the point one plate dives into the Earth hot viscous mantle is. In this semi-molten layer, solid rocks ooze and behave in unexpected ways, so understanding the overall dynamics of the core state can help determine the relationship between stresses along the fault before and after an earthquake.
The problem of the effect of moving loads on structures and their elements arises from the core of the earth by the shock force of boiling magma. Let a travelling wave spread along the X-axis with a constant speed along the surface of a piecewise homogeneous two-layer plate with a normal load of the form
f = F(x + V0t); f = f = 0. (1)
z 0 " J xy J xy v '
In this case, the condition F(Z) = 0 at
Z = 0.
In this problem, the initial condition is absent, since this problem is for a flat plate.
Due to the external influence of the form (1), the stressstrain states of the plate from the coordinate y does not depend.
The problem is reduced to solving the approximate equation for lateral displacement W points of the contact plane of the two-layer plate obtained in the first chapter [1; 3]
I d w ) „fA52w ) / a2 n „ f dew )
+Q5[ A^W) + Qe[a2dW) + Q7(3W) = F(x + V01),
(2)
where is the coefficient Q. are determined by formulas (2).
Since there are no initial conditions in the problem posed, it is necessary to look for a general simple solution to equation (2), passing to the moving coordinates associated with the moving coordinate system by the well-known Galileo's transformation i = x + \t
Then equation (2) goes over to the ordinary differential equation
( Q1 + V2 Q 2 + Q3 )) +
dZ (3)
+ (( Q 4 + V0 Q 5 + V2 Q 6 + Q7 ))=F (Z).
The general solution of equation (3) is sought in the form
W = W0exp ^ h , (4)
where ; - dimensionless frequency.
We introduce dimensionless parameters:
C = h =hl; p = p0; 2 =
h
Pi
Hi
(5)
i „ i
Di =-
0_2(1-V0 ) 1 _2 (l-v i )
where h0, h1 - the thicknesses of the upper and lower layers of the plate are respectively; V0, V1 - approximate Poisson's ratios of the upper and lower layers of the plate; |i0, i - approximate Lame coefficients; b1 - the speed of spreading cross waves.
The characteristic equation differential equation (3) has the form
A^6 + = 0, (6)
where are the coefficients A1, A2 and Q' equal to:
A1 = C 6Q'4 + C 4Q'5 + C 2Q'6 + Q'7;
(7)
A1 = C 4Q1 + C 2Q'2 + Q3
and
Q'i =(p + h)2; (8)
Q2 = -2((D0 + hD1 )(p + h) + ( -l)(p(l + h)-(D0p-h2D1 ))); Q3 =-4( -1)(Do + h2Di + 2hP2Do);
Q5 = --6 {pV ( (4Do (1 - Do ) +1) + (P2 -1) (4 + DO )) --h4 (2 (4D2 - 2D1 -1) - (P2 -1)1 (2 - D1 )) + +6h2 (p^ (4 (P22DO + D1 ) + (P2 -1) (2P2 (1 - Do ) - P2D1 (2 - Do ) +
+ (1 (1 + Do )))) +p2 +1) + 2P2h ^ ((2 + 4Do - D2 ) + + (h2(2P2 -P2D1 + 5D1 -D2))) + +P V ((P2 -1) (4 - 3Do ) + 2D1 (4 - D1 )) + 2h2Do (4 - D1 )]} ;
i
Q6 = J {p [-20q (1 - 3P2 + 4Do) + ( -1) (2
+h4[4Do (1 -2Di) + (P2 -l)Di (3-Dj] + (8)
+3h2 [(4PD0 ((P2 (1 + Di) + Di) - ( -1) (2 (P2 -1) Di (1 - Do) --(P2 (2 - Do - 4D0D1 ))) + (4D1 (1 + Do + P2 Do) -
- (p -1) (6Do D1 ( -1) - 6P2 Do + D1))] --4Do (1 + h2 (2P2 -1) (1 - D1) + P2 D1 + (1 - D1 ))]};
Q7 = J {p Do (4Do - 5 (P2 -1) + h4D1 ( - (P2 -1)) --3h2 [8P2 Do D1 - (P2 - 1))(2P2 + 1)Do D1 - 3P2 Do +
+ D1 (1 - Do)-3P2 Do + D1 (1 - Do ))]-- 4hP2 Do [(( -1) + 2D1) + h2 (2 (P2 -1) + (P2 +1) D1 )]}.
Since the first four roots of the algebraic equation (6) ^ = = = = 0,
To determine the constant coefficients, C we use
' i
the boundary conditions for the case V0 > a0, which have the form:
dW d2W ,, s W = 0; -= 0; —- = 0, (% = 0);
|W|„ <œ;
dW KW; d2W < W
d%% <X> d%2 <x>
(14)
(15)
Substituting a general solution of the inhomogeneous differential equation (9) into the boundary conditions (11), we obtain
C + С +■
Qb
= 0;
Cj + Cj (ao b-ß0 a) = 0;
y A. a + b
(16)
and the roots Ç5 and imaginary.
Therefore, the general solution of the homogeneous the coefficients equation (3) is equal to
4 + ^ K)b-2aa°P°] = °
From the condition of bounded oscillations at infinity, where
c2 = q = C4 =
wh„
-C ! + C 2 ^ + C 3 Ç2 + C 4 ^ +
(9)
Similarly, the general solution of the inhomogeneous equation (3) is
W0 = Wh + W rt,
0 homo part7
where Wpart - a particular solution of the inhomogeneous equation and we find depending on the type of function of the external influence.
If the right side of equation (3) is equal to
F(0 = Qe-<q sin(|30 (10)
then a particular solution to equation (3) is represented as
Thus, the solution to the problem of oscillation of an infinitely two-layer plate when exposed to a moving load has the form:
W = -
2—aa0 ßo -b
-1 (a2-ßo2 )+l|+
A [(a0-ß0 )b - 2aao ßo ]x
(17)
A2
It *
A2
+J a2 (ob-ßoa )sin
A2
'A *
W
[ л sin(ß0 ) + B cos .
(11)
Then, the general solution of differential equation (3) will be
W = C: + C2 k + C3 k2 + C4 k3 + C5cos
a2
л
A2 'a: k
л
(12)
-ak ;
a sin(ßo k) + b cos (ß0 k)),
where
a = A!(-ß20) - 12a0ß20 (a20-ß2))-A2 ( -6a20ß2 + ß0);
b = 2ao ßo (A 1 ( -10a2 ß2 + 3ß4 ) + 2A2 (2 - ß2 )).
The impacts emanating from the core of the Earth from occurring volcanism, creating travelling waves in the earth's crust, are described by the final formula (17) given above.
The mathematical concept of interpretation can be applied to the concept of occurring events in the core for determining the shock force P-waves, intensity S- and location when predicting natural disasters on Earth.
For an elastic two-layer plate, the sixth-order operator splits into the product of the second-longitudinal and fourth-order transverse-wave operators if the thicknesses of the plate components satisfy the derived equation containing the parameters of these components. As well as the mathematical concept of interpretation, the concepts of occurring events in the nucleus - determining the shock force P-waves, intensity S-waves and the place in predicting natural disasters on Earth can describe the concepts.
Thus, the above model can describe the propagating shock force of a wave occurring earthquake from volcanism. To understand the concept and prediction of natural disasters, it is necessary to consider the processes taking place in a single time, this mechanism of action allows you to approach the development of reliable principles for predicting the place and time of disasters in the earth's crust.
Q
2
2
2
x
+ C6 sin
+
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Computational nanotechnology
Vol. 7. № 3. 2020
Rakhimov R.Kh., Jalilov M.L., Makhsudov A.U. CONCLUSION
As a result of overvoltage of layers and pressures in the earth's core a system of cracks and voids appear in the layers, they expand, and various gases rush into them. Gases instantly heat up, heat and melt the earth's rocks, merging them into the core before the eruption of the volcano. P-waves, S-waves, and travelling waves appear, with the simultaneous origin of radioactive degassing on the surface of the earth's crust.
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Статья проверена программой Антиплагиат
Рецензент: Раджапов С.А., доктор физ.-мат. наук; ведущий научный сотрудник Физико-технического института НПО «Физика-Солнце» Академии наук Республики Узбекистан
Статья поступила в редакцию 15.07.2020, принята к публикации 20.08.2020 The article was received on 15.07.2020, accepted for publication 20.08.2020
ABOUT THE AUTHORS
Rustam Kh. Rakhimov, Dr. Sci. (Eng.); Head at the Laboratory No. 1 of the Institute of Materials Science "Physics-Sun" of the Republic of Uzbekistan. Tashkent, Republic of Uzbekistan. ORCID: https://orcid.org/0000-0001-6964-9260. E-mail: rustam-shsul@yandex.com Mmatmatisa L. Jalilov, Cand. Sci. (Eng.); Head at the Department "Computer Systems" of the Fergana branch of the Tashkent University of Information Technologies named after Muhammad Al-Khorazmiy. Fergana, Republic of Uzbekistan. ORCID: https://orcid.org/0000-0002-9471-4893. E-mail: mamatiso2015@yandex.ru Asatulla U. Makhsudov, senior researcher at the Physics-Technical Institute of Scientific and Production Association "Physics-Sun" of the Academy of Sciences of the Republic of Uzbekistan. Tashkent, Republic of Uzbekistan. ORCID: https://orcid.org/0000-0001-7935-3893. E-mail: asaduz50@rambler.ru
СВЕДЕНИЯ ОБ АВТОРАХ
Рахимов Рустам Хакимович, доктор технических наук, профессор; зав. лабораторией № 1 Института материаловедения Научно-производственного объединения «Физика-Солнце» Академии наук Республики Узбекистан. Ташкент, Республика Узбекистан. ORCID: https:// orcid.org/0000-0001-6964-9260. E-mail: rustam-shsul@ yandex.com
Джалилов Ммаматиса Латибджанович, кандидат технических наук; зав. кафедрой «Компьютерные системы» Ферганского филиала Ташкентского университета информационных технологий им. Мухаммада Ал-Хоразмий. Фергана, Республика Узбекистан. ORCID: https://orcid.org/0000-0002-9471-4893. E-mail: mamati-so2015@yandex.ru
Максудов Асатулла Урманович, старший научный сотрудник Научно-производственного объединения «Физика-Солнце» Академии наук Республики Узбекистан. Ташкент, Республика Узбекистан. ORCID: https://orcid. org/0000-0001-7935-3893, Е-mail: asaduz50@rambler.ru
